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+{"text":"\\section{Introduction}\nThe third observing run of the global gravitational wave network has not only produced a plethora of varied and unique astrophysics events \\cite{Abbott-AAP20-PopulationProperties, Abbott-AAP20-GWTC2Compact}, it has defined a milestone in quantum metrology: that the LIGO, VIRGO and GEO600 observatories are now all reliably improving their scientific output by incorporating squeezed quantum states \\cite{Tse-PRL19-QuantumEnhancedAdvanced, VirgoCollaboration-PRL19-IncreasingAstrophysical, Lough-PRL21-FirstDemonstration}. This marks the transition where optical squeezing, a widely researched, emerging quantum technology, has become an essential component producing new observational capability.\n\nFor advanced LIGO, observing run three provides the first peek into the future of quantum enhanced interferometry, revealing challenges and puzzles to be solved in the pursuit of ever more squeezing for ever greater observational range. Studying quantum noise in the LIGO interferometers is not simple. The audio-band data from the detectors contains background noise from many optical, mechanical and thermal sources, which must be isolated from the purely quantum contribution that responds to squeezing. All the while, the interferometers incorporate optical cavities, auxiliary optical fields, kg-scale suspended optics, and radiation pressure forces. The background noise and operational stability of the LIGO detectors is profoundly improved in observing run three \\cite{Buikema-PRD20-SensitivityPerformance}, enabling new precision observations of the interactions between squeezed states and the complex optomechanical detectors.\n\nQuantum radiation pressure noise (QRPN) is the most prominent new observation from squeezing \\cite{Yu-N20-QuantumCorrelations, Acernese-PRL20-QuantumBackaction}. QRPN results from the coupling of photon momentum from the amplitude quadrature of the light into the phase quadrature, as radiation force fluctuation integrates into mirror displacement uncertainty. When vacuum states enter the interferometer, rather than squeezed states, QRPN imposes the so-called standard quantum limit\n\\cite{Braginsky-S80-QuantumNondemolition, Braginsky-92-QuantumMeasurement, Braginsky-RMP96-QuantumNondemolition},\nbounding the performance of GW interferometers. Because the QRPN coupling between quadratures is coherent, squeezed states allow the SQL to be surpassed\n\\cite{Kimble-PRD01-ConversionConventional, Yu-N20-QuantumCorrelations}.\nBoth surpassing the SQL and increasing the observing range is possible by using a frequency-dependent squeezing (FDS) source implemented with a quantum filter cavity\n\\cite{Kimble-PRD01-ConversionConventional, McCuller-PRL20-FrequencyDependentSqueezing, Zhao-PRL20-FrequencyDependentSqueezed, Oelker-PRL16-AudioBandFrequencyDependent, Chelkowski-PRA05-ExperimentalCharacterization, Whittle-PRD20-OptimalDetuning, Khalili-PRD10-OptimalConfigurations, Evans-PRD13-RealisticFilter, Kwee-PRD14-DecoherenceDegradation}. LIGO is including such a source in the next observing run as part of its ``A+'' upgrade\\cite{Whittle-PRD20-OptimalDetuning, McCuller-PRL20-FrequencyDependentSqueezing}. To best utilize its filter cavity squeezing source, the frequency-dependence of LIGO's quantum response must be precisely understood.\n\nDegradations to squeezing from optical loss and ``phase noise'' fluctuations of the squeezing angle are also prominently observed in LIGO. Whereas QRPN's correlations cause frequency dependent effects, loss and phase noise are typically described as causing frequency independent, broadband changes to the quantum noise spectrum. This work analyzes the quantum response of both LIGO interferometers to injected squeezed states, indicating that QRPN and broadband degradations, taken independently, are insufficient to fully describe the observed quantum response to squeezing.\n\nThe first sections of this work expand the response and degradation model of squeezing to examine and explain the LIGO quantum noise data by decomposing it into independent, frequency-dependent parameters. The latter sections relate the parameter decomposition back to interferometer models, to navigate how squeezing interacts with cavities that have internal losses, transverse-mode selectivity, and radiation pressure interactions. The spectra at LIGO are explained using a set of broadly applicable analytical expressions, without the need for elaborate and specific computer simulations. The analytical models elucidate the physical basis of LIGO's squeezed state degradations, prioritizing transverse-mode quality using wavefront control of external relay optics\n\\cite{Cao-OEO20-EnhancingDynamic, Cao-AOA20-HighDynamic, Perreca-PRD20-AnalysisVisualization} to further improve quantum noise. This analysis also demonstrates the use of squeezing as a diagnostic tool\\cite{Mikhailov-PRA06-NoninvasiveMeasurements}, examining not only the cavities but also the radiation pressure interaction. These diagnostics show further evidence of the benefit of balanced homodyne detection \\cite{Fritschel-OEO14-BalancedHomodyne}, another planned component of the ``A+'' upgrade. The description of squeezing in this work expands the modeling of degradations in filter cavities \\cite{Kwee-PRD14-DecoherenceDegradation}, explicitly defining an intrinsic, non-statistical, form of dephasing. Finally, the derivations of the quantum response metrics in sec. \\ref{sec:derivations} show how to better utilize internal information inside interferometer simulations, simplifying the analysis of squeezing degradations for current and future gravitational wave detectors.\n\n\\section{Squeezing Response Metrics}\n\\label{sec:metrics}\n\nTo introduce the frequency-dependent squeezing metrics, it is worthwhile to first describe the metrics used for standard optical squeezing generated from an optical parametric amplifier (OPA), omitting any interferometer. For optical parametric amplifiers, the squeezing level is determined by three parameters. The first is the normalized nonlinear gain, $y$, which sets the squeezing level and scales from 0 for no squeezing to 1 for maximal squeezing at the threshold of amplifier oscillation. For LIGO, $y$ is determined from a calibration measurement of the parametric amplification \\cite{Xiao-PRL87-PrecisionMeasurement, Aoki-OEO06-Squeezing946nm, Takeno-OEO07-ObservationDB, Khalaidovski-CQG12-LongtermStable, Schnabel-OC04-SqueezedLight, Dwyer-OEO13-SqueezedQuadrature}. The second parameter is the optical efficiency $\\eta$ of states from their generation in the cavity all the way to their observation at readout. Losses that degrade squeezed states are indicated by $\\eta < 1$. Finally, there is the squeezing phase angle, $\\phi$, which determines the optical field quadrature with reduced noise and the quadrature with the noise increase mandated by Heisenberg uncertainty, anti-squeezing. By correlating the optical quadratures, variations in $\\phi$ continuously rotate between squeezing and anti-squeezing. These parameters relate to the observable noise as:\n\\begin{align}\n  N(\\phi) &= \\left( 1 - \\frac{4 \\eta y}{(1 + y)^2} \\right)\\cos^2(\\phi) + \\left( 1 + \\frac{4 \\eta y}{(1 - y)^2} \\right)\\sin^2(\\phi)\n\\end{align}\nThe noise, $N(\\phi)$, can be interpreted as the variance of a single homodyne observation of a single squeezed state, but for a continuous timeseries of measurements, $N$ can be considered as a power spectral density, relative to the density of shot-noise. Using relative noise units, $N=1$ corresponds to observing vacuum states rather than squeezing. While the nonlinear gain parameter $y$ may be physically measured and is common in experimental squeezing literature, theoretical work more commonly builds states from the squeezing operator, parameterized by $r$, which constructs an ideal, ``pure'' squeezed state that adjusts the noise power by $e^{\\pm 2r}$. State decoherence due to optical efficiency is then incorporated as a separate, secondary process. This is formally related to the previous expression using:\n\\begin{align}\n  N(\\phi)\n  &= \\eta\\left(\n    e^{-2r}\\cos^2(\\phi) + e^{+2r}\\sin^2(\\phi)\n    \\right) + \\left(1 - \\eta\\right)\n    \\label{eq:etaSQZ_basic1}\n  \\\\\n  e^{-2r}\n  &=\n    1 - \\frac{4y}{(1 + y)^2} \\label{eq:OPA_y_SQZ}\n    , \\hspace{2.5em} e^{+2r} = 1 + \\frac{4y}{(1 - y)^2}\n \n\\end{align}\nIn experiments, the squeezing angle drifts due to path length fluctuations and pump noise in the amplifier, but is monitored using additional coherent fields at shifted frequencies and stabilized by feedback control. This stabilization is imperfect, resulting in a root-mean-square (RMS) phase noise,  $\\ensuremath{\\phi^2_{\\text{rms}}}$, that mixes squeezing and antisqueezing. Using $\\hat{\\phi}$ to represent the statistical distribution of the squeezing angle, and $E[\\cdot]$ the expectation operation, phase noise can be incorporated as a tertiary process given the expectation values:\n\\begin{align}\n \\ensuremath{\\phi^2_{\\text{rms}}} &= E\\left[\\sin^2(\\delta \\hat{\\phi})\\right]\n             &\n \\ensuremath{\\phi} &= E\\left[\\hat{\\phi}\\right]\n  &\n\\delta \\hat{\\phi} &= \\hat{\\phi} - \\ensuremath{\\phi}\n\\end{align}\nresulting in the ensemble average noise $\\overline{N}$, relative shot noise.\n\\begin{align}\n  \\overline{N}(\\phi) &= E\\left[N(\\ensuremath{\\phi} + \\delta \\hat{ \\phi})\\right]\n              \\\\\n \n   &= \\eta\\left(1 - \\ensuremath{\\phi^2_{\\text{rms}}}\\right)\\left(e^{-2r}\\cos^2(\\ensuremath{\\phi}) + e^{+2r}\\sin^2(\\ensuremath{\\phi}) \\right)\n     \\nonumber\\\\ &\\hspace{1em}\n                   + \\eta\\ensuremath{\\phi^2_{\\text{rms}}} \\left(e^{+2r}\\cos^2(\\ensuremath{\\phi}) + e^{-2r}\\sin^2(\\ensuremath{\\phi}) \\right) + (1 - \\eta)\n    \\label{eq:etaSQZ_basic2}\n\\end{align}\nAgain, the relative noise $\\overline{N}$ is computed as a single value here, but represents a power spectral density that is experimentally measured at many frequencies. These equations, as they are typically used, represent a change to the quantum noise that is constant across all measured frequencies. Notably, the $\\ensuremath{\\phi^2_{\\text{rms}}}$ phase noise term, which caps at $1\/2$, enters as a weighting factor that averages the anti-squeezing noise increase with squeezing noise reduction, while $\\eta$ mixes squeezing with standard vacuum.\n\nIncorporating an interferometer such as LIGO requires extending these equations to handle frequency-dependent effects. The equations must include terms to represent multiple sources of loss entering before, during, and after the interferometer, as well as terms for the frequency-dependent scaling of the quantum noise due to QRPN and the interferometer's suspended mechanics. The extension of the metrics is described by the following equations and parameters:\n\\begin{align}\n  N(\\Omega)\n  &\\equiv \\Gamma(\\Omega) \\cdot \\Big( \\eta(\\Omega) S(\\Omega) + \\Lambda_\\ensuremath{\\text{IRO}}(\\Omega) \\Big) \n    \\label{eq:metric_N}\n  \\\\\n \n \n \n \n \n \n \n \n  S(\\Omega)\n  &\\equiv S_{\\!{-}}\\cos^2\\Big(\\ensuremath{\\phi} - \\theta(\\Omega)\\Big) + S_{\\!{+}} \\sin^2\\Big(\\ensuremath{\\phi} - \\theta(\\Omega)\\Big)\n    \\\\\n  S_{\\!\\pm} &\\equiv \\big(1 - \\Xi'(\\Omega)\\big)e^{\\pm2r} + \\Xi'(\\Omega) e^{\\mp2r}\n              \\label{eq:metric_S}\n  \\\\\n  \\Lambda_\\ensuremath{\\text{IRO}}(\\Omega) &\\equiv (1 - \\eta_\\ensuremath{\\text{I}})\\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}} + \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}}) + (1 - \\eta_\\ensuremath{\\text{O}}) \/ \\Gamma\n    \\label{eq:metric_Lambda}\n\\end{align}\nThese metrics are composed of the following variables:\n\\renewcommand{\\descriptionlabel}[1]{\\hspace{\\labelsep}{#1}:}\n\\begin{description}[noitemsep, nolistsep]\n \\item[$N(\\Omega)$] the power spectrum of quantum noise in the readout, relative to the vacuum power spectral density, $\\hbar \\omega \/ 2$, of broadband shot noise.\n \\item[$\\Gamma(\\Omega)$] The quantum noise gain of the interferometer optomechanics. While $N(\\Omega)$ is relative shot-noise, QRPN causes interferometers without injected squeezing to exceed shot noise at low frequencies, resulting in $\\Gamma > 1$. For optical systems with $\\Gamma\\ne1$, the system cannot be passive, and must apply internal squeezing\/antisqueezing to the optical fields.\n \\item[$e^{2r}, e^{-2r}$] The ``pure'' injected squeezing and anti-squeezing level, before including any degradations. This level is computed for optical parametric amplifier squeezers using \\cref{eq:OPA_y_SQZ}.\n \\item[$S_{\\!-}, S_{\\!+}$] The minimum and maximum relative noise change from squeezing at any squeezing angle, ignoring losses.\n \\item[$S(\\Omega)$] The potentially observable injected squeezing level, before applying losses or noise gain.\n  \\item[$\\ensuremath{\\phi}$] The frequency independent squeezing angle chosen between the source and readout. This is usually stabilized with a co-propagating coherent control field and feedback system.\n  \\item[$\\theta(\\Omega)$] the squeezing angle rotation due to the propagation through intervening optical system. In a GW interferometer, this can be due to a combination of cavity dispersion and optomechanical effects. Quantum filter cavities target this term to create frequency dependent squeeze rotation.\n  \\item[$\\eta_\\ensuremath{\\text{I}}(\\Omega)$, $\\eta_\\ensuremath{\\text{O}}(\\Omega)$, $\\eta_\\ensuremath{\\text{R}}(\\Omega)$] The individually budgeted transmission efficiencies of the squeezed field at input, reflection and output paths of the interferometer. $1 - \\eta_{\\text{I,R,O}}$ indicates optical power lost in that component.\n  \\item[$\\eta(\\Omega)$] The collective transmission efficiency of the squeezed field. This is usually the product of the efficiencies in each path, $\\eta = \\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}$, but can deviate from this when $\\Gamma \\ne 1$ and interferometer losses affect both $\\Gamma$ and $\\eta_\\ensuremath{\\text{R}}$.\n  \\item[$\\Lambda_\\ensuremath{\\text{IRO}}(\\Omega)$] The total transmission loss over the squeezing path that contaminates injected squeezed states with standard vacuum. When $\\Gamma \\approx 1$, then $\\Lambda_\\ensuremath{\\text{IRO}} \\approx 1 - \\eta$.\n  \\item[$\\Xi'(\\Omega)$] This is a squeezing-level dependent decoherence mechanism called dephasing. It\n    incorporates both statistical $\\ensuremath{\\phi^2_{\\text{rms}}}$ phase fluctuations and the fundamental degradation arises from optical losses with unbalanced cavities, denoted $\\Xi(\\Omega)$. It can also arise from QRPN with structural or viscous mechanical damping. \\Cref{sec:effective_dephasing} shows how to incorporate fundamental dephasing  $\\Xi(\\Omega)$, standard phase uncertainty, $\\ensuremath{\\phi^2_{\\text{rms}}}$, and cavity tuning fluctuations, $\\ensuremath{\\theta^2_{\\text{rms}}}(\\Omega)$, into $\\Xi'(\\Omega)$ to make a total effective dephasing factor. When small, these factors sum to approximate the effective total $\\Xi'$\n\\end{description}\n\nAfter the data analysis of the next section, these quantum response metrics are derived in \\cref{sec:derivations}. These squeezing metrics indicate three principle degradation mechanisms, all frequency-dependent. These are losses, where $\\Lambda_\\ensuremath{\\text{IRO}}(\\Omega)\\approx 1 {-} \\eta(\\Omega) > 0$; Mis-phasing, from $\\phi{-}\\theta(\\Omega)\\ne 0$; and de-phasing, $\\Xi(\\Omega) > 0$.\n\nThe interaction of squeezing with quantum radiation pressure noise is described within these terms. Broadband Squeezing naively forces a trade-off between increased measurement precision and increased quantum back-action. When squeezing is applied in the phase quadrature, it results in anti-squeezing of the amplitude quadrature. The amplitude quadrature then pushes the mirrors and increases QRPN; thus, the process of reducing imprecision seemingly increases back-action. In other terms, QRPN causes the interferometer's ``effective'' observed quadrature\\footnote{The observed quadrature in this context is with respect to the injected quantum states, be they squeezed or vacuum. It is dependent on the quadrature of the interferometer's homodyne readout, but does not refer specifically to it. The effective observed quadrature also does not refer to the specific quadrature that the interferometer signal is modulated into.} to transition from the phase quadrature at high frequencies to the amplitude quadtrature at low frequencies. In the context of these metrics, the observation quadrature is captured in the derivation of $\\theta(\\Omega)$. The associated back-action trade-off can be considered a mis-phasing degradation, allowing the SQL to be surpassed using the quantum quadrature correlations introduced by varying the squeezing angle\\cite{Yu-N20-QuantumCorrelations}. Frequency dependent squeezing, viewed as a modification of the squeezing source, can be considered as making $\\phi(\\Omega)$ frequency-dependent, tracking $\\theta(\\Omega)$. Alternatively, it can be viewed as a modification of the interferometer, to maintain $\\theta(\\Omega) \\approx 0$. While a quantum filter cavity is not explicitly treated in this work, the derivations of \\cref{sec:derivations} are setup to be able to include a filter cavity as a modification to the input path of the interferometer.\n\nWhile mis-phasing can be compensated using quantum filter cavities, the other two degradations are fundamental. For squeezed states, they establish the noise limit:\n\\begin{align}\n  N(\\Omega) &\\ge \\Gamma{\\cdot}\\left(2\\eta\\sqrt{\\Xi'(1-\\Xi')} + \\Lambda_\\ensuremath{\\text{IRO}}\\right),\n  &\n  e^{-2r} &= \\sqrt{\\Xi'(\\Omega)}\n  \\label{eq:N_limit}\n\\end{align}\nSetting the squeezing level as $\\sqrt{\\Xi'}$ solves for the optimal noise given the dephasing. Squeezing is then further degraded from losses, producing the noise limit. Notably, the optimal squeezing is generally frequency-dependent due to $\\Xi(\\Omega)$, indicating that for typical broadband squeezing sources, this bound cannot always be saturated at all frequencies.\n\n\\subsection{Ideal Interferometer Response}\n\\label{sec:ideal_IFO_metrics}\nBefore analyzing quantum noise data to utilize the squeezing metrics of \\crefrange{eq:metric_N}{eq:metric_Lambda}, it is worthwhile to first review the quantum noise features expected in the LIGO detector noise spectra\\cite{Kimble-PRD01-ConversionConventional, Aasi-CQG15-AdvancedLIGO}, under ideal conditions and without accounting for realistic effects present in the interferometer. The derivations later will then extend how the well-established equations below generalize to incorporate increasingly complex interferometer effects, both by extracting features from matrix-valued simulation models, as well as by extracting features from scalar boundary-value equations for cavities.\n\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_IFO, \n  label={SQZ_mm_IFO},\n  caption={\n    This simplified diagram of the interferometer layout shows the propagation of the source laser (solid red) and squeezed beam (dashed burgundy). At (a), the squeezed beam is sourced from a parametric amplifier cavity and circulated to the interferometer with a Faraday isolator. At (b), the squeezing field reflects from the interferometer. Depending on the frequency and transverse beam profile, the states partially transit the interferometer cavities, but also partially reflect promptly. The squeezing that enters the interferometer symmetrically is beam split inside the signal recycling cavity, coherently resonates in both arms, and recombines again at the beamsplitter, effectively experiencing the two branches as a single linear coupled cavity. Injected at a different port, the red laser field carries substantial laser power and is symmetrically split to pump the arm cavities. Differential length signals are sourced by modulating the circulating pump field, creating a phase-quadrature field that resonates in the same effective linear cavity as the squeezing. The signal is emitted at (b) follows with the reflected squeezing. The transverse beam profile (mode) of the signal and squeezing is then selected using the output mode cleaning cavity at (c). Ultimately, the signal and noise are read as timeseries in photodetectors at (d). This effect of coherent interference between prompt and cavity-circulated squeezing from this sequence is formulated, measured, and analyzed in the following sections.\n},\n}\n\n\nOther than shot noise imprecision, the dominant quantum effect in gravitational wave interferometers arises from radiation pressure noise. In an ideal, on-resonance interferometer, this noise is characterized by the interaction strength $\\ensuremath{\\mathcal{K}}(\\Omega)$ that correlates amplitude fluctuations entering the interferometer to phase fluctuations that are detected along with the signal. $\\ensuremath{\\mathcal{K}}$ is generated from the circulating arm power $P_\\ensuremath{\\text{A}}$ creating force noise that drives the mechanical susceptibility $\\chi(\\Omega)$. The susceptibility relates force to displacement on each of the four identical mirrors of mass $m$ in the GW arm cavities. The QRPN effect is enhanced by optical cavity gain $g(\\Omega)$ which resonantly enhances quantum fields entering the arm cavities and signal fields leaving them.\n\\begin{align}\n  \\mathcal{K}(\\Omega)\n  &= 16k\\frac{P_\\ensuremath{\\text{A}}}{c} g^2(\\Omega)\\chi(\\Omega),\n    &\n  g(\\Omega) &= \\frac{\\sqrt{{\\gamma_\\ensuremath{\\text{A}} c}\/{L_{\\ensuremath{\\text{a}}}}}}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n              \\label{eq:optical_gain_g}\n\\end{align}\nHere, $k$ is the wavenumber of the interferometer laser and $c$ the speed of light.\nThe arm cavity gain $g(\\Omega)$ is a function of the signal bandwidth $\\gamma_\\ensuremath{\\text{A}}$, derived later, and the interferometer arm length $L_\\ensuremath{\\text{a}}$. Unlike in past works, this expression of $\\ensuremath{\\mathcal{K}}(\\Omega)$ here is kept complex, holding the phase shift that arises from the interferometer cavity transfer function. The phase of $\\ensuremath{\\mathcal{K}}(\\Omega)$ is useful for later generalizations. $\\ensuremath{\\mathcal{K}}(\\Omega)$ adds the amplitude quadtrature noise power to the phase quadrature fluctuations directly reflected from the interferometer, setting the noise gain $\\Gamma(\\Omega)$\n\\begin{align}\n  \\Gamma(\\Omega) &= 1 + |\\mathcal{K}(\\Omega)|^2\n  &\n  \\theta(\\Omega) &= \\arctan(|\\ensuremath{\\mathcal{K}}(\\Omega)|)\n                   \\label{eq:GammaTheta_standard}\n\\end{align}\nThe relationship between $\\Gamma(\\Omega)$ and $\\theta(\\Omega)$ from $\\ensuremath{\\mathcal{K}}(\\Omega)$ is stated above as reference, but it will more appropriately handle the complex $\\ensuremath{\\mathcal{K}}(\\Omega)$ when it is derived later. The value $|\\ensuremath{\\mathcal{K}}(\\Omega_\\ensuremath{\\text{sql}})| \\equiv 1$ defines the crossover frequency $\\Omega_\\ensuremath{\\text{sql}}$ between noise contributions from shotnoise imprecision and QRPN, corresponding to $\\Gamma(\\Omega_\\ensuremath{\\text{sql}})= 2$ and $\\theta(\\Omega_\\ensuremath{\\text{sql}})=-45^\\circ$. For the $\\chi(\\Omega)$ susceptibility of a free test mass, the factor $\\ensuremath{\\mathcal{K}}(\\Omega)$ can be expressed using only frequency scales.\n\\begin{align}\n  \\mathcal{K}(\\Omega)\n  &= -\\frac{\\Omega^2_\\ensuremath{\\text{sql}}}{\\Omega^2}\\left( \\frac{\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\\right)^2,\n  &&\\text{given}&\n  \\chi(\\Omega) &\\equiv \\frac{-1}{m \\Omega^2}\n    \\label{eq:Kchi_standard}\n\\end{align}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/dual_d.pdf}\n\t\\caption{\n    This figure plots the total quantum and classical noise measured in the LIGO detectors in displacement amplitude spectral density units. The black trace plots a reference measurement of the total noise without injected squeezing at 0.25Hz resolution over 1.5Hr integration for LLO and 1.1Hr for LHO. The orange shotnoise measurement shows the displacement calibration, $\\sqrt{G(\\Omega)}$, in amplitude density units. Subtracting the shotnoise level from the reference yields the gray datapoints, which have been rebinned using a median statistic applied after the subtraction and with a logararithmic bin spacing. The subtraction primarily shows the classical noise but also contains QRPN. Multiple measurements are taken at varied squeezing angles, with 5 of 12 plotted for Livingston (LLO) and 5 of 34 plotted for Hanford (LHO), using the same median rebinning method as the gray subtraction. The variation in the data errorbars results from the binning span of each datapoint, $\\Delta F$, and the measurement integration time, $\\Delta T$. The measured spectra error relative to the total noise and proportional to $1\/\\sqrt{\\Delta F \\Delta T}$. The squeezing angle of $-3.9^\\circ$ and $1.4^\\circ$ datasets at LLO used ${\\sim}1$Hr integration, and the remainder used 15 min each. The squeezing angle $4.5^\\circ$ dataset at LHO used ${\\sim}1$Hr integration, while all others use 2 minutes each. The squeezing level $e^{\\pm2r}$ is constant over all angles, but different between the two sites. This accounts for the difference in the yellow, ${\\sim}30^{\\circ}$, dataset at each site.\n  }\n\t\\label{fig:data_h}\n\\end{figure*}\n\nFrequency independent losses are applied to squeezing before and after the interferometer using $\\eta=\\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{R}}\\eta_\\ensuremath{\\text{O}}$ where $\\eta_\\ensuremath{\\text{I}} < 1$, $\\eta_\\ensuremath{\\text{O}} < 1$. The ideal interferometer assumption of the formulas above enforce $\\eta_\\ensuremath{\\text{R}}=1$. Phase noise in squeezing is included in this ideal interferometer case using $\\Xi' = \\ensuremath{\\phi^2_{\\text{rms}}}$.\n\nThe above expressions relate the optical noise $N(\\Omega)$ of \\cref{eq:metric_N} to past models of the quantum strain sensitivity of GW interferometers\\cite{Kimble-PRD01-ConversionConventional, Buonanno-PRD01-QuantumNoise, Buonanno-PRD03-ScalingLaw}. Since $N(\\Omega)$ is relative to shot-noise, it must then be converted to strain or displacement using the optical cavity gain $g(\\Omega)$, by how it affects the GW signal through the calibration factor $G(\\Omega)$. This factor $G(\\Omega)$ relates strain modulations to optical field phase modulations in units of optical power.\n\\begin{align}\n  \\text{PSD}_{\\text{strain}}(\\Omega) &= G(\\Omega)N(\\Omega),\n                                       &\n  G(\\Omega) &= \\frac{\\hbar c}{\\eta_\\ensuremath{\\text{O}} L_{\\ensuremath{\\text{a}}}^2|g(\\Omega)|^{2} k P_\\ensuremath{\\text{A}} }\n              \\label{eq:optical_calibration_G}\n\\end{align}\nTogether, these relations allow one to succinctly calculate the effect of squeezing on the strain power spectrum in the case of an ideal interferometer. These factors and the calculations behind them will be revisited as non-idealities are introduced.\n\n\\section{Experimental Analysis and Results}\n\\label{sec:experiment}\n\nA goal of this paper is to use the squeezing response metrics of \\crefrange{eq:metric_N}{eq:metric_Lambda} to relate measurements of the instrument's noise spectrum to the parameters of the squeezer system, namely its degradations due to loss $1-\\eta$, radiation pressure from mis-phasing $\\phi{-}\\theta(\\Omega)$, and dephasings $\\Xi'(\\Omega)$. This section presents measurements from the LIGO interferometers that are best described using the established frequency-dependent metrics. The measurements then motivate the remaining discussion of the paper that construct simple interferometer models to describe this data in the context of the metrics. This section refers to and relates to the later sections to provide early experimental motivation for the discussions that follow. The reader may prefer instead to skip this section and first understand the models before returning to see their application to experimental data.\n\nThe main complexity in analyzing the LIGO data is that the detectors have additional classical noises, preventing a direct measurement of $N(\\Omega)$. The many frequency-dependent squeezing parameters must also be appropriately disentangled. To address both of these issues, the unknown squeezing parameters are fit simultaneously across multiple squeezing measurements. The classical noise contribution is determined by taking a reference dataset where the squeezer is disabled, such that $S(\\Omega)=1$, and then subtracting it from the datasets where squeezing is injected.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/dual_Q.pdf}\n  \\caption{\n    This figure shows the data of \\cref{fig:data_h} processed as per \\cref{sec:experiment} for each LIGO site. The processing subtracts away the classical noise determined from the unsqueezed reference dataset. The top panels show the relative noise change $Q_k(\\Omega_i)$ of \\cref{eq:Q_processing} computed using $\\Gamma_k(\\Omega_i)$ from the exact interferometer model of \\cref{sec:matrix_coupled_cavity} using the parameters of \\cref{tab:LIGO_params}. The top panel includes dots with errorbars for the processed data and lines for the best-fit $Q_k(\\Omega_i)$. The middle panel shows the best-fit frequency dependent loss as data points, with errorbars propagated through the fit. For LLO, two sets of loss datapoints are shown, corresponding to interferometer models with different readout angles $\\zeta$.\n    The loss plots also show $1{-}\\eta(\\Omega)$ as computed from the exact matrix model, along with a phenomenological fit against the model of \\cref{eq:Lambda_MM} of \\cref{sec:modeling_TMM}. The phenomenological fit assumes frequency independent losses from the input and output squeezing path with a frequency-dependent addition attributed to transverse mismatch. The bottom panels show the frequency-dependent fit to the observed squeezing angle $\\theta_k(\\Omega_i)$, using the convention of $\\theta(2\\pi{\\cdot}3\\text{kHz}) = 0$. It also plots $\\theta(\\Omega)$ as computed using the exact matrix model. For the LLO data, the $\\zeta=0^\\circ$ model is typically assumed for Michelson-like interferometers such as LIGO; However, the model at that readout angle implies losses at low frequencies that are not favored by the $\\eta(\\Omega)$ models explored in this paper. Alternatively, the $\\zeta\\approx -13^\\circ$ model is consistent with both the fitted losses and the fitted squeezing angles.\n  }\n  \\label{fig:data_Q}\n\\end{figure*}\n\nRepresentative strain spectra from the LIGO Livingston (LLO) and LIGO Hanford (LHO) observatory datasets are plotted in \\cref{fig:data_h}. The Livingston dataset is also reported in \\cite{Yu-N20-QuantumCorrelations}, which details the assumptions and error propagation for the classical noise components and calibration. Only statistical uncertainty is considered in this analysis, in order to propagate error to the parameter fits. The strain spectra of \\cref{fig:data_h} include a reference dataset where the squeezer is disabled, shown in black and at the highest frequency resolution. Additionally, the shotnoise ($N=1$) is plotted in orange, indicating the calibration $\\sqrt{G(\\Omega)}$ of \\cref{eq:optical_calibration_G}. The gray subtraction curve depicts the total classical noise contribution summed with the radiation pressure noise $G(\\Omega)\\ensuremath{\\mathcal{K}}^2(\\Omega)$. The gray dataset can equivalently be computed using a cross correlation of the two physical photodetectors at the interferometer readout\\cite{martynov-pra17-quantumcorrelation}. The equivalence of subtraction and cross correlation is used to precisely experimentally determine the shot-noise scale $G(\\Omega)$ from the displacement-calibrated data.\n\n\\subsection{Analysis}\n\nEach squeezing measurement, indexed by $k$, is indicated by $M_{\\text{ref}, k}(\\Omega_i)$, with a value at each frequency indexed by $i$. The reference dataset is denoted $M_{\\text{ref}}(\\Omega_i)$. The two are subtracted to cancel the stationary classical noise component. The calibration $G(\\Omega)$ is removed to result in the differential quantum noise measurement $D_k(\\Omega_i)$.\n\\begin{align}\n  D_k(\\Omega_i)\n  &\\equiv\n    \\frac{M_{\\text{sqz}, k}(\\Omega_i) - M_{\\text{ref}}(\\Omega_i)}{G(\\Omega_i)}\n    \\label{eq:D_processing}\n\\end{align}\nFor these datasets, the squeezing level $e^{\\pm 2r}$, is held constant and independently measured using the nonlinear gain technique\\cite{Dwyer-OEO13-SqueezedQuadrature} to derive $y$ of \\cref{eq:OPA_y_SQZ}. Each differential data $D_k(\\Omega_i)$ is taken at some squeezing angle $\\phi_k$, which is either fit (LLO) or derived from independent measurements (LHO). The parameters $\\eta_i$ and squeezing rotation $\\theta_i$ are independent at every frequency $\\Omega_i$ but fit simultaneously. All $\\phi_k$ are also fit simultaneously across all datasets. Nonlinear least squares fitting was performed using the Nelder-Mead simplex algorithm \\cite{Gao-COA12-ImplementingNelderMead} implemented in SciPy \\cite{Virtanen-NM20-SciPyFundamental}. The residual minimized by least squares fitting is\n\\begin{align}\n  \\mathcal{R} &= \\sum_{\\substack{i=0\\\\k=0}}^{\\mathcal{N}} \\left( \\frac{D_k(\\Omega_i) - \\overline{D}_k(\\Omega_i)}{\\Delta D_k(\\Omega_i)}\\right)^2\n\\end{align}\nThe measurement statistical uncertainty $\\Delta D$, dominated by the statistical uncertainty in power-spectrum estimation, was propagated through the datasets per \\cite{Yu-N20-QuantumCorrelations}. $\\overline{D}_k(\\Omega_i)$ is the model of the data that is a function of the fit parameters, $\\eta_i, \\theta_i, \\phi_k$ as well as independently measured parameters such as $e^{-2r}$. $\\Xi'(\\Omega_i)$ is not fit using this data since the squeezing level $e^{2r}$ is not varied across the datasets. This is discussed below. These given fit parameters affect are propagated through the squeezing metric functions create a model of this particular differential quantum noise measurement.\n\\begin{align}\n  \\overline{S}_k(\\Omega_i) &\\equiv\n  e^{-2r}\\cos^2\\Big(\\ensuremath{\\phi}_k - \\theta_i\\Big) + e^{+2r}\\sin^2\\Big(\\ensuremath{\\phi}_k - \\theta_i\\Big)\n  \\\\\n  \\overline{D}_k(\\Omega_i)\n  &\\equiv \\left.N(\\Omega_i)\\right|_{S = \\overline{S}_k(\\Omega_i)} - \\left.N(\\Omega_i)\\right|_{S = 1}\n\\end{align}\nWhich simplifies to\n\\begin{align}\n  \\overline{D}_k(\\Omega_i)\n  &= \\left(\\overline{S}_k(\\Omega_i) - 1\\right) \\eta_i \\Gamma(\\Omega_i)\n\\end{align}\nNotably, the individual efficiencies $\\eta_\\ensuremath{\\text{I}}, \\eta_\\ensuremath{\\text{R}}, \\eta_\\ensuremath{\\text{O}}$ cannot be individually measured and only the ``total'' efficiency $\\eta(\\Omega)$ is measurable using this differential method, where the classical noise is subtracted using a reference dataset with squeezing disabled. Additionally, the optical efficiency $\\eta$ can only be inferred given some knowledge or assumption of $\\Gamma(\\Omega)$. In effect, the product $\\eta \\Gamma$ is the primary measurable quantity, rather than its decomposition into separate $\\eta$ and $\\Gamma$ terms; However, for the purposes of modeling, decomposing the two is conceptually useful. Furthermore, to characterize physical losses, the efficiency $\\eta$ or loss $\\Lambda_\\ensuremath{\\text{IRO}}\\approx 1 {-} \\eta$ is easier to plot and interpret than the product $\\eta\\Gamma$.\n\nFor these reasons, the differential data $D_k(\\Omega_i)$ is further processed, creating the measurement $Q_k(\\Omega_i)$ with a form similar to \\cref{eq:etaSQZ_basic1}\n\\begin{align}\n  Q_k(\\Omega_i)\n  &\\equiv \n    \\frac{D_k}{\\Gamma(\\Omega_i)} + 1 \\approx S_k(\\Omega_i)\\eta_i + (1 - \\eta_i) + \\Delta Q\n    \\label{eq:Q_processing}\n\\end{align}\nThe LIGO squeezing data expressed in dB's of $Q_k(\\Omega_i)$ are plotted in the upper panels of \\cref{fig:data_Q}. The data and error bars are in discrete points, while the parameter fits to $Q_k$ using $\\eta_i$, $\\theta_i$ and $\\phi_k$ are the solid lines between the data points. The spectra in each set are calculated using the Welch method a median statistic at each frequency to average all of the frames through the integration time. This prevents biases due to instrumental glitches adding non-stationary classical noise. This technique is detailed in \\cite{Yu-N20-QuantumCorrelations}.\n\nAfter computing $Q_k(\\Omega_i)$ at full frequency resolution, the data is further rebinned to have logararithmic spacing by taking a median of the data points within the frequency range of each bin. This rebinning greatly improves the statistical uncertainty at high frequencies, where many points are collected. At lower frequencies, the relative error benefits less from binning; however, both the LLO and the LHO datasets use a long integration time for their reference measurement and at least one of the squeezing angle measurements. Using the median removes narrow-band lines visible in the strain spectra of \\cref{fig:data_h}. Fitting combines the few long-integration, low-error datasets with many short-integration, high-error sets at many variations of the operating parameters. The few low-error datasets reduce the absolute uncertainty in the resulting fit parameters, whereas the many variations reduce co-varying error that would otherwise result from modeling parameter degeneracies.\n\nThe relative statistical error in each bin of the original PSD $M_k(\\Omega_i)$ is approximately $(\\Delta F \\Delta T)^{-{1}\/{2}}$ given the integration time $\\Delta T$ of 2 minutes to 1 hour and bin-width $\\Delta F$ of 0.25Hz. This relative error is converted to absolute error and propagated through the processing steps of \\crefrange{eq:D_processing}{eq:Q_processing}. At low frequencies, the classical noise contribution to each $M_k$ is larger than the quantum noise. Although it is subtracted away to create $D_k(\\Omega_i)$, the classical noise increases the absolute error, and, along with less rebinning, results in the larger relative errors at low-frequency in \\cref{fig:data_Q}. After fitting the squeezing parameters, the Hessian of the reduced chi-square is computed from the Jacobian of the fit residuals with respect to the parameters. This Hessian represents the Fisher information, and the diagonals of its inverse provides the variances indicated by the plotted loss and angle parameter error bars.\n\nFor the LHO data, the fit parameters $\\phi_k$ are determined by mapping the demodulation angles of its coherent control feedback system\\cite{Tse-PRL19-QuantumEnhancedAdvanced, Dooley-OEO15-PhaseControl, Vahlbruch-PRL06-CoherentControl, Chelkowski-PRA07-CoherentControl} back to the squeezing angle. That mapping has 3 unknown parameters, an offset in demodulation angle, an offset in squeezing angle, and a nonlinear compression parameter, all of which are fit simultaneously in all datasets. This $\\phi_k$ mapping was not performed on the LLO data, as some systematic errors in the demodulation angle records bias the results. Despite fitting more independent parameters, the longer integration time of the LLO data gives it sufficiently low statistical uncertainty at frequencies below $\\Omega_\\ensuremath{\\text{sql}}$ that the model and parameter degeneracy between $\\phi_k$, $\\theta_i$ and $\\eta_i$ is not an issue.\n\n\\subsection{Results}\n\nThe middle panels of \\cref{fig:data_Q} show the fits to $\\eta_i$, though plotted as loss $1{-}\\eta_i$ to represent $\\Lambda_\\ensuremath{\\text{IRO}}$. Both datasets additionally include a red loss model curve fit, assembled using the equations in \\cref{sec:modeling}. The orange exact model curves use \\cref{sec:matrix_coupled_cavity}. The data and model curve fit shows a variation in the efficiency, where losses increase from low to high frequencies. This increase in loss can be attributed either to losses within the signal recyling cavity of the interferometer, or to a coherent effect resulting from transverse Gaussian beam parameter mismatch between the squeezer and interferometer cavities. At low frequencies, the optical efficiency is similar between the two LIGO sites, indicating that frequency independent component to the loss are consistent between the implementations at both LIGO sites. The differing high-frequency losses can reasonably be ascribed to variations in the optical beam telescopes of the squeezing system and are analyzed in \\cref{sec:modeling_TMM}.\n\nThe LLO middle panel of \\cref{fig:data_Q} shows two separate inferred loss $1 - \\eta_i$ datasets. These differ in their underlying model of $\\Gamma(\\Omega_i)$. The following section \\ref{sec:derivations} discusses how variations in $\\Gamma$ arise and describes the local oscillator angle $\\zeta$. The $\\zeta = 0$ data reflects the standard, ideal radiation pressure noise model of \\crefrange{eq:Kchi_standard}{eq:GammaTheta_standard}. This model is disfavored given the frequency dependency of $\\eta(\\Omega_i)$ derived using optical cavity models later in this paper. The $\\zeta = -13^\\circ$ model presents an alternative that is compatible with models of the optical efficiency. The need for this alternative indicates that squeezing metrics must account for variations in interferometer noise gain $\\Gamma$. Physically, these variations arise from the readout angle adjusting the prevalence of radiation pressure versus pondermotive squeezing. The $\\zeta = -13^\\circ$ model results in a smaller noise gain $\\Gamma$ at $40 \\text{Hz}$ than does the $\\zeta = 0^\\circ$ model. Since the lower $\\Gamma$ model is favored, this dataset provides some, moderate, evidence that LLO currently benefits from the quantum correlations introduced by the mirrors near $\\Omega_{\\text{SQL}}$, while experiencing lessened sensitivity elsewhere.\n\nThis data demonstrates that the readout angle has an effect on the interferometer sensitivity and the optimal local oscillator is not necessarily $\\xi = 0$ due to radiation pressure.  The quantum benefit of decreased $\\Gamma$ from the readout angle $\\xi$ is a method to achieve sub-SQL performance that is an alternative to injecting squeezing. Like squeezing, it has a frequency dependent enhancement known as the ``variational readout'' technique \\cite{Kimble-PRD01-ConversionConventional, Khalili-PRD07-QuantumVariational}, that a sensitivity increase from lowering $\\Gamma$ while minimizing the sensitivity decrease of the frequency-independent form. For LLO, the reduced sensitivity from $\\xi \\ne 0$ masquerades as a $5\\%$ loss of signal power, but does not actually affect the $\\eta$ or $\\Lambda_\\ensuremath{\\text{IRO}}$ contributions to the squeezing level.\n\nThe bottom panels of \\cref{fig:data_Q} show the fits of $\\theta_i$ of each dataset. The magnitude of $e^{\\pm 2r}$ provides a ``lever arm'' in the variation of $S_k(\\Omega_i)$ that strongly constrains the $\\phi_k{-}\\theta_i$ effective squeezing angle. These leveraged constraints result in small errorbars to the fitted $\\theta_i$. The LLO data are plotted with two models of the $\\theta_i$ based on the assumed local oscillator angle $\\zeta$. The $\\zeta=0^\\circ$ model follows the standard radiation pressure model of \\cref{eq:GammaTheta_standard} at low frequencies and includes a filter-cavity type rotation around the interferometer cavity bandwidth $\\gamma \\approx 2\\pi \\cdot 450\\text{ Hz}$. This rotation is modeled in \\cref{sec:modeling_cavities}. The $\\zeta=-13^{\\circ}$ model is computed using the coupled cavity model of \\cref{sec:matrix_coupled_cavity} and internally includes a weak optical spring effect along with the shifted readout angle $\\zeta$. Together, these effects modify the effective squeezing angle $\\theta$ away from \\cref{eq:GammaTheta_standard} at low frequencies, and agree well with the dataset. This agreement provides further evidence of the reduced radiation pressure noise gain $\\Gamma(\\Omega)$ in LLO that results from the effective LO readout angle $\\zeta$. A nonzero readout angle $\\zeta$ is reasonable to expect due to unequal optical losses in the LIGO interferometer arm cavities. The arm mismatch results in imperfect subtraction of the fringe-light amplitude quadrature at the beamsplitter, creating a static field that adds to the phase-quadrature light created from the Michelson offset and results in $\\zeta \\ne 0$. Past diagnostic measurements\nconclude that some power in the readout diodes must be in the amplitude quadrature, but until now could not determine the sign.\n\nAlthough the squeezing angle parameters $\\phi_k$ and $\\theta_i$ are fit, the frequency-dependent dephasing parameter $\\Xi_i$ cannot be reliably determined from these datasets given the accuracy to which $e^{\\pm 2r}$ is measured. Additionally, the squeezing level $e^{\\pm 2r}$ is not varied in this data, nor is it sufficiently large to resolve an influence from $\\Xi(\\Omega) < 10^{-3}$. This $\\sqrt{\\Xi} \\approx \\ensuremath{\\phi_{\\text{rms}}}$ is expected from independent measurements of phase jitter that propagate through the coherent control scheme of the squeezer system\\cite{Tse-PRL19-QuantumEnhancedAdvanced}. A large source of optically induced $\\Xi$ is not expected has the interferometer cavities are not sufficiently detuned. Measurements of the squeezing system indicate $\\ensuremath{\\phi_{\\text{rms}}} \\lesssim 30\\text{ mRad}$. Future LIGO measurements should include additional datasets that vary $r$ along a third indexing axis $j$ and should increase the injected squeezing level $e^{2r} > 30$ to measure, or at least bound, $\\Xi$ and its frequency-independent contribution $\\ensuremath{\\phi^2_{\\text{rms}}}$. The model fits described above are consistent with the data while assuming $\\Xi = \\ensuremath{\\phi^2_{\\text{rms}}} \\equiv 0$.\n\n\\section{Decomposition Derivation}\n\\label{sec:derivations}\n\nThe factors $\\eta(\\Omega), \\theta(\\Omega)$ and $\\Xi(\\Omega)$ from \\cref{sec:metrics} each describe an independent way for squeezing to degrade. $\\Gamma(\\Omega)$ indicates how the quantum noise scales above or below the shot noise level from squeezing and from quantum radiation pressure within the interferometer. They represent a natural extension of standard squeezing metrics that incorporates frequency dependence, and, as scalar functions, they are simple to plot and to relate with experimental measurements. This section delves into their derivation by employing matrices in the two photon formalism \\cite{Caves-PRA85-NewFormalism, Schumaker-PRA85-NewFormalism} to represent the operations of squeezing, adding loss, shifting the squeezing phase, reflecting from the interferometer, and final projection of the quantum state into the interferometer readout. The derived formulas can be used in frequency-domain simulation tools that compute noise spectra using matrix methods, so that the quantum response metrics can be provided in addition to opaquely propagating squeezing to an simulation result of $N(\\Omega)$. \n\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_cavities, \n  label=SQZ_mm_cavities, \n  caption={\n    The two-photon transformation matrices experienced by squeezing through the sequence of \\cref{fig:SQZ_mm_IFO}. The effective linear coupled cavity, including the optomechanical effect of radiation pressure, is collected and computed into the transformation $\\Tmat{H}_R$. The middle cavity is the signal recycling cavity and the rightmost cavity represents the coherent combination of both arms. Each cavity adds losses from each mirror. For simplicity, these are collected into round-trip cavity loss contributions, $\\Lambda_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{s}}}$, and $\\Lambda_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{s}}}$ that inject standard optical vacuum into the cavities, circulating and transforming into the loss terms $\\Tmat{T}_{\\ensuremath{\\text{R}},\\ensuremath{{\\,\\mu}}}$ while lowering the efficiency $\\eta_\\ensuremath{\\text{R}}$. Transformations of the squeezing at the input and output are included with the terms, $\\Tmat{H}_{\\ensuremath{\\text{I}}}$, $\\Tmat{H}_{\\ensuremath{\\text{O}}}$ and any additive vacuum contributions,  $\\Tmat{T}_{\\ensuremath{\\text{I}},\\ensuremath{{\\,\\mu}}}$, $\\Tmat{T}_{\\ensuremath{\\text{O}},\\ensuremath{{\\,\\mu}}}$.\n  },\n}\n\nTwo-photon matrices are an established method to represent transformations of the optical phase space of Guassian states in an input-output Heisenberg representation of the instrument\\cite{Danilishin-LRR12-QuantumMeasurement}. They are concise yet rigorous when measuring noise spectra from squeezed states using the quantum measurement process of Homodyne readout. Section II of \\cite{Danilishin-LRR19-AdvancedQuantum} provides a review of their usage in the context of gravitational-wave interferometers. Here, two-photon matrices are indicated by doublestruck-bold lettering, and are given strictly in the amplitude\/phase quadrature basis.\n\nEach matrix represents the transformation of the optical phase space of a single optical ``mode'' as it propagates through each physical element towards the readout. The term ``mode'' refers to a basis vector in a linear decomposition of optical field the transverse optical plane of many physical ports\\footnote{Simulations also typically span multiple optical frequencies, but this is not treated here.}. Each plane is further decomposed into transverse spatial modes using a Hermite or Laguerre-Gaussian basis. In this decomposition, each optical mode is indexed by the placeholder $\\mu$ and acts as a continuous transmission channel for optical quantum states. The phase space transformations of these continuous optical states is indexed by time or, more conveniently, frequency. Optical losses and mixing from transverse mismatch behave like beamsplitter operations, serving to couple multiple input modes, generally carrying vacuum states, to the mode of the readout where states are measured.\n\nThe mode of the injected squeezed states, and their specific transformations during beam propagation, must be distinguished from all of the lossy elements that couple in vacuum states. The squeezed states experience a sequence of transformations by the input elements, interferometer, and output elements, denoted $\\Tmat{H}_\\ensuremath{\\text{I}}(\\Omega)$,  $\\Tmat{H}_\\ensuremath{\\text{R}}(\\Omega)$,  $\\Tmat{H}_\\ensuremath{\\text{O}}(\\Omega)$. This sequence multiplies to formulate the total squeeze path propagation $\\Tmat{H}$. \n\\begin{align}\n  \\Tmat{H}(\\Omega) &= \n\\Tmat{H}_{\\text{O}}\\Tmat{H}_{\\text{R}}\\Tmat{H}_{\\text{I}}\n\\end{align}\nLossy optical paths mix the squeezed states with additional standard vacuum states. These are collected into sets of transformation matrices corresponding to each individual loss source, $\\{\\Tmat{T}_\\mu\\}$. See \\cref{fig:SQZ_mm_cavities}. The sets are grouped by their location along the squeezing path where the lossy element is incorporated. The beamsplitter-like operation that couples each loss is given by a $\\Tmat{\\Lambda}_\\mu$, indexed by its location and source along the squeezing path. Loss transformations $\\Tmat{\\Lambda}_\\mu$ are generally frequency-independent. $\\Tmat{\\Lambda}_{\\text{R}, i}$ are an exception, as they occur within the cavities of the interferometer and include some cavity response. The vacuum states associated with each loss then propagate along with the squeezed states and experience the remaining transformations that act on squeezing.\n\\begin{align}\n  \\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{H}_{\\text{O}}\\Tmat{H}_{\\text{R}}\\Tmat{\\Lambda}_{\\text{I},\\ensuremath{{\\,\\mu}}}\n    \\\\\n  \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{H}_{\\text{O}}\\Tmat{\\Lambda}_{\\text{R},\\ensuremath{{\\,\\mu}}}\n    \\\\\n  \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{\\Lambda}_{\\text{O},\\ensuremath{{\\,\\mu}}}\n  \\\\\n  \\left\\{\\Tmat{T}\\right\\} &= \\left\\{\\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}; \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}; \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}  \\right\\}\n\\end{align}\nTogether, all of the transformations of $\\Tmat{H}$ and  $\\{\\Tmat{T}\\}$ define the output states at the readout of the interferometer in terms of the input states entering through the squeezer and loss elements. The two quadrature observables of the optical states are given with the convention $\\hat{q}$ being the amplitude quadrature and $\\hat{p}$ being the phase, and they are indexed to distinguish their input port and transverse mode.\n\\begin{align}\n  \\begin{bmatrix}\\hat{q}_{\\text{out}}(\\Omega) \\\\ \\hat{p}_{\\text{out}}(\\Omega)\\end{bmatrix}\n  &=\n    \\Tmat{H} \\begin{bmatrix}\\hat{q}_{\\text{in}}(\\Omega) \\\\ \\hat{p}_{\\text{in}}(\\Omega)\\end{bmatrix} + \n  \\sum_{\\Tmat{T}_\\ensuremath{{\\,\\mu}}\\in \\left\\{\\Tmat{T}\\right\\}} \\Tmat{T}_\\ensuremath{{\\,\\mu}} \\begin{bmatrix}\\hat{q}_\\ensuremath{{\\,\\mu}}(\\Omega) \\\\ \\hat{p}_\\ensuremath{{\\,\\mu}}(\\Omega)\\end{bmatrix}\n  \\label{eq:qp_out}\n\\end{align}\nThe two-photon matrices $\\Tmat{H}$ and $\\Tmat{T}_\\ensuremath{{\\,\\mu}}$ must preserve commutation relations, namely $[\\hat{q}_\\text{out}, \\hat{p}_\\text{out}] = [\\hat{q}_\\ensuremath{{\\,\\mu}}, \\hat{p}_\\ensuremath{{\\,\\mu}}] = i\\hbar$. In doing so, the matrices ensure that losses within $\\Tmat{H}$ couple ancillary vacuum states that degrade squeezing.\n\nThe readout carries a continuous coherent optical field known as the ``local oscillator'' and the output states are read using homodyne readout. The phase of the local oscillator, $\\zeta$, defines the observed quadrature, $\\hat{m}$, for the homodyne measurement. Gravitational Wave interferometers typically use a ``Michelson offset''\\cite{Fricke-CQG12-DCReadout, Hild-CQG09-DCreadoutSignalrecycled, Ward-CQG08-DcReadout} in the paths adjacent their beamsplitter to operate slightly off of dark fringe. This offset couples a small portion of their pump carrier light to their output as the local oscillator field. This is a form of homodyne readout that fixes $\\zeta$ to measure in the phase quadrature, defined here to be when $\\zeta=0$. Imperfect interference at the beamsplitter can couple some amplitude quadrature and shift $\\zeta$ away from $0$. Balanced homodyne readout is an alternative implementation proposed for LIGO's ``A+'' upgrade and will allow $\\zeta$ to be freely chosen\\cite{Fritschel-OEO14-BalancedHomodyne}. Regardless of the implementation, the homodyne observable is $\\hat{m}$,\n\\begin{align}\n  \\hat{m} &=\n  \\ensuremath{\\Tvec{v}^{\\dagger}}\\begin{bmatrix}\\hat{q}_{\\text{out}}(\\Omega) \\\\ \\hat{p}_{\\text{out}}(\\Omega)\\end{bmatrix}\n           \n  &\n  \\ensuremath{\\Tvec{v}^{\\dagger}}(\\zeta)\n  &=\n    \\begin{bmatrix}\n      \\sin(\\zeta) & \\cos(\\zeta)\n    \\end{bmatrix}\n \\label{eq:homodyne_observable}\n\\end{align}\nHomodyne readout enforces a symmetrized expectation operator, denoted here with the subscript HR, for all measurements of the optical quantum states.\nFurther details of the measurement process are beyond the scope of this work, but the following quadratic expectations arise when computing the noise spectrum and are sufficient to simplify the homodyne expectation values of $\\hat{m}$.\n\\begin{align}\n  1 &= \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}^2}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}^2}_{\\text{HR}},\n      \\label{eq:vacuum_expectations}\n  \\hspace{1em}\n  0 = \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}\\hat{p}_\\ensuremath{{\\,\\mu}}}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}\\hat{q}_\\ensuremath{{\\,\\mu}}}_{\\text{HR}}\n      \\\\\n  0 &= \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}\\hat{q}_\\ensuremath{{\\,\\nu}}}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}\\hat{p}_\\ensuremath{{\\,\\nu}}}_{\\text{HR}} \\text{  for } \\nu \\ne \\mu\n      \\label{eq:vacuum_expectations2}\n\\end{align}\nAs a result of these expectations, the vector norm suffices to evaluate noise power using this matrix formalism.\nThe addition of squeezing can be seen either as a modification of the input states $\\hat{q}_{\\text{in}}$, $\\hat{q}_{\\text{in}}$, which violate \\cref{eq:vacuum_expectations}. This work uses the alternative picture, where an additional squeezing transformation occurring at the very start of the squeezing path $\\Tmat{H}$ that acts on $\\hat{q}_{\\text{in}}$, $\\hat{q}_{\\text{in}}$ that are also vacuum states. The squeezing transformation is defined by the squeezing level $r$ and the squeezing angle $\\phi$, which act via the matrices:\n\\begin{align}\n  \\Tmat{R}(\\phi)\n  &\\equiv\n  \\begin{bmatrix}\n    \\cos(\\phi) & {-}\\sin(\\phi)\\\\\n    \\sin(\\phi) & \\phantom{-}\\cos(\\phi)\n  \\end{bmatrix}\n &\n  \\Tmat{S}(r) &\\equiv\n  \\begin{bmatrix}\n    e^r & 0\\\\\n    0 & e^{-r}\n  \\end{bmatrix}\n\\end{align}\nWhen added to the squeezing path, the resulting quantum noise is calculated from the observable $\\hat{m}$.\n\\begin{align}\n  N(\\Omega) &= \\Braket{\\hat{m}^\\dagger\\hat{m}}_{\\text{HR}} = \n              \\left| \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}\\Tmat{R}(\\phi) \\Tmat{S}(r) \\right|^2 + \n\\sum_{\\Tmat{T}_\\ensuremath{{\\,\\mu}}\\in \\left\\{\\Tmat{T}\\right\\}} \\left| \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_\\ensuremath{{\\,\\mu}} \\right|^2\n                             \\label{eq:N_no_sqz_derivation}\n\\end{align}\nThe first term of which is one of the factors in \\cref{eq:metric_N}\n\\begin{align}\n  \\eta(\\Omega)\\cdot S(\\Omega, \\phi) \\cdot \\Gamma(\\Omega, \\zeta)\n \n \n \n \n \n   &= \\left|\\ensuremath{\\Tvec{v}^{\\dagger}}\n     \\Tmat{H}\\Tmat{R}(\\phi)\n     \\Tmat{S}(r)\n     \\right|^2\n    \\label{eq:decomposition_relation}\n\\end{align}\nAt this point, the factors can be separated because: $\\Tmat{R}\\Tmat{S}$ determines the factor $S(\\Omega, \\phi)$; $\\Tmat{H}$ has been ``reduced'' by loss, indicating when $\\eta(\\Omega) < 1$; and the benchmark noise level is defined by $\\Gamma(\\Omega)$, contained in the interferometer's optomechanical element $\\Tmat{H}_{\\text{R}}$.\n\nTo distinguish these terms, further manipulations are necessary. The first is to examine just the vector $\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}$ to determine how the later term $\\Tmat{R}\\Tmat{S}$ results in $S(\\Omega)$. Basis vectors for the two quadrature observables are defined, and the local oscillator is represented using them.\n\\begin{align}\n  \\ensuremath{\\Tvec{v}^{\\dagger}}(\\zeta)\n  &=\n  \\Tvec{e}_p^\\dagger\\Tmat{R}(\\zeta)\n  &\n    \\Tvec{e}_q &= \n    \\begin{bmatrix}\n      1 \\\\ 0\n    \\end{bmatrix}\n  &\n    \\Tvec{e}_p &= \n    \\begin{bmatrix}\n      0 \\\\ 1\n    \\end{bmatrix}\n\\end{align}\nThe basis vectors then allow the vector norm to be split into its two components $\\ensuremath{m_q}$ and $\\ensuremath{m_p}$, defining the \\textit{observed noise quadrature}.\n\\begin{align}\n  \\ensuremath{m_q}(\\Omega) &= \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H} \\Tvec{e}_q & \\ensuremath{m_p}(\\Omega) &= \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H} \\Tvec{e}_p\n\\end{align}\nThe vector $\\vec{m}$ contains the magnitude and angle of a projection of the quantum state $\\hat{q}_{\\text{in}}$, $\\hat{p}_{\\text{in}}$ at each frequency, but it also contains the complex phase shift from propagation delay in the interferometer and squeezing path. This later phase contribution does not affect noise calculations, but must be properly handled. Projecting it away requires maintaining phase information, and this is why the optomechanical factor $\\ensuremath{\\mathcal{K}}$ is complex in this work.\n\nThe squeezing angle rotation $\\Tmat{R}(\\phi)$ can be viewed through its left-multiplication, applying a rotation to the observed noise quadtrature rather than to the squeezing. In this picture, the angle $\\phi$ can align the observed quadtrature with either the squeezing or anti-squeezing quadrature. The rotation needed to do so determines $\\theta(\\Omega)$, again with the caveat that both $m_q$ and $m_p$ are complex. Their common phase carries the delay information, but their differential phase causes dephasing. In short, differential phase forces $\\vec{m}$ to project into both quadratures at any rotation $\\Tmat{R}(\\phi)$. This has the effect of always adding anti-squeezing to squeezing and vice-versa, resulting in the factor $\\Xi(\\Omega)$. The relations are fully derived in \\cref{sec:phase_noise_composition} using a singular value decomposition to identify the principle noise axes. It leads to the expressions\n\\begin{align}\n  \\theta(\\Omega) &\\approx -\\arctan\\left(\\Re\\!\\left\\{{\\frac{\\ensuremath{m_q}}{\\ensuremath{m_p}}}\\right\\}\\right)\n                   \\label{eq:theta_calculation}\n  \\\\\n  \\Xi(\\Omega) &= \\frac{1}{2} -  \\sqrt{\n        \\frac{\\left(|\\ensuremath{m_q}|^2 - |\\ensuremath{m_p}|^2\\right)^2 + 4 \\Re\\left\\{ \\ensuremath{m_q}\\ensuremath{\\conj{m}_p} \\right\\}}\n        {4\\left(|\\ensuremath{m_q}|^2 + |\\ensuremath{m_p}|^2\\right)^2}\n        } \n                   \\label{eq:Xi_calculation}\n\\end{align}\nThe observation vector $\\vec{m}$, and \\cref{eq:theta_calculation} generalizes the observed noise quadrature description of quantum radiation pressure noise. With it, the observed quadrature angle $\\theta(\\Omega)$ may be computed for any readout angle $\\zeta$ and for more complex interferometers $\\Tmat{H}_\\ensuremath{\\text{R}}$. The ideal interferometer example is demonstrated in \\cref{sec:ideal_IFO_example}\n\nThe phase and magnitude of of the previous argument allows one to determine $S(\\Omega)$ from the form of $\\Tmat{S}$ applied to $\\vec{m}\\Tmat{R}(\\phi)$. Factoring $S$ away, the magnitude of $\\vec{m}$ carries the efficiency of transmitting the squeezed state, along with the noise gain applied to it.\n\\begin{align}\n  \\eta(\\Omega) \\cdot \\Gamma(\\Omega, \\zeta)\n  &=\n    |\\ensuremath{m_q}|^2 + |\\ensuremath{m_p}|^2\n    \\label{eq:eta_gamma}\n\\end{align}\n$\\Gamma(\\Omega)$ expresses the total noise from the interferometer when squeezing is not applied, applying radiation pressure or optomechanical squeezing to both the squeezing path vacuum and internally loss-sourced vacuum. $\\eta \\Gamma$ is affected by all losses, but some of them affect $\\Gamma(\\Omega)$ as well. Using squeezing or a coherent field to probe $\\Tmat{H}$ always measures the product $\\eta \\Gamma$, so the noise gain factor $\\Gamma$ serves primarily as a benchmark. As a benchmark, it relates the dependence of $N(\\Omega)$ to $S(\\Omega)$ and separates the scaling by the efficiency $\\eta$ so that the physical losses may be determined. For this reason, there is freedom to define $\\Gamma$ to make it as independent from the losses as possible, so that it best serves as a benchmark. Here, it is defined using the simulated knowledge of the total noise from the interferometer elements alone:\n\\begin{align}\n \\Gamma(\\Omega) \n  &= \n\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}_{\\text{R}}\\right|^2 + \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{\\Lambda}_{\\text{R},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\label{eq:gamma_HR}\n\\end{align}\n$\\eta$ is then determined by dividing \\cref{eq:eta_gamma} by \\cref{eq:gamma_HR}. Under this definition of $\\Gamma$, $\\eta \\propto \\eta_\\ensuremath{\\text{I}}$ and $\\eta \\propto \\eta_\\ensuremath{\\text{O}}$. Losses within the interferometer affect $\\Gamma(\\Omega)$ slightly, and  $\\eta \\propto \\eta_\\ensuremath{\\text{R}}$ is only approximate. \\Cref{sec:radiation_pressure_calc} gives an example of how losses affect $\\eta$ and $\\Gamma$. The primary alternative definition is to use $\\Gamma = N\\big|_{S=1}$, but this definition makes $\\eta_\\ensuremath{\\text{O}}$ both less physically intuitive and also sensitive to interferometer parameters.\n\nSubtracting $\\eta\\Gamma$ from \\cref{eq:N_no_sqz_derivation} and factorizing by the optical paths provides the definition of the remaining efficiency terms.\n\\begin{align}\n  (1 - \\eta_\\ensuremath{\\text{O}})\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\\\\n  \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}}) \\Gamma\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\\\\n  \\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}(1 - \\eta_\\ensuremath{\\text{I}}) \\Gamma\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}\\right|^2\n\\end{align}\nWhich add together to create the loss term in \\cref{eq:metric_N}.\n\\begin{align}\n    \\Lambda_\\ensuremath{\\text{IRO}}\\Gamma &= \\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}(1 - \\eta_\\ensuremath{\\text{I}})\\Gamma\n    + \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}})\\Gamma\n    + (1 - \\eta_\\ensuremath{\\text{O}})\n \n \n \n \n \n \n \n\\end{align}\n\n\\subsection{Ideal Interferometer Example}\n\\label{sec:ideal_IFO_example}\n\nThe derivations are now extended to recreate and generalize the ideal noise model of \\cref{sec:ideal_IFO_metrics}, using \\cref{eq:Kchi_standard} for $\\ensuremath{\\mathcal{K}}$. The two-photon matrix corresponding to the interferometer in \\cref{fig:SQZ_mm_cavities} is given below for the lossless interferometer that is perfectly on resonance.\n\\begin{align}\n  \\Tmat{H}_\\ensuremath{\\text{R}}(\\Omega) &\\simeq\n  \\begin{bmatrix}\n    \\ensuremath{\\mathfrak{r}}(\\Omega) & 0\\\\\n    \\ensuremath{\\mathcal{K}}(\\Omega) & \\ensuremath{\\mathfrak{r}}(\\Omega)\n  \\end{bmatrix},\n  &\n  \\ensuremath{\\mathfrak{r}}(\\Omega) &\\simeq\n\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega},\n                 &\\Tmat{\\Lambda}_\\ensuremath{\\text{R}} &= \\Tmat{0}\n \n \n \n \n \n \n \n \n \n \n \n \n \n\\end{align}\nIn the ideal lossless case, the input and output paths also have perfect efficiency $\\eta_\\ensuremath{\\text{I}} \\simeq 1$ with $\\Tmat{H}_\\ensuremath{\\text{I}}(\\Omega) = \\eta_\\ensuremath{\\text{I}}\\Tmat{1},\\; \\Tmat{\\Lambda}_\\ensuremath{\\text{I}}(\\Omega) = \\sqrt{1-\\eta_\\ensuremath{\\text{I}}}\\Tmat{1}$ and similarly for the output. These can be used to compute $\\Tmat{H}$ and $\\vec{m}$.\n\\begin{align}\n  m_q &= \\cos(\\zeta)\\ensuremath{\\mathcal{K}}(\\Omega) + \\sin(\\zeta)\\ensuremath{\\mathfrak{r}}(\\Omega)\n  ,&\n  m_p &= \\cos(\\zeta)\\ensuremath{\\mathfrak{r}}(\\Omega)\n\\end{align}\nThe equations above maintain the correct phase information for this ideal case analysis. Interestingly, $\\ensuremath{\\mathcal{K}}$ and $\\ensuremath{\\mathfrak{r}}(\\Omega)$ have different magnitude responses resulting from different factors of $\\gamma_\\ensuremath{\\text{A}} \\pm i\\Omega$, yet their phase response is the same. This Kramers-Kronig coincedence ensures $\\Xi(\\Omega)=0$ as long as the $\\chi(\\Omega)$ contribution to $\\ensuremath{\\mathcal{K}}(\\Omega)$ is purely real. Thus, lossy mechanics will cause QRPN to dephase injected squeezing. This will not happen to any meaningful level for LIGO, but is noteworthy for optomechanics experiments operating on mechanical resonance.\n\nThe $\\vec{m}$ above also includes the effect of the readout angle. For $\\zeta=0$, it recovers \\crefrange{eq:optical_gain_g}{eq:Kchi_standard}. More generally, it gives\n\\begin{align}\n  \\Gamma(\\Omega) &= |\\cos(\\zeta)\\ensuremath{\\mathcal{K}}(\\Omega)|^2 + \\sin(2\\zeta)|\\ensuremath{\\mathcal{K}}(\\Omega)| + 1\n  \\\\\n  \\theta(\\Omega) &= \\arctan\\big(|\\ensuremath{\\mathcal{K}}(\\Omega)| - \\tan(\\zeta)\\big)\n\\end{align}\nThe exact expressions above can be simplified to better relate them to the LIGO data. Firstly, the squeezing angle is modified to be 0 at high frequencies, to match the conventions of the data. This modified angle is $\\theta'(\\Omega) = \\theta(\\Omega) - \\theta(\\Omega{\\gg}\\gamma_\\ensuremath{\\text{A}})$. Secondly, small shifts of the homodyne angle are linearized.\n\\begin{align}\n  \\Gamma'(\\Omega) &\\approx\n                  \n                  \n                   \\big(1 - |K(\\Omega)|\\big)^2 + 2\\big(1 + \\zeta\\big)|K(\\Omega)|\n \n  \\\\\n  \\theta'(\\Omega) &\\approx \\arctan\\big(|\\ensuremath{\\mathcal{K}}(\\Omega)|) - \\zeta \\frac{|\\ensuremath{\\mathcal{K}}(\\Omega)|^2}{1 + |\\ensuremath{\\mathcal{K}}(\\Omega)|^2}\n\\end{align}\nThe linearized $\\Gamma'$ shows that , when $\\zeta=-13^\\circ=-0.23$, for frequencies near $\\Omega_\\ensuremath{\\text{sql}}$, $\\ensuremath{\\mathcal{K}}(\\Omega_\\ensuremath{\\text{sql}}) \\simeq 1$, the interferometer quantum noise is reduced by about 23\\% with respect to a nominal $\\xi=0$ readout. This change is shown in the blue vs. grey plotted data for the Livingston loss plot in \\cref{fig:data_Q} of $1-\\eta$. There, $\\eta$ changes as the $\\Gamma$ model changes since only $\\eta\\Gamma$ can be measured due to subtracting an unsqueezed reference dataset. The $23\\%$ noise reduction corresponds to approximately 1dB improvement from pondermotive quantum correlations. The angle formula above indicates that for frequencies $\\Omega\\lesssim\\Omega_\\ensuremath{\\text{sql}}$, the local oscillator also adds some additional shift to $\\theta$ at low frequency, which is also observed in the LLO angle fits.\n\nThis analysis gives an example of how the derivations of this section are applied to extend the existing ideal interferometer models towards the real instruments. Exact models including more optical physics are yet more analytically opaque, but give a more complete complete picture if implemented numerically. \\Cref{sec:matrix_coupled_cavity} shows the full matrix solution, including the cavities, to recover these equations while also handling cavity length offset detunings. It also includes transverse modal mismatch in its description. \\Cref{sec:radiation_pressure_calc} gives the minimal extension of this ideal lossless interferometer to incorporate transverse mismatch, showing how the noise gain, $\\Gamma$,  and rotation angle $\\theta$ change specifically from mismatch. In particular, it shows that relating a measurement of $\\Omega_\\ensuremath{\\text{sql}}$ using squeezing back to the arm power $P_\\ensuremath{\\text{A}}$ using \\cref{eq:optical_gain_g} and  \\cref{eq:Kchi_standard} is biased by transverse mismatch.\n\n\\section{Cavity Modeling and Metrics}\n\\label{sec:modeling}\n\nThe previous section derives the general form of the squeezing metrics using matrices of the two photon formalism.\nFor passive systems, the optical transfer function, $\\ensuremath{\\mathfrak{h}}(\\Omega)$, given at every sideband frequency, is sufficient to characterize the response to externally-supplied squeezing. The conceptual simplification and restriction to using only transfer functions is useful for interferometer modeling. Transfer functions, being complex scalar functions, are suitable for analytic calculations of cavity response and can be decomposed into rational function forms to inspect the rational roots, zeros and poles, and the overall gain of the response.\n\nThis section analyzes the coupled cavity system of the interferometer, depicted in \\cref{fig:SQZ_mm_cavities}, through its decomposition into roots. More complicated transverse modal simulations analyze the frequency response of the interferometer cavities for each optical mode to every other mode. Modal simulations thus output a matrix of transfer functions, $\\mat{H}(\\Omega)$, which is difficult to analytically manipulate, but \\cref{sec:modeling_TMM} shows how it can be projected back to a single scalar transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$ and further simplified into the squeezing metrics.\n\nThe transfer function techniques of this section elucidate new squeezing results by avoiding the combined complexity of both two-photon and modal vector spaces. The full generality of two-photon matrices is only required for active systems that introduce internal squeezing, parametric gain or radiation pressure. Passive systems have the property that $\\hat{q}_\\text{out}$, $\\hat{p}_\\text{out}$ also obey the expectations of \\cref{eq:vacuum_expectations,eq:vacuum_expectations2}. Following the notation of \\cref{sec:derivations}, this results in the following condition.\n\\begin{align}\n  \\Tmat{1} &= \\Tmat{H}\\Tmat{H}^\\dagger\n  +\\sum_{\\ensuremath{{\\,\\mu}}}\\Tmat{T}_{\\ensuremath{{\\,\\mu}}}\\Tmat{T}^\\dagger_{\\ensuremath{{\\,\\mu}}}\n             \\label{eq:passivity_condition}\n\\end{align}\nAdditionally, $\\Gamma = 1$ is implied by that condition.\nWithout parametric gain, photons at upper and lower sideband frequencies are never correlated by a passive system. By the passivity condition and manipulations between sideband and quadrature basis, \\cref{sec:passive_derivations} derives the squeezing metrics purely in terms of the transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$.\n\\begin{align}\n  \\theta(\\Omega)\n  &=\n    \\big(\\arg\\big(\\ensuremath{\\mathfrak{h}}(+\\Omega)\\big) + \\arg\\big(\\ensuremath{\\mathfrak{h}}(-\\Omega)\\big)\\big)\/2\n    \\label{eq:theta_passive}\n  \\\\\n  \\eta(\\Omega)\n  &=\n    \\big( |\\ensuremath{\\mathfrak{h}}(+\\Omega)|^2 + |\\ensuremath{\\mathfrak{h}}(-\\Omega)|^2\\big)\/2\n    \\label{eq:eta_passive}\n  \\\\\n  \\Xi(\\Omega)\n  &=\n    \\big( |\\ensuremath{\\mathfrak{h}}(+\\Omega)| - |\\ensuremath{\\mathfrak{h}}(-\\Omega)| \\big)^2\/4\\eta\n    \\label{eq:Xi_passive}\n\\end{align}\nQuantum filter cavities are a method to use an entirely optical system to reduce the radiation pressure associated with squeezed light\\cite{McCuller-PRL20-FrequencyDependentSqueezing, Zhao-PRL20-FrequencyDependentSqueezed}. They are passive cavities, and provide a useful example to study these squeezing metric formulas.\nThe first of these, \\cref{eq:theta_passive}, is a well-established formula for the filter cavity design. It indicates that for cavities with an asymmetric phase response, usually due to being off-resonance or ``detuned'', that the squeezing field picks up a frequency-dependent quadrature rotation. Such a rotation applied in $\\Tmat{H}_\\ensuremath{\\text{I}}$ can be generated by a cavity with transfer function $\\ensuremath{\\mathfrak{h}}_\\ensuremath{\\text{I}}(\\Omega)$ before the interferometer. This cavity rotation compensates the $\\theta(\\Omega)$ due to $\\Tmat{H}_\\ensuremath{\\text{R}}$. Together, the product $\\Tmat{H}_\\ensuremath{\\text{R}}\\Tmat{H}_\\ensuremath{\\text{I}}$ has $\\theta(\\Omega) = 0$, allowing a single choice of squeezing angle $\\phi$ to optimize $N(\\Omega)$ at all frequencies.\n\nThe formulas \\cref{eq:eta_passive} and \\cref{eq:Xi_passive} indicate how losses represented in a transfer function translate to loss-like and dephasing degradations from cavity reflections. For filter cavities, these degradations are investigated in \\cite{Kwee-PRD14-DecoherenceDegradation}, but this new factorization into scalar functions clarifies the discussion. The efficiency $\\eta(\\Omega)$ behaves as expected, an average of the loss in each sideband. The form of $\\Xi(\\Omega)$ is less expected, showing how the combination of loss and detuning in filter cavities creates noise that scales with the squeezing level. A simple picture for the dephasing effect is that when optical quadratures are squeezed, the noise power in both upper and lower sidebands is strictly increased. The sideband correlations allow the increased noise to subtract away for squeezed quadrature measurements but to add for measurements in the anti-squeezed quadrature. The asymmetric losses of detuned cavities preserve the noise increase on one sideband, while degrading the correlations. This ruins the subtraction for the squeezed quadrature and introduces $\\Xi(\\Omega) > 0$. This source of noise is $e^{\\pm r}$ squeezing level dependent but entirely unrelated to fluctuations of the squeezing phase $\\ensuremath{\\phi_{\\text{rms}}}$.\n\n\\subsection{Single Cavity Model for Interferometers}\n\\label{sec:modeling_cavities}\n\nThis section analyzes the effect of the interferometer cavities on squeezing. It starts by considering an interferometer with only one cavity - either in the Michelson arms or from a mirror at the output port, but not both. It represents the first generation of GW detectors. This single cavity scenario is also similar to a quantum filter cavity, in the regime of small detuning\\cite{Komori-PRD20-DemonstrationAmplitude, Corbitt-PRD04-OpticalCavities, Khalili-PRD08-IncreasingFuture}. Advanced LIGO uses a coupled cavity system, depicted in \\cref{fig:SQZ_mm_cavities}, and the transfer function equations for the reflection from the resonant sideband extraction cavity is extended in the next subsection to include the loss and detuning of the additional cavity.\n\nA single cavity operated near resonance may be described using the scale parameters of the cavity bandwidth $\\gamma_\\ensuremath{\\text{A}}$, loss rate $\\lambda_\\ensuremath{\\text{A}}$ and detuning frequency $\\delta_\\ensuremath{\\text{A}}$, which are computed from the physical parameters of the mirror transmissivity $T_\\ensuremath{\\text{a}}$, round-trip loss $\\Lambda_\\ensuremath{\\text{a}}$, cavity length $L_\\ensuremath{\\text{a}}$, and microscopic length detuning $\\Delta L_\\ensuremath{\\text{a}}$.\n\\begin{align}\n  \\gamma_\\ensuremath{\\text{A}}  &= \\frac{c T_\\ensuremath{\\text{a}}}{4 L_\\ensuremath{\\text{a}}}\n  & \n  \\lambda_\\ensuremath{\\text{A}} &= \\frac{c \\Lambda_\\ensuremath{\\text{a}}}{4 L_\\ensuremath{\\text{a}}}\n  &\n  \\delta_\\ensuremath{\\text{A}}  &= -ck\\frac{\\Delta L_\\ensuremath{\\text{a}}}{L_\\ensuremath{\\text{a}}}\n                 \\label{eq:cavity_relations}\n\\end{align}\nThese relations are accurate in the high-finesse limit $T_\\ensuremath{\\text{a}} \\ll 1$, and combine to give the transfer function of the frequency-dependent cavity reflection.\n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_1(\\Omega)\n \n \n  &\\approx\n    -\\frac{(\\gamma_\\ensuremath{\\text{A}} - \\lambda_\\ensuremath{\\text{A}}) - i(\\Omega - \\delta_\\ensuremath{\\text{A}})}{(\\gamma_\\ensuremath{\\text{A}} + \\lambda_\\ensuremath{\\text{A}}) + i(\\Omega - \\delta_\\ensuremath{\\text{A}})}\n    \\label{eq:single_cavity}\n\\end{align}\nNotably, the sign of the reflectivity for a high-finesse cavity on resonance $\\ensuremath{\\mathfrak{r}}_1(\\Omega \\ll \\gamma_\\ensuremath{\\text{A}})=-1$,\nbut outside of resonance $\\ensuremath{\\mathfrak{r}}_1(\\Omega \\gg \\gamma_\\ensuremath{\\text{A}})=1$. This sign determines constructive or destructive interference in transverse mismatch loss analyzed in the next section. The internal losses of the cavity $\\Lambda_\\ensuremath{\\text{a}}$ become cavity-enhanced in the reflection, causing squeezing to experience losses of $\\Lambda_\\ensuremath{\\text{A}}$.\n\\begin{align}\n  \\Lambda_\\ensuremath{\\text{A}}\n  &\\equiv 1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1\\\\ |\\Omega| \\ll \\gamma_\\ensuremath{\\text{A}}}}\n \n \n \n  \\approx \\frac{4\\lambda_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}}\n                            \\approx \\frac{4\\Lambda_\\ensuremath{\\text{a}}}{T_\\ensuremath{\\text{a}}}\n\\end{align}\nFurthermore, detuning the cavity off of resonance causes a rotation of reflected squeezing. For small detunings, the rotation can be approximated.\n\\begin{align}\n  \\theta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1\\\\ k\\Delta L_\\ensuremath{\\text{a}} \\ll T_\\ensuremath{\\text{a}}}}\n  &\\approx\n  \\frac{2\\delta_\\ensuremath{\\text{A}}\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\approx\n   -k \\Delta L_\\ensuremath{\\text{a}}\\frac{8}{T_\\ensuremath{\\text{a}}}\\frac{\\gamma_\\ensuremath{\\text{A}}^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n\\end{align}\nFluctuations in $\\Delta L_\\ensuremath{\\text{a}}$ or $\\delta_\\ensuremath{\\text{A}}$ lead to a phase noise analogous to $\\ensuremath{\\phi_{\\text{rms}}}$, but with the frequency dependence from the above equation\\cite{Kwee-PRD14-DecoherenceDegradation}. Additionally, losses in the cavity lead to intrinsic dephasing $\\Xi(\\Omega)$, calculated below. This calculation is valid at any detuning $\\delta_\\ensuremath{\\text{A}}$, even those larger than the cavity width $\\gamma_\\ensuremath{\\text{A}}$. Its validity only requires being in the overcoupled cavity regime, where losses $\\lambda_\\ensuremath{\\text{A}} \\lesssim \\gamma_\\ensuremath{\\text{A}}\/2$.\n\\begin{align}\n  \\Xi(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1}}\n  &\\approx\n    \\left(\n  \\frac{4\\gamma_\\ensuremath{\\text{A}}\\lambda_\\ensuremath{\\text{A}}\\delta_\\ensuremath{\\text{A}}\\Omega}{\\left( \\gamma_\\ensuremath{\\text{A}}^2 + (\\Omega - \\delta_\\ensuremath{\\text{A}})^2 \\right)\\left( \\gamma_\\ensuremath{\\text{A}}^2 +  (\\Omega + \\delta_\\ensuremath{\\text{A}})^2 \\right)}\\right)^2\n\\end{align}\nWhen plotted, this expression for $\\Xi(\\Omega)$ has a Lorentzian-like profile, with a peak at $\\Omega_{\\Xi\\text{max}}$. Above $|\\delta_\\ensuremath{\\text{A}}| \\gtrsim \\gamma_\\ensuremath{\\text{A}}$, where the cavity resonance acts entirely either on upper or lower sidebands, the peak dephasing reaches a maximum. At small detunings, $|\\delta_\\ensuremath{\\text{A}}| \\lesssim \\gamma_\\ensuremath{\\text{A}}$, the sideband loss asymmetry scales with the detuning.\n\\begin{align}\n  \\Omega_{\\Xi{\\text{max}}} &\\approx \\sqrt{\\gamma^2_\\ensuremath{\\text{A}}\/4 + \\delta^2 },\n  & \\Xi_{\\text{max}} \n  &\\approx \n    \\frac{\\lambda^2_\\ensuremath{\\text{A}}}{\\gamma^2_\\ensuremath{\\text{A}}}{\\cdot}\\frac{8\\delta^2_\\ensuremath{\\text{A}}}{5\\gamma^2_\\ensuremath{\\text{A}} + 8\\delta^2_\\ensuremath{\\text{A}}}\n    \\label{eq:detuning_dephasing}\n\\end{align}\nThis single cavity model is also useful for analyzing quantum filter cavities and, like the $\\Xi$ metric itself, these peak values have not been calculated in past frequency-dependent squeezing work. Conventional squeezing phase uncertainty, $\\ensuremath{\\phi_{\\text{rms}}}$, can be cast into the units RMS radians of phase deviation, leading to the noise suppression limit for squeezing $S(\\Omega)\\ge 2\\ensuremath{\\phi_{\\text{rms}}}$, by \\cref{eq:N_limit}. For highly detuned cavities such as quantum filter cavities, $\\sqrt{\\Xi(\\Omega)} \\approx \\Lambda_\\ensuremath{\\text{fc}} \/ T_\\ensuremath{\\text{fc}}$. Using the parameters of the A+ filter cavity \\cite{Whittle-PRD20-OptimalDetuning}, $\\Lambda_{\\ensuremath{\\text{fc}}}{\\approx}60\\text{ppm}$ and $T_{\\ensuremath{\\text{fc}}} {=} 1000 \\text{ppm}$ indicates that optical dephasing is of order $60\\text{mRad}$. For an optimal filter cavity with low losses\\cite{Whittle-PRD20-OptimalDetuning}, this dephasing maximum occurs at $\\Omega_{\\Xi{\\text{max}}} = \\sqrt{5\/8}\\Omega_{\\text{SQL}}$. This level of dephasing is commensurate with or even exceeds the expected residual phase uncertainty $\\ensuremath{\\phi_{\\text{rms}}} < 30\\text{mRad}$.\n\nOptical dephasing from the LIGO interferometer cavities is not expected to be large for as they are stably operated on-resonance; however, detuned configurations of LIGO\\cite{Ganapathy-PRD21-TuningAdvanced} are limited by dephasing from the unbalanced response and optical losses in the signal recycling cavity.\n\n\\subsection{Double Cavity Model for Interferometers}\n\nFor interferometers using resonant sideband extraction, like LIGO, the arm cavities have a length $L_\\ensuremath{\\text{a}}$, an input transmissivity of $T_\\ensuremath{\\text{a}}$, and are each on resonance to store circulating laser power. The signal recycling cavity (SRC) has a length $L_\\ensuremath{\\text{s}}$ and a signal recycling mirror (SRM) of transmissivity $T_\\ensuremath{\\text{s}}$. The SRM forms a cavity with respect to the arm input mirror that resonantly increases the effective transmissivity experienced by the arm cavities to be larger than $T_\\ensuremath{\\text{a}}$, broadening the signal bandwidth. While the SRC is resonant with respect to the arm input mirror, it is anti-resonant with respect to the arm cavity, due to the negative sign of \\cref{eq:single_cavity}. The anti-resonance leads to the opposite sign in the reflection transfer function below, \\cref{eq:double_cavity}. The discrepancy in resonance vs. anti-resonance viewpoints is why the signal recycling cavity is also called the signal extraction cavity in GW literature.\n\nThe coupled cavity forms two bandwidth scales for the system, $\\gamma_\\ensuremath{\\text{A}}$, the modified effective arm bandwidth, and $\\gamma_\\ensuremath{\\text{S}}$, the bandwidth of the signal recycling cavity. The arm and signal cavities have their respective round-trip losses $\\Lambda_\\ensuremath{\\text{a}}$ and $\\Lambda_\\ensuremath{\\text{s}}$, as well as length detunings $\\Delta L_\\ensuremath{\\text{a}}$, $\\Delta L_\\ensuremath{\\text{s}}$. In practice, the arm length detuning is expected to be negligible to maximize the power storage, but the signal recycling cavity detuning can be varied by modifying a bias in the control system that stabilizes $\\Delta L_\\ensuremath{\\text{s}}$.\n\nThe scale parameters for the cavity transfer function are approximated from the physical parameters:\n\\begin{align}\n  u_\\ensuremath{\\text{a}} &= 1 - \\sqrt{1 - T_\\ensuremath{\\text{a}}}\n  &\n  u_\\ensuremath{\\text{s}} &= 1 - \\sqrt{1 - T_\\ensuremath{\\text{s}}}\n           \\label{eq:u_factors}\n  \\\\\n  \\gamma_\\ensuremath{\\text{A}}  &= \\frac{c u_\\ensuremath{\\text{a}}}{2 L_\\ensuremath{\\text{a}}} \\cdot\\frac{2 - u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{s}}}\n  & \n    \\gamma_\\ensuremath{\\text{S}}   &= \\frac{c u_\\ensuremath{\\text{s}}}{2 L_\\ensuremath{\\text{s}}}\n                    \\label{eq:double_cavity_gamma}\n  \\\\\n  \\lambda_\\ensuremath{\\text{A}} &= \\frac{c}{L_\\ensuremath{\\text{a}}} \\left(\\frac{\\Lambda_\\ensuremath{\\text{a}}}{4} - \\frac{u_\\ensuremath{\\text{a}} \\Lambda_\\ensuremath{\\text{s}}}{2 u_\\ensuremath{\\text{s}}^2} + \\frac{u_\\ensuremath{\\text{a}} \\Lambda_\\ensuremath{\\text{s}}}{4 u_\\ensuremath{\\text{s}}}\\right)\n  &\n    \\lambda_\\ensuremath{\\text{S}}  &= \\frac{c\\Lambda_\\ensuremath{\\text{s}}}{4L_\\ensuremath{\\text{s}}}\\left(1 - \\frac{u_\\ensuremath{\\text{s}}}{2} \\right)\n  \\\\\n  \\delta_\\ensuremath{\\text{A}} &= -ck\\frac{\\Delta L_\\ensuremath{\\text{a}}}{L_\\ensuremath{\\text{a}}}  - \\frac{\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{S}}}\\delta_\\ensuremath{\\text{S}}\n  &\n  \\delta_\\ensuremath{\\text{S}} &= -ck\\frac{\\Delta L_\\ensuremath{\\text{s}}}{L_\\ensuremath{\\text{s}}} \n           \\label{eq:delta_factors}\n\\end{align}\nThese approximations are valid for the LIGO mirror parameters, see \\cref{tab:LIGO_params}, and model the loss and detuning to {\\color{red} $5\\%$} accuracy. They are derived in \\cref{sec:scalar_coupled_cavity} from Taylor expansions, solving roots, and selectively removing terms. Expanding in the $u$ factors of \\cref{eq:u_factors} gives lower error than expanding in transmissivity or reflectivity factors directly, due to the low effective finesse of the coupled cavity system and the high transmissivity of the SRM.\n\\begin{table}\n\\centering\n\\caption{Parameters of LIGO for data fitting and modeling}\n\\begin{ruledtabular}\n\\begin{tabular}{l|c|cc}\nParameter & Symbol & LLO Value & LHO Value \\\\\n\\hline\narm input transmissivity & $T_\\ensuremath{\\text{a}}$&0.0148 & 0.0142 \\\\\narm length & $L_\\ensuremath{\\text{a}}$ & \\multicolumn{2}{c}{3995 m}\\\\\narm round-trip loss & $\\Lambda_{\\ensuremath{\\text{a}}}$ & \\multicolumn{2}{c}{${\\sim}80$ppm}\\\\\nSRM transmission& $T_\\ensuremath{\\text{s}}$&\\multicolumn{2}{c}{0.325}  \\\\\nSRC length & $L_\\ensuremath{\\text{s}}$ &\\multicolumn{2}{c}{55 m}\\\\\nSRC round-trip loss & $\\Lambda_\\ensuremath{\\text{s}}$ & \\multicolumn{2}{c}{$\\lesssim 3000$ppm}\\\\\n  \\hline\nMirror mass& $m$ &  \\multicolumn{2}{c}{39.9kg}\\\\\nArm power& $P_\\ensuremath{\\text{A}}$ & $200{\\pm} 10\\text{ kW}$  &  $190{\\pm}10\\text{ kW}$ \\\\\nQRPN crossover& $\\Omega_{\\ensuremath{\\text{sql}}}$ & $2\\pi\\cdot 33$ Hz & $2\\pi\\cdot 30$ Hz \\\\\n  \\hline\narm signal band& $\\gamma_\\ensuremath{\\text{A}}$& $2\\pi\\cdot450$ Hz &  $2\\pi\\cdot410$ Hz \\\\\nSRC band& $\\gamma_\\ensuremath{\\text{S}}$& \\multicolumn{2}{c}{$2\\pi\\cdot80$kHz} \\\\\nArm length detuning & $\\Delta L_{\\ensuremath{\\text{a}}}$ & \\multicolumn{2}{c}{0nm}\\\\\nSRC length detuning & $\\Delta L_{\\ensuremath{\\text{s}}}$ & 1.02nm & 1.23nm\\\\\n  \\hline\narm resonant loss& $\\Lambda_\\ensuremath{\\text{A}}$ & \\multicolumn{2}{c}{$\\lesssim 2000$PPM}\\\\\nSRC resonant loss& $\\Lambda_\\ensuremath{\\text{S}}$ & \\multicolumn{2}{c}{${\\sim} 1\\%$ to $3\\%$}\\\\\narm\/SRC detuning& $\\delta_{\\ensuremath{\\text{A}}}$ & $2\\pi\\cdot 10.1$Hz & $2\\pi\\cdot 11.2$Hz \\\\\n  \\hline\nInjected squeezing & $e^{\\pm 2r}$ & {${\\pm}9.7\\text{ dB}$}  &  {${\\pm}8.7\\text{ dB}$} \\\\\nSQZ-OMC mismatch& $\\Upsilon_{\\ensuremath{\\text{O}}}$ & $2\\%$ & $4\\%$ \\\\\nReflection mismatch (fit)& $\\Upsilon_{\\ensuremath{\\text{R}}}$ & $12\\%$ & $35\\%$ \\\\\nAdditional SQZ loss (fit)& $\\Lambda_{\\text{IO}}=1{-}\\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{O}}$ & $31\\%$ & $34\\%$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{tab:LIGO_params}\n\\end{table}\nThe scale factors result in the following reflectivity transfer function.\n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_2(\\Omega)\n  &=\n    \\frac{(\\gamma_\\ensuremath{\\text{A}} - \\lambda_\\ensuremath{\\text{A}}) - i(\\Omega - \\delta_\\ensuremath{\\text{A}})}{(\\gamma_\\ensuremath{\\text{A}} + \\lambda_\\ensuremath{\\text{A}}) + i(\\Omega - \\delta_\\ensuremath{\\text{A}})}\n   \n    {\\cdot}\\frac{(\\gamma_\\ensuremath{\\text{S}} - \\lambda_\\ensuremath{\\text{S}}) - i(\\Omega - \\delta_\\ensuremath{\\text{S}})}{(\\gamma_\\ensuremath{\\text{S}} + \\lambda_\\ensuremath{\\text{S}}) + i(\\Omega - \\delta_\\ensuremath{\\text{S}})}\n    \\label{eq:double_cavity}\n\\end{align}\nNotably, this reflectivity is $\\ensuremath{\\mathfrak{r}}_2(\\pm\\Omega \\ll \\gamma)=1$ and $\\ensuremath{\\mathfrak{r}}_2(\\gamma_\\ensuremath{\\text{A}} \\ll \\pm\\Omega \\ll \\gamma_\\ensuremath{\\text{S}})=-1$ which has an opposite overall sign to that of single cavity interferometers. On reflection, the squeezing field experiences different cavity enhanced losses depending on the frequency.\n\\begin{align}\n  \\Lambda_{\\ensuremath{\\text{S}}} &\\equiv \n1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ \\gamma_\\ensuremath{\\text{A}} \\ll |\\Omega| \\ll \\gamma_\\ensuremath{\\text{S}}}}\n&&\\hspace{-3em}\\approx \\frac{2 - u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{s}}}\\Lambda_\\ensuremath{\\text{s}}\n  \\\\\n  \\Lambda_{\\ensuremath{\\text{A}}} &\\equiv \n1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ |\\Omega| \\ll \\gamma_\\ensuremath{\\text{A}}}}\n&&\\hspace{-3em}\\approx \\frac{4\\lambda_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}}+\\Lambda_\\ensuremath{\\text{s}} \\approx \\frac{u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{a}}}\\Lambda_\\ensuremath{\\text{a}}\n\\end{align}\nThe dataset of \\cref{sec:experiment} shows frequency dependent losses, where the loss increases $12\\%$ for LLO and $33\\%$ for LHO. Assuming the losses result from the equations above, this corresponds to round-trip losses in the LIGO signal recycling cavities, $\\Lambda_\\ensuremath{\\text{s}}$, of $1.1\\%$ to $3.2\\%$, which is not realistic. Most mechanisms that introduce loss in the SRC would also introduce it into the power recycling cavity in an obvious manner. The current power recycling factors exclude this possibility, and independent measurements of $\\gamma_\\ensuremath{\\text{A}}$ bound $\\Lambda_\\ensuremath{\\text{s}}$ losses to ${\\le} 3000$ppm. The next section investigates how transverse mismatch can result in this level of observed losses.\n\nIn addition to the losses, \\cref{eq:double_cavity} can be used to determine the cavity-induced squeeze state rotation from the detuning of the signal recycling cavity.\n\\begin{align}\n  \\theta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ \\delta_\\ensuremath{\\text{S}} \\ll \\gamma_\\ensuremath{\\text{S}} \\\\ \\Delta L_\\ensuremath{\\text{a}} = 0}}\n  &\\approx\n  \\frac{2\\delta_\\ensuremath{\\text{A}}\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    +\n  \\frac{2\\delta_\\ensuremath{\\text{S}}\\gamma_\\ensuremath{\\text{S}}}{\\gamma_\\ensuremath{\\text{S}}^2 + \\Omega^2}\n \n \n \n \n \n \n \n \n    \\\\\n  &\\approx\n    k\\Delta L_\\ensuremath{\\text{s}}\\frac{4}{u_\\ensuremath{\\text{s}}}\n    \\left(\n    \\frac{\\gamma^2_\\ensuremath{\\text{S}}}{\\gamma_\\ensuremath{\\text{S}}^2 + \\Omega^2}\n    -\n    \\frac{\\gamma^2_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n \\right)\n    \\label{eq:double_cav_theta}\n \n\\end{align}\nThis indicates the surpising result that detuning the SRC length does not affect the squeezing within the effective\narm bandwidth to first order. Instead, it adds the squeezing rotation in the middle band above the arm bandwidth but below the SRC bandwidth. In the data analysis of \\cref{sec:experiment} and \\cref{fig:data_Q}, the convention for $\\theta(\\Omega)$ is set to be 0 at ``high'' frequencies in this intermediate cavity band, in which case it appears to cause a rotation around $\\gamma_\\ensuremath{\\text{A}}$. This convention used for the data corresponds to omitting the first, $\\gamma_\\ensuremath{\\text{S}}$-scaled term of \\cref{eq:double_cav_theta}.\n\n\\section{Transverse Mismatch Model}\n\\label{sec:modeling_TMM}\n\nSqueezing, as it is typically implemented for GW interferometers, modifies the quantum states in a single optical mode. For LIGO, this mode is the fundamental Gaussian beam resonating in the parametric amplifier cavity serving as the squeezed state source. The cavity geometry establishes a specific complex Gaussian beam parameter that defines a modal basis decomposition into Hermite Gaussian (HG) or Laguerre Guassian (LG) modes. That basis is transformed and redefined during the beam propagation through free space and through telescope lenses on its way to and from the interferometer. The cavities of the interferometer each define their own resonating beam parameters and respective HG or LG basis of optical modes.\n\nIn practice, the telescopes propagating the squeezed beam to and from the interferometer imperfectly match the complex beam parameters, so basis transformations must occur that mix the optical modes. The mismatch of complex beam parameters is called here ``transverse mismatch''. Non-fundamental HG or LG transverse modes do not enter the OPA cavity, and so carry standard vacuum rather than squeezing. Basis mixing from transverse mismatch thus leads to losses; however, unlike typical losses such basis transformations are coherent and unitary, which leads to the constructive and destructive interference effects studied in this section.\n\nThe interferometer transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$ is a single scalar function representing the frequency dependence of the squeezing channel from source to readout, but the optical fields physically have many more channels. The cavities visited by the squeezed states each have a transfer function matrix in their local basis, given by $\\mat{H}_{\\ensuremath{\\text{I}}}$, $\\mat{H}_{\\ensuremath{\\text{R}}}$, $\\mat{H}_{\\ensuremath{\\text{O}}}$ for the squeezing input, interferometer reflection, and system output respectively. The diagonals of these matrices indicate the frequency response during traversal for every transverse optical mode. The off-diagonals represent the coupling response between modes that result from scattering and optical wavefront errors.\n\nBetween the cavities, $\\mat{U}$ matrices represent the basis transformations due to transverse mismatch. Here, $\\vec{e}_{\\text{sqz}}$, $\\vec{e}_{\\text{read}}$ are basis vectors for projecting from the single optical mode of the emitted squeezed states and to the single mode of the optical homodyne readout defined by its local oscillator field.\n\\begin{align}\n\\ensuremath{\\mathfrak{h}}(\\Omega) &= {\\vec{e}_{\\text{read}}}^{\\,\\dagger} \\mat{H}(\\Omega)\\vec{e}_{\\text{sqz}}\n                  \\label{eq:TF_projection}\n  \\\\\n\\mat{H}(\\Omega) &=\\mat{H}_\\ensuremath{\\text{O}}\\mat{U}_{\\ensuremath{\\text{O}}, \\ensuremath{\\text{R}}}\\mat{H}_\\ensuremath{\\text{R}}\\mat{U}_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{I}}}\\mat{H}_\\ensuremath{\\text{I}}\n                  \\label{eq:TF_matrix}\n\\end{align}\n\\Cref{eq:TF_projection} and \\cref{eq:TF_matrix} give the general, basis independent, form to compose the effective transfer function for the squeezed field using a multi-modal simulation of a passive interferometer. This is complicated in the general case, but the following analysis develops a simpler, though general, model for how transverse mismatch manifests as squeezing losses.\n\nTransverse mismatch is often physically measured as a loss of coupling efficiency, $\\ensuremath{\\Upsilon}$, of an external Gaussian beam to a cavity measured as a change in optical power. Realistically, more than two transverse modes are necessary to maintain realistic and unitary basis transformations, but, for small mismatches of complex beam parameters, $\\Upsilon < 10\\%$. In this case, only the two lowest modes in the Laguerre-Gauss basis have significant cross-coupling. For low losses, the fundamental Gaussian mode, LG0, loses most of its power to the radially symmetric LG1 mode, assuming low astigmatism and omitting azimuthal indices. This motivates the following simplistic two-mode model to analyze the effect of losses on $\\ensuremath{\\mathfrak{h}}(\\Omega)$. In this model $\\mat{U}$ gives the unitary, though not perfectly physical, basis transformation:\n\\begin{align}\n  \\mat{U}(\\ensuremath{\\Upsilon}, \\psi, \\phi)\n  &\\equiv\n    e^{i\\phi}\\begin{bmatrix}\n    \\sqrt{\\ensuremath{1{-}\\MM}} & -e^{i\\psi}\\sqrt{\\ensuremath{\\Upsilon}}\n    \\\\\n    e^{-i\\psi}\\sqrt{\\ensuremath{\\Upsilon}} & \\sqrt{\\ensuremath{1{-}\\MM}}\n  \\end{bmatrix}\n                           \\label{eq:U_def}\n\\end{align}\nThis unitary transformation includes two unknown phase parameters. The first, $\\psi$, is the phase of the mismatch, which characterizes whether beam size error or wavefront phasing error dominates the overlap integral of the external LG0 and cavity LG1 modes. The second, $\\phi$, is the mismatch phase error from the external LG0 to the cavity LG0. The $\\phi$ term is included above to fully express the unitary freedom of $\\mat{U}$, but is indistinguishable from path length offsets, physically controlled to be $0$, and ignored in further expressions.\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_chain, \n  caption={\n    Propagation of the squeezed beam and unsqueezed higher order transverse beam modes from source to readout. The stages (a)-(d) correspond to the components in \\cref{fig:SQZ_mm_IFO}, depicting the matrix math of \\crefrange{eq:H_refl}{eq:h_chain}. $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ represents the transverse mismatch loss of the squeezing to interferometer, and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$ is the mismatch of the squeezing to readout via the output mode cleaner. These mismatches cause beamsplitter-like mixing between the LG0 and LG1+ modes through \\cref{eq:U_def}. $\\psi_\\ensuremath{\\text{I}}$,  $\\psi_\\ensuremath{\\text{O}}$,  $\\psi_{\\text{G}}$ are unmeasured phasing terms of the interferometer and output mismatch and of the Gouy-phase advance from the beam propagating to the output mode cleaner.\n  },\n  label=SQZ_mm_chain,\n}\n\nIn the case of a GW interferometer with an output mode cleaner, there are two mode matching efficiencies expressed as individually measurable parameters. The first is the coupling efficiency (in power) and phasing associated between the squeezer and interferometer $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}, \\ensuremath{\\psi_\\text{I}}$. The second are parameters for efficiency and phasing between the squeezer and output mode cleaner, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}, \\ensuremath{\\psi_\\text{O}}$, which defines the mode of the interferometer's homodyne readout. Both cases represent a basis change from the Laguerre-Gauss modes of the squeezer OPA cavity into the basis of each respective cavity. In constructing $\\mat{H}(\\Omega)$, however, the squeezing is transformed to the interferometer basis, reflects, and then transforms back to the squeezing basis. This corresponds to the operations of \\cref{fig:SQZ_mm_chain}. There are also parameters to express the coupling efficiency and phase, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{F}}, \\ensuremath{\\psi_\\text{F}}$, between the interferometer cavity and the OMC cavity. The $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{F}}$ parameter is less natural to analyze squeezing is not independent from  $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$. It is considered at the end of this section, as it can also be independently measured.\n\n\\cref{fig:SQZ_mm_chain} is implemented into \\cref{eq:TF_matrix} through this simplistic two-mode representation by assuming that the interferometer reflection transfer function $\\ensuremath{\\mathfrak{r}}(\\Omega)$ applies to the LG0 mode in the interferometer basis. The LG1 mode picks up the reflection transfer function $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}$, which is approximately ${\\sim}1$ due to high order modes being non-resonant in the interferometer cavities and thus directly reflecting. %\n\\begin{align}\n  \\mat{H}_\\ensuremath{\\text{R}} &= \n  \\begin{bmatrix}\n    \\ensuremath{\\mathfrak{r}}(\\Omega) & 0\n    \\\\\n    0 & \\ensuremath{\\mathfrak{r}_{\\text{hom}}}(\\Omega)\n  \\end{bmatrix},\n  &\n  \\mat{G} &= \n    \\begin{bmatrix}\n    1 & 0\n    \\\\\n    0 & e^{i\\psi_G}\n  \\end{bmatrix}\n        \\label{eq:H_refl}\n        \\\\\n  \\ensuremath{\\mathfrak{r}}(\\Omega) &= \\ensuremath{\\mathfrak{r}}_2(\\Omega) \\text{ or } \\ensuremath{\\mathfrak{r}}_1(\\Omega)\n  &\n  \\ensuremath{\\mathfrak{r}_{\\text{hom}}}(\\Omega) &= e^{i\\theta_{\\text{hom}}} \\approx 1\n\\end{align}\nThe reflection term $\\ensuremath{\\mathfrak{r}}(\\Omega)$ can use either the single, \\cref{eq:single_cavity}, or double,  \\cref{eq:double_cavity}, cavity forms. LIGO, using resonant sideband extraction, uses $\\ensuremath{\\mathfrak{r}}_2(\\Omega)$. Frequencies where the reflection takes a negative sign will be shown to experience destructive interference from modal basis changes, increasing squeezing losses. The $\\mat{G}$ matrix includes a phasing factor due to additional Gouy phase of higher-order-modes. This factor is degenerate with the mismatch phasings $\\psi_\\ensuremath{\\text{I}}$ and $\\psi_\\ensuremath{\\text{O}}$ in observable effects. These matrices are composed per \\cref{fig:SQZ_mm_chain} to formulate the overall transfer function of the squeezed field.\n\\begin{align}\n  \\mat{H}(\\Omega) &= \n\\underbrace{\\mat{U}(\\ensuremath{\\Upsilon_{\\text{O}}}, \\ensuremath{\\psi_\\text{O}})\n\\mat{G}\n\\mat{U}^\\dagger(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})}_{\\mat{U}_{\\text{O,R}}}\n\\mat{H}_\\ensuremath{\\text{R}}\n                    \\underbrace{\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})\n                    }_{\\mat{U}_{\\text{R,I}}}\n                    \\label{eq:H_chain}\n                    \\\\\n  \\ensuremath{\\mathfrak{h}}(\\Omega) &= \n  \\begin{bmatrix}\n    1 \\\\ 0\n  \\end{bmatrix}^T\n  \\mat{H}(\\Omega)\n  \\begin{bmatrix}\n    1 \\\\ 0\n  \\end{bmatrix}\n                    \\label{eq:h_chain}\n  ,\n  \\text{   and using }\n\\mat{H}_\\ensuremath{\\text{O}} = \\mat{H}_\\ensuremath{\\text{I}} = \\mat{1}\n\\end{align}\nIgnoring intra-cavity losses and detunings, the two reflection forms $\\ensuremath{\\mathfrak{r}}_1$, $\\ensuremath{\\mathfrak{r}}_2$ can be simplified to give their respective transfer functions $\\ensuremath{\\mathfrak{h}}_1$, $\\ensuremath{\\mathfrak{h}}_2$.\n\nFor quantum noise below $\\Omega < \\gamma_\\ensuremath{\\text{S}}$, the double cavity reflectivity $\\ensuremath{\\mathfrak{r}}_2(\\Omega)$ behaves like a single cavity, using the $\\gamma_\\ensuremath{\\text{A}}$ of \\cref{eq:double_cavity_gamma} and with the opposite reflection sign as \\cref{eq:single_cavity}. \n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_2(\\Omega) &\\approx +\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n  &\\Rightarrow&&\n  \\ensuremath{\\mathfrak{h}}_2(\\Omega) &= \\sqrt{\\textstyle \\ensuremath{1{-}\\MMO}}\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\ensuremath{\\alpha}\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n                   \\label{eq:h2_TMM}\n                  \\\\\n  \\ensuremath{\\mathfrak{r}}_1(\\Omega) &= -\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n  &\\Rightarrow&&\n  \\ensuremath{\\mathfrak{h}}_1(\\Omega) &= \\sqrt{\\textstyle \\ensuremath{1{-}\\MMO}}\\frac{i\\Omega - \\ensuremath{\\alpha}\\gamma_\\ensuremath{\\text{A}}}{i\\Omega + \\gamma_\\ensuremath{\\text{A}}}\n\\end{align}\nUsing the factor\n\\begin{align}\n  \\ensuremath{\\alpha} &\\equiv 1 - 2\\ensuremath{\\Upsilon_{\\text{I}}} + 2\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}e^{i\\ensuremath{\\psi_\\text{R}}}\n             \\intertext{where:}\n  \\ensuremath{\\beta} &\\equiv \\textstyle\\sqrt{\\frac{\\ensuremath{1{-}\\MMI}}{\\ensuremath{1{-}\\MMO}}} \\approx 1\n             \\\\\n  \\ensuremath{\\psi_\\text{R}} &\\equiv \\ensuremath{\\psi_\\text{O}} + \\psi_G - \\ensuremath{\\psi_\\text{I}}\n\\end{align}\nThe phasing factor $\\ensuremath{\\psi_\\text{R}}$ shows that the unknown mismatch phasings combine to a single unknown overall phase. This overall phase determines the extent to which the separate beam mismatches of $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$ coherently stack or cancel with each-other. The factor $\\ensuremath{\\alpha}$ is the total squeezer LG0 to readout LG0 coupling factor for the effective mode mismatch of the full system, specifically when the interferometer reflection $\\ensuremath{\\mathfrak{r}}(\\Omega) = -1$. As an effective mismatch, it can be related back to the diagonal elements of \\cref{eq:U_def} to give an effective mismatch loss on reflection, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{R}}$.\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{R}}} &= 1 - \\textstyle |\\ensuremath{\\alpha}|^2 \\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} - 4\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}\\cos(\\ensuremath{\\psi_\\text{R}})\n         \\label{eq:MMR_def}\n\\end{align}\nThis effective mismatch loss becomes apparent after computing the full system efficiency $\\eta(\\Omega)$ (\\cref{eq:eta_passive}) using $\\ensuremath{\\mathfrak{h}}_1$ and $\\ensuremath{\\mathfrak{h}}_2$.\n\\begin{align}\n  \\eta_\\ensuremath{\\text{R}}(\\Omega)\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &=\n    \\left( \\ensuremath{1{-}\\MMO} \\right)\\frac{\\gamma_\\ensuremath{\\text{A}}^2 + \\left(\\ensuremath{1{-}\\MMR}\\right)\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\label{eq:effective_MMR_loss2}\n                  \\\\\n  \\eta_\\ensuremath{\\text{R}}(\\Omega)\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_1}\n  &=\n    \\left( \\ensuremath{1{-}\\MMO} \\right)\\frac{\\Omega^2 + \\left(\\ensuremath{1{-}\\MMR}\\right)\\gamma_\\ensuremath{\\text{A}}^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\label{eq:effective_MMR_loss1}\n\\end{align}\nFor the double cavity system of LIGO, \\cref{fig:data_Q} is presented using the loss rather than efficiency. To relate to the measurement, the loss attributable to mode mismatch is then written\n\\begin{align}\n  \\Lambda_\\ensuremath{\\Upsilon}(\\Omega) \\equiv 1 - \\eta_\\ensuremath{\\text{R}}\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &\\approx\n    \\ensuremath{\\Upsilon_{\\text{O}}} + \\frac{\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\\ensuremath{\\Upsilon_{\\text{R}}}\n    \\label{eq:Lambda_MM}\n\\end{align}\nMode mismatches between the squeezer and OMC were directly measured during the LIGO squeezer installation to be $2\\%-4\\%$, and mismatches from the squeezer and interferometer were indirectly measured but are expected to be of a similar level. The large factors in \\cref{eq:MMR_def} indicate that the independent mismatch measurements are compatible with the observed frequency dependence and levels of the losses to squeezing. The effective mismatch loss $\\ensuremath{\\Upsilon_{\\text{R}}}$ has the following bounds with respect to the independent mismatch measurements.\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{R}}} &\\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when } \\ensuremath{\\Upsilon_{\\text{O}}} = 0\n                          \\\\\n  0 \\le \\ensuremath{\\Upsilon_{\\text{R}}} &\\le 8\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when } \\ensuremath{\\Upsilon_{\\text{I}}} = \\ensuremath{\\Upsilon_{\\text{O}}}\n                          \\\\\n  \\ensuremath{\\Upsilon_{\\text{R}}} &\\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when averaged over } \\ensuremath{\\psi_\\text{R}}\n                         \\label{eq:TMM_bound_avg}\n\\end{align}\nIt is worth noting here how the realistic interferometer differs from this simple two-mode model. The primary key difference is that real mismatch occurs with more transverse modes. Expanding this matrix model to include more modes primarily adds more $\\cos(\\ensuremath{\\psi_\\text{R}})$-type factors to the last term of \\cref{eq:MMR_def}. These factors will tend to average coherent additive mismatch between the squeezer and the OMC away, leaving only the squeezer to interferometer terms. Additionally, not only is there beam parameter mismatch from imperfect beam-matching telescopes, but there is also some amount of misalignment, statically or in RMS drift. Mismatch into modes of different order picks up different factors of $\\ensuremath{\\psi_\\text{G}}$. Together, including more modes leaves the bounds above intact, but makes \\cref{eq:TMM_bound_avg} more representative given the expanded dimensionality of mismatch-space to average away $\\cos(\\ensuremath{\\psi_\\text{R}})$.\n\nThe other notable difference in realistic instruments is that the high order modes pick up small phase shifts of reflection, as the cavities are not perfectly out of resonance at all high order modes. This corresponds to $\\ensuremath{\\mathfrak{r}}_{\\text{hom}} \\ne 1$. The signal recycling mirror is sufficiently low transmissivity that the finesse is low and, even when off-resonance, higher order modes pick up a small but slowly varying phase shift. This has the property of mixing the frequency dependent losses resulting from $\\ensuremath{\\mathfrak{h}}_1$ and $\\ensuremath{\\mathfrak{h}}_2$, resulting in a slightly more varied frequency-dependence that is captured in the full model of \\cref{sec:matrix_coupled_cavity}.\n\n\nWhile the phasing of the mismatch, $\\ensuremath{\\psi_\\text{R}}$, is not directly measurable, it manifests in an observable way. It adds to the complex phase of $\\ensuremath{\\alpha}$ to cause a slight rotation of the squeezing phase, making the cavity appear as if it is detuned. the frequency dependence and magnitude of this rotation is given by (c.f. \\cref{eq:theta_passive}),\n\\begin{align}\n  \\theta_\\ensuremath{\\Upsilon}(\\Omega) \\equiv \\theta\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &\\approx\n    \\frac{-\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}2\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}\\sin(\\ensuremath{\\psi_\\text{R}})\n\\end{align}\nwhich adds to the rotation from cavity length detuning \\cref{eq:double_cav_theta}. The addition of this term with $\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}} \\ensuremath{\\Upsilon_{\\text{O}}}}$ unknown confounds the ability to use the data of \\cref{fig:data_Q} to constrain $\\ensuremath{\\psi_\\text{R}}$. There is a small discrepancy between the length-detuning induced optical spring observed in the interferometer calibration \\cite{Cahillane-PRD17-CalibrationUncertainty, Sun-CQG20-CharacterizationSystematic} and the detuning inferred from the data. The additional mismatch phase shift helps explain that such a discrepancy is possible, but the two should be studied in more detail. Note that the small Gouy phase shift from $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}$ can be significant for this small detuning effect. The expression above is primarily provided to indicate the magnitude of variation as a function of $\\sin(\\ensuremath{\\psi_\\text{R}})$, so that future observations can better constrain $\\ensuremath{\\psi_\\text{R}}$ by comparing squeezing measurements of $\\theta(\\Omega)$ with calibration measurements of the optical spring arising from $\\delta_\\ensuremath{\\text{S}}$.\n\nThe asymmetric contribution of $\\ensuremath{\\alpha}$ in \\cref{eq:h2_TMM} also causes mode mismatch to contribute to optical dephasing, $\\Xi(\\Omega)$ (c.f. \\cref{eq:Xi_passive}). The dependence on $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}, \\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{R}}$ is complex and does not have single dominating contributions, so an analytic expression is not computed here. Using the exact models of \\cref{sec:matrix_coupled_cavity}, that mimic the datasets of \\cref{sec:experiment} give a contribution of $\\sqrt{\\Xi}$ that peaks at $\\gamma_\\ensuremath{\\text{A}}$ and is 10-20\\text{mRad} for the Livingston LLO model, and 10-50mRad for the Hanford LHO model, with a range due to imperfect knowledge of the mismatch parameters.\n\nThe transverse mismatch calculations so far use the parameters $\\ensuremath{\\Upsilon_{\\text{O}}}$, which is directly measurable, and $\\ensuremath{\\Upsilon_{\\text{I}}}$, which is independent, but $\\ensuremath{\\Upsilon_{\\text{I}}}$ can not easily be measured using invasive direct measurements due to the fragile operating state of the GW interferometer. Another mismatch parameter exists for the signal beam traveling with the Michelson fringe-offset light. This beam experiences a separate mode matching efficiency, $\\ensuremath{\\Upsilon_{\\text{F}}}$, denoting the mismatch loss between the interferometer and the OMC. $\\ensuremath{\\Upsilon_{\\text{F}}}$ can be calculated from the original parameters by following the red signal path depicted in \\cref{fig:SQZ_mm_chain}.\n\\begin{align}\ne^{i\\phi_\\ensuremath{\\text{F}}}\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{F}}}, \\ensuremath{\\psi_\\text{S}})\n  &=\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{O}}}, \\ensuremath{\\psi_\\text{O}})\n\\mat{G}\n\\mat{U}^\\dagger(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})\n\\end{align}\nExpanding this form results in the following relations\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{F}}} &\\approx \\ensuremath{\\Upsilon_{\\text{O}}} + \\ensuremath{\\Upsilon_{\\text{I}}} - 2\\sqrt{\\ensuremath{\\Upsilon_{\\text{O}}}\\ensuremath{\\Upsilon_{\\text{I}}}}\\cos(\\ensuremath{\\psi_\\text{R}})\n         \\\\\n   \\ensuremath{\\Upsilon_{\\text{F}}} &\\approx \\ensuremath{\\Upsilon_{\\text{R}}}\/2 + \\ensuremath{\\Upsilon_{\\text{O}}} - \\ensuremath{\\Upsilon_{\\text{I}}}\n\\end{align}\nExperimentally, $\\ensuremath{\\Upsilon_{\\text{F}}}$ can be determined or estimated more directly than $\\ensuremath{\\Upsilon_{\\text{I}}}$ by using signal fields from the arms, though can be confused with projection loss when the local oscillator readout angle $\\xi\\ne 0$ (c.f. \\cref{eq:homodyne_observable}). These formulas provide the set of relations to estimate each of the mode mismatch parameters from the others, and potentially the overall mismatch phase $\\ensuremath{\\psi_\\text{R}}$ as well. These relations are calculated using the assumptions of this section: the two-mode approximation and that $\\ensuremath{\\mathfrak{r}}_{\\text{hom}} \\simeq 1$.\n\nTogether, the relations of this section give insight in to how the physical mismatch parameters, $\\ensuremath{\\Upsilon_{\\text{I}}}$, $\\ensuremath{\\Upsilon_{\\text{O}}}$ and $\\ensuremath{\\Upsilon_{\\text{F}}}$ contribute to squeezing degradations. $\\ensuremath{\\Upsilon_{\\text{R}}}$ is a new form of effective mismatch parameter that is directly measurable from squeezing data, using the analysis of \\cref{sec:experiment}. It indicates how squeezing changes with frequency due to \\crefrange{eq:effective_MMR_loss2}{eq:effective_MMR_loss1}. Together, the complex, coherent interactions of transverse modal mixing on squeezed state can be concisely characterized in cavity-enhanced interferometers.\n\n\\subsection{Implications for Frequency Dependent Squeezing}\nThis analysis of the transverse mismatch applies to the reflection of squeezing off of any form of cavity. Namely, the detuned filter cavity for frequency-dependent rotation of squeezing in the LIGO A+ upgrade. This cavity will be installed on the input, $\\Tmat{H}_\\ensuremath{\\text{I}}$ section of the squeezing transformation sequence. The filter cavity mismatch loss $\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}$ will behave analogously to $\\ensuremath{\\Upsilon_{\\text{I}}}$, introducing losses of ${\\sim}4\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}$ at frequencies resonating in the cavity. The mismatch loss adds to those caused by the internal round-trip cavity loss $\\Lambda_{\\ensuremath{\\text{fc}}}$, creating the effective loss $\\Lambda_{\\ensuremath{\\text{FC}}} \\approx 4\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}} + 4\\lambda_{\\ensuremath{\\text{FC}}}\/\\gamma_{\\ensuremath{\\text{FC}}}$ using \\cref{eq:cavity_relations}.\n\nThe intra-cavity losses then set the scale for how much transverse mismatch is allowable before mismatch dominates the squeezing degradation, $\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}} < \\Lambda_\\ensuremath{\\text{fc}} \/ T_\\ensuremath{\\text{fc}}$. More importantly, they add to the dephasing from the detuned cavity, by creating an effective $\\lambda'_\\ensuremath{\\text{FC}} = \\lambda_{\\ensuremath{\\text{FC}}} + \\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}\\gamma_{\\ensuremath{\\text{FC}}}$ which can be used in \\cref{eq:detuning_dephasing}. The dephasing will set the limit to the allowable injected squeezing $e^{\\pm 2r}$ level as it introduces anti-squeezing at critical frequencies in the spectrum for astrophysics.\n\n\\section{Conclusions}\n\nBefore this work, the squeezing level in the LIGO interferometers was routinely estimated using primarily high-frequency measurements. This was done to utilize a frequency band where the classical noise contributions were small, while also giving a large bandwidth over which to improve the $\\Delta F \\Delta t$ statistical error in noise estimates. In doing so, LIGO recorded a biased view of the state of squeezing performance between the two instruments. The data analysis of this work has revealed several critical features to better understand and ultimately improve the quantum noise in LIGO.\n\nFirst, it indicates that the two sites have similar optical losses in their injection and readout components, as seen from the low-frequency losses of \\cref{fig:data_Q}. There is still a small excess of losses over the predictions given in \\cite{Tse-PRL19-QuantumEnhancedAdvanced}, but substantially less than implied when estimating the losses using high frequency observations. The most culpable loss components in the LIGO interferometers are being upgraded for the next observing run.\n\nSecond, this data analysis indicates that squeezing is degraded particularly at high frequencies, and the modeling and derivations provide the mechanism of transverse optical mode mismatch, external to the cavities, as a plausible physical explanation. This will be addressed in LIGO through the addition of active wavefront control to better match the beam profiles between the squeezer's parametric amplifier, new filter cavity installation, interferometer, and output mode cleaner.\n\nThird, the quantum radiation pressure noise is now not only measured, but employed as a diagnostic tool along with squeezing. QRPN indicates that the effective local oscillator angle in the Michelson fringe offset light at LLO is a specific, nonzero, value. This indicates that to power up the detector further, while maintaining a constant level of fringe light, the angle will grow larger and cause more pronounced degradation of the sensitivity by projecting out of the signal's quadrature. Ultimately, the LO angle should become configurable using balanced homodyne detection, another planned upgrade as part of ``A+''.\n\nFinally, this work carefully derives useful formula to manipulate the quantum squeezing response metrics. These are useful to reason and rationalize the interactions of squeezing with ever more complex detectors, both for gravitational-wave interferometers, and more generally as squeezing-enhanced optical metrology becomes more commonplace. The design of a future generation of gravitational wave detectors must be optimized specifically to maintain exceptional levels of squeezing compared to today. The quantum response metrics derived in this paper will aid that design work by simplifying our interpretation of squeezing with simulations. With these diagnostics and the data from observing run 3, LIGO is now better prepared to install and characterize frequency dependent squeezing in its ``A+'' upgrade not as a demonstration, but for stable, long-term improvement of the quantum enhanced observatories to detect astrophysical events.\n\n\\begin{acknowledgments}\n LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation, and operates under Cooperative Agreement No. PHY-0757058. Advanced LIGO was built under Grant No. PHY-0823459. The authors gratefully acknowledge the National Science Foundation Graduate Research Fellowship under Grant No. 1122374. \n\\end{acknowledgments}\n\n\\input{fdep_main.bbl}\n\n\n\n\n\n\\section{Introduction}\nThe third observing run of the global gravitational wave network has not only produced a plethora of varied and unique astrophysics events \\cite{Abbott-AAP20-PopulationProperties, Abbott-AAP20-GWTC2Compact}, it has defined a milestone in quantum metrology: that the LIGO, VIRGO and GEO600 observatories are now all reliably improving their scientific output by incorporating squeezed quantum states \\cite{Tse-PRL19-QuantumEnhancedAdvanced, VirgoCollaboration-PRL19-IncreasingAstrophysical, Lough-PRL21-FirstDemonstration}. This marks the transition where optical squeezing, a widely researched, emerging quantum technology, has become an essential component producing new observational capability.\n\nFor advanced LIGO, observing run three provides the first peek into the future of quantum enhanced interferometry, revealing challenges and puzzles to be solved in the pursuit of ever more squeezing for ever greater observational range. Studying quantum noise in the LIGO interferometers is not simple. The audio-band data from the detectors contains background noise from many optical, mechanical and thermal sources, which must be isolated from the purely quantum contribution that responds to squeezing. All the while, the interferometers incorporate optical cavities, auxiliary optical fields, kg-scale suspended optics, and radiation pressure forces. The background noise and operational stability of the LIGO detectors is profoundly improved in observing run three \\cite{Buikema-PRD20-SensitivityPerformance}, enabling new precision observations of the interactions between squeezed states and the complex optomechanical detectors.\n\nQuantum radiation pressure noise (QRPN) is the most prominent new observation from squeezing \\cite{Yu-N20-QuantumCorrelations, Acernese-PRL20-QuantumBackaction}. QRPN results from the coupling of photon momentum from the amplitude quadrature of the light into the phase quadrature, as radiation force fluctuation integrates into mirror displacement uncertainty. When vacuum states enter the interferometer, rather than squeezed states, QRPN imposes the so-called standard quantum limit\n\\cite{Braginsky-S80-QuantumNondemolition, Braginsky-92-QuantumMeasurement, Braginsky-RMP96-QuantumNondemolition},\nbounding the performance of GW interferometers. Because the QRPN coupling between quadratures is coherent, squeezed states allow the SQL to be surpassed\n\\cite{Kimble-PRD01-ConversionConventional, Yu-N20-QuantumCorrelations}.\nBoth surpassing the SQL and increasing the observing range is possible by using a frequency-dependent squeezing (FDS) source implemented with a quantum filter cavity\n\\cite{Kimble-PRD01-ConversionConventional, McCuller-PRL20-FrequencyDependentSqueezing, Zhao-PRL20-FrequencyDependentSqueezed, Oelker-PRL16-AudioBandFrequencyDependent, Chelkowski-PRA05-ExperimentalCharacterization, Whittle-PRD20-OptimalDetuning, Khalili-PRD10-OptimalConfigurations, Evans-PRD13-RealisticFilter, Kwee-PRD14-DecoherenceDegradation}. LIGO is including such a source in the next observing run as part of its ``A+'' upgrade\\cite{Whittle-PRD20-OptimalDetuning, McCuller-PRL20-FrequencyDependentSqueezing}. To best utilize its filter cavity squeezing source, the frequency-dependence of LIGO's quantum response must be precisely understood.\n\nDegradations to squeezing from optical loss and ``phase noise'' fluctuations of the squeezing angle are also prominently observed in LIGO. Whereas QRPN's correlations cause frequency dependent effects, loss and phase noise are typically described as causing frequency independent, broadband changes to the quantum noise spectrum. This work analyzes the quantum response of both LIGO interferometers to injected squeezed states, indicating that QRPN and broadband degradations, taken independently, are insufficient to fully describe the observed quantum response to squeezing.\n\nThe first sections of this work expand the response and degradation model of squeezing to examine and explain the LIGO quantum noise data by decomposing it into independent, frequency-dependent parameters. The latter sections relate the parameter decomposition back to interferometer models, to navigate how squeezing interacts with cavities that have internal losses, transverse-mode selectivity, and radiation pressure interactions. The spectra at LIGO are explained using a set of broadly applicable analytical expressions, without the need for elaborate and specific computer simulations. The analytical models elucidate the physical basis of LIGO's squeezed state degradations, prioritizing transverse-mode quality using wavefront control of external relay optics\n\\cite{Cao-OEO20-EnhancingDynamic, Cao-AOA20-HighDynamic, Perreca-PRD20-AnalysisVisualization} to further improve quantum noise. This analysis also demonstrates the use of squeezing as a diagnostic tool\\cite{Mikhailov-PRA06-NoninvasiveMeasurements}, examining not only the cavities but also the radiation pressure interaction. These diagnostics show further evidence of the benefit of balanced homodyne detection \\cite{Fritschel-OEO14-BalancedHomodyne}, another planned component of the ``A+'' upgrade. The description of squeezing in this work expands the modeling of degradations in filter cavities \\cite{Kwee-PRD14-DecoherenceDegradation}, explicitly defining an intrinsic, non-statistical, form of dephasing. Finally, the derivations of the quantum response metrics in sec. \\ref{sec:derivations} show how to better utilize internal information inside interferometer simulations, simplifying the analysis of squeezing degradations for current and future gravitational wave detectors.\n\n\\section{Squeezing Response Metrics}\n\\label{sec:metrics}\n\nTo introduce the frequency-dependent squeezing metrics, it is worthwhile to first describe the metrics used for standard optical squeezing generated from an optical parametric amplifier (OPA), omitting any interferometer. For optical parametric amplifiers, the squeezing level is determined by three parameters. The first is the normalized nonlinear gain, $y$, which sets the squeezing level and scales from 0 for no squeezing to 1 for maximal squeezing at the threshold of amplifier oscillation. For LIGO, $y$ is determined from a calibration measurement of the parametric amplification \\cite{Xiao-PRL87-PrecisionMeasurement, Aoki-OEO06-Squeezing946nm, Takeno-OEO07-ObservationDB, Khalaidovski-CQG12-LongtermStable, Schnabel-OC04-SqueezedLight, Dwyer-OEO13-SqueezedQuadrature}. The second parameter is the optical efficiency $\\eta$ of states from their generation in the cavity all the way to their observation at readout. Losses that degrade squeezed states are indicated by $\\eta < 1$. Finally, there is the squeezing phase angle, $\\phi$, which determines the optical field quadrature with reduced noise and the quadrature with the noise increase mandated by Heisenberg uncertainty, anti-squeezing. By correlating the optical quadratures, variations in $\\phi$ continuously rotate between squeezing and anti-squeezing. These parameters relate to the observable noise as:\n\\begin{align}\n  N(\\phi) &= \\left( 1 - \\frac{4 \\eta y}{(1 + y)^2} \\right)\\cos^2(\\phi) + \\left( 1 + \\frac{4 \\eta y}{(1 - y)^2} \\right)\\sin^2(\\phi)\n\\end{align}\nThe noise, $N(\\phi)$, can be interpreted as the variance of a single homodyne observation of a single squeezed state, but for a continuous timeseries of measurements, $N$ can be considered as a power spectral density, relative to the density of shot-noise. Using relative noise units, $N=1$ corresponds to observing vacuum states rather than squeezing. While the nonlinear gain parameter $y$ may be physically measured and is common in experimental squeezing literature, theoretical work more commonly builds states from the squeezing operator, parameterized by $r$, which constructs an ideal, ``pure'' squeezed state that adjusts the noise power by $e^{\\pm 2r}$. State decoherence due to optical efficiency is then incorporated as a separate, secondary process. This is formally related to the previous expression using:\n\\begin{align}\n  N(\\phi)\n  &= \\eta\\left(\n    e^{-2r}\\cos^2(\\phi) + e^{+2r}\\sin^2(\\phi)\n    \\right) + \\left(1 - \\eta\\right)\n    \\label{eq:etaSQZ_basic1}\n  \\\\\n  e^{-2r}\n  &=\n    1 - \\frac{4y}{(1 + y)^2} \\label{eq:OPA_y_SQZ}\n    , \\hspace{2.5em} e^{+2r} = 1 + \\frac{4y}{(1 - y)^2}\n \n\\end{align}\nIn experiments, the squeezing angle drifts due to path length fluctuations and pump noise in the amplifier, but is monitored using additional coherent fields at shifted frequencies and stabilized by feedback control. This stabilization is imperfect, resulting in a root-mean-square (RMS) phase noise,  $\\ensuremath{\\phi^2_{\\text{rms}}}$, that mixes squeezing and antisqueezing. Using $\\hat{\\phi}$ to represent the statistical distribution of the squeezing angle, and $E[\\cdot]$ the expectation operation, phase noise can be incorporated as a tertiary process given the expectation values:\n\\begin{align}\n \\ensuremath{\\phi^2_{\\text{rms}}} &= E\\left[\\sin^2(\\delta \\hat{\\phi})\\right]\n             &\n \\ensuremath{\\phi} &= E\\left[\\hat{\\phi}\\right]\n  &\n\\delta \\hat{\\phi} &= \\hat{\\phi} - \\ensuremath{\\phi}\n\\end{align}\nresulting in the ensemble average noise $\\overline{N}$, relative shot noise.\n\\begin{align}\n  \\overline{N}(\\phi) &= E\\left[N(\\ensuremath{\\phi} + \\delta \\hat{ \\phi})\\right]\n              \\\\\n \n   &= \\eta\\left(1 - \\ensuremath{\\phi^2_{\\text{rms}}}\\right)\\left(e^{-2r}\\cos^2(\\ensuremath{\\phi}) + e^{+2r}\\sin^2(\\ensuremath{\\phi}) \\right)\n     \\nonumber\\\\ &\\hspace{1em}\n                   + \\eta\\ensuremath{\\phi^2_{\\text{rms}}} \\left(e^{+2r}\\cos^2(\\ensuremath{\\phi}) + e^{-2r}\\sin^2(\\ensuremath{\\phi}) \\right) + (1 - \\eta)\n    \\label{eq:etaSQZ_basic2}\n\\end{align}\nAgain, the relative noise $\\overline{N}$ is computed as a single value here, but represents a power spectral density that is experimentally measured at many frequencies. These equations, as they are typically used, represent a change to the quantum noise that is constant across all measured frequencies. Notably, the $\\ensuremath{\\phi^2_{\\text{rms}}}$ phase noise term, which caps at $1\/2$, enters as a weighting factor that averages the anti-squeezing noise increase with squeezing noise reduction, while $\\eta$ mixes squeezing with standard vacuum.\n\nIncorporating an interferometer such as LIGO requires extending these equations to handle frequency-dependent effects. The equations must include terms to represent multiple sources of loss entering before, during, and after the interferometer, as well as terms for the frequency-dependent scaling of the quantum noise due to QRPN and the interferometer's suspended mechanics. The extension of the metrics is described by the following equations and parameters:\n\\begin{align}\n  N(\\Omega)\n  &\\equiv \\Gamma(\\Omega) \\cdot \\Big( \\eta(\\Omega) S(\\Omega) + \\Lambda_\\ensuremath{\\text{IRO}}(\\Omega) \\Big) \n    \\label{eq:metric_N}\n  \\\\\n \n \n \n \n \n \n \n \n  S(\\Omega)\n  &\\equiv S_{\\!{-}}\\cos^2\\Big(\\ensuremath{\\phi} - \\theta(\\Omega)\\Big) + S_{\\!{+}} \\sin^2\\Big(\\ensuremath{\\phi} - \\theta(\\Omega)\\Big)\n    \\\\\n  S_{\\!\\pm} &\\equiv \\big(1 - \\Xi'(\\Omega)\\big)e^{\\pm2r} + \\Xi'(\\Omega) e^{\\mp2r}\n              \\label{eq:metric_S}\n  \\\\\n  \\Lambda_\\ensuremath{\\text{IRO}}(\\Omega) &\\equiv (1 - \\eta_\\ensuremath{\\text{I}})\\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}} + \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}}) + (1 - \\eta_\\ensuremath{\\text{O}}) \/ \\Gamma\n    \\label{eq:metric_Lambda}\n\\end{align}\nThese metrics are composed of the following variables:\n\\renewcommand{\\descriptionlabel}[1]{\\hspace{\\labelsep}{#1}:}\n\\begin{description}[noitemsep, nolistsep]\n \\item[$N(\\Omega)$] the power spectrum of quantum noise in the readout, relative to the vacuum power spectral density, $\\hbar \\omega \/ 2$, of broadband shot noise.\n \\item[$\\Gamma(\\Omega)$] The quantum noise gain of the interferometer optomechanics. While $N(\\Omega)$ is relative shot-noise, QRPN causes interferometers without injected squeezing to exceed shot noise at low frequencies, resulting in $\\Gamma > 1$. For optical systems with $\\Gamma\\ne1$, the system cannot be passive, and must apply internal squeezing\/antisqueezing to the optical fields.\n \\item[$e^{2r}, e^{-2r}$] The ``pure'' injected squeezing and anti-squeezing level, before including any degradations. This level is computed for optical parametric amplifier squeezers using \\cref{eq:OPA_y_SQZ}.\n \\item[$S_{\\!-}, S_{\\!+}$] The minimum and maximum relative noise change from squeezing at any squeezing angle, ignoring losses.\n \\item[$S(\\Omega)$] The potentially observable injected squeezing level, before applying losses or noise gain.\n  \\item[$\\ensuremath{\\phi}$] The frequency independent squeezing angle chosen between the source and readout. This is usually stabilized with a co-propagating coherent control field and feedback system.\n  \\item[$\\theta(\\Omega)$] the squeezing angle rotation due to the propagation through intervening optical system. In a GW interferometer, this can be due to a combination of cavity dispersion and optomechanical effects. Quantum filter cavities target this term to create frequency dependent squeeze rotation.\n  \\item[$\\eta_\\ensuremath{\\text{I}}(\\Omega)$, $\\eta_\\ensuremath{\\text{O}}(\\Omega)$, $\\eta_\\ensuremath{\\text{R}}(\\Omega)$] The individually budgeted transmission efficiencies of the squeezed field at input, reflection and output paths of the interferometer. $1 - \\eta_{\\text{I,R,O}}$ indicates optical power lost in that component.\n  \\item[$\\eta(\\Omega)$] The collective transmission efficiency of the squeezed field. This is usually the product of the efficiencies in each path, $\\eta = \\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}$, but can deviate from this when $\\Gamma \\ne 1$ and interferometer losses affect both $\\Gamma$ and $\\eta_\\ensuremath{\\text{R}}$.\n  \\item[$\\Lambda_\\ensuremath{\\text{IRO}}(\\Omega)$] The total transmission loss over the squeezing path that contaminates injected squeezed states with standard vacuum. When $\\Gamma \\approx 1$, then $\\Lambda_\\ensuremath{\\text{IRO}} \\approx 1 - \\eta$.\n  \\item[$\\Xi'(\\Omega)$] This is a squeezing-level dependent decoherence mechanism called dephasing. It\n    incorporates both statistical $\\ensuremath{\\phi^2_{\\text{rms}}}$ phase fluctuations and the fundamental degradation arises from optical losses with unbalanced cavities, denoted $\\Xi(\\Omega)$. It can also arise from QRPN with structural or viscous mechanical damping. \\Cref{sec:effective_dephasing} shows how to incorporate fundamental dephasing  $\\Xi(\\Omega)$, standard phase uncertainty, $\\ensuremath{\\phi^2_{\\text{rms}}}$, and cavity tuning fluctuations, $\\ensuremath{\\theta^2_{\\text{rms}}}(\\Omega)$, into $\\Xi'(\\Omega)$ to make a total effective dephasing factor. When small, these factors sum to approximate the effective total $\\Xi'$\n\\end{description}\n\nAfter the data analysis of the next section, these quantum response metrics are derived in \\cref{sec:derivations}. These squeezing metrics indicate three principle degradation mechanisms, all frequency-dependent. These are losses, where $\\Lambda_\\ensuremath{\\text{IRO}}(\\Omega)\\approx 1 {-} \\eta(\\Omega) > 0$; Mis-phasing, from $\\phi{-}\\theta(\\Omega)\\ne 0$; and de-phasing, $\\Xi(\\Omega) > 0$.\n\nThe interaction of squeezing with quantum radiation pressure noise is described within these terms. Broadband Squeezing naively forces a trade-off between increased measurement precision and increased quantum back-action. When squeezing is applied in the phase quadrature, it results in anti-squeezing of the amplitude quadrature. The amplitude quadrature then pushes the mirrors and increases QRPN; thus, the process of reducing imprecision seemingly increases back-action. In other terms, QRPN causes the interferometer's ``effective'' observed quadrature\\footnote{The observed quadrature in this context is with respect to the injected quantum states, be they squeezed or vacuum. It is dependent on the quadrature of the interferometer's homodyne readout, but does not refer specifically to it. The effective observed quadrature also does not refer to the specific quadrature that the interferometer signal is modulated into.} to transition from the phase quadrature at high frequencies to the amplitude quadtrature at low frequencies. In the context of these metrics, the observation quadrature is captured in the derivation of $\\theta(\\Omega)$. The associated back-action trade-off can be considered a mis-phasing degradation, allowing the SQL to be surpassed using the quantum quadrature correlations introduced by varying the squeezing angle\\cite{Yu-N20-QuantumCorrelations}. Frequency dependent squeezing, viewed as a modification of the squeezing source, can be considered as making $\\phi(\\Omega)$ frequency-dependent, tracking $\\theta(\\Omega)$. Alternatively, it can be viewed as a modification of the interferometer, to maintain $\\theta(\\Omega) \\approx 0$. While a quantum filter cavity is not explicitly treated in this work, the derivations of \\cref{sec:derivations} are setup to be able to include a filter cavity as a modification to the input path of the interferometer.\n\nWhile mis-phasing can be compensated using quantum filter cavities, the other two degradations are fundamental. For squeezed states, they establish the noise limit:\n\\begin{align}\n  N(\\Omega) &\\ge \\Gamma{\\cdot}\\left(2\\eta\\sqrt{\\Xi'(1-\\Xi')} + \\Lambda_\\ensuremath{\\text{IRO}}\\right),\n  &\n  e^{-2r} &= \\sqrt{\\Xi'(\\Omega)}\n  \\label{eq:N_limit}\n\\end{align}\nSetting the squeezing level as $\\sqrt{\\Xi'}$ solves for the optimal noise given the dephasing. Squeezing is then further degraded from losses, producing the noise limit. Notably, the optimal squeezing is generally frequency-dependent due to $\\Xi(\\Omega)$, indicating that for typical broadband squeezing sources, this bound cannot always be saturated at all frequencies.\n\n\\subsection{Ideal Interferometer Response}\n\\label{sec:ideal_IFO_metrics}\nBefore analyzing quantum noise data to utilize the squeezing metrics of \\crefrange{eq:metric_N}{eq:metric_Lambda}, it is worthwhile to first review the quantum noise features expected in the LIGO detector noise spectra\\cite{Kimble-PRD01-ConversionConventional, Aasi-CQG15-AdvancedLIGO}, under ideal conditions and without accounting for realistic effects present in the interferometer. The derivations later will then extend how the well-established equations below generalize to incorporate increasingly complex interferometer effects, both by extracting features from matrix-valued simulation models, as well as by extracting features from scalar boundary-value equations for cavities.\n\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_IFO, \n  label={SQZ_mm_IFO},\n  caption={\n    This simplified diagram of the interferometer layout shows the propagation of the source laser (solid red) and squeezed beam (dashed burgundy). At (a), the squeezed beam is sourced from a parametric amplifier cavity and circulated to the interferometer with a Faraday isolator. At (b), the squeezing field reflects from the interferometer. Depending on the frequency and transverse beam profile, the states partially transit the interferometer cavities, but also partially reflect promptly. The squeezing that enters the interferometer symmetrically is beam split inside the signal recycling cavity, coherently resonates in both arms, and recombines again at the beamsplitter, effectively experiencing the two branches as a single linear coupled cavity. Injected at a different port, the red laser field carries substantial laser power and is symmetrically split to pump the arm cavities. Differential length signals are sourced by modulating the circulating pump field, creating a phase-quadrature field that resonates in the same effective linear cavity as the squeezing. The signal is emitted at (b) follows with the reflected squeezing. The transverse beam profile (mode) of the signal and squeezing is then selected using the output mode cleaning cavity at (c). Ultimately, the signal and noise are read as timeseries in photodetectors at (d). This effect of coherent interference between prompt and cavity-circulated squeezing from this sequence is formulated, measured, and analyzed in the following sections.\n},\n}\n\n\nOther than shot noise imprecision, the dominant quantum effect in gravitational wave interferometers arises from radiation pressure noise. In an ideal, on-resonance interferometer, this noise is characterized by the interaction strength $\\ensuremath{\\mathcal{K}}(\\Omega)$ that correlates amplitude fluctuations entering the interferometer to phase fluctuations that are detected along with the signal. $\\ensuremath{\\mathcal{K}}$ is generated from the circulating arm power $P_\\ensuremath{\\text{A}}$ creating force noise that drives the mechanical susceptibility $\\chi(\\Omega)$. The susceptibility relates force to displacement on each of the four identical mirrors of mass $m$ in the GW arm cavities. The QRPN effect is enhanced by optical cavity gain $g(\\Omega)$ which resonantly enhances quantum fields entering the arm cavities and signal fields leaving them.\n\\begin{align}\n  \\mathcal{K}(\\Omega)\n  &= 16k\\frac{P_\\ensuremath{\\text{A}}}{c} g^2(\\Omega)\\chi(\\Omega),\n    &\n  g(\\Omega) &= \\frac{\\sqrt{{\\gamma_\\ensuremath{\\text{A}} c}\/{L_{\\ensuremath{\\text{a}}}}}}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n              \\label{eq:optical_gain_g}\n\\end{align}\nHere, $k$ is the wavenumber of the interferometer laser and $c$ the speed of light.\nThe arm cavity gain $g(\\Omega)$ is a function of the signal bandwidth $\\gamma_\\ensuremath{\\text{A}}$, derived later, and the interferometer arm length $L_\\ensuremath{\\text{a}}$. Unlike in past works, this expression of $\\ensuremath{\\mathcal{K}}(\\Omega)$ here is kept complex, holding the phase shift that arises from the interferometer cavity transfer function. The phase of $\\ensuremath{\\mathcal{K}}(\\Omega)$ is useful for later generalizations. $\\ensuremath{\\mathcal{K}}(\\Omega)$ adds the amplitude quadtrature noise power to the phase quadrature fluctuations directly reflected from the interferometer, setting the noise gain $\\Gamma(\\Omega)$\n\\begin{align}\n  \\Gamma(\\Omega) &= 1 + |\\mathcal{K}(\\Omega)|^2\n  &\n  \\theta(\\Omega) &= \\arctan(|\\ensuremath{\\mathcal{K}}(\\Omega)|)\n                   \\label{eq:GammaTheta_standard}\n\\end{align}\nThe relationship between $\\Gamma(\\Omega)$ and $\\theta(\\Omega)$ from $\\ensuremath{\\mathcal{K}}(\\Omega)$ is stated above as reference, but it will more appropriately handle the complex $\\ensuremath{\\mathcal{K}}(\\Omega)$ when it is derived later. The value $|\\ensuremath{\\mathcal{K}}(\\Omega_\\ensuremath{\\text{sql}})| \\equiv 1$ defines the crossover frequency $\\Omega_\\ensuremath{\\text{sql}}$ between noise contributions from shotnoise imprecision and QRPN, corresponding to $\\Gamma(\\Omega_\\ensuremath{\\text{sql}})= 2$ and $\\theta(\\Omega_\\ensuremath{\\text{sql}})=-45^\\circ$. For the $\\chi(\\Omega)$ susceptibility of a free test mass, the factor $\\ensuremath{\\mathcal{K}}(\\Omega)$ can be expressed using only frequency scales.\n\\begin{align}\n  \\mathcal{K}(\\Omega)\n  &= -\\frac{\\Omega^2_\\ensuremath{\\text{sql}}}{\\Omega^2}\\left( \\frac{\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\\right)^2,\n  &&\\text{given}&\n  \\chi(\\Omega) &\\equiv \\frac{-1}{m \\Omega^2}\n    \\label{eq:Kchi_standard}\n\\end{align}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/dual_d.pdf}\n\t\\caption{\n    This figure plots the total quantum and classical noise measured in the LIGO detectors in displacement amplitude spectral density units. The black trace plots a reference measurement of the total noise without injected squeezing at 0.25Hz resolution over 1.5Hr integration for LLO and 1.1Hr for LHO. The orange shotnoise measurement shows the displacement calibration, $\\sqrt{G(\\Omega)}$, in amplitude density units. Subtracting the shotnoise level from the reference yields the gray datapoints, which have been rebinned using a median statistic applied after the subtraction and with a logararithmic bin spacing. The subtraction primarily shows the classical noise but also contains QRPN. Multiple measurements are taken at varied squeezing angles, with 5 of 12 plotted for Livingston (LLO) and 5 of 34 plotted for Hanford (LHO), using the same median rebinning method as the gray subtraction. The variation in the data errorbars results from the binning span of each datapoint, $\\Delta F$, and the measurement integration time, $\\Delta T$. The measured spectra error relative to the total noise and proportional to $1\/\\sqrt{\\Delta F \\Delta T}$. The squeezing angle of $-3.9^\\circ$ and $1.4^\\circ$ datasets at LLO used ${\\sim}1$Hr integration, and the remainder used 15 min each. The squeezing angle $4.5^\\circ$ dataset at LHO used ${\\sim}1$Hr integration, while all others use 2 minutes each. The squeezing level $e^{\\pm2r}$ is constant over all angles, but different between the two sites. This accounts for the difference in the yellow, ${\\sim}30^{\\circ}$, dataset at each site.\n  }\n\t\\label{fig:data_h}\n\\end{figure*}\n\nFrequency independent losses are applied to squeezing before and after the interferometer using $\\eta=\\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{R}}\\eta_\\ensuremath{\\text{O}}$ where $\\eta_\\ensuremath{\\text{I}} < 1$, $\\eta_\\ensuremath{\\text{O}} < 1$. The ideal interferometer assumption of the formulas above enforce $\\eta_\\ensuremath{\\text{R}}=1$. Phase noise in squeezing is included in this ideal interferometer case using $\\Xi' = \\ensuremath{\\phi^2_{\\text{rms}}}$.\n\nThe above expressions relate the optical noise $N(\\Omega)$ of \\cref{eq:metric_N} to past models of the quantum strain sensitivity of GW interferometers\\cite{Kimble-PRD01-ConversionConventional, Buonanno-PRD01-QuantumNoise, Buonanno-PRD03-ScalingLaw}. Since $N(\\Omega)$ is relative to shot-noise, it must then be converted to strain or displacement using the optical cavity gain $g(\\Omega)$, by how it affects the GW signal through the calibration factor $G(\\Omega)$. This factor $G(\\Omega)$ relates strain modulations to optical field phase modulations in units of optical power.\n\\begin{align}\n  \\text{PSD}_{\\text{strain}}(\\Omega) &= G(\\Omega)N(\\Omega),\n                                       &\n  G(\\Omega) &= \\frac{\\hbar c}{\\eta_\\ensuremath{\\text{O}} L_{\\ensuremath{\\text{a}}}^2|g(\\Omega)|^{2} k P_\\ensuremath{\\text{A}} }\n              \\label{eq:optical_calibration_G}\n\\end{align}\nTogether, these relations allow one to succinctly calculate the effect of squeezing on the strain power spectrum in the case of an ideal interferometer. These factors and the calculations behind them will be revisited as non-idealities are introduced.\n\n\\section{Experimental Analysis and Results}\n\\label{sec:experiment}\n\nA goal of this paper is to use the squeezing response metrics of \\crefrange{eq:metric_N}{eq:metric_Lambda} to relate measurements of the instrument's noise spectrum to the parameters of the squeezer system, namely its degradations due to loss $1-\\eta$, radiation pressure from mis-phasing $\\phi{-}\\theta(\\Omega)$, and dephasings $\\Xi'(\\Omega)$. This section presents measurements from the LIGO interferometers that are best described using the established frequency-dependent metrics. The measurements then motivate the remaining discussion of the paper that construct simple interferometer models to describe this data in the context of the metrics. This section refers to and relates to the later sections to provide early experimental motivation for the discussions that follow. The reader may prefer instead to skip this section and first understand the models before returning to see their application to experimental data.\n\nThe main complexity in analyzing the LIGO data is that the detectors have additional classical noises, preventing a direct measurement of $N(\\Omega)$. The many frequency-dependent squeezing parameters must also be appropriately disentangled. To address both of these issues, the unknown squeezing parameters are fit simultaneously across multiple squeezing measurements. The classical noise contribution is determined by taking a reference dataset where the squeezer is disabled, such that $S(\\Omega)=1$, and then subtracting it from the datasets where squeezing is injected.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/dual_Q.pdf}\n  \\caption{\n    This figure shows the data of \\cref{fig:data_h} processed as per \\cref{sec:experiment} for each LIGO site. The processing subtracts away the classical noise determined from the unsqueezed reference dataset. The top panels show the relative noise change $Q_k(\\Omega_i)$ of \\cref{eq:Q_processing} computed using $\\Gamma_k(\\Omega_i)$ from the exact interferometer model of \\cref{sec:matrix_coupled_cavity} using the parameters of \\cref{tab:LIGO_params}. The top panel includes dots with errorbars for the processed data and lines for the best-fit $Q_k(\\Omega_i)$. The middle panel shows the best-fit frequency dependent loss as data points, with errorbars propagated through the fit. For LLO, two sets of loss datapoints are shown, corresponding to interferometer models with different readout angles $\\zeta$.\n    The loss plots also show $1{-}\\eta(\\Omega)$ as computed from the exact matrix model, along with a phenomenological fit against the model of \\cref{eq:Lambda_MM} of \\cref{sec:modeling_TMM}. The phenomenological fit assumes frequency independent losses from the input and output squeezing path with a frequency-dependent addition attributed to transverse mismatch. The bottom panels show the frequency-dependent fit to the observed squeezing angle $\\theta_k(\\Omega_i)$, using the convention of $\\theta(2\\pi{\\cdot}3\\text{kHz}) = 0$. It also plots $\\theta(\\Omega)$ as computed using the exact matrix model. For the LLO data, the $\\zeta=0^\\circ$ model is typically assumed for Michelson-like interferometers such as LIGO; However, the model at that readout angle implies losses at low frequencies that are not favored by the $\\eta(\\Omega)$ models explored in this paper. Alternatively, the $\\zeta\\approx -13^\\circ$ model is consistent with both the fitted losses and the fitted squeezing angles.\n  }\n  \\label{fig:data_Q}\n\\end{figure*}\n\nRepresentative strain spectra from the LIGO Livingston (LLO) and LIGO Hanford (LHO) observatory datasets are plotted in \\cref{fig:data_h}. The Livingston dataset is also reported in \\cite{Yu-N20-QuantumCorrelations}, which details the assumptions and error propagation for the classical noise components and calibration. Only statistical uncertainty is considered in this analysis, in order to propagate error to the parameter fits. The strain spectra of \\cref{fig:data_h} include a reference dataset where the squeezer is disabled, shown in black and at the highest frequency resolution. Additionally, the shotnoise ($N=1$) is plotted in orange, indicating the calibration $\\sqrt{G(\\Omega)}$ of \\cref{eq:optical_calibration_G}. The gray subtraction curve depicts the total classical noise contribution summed with the radiation pressure noise $G(\\Omega)\\ensuremath{\\mathcal{K}}^2(\\Omega)$. The gray dataset can equivalently be computed using a cross correlation of the two physical photodetectors at the interferometer readout\\cite{martynov-pra17-quantumcorrelation}. The equivalence of subtraction and cross correlation is used to precisely experimentally determine the shot-noise scale $G(\\Omega)$ from the displacement-calibrated data.\n\n\\subsection{Analysis}\n\nEach squeezing measurement, indexed by $k$, is indicated by $M_{\\text{ref}, k}(\\Omega_i)$, with a value at each frequency indexed by $i$. The reference dataset is denoted $M_{\\text{ref}}(\\Omega_i)$. The two are subtracted to cancel the stationary classical noise component. The calibration $G(\\Omega)$ is removed to result in the differential quantum noise measurement $D_k(\\Omega_i)$.\n\\begin{align}\n  D_k(\\Omega_i)\n  &\\equiv\n    \\frac{M_{\\text{sqz}, k}(\\Omega_i) - M_{\\text{ref}}(\\Omega_i)}{G(\\Omega_i)}\n    \\label{eq:D_processing}\n\\end{align}\nFor these datasets, the squeezing level $e^{\\pm 2r}$, is held constant and independently measured using the nonlinear gain technique\\cite{Dwyer-OEO13-SqueezedQuadrature} to derive $y$ of \\cref{eq:OPA_y_SQZ}. Each differential data $D_k(\\Omega_i)$ is taken at some squeezing angle $\\phi_k$, which is either fit (LLO) or derived from independent measurements (LHO). The parameters $\\eta_i$ and squeezing rotation $\\theta_i$ are independent at every frequency $\\Omega_i$ but fit simultaneously. All $\\phi_k$ are also fit simultaneously across all datasets. Nonlinear least squares fitting was performed using the Nelder-Mead simplex algorithm \\cite{Gao-COA12-ImplementingNelderMead} implemented in SciPy \\cite{Virtanen-NM20-SciPyFundamental}. The residual minimized by least squares fitting is\n\\begin{align}\n  \\mathcal{R} &= \\sum_{\\substack{i=0\\\\k=0}}^{\\mathcal{N}} \\left( \\frac{D_k(\\Omega_i) - \\overline{D}_k(\\Omega_i)}{\\Delta D_k(\\Omega_i)}\\right)^2\n\\end{align}\nThe measurement statistical uncertainty $\\Delta D$, dominated by the statistical uncertainty in power-spectrum estimation, was propagated through the datasets per \\cite{Yu-N20-QuantumCorrelations}. $\\overline{D}_k(\\Omega_i)$ is the model of the data that is a function of the fit parameters, $\\eta_i, \\theta_i, \\phi_k$ as well as independently measured parameters such as $e^{-2r}$. $\\Xi'(\\Omega_i)$ is not fit using this data since the squeezing level $e^{2r}$ is not varied across the datasets. This is discussed below. These given fit parameters affect are propagated through the squeezing metric functions create a model of this particular differential quantum noise measurement.\n\\begin{align}\n  \\overline{S}_k(\\Omega_i) &\\equiv\n  e^{-2r}\\cos^2\\Big(\\ensuremath{\\phi}_k - \\theta_i\\Big) + e^{+2r}\\sin^2\\Big(\\ensuremath{\\phi}_k - \\theta_i\\Big)\n  \\\\\n  \\overline{D}_k(\\Omega_i)\n  &\\equiv \\left.N(\\Omega_i)\\right|_{S = \\overline{S}_k(\\Omega_i)} - \\left.N(\\Omega_i)\\right|_{S = 1}\n\\end{align}\nWhich simplifies to\n\\begin{align}\n  \\overline{D}_k(\\Omega_i)\n  &= \\left(\\overline{S}_k(\\Omega_i) - 1\\right) \\eta_i \\Gamma(\\Omega_i)\n\\end{align}\nNotably, the individual efficiencies $\\eta_\\ensuremath{\\text{I}}, \\eta_\\ensuremath{\\text{R}}, \\eta_\\ensuremath{\\text{O}}$ cannot be individually measured and only the ``total'' efficiency $\\eta(\\Omega)$ is measurable using this differential method, where the classical noise is subtracted using a reference dataset with squeezing disabled. Additionally, the optical efficiency $\\eta$ can only be inferred given some knowledge or assumption of $\\Gamma(\\Omega)$. In effect, the product $\\eta \\Gamma$ is the primary measurable quantity, rather than its decomposition into separate $\\eta$ and $\\Gamma$ terms; However, for the purposes of modeling, decomposing the two is conceptually useful. Furthermore, to characterize physical losses, the efficiency $\\eta$ or loss $\\Lambda_\\ensuremath{\\text{IRO}}\\approx 1 {-} \\eta$ is easier to plot and interpret than the product $\\eta\\Gamma$.\n\nFor these reasons, the differential data $D_k(\\Omega_i)$ is further processed, creating the measurement $Q_k(\\Omega_i)$ with a form similar to \\cref{eq:etaSQZ_basic1}\n\\begin{align}\n  Q_k(\\Omega_i)\n  &\\equiv \n    \\frac{D_k}{\\Gamma(\\Omega_i)} + 1 \\approx S_k(\\Omega_i)\\eta_i + (1 - \\eta_i) + \\Delta Q\n    \\label{eq:Q_processing}\n\\end{align}\nThe LIGO squeezing data expressed in dB's of $Q_k(\\Omega_i)$ are plotted in the upper panels of \\cref{fig:data_Q}. The data and error bars are in discrete points, while the parameter fits to $Q_k$ using $\\eta_i$, $\\theta_i$ and $\\phi_k$ are the solid lines between the data points. The spectra in each set are calculated using the Welch method a median statistic at each frequency to average all of the frames through the integration time. This prevents biases due to instrumental glitches adding non-stationary classical noise. This technique is detailed in \\cite{Yu-N20-QuantumCorrelations}.\n\nAfter computing $Q_k(\\Omega_i)$ at full frequency resolution, the data is further rebinned to have logararithmic spacing by taking a median of the data points within the frequency range of each bin. This rebinning greatly improves the statistical uncertainty at high frequencies, where many points are collected. At lower frequencies, the relative error benefits less from binning; however, both the LLO and the LHO datasets use a long integration time for their reference measurement and at least one of the squeezing angle measurements. Using the median removes narrow-band lines visible in the strain spectra of \\cref{fig:data_h}. Fitting combines the few long-integration, low-error datasets with many short-integration, high-error sets at many variations of the operating parameters. The few low-error datasets reduce the absolute uncertainty in the resulting fit parameters, whereas the many variations reduce co-varying error that would otherwise result from modeling parameter degeneracies.\n\nThe relative statistical error in each bin of the original PSD $M_k(\\Omega_i)$ is approximately $(\\Delta F \\Delta T)^{-{1}\/{2}}$ given the integration time $\\Delta T$ of 2 minutes to 1 hour and bin-width $\\Delta F$ of 0.25Hz. This relative error is converted to absolute error and propagated through the processing steps of \\crefrange{eq:D_processing}{eq:Q_processing}. At low frequencies, the classical noise contribution to each $M_k$ is larger than the quantum noise. Although it is subtracted away to create $D_k(\\Omega_i)$, the classical noise increases the absolute error, and, along with less rebinning, results in the larger relative errors at low-frequency in \\cref{fig:data_Q}. After fitting the squeezing parameters, the Hessian of the reduced chi-square is computed from the Jacobian of the fit residuals with respect to the parameters. This Hessian represents the Fisher information, and the diagonals of its inverse provides the variances indicated by the plotted loss and angle parameter error bars.\n\nFor the LHO data, the fit parameters $\\phi_k$ are determined by mapping the demodulation angles of its coherent control feedback system\\cite{Tse-PRL19-QuantumEnhancedAdvanced, Dooley-OEO15-PhaseControl, Vahlbruch-PRL06-CoherentControl, Chelkowski-PRA07-CoherentControl} back to the squeezing angle. That mapping has 3 unknown parameters, an offset in demodulation angle, an offset in squeezing angle, and a nonlinear compression parameter, all of which are fit simultaneously in all datasets. This $\\phi_k$ mapping was not performed on the LLO data, as some systematic errors in the demodulation angle records bias the results. Despite fitting more independent parameters, the longer integration time of the LLO data gives it sufficiently low statistical uncertainty at frequencies below $\\Omega_\\ensuremath{\\text{sql}}$ that the model and parameter degeneracy between $\\phi_k$, $\\theta_i$ and $\\eta_i$ is not an issue.\n\n\\subsection{Results}\n\nThe middle panels of \\cref{fig:data_Q} show the fits to $\\eta_i$, though plotted as loss $1{-}\\eta_i$ to represent $\\Lambda_\\ensuremath{\\text{IRO}}$. Both datasets additionally include a red loss model curve fit, assembled using the equations in \\cref{sec:modeling}. The orange exact model curves use \\cref{sec:matrix_coupled_cavity}. The data and model curve fit shows a variation in the efficiency, where losses increase from low to high frequencies. This increase in loss can be attributed either to losses within the signal recyling cavity of the interferometer, or to a coherent effect resulting from transverse Gaussian beam parameter mismatch between the squeezer and interferometer cavities. At low frequencies, the optical efficiency is similar between the two LIGO sites, indicating that frequency independent component to the loss are consistent between the implementations at both LIGO sites. The differing high-frequency losses can reasonably be ascribed to variations in the optical beam telescopes of the squeezing system and are analyzed in \\cref{sec:modeling_TMM}.\n\nThe LLO middle panel of \\cref{fig:data_Q} shows two separate inferred loss $1 - \\eta_i$ datasets. These differ in their underlying model of $\\Gamma(\\Omega_i)$. The following section \\ref{sec:derivations} discusses how variations in $\\Gamma$ arise and describes the local oscillator angle $\\zeta$. The $\\zeta = 0$ data reflects the standard, ideal radiation pressure noise model of \\crefrange{eq:Kchi_standard}{eq:GammaTheta_standard}. This model is disfavored given the frequency dependency of $\\eta(\\Omega_i)$ derived using optical cavity models later in this paper. The $\\zeta = -13^\\circ$ model presents an alternative that is compatible with models of the optical efficiency. The need for this alternative indicates that squeezing metrics must account for variations in interferometer noise gain $\\Gamma$. Physically, these variations arise from the readout angle adjusting the prevalence of radiation pressure versus pondermotive squeezing. The $\\zeta = -13^\\circ$ model results in a smaller noise gain $\\Gamma$ at $40 \\text{Hz}$ than does the $\\zeta = 0^\\circ$ model. Since the lower $\\Gamma$ model is favored, this dataset provides some, moderate, evidence that LLO currently benefits from the quantum correlations introduced by the mirrors near $\\Omega_{\\text{SQL}}$, while experiencing lessened sensitivity elsewhere.\n\nThis data demonstrates that the readout angle has an effect on the interferometer sensitivity and the optimal local oscillator is not necessarily $\\xi = 0$ due to radiation pressure.  The quantum benefit of decreased $\\Gamma$ from the readout angle $\\xi$ is a method to achieve sub-SQL performance that is an alternative to injecting squeezing. Like squeezing, it has a frequency dependent enhancement known as the ``variational readout'' technique \\cite{Kimble-PRD01-ConversionConventional, Khalili-PRD07-QuantumVariational}, that a sensitivity increase from lowering $\\Gamma$ while minimizing the sensitivity decrease of the frequency-independent form. For LLO, the reduced sensitivity from $\\xi \\ne 0$ masquerades as a $5\\%$ loss of signal power, but does not actually affect the $\\eta$ or $\\Lambda_\\ensuremath{\\text{IRO}}$ contributions to the squeezing level.\n\nThe bottom panels of \\cref{fig:data_Q} show the fits of $\\theta_i$ of each dataset. The magnitude of $e^{\\pm 2r}$ provides a ``lever arm'' in the variation of $S_k(\\Omega_i)$ that strongly constrains the $\\phi_k{-}\\theta_i$ effective squeezing angle. These leveraged constraints result in small errorbars to the fitted $\\theta_i$. The LLO data are plotted with two models of the $\\theta_i$ based on the assumed local oscillator angle $\\zeta$. The $\\zeta=0^\\circ$ model follows the standard radiation pressure model of \\cref{eq:GammaTheta_standard} at low frequencies and includes a filter-cavity type rotation around the interferometer cavity bandwidth $\\gamma \\approx 2\\pi \\cdot 450\\text{ Hz}$. This rotation is modeled in \\cref{sec:modeling_cavities}. The $\\zeta=-13^{\\circ}$ model is computed using the coupled cavity model of \\cref{sec:matrix_coupled_cavity} and internally includes a weak optical spring effect along with the shifted readout angle $\\zeta$. Together, these effects modify the effective squeezing angle $\\theta$ away from \\cref{eq:GammaTheta_standard} at low frequencies, and agree well with the dataset. This agreement provides further evidence of the reduced radiation pressure noise gain $\\Gamma(\\Omega)$ in LLO that results from the effective LO readout angle $\\zeta$. A nonzero readout angle $\\zeta$ is reasonable to expect due to unequal optical losses in the LIGO interferometer arm cavities. The arm mismatch results in imperfect subtraction of the fringe-light amplitude quadrature at the beamsplitter, creating a static field that adds to the phase-quadrature light created from the Michelson offset and results in $\\zeta \\ne 0$. Past diagnostic measurements\nconclude that some power in the readout diodes must be in the amplitude quadrature, but until now could not determine the sign.\n\nAlthough the squeezing angle parameters $\\phi_k$ and $\\theta_i$ are fit, the frequency-dependent dephasing parameter $\\Xi_i$ cannot be reliably determined from these datasets given the accuracy to which $e^{\\pm 2r}$ is measured. Additionally, the squeezing level $e^{\\pm 2r}$ is not varied in this data, nor is it sufficiently large to resolve an influence from $\\Xi(\\Omega) < 10^{-3}$. This $\\sqrt{\\Xi} \\approx \\ensuremath{\\phi_{\\text{rms}}}$ is expected from independent measurements of phase jitter that propagate through the coherent control scheme of the squeezer system\\cite{Tse-PRL19-QuantumEnhancedAdvanced}. A large source of optically induced $\\Xi$ is not expected has the interferometer cavities are not sufficiently detuned. Measurements of the squeezing system indicate $\\ensuremath{\\phi_{\\text{rms}}} \\lesssim 30\\text{ mRad}$. Future LIGO measurements should include additional datasets that vary $r$ along a third indexing axis $j$ and should increase the injected squeezing level $e^{2r} > 30$ to measure, or at least bound, $\\Xi$ and its frequency-independent contribution $\\ensuremath{\\phi^2_{\\text{rms}}}$. The model fits described above are consistent with the data while assuming $\\Xi = \\ensuremath{\\phi^2_{\\text{rms}}} \\equiv 0$.\n\n\\section{Decomposition Derivation}\n\\label{sec:derivations}\n\nThe factors $\\eta(\\Omega), \\theta(\\Omega)$ and $\\Xi(\\Omega)$ from \\cref{sec:metrics} each describe an independent way for squeezing to degrade. $\\Gamma(\\Omega)$ indicates how the quantum noise scales above or below the shot noise level from squeezing and from quantum radiation pressure within the interferometer. They represent a natural extension of standard squeezing metrics that incorporates frequency dependence, and, as scalar functions, they are simple to plot and to relate with experimental measurements. This section delves into their derivation by employing matrices in the two photon formalism \\cite{Caves-PRA85-NewFormalism, Schumaker-PRA85-NewFormalism} to represent the operations of squeezing, adding loss, shifting the squeezing phase, reflecting from the interferometer, and final projection of the quantum state into the interferometer readout. The derived formulas can be used in frequency-domain simulation tools that compute noise spectra using matrix methods, so that the quantum response metrics can be provided in addition to opaquely propagating squeezing to an simulation result of $N(\\Omega)$. \n\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_cavities, \n  label=SQZ_mm_cavities, \n  caption={\n    The two-photon transformation matrices experienced by squeezing through the sequence of \\cref{fig:SQZ_mm_IFO}. The effective linear coupled cavity, including the optomechanical effect of radiation pressure, is collected and computed into the transformation $\\Tmat{H}_R$. The middle cavity is the signal recycling cavity and the rightmost cavity represents the coherent combination of both arms. Each cavity adds losses from each mirror. For simplicity, these are collected into round-trip cavity loss contributions, $\\Lambda_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{s}}}$, and $\\Lambda_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{s}}}$ that inject standard optical vacuum into the cavities, circulating and transforming into the loss terms $\\Tmat{T}_{\\ensuremath{\\text{R}},\\ensuremath{{\\,\\mu}}}$ while lowering the efficiency $\\eta_\\ensuremath{\\text{R}}$. Transformations of the squeezing at the input and output are included with the terms, $\\Tmat{H}_{\\ensuremath{\\text{I}}}$, $\\Tmat{H}_{\\ensuremath{\\text{O}}}$ and any additive vacuum contributions,  $\\Tmat{T}_{\\ensuremath{\\text{I}},\\ensuremath{{\\,\\mu}}}$, $\\Tmat{T}_{\\ensuremath{\\text{O}},\\ensuremath{{\\,\\mu}}}$.\n  },\n}\n\nTwo-photon matrices are an established method to represent transformations of the optical phase space of Guassian states in an input-output Heisenberg representation of the instrument\\cite{Danilishin-LRR12-QuantumMeasurement}. They are concise yet rigorous when measuring noise spectra from squeezed states using the quantum measurement process of Homodyne readout. Section II of \\cite{Danilishin-LRR19-AdvancedQuantum} provides a review of their usage in the context of gravitational-wave interferometers. Here, two-photon matrices are indicated by doublestruck-bold lettering, and are given strictly in the amplitude\/phase quadrature basis.\n\nEach matrix represents the transformation of the optical phase space of a single optical ``mode'' as it propagates through each physical element towards the readout. The term ``mode'' refers to a basis vector in a linear decomposition of optical field the transverse optical plane of many physical ports\\footnote{Simulations also typically span multiple optical frequencies, but this is not treated here.}. Each plane is further decomposed into transverse spatial modes using a Hermite or Laguerre-Gaussian basis. In this decomposition, each optical mode is indexed by the placeholder $\\mu$ and acts as a continuous transmission channel for optical quantum states. The phase space transformations of these continuous optical states is indexed by time or, more conveniently, frequency. Optical losses and mixing from transverse mismatch behave like beamsplitter operations, serving to couple multiple input modes, generally carrying vacuum states, to the mode of the readout where states are measured.\n\nThe mode of the injected squeezed states, and their specific transformations during beam propagation, must be distinguished from all of the lossy elements that couple in vacuum states. The squeezed states experience a sequence of transformations by the input elements, interferometer, and output elements, denoted $\\Tmat{H}_\\ensuremath{\\text{I}}(\\Omega)$,  $\\Tmat{H}_\\ensuremath{\\text{R}}(\\Omega)$,  $\\Tmat{H}_\\ensuremath{\\text{O}}(\\Omega)$. This sequence multiplies to formulate the total squeeze path propagation $\\Tmat{H}$. \n\\begin{align}\n  \\Tmat{H}(\\Omega) &= \n\\Tmat{H}_{\\text{O}}\\Tmat{H}_{\\text{R}}\\Tmat{H}_{\\text{I}}\n\\end{align}\nLossy optical paths mix the squeezed states with additional standard vacuum states. These are collected into sets of transformation matrices corresponding to each individual loss source, $\\{\\Tmat{T}_\\mu\\}$. See \\cref{fig:SQZ_mm_cavities}. The sets are grouped by their location along the squeezing path where the lossy element is incorporated. The beamsplitter-like operation that couples each loss is given by a $\\Tmat{\\Lambda}_\\mu$, indexed by its location and source along the squeezing path. Loss transformations $\\Tmat{\\Lambda}_\\mu$ are generally frequency-independent. $\\Tmat{\\Lambda}_{\\text{R}, i}$ are an exception, as they occur within the cavities of the interferometer and include some cavity response. The vacuum states associated with each loss then propagate along with the squeezed states and experience the remaining transformations that act on squeezing.\n\\begin{align}\n  \\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{H}_{\\text{O}}\\Tmat{H}_{\\text{R}}\\Tmat{\\Lambda}_{\\text{I},\\ensuremath{{\\,\\mu}}}\n    \\\\\n  \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{H}_{\\text{O}}\\Tmat{\\Lambda}_{\\text{R},\\ensuremath{{\\,\\mu}}}\n    \\\\\n  \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{\\Lambda}_{\\text{O},\\ensuremath{{\\,\\mu}}}\n  \\\\\n  \\left\\{\\Tmat{T}\\right\\} &= \\left\\{\\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}; \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}; \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}  \\right\\}\n\\end{align}\nTogether, all of the transformations of $\\Tmat{H}$ and  $\\{\\Tmat{T}\\}$ define the output states at the readout of the interferometer in terms of the input states entering through the squeezer and loss elements. The two quadrature observables of the optical states are given with the convention $\\hat{q}$ being the amplitude quadrature and $\\hat{p}$ being the phase, and they are indexed to distinguish their input port and transverse mode.\n\\begin{align}\n  \\begin{bmatrix}\\hat{q}_{\\text{out}}(\\Omega) \\\\ \\hat{p}_{\\text{out}}(\\Omega)\\end{bmatrix}\n  &=\n    \\Tmat{H} \\begin{bmatrix}\\hat{q}_{\\text{in}}(\\Omega) \\\\ \\hat{p}_{\\text{in}}(\\Omega)\\end{bmatrix} + \n  \\sum_{\\Tmat{T}_\\ensuremath{{\\,\\mu}}\\in \\left\\{\\Tmat{T}\\right\\}} \\Tmat{T}_\\ensuremath{{\\,\\mu}} \\begin{bmatrix}\\hat{q}_\\ensuremath{{\\,\\mu}}(\\Omega) \\\\ \\hat{p}_\\ensuremath{{\\,\\mu}}(\\Omega)\\end{bmatrix}\n  \\label{eq:qp_out}\n\\end{align}\nThe two-photon matrices $\\Tmat{H}$ and $\\Tmat{T}_\\ensuremath{{\\,\\mu}}$ must preserve commutation relations, namely $[\\hat{q}_\\text{out}, \\hat{p}_\\text{out}] = [\\hat{q}_\\ensuremath{{\\,\\mu}}, \\hat{p}_\\ensuremath{{\\,\\mu}}] = i\\hbar$. In doing so, the matrices ensure that losses within $\\Tmat{H}$ couple ancillary vacuum states that degrade squeezing.\n\nThe readout carries a continuous coherent optical field known as the ``local oscillator'' and the output states are read using homodyne readout. The phase of the local oscillator, $\\zeta$, defines the observed quadrature, $\\hat{m}$, for the homodyne measurement. Gravitational Wave interferometers typically use a ``Michelson offset''\\cite{Fricke-CQG12-DCReadout, Hild-CQG09-DCreadoutSignalrecycled, Ward-CQG08-DcReadout} in the paths adjacent their beamsplitter to operate slightly off of dark fringe. This offset couples a small portion of their pump carrier light to their output as the local oscillator field. This is a form of homodyne readout that fixes $\\zeta$ to measure in the phase quadrature, defined here to be when $\\zeta=0$. Imperfect interference at the beamsplitter can couple some amplitude quadrature and shift $\\zeta$ away from $0$. Balanced homodyne readout is an alternative implementation proposed for LIGO's ``A+'' upgrade and will allow $\\zeta$ to be freely chosen\\cite{Fritschel-OEO14-BalancedHomodyne}. Regardless of the implementation, the homodyne observable is $\\hat{m}$,\n\\begin{align}\n  \\hat{m} &=\n  \\ensuremath{\\Tvec{v}^{\\dagger}}\\begin{bmatrix}\\hat{q}_{\\text{out}}(\\Omega) \\\\ \\hat{p}_{\\text{out}}(\\Omega)\\end{bmatrix}\n           \n  &\n  \\ensuremath{\\Tvec{v}^{\\dagger}}(\\zeta)\n  &=\n    \\begin{bmatrix}\n      \\sin(\\zeta) & \\cos(\\zeta)\n    \\end{bmatrix}\n \\label{eq:homodyne_observable}\n\\end{align}\nHomodyne readout enforces a symmetrized expectation operator, denoted here with the subscript HR, for all measurements of the optical quantum states.\nFurther details of the measurement process are beyond the scope of this work, but the following quadratic expectations arise when computing the noise spectrum and are sufficient to simplify the homodyne expectation values of $\\hat{m}$.\n\\begin{align}\n  1 &= \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}^2}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}^2}_{\\text{HR}},\n      \\label{eq:vacuum_expectations}\n  \\hspace{1em}\n  0 = \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}\\hat{p}_\\ensuremath{{\\,\\mu}}}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}\\hat{q}_\\ensuremath{{\\,\\mu}}}_{\\text{HR}}\n      \\\\\n  0 &= \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}\\hat{q}_\\ensuremath{{\\,\\nu}}}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}\\hat{p}_\\ensuremath{{\\,\\nu}}}_{\\text{HR}} \\text{  for } \\nu \\ne \\mu\n      \\label{eq:vacuum_expectations2}\n\\end{align}\nAs a result of these expectations, the vector norm suffices to evaluate noise power using this matrix formalism.\nThe addition of squeezing can be seen either as a modification of the input states $\\hat{q}_{\\text{in}}$, $\\hat{q}_{\\text{in}}$, which violate \\cref{eq:vacuum_expectations}. This work uses the alternative picture, where an additional squeezing transformation occurring at the very start of the squeezing path $\\Tmat{H}$ that acts on $\\hat{q}_{\\text{in}}$, $\\hat{q}_{\\text{in}}$ that are also vacuum states. The squeezing transformation is defined by the squeezing level $r$ and the squeezing angle $\\phi$, which act via the matrices:\n\\begin{align}\n  \\Tmat{R}(\\phi)\n  &\\equiv\n  \\begin{bmatrix}\n    \\cos(\\phi) & {-}\\sin(\\phi)\\\\\n    \\sin(\\phi) & \\phantom{-}\\cos(\\phi)\n  \\end{bmatrix}\n &\n  \\Tmat{S}(r) &\\equiv\n  \\begin{bmatrix}\n    e^r & 0\\\\\n    0 & e^{-r}\n  \\end{bmatrix}\n\\end{align}\nWhen added to the squeezing path, the resulting quantum noise is calculated from the observable $\\hat{m}$.\n\\begin{align}\n  N(\\Omega) &= \\Braket{\\hat{m}^\\dagger\\hat{m}}_{\\text{HR}} = \n              \\left| \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}\\Tmat{R}(\\phi) \\Tmat{S}(r) \\right|^2 + \n\\sum_{\\Tmat{T}_\\ensuremath{{\\,\\mu}}\\in \\left\\{\\Tmat{T}\\right\\}} \\left| \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_\\ensuremath{{\\,\\mu}} \\right|^2\n                             \\label{eq:N_no_sqz_derivation}\n\\end{align}\nThe first term of which is one of the factors in \\cref{eq:metric_N}\n\\begin{align}\n  \\eta(\\Omega)\\cdot S(\\Omega, \\phi) \\cdot \\Gamma(\\Omega, \\zeta)\n \n \n \n \n \n   &= \\left|\\ensuremath{\\Tvec{v}^{\\dagger}}\n     \\Tmat{H}\\Tmat{R}(\\phi)\n     \\Tmat{S}(r)\n     \\right|^2\n    \\label{eq:decomposition_relation}\n\\end{align}\nAt this point, the factors can be separated because: $\\Tmat{R}\\Tmat{S}$ determines the factor $S(\\Omega, \\phi)$; $\\Tmat{H}$ has been ``reduced'' by loss, indicating when $\\eta(\\Omega) < 1$; and the benchmark noise level is defined by $\\Gamma(\\Omega)$, contained in the interferometer's optomechanical element $\\Tmat{H}_{\\text{R}}$.\n\nTo distinguish these terms, further manipulations are necessary. The first is to examine just the vector $\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}$ to determine how the later term $\\Tmat{R}\\Tmat{S}$ results in $S(\\Omega)$. Basis vectors for the two quadrature observables are defined, and the local oscillator is represented using them.\n\\begin{align}\n  \\ensuremath{\\Tvec{v}^{\\dagger}}(\\zeta)\n  &=\n  \\Tvec{e}_p^\\dagger\\Tmat{R}(\\zeta)\n  &\n    \\Tvec{e}_q &= \n    \\begin{bmatrix}\n      1 \\\\ 0\n    \\end{bmatrix}\n  &\n    \\Tvec{e}_p &= \n    \\begin{bmatrix}\n      0 \\\\ 1\n    \\end{bmatrix}\n\\end{align}\nThe basis vectors then allow the vector norm to be split into its two components $\\ensuremath{m_q}$ and $\\ensuremath{m_p}$, defining the \\textit{observed noise quadrature}.\n\\begin{align}\n  \\ensuremath{m_q}(\\Omega) &= \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H} \\Tvec{e}_q & \\ensuremath{m_p}(\\Omega) &= \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H} \\Tvec{e}_p\n\\end{align}\nThe vector $\\vec{m}$ contains the magnitude and angle of a projection of the quantum state $\\hat{q}_{\\text{in}}$, $\\hat{p}_{\\text{in}}$ at each frequency, but it also contains the complex phase shift from propagation delay in the interferometer and squeezing path. This later phase contribution does not affect noise calculations, but must be properly handled. Projecting it away requires maintaining phase information, and this is why the optomechanical factor $\\ensuremath{\\mathcal{K}}$ is complex in this work.\n\nThe squeezing angle rotation $\\Tmat{R}(\\phi)$ can be viewed through its left-multiplication, applying a rotation to the observed noise quadtrature rather than to the squeezing. In this picture, the angle $\\phi$ can align the observed quadtrature with either the squeezing or anti-squeezing quadrature. The rotation needed to do so determines $\\theta(\\Omega)$, again with the caveat that both $m_q$ and $m_p$ are complex. Their common phase carries the delay information, but their differential phase causes dephasing. In short, differential phase forces $\\vec{m}$ to project into both quadratures at any rotation $\\Tmat{R}(\\phi)$. This has the effect of always adding anti-squeezing to squeezing and vice-versa, resulting in the factor $\\Xi(\\Omega)$. The relations are fully derived in \\cref{sec:phase_noise_composition} using a singular value decomposition to identify the principle noise axes. It leads to the expressions\n\\begin{align}\n  \\theta(\\Omega) &\\approx -\\arctan\\left(\\Re\\!\\left\\{{\\frac{\\ensuremath{m_q}}{\\ensuremath{m_p}}}\\right\\}\\right)\n                   \\label{eq:theta_calculation}\n  \\\\\n  \\Xi(\\Omega) &= \\frac{1}{2} -  \\sqrt{\n        \\frac{\\left(|\\ensuremath{m_q}|^2 - |\\ensuremath{m_p}|^2\\right)^2 + 4 \\Re\\left\\{ \\ensuremath{m_q}\\ensuremath{\\conj{m}_p} \\right\\}}\n        {4\\left(|\\ensuremath{m_q}|^2 + |\\ensuremath{m_p}|^2\\right)^2}\n        } \n                   \\label{eq:Xi_calculation}\n\\end{align}\nThe observation vector $\\vec{m}$, and \\cref{eq:theta_calculation} generalizes the observed noise quadrature description of quantum radiation pressure noise. With it, the observed quadrature angle $\\theta(\\Omega)$ may be computed for any readout angle $\\zeta$ and for more complex interferometers $\\Tmat{H}_\\ensuremath{\\text{R}}$. The ideal interferometer example is demonstrated in \\cref{sec:ideal_IFO_example}\n\nThe phase and magnitude of of the previous argument allows one to determine $S(\\Omega)$ from the form of $\\Tmat{S}$ applied to $\\vec{m}\\Tmat{R}(\\phi)$. Factoring $S$ away, the magnitude of $\\vec{m}$ carries the efficiency of transmitting the squeezed state, along with the noise gain applied to it.\n\\begin{align}\n  \\eta(\\Omega) \\cdot \\Gamma(\\Omega, \\zeta)\n  &=\n    |\\ensuremath{m_q}|^2 + |\\ensuremath{m_p}|^2\n    \\label{eq:eta_gamma}\n\\end{align}\n$\\Gamma(\\Omega)$ expresses the total noise from the interferometer when squeezing is not applied, applying radiation pressure or optomechanical squeezing to both the squeezing path vacuum and internally loss-sourced vacuum. $\\eta \\Gamma$ is affected by all losses, but some of them affect $\\Gamma(\\Omega)$ as well. Using squeezing or a coherent field to probe $\\Tmat{H}$ always measures the product $\\eta \\Gamma$, so the noise gain factor $\\Gamma$ serves primarily as a benchmark. As a benchmark, it relates the dependence of $N(\\Omega)$ to $S(\\Omega)$ and separates the scaling by the efficiency $\\eta$ so that the physical losses may be determined. For this reason, there is freedom to define $\\Gamma$ to make it as independent from the losses as possible, so that it best serves as a benchmark. Here, it is defined using the simulated knowledge of the total noise from the interferometer elements alone:\n\\begin{align}\n \\Gamma(\\Omega) \n  &= \n\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}_{\\text{R}}\\right|^2 + \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{\\Lambda}_{\\text{R},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\label{eq:gamma_HR}\n\\end{align}\n$\\eta$ is then determined by dividing \\cref{eq:eta_gamma} by \\cref{eq:gamma_HR}. Under this definition of $\\Gamma$, $\\eta \\propto \\eta_\\ensuremath{\\text{I}}$ and $\\eta \\propto \\eta_\\ensuremath{\\text{O}}$. Losses within the interferometer affect $\\Gamma(\\Omega)$ slightly, and  $\\eta \\propto \\eta_\\ensuremath{\\text{R}}$ is only approximate. \\Cref{sec:radiation_pressure_calc} gives an example of how losses affect $\\eta$ and $\\Gamma$. The primary alternative definition is to use $\\Gamma = N\\big|_{S=1}$, but this definition makes $\\eta_\\ensuremath{\\text{O}}$ both less physically intuitive and also sensitive to interferometer parameters.\n\nSubtracting $\\eta\\Gamma$ from \\cref{eq:N_no_sqz_derivation} and factorizing by the optical paths provides the definition of the remaining efficiency terms.\n\\begin{align}\n  (1 - \\eta_\\ensuremath{\\text{O}})\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\\\\n  \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}}) \\Gamma\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\\\\n  \\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}(1 - \\eta_\\ensuremath{\\text{I}}) \\Gamma\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}\\right|^2\n\\end{align}\nWhich add together to create the loss term in \\cref{eq:metric_N}.\n\\begin{align}\n    \\Lambda_\\ensuremath{\\text{IRO}}\\Gamma &= \\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}(1 - \\eta_\\ensuremath{\\text{I}})\\Gamma\n    + \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}})\\Gamma\n    + (1 - \\eta_\\ensuremath{\\text{O}})\n \n \n \n \n \n \n \n\\end{align}\n\n\\subsection{Ideal Interferometer Example}\n\\label{sec:ideal_IFO_example}\n\nThe derivations are now extended to recreate and generalize the ideal noise model of \\cref{sec:ideal_IFO_metrics}, using \\cref{eq:Kchi_standard} for $\\ensuremath{\\mathcal{K}}$. The two-photon matrix corresponding to the interferometer in \\cref{fig:SQZ_mm_cavities} is given below for the lossless interferometer that is perfectly on resonance.\n\\begin{align}\n  \\Tmat{H}_\\ensuremath{\\text{R}}(\\Omega) &\\simeq\n  \\begin{bmatrix}\n    \\ensuremath{\\mathfrak{r}}(\\Omega) & 0\\\\\n    \\ensuremath{\\mathcal{K}}(\\Omega) & \\ensuremath{\\mathfrak{r}}(\\Omega)\n  \\end{bmatrix},\n  &\n  \\ensuremath{\\mathfrak{r}}(\\Omega) &\\simeq\n\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega},\n                 &\\Tmat{\\Lambda}_\\ensuremath{\\text{R}} &= \\Tmat{0}\n \n \n \n \n \n \n \n \n \n \n \n \n \n\\end{align}\nIn the ideal lossless case, the input and output paths also have perfect efficiency $\\eta_\\ensuremath{\\text{I}} \\simeq 1$ with $\\Tmat{H}_\\ensuremath{\\text{I}}(\\Omega) = \\eta_\\ensuremath{\\text{I}}\\Tmat{1},\\; \\Tmat{\\Lambda}_\\ensuremath{\\text{I}}(\\Omega) = \\sqrt{1-\\eta_\\ensuremath{\\text{I}}}\\Tmat{1}$ and similarly for the output. These can be used to compute $\\Tmat{H}$ and $\\vec{m}$.\n\\begin{align}\n  m_q &= \\cos(\\zeta)\\ensuremath{\\mathcal{K}}(\\Omega) + \\sin(\\zeta)\\ensuremath{\\mathfrak{r}}(\\Omega)\n  ,&\n  m_p &= \\cos(\\zeta)\\ensuremath{\\mathfrak{r}}(\\Omega)\n\\end{align}\nThe equations above maintain the correct phase information for this ideal case analysis. Interestingly, $\\ensuremath{\\mathcal{K}}$ and $\\ensuremath{\\mathfrak{r}}(\\Omega)$ have different magnitude responses resulting from different factors of $\\gamma_\\ensuremath{\\text{A}} \\pm i\\Omega$, yet their phase response is the same. This Kramers-Kronig coincedence ensures $\\Xi(\\Omega)=0$ as long as the $\\chi(\\Omega)$ contribution to $\\ensuremath{\\mathcal{K}}(\\Omega)$ is purely real. Thus, lossy mechanics will cause QRPN to dephase injected squeezing. This will not happen to any meaningful level for LIGO, but is noteworthy for optomechanics experiments operating on mechanical resonance.\n\nThe $\\vec{m}$ above also includes the effect of the readout angle. For $\\zeta=0$, it recovers \\crefrange{eq:optical_gain_g}{eq:Kchi_standard}. More generally, it gives\n\\begin{align}\n  \\Gamma(\\Omega) &= |\\cos(\\zeta)\\ensuremath{\\mathcal{K}}(\\Omega)|^2 + \\sin(2\\zeta)|\\ensuremath{\\mathcal{K}}(\\Omega)| + 1\n  \\\\\n  \\theta(\\Omega) &= \\arctan\\big(|\\ensuremath{\\mathcal{K}}(\\Omega)| - \\tan(\\zeta)\\big)\n\\end{align}\nThe exact expressions above can be simplified to better relate them to the LIGO data. Firstly, the squeezing angle is modified to be 0 at high frequencies, to match the conventions of the data. This modified angle is $\\theta'(\\Omega) = \\theta(\\Omega) - \\theta(\\Omega{\\gg}\\gamma_\\ensuremath{\\text{A}})$. Secondly, small shifts of the homodyne angle are linearized.\n\\begin{align}\n  \\Gamma'(\\Omega) &\\approx\n                  \n                  \n                   \\big(1 - |K(\\Omega)|\\big)^2 + 2\\big(1 + \\zeta\\big)|K(\\Omega)|\n \n  \\\\\n  \\theta'(\\Omega) &\\approx \\arctan\\big(|\\ensuremath{\\mathcal{K}}(\\Omega)|) - \\zeta \\frac{|\\ensuremath{\\mathcal{K}}(\\Omega)|^2}{1 + |\\ensuremath{\\mathcal{K}}(\\Omega)|^2}\n\\end{align}\nThe linearized $\\Gamma'$ shows that , when $\\zeta=-13^\\circ=-0.23$, for frequencies near $\\Omega_\\ensuremath{\\text{sql}}$, $\\ensuremath{\\mathcal{K}}(\\Omega_\\ensuremath{\\text{sql}}) \\simeq 1$, the interferometer quantum noise is reduced by about 23\\% with respect to a nominal $\\xi=0$ readout. This change is shown in the blue vs. grey plotted data for the Livingston loss plot in \\cref{fig:data_Q} of $1-\\eta$. There, $\\eta$ changes as the $\\Gamma$ model changes since only $\\eta\\Gamma$ can be measured due to subtracting an unsqueezed reference dataset. The $23\\%$ noise reduction corresponds to approximately 1dB improvement from pondermotive quantum correlations. The angle formula above indicates that for frequencies $\\Omega\\lesssim\\Omega_\\ensuremath{\\text{sql}}$, the local oscillator also adds some additional shift to $\\theta$ at low frequency, which is also observed in the LLO angle fits.\n\nThis analysis gives an example of how the derivations of this section are applied to extend the existing ideal interferometer models towards the real instruments. Exact models including more optical physics are yet more analytically opaque, but give a more complete complete picture if implemented numerically. \\Cref{sec:matrix_coupled_cavity} shows the full matrix solution, including the cavities, to recover these equations while also handling cavity length offset detunings. It also includes transverse modal mismatch in its description. \\Cref{sec:radiation_pressure_calc} gives the minimal extension of this ideal lossless interferometer to incorporate transverse mismatch, showing how the noise gain, $\\Gamma$,  and rotation angle $\\theta$ change specifically from mismatch. In particular, it shows that relating a measurement of $\\Omega_\\ensuremath{\\text{sql}}$ using squeezing back to the arm power $P_\\ensuremath{\\text{A}}$ using \\cref{eq:optical_gain_g} and  \\cref{eq:Kchi_standard} is biased by transverse mismatch.\n\n\\section{Cavity Modeling and Metrics}\n\\label{sec:modeling}\n\nThe previous section derives the general form of the squeezing metrics using matrices of the two photon formalism.\nFor passive systems, the optical transfer function, $\\ensuremath{\\mathfrak{h}}(\\Omega)$, given at every sideband frequency, is sufficient to characterize the response to externally-supplied squeezing. The conceptual simplification and restriction to using only transfer functions is useful for interferometer modeling. Transfer functions, being complex scalar functions, are suitable for analytic calculations of cavity response and can be decomposed into rational function forms to inspect the rational roots, zeros and poles, and the overall gain of the response.\n\nThis section analyzes the coupled cavity system of the interferometer, depicted in \\cref{fig:SQZ_mm_cavities}, through its decomposition into roots. More complicated transverse modal simulations analyze the frequency response of the interferometer cavities for each optical mode to every other mode. Modal simulations thus output a matrix of transfer functions, $\\mat{H}(\\Omega)$, which is difficult to analytically manipulate, but \\cref{sec:modeling_TMM} shows how it can be projected back to a single scalar transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$ and further simplified into the squeezing metrics.\n\nThe transfer function techniques of this section elucidate new squeezing results by avoiding the combined complexity of both two-photon and modal vector spaces. The full generality of two-photon matrices is only required for active systems that introduce internal squeezing, parametric gain or radiation pressure. Passive systems have the property that $\\hat{q}_\\text{out}$, $\\hat{p}_\\text{out}$ also obey the expectations of \\cref{eq:vacuum_expectations,eq:vacuum_expectations2}. Following the notation of \\cref{sec:derivations}, this results in the following condition.\n\\begin{align}\n  \\Tmat{1} &= \\Tmat{H}\\Tmat{H}^\\dagger\n  +\\sum_{\\ensuremath{{\\,\\mu}}}\\Tmat{T}_{\\ensuremath{{\\,\\mu}}}\\Tmat{T}^\\dagger_{\\ensuremath{{\\,\\mu}}}\n             \\label{eq:passivity_condition}\n\\end{align}\nAdditionally, $\\Gamma = 1$ is implied by that condition.\nWithout parametric gain, photons at upper and lower sideband frequencies are never correlated by a passive system. By the passivity condition and manipulations between sideband and quadrature basis, \\cref{sec:passive_derivations} derives the squeezing metrics purely in terms of the transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$.\n\\begin{align}\n  \\theta(\\Omega)\n  &=\n    \\big(\\arg\\big(\\ensuremath{\\mathfrak{h}}(+\\Omega)\\big) + \\arg\\big(\\ensuremath{\\mathfrak{h}}(-\\Omega)\\big)\\big)\/2\n    \\label{eq:theta_passive}\n  \\\\\n  \\eta(\\Omega)\n  &=\n    \\big( |\\ensuremath{\\mathfrak{h}}(+\\Omega)|^2 + |\\ensuremath{\\mathfrak{h}}(-\\Omega)|^2\\big)\/2\n    \\label{eq:eta_passive}\n  \\\\\n  \\Xi(\\Omega)\n  &=\n    \\big( |\\ensuremath{\\mathfrak{h}}(+\\Omega)| - |\\ensuremath{\\mathfrak{h}}(-\\Omega)| \\big)^2\/4\\eta\n    \\label{eq:Xi_passive}\n\\end{align}\nQuantum filter cavities are a method to use an entirely optical system to reduce the radiation pressure associated with squeezed light\\cite{McCuller-PRL20-FrequencyDependentSqueezing, Zhao-PRL20-FrequencyDependentSqueezed}. They are passive cavities, and provide a useful example to study these squeezing metric formulas.\nThe first of these, \\cref{eq:theta_passive}, is a well-established formula for the filter cavity design. It indicates that for cavities with an asymmetric phase response, usually due to being off-resonance or ``detuned'', that the squeezing field picks up a frequency-dependent quadrature rotation. Such a rotation applied in $\\Tmat{H}_\\ensuremath{\\text{I}}$ can be generated by a cavity with transfer function $\\ensuremath{\\mathfrak{h}}_\\ensuremath{\\text{I}}(\\Omega)$ before the interferometer. This cavity rotation compensates the $\\theta(\\Omega)$ due to $\\Tmat{H}_\\ensuremath{\\text{R}}$. Together, the product $\\Tmat{H}_\\ensuremath{\\text{R}}\\Tmat{H}_\\ensuremath{\\text{I}}$ has $\\theta(\\Omega) = 0$, allowing a single choice of squeezing angle $\\phi$ to optimize $N(\\Omega)$ at all frequencies.\n\nThe formulas \\cref{eq:eta_passive} and \\cref{eq:Xi_passive} indicate how losses represented in a transfer function translate to loss-like and dephasing degradations from cavity reflections. For filter cavities, these degradations are investigated in \\cite{Kwee-PRD14-DecoherenceDegradation}, but this new factorization into scalar functions clarifies the discussion. The efficiency $\\eta(\\Omega)$ behaves as expected, an average of the loss in each sideband. The form of $\\Xi(\\Omega)$ is less expected, showing how the combination of loss and detuning in filter cavities creates noise that scales with the squeezing level. A simple picture for the dephasing effect is that when optical quadratures are squeezed, the noise power in both upper and lower sidebands is strictly increased. The sideband correlations allow the increased noise to subtract away for squeezed quadrature measurements but to add for measurements in the anti-squeezed quadrature. The asymmetric losses of detuned cavities preserve the noise increase on one sideband, while degrading the correlations. This ruins the subtraction for the squeezed quadrature and introduces $\\Xi(\\Omega) > 0$. This source of noise is $e^{\\pm r}$ squeezing level dependent but entirely unrelated to fluctuations of the squeezing phase $\\ensuremath{\\phi_{\\text{rms}}}$.\n\n\\subsection{Single Cavity Model for Interferometers}\n\\label{sec:modeling_cavities}\n\nThis section analyzes the effect of the interferometer cavities on squeezing. It starts by considering an interferometer with only one cavity - either in the Michelson arms or from a mirror at the output port, but not both. It represents the first generation of GW detectors. This single cavity scenario is also similar to a quantum filter cavity, in the regime of small detuning\\cite{Komori-PRD20-DemonstrationAmplitude, Corbitt-PRD04-OpticalCavities, Khalili-PRD08-IncreasingFuture}. Advanced LIGO uses a coupled cavity system, depicted in \\cref{fig:SQZ_mm_cavities}, and the transfer function equations for the reflection from the resonant sideband extraction cavity is extended in the next subsection to include the loss and detuning of the additional cavity.\n\nA single cavity operated near resonance may be described using the scale parameters of the cavity bandwidth $\\gamma_\\ensuremath{\\text{A}}$, loss rate $\\lambda_\\ensuremath{\\text{A}}$ and detuning frequency $\\delta_\\ensuremath{\\text{A}}$, which are computed from the physical parameters of the mirror transmissivity $T_\\ensuremath{\\text{a}}$, round-trip loss $\\Lambda_\\ensuremath{\\text{a}}$, cavity length $L_\\ensuremath{\\text{a}}$, and microscopic length detuning $\\Delta L_\\ensuremath{\\text{a}}$.\n\\begin{align}\n  \\gamma_\\ensuremath{\\text{A}}  &= \\frac{c T_\\ensuremath{\\text{a}}}{4 L_\\ensuremath{\\text{a}}}\n  & \n  \\lambda_\\ensuremath{\\text{A}} &= \\frac{c \\Lambda_\\ensuremath{\\text{a}}}{4 L_\\ensuremath{\\text{a}}}\n  &\n  \\delta_\\ensuremath{\\text{A}}  &= -ck\\frac{\\Delta L_\\ensuremath{\\text{a}}}{L_\\ensuremath{\\text{a}}}\n                 \\label{eq:cavity_relations}\n\\end{align}\nThese relations are accurate in the high-finesse limit $T_\\ensuremath{\\text{a}} \\ll 1$, and combine to give the transfer function of the frequency-dependent cavity reflection.\n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_1(\\Omega)\n \n \n  &\\approx\n    -\\frac{(\\gamma_\\ensuremath{\\text{A}} - \\lambda_\\ensuremath{\\text{A}}) - i(\\Omega - \\delta_\\ensuremath{\\text{A}})}{(\\gamma_\\ensuremath{\\text{A}} + \\lambda_\\ensuremath{\\text{A}}) + i(\\Omega - \\delta_\\ensuremath{\\text{A}})}\n    \\label{eq:single_cavity}\n\\end{align}\nNotably, the sign of the reflectivity for a high-finesse cavity on resonance $\\ensuremath{\\mathfrak{r}}_1(\\Omega \\ll \\gamma_\\ensuremath{\\text{A}})=-1$,\nbut outside of resonance $\\ensuremath{\\mathfrak{r}}_1(\\Omega \\gg \\gamma_\\ensuremath{\\text{A}})=1$. This sign determines constructive or destructive interference in transverse mismatch loss analyzed in the next section. The internal losses of the cavity $\\Lambda_\\ensuremath{\\text{a}}$ become cavity-enhanced in the reflection, causing squeezing to experience losses of $\\Lambda_\\ensuremath{\\text{A}}$.\n\\begin{align}\n  \\Lambda_\\ensuremath{\\text{A}}\n  &\\equiv 1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1\\\\ |\\Omega| \\ll \\gamma_\\ensuremath{\\text{A}}}}\n \n \n \n  \\approx \\frac{4\\lambda_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}}\n                            \\approx \\frac{4\\Lambda_\\ensuremath{\\text{a}}}{T_\\ensuremath{\\text{a}}}\n\\end{align}\nFurthermore, detuning the cavity off of resonance causes a rotation of reflected squeezing. For small detunings, the rotation can be approximated.\n\\begin{align}\n  \\theta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1\\\\ k\\Delta L_\\ensuremath{\\text{a}} \\ll T_\\ensuremath{\\text{a}}}}\n  &\\approx\n  \\frac{2\\delta_\\ensuremath{\\text{A}}\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\approx\n   -k \\Delta L_\\ensuremath{\\text{a}}\\frac{8}{T_\\ensuremath{\\text{a}}}\\frac{\\gamma_\\ensuremath{\\text{A}}^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n\\end{align}\nFluctuations in $\\Delta L_\\ensuremath{\\text{a}}$ or $\\delta_\\ensuremath{\\text{A}}$ lead to a phase noise analogous to $\\ensuremath{\\phi_{\\text{rms}}}$, but with the frequency dependence from the above equation\\cite{Kwee-PRD14-DecoherenceDegradation}. Additionally, losses in the cavity lead to intrinsic dephasing $\\Xi(\\Omega)$, calculated below. This calculation is valid at any detuning $\\delta_\\ensuremath{\\text{A}}$, even those larger than the cavity width $\\gamma_\\ensuremath{\\text{A}}$. Its validity only requires being in the overcoupled cavity regime, where losses $\\lambda_\\ensuremath{\\text{A}} \\lesssim \\gamma_\\ensuremath{\\text{A}}\/2$.\n\\begin{align}\n  \\Xi(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1}}\n  &\\approx\n    \\left(\n  \\frac{4\\gamma_\\ensuremath{\\text{A}}\\lambda_\\ensuremath{\\text{A}}\\delta_\\ensuremath{\\text{A}}\\Omega}{\\left( \\gamma_\\ensuremath{\\text{A}}^2 + (\\Omega - \\delta_\\ensuremath{\\text{A}})^2 \\right)\\left( \\gamma_\\ensuremath{\\text{A}}^2 +  (\\Omega + \\delta_\\ensuremath{\\text{A}})^2 \\right)}\\right)^2\n\\end{align}\nWhen plotted, this expression for $\\Xi(\\Omega)$ has a Lorentzian-like profile, with a peak at $\\Omega_{\\Xi\\text{max}}$. Above $|\\delta_\\ensuremath{\\text{A}}| \\gtrsim \\gamma_\\ensuremath{\\text{A}}$, where the cavity resonance acts entirely either on upper or lower sidebands, the peak dephasing reaches a maximum. At small detunings, $|\\delta_\\ensuremath{\\text{A}}| \\lesssim \\gamma_\\ensuremath{\\text{A}}$, the sideband loss asymmetry scales with the detuning.\n\\begin{align}\n  \\Omega_{\\Xi{\\text{max}}} &\\approx \\sqrt{\\gamma^2_\\ensuremath{\\text{A}}\/4 + \\delta^2 },\n  & \\Xi_{\\text{max}} \n  &\\approx \n    \\frac{\\lambda^2_\\ensuremath{\\text{A}}}{\\gamma^2_\\ensuremath{\\text{A}}}{\\cdot}\\frac{8\\delta^2_\\ensuremath{\\text{A}}}{5\\gamma^2_\\ensuremath{\\text{A}} + 8\\delta^2_\\ensuremath{\\text{A}}}\n    \\label{eq:detuning_dephasing}\n\\end{align}\nThis single cavity model is also useful for analyzing quantum filter cavities and, like the $\\Xi$ metric itself, these peak values have not been calculated in past frequency-dependent squeezing work. Conventional squeezing phase uncertainty, $\\ensuremath{\\phi_{\\text{rms}}}$, can be cast into the units RMS radians of phase deviation, leading to the noise suppression limit for squeezing $S(\\Omega)\\ge 2\\ensuremath{\\phi_{\\text{rms}}}$, by \\cref{eq:N_limit}. For highly detuned cavities such as quantum filter cavities, $\\sqrt{\\Xi(\\Omega)} \\approx \\Lambda_\\ensuremath{\\text{fc}} \/ T_\\ensuremath{\\text{fc}}$. Using the parameters of the A+ filter cavity \\cite{Whittle-PRD20-OptimalDetuning}, $\\Lambda_{\\ensuremath{\\text{fc}}}{\\approx}60\\text{ppm}$ and $T_{\\ensuremath{\\text{fc}}} {=} 1000 \\text{ppm}$ indicates that optical dephasing is of order $60\\text{mRad}$. For an optimal filter cavity with low losses\\cite{Whittle-PRD20-OptimalDetuning}, this dephasing maximum occurs at $\\Omega_{\\Xi{\\text{max}}} = \\sqrt{5\/8}\\Omega_{\\text{SQL}}$. This level of dephasing is commensurate with or even exceeds the expected residual phase uncertainty $\\ensuremath{\\phi_{\\text{rms}}} < 30\\text{mRad}$.\n\nOptical dephasing from the LIGO interferometer cavities is not expected to be large for as they are stably operated on-resonance; however, detuned configurations of LIGO\\cite{Ganapathy-PRD21-TuningAdvanced} are limited by dephasing from the unbalanced response and optical losses in the signal recycling cavity.\n\n\\subsection{Double Cavity Model for Interferometers}\n\nFor interferometers using resonant sideband extraction, like LIGO, the arm cavities have a length $L_\\ensuremath{\\text{a}}$, an input transmissivity of $T_\\ensuremath{\\text{a}}$, and are each on resonance to store circulating laser power. The signal recycling cavity (SRC) has a length $L_\\ensuremath{\\text{s}}$ and a signal recycling mirror (SRM) of transmissivity $T_\\ensuremath{\\text{s}}$. The SRM forms a cavity with respect to the arm input mirror that resonantly increases the effective transmissivity experienced by the arm cavities to be larger than $T_\\ensuremath{\\text{a}}$, broadening the signal bandwidth. While the SRC is resonant with respect to the arm input mirror, it is anti-resonant with respect to the arm cavity, due to the negative sign of \\cref{eq:single_cavity}. The anti-resonance leads to the opposite sign in the reflection transfer function below, \\cref{eq:double_cavity}. The discrepancy in resonance vs. anti-resonance viewpoints is why the signal recycling cavity is also called the signal extraction cavity in GW literature.\n\nThe coupled cavity forms two bandwidth scales for the system, $\\gamma_\\ensuremath{\\text{A}}$, the modified effective arm bandwidth, and $\\gamma_\\ensuremath{\\text{S}}$, the bandwidth of the signal recycling cavity. The arm and signal cavities have their respective round-trip losses $\\Lambda_\\ensuremath{\\text{a}}$ and $\\Lambda_\\ensuremath{\\text{s}}$, as well as length detunings $\\Delta L_\\ensuremath{\\text{a}}$, $\\Delta L_\\ensuremath{\\text{s}}$. In practice, the arm length detuning is expected to be negligible to maximize the power storage, but the signal recycling cavity detuning can be varied by modifying a bias in the control system that stabilizes $\\Delta L_\\ensuremath{\\text{s}}$.\n\nThe scale parameters for the cavity transfer function are approximated from the physical parameters:\n\\begin{align}\n  u_\\ensuremath{\\text{a}} &= 1 - \\sqrt{1 - T_\\ensuremath{\\text{a}}}\n  &\n  u_\\ensuremath{\\text{s}} &= 1 - \\sqrt{1 - T_\\ensuremath{\\text{s}}}\n           \\label{eq:u_factors}\n  \\\\\n  \\gamma_\\ensuremath{\\text{A}}  &= \\frac{c u_\\ensuremath{\\text{a}}}{2 L_\\ensuremath{\\text{a}}} \\cdot\\frac{2 - u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{s}}}\n  & \n    \\gamma_\\ensuremath{\\text{S}}   &= \\frac{c u_\\ensuremath{\\text{s}}}{2 L_\\ensuremath{\\text{s}}}\n                    \\label{eq:double_cavity_gamma}\n  \\\\\n  \\lambda_\\ensuremath{\\text{A}} &= \\frac{c}{L_\\ensuremath{\\text{a}}} \\left(\\frac{\\Lambda_\\ensuremath{\\text{a}}}{4} - \\frac{u_\\ensuremath{\\text{a}} \\Lambda_\\ensuremath{\\text{s}}}{2 u_\\ensuremath{\\text{s}}^2} + \\frac{u_\\ensuremath{\\text{a}} \\Lambda_\\ensuremath{\\text{s}}}{4 u_\\ensuremath{\\text{s}}}\\right)\n  &\n    \\lambda_\\ensuremath{\\text{S}}  &= \\frac{c\\Lambda_\\ensuremath{\\text{s}}}{4L_\\ensuremath{\\text{s}}}\\left(1 - \\frac{u_\\ensuremath{\\text{s}}}{2} \\right)\n  \\\\\n  \\delta_\\ensuremath{\\text{A}} &= -ck\\frac{\\Delta L_\\ensuremath{\\text{a}}}{L_\\ensuremath{\\text{a}}}  - \\frac{\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{S}}}\\delta_\\ensuremath{\\text{S}}\n  &\n  \\delta_\\ensuremath{\\text{S}} &= -ck\\frac{\\Delta L_\\ensuremath{\\text{s}}}{L_\\ensuremath{\\text{s}}} \n           \\label{eq:delta_factors}\n\\end{align}\nThese approximations are valid for the LIGO mirror parameters, see \\cref{tab:LIGO_params}, and model the loss and detuning to {\\color{red} $5\\%$} accuracy. They are derived in \\cref{sec:scalar_coupled_cavity} from Taylor expansions, solving roots, and selectively removing terms. Expanding in the $u$ factors of \\cref{eq:u_factors} gives lower error than expanding in transmissivity or reflectivity factors directly, due to the low effective finesse of the coupled cavity system and the high transmissivity of the SRM.\n\\begin{table}\n\\centering\n\\caption{Parameters of LIGO for data fitting and modeling}\n\\begin{ruledtabular}\n\\begin{tabular}{l|c|cc}\nParameter & Symbol & LLO Value & LHO Value \\\\\n\\hline\narm input transmissivity & $T_\\ensuremath{\\text{a}}$&0.0148 & 0.0142 \\\\\narm length & $L_\\ensuremath{\\text{a}}$ & \\multicolumn{2}{c}{3995 m}\\\\\narm round-trip loss & $\\Lambda_{\\ensuremath{\\text{a}}}$ & \\multicolumn{2}{c}{${\\sim}80$ppm}\\\\\nSRM transmission& $T_\\ensuremath{\\text{s}}$&\\multicolumn{2}{c}{0.325}  \\\\\nSRC length & $L_\\ensuremath{\\text{s}}$ &\\multicolumn{2}{c}{55 m}\\\\\nSRC round-trip loss & $\\Lambda_\\ensuremath{\\text{s}}$ & \\multicolumn{2}{c}{$\\lesssim 3000$ppm}\\\\\n  \\hline\nMirror mass& $m$ &  \\multicolumn{2}{c}{39.9kg}\\\\\nArm power& $P_\\ensuremath{\\text{A}}$ & $200{\\pm} 10\\text{ kW}$  &  $190{\\pm}10\\text{ kW}$ \\\\\nQRPN crossover& $\\Omega_{\\ensuremath{\\text{sql}}}$ & $2\\pi\\cdot 33$ Hz & $2\\pi\\cdot 30$ Hz \\\\\n  \\hline\narm signal band& $\\gamma_\\ensuremath{\\text{A}}$& $2\\pi\\cdot450$ Hz &  $2\\pi\\cdot410$ Hz \\\\\nSRC band& $\\gamma_\\ensuremath{\\text{S}}$& \\multicolumn{2}{c}{$2\\pi\\cdot80$kHz} \\\\\nArm length detuning & $\\Delta L_{\\ensuremath{\\text{a}}}$ & \\multicolumn{2}{c}{0nm}\\\\\nSRC length detuning & $\\Delta L_{\\ensuremath{\\text{s}}}$ & 1.02nm & 1.23nm\\\\\n  \\hline\narm resonant loss& $\\Lambda_\\ensuremath{\\text{A}}$ & \\multicolumn{2}{c}{$\\lesssim 2000$PPM}\\\\\nSRC resonant loss& $\\Lambda_\\ensuremath{\\text{S}}$ & \\multicolumn{2}{c}{${\\sim} 1\\%$ to $3\\%$}\\\\\narm\/SRC detuning& $\\delta_{\\ensuremath{\\text{A}}}$ & $2\\pi\\cdot 10.1$Hz & $2\\pi\\cdot 11.2$Hz \\\\\n  \\hline\nInjected squeezing & $e^{\\pm 2r}$ & {${\\pm}9.7\\text{ dB}$}  &  {${\\pm}8.7\\text{ dB}$} \\\\\nSQZ-OMC mismatch& $\\Upsilon_{\\ensuremath{\\text{O}}}$ & $2\\%$ & $4\\%$ \\\\\nReflection mismatch (fit)& $\\Upsilon_{\\ensuremath{\\text{R}}}$ & $12\\%$ & $35\\%$ \\\\\nAdditional SQZ loss (fit)& $\\Lambda_{\\text{IO}}=1{-}\\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{O}}$ & $31\\%$ & $34\\%$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{tab:LIGO_params}\n\\end{table}\nThe scale factors result in the following reflectivity transfer function.\n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_2(\\Omega)\n  &=\n    \\frac{(\\gamma_\\ensuremath{\\text{A}} - \\lambda_\\ensuremath{\\text{A}}) - i(\\Omega - \\delta_\\ensuremath{\\text{A}})}{(\\gamma_\\ensuremath{\\text{A}} + \\lambda_\\ensuremath{\\text{A}}) + i(\\Omega - \\delta_\\ensuremath{\\text{A}})}\n   \n    {\\cdot}\\frac{(\\gamma_\\ensuremath{\\text{S}} - \\lambda_\\ensuremath{\\text{S}}) - i(\\Omega - \\delta_\\ensuremath{\\text{S}})}{(\\gamma_\\ensuremath{\\text{S}} + \\lambda_\\ensuremath{\\text{S}}) + i(\\Omega - \\delta_\\ensuremath{\\text{S}})}\n    \\label{eq:double_cavity}\n\\end{align}\nNotably, this reflectivity is $\\ensuremath{\\mathfrak{r}}_2(\\pm\\Omega \\ll \\gamma)=1$ and $\\ensuremath{\\mathfrak{r}}_2(\\gamma_\\ensuremath{\\text{A}} \\ll \\pm\\Omega \\ll \\gamma_\\ensuremath{\\text{S}})=-1$ which has an opposite overall sign to that of single cavity interferometers. On reflection, the squeezing field experiences different cavity enhanced losses depending on the frequency.\n\\begin{align}\n  \\Lambda_{\\ensuremath{\\text{S}}} &\\equiv \n1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ \\gamma_\\ensuremath{\\text{A}} \\ll |\\Omega| \\ll \\gamma_\\ensuremath{\\text{S}}}}\n&&\\hspace{-3em}\\approx \\frac{2 - u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{s}}}\\Lambda_\\ensuremath{\\text{s}}\n  \\\\\n  \\Lambda_{\\ensuremath{\\text{A}}} &\\equiv \n1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ |\\Omega| \\ll \\gamma_\\ensuremath{\\text{A}}}}\n&&\\hspace{-3em}\\approx \\frac{4\\lambda_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}}+\\Lambda_\\ensuremath{\\text{s}} \\approx \\frac{u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{a}}}\\Lambda_\\ensuremath{\\text{a}}\n\\end{align}\nThe dataset of \\cref{sec:experiment} shows frequency dependent losses, where the loss increases $12\\%$ for LLO and $33\\%$ for LHO. Assuming the losses result from the equations above, this corresponds to round-trip losses in the LIGO signal recycling cavities, $\\Lambda_\\ensuremath{\\text{s}}$, of $1.1\\%$ to $3.2\\%$, which is not realistic. Most mechanisms that introduce loss in the SRC would also introduce it into the power recycling cavity in an obvious manner. The current power recycling factors exclude this possibility, and independent measurements of $\\gamma_\\ensuremath{\\text{A}}$ bound $\\Lambda_\\ensuremath{\\text{s}}$ losses to ${\\le} 3000$ppm. The next section investigates how transverse mismatch can result in this level of observed losses.\n\nIn addition to the losses, \\cref{eq:double_cavity} can be used to determine the cavity-induced squeeze state rotation from the detuning of the signal recycling cavity.\n\\begin{align}\n  \\theta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ \\delta_\\ensuremath{\\text{S}} \\ll \\gamma_\\ensuremath{\\text{S}} \\\\ \\Delta L_\\ensuremath{\\text{a}} = 0}}\n  &\\approx\n  \\frac{2\\delta_\\ensuremath{\\text{A}}\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    +\n  \\frac{2\\delta_\\ensuremath{\\text{S}}\\gamma_\\ensuremath{\\text{S}}}{\\gamma_\\ensuremath{\\text{S}}^2 + \\Omega^2}\n \n \n \n \n \n \n \n \n    \\\\\n  &\\approx\n    k\\Delta L_\\ensuremath{\\text{s}}\\frac{4}{u_\\ensuremath{\\text{s}}}\n    \\left(\n    \\frac{\\gamma^2_\\ensuremath{\\text{S}}}{\\gamma_\\ensuremath{\\text{S}}^2 + \\Omega^2}\n    -\n    \\frac{\\gamma^2_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n \\right)\n    \\label{eq:double_cav_theta}\n \n\\end{align}\nThis indicates the surpising result that detuning the SRC length does not affect the squeezing within the effective\narm bandwidth to first order. Instead, it adds the squeezing rotation in the middle band above the arm bandwidth but below the SRC bandwidth. In the data analysis of \\cref{sec:experiment} and \\cref{fig:data_Q}, the convention for $\\theta(\\Omega)$ is set to be 0 at ``high'' frequencies in this intermediate cavity band, in which case it appears to cause a rotation around $\\gamma_\\ensuremath{\\text{A}}$. This convention used for the data corresponds to omitting the first, $\\gamma_\\ensuremath{\\text{S}}$-scaled term of \\cref{eq:double_cav_theta}.\n\n\\section{Transverse Mismatch Model}\n\\label{sec:modeling_TMM}\n\nSqueezing, as it is typically implemented for GW interferometers, modifies the quantum states in a single optical mode. For LIGO, this mode is the fundamental Gaussian beam resonating in the parametric amplifier cavity serving as the squeezed state source. The cavity geometry establishes a specific complex Gaussian beam parameter that defines a modal basis decomposition into Hermite Gaussian (HG) or Laguerre Guassian (LG) modes. That basis is transformed and redefined during the beam propagation through free space and through telescope lenses on its way to and from the interferometer. The cavities of the interferometer each define their own resonating beam parameters and respective HG or LG basis of optical modes.\n\nIn practice, the telescopes propagating the squeezed beam to and from the interferometer imperfectly match the complex beam parameters, so basis transformations must occur that mix the optical modes. The mismatch of complex beam parameters is called here ``transverse mismatch''. Non-fundamental HG or LG transverse modes do not enter the OPA cavity, and so carry standard vacuum rather than squeezing. Basis mixing from transverse mismatch thus leads to losses; however, unlike typical losses such basis transformations are coherent and unitary, which leads to the constructive and destructive interference effects studied in this section.\n\nThe interferometer transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$ is a single scalar function representing the frequency dependence of the squeezing channel from source to readout, but the optical fields physically have many more channels. The cavities visited by the squeezed states each have a transfer function matrix in their local basis, given by $\\mat{H}_{\\ensuremath{\\text{I}}}$, $\\mat{H}_{\\ensuremath{\\text{R}}}$, $\\mat{H}_{\\ensuremath{\\text{O}}}$ for the squeezing input, interferometer reflection, and system output respectively. The diagonals of these matrices indicate the frequency response during traversal for every transverse optical mode. The off-diagonals represent the coupling response between modes that result from scattering and optical wavefront errors.\n\nBetween the cavities, $\\mat{U}$ matrices represent the basis transformations due to transverse mismatch. Here, $\\vec{e}_{\\text{sqz}}$, $\\vec{e}_{\\text{read}}$ are basis vectors for projecting from the single optical mode of the emitted squeezed states and to the single mode of the optical homodyne readout defined by its local oscillator field.\n\\begin{align}\n\\ensuremath{\\mathfrak{h}}(\\Omega) &= {\\vec{e}_{\\text{read}}}^{\\,\\dagger} \\mat{H}(\\Omega)\\vec{e}_{\\text{sqz}}\n                  \\label{eq:TF_projection}\n  \\\\\n\\mat{H}(\\Omega) &=\\mat{H}_\\ensuremath{\\text{O}}\\mat{U}_{\\ensuremath{\\text{O}}, \\ensuremath{\\text{R}}}\\mat{H}_\\ensuremath{\\text{R}}\\mat{U}_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{I}}}\\mat{H}_\\ensuremath{\\text{I}}\n                  \\label{eq:TF_matrix}\n\\end{align}\n\\Cref{eq:TF_projection} and \\cref{eq:TF_matrix} give the general, basis independent, form to compose the effective transfer function for the squeezed field using a multi-modal simulation of a passive interferometer. This is complicated in the general case, but the following analysis develops a simpler, though general, model for how transverse mismatch manifests as squeezing losses.\n\nTransverse mismatch is often physically measured as a loss of coupling efficiency, $\\ensuremath{\\Upsilon}$, of an external Gaussian beam to a cavity measured as a change in optical power. Realistically, more than two transverse modes are necessary to maintain realistic and unitary basis transformations, but, for small mismatches of complex beam parameters, $\\Upsilon < 10\\%$. In this case, only the two lowest modes in the Laguerre-Gauss basis have significant cross-coupling. For low losses, the fundamental Gaussian mode, LG0, loses most of its power to the radially symmetric LG1 mode, assuming low astigmatism and omitting azimuthal indices. This motivates the following simplistic two-mode model to analyze the effect of losses on $\\ensuremath{\\mathfrak{h}}(\\Omega)$. In this model $\\mat{U}$ gives the unitary, though not perfectly physical, basis transformation:\n\\begin{align}\n  \\mat{U}(\\ensuremath{\\Upsilon}, \\psi, \\phi)\n  &\\equiv\n    e^{i\\phi}\\begin{bmatrix}\n    \\sqrt{\\ensuremath{1{-}\\MM}} & -e^{i\\psi}\\sqrt{\\ensuremath{\\Upsilon}}\n    \\\\\n    e^{-i\\psi}\\sqrt{\\ensuremath{\\Upsilon}} & \\sqrt{\\ensuremath{1{-}\\MM}}\n  \\end{bmatrix}\n                           \\label{eq:U_def}\n\\end{align}\nThis unitary transformation includes two unknown phase parameters. The first, $\\psi$, is the phase of the mismatch, which characterizes whether beam size error or wavefront phasing error dominates the overlap integral of the external LG0 and cavity LG1 modes. The second, $\\phi$, is the mismatch phase error from the external LG0 to the cavity LG0. The $\\phi$ term is included above to fully express the unitary freedom of $\\mat{U}$, but is indistinguishable from path length offsets, physically controlled to be $0$, and ignored in further expressions.\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_chain, \n  caption={\n    Propagation of the squeezed beam and unsqueezed higher order transverse beam modes from source to readout. The stages (a)-(d) correspond to the components in \\cref{fig:SQZ_mm_IFO}, depicting the matrix math of \\crefrange{eq:H_refl}{eq:h_chain}. $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ represents the transverse mismatch loss of the squeezing to interferometer, and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$ is the mismatch of the squeezing to readout via the output mode cleaner. These mismatches cause beamsplitter-like mixing between the LG0 and LG1+ modes through \\cref{eq:U_def}. $\\psi_\\ensuremath{\\text{I}}$,  $\\psi_\\ensuremath{\\text{O}}$,  $\\psi_{\\text{G}}$ are unmeasured phasing terms of the interferometer and output mismatch and of the Gouy-phase advance from the beam propagating to the output mode cleaner.\n  },\n  label=SQZ_mm_chain,\n}\n\nIn the case of a GW interferometer with an output mode cleaner, there are two mode matching efficiencies expressed as individually measurable parameters. The first is the coupling efficiency (in power) and phasing associated between the squeezer and interferometer $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}, \\ensuremath{\\psi_\\text{I}}$. The second are parameters for efficiency and phasing between the squeezer and output mode cleaner, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}, \\ensuremath{\\psi_\\text{O}}$, which defines the mode of the interferometer's homodyne readout. Both cases represent a basis change from the Laguerre-Gauss modes of the squeezer OPA cavity into the basis of each respective cavity. In constructing $\\mat{H}(\\Omega)$, however, the squeezing is transformed to the interferometer basis, reflects, and then transforms back to the squeezing basis. This corresponds to the operations of \\cref{fig:SQZ_mm_chain}. There are also parameters to express the coupling efficiency and phase, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{F}}, \\ensuremath{\\psi_\\text{F}}$, between the interferometer cavity and the OMC cavity. The $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{F}}$ parameter is less natural to analyze squeezing is not independent from  $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$. It is considered at the end of this section, as it can also be independently measured.\n\n\\cref{fig:SQZ_mm_chain} is implemented into \\cref{eq:TF_matrix} through this simplistic two-mode representation by assuming that the interferometer reflection transfer function $\\ensuremath{\\mathfrak{r}}(\\Omega)$ applies to the LG0 mode in the interferometer basis. The LG1 mode picks up the reflection transfer function $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}$, which is approximately ${\\sim}1$ due to high order modes being non-resonant in the interferometer cavities and thus directly reflecting. %\n\\begin{align}\n  \\mat{H}_\\ensuremath{\\text{R}} &= \n  \\begin{bmatrix}\n    \\ensuremath{\\mathfrak{r}}(\\Omega) & 0\n    \\\\\n    0 & \\ensuremath{\\mathfrak{r}_{\\text{hom}}}(\\Omega)\n  \\end{bmatrix},\n  &\n  \\mat{G} &= \n    \\begin{bmatrix}\n    1 & 0\n    \\\\\n    0 & e^{i\\psi_G}\n  \\end{bmatrix}\n        \\label{eq:H_refl}\n        \\\\\n  \\ensuremath{\\mathfrak{r}}(\\Omega) &= \\ensuremath{\\mathfrak{r}}_2(\\Omega) \\text{ or } \\ensuremath{\\mathfrak{r}}_1(\\Omega)\n  &\n  \\ensuremath{\\mathfrak{r}_{\\text{hom}}}(\\Omega) &= e^{i\\theta_{\\text{hom}}} \\approx 1\n\\end{align}\nThe reflection term $\\ensuremath{\\mathfrak{r}}(\\Omega)$ can use either the single, \\cref{eq:single_cavity}, or double,  \\cref{eq:double_cavity}, cavity forms. LIGO, using resonant sideband extraction, uses $\\ensuremath{\\mathfrak{r}}_2(\\Omega)$. Frequencies where the reflection takes a negative sign will be shown to experience destructive interference from modal basis changes, increasing squeezing losses. The $\\mat{G}$ matrix includes a phasing factor due to additional Gouy phase of higher-order-modes. This factor is degenerate with the mismatch phasings $\\psi_\\ensuremath{\\text{I}}$ and $\\psi_\\ensuremath{\\text{O}}$ in observable effects. These matrices are composed per \\cref{fig:SQZ_mm_chain} to formulate the overall transfer function of the squeezed field.\n\\begin{align}\n  \\mat{H}(\\Omega) &= \n\\underbrace{\\mat{U}(\\ensuremath{\\Upsilon_{\\text{O}}}, \\ensuremath{\\psi_\\text{O}})\n\\mat{G}\n\\mat{U}^\\dagger(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})}_{\\mat{U}_{\\text{O,R}}}\n\\mat{H}_\\ensuremath{\\text{R}}\n                    \\underbrace{\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})\n                    }_{\\mat{U}_{\\text{R,I}}}\n                    \\label{eq:H_chain}\n                    \\\\\n  \\ensuremath{\\mathfrak{h}}(\\Omega) &= \n  \\begin{bmatrix}\n    1 \\\\ 0\n  \\end{bmatrix}^T\n  \\mat{H}(\\Omega)\n  \\begin{bmatrix}\n    1 \\\\ 0\n  \\end{bmatrix}\n                    \\label{eq:h_chain}\n  ,\n  \\text{   and using }\n\\mat{H}_\\ensuremath{\\text{O}} = \\mat{H}_\\ensuremath{\\text{I}} = \\mat{1}\n\\end{align}\nIgnoring intra-cavity losses and detunings, the two reflection forms $\\ensuremath{\\mathfrak{r}}_1$, $\\ensuremath{\\mathfrak{r}}_2$ can be simplified to give their respective transfer functions $\\ensuremath{\\mathfrak{h}}_1$, $\\ensuremath{\\mathfrak{h}}_2$.\n\nFor quantum noise below $\\Omega < \\gamma_\\ensuremath{\\text{S}}$, the double cavity reflectivity $\\ensuremath{\\mathfrak{r}}_2(\\Omega)$ behaves like a single cavity, using the $\\gamma_\\ensuremath{\\text{A}}$ of \\cref{eq:double_cavity_gamma} and with the opposite reflection sign as \\cref{eq:single_cavity}. \n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_2(\\Omega) &\\approx +\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n  &\\Rightarrow&&\n  \\ensuremath{\\mathfrak{h}}_2(\\Omega) &= \\sqrt{\\textstyle \\ensuremath{1{-}\\MMO}}\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\ensuremath{\\alpha}\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n                   \\label{eq:h2_TMM}\n                  \\\\\n  \\ensuremath{\\mathfrak{r}}_1(\\Omega) &= -\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n  &\\Rightarrow&&\n  \\ensuremath{\\mathfrak{h}}_1(\\Omega) &= \\sqrt{\\textstyle \\ensuremath{1{-}\\MMO}}\\frac{i\\Omega - \\ensuremath{\\alpha}\\gamma_\\ensuremath{\\text{A}}}{i\\Omega + \\gamma_\\ensuremath{\\text{A}}}\n\\end{align}\nUsing the factor\n\\begin{align}\n  \\ensuremath{\\alpha} &\\equiv 1 - 2\\ensuremath{\\Upsilon_{\\text{I}}} + 2\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}e^{i\\ensuremath{\\psi_\\text{R}}}\n             \\intertext{where:}\n  \\ensuremath{\\beta} &\\equiv \\textstyle\\sqrt{\\frac{\\ensuremath{1{-}\\MMI}}{\\ensuremath{1{-}\\MMO}}} \\approx 1\n             \\\\\n  \\ensuremath{\\psi_\\text{R}} &\\equiv \\ensuremath{\\psi_\\text{O}} + \\psi_G - \\ensuremath{\\psi_\\text{I}}\n\\end{align}\nThe phasing factor $\\ensuremath{\\psi_\\text{R}}$ shows that the unknown mismatch phasings combine to a single unknown overall phase. This overall phase determines the extent to which the separate beam mismatches of $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$ coherently stack or cancel with each-other. The factor $\\ensuremath{\\alpha}$ is the total squeezer LG0 to readout LG0 coupling factor for the effective mode mismatch of the full system, specifically when the interferometer reflection $\\ensuremath{\\mathfrak{r}}(\\Omega) = -1$. As an effective mismatch, it can be related back to the diagonal elements of \\cref{eq:U_def} to give an effective mismatch loss on reflection, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{R}}$.\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{R}}} &= 1 - \\textstyle |\\ensuremath{\\alpha}|^2 \\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} - 4\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}\\cos(\\ensuremath{\\psi_\\text{R}})\n         \\label{eq:MMR_def}\n\\end{align}\nThis effective mismatch loss becomes apparent after computing the full system efficiency $\\eta(\\Omega)$ (\\cref{eq:eta_passive}) using $\\ensuremath{\\mathfrak{h}}_1$ and $\\ensuremath{\\mathfrak{h}}_2$.\n\\begin{align}\n  \\eta_\\ensuremath{\\text{R}}(\\Omega)\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &=\n    \\left( \\ensuremath{1{-}\\MMO} \\right)\\frac{\\gamma_\\ensuremath{\\text{A}}^2 + \\left(\\ensuremath{1{-}\\MMR}\\right)\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\label{eq:effective_MMR_loss2}\n                  \\\\\n  \\eta_\\ensuremath{\\text{R}}(\\Omega)\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_1}\n  &=\n    \\left( \\ensuremath{1{-}\\MMO} \\right)\\frac{\\Omega^2 + \\left(\\ensuremath{1{-}\\MMR}\\right)\\gamma_\\ensuremath{\\text{A}}^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\label{eq:effective_MMR_loss1}\n\\end{align}\nFor the double cavity system of LIGO, \\cref{fig:data_Q} is presented using the loss rather than efficiency. To relate to the measurement, the loss attributable to mode mismatch is then written\n\\begin{align}\n  \\Lambda_\\ensuremath{\\Upsilon}(\\Omega) \\equiv 1 - \\eta_\\ensuremath{\\text{R}}\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &\\approx\n    \\ensuremath{\\Upsilon_{\\text{O}}} + \\frac{\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\\ensuremath{\\Upsilon_{\\text{R}}}\n    \\label{eq:Lambda_MM}\n\\end{align}\nMode mismatches between the squeezer and OMC were directly measured during the LIGO squeezer installation to be $2\\%-4\\%$, and mismatches from the squeezer and interferometer were indirectly measured but are expected to be of a similar level. The large factors in \\cref{eq:MMR_def} indicate that the independent mismatch measurements are compatible with the observed frequency dependence and levels of the losses to squeezing. The effective mismatch loss $\\ensuremath{\\Upsilon_{\\text{R}}}$ has the following bounds with respect to the independent mismatch measurements.\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{R}}} &\\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when } \\ensuremath{\\Upsilon_{\\text{O}}} = 0\n                          \\\\\n  0 \\le \\ensuremath{\\Upsilon_{\\text{R}}} &\\le 8\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when } \\ensuremath{\\Upsilon_{\\text{I}}} = \\ensuremath{\\Upsilon_{\\text{O}}}\n                          \\\\\n  \\ensuremath{\\Upsilon_{\\text{R}}} &\\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when averaged over } \\ensuremath{\\psi_\\text{R}}\n                         \\label{eq:TMM_bound_avg}\n\\end{align}\nIt is worth noting here how the realistic interferometer differs from this simple two-mode model. The primary key difference is that real mismatch occurs with more transverse modes. Expanding this matrix model to include more modes primarily adds more $\\cos(\\ensuremath{\\psi_\\text{R}})$-type factors to the last term of \\cref{eq:MMR_def}. These factors will tend to average coherent additive mismatch between the squeezer and the OMC away, leaving only the squeezer to interferometer terms. Additionally, not only is there beam parameter mismatch from imperfect beam-matching telescopes, but there is also some amount of misalignment, statically or in RMS drift. Mismatch into modes of different order picks up different factors of $\\ensuremath{\\psi_\\text{G}}$. Together, including more modes leaves the bounds above intact, but makes \\cref{eq:TMM_bound_avg} more representative given the expanded dimensionality of mismatch-space to average away $\\cos(\\ensuremath{\\psi_\\text{R}})$.\n\nThe other notable difference in realistic instruments is that the high order modes pick up small phase shifts of reflection, as the cavities are not perfectly out of resonance at all high order modes. This corresponds to $\\ensuremath{\\mathfrak{r}}_{\\text{hom}} \\ne 1$. The signal recycling mirror is sufficiently low transmissivity that the finesse is low and, even when off-resonance, higher order modes pick up a small but slowly varying phase shift. This has the property of mixing the frequency dependent losses resulting from $\\ensuremath{\\mathfrak{h}}_1$ and $\\ensuremath{\\mathfrak{h}}_2$, resulting in a slightly more varied frequency-dependence that is captured in the full model of \\cref{sec:matrix_coupled_cavity}.\n\n\nWhile the phasing of the mismatch, $\\ensuremath{\\psi_\\text{R}}$, is not directly measurable, it manifests in an observable way. It adds to the complex phase of $\\ensuremath{\\alpha}$ to cause a slight rotation of the squeezing phase, making the cavity appear as if it is detuned. the frequency dependence and magnitude of this rotation is given by (c.f. \\cref{eq:theta_passive}),\n\\begin{align}\n  \\theta_\\ensuremath{\\Upsilon}(\\Omega) \\equiv \\theta\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &\\approx\n    \\frac{-\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}2\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}\\sin(\\ensuremath{\\psi_\\text{R}})\n\\end{align}\nwhich adds to the rotation from cavity length detuning \\cref{eq:double_cav_theta}. The addition of this term with $\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}} \\ensuremath{\\Upsilon_{\\text{O}}}}$ unknown confounds the ability to use the data of \\cref{fig:data_Q} to constrain $\\ensuremath{\\psi_\\text{R}}$. There is a small discrepancy between the length-detuning induced optical spring observed in the interferometer calibration \\cite{Cahillane-PRD17-CalibrationUncertainty, Sun-CQG20-CharacterizationSystematic} and the detuning inferred from the data. The additional mismatch phase shift helps explain that such a discrepancy is possible, but the two should be studied in more detail. Note that the small Gouy phase shift from $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}$ can be significant for this small detuning effect. The expression above is primarily provided to indicate the magnitude of variation as a function of $\\sin(\\ensuremath{\\psi_\\text{R}})$, so that future observations can better constrain $\\ensuremath{\\psi_\\text{R}}$ by comparing squeezing measurements of $\\theta(\\Omega)$ with calibration measurements of the optical spring arising from $\\delta_\\ensuremath{\\text{S}}$.\n\nThe asymmetric contribution of $\\ensuremath{\\alpha}$ in \\cref{eq:h2_TMM} also causes mode mismatch to contribute to optical dephasing, $\\Xi(\\Omega)$ (c.f. \\cref{eq:Xi_passive}). The dependence on $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}, \\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{R}}$ is complex and does not have single dominating contributions, so an analytic expression is not computed here. Using the exact models of \\cref{sec:matrix_coupled_cavity}, that mimic the datasets of \\cref{sec:experiment} give a contribution of $\\sqrt{\\Xi}$ that peaks at $\\gamma_\\ensuremath{\\text{A}}$ and is 10-20\\text{mRad} for the Livingston LLO model, and 10-50mRad for the Hanford LHO model, with a range due to imperfect knowledge of the mismatch parameters.\n\nThe transverse mismatch calculations so far use the parameters $\\ensuremath{\\Upsilon_{\\text{O}}}$, which is directly measurable, and $\\ensuremath{\\Upsilon_{\\text{I}}}$, which is independent, but $\\ensuremath{\\Upsilon_{\\text{I}}}$ can not easily be measured using invasive direct measurements due to the fragile operating state of the GW interferometer. Another mismatch parameter exists for the signal beam traveling with the Michelson fringe-offset light. This beam experiences a separate mode matching efficiency, $\\ensuremath{\\Upsilon_{\\text{F}}}$, denoting the mismatch loss between the interferometer and the OMC. $\\ensuremath{\\Upsilon_{\\text{F}}}$ can be calculated from the original parameters by following the red signal path depicted in \\cref{fig:SQZ_mm_chain}.\n\\begin{align}\ne^{i\\phi_\\ensuremath{\\text{F}}}\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{F}}}, \\ensuremath{\\psi_\\text{S}})\n  &=\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{O}}}, \\ensuremath{\\psi_\\text{O}})\n\\mat{G}\n\\mat{U}^\\dagger(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})\n\\end{align}\nExpanding this form results in the following relations\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{F}}} &\\approx \\ensuremath{\\Upsilon_{\\text{O}}} + \\ensuremath{\\Upsilon_{\\text{I}}} - 2\\sqrt{\\ensuremath{\\Upsilon_{\\text{O}}}\\ensuremath{\\Upsilon_{\\text{I}}}}\\cos(\\ensuremath{\\psi_\\text{R}})\n         \\\\\n   \\ensuremath{\\Upsilon_{\\text{F}}} &\\approx \\ensuremath{\\Upsilon_{\\text{R}}}\/2 + \\ensuremath{\\Upsilon_{\\text{O}}} - \\ensuremath{\\Upsilon_{\\text{I}}}\n\\end{align}\nExperimentally, $\\ensuremath{\\Upsilon_{\\text{F}}}$ can be determined or estimated more directly than $\\ensuremath{\\Upsilon_{\\text{I}}}$ by using signal fields from the arms, though can be confused with projection loss when the local oscillator readout angle $\\xi\\ne 0$ (c.f. \\cref{eq:homodyne_observable}). These formulas provide the set of relations to estimate each of the mode mismatch parameters from the others, and potentially the overall mismatch phase $\\ensuremath{\\psi_\\text{R}}$ as well. These relations are calculated using the assumptions of this section: the two-mode approximation and that $\\ensuremath{\\mathfrak{r}}_{\\text{hom}} \\simeq 1$.\n\nTogether, the relations of this section give insight in to how the physical mismatch parameters, $\\ensuremath{\\Upsilon_{\\text{I}}}$, $\\ensuremath{\\Upsilon_{\\text{O}}}$ and $\\ensuremath{\\Upsilon_{\\text{F}}}$ contribute to squeezing degradations. $\\ensuremath{\\Upsilon_{\\text{R}}}$ is a new form of effective mismatch parameter that is directly measurable from squeezing data, using the analysis of \\cref{sec:experiment}. It indicates how squeezing changes with frequency due to \\crefrange{eq:effective_MMR_loss2}{eq:effective_MMR_loss1}. Together, the complex, coherent interactions of transverse modal mixing on squeezed state can be concisely characterized in cavity-enhanced interferometers.\n\n\\subsection{Implications for Frequency Dependent Squeezing}\nThis analysis of the transverse mismatch applies to the reflection of squeezing off of any form of cavity. Namely, the detuned filter cavity for frequency-dependent rotation of squeezing in the LIGO A+ upgrade. This cavity will be installed on the input, $\\Tmat{H}_\\ensuremath{\\text{I}}$ section of the squeezing transformation sequence. The filter cavity mismatch loss $\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}$ will behave analogously to $\\ensuremath{\\Upsilon_{\\text{I}}}$, introducing losses of ${\\sim}4\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}$ at frequencies resonating in the cavity. The mismatch loss adds to those caused by the internal round-trip cavity loss $\\Lambda_{\\ensuremath{\\text{fc}}}$, creating the effective loss $\\Lambda_{\\ensuremath{\\text{FC}}} \\approx 4\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}} + 4\\lambda_{\\ensuremath{\\text{FC}}}\/\\gamma_{\\ensuremath{\\text{FC}}}$ using \\cref{eq:cavity_relations}.\n\nThe intra-cavity losses then set the scale for how much transverse mismatch is allowable before mismatch dominates the squeezing degradation, $\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}} < \\Lambda_\\ensuremath{\\text{fc}} \/ T_\\ensuremath{\\text{fc}}$. More importantly, they add to the dephasing from the detuned cavity, by creating an effective $\\lambda'_\\ensuremath{\\text{FC}} = \\lambda_{\\ensuremath{\\text{FC}}} + \\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}\\gamma_{\\ensuremath{\\text{FC}}}$ which can be used in \\cref{eq:detuning_dephasing}. The dephasing will set the limit to the allowable injected squeezing $e^{\\pm 2r}$ level as it introduces anti-squeezing at critical frequencies in the spectrum for astrophysics.\n\n\\section{Conclusions}\n\nBefore this work, the squeezing level in the LIGO interferometers was routinely estimated using primarily high-frequency measurements. This was done to utilize a frequency band where the classical noise contributions were small, while also giving a large bandwidth over which to improve the $\\Delta F \\Delta t$ statistical error in noise estimates. In doing so, LIGO recorded a biased view of the state of squeezing performance between the two instruments. The data analysis of this work has revealed several critical features to better understand and ultimately improve the quantum noise in LIGO.\n\nFirst, it indicates that the two sites have similar optical losses in their injection and readout components, as seen from the low-frequency losses of \\cref{fig:data_Q}. There is still a small excess of losses over the predictions given in \\cite{Tse-PRL19-QuantumEnhancedAdvanced}, but substantially less than implied when estimating the losses using high frequency observations. The most culpable loss components in the LIGO interferometers are being upgraded for the next observing run.\n\nSecond, this data analysis indicates that squeezing is degraded particularly at high frequencies, and the modeling and derivations provide the mechanism of transverse optical mode mismatch, external to the cavities, as a plausible physical explanation. This will be addressed in LIGO through the addition of active wavefront control to better match the beam profiles between the squeezer's parametric amplifier, new filter cavity installation, interferometer, and output mode cleaner.\n\nThird, the quantum radiation pressure noise is now not only measured, but employed as a diagnostic tool along with squeezing. QRPN indicates that the effective local oscillator angle in the Michelson fringe offset light at LLO is a specific, nonzero, value. This indicates that to power up the detector further, while maintaining a constant level of fringe light, the angle will grow larger and cause more pronounced degradation of the sensitivity by projecting out of the signal's quadrature. Ultimately, the LO angle should become configurable using balanced homodyne detection, another planned upgrade as part of ``A+''.\n\nFinally, this work carefully derives useful formula to manipulate the quantum squeezing response metrics. These are useful to reason and rationalize the interactions of squeezing with ever more complex detectors, both for gravitational-wave interferometers, and more generally as squeezing-enhanced optical metrology becomes more commonplace. The design of a future generation of gravitational wave detectors must be optimized specifically to maintain exceptional levels of squeezing compared to today. The quantum response metrics derived in this paper will aid that design work by simplifying our interpretation of squeezing with simulations. With these diagnostics and the data from observing run 3, LIGO is now better prepared to install and characterize frequency dependent squeezing in its ``A+'' upgrade not as a demonstration, but for stable, long-term improvement of the quantum enhanced observatories to detect astrophysical events.\n\n\\begin{acknowledgments}\n LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation, and operates under Cooperative Agreement No. PHY-0757058. Advanced LIGO was built under Grant No. PHY-0823459. The authors gratefully acknowledge the National Science Foundation Graduate Research Fellowship under Grant No. 1122374. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe $S$ matrix describing electron scattering at ultrarelativistic pointlike \ncharges was shown to be determined by the \ngauge phase leading to the Dirac equation represented in the temporal gauge\n\\cite{Eichmann}. \nWe found that it naturally exhibits the same form as the well known eikonal\nexpression, as is expected by Lorentz invariance. \n\nThe gauge phase leading to the temporal gauge reads\n\\[\n\\phi(x)=e^{-i\\int_{-\\infty}^{t}A_0(x)dt'}\n\\]\nThe consideration of asymptotic states corresponds to sending the upper\nbound of the integral to infinity. \nThe infinite time integral  over the scalar part of the\nelectromagnetic potential in the exponent \nhas to be understood as the principal value of the integral. \nIt can be decomposed into a finite term and an infinite quantity,\nexpressing the familiar divergence of phases in Coulomb scattering. The\ninfinite term can be removed by a gauge transformation which converts the\nCoulomb boundary conditions of the original problem into a modified \nshort-range potential allowing for asymptotic plane wave solutions. \n\nThe ultrarelativistic limit of the gauge transformed potential $A_0'(x)$ \nreads \\cite{Aichelburg,Baltz3,Segev1,Eichmann}\n\\begin{equation}\n\\label{potlim}\n\\lim_{\\gamma \\to \\infty} A_0'(x) = Z\\alpha \\delta (z-t) \\ln x_\\perp^2 \n\\end{equation}\nand hence we obtain for the $S$ operator in coordinate space \n($\\hat{\\gamma}_-=\\hat{\\gamma}_0-\\hat{\\gamma}_3$ is the Dirac matrix structure of\nthe interaction) \n\\begin{equation}\n\\label{sopcoo}\nS=e^{-iZ\\alpha \\ln x_\\perp^2 }\\hat{\\gamma}_-\n\\end{equation}\nThe obtained $S$ operator is a\nunitary operatorb due to its conformity to the gauge phase. \nIt agrees with the first term of the Magnus expansion of   \nthe time-evolution-operator \\cite{Magnus}, \nsince the considered gauge-transformed \ninteraction was assumed to be compressed to infinitely short times. \n\nThis result was proven to be of completely\nperturbative nature \\cite{Eichmann}.\nNote, however, \nthat the perturbative derivation did not require the explicit \nFourier transform of the transverse part of (\\ref{potlim}),\nwhich is an\nill defined object. \nFor that reason the deduction of the small-coupling limit \n($Z\\alpha\\to 0$) of (\\ref{sopcoo}) in momentum space \nmust be treated with special care. \n\nIn a rigorous distributional sense \nit can be defined as the {\\it weak limit} $\\lambda\\to 0$\n\\cite{Ferrari}\n\\begin{equation} \n\\label{ftlog}\n\\int d^2x_\\perp e^{-i\\vec{k}_\\perp\\vec{x}_\\perp}\\ln x_\\perp^2\n=\\lim_{\\lambda \\to 0}4\\pi\\left(\\frac{-1}{k_\\perp^2+ \\lambda^2}-\\pi\\delta^2\n(k_\\perp)\\ln\\left(\\frac{\\lambda^2}{\\mu^2}\\right)\\right)\n\\end{equation}\nwith $\\lambda =\n\\omega\/\\gamma$, $\\mu=2\/e^C$.\n'Weak limit' means \nthat the limit $\\lambda\\to 0$ \nhas to be taken after having integrated the expression with a test function. \nThe second term on the RHS arises from the gauge transformation applied to\nthe potential and thus accounts for the Coulomb distortions. \n\nDiscarding the second term and taking the limit $\\lambda\\to 0$ \ndirectly would \naccidentally yield an expression for the \nFourier transform of (\\ref{potlim}), being identical to \nthe Fourier transform of the ungauged\npotential in the limit $\\gamma\\to \\infty$ \n\\[                                      \n\\int d^4x \\;e^{ikx}A_0=-(2\\pi)^2 Z \\alpha \\delta (k_-)\\frac{2}{k^2_\\perp}\n\\]\n\nThis error is made if one intends to extract the correct small-coupling\nlimit from a naive Taylor expansion \nof the Fourier transformed $T$ matrix \nwhose linear term reads\n\\begin{equation}\n\\label{ttaylor}\n\\lim_{Z\\alpha\\to 0}T(k)\\approx (2\\pi)^2\\delta (k_-) iZ \\alpha\n\\frac{2}{k^2_\\perp} \\overline{u}(p')\\hat{\\gamma}_-u(p)\n\\end{equation}\nHere $u(p)$ is the electron unit spinor,\n$p$ and $p'$ are the initial and final  momenta of the electron,\nrepectively, $k=p'-p$.\n\nSince, however,  Taylor expansion and Fourier transformation \ndo not commute in this case, \nthe Taylor expansion of the Fourier transformed $T$ matrix for this purpose\nis not justified.\\footnote{Note, that with the above mentioned\ndistribution-theoretical precautions, it is possible to obtain the correct\nresult via Taylor expansion \\cite{Grignani}.}\nThe correct small-coupling limit of the scattering amplitude in\nmomentum space can thus not be found by a naive \nTaylor expansion of the Fourier transformed $T$ matrix and does not agree \nwith first-order\nperturbation theory. According to (\\ref{ftlog}) this is simply \nbased on the fact, that \nthe gauge transformed potential correctly\naccounts for Coulomb boundary conditions. \n\nIn the following \nwe want to investigate, how the correct treatment of Coulomb boundary\nconditions in all orders of \nperturbation theory influences the cross section of\nthe scattering process. \n\n\n\n\n\n\n\n\n\\section{Implications on the cross section}\n\nWe consider the exact amplitude for electron scattering at\nan ultrarelativistic pointlike charge, moving in $+z$ direction. \nWe state it in terms of the invariant squared momentum transfer \n$t\\approx-k_\\perp^2$ for\n$\\gamma\\to \\infty$\\footnote{Note the striking\nsimilarity between (\\ref{ultraamp}) and the nonrelativistic (Schr\\\"odinger) \namplitude\n\\[\nf(\\theta)=-\\frac{1}{2k^2\\sin^2\\frac{\\theta}{2}}\\frac{\\Gamma\n\\left(1+\\frac{i}{k}\\right)}{\\Gamma\n\\left(1-\\frac{i}{k}\\right)}e^{-\\frac{i}{k}\\ln \\sin^2 \\frac{\\theta}{2} }\n\\]\nwith the squared momentum transfer being proportional to $\\sin^2 \\theta \/2$.\n}\n\\begin{eqnarray}\n\\label{ultraamp}\nA&=& 2\\pi\\delta (k_-) F_{p',p}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\n\\overline{u}(p')\\hat{\\gamma}_-u(p)\\nonumber \\\\\n&=&-i8 \\pi^2 Z\\alpha\n\\delta (k_-)\\frac{1}{t}\n\\frac{\\Gamma(-i\\alpha Z)}{\\Gamma(i\\alpha Z)}\ne^{-iZ\\alpha \\ln (-t\/4)}\\overline{u}(p')\\hat{\\gamma}_-u(p)\n\\end{eqnarray}\n$F_{p',p}$ abbreviates the Fourier transform with respect to the transverse\ncoordinates, taken at the momentum \n$\\vec{k}_\\perp=(\\vec{p'}_\\perp-\\vec{p}_\\perp)$.\nThe cross section for this scattering process is found to be exactly the Mott\nformula for Coulomb scattering of ultrarelativistic electrons \nat a static source, \nLorentz-transformed into the electron's rest frame. \nSuch kind of agreement between the exact result and the first order\nperturbation theory is also found in the nonrelativistic case, known\nas one of the peculiarities of the Coulomb field.\n\nThe well established \neikonalization of the scattering amplitude and thus the \nreduction to Mott's result imply, that in the high-energy limit \nthe electron and the positron Coulomb scattering cross section\nbecome identical. This behaviour of the cross section at\nultrarelativistic energies \nis required by the Pomeranchuk theorem \\cite{Itzykson}. \n\nOne can draw an analogy to pomeron exchange in hadron physics: \nThe scattering process can be described in terms of the single exchange of\nan ''effective photon`` according to the modified potential\n\\begin{equation}\n\\label{modpot}\nV_0(x)= V_3(x)\n=\\delta (z-t)\\left(\\left(\\frac{1}{x_\\perp}\\right)^{2iZ\\alpha}-1\\right)\n\\end{equation}\n\nIn the field of\ntwo ultrarelativistic colliding pointlike nuclei, \nthe exact scattering amplitude \nwas shown to retain the structure of the\nsecond-order perturbative result, due to the causal decoupling property\n\\cite{Eichmann}. \nEach interaction can be described by the modified potential (\\ref{modpot}).\nIn the following we consider the symmetric collision of two \nions with charge $Ze$, the extention to asymmetric collisions is trivial. \n\nAccounting for both time orderings, the amplitude reads\n\\begin{eqnarray}\nA^{tot}_{p'p}&=&\\int \\frac{d^2k_\\perp}{(2\\pi)^2}\nF_{k,p}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\nF_{p',k}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\ne^{i(\\vec{p'}_\\perp-\\vec{k}_\\perp)\\cdot \\vec{b}}\n\\nonumber \\\\\n&&\\left(\\overline{u}(p')\\frac{-\\hat{\\vec{\\alpha}}_\\perp \n\\cdot \\vec{k}_\\perp + \n\\gamma_0\nm}{p'_+ p_- - {k}_\\perp^2 -m^2 +i\\epsilon} \n\\hat{\\gamma}_+u(p)\\right.\n\\nonumber\\\\\n\\label{Ttot}\n&&\\left.\\overline{u}(p')\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot (\\vec{p}_\\perp +\\vec{p'}_\\perp -\\vec{k}_\\perp) +\n\\gamma_0\nm}{p'_- p_+ -\n(\\vec{p}_\\perp +\\vec{p'}_\\perp -{k}_\\perp)^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_-u(p)\\right)\n\\end{eqnarray}\nHere the trajectory of one ion is shifted by the impact\nparameter $\\vec{b}$, which is accounted for by the factor\n$e^{i(\\vec{p'}_\\perp-\\vec{k}_\\perp)\\cdot \\vec{b}}$.\n\nWe now use the crossing invariance of the amplitude to apply the\nobtained result to electron-positron pair production. The initial electron\nfour momentum $p$ has then to be replaced by the negative final positron\nmomentum $p\\to -p^p$. The final electron momentum will be\ndenoted by $p'=p^e$.\nWith (\\ref{Ttot}) we obtain for the pair production probability\n\\begin{eqnarray}\n\\frac{d\\sigma}{d^2b} &=&  |A^{tot}_{p'p}|^2\n\\frac{md^3p^e}{(2\\pi)^3E^e}\\frac{md^3p^p}{(2\\pi)^3E^p}\n\\nonumber \\\\\n&=&\n\\frac{md^3p^e}{(2\\pi)^3E^e}\\frac{md^3p^p}{(2\\pi)^3E^p}\n\\int \\frac{d^2k_\\perp}{(2\\pi)^2}\\int \\frac{d^2k'_\\perp}{(2\\pi)^2}\nF_{k,-p^p}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\nF_{p^e,k}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\\nonumber\\\\\n&&F^\\ast_{k',-p^p}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\nF^\\ast_{p^e,k'}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\ne^{i(\\vec{k'}_\\perp-\\vec{k}_\\perp)\\cdot \\vec{b}}\\nonumber\n\\\\\n&&\\left(\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot \\vec{k}_\\perp +\n\\gamma_0\nm}{-p^e_+ p^p_- - {k}_\\perp^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_+u(-p^p)\\right.\\nonumber\\\\\n&&\\left.\n+\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot (-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -\\vec{k}_\\perp) +\n\\gamma_0\nm}{-p^e_- p^p_+ -\n(-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -{k}_\\perp)^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_-u(-p^p)\\right)\\nonumber\\\\\n&&\\times\\left(\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot \\vec{k'}_\\perp +\n\\gamma_0\nm}{-p^e_+ p^p_- - {k'}_\\perp^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_+u(-p^p)\\right.\\nonumber\\\\\n\\label{csb}\n&&\\left.\n+\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot (-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -\\vec{k'}_\\perp) +\n\\gamma_0\nm}{-p^e_- p^p_+ -\n(-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -{k'}_\\perp)^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_-u(-p^p)\\right)^\\ast\n\\end{eqnarray}\n\nThe integration over the impact parameter yields the pair production cross\nsection. \nDue to the $\\delta^2(\\vec{k'}_\\perp-\\vec{k}_\\perp)$-function occuring in \nthe $\\vec{b}$ integration, \na further momentum integral can be perfomed, leaving \n\\begin{eqnarray}\nd\\sigma&=&\\frac{md^3p^e}{(2\\pi)^3E^e}\\frac{md^3p^p}{(2\\pi)^3E^p}\\int\n\\frac{d^2k_\\perp}{(2\\pi)^2}\n|F_{k,-p^p}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)|^2\n|F_{p^e,k}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)|^2\\nonumber\n\\\\\n&&\\left|\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot \\vec{k}_\\perp +\n\\gamma_0\nm}{-p^e_+ p^p_- - {k}_\\perp^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_+u(-p^p)\\right.\\nonumber\\\\\n\\label{cs}\n&&\\left.+\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot (-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -\\vec{k}_\\perp) +\n\\gamma_0\nm}{-p^e_- p^p_+ -\n(-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -{k}_\\perp)^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_-u(-p^p)\\right|^2\n\\end{eqnarray}\nThus, upon integration over the whole impact parameter plane, \nthe phases in the individual scattering amplitudes (see (\\ref{ultraamp})) \ncancel. Consequently, in the limit $\\gamma\\to \\infty$ the \ncross section is found to reduce \nto the lowest-order\nperturbation theory, \nthe two-photon result. \nThis behaviour does not only naturally explain \\cite{Segev2} \nthe experimentally \nobserved quadratic\ndependence on the target's and the projectile's charge \n\\cite{Vane}, \nbut also implies,\nthat no asymmetries should occur in \nelectron and positron spectra.  \n\nEquation (\\ref{cs}) is strictly valid only for pointlike ions. \nThe focus on\nelectromagnetic reactions in peripheral heavy-ion collisions \nimplies a restricted range of impact parameters with a lower bound \n$b=r_A+r_B$, $r_A$ and $r_B$ being the radii of the ions.\nTherefore \nin experiments which are triggered on peripheral collisions, effects of the \nCoulomb distortions \ndescribed \nby the phase in (\\ref{ultraamp}) will be visible. \n\nThe eikonal approximation (and thus the cross section) is known to become\nenergy-independent in the ultrarelativistic limit \\cite{Cheng-Wu}. \nThis dependence  \nis restored by accounting for the correct transverse \nmomentum range, which is restricted by the validity of (\\ref{potlim}). This\ncondition reads \\cite{Baltz,Eichmann}\n\\begin{equation}\n\\label{applcond}\n|\\vec{k}_\\perp|\\gg \\frac{\\omega}{\\gamma}\n\\end{equation}\nSuch a low energy cut off is also necessary to cure the divergence in\n(\\ref{Ttot}). \n\n\\section{Equivalent Photon Approximation}\nWe intend to study the behaviour of the cross section,\nboth impact parameter dependent and impact parameter integrated, in the\nWeizs\\\"acker-Williams method of equivalent photons. \nThis approximation uses the similarity between the fields of a fast\nmoving charge and a swarm of real photons moving in beam direction. It\napproximately correstponds to the first order Born approximation in the\ntemporal gauge: Only the transverse part of the interaction is considered -- \nthe longitudinal part is suppressed by $1\/\\gamma^2$ -- and the vertex\nfunction is evaluated on-shell at $k^2=0$, i.e. for an assumed real photon.\nRewriting the exact  \ncross section in terms of the real photon cross section, the\nwhole information about the scattering potential, which can be the retarded\nCoulomb potential or the modified potential (\\ref{modpot}), respectively, is\nthen contained in the distribution function of the equivalent photons\n$n(\\omega)$. \nRoughly speaking, this photon distribution function is determined by the\nsquared absolut value of the Fourier-transformed potential (in temporal\ngauge).\nThe obvious \nadvantage of casting the exchange of effective photons according to\n(\\ref{modpot}) in the Weizs\\\"acker-Williams form is, that \nany difference between the second-order\nperturbative result and the exact calculation will be solely generated by \ndifferences between the equivalent photon distributions.\n\nThe Weizs\\\"acker-Williams approximation is applicable if the exchanged\nmomentum meets the following conditions \\cite{LandauIV} \n\\begin{equation}\n\\label{wwbed1}\n\\frac{\\omega}{\\gamma^2}\\ll |\\vec{k}_\\perp|\\ll m\n\\end{equation}\nand\n\\begin{equation}\n\\label{wwbed2}\n|\\vec{k}_\\perp|\\ll \\omega \\ll m \\gamma\n\\end{equation}\nThe upper bounds mainly stem from the requirement, that $|k^2|$\n is negligible compared to $2p\\cdot k \\ge m^2$, such that the intermediate\n propagators of the scattered particles \ncan be approximated by those describing the interaction with real photons. \nThe particle's rest mass in (\\ref{wwbed1}) is however \na conservative upper\nbound and the equivalent photon method is not strictly invalid for $|k^2|\n\\sim m^2$. \nNote, that for the approximative\ncalculations in \\cite{Bottcher}, the transverse\nmass of the scattered particle was taken as the upper bound for\n(\\ref{wwbed1}).\n\n\nSince the exact amplitude takes the eikonal form, we must point out the\nfollowing: \nThe expansion of the ultrarelativistic\nscattering amplitude in powers of the transferred\nmomentum yields, as the leading term, the eikonal expression (describing the\nminimal deflection from the initial straight-line trajectory)\n\\cite{Abarbanel,Sugar}. Its\nperturbation-theoretical derivation requires that the\nquadratic terms $k^2$ are negligible \nrelative to the terms $2p_i\\cdot k$ in the denominators\nof the propagators, where $k$ is any partial sum of the internal momenta\n\\cite{Levy}.\nThe exact validity of the eikonal formula at infinite energies \ntherefore shows, that the \ntransferred momentum $|k^2|$ does not exceed $m^2$ (irrespective of the\nvalue\nof $m$). \nThis agrees with the theoretical observation, that at high energies\nparticles\nare predominantly scattered into a cone with opening angle $\\theta \\sim 1\/\n\\gamma$, corresponding to momentum transfers $|k^2|\\sim m^2$. The main\ncontributions to the cross section are thus expected from spatial distances\nlarger or equal the Compton wavelength of the particle.\\footnote{From the\nasymmetry of electron and positron spectra\nproduced in $S$(200 GeV\/n)+Au collisions , the\nmean transverse distance from the target ion was deduced to be approximately\ntwo Compton wavelengths \\cite{Vane}. The collision energy corresponds to\n$\\gamma \\approx 10$ in the center of speed system.}\n\nMoreover, the longitudinal part of the interaction vanishes identically in\nthe\nlimit $\\gamma\\to \\infty$.\nHence, the applicability conditions of the Weizs\\\"acker-Williams\napproximation\nare trivially fulfilled in the limit $\\gamma\\to \\infty$.\\footnote{Just as\nthe\neikonal formula, the Weizs\\\"acker-Williams approximation can be viewed as\nthe leading term of an expansion in powers of $k^2\/m^2$ \\cite{Brodsky}. The\nvalidity of the eikonal expression then automatically implies the validity\nof\nthe Weizs\\\"acker-Williams method.}\n\nWe have the freedom to apply this method to the interaction of the electron\nwith both nuclei, \ngiving two possibilities (see Figure \\ref{fig01})\n\\begin{figure}[h]\n\\begin{center}\n\\begin{minipage}{16cm}\n\\parbox{7cm}{\\psfig{figure=comptww.eps,width=6cm}}\\hspace{1cm}\n\\parbox{7cm}{\\psfig{figure=bremsww.eps,width=6cm}}\n\\end{minipage}\n\\end{center}\n\\caption{\\label{fig01}The two possible distinct processes, that can be used\nto describe electron-positron production in heavy-ion collision. One or both  \nultrarelativistic ions can be replaced by an equivalent photon distribution.\nIf the bremsstrahlung process b) is calculated in one ion's rest frame, the\nelectron must be assumed ultrarelativistic, to yield agreement with\na).}\n\\end{figure}\n\nThe two possible calculation schemes (Figure \\ref{fig01})\nagree, since in b) \nthe bremsstrahlung emission and the scattering at the external\npotential decouple. This is due to the fact, that the region in which the\nultrarelativistic electron ''feels`` the external field is assumed to be pointlike and\nany frequency of the emitted photon is ''soft`` compared to the timescale of\nthe scattering.\nCoulomb effects arise, if one explicitly accounts for the finite\ninteraction time, either in the scattering process by correcting the eikonal\nformula\n\\label{eikexp} \nor by keeping the eikonal amplitude for the scattering process\nbut assuming a Rutherford-deflected trajectory for the photon emission\n\\cite{LandauIV}. Corrections to the eikonal formula account for higher\norders of e.g. the Magnus expansion \\cite{Magnus},\nwhich is an expansion in the interaction time $\\tau$\naround the\ninstantaneous interaction $\\tau\\sim 1\/\\gamma\\to 0$.\nIn general these Coulomb effects vanish, if the\nenergy of the emitted photon is too small to resolve details of the\nscattering process, and the recoil of the electron is negligible.  \n\nTo apply the Weizs\\\"acker-Williams method to the bremsstrahlung photon, \nthe recoil of the\nbremsstrahlung photon\nmust, however, be assumed\nnegligible. \nThe small momentum\ntransfer \nis in turn ensured by the\neikonalization of the scattering process. \n\nThe equivalent single-photon distributions $n_{A\/B}(\\omega)$ \nof the ions $A$\nand $B$, are determined from the effective potential\n(\\ref{modpot}). \nThe photon distribution reads \n\\begin{equation}\nn(\\omega)=\\frac{1}{4\\pi^2 \\alpha \\omega} \n\\int_{\\omega\/\\gamma}^{m} k_\\perp dk_\\perp \n\\left|k_\\perp\\pi\nZ\\alpha\\left(\\frac{4}{k_\\perp^2}\\right)^{1-iZ\\alpha}\\frac{\\Gamma(-i\\alpha\nZ)}{\\Gamma(i\\alpha Z)}\\right|^2=\n\\frac{2Z^2 \\alpha}{\\pi\\omega}\n\\ln\\left(\\frac{m\\gamma}{\\omega}\\right)\n\\end{equation}\nThe lower bound of the integral is taken from the condition\n(\\ref{applcond}).\nThe upper bound, the electron rest mass, is imposed by (\\ref{wwbed1}).\nThe prefactor arises from properly rewriting the cross\nsection (\\ref{cs}) in terms of the real photon cross section (i.e. the\nCompton cross section) and photon\ndistribution functions. \n\nThis photon distribution \ncoincides with the equivalent-photon distribution obtained from the\nCoulomb potential to logarithmic accuracy \\cite{LandauIV} \nand is not changed by Coulomb\neffects. \n\n\n\n\n\n\n\n\n\n\n\\section{Impact parameter dependent cross section}\nThe impact parameter dependent equivalent photon method for the exact\ncalculation, using the modified potential (\\ref{modpot}), \ncan be derived similarly to\n\\cite{Vidovic}. \nTo this end \nwe have to modify the integrands in (\\ref{csb}) such that they  \naccount for the limited momentum range ((\\ref{applcond}) and\n(\\ref{wwbed1})). \n\n\nThe cut off of low transverse momenta according to (\\ref{applcond}) can be\nachieved by the following replacement in \n(\\ref{ultraamp})\\footnote{Note, that for the\nSchr\\\"odinger case the exact validity of the eikonal formula \ncan be proven for a certain off shell domain of the \nmomentum transfer for the whole energy plane \\cite{Banyai}.} \n\\begin{equation}\n\\label{text}\nt=-k_\\perp^2 \\to -\\frac{\\omega^2}{\\gamma^2}-k_\\perp^2\n\\end{equation}\nThis substitution suppresses small transverse momenta less strongly than the\nstrict cut off at $k_\\perp=\\omega \/\\gamma$. It assumes the sufficient\naccuracy of the classical\neikonal amplitude for\n$1\/\\gamma$ in the near vicinity of $1\/\\gamma =0$, which is guaranteed by the\npossibility of continuous extentions of the eikonal formula towards finite\n$\\gamma$ and large $t$. \nA Yukawa-type damping of transverse distances corresponding to the cut off\ntransverse momenta yields additional terms that change the character of the\namplitude significantly and can not be motivated physically. \n\nIn accordance with the exact validity of the eikonal formula, the physical\nsituation is such, that the transferred momenta are restricted by  the\ncondition $|k^2|\\ll m^2$. They are, however, \nnaturally cut off, if one introduces \na form factor to account\nfor the finite extent of the nuclei. \nThus, large momenta \nhave to be cut off at $k_\\perp\\approx 1\/r_>$, where $r_>$ is the larger \nvalue of either \nthe nuclear radius or the Compton wavelength of the scattered\nparticle \\cite{Jackson}. In this respect, the electron is an\nexception, since all other particles have Compton wavelengths smaller or\ncomparable to the nuclear size.\nTo present the calculations in a uniform manner, we use the form factor of\nthe nucleus to cut off the large momenta. \n\nThe impact parameter dependent cross section for particle production,\ndescribed in the equivalent photon method reads \\cite{Vidovic}\n\\begin{equation}\n\\label{csbww}\n\\frac{d\\sigma}{d^2b}=\\int d\\omega_1 \\int d\\omega_2\n\\left[n_\\|(\\omega_1,\\omega_2,\\vec{b})\n\\sigma^{\\gamma\\gamma}_\\|(\\omega_1,\\omega_2) +\nn_\\perp(\\omega_1,\\omega_2,\\vec{b})\n\\sigma^{\\gamma\\gamma}_\\perp(\\omega_1,\\omega_2)\\right]\n\\end{equation}\nwith the two-photon distribution functions\n$n_{\\|\/\\perp}(\\omega_1,\\omega_2,\\vec{b})$. The elementary two-photon cross\nsections and the two-photon distribution functions explicitly account for\nthe parallel or orthogonal orientation of the photon-polarizations, denoted\nby the indices $\\|$ and $\\perp$, respectively. \nSince the integration over the impact\nparameter plane implies an averaging over the photon polarizations, the\nexplicit occurrence of the photon polarizations in the impact parameter\ndependent cross section is expected.\nThe functions\n$n_{\\|\/\\perp}(\\omega_1,\\omega_2,\\vec{b})$ can be expressed in terms of\nsingle-photon distribution functions $n(\\omega,b)$, \ndepending on the transverse separation:\n\\begin{eqnarray}\n\\label{n2par}\nn_\\|(\\omega_1,\\omega_2,\\vec{b})&=&\\int d^2x_\\perp\nn(\\omega_1,\\vec{x}_\\perp-\\vec{b})\\,n(\\omega_2,\\vec{x}_\\perp)\n\\left(\\frac{(\\vec{x}_\\perp-\\vec{b})\\cdot\n\\vec{x}_\\perp}{|\\vec{x}_\\perp-\\vec{b}||\\vec{x}_\\perp|}\\right)\\\\\n\\label{n2senk}\nn_\\perp(\\omega_1,\\omega_2,\\vec{b})&=&\\int d^2x_\\perp\nn(\\omega_1,\\vec{x}_\\perp-\\vec{b})\\,n(\\omega_2,\\vec{x}_\\perp)\n\\left(\\frac{(\\vec{x}_\\perp-\\vec{b})\\times \n\\vec{x}_\\perp}{|\\vec{x}_\\perp-\\vec{b}||\\vec{x}_\\perp|}\\right)\n\\end{eqnarray}\nwith\n\\begin{equation} \n\\label{photdist}\nn(\\omega,b)=\\frac{Z^2\\alpha}{\\pi^2\\omega}\\left|\\int_0^\\infty dk_\\perp k_\\perp^2\n\\frac{F(k_\\perp^2+\\omega^2\/\\gamma^2)}{(k_\\perp^2+\\omega^2\n\/\\gamma^2)^{1-iZ\\alpha}}J_1(bk_\\perp)\\right|^2\n\\end{equation}\n$J_1$ is a Bessel function. The function $F$ denotes the chosen form\nfactor of the nucleus. \n\nFor a pointlike charge ($F\\equiv 1$), the photon distribution function can\nbe calculated analytically. We obtain for the Coulomb potential and the\nmodified potential \n\\begin{equation}\n\\label{photdistpt}\nn(\\omega,b)=\\left\\{\\begin{array}{ll}\n\\displaystyle \\frac{Z^2\\alpha}{\\pi^2}\\frac{\\omega}{\\gamma^2}\\left[K_1\\left(\\frac{\\omega\nb}{\\gamma}\\right)\\right]^2\\;\\;\\;&{\\rm retarded\\;Coulomb\\;potential}\\\\\n&\\\\\n\\displaystyle\n\\frac{Z^2\\alpha}{\\pi^2}\\frac{\\omega}{\\gamma^2}\\left[\\frac{K_{1+iZ\\alpha}(\\omega\nb\/\\gamma)}{\\Gamma(1-iZ\\alpha)}\\right]^2&{\\rm\nmodified\\;potential\\;(\\ref{modpot})}\n\\end{array}\\right.\n\\end{equation}\n$K_\\nu$ is a modified Bessel function. For small arguments of the Bessel\nfunction one can use the asymptotic expression \\cite{Abramowitz}\n\\begin{equation}\n\\label{kasym}\nK_\\nu (z)\\sim \\frac{1}{2}\n\\Gamma (\\nu )\n(\\frac{1}{2} z)^{-\\nu }\n\\end{equation}\nTherefore, for $\\omega b \\ll \\gamma$ and assumed\npoint-like charges the photon\ndistribution functions (\\ref{photdistpt}) nearly completely agree. \n\\newpage \n\\begin{figure}[h]\n\\centerline{\\psfig{figure=photvert.eps,width=17cm}}\n\\caption{\\label{fig02}The single-photon distribution function for various \nphoton energies (indicated in the plot) as\na function of the transverse distance from the ultrarelativistic charge. The\ncalculations are done for lead ions ($Z=82$) at\nLHC energies ($\\gamma\\approx 3000$). }\n\\end{figure}\n\nWe have numerically evaluated the photon distribution function\n(\\ref{photdist}) for an extended nucleus, using a gaussian form factor\n$F(Q^2)=e^{-Q^2\/(2Q_0^2)}$ with $Q_0=60MeV$ which describes the $Pb$ nucleus \n\\cite{Drees}. \nFigure \\ref{fig02} shows a comparison between the photon distribution\nfunctions for both, point like nuclei ($F\\equiv 1$) and extended nuclei\nusin either \nthe\nretarded Coulomb potential or the modified potential (\\ref{modpot}),\nwhich represents the exact calculation. \nAt small distances up to a few multiples\nof the nuclear radius, the modified potential gives a smaller number of\nequivalent \nphotons than the pure Coulomb potential. For large distances or large photon\nenergies (\\ref{kasym}) looses its validity and the modified potential gives\na larger number of photons than the Coulomb potential (see Figure\n\\ref{fig03}). \n\nThe significance of these results can be assessed \nin single-photon induced processes,\nsuch as the electromagnetic dissociation of nuclei in peripheral heavy-ion\ncollisions. In the Weizs\\\"acker-Williams approach \nthe dissociation probability at a given impact parameter is\ncalculated by integrating \nthe product of the measured \nphoton-nucleus dissociation cross section of the target,\n$\\sigma_T^\\gamma(\\omega)$, \nand the\nequivalent photon distribution, $n_P(\\omega,b)$, \nof the projectile (\\ref{photdist}) \nover the photon energy \\cite{Vidovic2,Norbury0}\n\\[ \nP(b)=\\int d\\omega \\,n_P(\\omega,b)\\, \\sigma_T^\\gamma(\\omega)\n\\]\nAs shown in \\cite{Vidovic2} for $Pb+Pb$ collisions at LHC energies, \nthe dissociation \nprobability violates unitarity at small impact parameters up to 25 fm. At\nsmall impact parameters, however, the photon distribution of the modified\npotential is suppressed relative to that of the pure Coulomb potential and\nyields a reduction of the probabiltiy, whereas at large impact parameters,\nthe dissociation probability is enhanced. The reduction visible in Figure\n\\ref{fig02} taken alone is too small to cure the unitarity violation.\nPossibly this can be achieved if in addition \nmulti-phonon excitations \\cite{Bertulani,Norbury1} are taken into account,  \nwhich also reduce the dissociation  probability at small impact parameters. \n\\begin{figure}[hbtp]\n\\centerline{\\psfig{figure=ratio.eps,width=17cm}}\n\\caption{\\label{fig03}The ratio of the equivalent photon numbers of the\npure Coulomb potential, $n_C(\\omega,x_\\perp)$, and the photon numbers of the\neffective potential, $n_{eff}(\\omega,x_\\perp)$. At small transverse\ndistances, one finds a deviation of up to $70\\%$ for small photon energies.\nFar outside the nucleus, the photon distribution functions are determined by\n(\\protect\\ref{photdistpt}). For large distances one asymptotically finds a\ndeviation in the order of $40\\%$, independent of the photon energy.}\n\\end{figure}\n\n\nThe impact parameter dependent two-photon distribution functions also have\nto be corrected according to (\\ref{photdist}), thus correcting the\npair production probability (\\ref{csbww}), i.e. (\\ref{csb}), for Coulomb\neffects. Due to the complicated convolution of the single-photon\ndistribution functions in (\\ref{n2par}) and (\\ref{n2senk}), effects on the\ntwo-photon distribution function are not obvious. For large distances\n$x_\\perp \\gg b$, however, one directly finds an enhancement of the\nequivalent two-photon numbers. \n\n\n\n\n\n\\section{Summary}\nIt was shown by either \nsumming the perturbation series \\cite{Eichmann} or by\nmatching plane waves at the delta function potential on the light front\n\\cite{Segev1,Baltz,Eichmann}, that the eikonal expression for the scattering\namplitude becomes exact in the ultrarelativistic limit ($\\gamma\\to \\infty$).\nThis allows to neglect the squared momentum transfer $k^2$ \nrelative to the term $2p_i\\cdot k$ in the denominator of the propagator of the\nscattered particle. As a consequence \nthe applicability\nconditions of the Weizs\\\"acker-Williams method are fulfilled automatically \n-- irrespective of the mass of the scattered particle. \n\nFurthermore, the exact validity of the eikonal formula for ultrarelativistic\nscattering processes confirm the Pomeranchuk theorem,\nstating that\nthe cross sections for antiparticle and particle scattering at a given\ntarget become identical in the ultrarelativistic limit. In analogy to\npomeron exchange in hadron physics, on can describe the exact interaction as\nthe exchange of an effective photon, according to a modified, effective\npotential given by (\\ref{modpot}). \nThe cross section, as a peculiarity of the Coulomb interaction, becomes\nidentical to the Mott result. The exact \npair production cross section in the field\nof two ultrarelativistic colliding (pointlike) ions also reduces to the\nsecond order perturbative result \\cite{Baltz} which was evaluated in \n\\cite{Bottcher}. This allows for two\nconclusions: i) The production rate scales with the square of the target and\nthe projectile charge \\cite{Vane,Segev2} ii) Asymmetries in the electron and\npositron spectra should not occur. \n\nNote, however, that the presented formalism is valid only if the produced\nparticles are fas with respect to both nuclei. \nTherefore, the observed \\cite{Vane} asymmetry at small electron and \npositron momenta remains unaffected by these considerations. \n\nWe applied the Weizs\\\"acker-Williams approach to pair production using the\nmodified potential (\\ref{modpot}), correctly accounting for the Coulomb\nboundary conditions. The impact parameter dependent single-photon\ndistribution calculated with the modified potential shows \ndeviations from the equivalent photon distribution function obtained from\nthe retarded Coulomb potential in order of up to $70\\%$ at small separations \nand approximately $40\\%$ at large separations from the ion. \n\nIn combination with multi-phonon excitation, the correct treatment of\nCoulomb distortions possibly solves the problem of unitarity violation in\nphotodissociation processes in ultrarelativistic heavy-ion collisions. \n\nThe pair production probability is also subject to changes due to the\nmodified photon numbers at given impact parameters and photon energies. \nThe perturbation theoretical probability, as calculated here, rather\nrepresents the average number of produced pairs and exceeds unity at\nsufficiently small\nimpact parameters. The ''true`` pair\nproduction probability has to be corrected by the vacuum-to-vacuum\namplitude, which in turn can be calculated from the perturbative pair\nproduction probability \\cite{Best,Rhoades-Brown}. This nontrivial influence\non the pair production cross section is subject of further studies. \n\n\\section*{Acknowledgements}\nThis work was supported by\n{\\it Deutsche Forschungsgemeinschaft} DFG within the project Gr-243\/44-2.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\\subsection{Motivation and previous work}\n\n\\IEEEPARstart{D}{riven} by the continuously increasing demands for high system throughput, low latency, ultra reliability, improved fairness and near-instant connectivity, \\gls{5G} wireless communication networks are being standardized \\cite{Shafi17} while, at the same time, insights and innovations from industry and academia are paving the road for the coming of the \\gls{6G} \\cite{David18}. As stated by Marzetta \\emph{et al.} in \\cite[Chapter 1]{Marzetta16}, there are three basic pillars at the physical layer that can be used to sustain the spectral and energy efficiencies that these networks are expected to provide: (i) employing massive \\gls{MIMO}, (ii) using \\gls{UDN} deployments, and (iii) exploiting new frequency bands.\n\nMassive \\gls{MIMO} systems, equipped with a large number of antenna elements, are intended to be used as \\gls{MU-MIMO} arrangements in which the number of antenna elements at each \\gls{AP} is much larger than the number of \\glspl{MS} simultaneously served over the same time\/frequency resources. The operation of massive \\gls{MIMO} schemes is based on the availability of \\gls{CSI} acquired through \\gls{TDD} operation and the use of \\gls{UL} pilot signals. Such a setting allows for very high spectral and energy efficiencies using simple linear signal processing in the form of conjugate beamforming or \\gls{ZF} \\cite{Marzetta10,Marzetta16}.\n\nIn \\glspl{UDN}, a large number of \\glspl{AP} deployed within a given coverage area cooperate to jointly transmit\/receive to\/from a (relatively) reduced number of \\glspl{MS} thanks to the availability of high-performance low-latency fronthaul links connecting the \\glspl{AP} to a central coordinating node. Coordination among \\glspl{AP} can effectively control (or even eliminate) intercellular interference in an approach that was first referred to as network \\gls{MIMO} \\cite{Karakayali06,Gesbert10}, later led to the concept of \\gls{CoMP} transmission \\cite{Irmer11} and, more recently, to that of \\gls{C-RAN} \\cite{Checko15}. In a \\gls{C-RAN}, the \\glspl{AP}, which are treated as a distributed \\gls{MIMO} system, are connected to a cloud-computing based \\gls{CPU} in charge, among many others, of the baseband processing tasks of all \\glspl{AP}. Conceptually similar to the \\gls{C-RAN} architecture, but explicitly relying on assumptions specific of the massive \\gls{MIMO} regime, distributed massive \\gls{MIMO}-based \\glspl{UDN} have been recently termed as \\emph{cell-free massive \\gls{MIMO}} networks \\cite{Ngo15,Ngo17}. In these networks, a massive number of \\glspl{AP} connected to a \\gls{CPU} are distributed across the coverage area and, as in the cellular collocated massive \\gls{MIMO} schemes, exploit the channel hardening and favorable propagation properties to coherently serve a large number of \\glspl{MS} over the same time\/frequency resources. Typically using simple linear signal processing schemes, they are claimed to provide uniformly good \\gls{QoS} to the whole set of served \\glspl{MS} irrespective of their particular location in the coverage area.\n\nSince the microwave radio spectrum (from 300 MHz to 6 GHz) is highly congested, the use of massive antenna systems and network densification alone may not be sufficient to meet the \\gls{QoS} demands in next generation wireless communications networks. Thus, another promising physical layer solution that is expected to play a pivotal role in \\gls{5G} and beyond \\gls{5G} communication systems is to increase the available spectrum by exploring new less-congested frequency bands. In particular, there has been a growing interest in exploiting the so-called \\gls{mmWave} bands \\cite{Rappaport13,Boccardi14,Akdeniz14,Rappaport17}. The available spectrum at these frequencies is orders of magnitude higher than that available at the microwave bands and, moreover, the very small wavelengths of \\glspl{mmWave}, combined with the technological advances in low-power CMOS \\gls{RF} miniaturization, allow for the integration of a large number of antenna elements into small form factors. Large antenna arrays can then be used to effectively implement \\gls{mmWave} massive \\gls{MIMO} schemes (see, for instance, \\cite{Gao18,Busari18} and references therein) that, with appropriate beamforming, can more than compensate for the orders-of-magnitude increase in free-space path-loss produced by the use of higher frequencies.\n\nThe performance of cell-free massive \\gls{MIMO} using conventional sub-6 GHz frequency bands and assuming infinite-capacity fronthaul links has been extensively studied in, for instance, \\cite{Ngo17,Nayebi17,Nguyen17,Ngo18}.\nCell-free massive \\gls{MIMO} networks using capacity-constrained fronthaul links have also been considered in \\cite{Bashar18b,Boroujerdi18} but assuming, again, the use of fully digital precoders in conventional sub-6 GHz frequency bands.\nSub-6 GHz massive \\gls{MIMO} systems are often assumed to implement a fully-digital baseband signal processing requiring a dedicated \\gls{RF} chain for each antenna element. The present status of \\gls{mmWave} technology, however, characterized by high-power consumption levels and high production costs, precludes the fully-digital implementation of massive \\gls{MIMO} architectures, and typically forces \\gls{mmWave} systems to rely on hybrid digital-analog signal processing architectures. In these hybrid transceiver architectures, a large antenna array connects to a limited number of \\gls{RF} chains via high-dimensional \\gls{RF} precoders, typically implemented using analog phase shifters and\/or analog switches, and low-dimensional baseband digital precoders are then used at the output of the \\gls{RF} chains \\cite{Ayach14,Gao16,Molisch17}. The network of phase shifters connecting the array of antennas to the \\gls{RF} chains determines whether the structure is fully or partially connected \\cite{Park17TWC}. Thus, the assumptions, methods and analytical expressions in \\cite{Ngo17,Nayebi17,Nguyen17,Ngo18,Bashar18b,Boroujerdi18} cannot by applied directly when assuming the use of \\gls{mmWave} frequency bands. Despite its evident potential, as far as we know, besides \\cite{Alonzo17,Alonzo18} there is no other research work on cell-free \\gls{mmWave} massive \\gls{MIMO} systems and, furthermore, the authors of these works did not face one of the main challenges in the implementation of cooperative \\glspl{UDN}, that is, the fact that these systems require of a substantial information exchange between the \\glspl{AP} and the \\gls{CPU} via capacity-constrained fronthaul links. Moreover, they also considered the use of oversimplified \\gls{mmWave} channel models and \\gls{RF} precoding stages, without constraining the available number of \\gls{RF}-chains at each \\gls{AP}.\n\n\\subsection{Aim and contributions}\n\nMotivated by the above considerations, our main aim in this paper is to address the design and performance evaluation of realistic cell-free \\gls{mmWave} massive \\gls{MIMO} systems using hybrid precoders and assuming the availability of capacity-constrained fronthaul links connecting the \\glspl{AP} and the \\gls{CPU}. The main contributions of our work can be summarized as follows:\n\n\\begin{itemize}[noitemsep,wide=0pt, leftmargin=\\dimexpr\\labelwidth + 2\\labelsep\\relax]\n\n\\item The performance of both the \\gls{DL} and \\gls{UL} of cell-free \\gls{mmWave} massive \\gls{MIMO} systems is considered with particular emphasis on the per-user rate, rather than the system sum-rate, by posing max-min fairness resource allocation problems that take into account the effects of imperfect channel estimation, power control, non-orthogonality of pilot sequences, and fronthaul capacity constraints. Instead of assuming the use of rather simple uniform quantization processes when forwarding information on the capacity-constrained fronthauls, the proposed optimization problems assume the use of large-block lattice quantization codes able to approximate a Gaussian quantization noise distribution. Optimal solutions to these problems are proposed that combine the use of block coordinate descent methods with sequential linear programs\n\n\\item A hybrid beamforming implementation is proposed where the \\gls{RF} high-dimensionality phase shifter-based precoding\/decoding stage is based on large-scale second-order statistics of the propagation channel, and hence does not need the estimation of high-dimensionality instantaneous \\gls{CSI}. The low-dimensionality baseband \\gls{MU-MIMO} precoding\/decoding stage can then be easily implemented by standard signal processing schemes using small-scale estimated \\gls{CSI}. As will be shown in the numerical results section, such a reduced complexity hybrid precoding scheme, when combined with appropriate user selection, performs very well in the fronthaul capacity-constrained \\gls{UDN} \\gls{mmWave}-based scenarios under consideration.\n\n\\item A suboptimal pilot allocation strategy is proposed that, based on the idea of clustering by dissimilarity, avoids the computational complexity of the optimal pilot allocation scheme. The performance of the proposed \\emph{dissimilarity cluster-based pilot assignment algorithm} is compared with that of both the \\emph{pure random pilot allocation approach} and the \\emph{balanced random pilot strategy}.\n\n\\item For those cases in which the number of active \\glspl{MS} in the network is greater than the number of available \\gls{RF} chains at a particular \\gls{AP}, a \\gls{MS} selection algorithm is proposed that aims at maximizing the minimum average sum-energy (i.e., Frobenius norm) of the equivalent channel between the \\glspl{AP} and any of the active \\glspl{MS}, constrained by the fact that each \\gls{AP} can only beamform to a number of \\glspl{MS} less or equal than the number of available \\gls{RF} chains.\n\n\\end{itemize}\n\n\\subsection{Paper organization and notational remarks}\n\nThe remainder of this paper is organized as follows. In Section \\ref{sec:System_model} the proposed cell-free \\gls{mmWave} massive \\gls{MIMO} system is introduced. Different subsections are devoted to the description of the channel model, the large-scale and small-scale training phases, the channel estimation process, and the \\gls{DL} and \\gls{UL} payload transmission phases. The achievable \\gls{DL} and \\gls{UL} rates are presented in Section \\ref{sec:Achievable_rates} and further developed in Appendices \\ref{app:Appendix_1} and \\ref{app:Appendix_2}. Section \\ref{sec:fronthaul_capacity} is dedicated to the calculation of the capacity consumption of both the \\gls{DL} and \\gls{UL} fronthaul links. The pilot assignment, power allocation and quantization optimization processes are described in Sections \\ref{sec:pilot_assignment} and \\ref{sec:power allocation_quantization}. Numerical results and discussions are provided in Section \\ref{sec:numerical_results} and, finally, concluding remarks are summarized in Section \\ref{sec:Conclusion}.\n\n\\emph{Notation}: Vectors and matrices are denoted by lower-case and upper-case boldface symbols. The $q$-dimensional identity matrix is represented by $\\boldsymbol{I}_q$. The operator $\\det(\\bs{X})$ represents the determinant of matrix $\\bs{X}$, $\\tr(\\bs{X})$ denotes its trace, $\\|\\bs{X}\\|_F$ is its Frobenius norm, whereas $\\bs{X}^{-1}$, $\\bs{X}^T$, $\\bs{X}^*$ and $\\bs{X}^H$ denote its inverse, transpose, conjugate and conjugate transpose (also known as Hermitian), respectively. With a slight abuse of notation, the operator $\\diag(\\boldsymbol{x})$ is used to denote a diagonal matrix with the entries of vector $\\boldsymbol{x}$ on its main diagonal, and the operator $\\diag(\\boldsymbol{X})$ is used to denote a vector containing the entries in the main diagonal of matrix $\\bs{X}$. The expectation operator is denoted by $\\mathbb{E}\\{\\cdot\\}$. Finally, $\\mathcal{CN}(\\bs{m},\\bs{R})$ denotes a circularly symmetric complex Gaussian vector distributions with mean $\\bs{m}$ and covariance $\\bs{R}$, $\\mathcal{N}(0,\\sigma^2)$ denotes a real valued zero-mean Gaussian random variable with standard deviation $\\sigma$, and $\\mathcal{U}[a,b]$ represents a random variable uniformly distributed in the range $[a,b]$.\n\n\\section{System model}\n\\label{sec:System_model}\n\nLet us consider a cell-free massive \\gls{MIMO} system where a \\gls{CPU} coordinates the communication between $M$ \\glspl{AP} and $K$ single-antenna \\glspl{MS} randomly distributed in a large area. Each of the \\glspl{AP} communicates with the \\gls{CPU} via error-free fronthaul links with \\gls{DL} and \\gls{UL} capacities ${C_F}_d$ and ${C_F}_u$, respectively. Baseband processing of the transmitted\/received signals is performed at the \\gls{CPU}, while the \\gls{RF} operations are carried out at the \\glspl{AP}. Each \\gls{AP} is equipped with an array of $N > K$ antennas and $L \\leq N$ \\gls{RF} chains. A fully-connected architecture is considered where each \\gls{RF} chain is connected to the whole set of antenna elements using $N$ analog phase shifters. Without loss of essential generality, it is assumed in this paper that the number of active \\gls{RF} chains at each of the \\glspl{AP} in the network is equal to $L_A=\\min\\{K,L\\}$. That is, if $K \\leq L$, all \\glspl{AP} in the cell-free network provide service to the whole set of \\glspl{MS} and if $K > L$, instead, each \\gls{AP} can only provide service to $L$ out of the $K$ \\glspl{MS} in the network and, thus, an algorithm must be devised to decide which are the \\glspl{MS} to be beamformed by each of the \\glspl{AP}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=7.8cm]{TDD_frame}\n  \\caption{Allocation of the samples in large-scale and short-scale coherence intervals.}\\label{fig:TDD_frame}\n\\end{figure}\n\nThe propagation channels linking the \\glspl{AP} to the \\glspl{MS} are typically characterized by small-scale parameters that are (almost) static over a coherence time-frequency interval of $\\tau_c$ time-frequency samples (see \\cite[Chapter 2]{Marzetta16}), and large-scale parameters (i.e., path loss propagation losses and covariance matrices) that can be safely assumed to be static over a time-frequency interval ${\\tau_L}_c \\gg \\tau_c$. As shown in the following subsections, these channel characteristics can be leveraged to simplify both the channel estimation and the precoding\/combining processes. In particular, \\gls{DL} and \\gls{UL} transmissions between \\glspl{AP} and \\glspl{MS} are organized in a half-duplex \\gls{TDD} operation whereby each coherence interval is split into three phases, namely, the \\gls{UL} training phase, the \\gls{DL} payload data transmission phase and the \\gls{UL} payload data transmission phase, and every \\emph{large-scale coherence interval} ${\\tau_L}_c$ the system performs an estimation of the large-scale parameters of the channel (see Fig. \\ref{fig:TDD_frame}). In the \\gls{UL} training phase, all \\glspl{MS} transmit \\gls{UL} training orthogonal pilots allowing the \\glspl{AP} to estimate the propagation channels to every \\gls{MS} in the network\\footnote{Note that channel reciprocity can be exploited in \\gls{TDD} systems and therefore only \\gls{UL} pilots need to be transmitted.}. Subsequently, these channel estimates are used to detect the signals transmitted from the \\glspl{MS} in the \\gls{UL} payload data transmission phase and to compute the precoding filters governing the \\gls{DL} payload data transmission. Not shown are guard intervals between \\gls{UL} and \\gls{DL} transmissions.\n\n\\subsection{Channel Model}\n\\label{subsec:Channel_model}\n\nMmWave propagation is characterized by very high distance-based propagation losses that lead to sparse scattering multipath propagation. Furthermore, the use of mmWave transmitters and receivers with large tightly-packet antenna arrays results in high antenna correlation levels. These characteristics make most of the statistical channel models used in conventional sub-6 GHz \\gls{MIMO} research work inaccurate when dealing with mmWave scenarios. Thus, a modified version of the discrete-time narrowband clustered channel model proposed by Akdeniz \\emph{et al.} in \\cite{Akdeniz14} and further extended by Samimi and Rappaport in \\cite{Samimi14} will be used in this paper to capture the peculiarities of mmWave channels.\n\nThe link between the $m$th \\gls{AP} and the $k$th \\gls{MS} will be considered to be in one out of three possible conditions: outage, \\gls{LOS} or \\gls{NLOS} with probabilities:\n\\begin{subequations}\n\\begin{equation}\n   p_{out}(d_{mk})=\\max\\left(0,1-e^{-a_{out}d_{mk}+b_{out}}\\right),\n\\end{equation}\n\\begin{equation}\n   p_{\\gls{LOS}}(d_{mk})=\\left(1-p_{out}(d_{mk})\\right) e^{-a_{\\gls{LOS}}d_{mk}},\n\\end{equation}\n\\begin{equation}\n   p_{\\gls{NLOS}}(d_{mk})=1-p_{out}(d_{mk})-p_{\\gls{LOS}}(d_{mk}),\n\\end{equation}\n\\end{subequations}\nrespectively, where $d_{mk}$ is the distance (in meters) between the \\gls{AP} and the \\gls{MS}, and, according to \\cite[Table I]{Akdeniz14}, $1\/a_{out}=30$~m, $b_{out}=5.2$, and $1\/a_{\\gls{LOS}}=67.1$~m. Those links that are in outage will be characterized with infinite propagation losses, while for the links that are not in outage, the propagation losses will be characterized using a standard linear model with shadowing as\n\\begin{equation}\n   \\PL(d_{mk})[dB]=\\alpha+10 \\beta \\log_{10}(d_{mk})+\\chi_{mk},\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are the least square fits of floating intercept and slope and depend on the carrier frequency and on whether the link is in \\gls{LOS} or \\gls{NLOS} (see \\cite[Table I]{Akdeniz14}), and $\\chi_{mk}$ denotes the large-scale shadow fading component, which is modelled as a zero mean spatially correlated normal random variable with standard deviation $\\sigma_\\chi$ (again, see \\cite[Table I]{Akdeniz14} to obtain the typical values of $\\sigma_\\chi$ for \\gls{LOS} and \\gls{NLOS} links) whose spatial correlation model is described in \\cite[(54)-(55)]{Ngo17}.\n\nThe \\gls{UL} channel vector $\\bs{h}_{mk} \\in \\mathbb{C}^{N \\times 1}$ between \\gls{MS} $k$ and \\gls{AP} $m$ will be modelled as the sum of the contributions of $C_{mk}$ scattering clusters, each contributing $P_{mk}$ propagation paths as\n\\begin{equation}\n   \\bs{h}_{mk}=\\sum_{c=1}^{C_{mk}}\\sum_{p=1}^{P_{mk}} \\alpha_{mk,cp} \\bs{a}\\left(\\theta_{mk,cp},\\phi_{mk,cp}\\right),\n\\end{equation}\nwhere $\\alpha_{mk,cp}$ is the complex small-scale fading gain on the $p$th path of cluster $c$, and $\\bs{a}\\left(\\theta_{mk,cp},\\phi_{mk,cp}\\right)$ represents the \\gls{AP} normalized array response vector at the azimuth and elevation angles $\\theta_{mk,cp}$ and $\\phi_{mk,cp}$, respectively. These angles, as stated by Akdeniz \\emph{et al.} in \\cite[Section III.E]{Akdeniz14} can be generated as wrapped Gaussians around the cluster central angles with standard deviation given by the \\gls{rms} angular spreads for the cluster. The azimuth cluster central angles are uniformly distributed in the range $[-\\pi,\\pi]$ and the elevation cluster central angles are set to the \\gls{LOS} elevation angle. Moreover, the cluster \\gls{rms} angular spreads are exponentially distributed with a mean equal to $1\/\\lambda_{\\gls{rms}}$ that depends on the carrier frequency and on whether we are considering the azimuth or elevation directions (see \\cite[Table I]{Akdeniz14}). The number of clusters is distributed as a random variable of the form\n\\begin{equation}\n   C_{mk} \\sim \\max\\left\\{\\text{Poisson}(\\sigma_C),1\\right\\},\n\\end{equation}\nwhere $\\sigma_C$ is set to the empirical mean of $C_{mk}$. The small-scale fading gains are distributed as\n\\begin{equation}\n   \\alpha_{mk,cp} \\sim \\mathcal{CN}\\left(0,\\gamma_{mk,c}10^{-\\PL(d_{mk})\/10}\\right),\n\\end{equation}\nwhere the cluster $c$ is assumed to contribute with a fraction of power given by\n\\begin{equation}\n   \\gamma_{mk,c}=\\frac{N \\gamma'_{mk,c}}{P_{mk}\\sum_{j=1}^{C_{mk}} \\gamma'_{mk,j}},\n\\end{equation}\nwith\n\\begin{equation}\n   \\gamma'_{mk,j}=U_{mk,j}^{r_\\tau-1} 10^{Z_{mk,j}\/10},\n\\end{equation}\n$U_{mk,j} \\sim \\mathcal{U}[0,1]$, $Z_{mk,j} \\sim \\mathcal{N}(0,\\zeta^2)$, and the constants $r_\\tau$ and $\\zeta^2$ being treated as model parameters (see \\cite[Table I]{Akdeniz14}).\n\nAlthough the small-scale fading gains $\\alpha_{mk,cp}$ are assumed to be static throughout the coherence interval and then change independently (i.e., block fading), the spatial covariance matrices\n\\begin{equation}\n\\begin{split}\n   \\bs{R}_{mk}=&\\mathbb{E}\\left\\{\\bs{h}_{mk} \\bs{h}_{mk}^H \\right\\} \\\\\n              =&10^{-\\PL(d_{mk})\/10}\\sum_{c=1}^{C_{mk}}\\gamma_{mk,c} \\\\\n               &\\times \\sum_{p=1}^{P_{mk}} \\bs{a}\\left(\\theta_{mk,cp},\\phi_{mk,cp}\\right)\\bs{a}^H\\left(\\theta_{mk,cp},\\phi_{mk,cp}\\right),\n\\end{split}\n\\end{equation}\nare assumed to vary at a much smaller pace (i.e., ${\\tau_L}_c \\gg \\tau_c$).\n\n\\subsection{Large-scale training phase}\n\\label{subsec:large-scale-training}\n\n\\subsubsection{\\gls{RF} precoder\/combiner design}\n\nIn order to exploit the \\gls{UL}\/\\gls{DL} channel reciprocity using the \\gls{TDD} frame structure shown in Fig. \\ref{fig:TDD_frame}, it is assumed in this paper that the $N \\times L_A$ \\gls{RF} matrix $\\bs{W}_m^{RF}$, describing the effects of the active analog phase shifters at the $m$th \\gls{AP}, is common to the \\gls{DL} (\\gls{RF} precoding phase) and \\gls{UL} (\\gls{RF} combining phase). Furthermore, denoting by $\\mathcal{K}_m=\\left\\{\\kappa_{m 1}, \\ldots, \\kappa_{m L_A}\\right\\}$ the set of $L_A$ \\glspl{MS} beamformed by the $m$th \\gls{AP}, it is assumed that $\\bs{W}_m^{RF}$ is a function of only the spatial channel covariance matrices $\\left\\{\\bs{R}_{mk}\\right\\}_{k\\in\\mathcal{K}_m}$, known at the $m$th \\gls{AP} through spatial channel covariance estimation for hybrid analog-digital \\gls{MIMO} precoding architectures (see e.g. \\cite{Adhikary14,Mendez15,Park16,Park17}).\n\nUsing eigen-decomposition, the covariance matrix of the propagation channel linking \\gls{MS} k and \\gls{AP} $m$ can be expressed as $\\bs{R}_{mk}=\\bs{U}_{mk}\\bs{\\Lambda}_{mk}\\bs{U}_{mk}^H$, where $\\bs{\\Lambda}_{mk}=\\diag\\left(\\left[\\lambda_{mk,1}\\,\\ldots\\,\\lambda_{mk,r_{mk}}\\right]\\right)$ contains the $r_{mk}$ non-null eigenvalues of $\\bs{R}_{mk}$, and $\\bs{U}_{mk}$ is the $N \\times r_{mk}$ matrix of the corresponding eigenvectors. Hence, assuming the use of (constrained) statistical eigen beamforming \\cite{Park17TSC,Mai18}, the analog \\gls{RF} precoder\/combiner can be designed as\n\\begin{equation}\n\\begin{split}\n   \\bs{W}_m^{RF}=&\\begin{bmatrix}\n                    \\bs{w}_{m \\kappa_{m 1}}^{RF} & \\ldots & \\bs{w}_{m \\kappa_{m L_A}}^{RF}\n                 \\end{bmatrix}\\\\\n                =&\\begin{bmatrix}\n                    e^{-j\\angle\\bs{u}_{m \\kappa_{m 1},\\max}} & \\ldots & e^{-j\\angle\\bs{u}_{m \\kappa_{m L_A},\\max}}\n                 \\end{bmatrix},\n\\end{split}\n\\end{equation}\nwhere $\\bs{u}_{mk,\\max}$ is the dominant eigenvector of $\\bs{R}_{mk}$ associated to the maximum eigenvalue $\\lambda_{mk,\\max}$, and the function $\\angle\\bs{x}$ returns the phase angles, in radians, for each element of the complex vector $\\bs{x}$. Note that using the \\gls{RF} precoding\/combining matrix, the equivalent channel vector between \\gls{MS} k and \\gls{AP} $m$, including the \\gls{RF} precoding\/decoding matrix, is defined as\n\\begin{equation}\n   \\bs{g}_{mk} = {\\bs{W}_m^{RF}}^T \\bs{h}_{mk}\\ \\in\\ \\mathbb{C}^{L_A \\times 1},\n\\end{equation}\nwhose dimension is much less than the number of antennas of the massive \\gls{MIMO} array used at the $m$th \\gls{AP}, thus largely simplifying the small-scale training phase.\n\n\\subsubsection{Selection of \\glspl{MS} to beamform from each \\gls{AP}}\n\nAs previously stated, in those highly probable cases in which the number of active \\glspl{MS} in the network is greater than the number of available \\gls{RF} chains at each \\gls{AP} (i.e., $K > L$), the $m$th \\gls{AP}, with $m\\in\\{1,\\ldots,M\\}$, can only beamform to a group of $L$ out of the $K$ \\glspl{MS} in the network, which are indexed by the set $\\mathcal{K}_m=\\left\\{\\kappa_{m1},\\ldots,\\kappa_{mL}\\right\\}$. As the \\gls{RF} beamforming matrices at the \\glspl{AP} are a function of only the large-scale spatial channel covariance matrices and are common to both the \\gls{UL} and the \\gls{DL}, the selection of the sets of \\glspl{MS} to beamform from each \\gls{AP} must also be based only on the available large-scale \\gls{CSI}. Inspired by the Frobenius norm-based suboptimal user selection algorithm proposed by Shen \\emph{et al.} in \\cite{Shen06}, a selection algorithm is proposed that aims at maximizing the sum of the average energy (i.e., average Frobenius norm) of the equivalent channels (including the corresponding beamformer) between the $M$ \\glspl{AP} and the $K$ \\glspl{MS} with the constraints that, first, the minimum average energy of the equivalent channel between the $M$ \\glspl{AP} and any of the active \\glspl{MS} must be maximized and, second, that each \\gls{AP} can only beamform to $L$ \\glspl{MS}. Note that this optimization problem, which tends to provide some degree of (average) max-min fairness among \\glspl{MS}, can be efficiently solved by using an iterative reverse-delete algorithm (similar to that used in graph theory to obtain a minimum spanning tree from a given connected, edge-weighted graph). In particular, at the beginning of the $i$th iteration of the algorithm the cell-free network is represented by a very simple edge-weighted directed graph with $M$ source nodes and $K$ sink nodes, where the $m$th source node, representing the $m$th \\gls{AP}, is connected to a group $\\mathcal{K}_m^{(i)}$ of sink nodes, representing the active \\glspl{MS} beamformed by the $m$th \\gls{AP}. The connection (edge) between the $m$th source node and the $l$th sink node in $\\mathcal{K}_m^{(i)}$ is weighted by the average Frobenius norm of the equivalent channel linking the $m$th \\gls{AP} and \\gls{MS} $l\\in \\mathcal{K}_m^{(i)}$, that can be obtained as\n\\begin{equation}\n   \\xi_{ml}=\\mathbb{E}\\left\\{\\left\\|{\\bs{w}_{ml}^{RF}}^T\\bs{h}_{ml}\\right\\|_F^2\\right\\}={\\bs{w}_{ml}^{RF}}^T \\bs{R}_{ml} \\bs{w}_{ml}^{RF}.\n\\end{equation}\nThe average sum energy of the equivalent channels between the $M$ \\glspl{AP} and \\gls{MS} $k$ at the beginning of the $i$th iteration is\n\\begin{equation}\n   \\mathcal{E}_k^{(i)} = \\sum_{m\\in\\mathcal{M}_k^{(i)}} \\xi_{mk},\n\\end{equation}\nwhere $\\mathcal{M}_k^{(i)}$ is the set of \\glspl{AP} beamforming to \\gls{MS} $k$ at the beginning of the $i$th iteration. During this iteration, the reverse-delete algorithm removes the edge (i.e., the \\gls{RF} chain and associated beamformer) that, first, goes out of one of those \\glspl{AP} still beamforming to more than $L$ \\glspl{MS} and, second, has the minimum weight maximizing the minimum average sum energy after removal. The algorithm begins with a fully connected graph and stops when all \\glspl{AP} beamform to exactly $L$ \\glspl{MS}. Hence, note that $M(K-L)$ iterations are needed to select the sets $\\mathcal{K}_m$ for $m\\in\\{1,\\ldots,M\\}$.\n\n\\subsection{Small-scale training phase}\n\\label{subsec:small-scale-training}\nCommunication in any coherence interval of a \\gls{TDD}-based massive \\gls{MIMO} system invariably starts with the \\glspl{MS} sending the pilot sequences to allow the channel to be estimated at the \\glspl{AP}. Let $\\tau_p$ denote the \\gls{UL} training phase duration (measured in samples on a time-frequency grid) per coherence interval. During the \\gls{UL} training phase, all $K$ \\glspl{MS} simultaneously transmit pilot sequences of $\\tau_p$ samples to the \\glspl{AP} and thus, the $L_A \\times \\tau_p$ received \\gls{UL} signal matrix at the $m$th \\gls{AP} is given by\n\\begin{equation}\n   {\\bs{Y}_p}_m=\\sqrt{\\tau_p P_p}\\sum_{k'=1}^K \\bs{g}_{mk'} \\bs{\\varphi}_{k'}^T+{\\bs{N}_p}_m,\n\\end{equation}\nwhere $P_p$ is the transmit power of each pilot symbol, $\\bs{\\varphi}_k$ denotes the $\\tau_p\\times 1$ training sequence assigned to \\gls{MS} $k$, with $\\|\\bs{\\varphi}_k\\|_F^2=1$, and ${\\bs{N}_p}_m$ is an $L_A\\times\\tau_p$ matrix of i.i.d. additive noise samples with each entry distributed as\\footnote{Note that in the \\gls{UL} of a fully-connected hybrid beamforming architecture each reception chain is composed of $N$ antenna elements, each connected to a low-noise amplifier (LNA) characterized by a power gain $G_{\\textrm{LNA}}$ and a noise temperature $T_{\\text{LNA}}$. Each of the $N$ LNAs feeds an analog passive phase shifter characterized by an insertion loss $L_{\\text{PS}}$. The outputs of the $N$ phase shifters are introduced to a power combiner whose insertion losses are typically proportional to the number of inputs, that is, $L_{\\text{PC}}=N L_{\\text{PC}_{in}}$. Finally, the output of the power combiner is introduced to an $\\gls{RF}$ chain characterized by a power gain $G_{\\text{RF}}$ and a noise temperature $T_{\\text{RF}}$. Thus, the equivalent noise temperature of each receive chain can be obtained as $T_u=N \\left(T_0+T_{\\text{LNA}}+\\frac{T_0(L_{\\text{PS}}L_{\\text{PC}_{in}}-1)}{G_{\\text{LNA}}}+\\frac{T_{\\text{RF}}L_{\\text{PS}}L_{\\text{PC}_{in}}}{G_{\\text{LNA}}}\\right)$.}  $\\mathcal{CN}(0,\\sigma_u^2(N))$. Ideally, training sequences should be chosen to be mutually orthogonal, however, since in most practical scenarios it holds that $K>\\tau_p$, a given training sequence is assigned to more than one \\gls{MS}, thus resulting in the so-called pilot contamination, a widely studied phenomenon in the context of collocated massive \\gls{MIMO} systems \\cite{Elijah16}.\n\n\\subsection{Channel estimation}\n\\label{channel_estimation}\n\nChannel estimation is known to play a central role in the performance of massive \\gls{MIMO} schemes \\cite{Lu14} and also in the specific context of cell-free architectures \\cite{Ngo17}.\nThe \\gls{MMSE} estimation filter for the channel between the $k$th active \\gls{MS} and the $m$th \\gls{AP} can be calculated as\n\\begin{equation}\n\\begin{split}\n   \\bs{D}_{mk}&=\\arg\\min_{\\bs{D}}\\mathbb{E}\\left\\{\\left\\|\\bs{g}_{mk}-\\bs{D} {\\bs{Y}_p}_m \\bs{\\varphi}_k^*\\right\\|^2\\right\\} \\\\\n                          &= \\sqrt{\\tau_p P_p} \\bs{R}_{mk}^{RF} \\bs{Q}_{mk}^{-1},\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\n   \\bs{R}_{mk}^{RF} = \\mathbb{E}\\left\\{\\bs{g}_{mk}  \\bs{g}_{mk}^H\\right\\}={\\bs{W}_m^{RF}}^T  \\bs{R}_{mk} {\\bs{W}_m^{RF}}^*,\n\\end{equation}\nand\n\\begin{equation}\n   \\bs{Q}_{mk}=\\tau_p P_p\\sum_{k'=1}^K \\bs{R}_{mk'}^{RF} \\left|\\bs{\\varphi}_{k'}^T \\bs{\\varphi}_k^*\\right|^2 + \\sigma_u^2(N) \\bs{I}_{L_A}.\n\\end{equation}\nHence, the corresponding estimated channel vector can be expressed as\n\\begin{equation}\n   \\hat{\\bs{g}}_{mk}=\\bs{D}_{mk} {\\bs{Y}_p}_m \\bs{\\varphi}_k^* = \\sqrt{\\tau_p P_p} \\bs{R}_{mk}^{RF} \\bs{Q}_{mk}^{-1} {\\bs{Y}_p}_m \\bs{\\varphi}_k^*.\n\\end{equation}\nThe \\gls{MMSE} channel vector estimates can be shown to be distributed as ${\\hat{\\bs{g}}}_{mk} \\sim \\mathcal{CN}\\left(\\bs{0},{\\hat{\\bs{R}}}_{mk}^{RF}\\right)$, where\n\\begin{equation}\n   \\hat{\\bs{R}}_{mk}^{RF}\\triangleq \\tau_p P_p \\bs{R}_{mk}^{RF}\\bs{Q}_{mk}^{-1}{\\bs{R}_{mk}^{RF}}^H.\n\\end{equation}\nFurthermore, the channel vector $\\bs{g}_{mk}$ can be decomposed as $\\bs{g}_{mk}=\\hat{\\bs{g}}_{mk}+\\tilde{\\bs{g}}_{mk}$, where $\\tilde{\\bs{g}}_{mk}$ is the \\gls{MMSE} channel estimation error, which is statistically independent of both $\\bs{g}_{mk}$ and $\\hat{\\bs{g}}_{mk}$.\n\n\\subsection{Downlink payload data transmission}\n\nLet us define $\\bs{s}_d=\\left[{s_d}_1 \\ldots {s_d}_K\\right]^T$ as the $K \\times 1$ vector of symbols jointly (cooperatively) transmitted from the \\glspl{AP} to the \\glspl{MS}, such that $E\\left\\{\\bs{s}_d\\bs{s}_d^H\\right\\}=\\bs{I}_K$. Let us also define $\\bs{x}_m=\\mathcal{P}_m\\left(\\bs{s}_d\\right)$ as the $N \\times 1$ vector of signals transmitted from the $m$th \\gls{AP}, where $\\mathcal{P}_m\\left(\\bs{s}_d\\right)$ is used to denote the mathematical operations (linear and\/or non-linear) used to obtain $\\bs{x}_m$ from $\\bs{s}_d$. Note that this vector must comply with a power constraint $\\mathbb{E}\\left\\{\\left\\|\\bs{x}_m\\right\\|_F^2\\right\\}\\leq \\overline{P}_m$, where $\\overline{P}_m$ is the maximum average transmit power available at \\gls{AP} $m$. Using this notation, the signal received by \\gls{MS} $k$ can be expressed as\n\\begin{equation}\n   {y_d}_k=\\sum_{m=1}^M \\bs{h}_{mk}^T \\bs{x}_m + {n_d}_k,\n\\label{eq:yk}\n\\end{equation}\nwhere ${n_d}_k \\sim \\mathcal{CN}(0,\\sigma_d^2)$ is the Gaussian noise sample at \\gls{MS} $k$. The vector $\\bs{y}_d = \\left[{y_d}_1\\, \\ldots\\, {y_d}_K\\right]^T$ containing the signals received by the $K$ scheduled \\glspl{MS} in the network can then be expressed as\n\\begin{equation}\n   \\bs{y}_d=\\sum_{m=1}^M \\bs{H}_m^T \\bs{x}_m + \\bs{n}_d,\n\\end{equation}\nwhere $\\bs{H}_m=\\left[\\bs{h}_{m1}\\, \\ldots\\, \\bs{h}_{mK}\\right]$ and $\\bs{n}_d=\\left[{n_d}_1\\, \\ldots\\, {n_d}_K\\right]^T$.\n\nThe mathematical operations that symbol vector $\\bs{s}_d$ undergoes before being transmitted, generically represented as $\\bs{x}_m = \\mathcal{P}_m(\\bs{s}_d)$, for all $m \\in \\{1,\\ldots,M\\}$, include, first, a baseband precoding task at the \\gls{CPU}, second, a compressing process of all or part of the data that must be sent from the \\gls{CPU} to the \\glspl{AP} through the fronthaul links and, third, an \\gls{RF} precoding task at each of the \\glspl{AP}. Let us denote by ${\\mathcal{Q}_d}_m(\\bs{x})$ and ${\\mathcal{Q}_d}_m^{-1}(\\bs{x})$ the quantization and unquantization mathematical operations performed by the \\gls{CAP}-based \\gls{CPU}-\\gls{AP} functional split on a vector of signal samples $\\bs{x}$ to be transmitted by the $m$th \\gls{AP}. Due to the distortion introduced by the quantization\/unquantization processes, we have that \\cite{Zamir96,Femenias18}\n\\begin{equation}\n   \\hat{\\mathcal{Q}}_{dm}(\\bs{x})\\triangleq {\\mathcal{Q}_d}_m^{-1}({\\mathcal{Q}_d}_m(\\bs{x}))=\\bs{x}+{\\bs{q}_d}_m,\n\\end{equation}\nwhere ${\\bs{q}_d}_m$ is the quantization noise vector, which is assumed to be statistically distributed as ${\\bs{q}_d}_m \\sim \\mathcal{CN}\\left(\\bs{0},\\sigma_{q_{dm}}^2 \\bs{I}\\right)$. As shown by Zamir \\emph{et al.} in \\cite{Zamir96}, this assumption is supported by the fact that large-block lattice quantization codes are able to approximate a Gaussian quantization noise distribution. Thus, the mathematical operations describing the \\gls{CPU}-\\gls{AP} functional split considered in this paper can be summarized as\n\\begin{equation}\n\\begin{split}\n   \\bs{x}_m=\\mathcal{P}_m(\\bs{s}_d)&=\\bs{W}_m^{RF} \\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB} \\bs{\\Upsilon}^{1\/2} \\bs{s}_d\\right) \\\\\n                                   &=\\bs{W}_m^{RF} \\left(\\bs{W}_{d\\,m}^{BB} \\bs{\\Upsilon}^{1\/2} \\bs{s}_d + {\\bs{q}_d}_m\\right),\n\\end{split}\n\\label{eq:bsxm}\n\\end{equation}\nwhere $\\bs{W}_d^{BB} = \\left[{\\bs{W}_{d\\,1}^{BB}}^T\\ \\ldots\\ {\\bs{W}_{d\\,M}^{BB}}^T\\right]^T\\ \\in\\ \\mathbb{C}^{M L_A \\times K}$, with $\\bs{W}_{d\\,m}^{BB}=\\left[\\bs{w}_{dm1}^{BB}\\ \\ldots\\ \\bs{w}_{dmK}^{BB}\\right]\\ \\in\\ \\mathbb{C}^{L_A \\times K}$ denoting the baseband precoding matrix affecting the signal transmitted by the $m$th \\gls{AP}, and $\\bs{\\Upsilon}=\\diag\\left([\\upsilon_1\\,\\ldots\\,\\upsilon_K]\\right)$ is a $K \\times K$ diagonal matrix containing the power control coefficients in its main diagonal, which are chosen to satisfy the following necessary power constraint at the $m$th \\gls{AP}\n\\begin{equation}\n\\begin{split}\n   \\mathbb{E}\\left\\{\\left\\|\\bs{x}_m\\right\\|_F^2\\right\\} &= \\sum_{k=1}^K \\upsilon_k \\theta_{mk}^{BB\/RF}+{\\sigma_q^2}_{dm} \\left\\|\\bs{W}_m^{RF}\\right\\|_F^2 \\\\\n                                                        &= \\sum_{k=1}^K \\upsilon_k \\theta_{mk}^{BB\/RF} + {\\sigma_q^2}_{dm} L_A N \\leq \\overline{P}_m,\n\\end{split}\n\\label{eq:Power_constraint}\n\\end{equation}\nwhere we have used the definition\n\\begin{equation}\n   \\theta_{mk}^{BB\/RF}=\\mathbb{E}\\left\\{\\left\\|\\bs{W}_m^{RF} \\bs{w}_{dmk}^{BB}\\right\\|_F^2\\right\\}.\n\\end{equation}\n\nUsing the proposed hybrid \\gls{CAP} approach, the signal received by the $K$ \\glspl{MS} can be rewritten as\n\\begin{equation}\n\\begin{split}\n   \\bs{y}_d=&\\sum_{m=1}^M \\bs{H}_m^T \\bs{W}_m^{RF} \\bs{W}_{d\\,m}^{BB} \\bs{\\Upsilon}^{1\/2} \\bs{s}_d \\\\\n            &+ \\sum_{m=1}^M \\bs{H}_m^T \\bs{W}_m^{RF} {\\bs{q}_d}_m + \\bs{n}_d \\\\\n           =&\\ \\bs{G}^T \\bs{W}_d^{BB}\\bs{\\Upsilon}^{1\/2} \\bs{s}_d + \\bs{\\eta}_d,\n\\end{split}\n\\label{eq:yd}\n\\end{equation}\nwhere $\\bs{G}=[\\bs{G}_1^T\\,\\ldots\\,\\bs{G}_M^T]^T$, with $\\bs{G}_m={\\bs{W}_m^{RF}}^T \\bs{H}_m$, representing the equivalent \\gls{MIMO} channel matrix between\nthe $K$ \\glspl{MS} and the $M$ \\glspl{AP}, including the RF precoding\/decoding matrices, and\n\\begin{equation}\n   \\bs{\\eta}_d  =\\bs{G}^T \\bs{q}_d + \\bs{n}_d,\n\\end{equation}\nwith $\\bs{q}_d=[{\\bs{q}_d}_1^T \\ldots {\\bs{q}_d}_M^T]^T$, includes the thermal noise as well as the quantization noise samples received from all the \\glspl{AP} in the network. Now, using the classical \\gls{ZF} \\gls{MU-MIMO} baseband precoder to harness the spatial multiplexing, we have that\n\\begin{equation}\n   \\bs{W}_d^{BB}=\\hat{\\bs{G}}^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1}\n\\end{equation}\nor, equivalently,\n\\begin{equation}\n   \\bs{W}_{d\\,m}^{BB} = \\hat{\\bs{G}}_m^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1}\\ \\forall m,\n\\end{equation}\nwhere we have assumed that $\\bs{G}=\\hat{\\bs{G}}+\\tilde{\\bs{G}}$ and $\\bs{G}_m=\\hat{\\bs{G}}_m+\\tilde{\\bs{G}}_m$. Consequently, the signal received by the $k$th \\gls{MS} can be expressed as\n\\begin{equation}\n\\begin{split}\n   {y_d}_k = &\\bs{g}_k^T \\hat{\\bs{G}}^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1} \\bs{\\Upsilon}^{1\/2}\\bs{s}_d + {\\eta_d}_k \\\\\n           = &\\left(\\hat{\\bs{g}}_k^T + \\tilde{\\bs{g}}_k^T\\right) \\hat{\\bs{G}}^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1} \\bs{\\Upsilon}^{1\/2}\\bs{s}_d + {\\eta_d}_k \\\\\n           = &\\sqrt{\\upsilon_k} {s_d}_k + \\tilde{\\bs{g}}_k^T \\hat{\\bs{G}}^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1} \\bs{\\Upsilon}^{1\/2}\\bs{s}_d + {\\eta_d}_k\n\\end{split}\n\\label{eq:ydk}\n\\end{equation}\nwhere ${\\eta_d}_k = \\bs{g}_k^T \\bs{q}_d + {n_d}_k$. The first term denotes the useful received signal, the second term contains the interference terms due to the use of imperfect \\gls{CSI} (pilot contamination), and the third term encompass both the quantification and thermal noise samples.\n\n\\subsection{Uplink payload data transmission}\n\nIn the \\gls{UL}, the vector of received signals at the output of the $L_A$ \\gls{RF} chains (including the \\gls{RF} phase shifters) of the $m$th \\gls{AP} is given by\n\\begin{equation}\n\\begin{split}\n   {\\bs{r}_u}_m=&\\sqrt{P_u}\\sum_{k'=1}^K \\bs{g}_{mk'} \\sqrt{\\omega_{k'}} {s_u}_{k'} + {\\bs{n}_u}_m \\\\\n               =&\\sqrt{P_u}\\bs{G}_m \\bs{\\Omega}^{1\/2} \\bs{s}_u + {\\bs{n}_u}_m,\n\\end{split}\n\\end{equation}\nwhere $P_u$ is the maximum average \\gls{UL} transmit power available at any of the active \\glspl{MS}, $\\bs{s}_u=[{s_u}_1\\,\\ldots\\,{s_u}_K]^T$ denotes the vector of symbols transmitted by the $K$ active \\gls{MS}, $\\bs{\\Omega}=\\diag([\\omega_1\\,\\ldots\\,\\omega_K])$, with $0 \\leq \\omega_k \\leq 1$, is a matrix containing the power control coefficients used at the \\glspl{MS}, and ${\\bs{n}_u}_m \\sim \\mathcal{CN}(\\bs{0},\\sigma_u^2(N) \\bs{I}_{L_A})$ is the vector of additive thermal noise samples at the output of the $L_A$ \\gls{RF} chains of the $m$th \\gls{AP}. The received vector of signals at each of the \\glspl{AP} in the network is quantized and forwarded to the \\gls{CPU} via the \\gls{UL} fronthaul links, where they are unquantized and jointly processed using a set of baseband combining vectors. Using a similar approach to that employed to model the \\gls{DL} transmission, the received vector of (unquantized) samples from the $m$th \\gls{AP} can be expressed as\n\\begin{equation}\n   {\\bs{z}_u}_m=\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right)={\\bs{r}_u}_m + {\\bs{q}_u}_m,\n\\end{equation}\nwhere ${\\bs{q}_u}_m$ is the quantization noise vector, which is assumed to be statistically distributed as ${\\bs{q}_u}_m \\sim \\mathcal{CN}\\left(\\bs{0},\\sigma_{q_{u m}}^2 \\bs{I}_{L_A}\\right)$. Now, assuming the use of \\gls{ZF} \\gls{MIMO} detection, the \\gls{CPU} uses the detection matrix\n\\begin{equation}\n   \\bs{W}_u^{BB}=\\left(\\hat{\\bs{G}}^H \\hat{\\bs{G}}\\right)^{-1}\\hat{\\bs{G}}^H = {\\bs{W}_d^{BB}}^T\n\\end{equation}\nor, equivalently\n\\begin{equation}\n   \\bs{W}_{u m}^{BB}=\\left(\\hat{\\bs{G}}^H \\hat{\\bs{G}}\\right)^{-1}\\hat{\\bs{G}}_m^H = {\\bs{W}_{d m}^{BB}}^T,\\ \\forall m,\n\\end{equation}\nto jointly process the vector $\\bs{z}_u=\\left[{\\bs{z}_u}_1^T\\,\\ldots\\,{\\bs{z}_u}_M^T\\right]^T$ and obtain the vector of detected samples\n\\begin{equation}\n\\begin{split}\n   \\bs{y}_u=&\\bs{W}_u^{BB}\\bs{z}_u=\\sqrt{P_u}\\bs{W}_u^{BB} \\bs{G} \\bs{\\Omega}^{1\/2} \\bs{s}_u + \\bs{\\eta}_u \\\\\n           =&\\sqrt{P_u} \\bs{\\Omega}^{1\/2} \\bs{s}_u + \\sqrt{P_u}\\bs{W}_u^{BB} \\tilde{\\bs{G}} \\bs{\\Omega}^{1\/2} \\bs{s}_u + \\bs{\\eta}_u,\n\\end{split}\n\\end{equation}\nwhere $\\bs{\\eta}_u=\\bs{W}_u^{BB}\\left(\\bs{q}_u + \\bs{n}_u\\right)$. Again, the first term denotes the useful received signal, the second term contains the interference terms due to the use of imperfect \\gls{CSI}, and the third term includes both the quantification and thermal noise samples. The detected sample corresponding to the symbol transmitted by the $k$th \\gls{MS} can then be obtained as\n\\begin{equation}\n   {y_u}_k =\\sqrt{P_u} \\omega_k^{1\/2} {s_u}_k + \\sqrt{P_u}\\left[\\bs{W}_u^{BB} \\tilde{\\bs{G}} \\bs{\\Omega}^{1\/2} \\bs{s}_u\\right]_k + {\\eta_u}_k,\n\\label{eq:yuk}\n\\end{equation}\nwhere $[\\bs{x}]_k$ denotes the $k$th entry of vector $\\bs{x}$.\n\n\\section{Achievable rates}\n\\label{sec:Achievable_rates}\n\nAnalysis techniques similar to those applied, for instance, in \\cite{Hassibi03,Yang13,Interdonato16,Marzetta16,Ngo17,Nayebi17}, are used in this section to derive \\gls{DL} and \\gls{UL} achievable rates. In particular, the sum of the second and third terms on the \\gls{RHS} of \\eqref{eq:ydk}, for the \\gls{DL} case, and \\eqref{eq:yuk}, for the \\gls{UL} case, are treated as \\emph{effective noise}. The additive terms constituting the \\emph{effective noise} are, in both \\gls{DL} and \\gls{UL} cases, mutually uncorrelated, and uncorrelated with ${s_d}_k$ and ${s_u}_k$, respectively. Therefore, both the desired signal and the so-called \\emph{effective noise} are uncorrelated. Now, recalling the fact that uncorrelated Gaussian noise represents the worst case, from a capacity point of view, and that the complex-valued fast fading random variables characterizing the propagation channels between different pairs of \\gls{AP}-\\gls{MS} connections are independent, the \\gls{DL} and \\gls{UL} achievable rates (measured in bits per second per Hertz) for \\gls{MS} $k$ can be obtained as stated in the following theorems:\n\\begin{theorem}[Downlink achievable rate] An achievable rate of \\gls{MS} k using the analog precoders $\\bs{W}_m^{RF}$, for all $m\\in\\{1,\\ldots,M\\}$, and the \\gls{ZF} baseband precoder $\\bs{W}_d^{BB} = \\hat{\\bs{G}}^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1}$ is ${R_d}_k=\\log_2\\left(1+{\\SINR_d}_k\\right)$, with\n\\begin{equation}\n   {\\SINR_d}_k=\\frac{\\upsilon_k}{\\sum_{k'=1}^K \\upsilon_{k'} \\varpi_{kk'} + \\sigma_{\\eta_{dk}}^2},\n   \\label{eq:SINRdk}\n\\end{equation}\nwhere\n\\begin{equation}\n   \\sigma_{\\eta_{dk}}^2=\\sum_{m=1}^M {\\sigma_q^2}_{dm} \\tr\\left(\\bs{R}_{mk}^{RF}\\right) + \\sigma_d^2,\n\\end{equation}\nand\n\\begin{equation}\n   \\varpi_{kk'}=\\left[\\diag\\left(\\mathbb{E}\\left\\{{\\bs{W}_d^{BB}}^H \\tilde{\\bs{g}}_k^* \\tilde{\\bs{g}}_k^T\\bs{W}_d^{BB}\\right\\}\\right)\\right]_{k'}.\n\\end{equation}\n\\label{theo:theorem_DL}\n\\end{theorem}\n\n\\begin{proof}\nSee Appendix \\ref{app:Appendix_1}.\n\\end{proof}\n\n\\begin{theorem}[Uplink achievable rate] An achievable \\gls{UL} rate for the $k$th \\gls{MS} in the Cell-Free Massive \\gls{MIMO} system with limited capacity fronthaul links and using \\gls{ZF} \\gls{MIMO} detection, for any $M$, $N$ and $K$, is given by ${R_u}_k=\\log_2\\left(1+{\\SINR_u}_k\\right)$, with\n\\begin{equation}\n   {\\SINR_u}_k=\\frac{P_u \\omega_k}{P_u \\sum_{k'=1}^K \\omega_{k'} \\delta_{kk'} + \\sigma_{\\eta_{uk}}^2},\n   \\label{eq:SINRuk}\n\\end{equation}\nwhere\n\\begin{equation}\n   \\delta_{kk'}=\\left[\\diag\\left(\\mathbb{E}\\left\\{\\tilde{\\bs{G}}^H{\\bs{w}_{uk}^{BB}}^H \\bs{w}_{uk}^{BB} \\tilde{\\bs{G}}\\right\\}\\right)\\right]_{k'}\n\\end{equation}\nwith $\\bs{w}_{uk}^{BB}$ denoting the $k$th row of $\\bs{W}_u^{BB}$, or, equivalently,\n\\begin{equation}\n   \\delta_{kk'}=\\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_u^{BB} \\tilde{\\bs{g}}_{k'} \\tilde{\\bs{g}}_{k'}^H{\\bs{W}_u^{BB}}^H\\right\\}\\right)\\right]_k,\n\\end{equation}\nand\n\\begin{equation}\n   \\sigma_{\\eta_{uk}}^2=\\sum_{m=1}^M\\left({\\sigma_q^2}_{um} +\\sigma_u^2(N)\\right){\\nu_u}_{mk},\n\\end{equation}\nwith\n\\begin{equation}\n   {\\nu_u}_{mk}=\\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_{u\\,m}^{BB} {\\bs{W}_{u\\,m}^{BB}}^H\\right\\}\\right)\\right]_k.\n\\end{equation}\n\\label{theo:theorem_UL}\n\\end{theorem}\n\n\\begin{proof}\nSee Appendix \\ref{app:Appendix_2}.\n\\end{proof}\n\n\\section{Fronthaul capacity consumption}\n\\label{sec:fronthaul_capacity}\n\nThe \\gls{DL} quantization process performed at the $m$th \\gls{AP} can be expressed as\n\\begin{equation}\n   \\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)=\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d + {\\bs{q}_d}_m.\n\\end{equation}\nFrom standard random coding arguments \\cite{Cover06}, vector $\\bs{s}_d$ can be safely assumed to be distributed as $\\bs{s}_d\\sim\\mathcal{CN}(0,\\bs{I}_K)$ and thus, the quantized vector $\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)$ is distributed as $\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)\\sim\\mathcal{CN}\\left(\\bs{0},\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon} {\\bs{W}_{d\\,m}^{BB}}^H+{\\sigma_q^2}_{dm}\\bs{I}_{L_A}\\right)$. Furthermore, as the differential entropy of a vector $\\bs{x}\\sim\\mathcal{CN}(\\bs{\\omega},\\bs{\\Theta})$ is given by $\\mathcal{H}(\\bs{x})=\\log\\det(\\pi e \\bs{\\Theta})$ \\cite{Cover06}, the required average rate to transfer the quantized vector $\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)$ on the corresponding \\gls{DL} fronthaul link can be obtained as (in bps\/Hz)\n\\begin{equation}\n\\begin{split}\n   \\hat{C}_{dm} &= \\mathbb{E}\\left\\{I\\left(\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right);\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)\\right\\} \\\\\n                &= \\mathbb{E}\\left\\{\\mathcal{H}\\left(\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)\\right)\\right\\} \\\\\n                &\\quad-\\mathbb{E}\\left\\{\\mathcal{H}\\left(\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)\\bigr|\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right) \\right\\} \\\\\n                &= \\mathbb{E}\\left\\{\\log_2\\det\\left(\\frac{1}{{\\sigma_q^2}_{dm}} \\bs{W}_{d\\,m}^{BB} \\bs{\\Upsilon} {\\bs{W}_{d\\,m}^{BB}}^H + \\bs{I}_{L_A}\\right)\\right\\},\n\\end{split}\n\\end{equation}\nwhere $I(\\hat{\\bs{x}};\\bs{x})$ is used to denote the mutual information between vectors $\\hat{\\bs{x}}$ and $\\bs{x}$, and $\\mathcal{H}(\\hat{\\bs{x}}|\\bs{x})$ is the differential entropy of $\\hat{\\bs{x}}$ conditioned on $\\bs{x}$. Since the determinant is a log-concave function on the set of positive semidefinite matrices, it follows from Jensen's inequality that\n\\begin{equation}\n\\begin{split}\n   \\hat{C}_{dm} &\\leq \\log_2\\det\\left(\\frac{1}{{\\sigma_q^2}_{dm}} \\mathbb{E}\\left\\{\\bs{W}_{d\\,m}^{BB} \\bs{\\Upsilon} {\\bs{W}_{d\\,m}^{BB}}^H\\right\\} + \\bs{I}_{L_A}\\right) \\\\\n                &= \\log_2\\det\\left(\\frac{1}{{\\sigma_q^2}_{dm}} \\sum_{k=1}^K \\upsilon_k \\bs{R}_{mk}^{BB} + \\bs{I}_{L_A}\\right),\n\\end{split}\n\\end{equation}\nwhere $\\bs{R}_{mk}^{BB} = \\mathbb{E}\\left\\{\\bs{w}_{mk}^{BB} {\\bs{w}_{mk}^{BB}}^H\\right\\}$.\n\nAnalogously, the \\gls{UL} quantization process performed at the $m$th \\gls{AP} is given by $\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right)={\\bs{r}_u}_m + {\\bs{q}_u}_m$. Thus, using arguments similar to those used in the \\gls{DL} case, the required average rate to transfer the quantized vector $\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right)$ on the corresponding \\gls{UL} fronthaul link can be upper bounded as (in bps\/Hz)\n\\begin{equation}\n\\begin{split}\n   &\\hat{C}_{um} = \\mathbb{E}\\left\\{I\\left(\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right);{\\bs{r}_u}_m\\right)\\right\\} \\\\\n   &\\ = \\mathbb{E}\\left\\{\\mathcal{H}\\left(\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right)\\right)\\right\\}-\\mathbb{E}\\left\\{\\mathcal{H}\\left(\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right)\\bigr|{\\bs{r}_u}_m\\right) \\right\\} \\\\\n   &\\ \\leq \\log_2\\det\\left(\\frac{P_u}{{\\sigma_q^2}_{um}} \\sum_{k=1}^K \\omega_k \\bs{R}_{mk}^{RF} + \\left(\\frac{\\sigma_u^2(N)}{{\\sigma_q^2}_{um}} +1\\right)\\bs{I}_{L_A}\\right).\n\\end{split}\n\\end{equation}\n\n\n\n\\section{Pilot assignment}\n\\label{sec:pilot_assignment}\n\nTo warrant an appropriate system performance, the \\gls{RRM} unit must efficiently manage both the pilot assignment and the \\gls{UL} and \\gls{DL} power control. As the pilots are not power controlled, pilot assignment and power control can be conducted independently. Since the length of the pilot sequences is limited to $\\tau_p$, there only exist $\\tau_p$ orthogonal pilot sequences. In a network with $K\\leq \\tau_p$ \\glspl{MS}, an optimal pilot assignment strategy simply allocates $K$ orthogonal pilots to the $K$ \\glspl{MS}. The real pilot assignment problem arises when $K > \\tau_p$. In this case, fully orthogonal pilot assignment is no longer possible and hence, other pilot assignment strategies must be devised.\n\nOn the one hand, designing an optimal pilot assignment strategy aiming at maximizing the minimum rate allocated to the active \\glspl{MS} in the network is a very difficult combinatorial problem, computationally unmanageable in most network setups of practical interest \\cite{Ngo17}. On the other hand, using straightforward strategies such as, for instance, the pure \\gls{RPA} scheme \\cite{Ahmadi16}, where each \\gls{MS} is randomly assigned one pilot sequence out of the set of $\\tau_p$ orthogonal pilot sequences, or the \\gls{BRPA} scheme, where each \\gls{MS} is allocated a pilot sequence that is sequentially and cyclically selected from the ordered set of available orthogonal pilots, provides poor performance results. In order to avoid the computational complexity of the optimal strategies while improving the performance of the baseline \\gls{RPA} or \\gls{BRPA} approaches, a suboptimal solution is proposed in this paper that is based on the idea of \\emph{clustering by dissimilarity}. This suboptimal approach, that will be termed as the \\gls{DCPA} strategy, is motivated by the following key observation:\n\n\\textbf{Key observation}: \\emph{In those scenarios where $K > \\tau_p$, cell-free communication is severely impaired whenever \\glspl{MS} showing very similar large-scale propagation patterns to the set of \\glspl{AP} (that is, \\glspl{MS} typically located nearby) are allocated the same pilot sequence. In this case, the inter-\\gls{MS} interference leads to very poor channel estimates at all \\glspl{AP} and, eventually, to low \\glspl{SINR}.}\n\nThe clustering algorithm proposed in this work basically ensures that pilot sequences are only reused by \\glspl{MS} showing \\emph{dissimilar} large-scale propagation patterns to the \\glspl{AP} (that is, \\glspl{MS} typically located sufficiently apart). Two key aspects regarding the clustering operation are thus, on the one hand, to decide which should be the large-scale propagation pattern that ought to be used to represent a given \\gls{MS} and, on the other hand, to decide what metric should be used to measure \\emph{similarity} among the large-scale propagation patterns characterizing different \\glspl{MS}. To this end, and resting upon the premise that the \\gls{CPU} has perfect knowledge of the large-scale gains, let $\\bs{\\xi}_k=\\left[\\xi_{1k}\\,\\ldots\\,\\xi_{Mk}\\right]^T$ denote the $M\\times 1$ vector containing the average Frobenius norms of the equivalent channels linking the $k$th \\gls{MS} to all $M$ \\glspl{AP} in the cell-free network. Vector $\\bs{\\xi}_k$ can be considered as an effective \\emph{fingerprint} characterizing the location of \\gls{MS} $k$. Now, although no single definition of a similarity measure exists, the so-called \\emph{cosine similarity} measure is one of the most commonly used similarity metrics when dealing with real-valued vectors. Hence, as the \\emph{fingerprint} vectors characterizing the different \\glspl{MS} are non-negative real-valued, the cosine similarity measure between two \\emph{fingerprint} vectors $\\bs{\\xi}_k$ and $\\bs{\\xi}_{k'}$, defined as\n\\begin{equation}\n   f_D\\left(\\bs{\\xi}_k, \\bs{\\xi}_{k'}\\right)=\\frac{\\bs{\\xi}_k^T\\bs{\\xi}_{k'}}{\\|\\bs{\\xi}_k\\|_2 \\|\\bs{\\xi}_{k'} \\|_2},\n\\end{equation}\nwill be used as a proper similarity metric in our work. The resulting similarity values range from 0, meaning orthogonality (perfect dissimilarity), to 1, meaning exact match (perfect similarity).\n\nThe proposed \\gls{DCPA} algorithm proceeds as follows. In a first step, it calculates the fingerprint of an imaginary \\gls{MS} centroid, defined as\n\\begin{equation}\n   \\bs{\\xi}_C=\\frac{1}{K}\\sum_{k=1}^K \\bs{\\xi}_k.\n\\end{equation}\nThen, it moves onward to the calculation of the cosine similarity measures among the fingerprint vectors characterizing the $K$ \\glspl{MS} in the network and the fingerprint of the centroid, that is, the algorithm proceeds to the calculation of $f_D\\left(\\bs{\\xi}_k, \\bs{\\xi}_C\\right)$, for all $k\\in\\{1,\\ldots,K\\}$. The \\glspl{MS} are then sorted in descending order of similarity with the centroid, that is, the algorithm obtains the ordered set of subindices $\\mathcal{O}=\\left\\{o_1, o_2, \\ldots, o_K\\right\\}$, such that $f_D\\left(\\bs{\\xi}_{o_1}, \\bs{\\xi}_C\\right) \\leq f_D\\left(\\bs{\\xi}_{o_2}, \\bs{\\xi}_C\\right)\\leq \\cdots \\leq f_D\\left(\\bs{\\xi}_{o_K}, \\bs{\\xi}_C\\right)$. Once the \\glspl{MS} have been sorted, the algorithm constructs $\\tau_p$ clusters of \\glspl{MS}, namely $\\mathcal{K}_1, \\ldots, \\mathcal{K}_{\\tau_p}$, with\n\\begin{equation}\n\\begin{split}\n   \\mathcal{K}_t=&\\mathcal{O}\\left(t:\\tau_p:K\\right) \\\\\n                =&\\left\\{o_t, o_{t+\\tau_p}, o_{t+2 \\tau_p},\\ldots\\right\\},\\ \\forall t\\in\\{1,\\ldots,\\tau_p\\},\n\\end{split}\n\\end{equation}\nand all \\glspl{MS} in cluster $\\mathcal{K}_t$, which are located far from each other, are allocated the same pilot code $\\bs{\\varphi}_t$. Note that the application of this algorithm ensures that, as far as it is possible, two \\glspl{MS} having similar large-scale propagation fingerprints are allocated different pilot codes and, thus, they do not interfere to each other during the \\gls{UL} channel estimation process. In other words, it aims at minimizing the residual interuser interference terms in both \\eqref{eq:ydk} and \\eqref{eq:yuk}.\n\n\\section{Max-min power allocation and optimal quantization}\n\\label{sec:power allocation_quantization}\n\n\\subsection{Downlink power control and quantization}\n\nIn line with previous research works on cell-free architectures \\cite{Ngo15,Ngo17,Nayebi17,Bashar18b}, our aim in this subsection is to find the power control coefficients $\\upsilon_k$, for all $k\\in\\{1,\\ldots,K\\}$, and the quantization noise variances ${\\sigma_q^2}_{dm}$, for all $m\\in\\{1,\\ldots,M\\}$, that maximize the minimum of the achievable \\gls{DL} rates of all \\glspl{MS} while satisfying the average transmit power and \\gls{DL} fronthaul capacity constraints at each \\gls{AP}. Mathematically, this optimization problem can be formulated as\n\\begin{equation}\n\\begin{split}\n   &\\max_{\\substack{\\bs{\\Upsilon} \\succeq 0 \\\\ {\\bs{\\sigma}_q}_d \\succeq 0}}\\ \\min_{k\\in\\{1,\\ldots,K\\}} \\frac{\\upsilon_k}{\\sum_{k'=1}^K \\upsilon_{k'} \\varpi_{kk'} + \\sigma_{\\eta_{dk}}^2} \\\\\n   &\\textrm{s.t. } \\sum_{k=1}^K \\upsilon_k \\theta_{mk}^{BB\/RF} \\leq \\overline{P}_m - {\\sigma_q^2}_{dm} L_A N,\\,\\forall\\,m, \\\\\n   &\\phantom{\\textrm{s.t. }} \\log_2\\det\\left(\\sum_{k=1}^K \\frac{\\upsilon_k}{{\\sigma_q^2}_{dm}} \\bs{R}_{mk}^{BB} + \\bs{I}_{L_A}\\right) \\leq {C_F}_d,\\,\\forall\\,m,\n\\end{split}\n\\label{eq:opt_problem_DL}\n\\end{equation}\nwhere we have used the definition ${\\bs{\\sigma}_q}_d=[{\\sigma_q}_{d1}\\,\\ldots\\,{\\sigma_q}_{dM}]^T$.\n\nOptimization problem \\eqref{eq:opt_problem_DL} is characterized by continuous objective and constraint functions of interdependent block variables, namely, $\\bs{\\Upsilon}$ and ${\\bs{\\sigma}_q}_d$.  A widely used approach for solving optimization problems of this class is the so-called \\gls{BCD} method. This is an iterative optimization approach that, at each iteration and in a cyclic order, optimizes one of the blocks while the remaining variables are held fixed \\cite{Tseng01,Beck13}. Convergence of the \\gls{BCD} method is ensured whenever each of the subproblems to be optimized in each iteration can be exactly solved to its unique optimal solution.\n\nThe first important fact to note is that, given a power allocation matrix $\\bs{\\Upsilon}^{(i-1)}$ obtained at the $(i-1)$th iteration, and as the achievable user rates monotonically increase with the capacity of the fronthaul links between the \\glspl{AP} and the \\gls{CPU}, the optimal solution for the acceptable fronthaul quantization noise in the $i$th iteration is achieved when the fronthaul capacity constraints are satisfied with equality, that is, when\n\\begin{equation}\n   \\det\\left(\\sum_{k=1}^K \\frac{\\upsilon_k^{(i-1)}}{{\\sigma_q^2}_{dm}^{(i)}} \\bs{R}_{mk}^{BB} + \\bs{I}_{L_A}\\right) = 2^{{C_F}_d},\\,\\forall\\,m.\n   \\label{eq:trasc_funct}\n\\end{equation}\nNote that ${\\sigma_q^2}_{dm}^{(i)}$ cannot be expressed in a closed-form algebraic expression as it only admits a solution in the form of a transcendental function\n\\begin{equation}\n   {\\sigma_q^2}_{dm}^{(i)}=F_d\\left(\\bs{\\Upsilon}^{(i-1)}, \\left\\{\\bs{R}_{mk}^{BB}\\right\\}_{k=1}^K, {C_F}_d\\right)\n\\end{equation}\nthat can be numerically solved by applying mathematical software tools to \\eqref{eq:trasc_funct}.\n\nOnce the optimal block of variables ${\\bs{\\sigma}_q}_d^{(i)}$ have been obtained, the optimization problem in \\eqref{eq:opt_problem_DL} can be rewritten in terms of the power allocation matrix $\\bs{\\Upsilon}^{(i)}$ as\n\\begin{equation}\n\\begin{split}\n   &\\max_{\\bs{\\Upsilon}^{(i)} \\succeq 0}\\ \\min_{k\\in\\{1,\\ldots,K\\}} \\frac{\\upsilon_k^{(i)}}{\\displaystyle{\\small{\\sum_{k'=1}^K} \\upsilon_{k'}^{(i)} \\gamma_{kk'} + \\small{\\sum_{m=1}^M} {\\sigma_q^2}_{dm}^{(i)} \\tr\\left(\\bs{R}_{mk}^{RF}\\right) + \\sigma_d^2}} \\\\\n   &\\textrm{s.t. } \\sum_{k=1}^K \\upsilon_k^{(i)} \\theta_{mk}^{BB\/RF} \\leq \\overline{P}_m - N L_A {\\sigma_q^2}_{dm}^{(i)},\\,\\forall\\,m.\n\\end{split}\n\\label{eq:opt_problem_DL_GOPA}\n\\end{equation}\nNote that this is a convergent quasi-linear optimization problem that can be solved using conventional standard convex optimization methods \\cite{Ngo17,Nayebi17}.\n\n\n\n\\subsection{Uplink power control and quantization}\n\nIn this subsection we aim at finding the power control coefficients $\\omega_k$, for all $k\\in\\{1,\\ldots,K\\}$, and quantization noise variances ${\\sigma_q^2}_{um}$, for all $m\\in\\{1,\\ldots,M\\}$, that maximize the minimum of the achievable ulink rates of all \\glspl{MS} while satisfying the power control coefficient constraints at each \\gls{MS} and the \\gls{UL} fronthaul capacity constraints at each \\gls{AP}. This optimization problem can be formulated as\n\\begin{equation}\n\\begin{split}\n   &\\max_{\\substack{\\bs{\\omega} \\succeq 0 \\\\ {\\bs{\\sigma}_q}_u \\succeq 0}}\\ \\min_{k\\in\\{1,\\ldots,K\\}} \\frac{P_u \\omega_k}{P_u \\sum_{k'=1}^K \\omega_{k'} \\delta_{kk'} + \\sigma_{\\eta_{uk}}^2} \\\\\n   &\\textrm{s.t. } 0 \\leq \\omega_k \\leq 1,\\,\\forall\\,k, \\\\\n   &\\phantom{\\textrm{s.t. }} \\det\\left(\\frac{P_u}{{\\sigma_q^2}_{um}} \\sum_{k=1}^K \\omega_k \\bs{R}_{mk}^{RF} + \\vartheta_m\\bs{I}_{L_A}\\right) \\leq 2^{{C_F}_u},\\,\\forall\\,m,\n\\end{split}\n\\label{eq:opt_problem_UL}\n\\end{equation}\nwhere ${\\bs{\\sigma}_q}_u=[{\\sigma_q}_{u1}\\,\\ldots\\,{\\sigma_q}_{uM}]^T$, and we have used the definition $\\vartheta_m= 1+\\sigma_u^2(N)\/{\\sigma_q^2}_{um}$. As for the \\gls{DL} case, problem \\eqref{eq:opt_problem_UL} admits the use of the \\gls{BCD} method where, in each iteration, the nonconvex transcendental function ${\\sigma_q^2}_{um}=F_u\\left(\\bs{\\Omega}, \\left\\{\\bs{R}_{mk}^{RF}\\right\\}_{k=1}^K, P_u, {C_F}_u\\right)$ is approximated by a constant calculated using the power allocation vector obtained in the previous iteration of the algorithm. That is, in the $i$th iteration of the \\gls{UL} optimal power allocation approach, the algorithm solves the optimization problem\n\\begin{equation}\n\\begin{split}\n   &\\max_{\\bs{\\Omega}^{(i)} \\succeq 0}\\ \\min_{k\\in\\{1,\\ldots,K\\}} \\frac{P_u \\omega_k^{(i)}}{P_u \\sum_{k'=1}^K \\omega_{k'}^{(i)} \\delta_{kk'} + {\\sigma_{\\eta_{uk}}^{\\,2(i)}}}, \\\\\n   &\\textrm{s.t. } 0 \\leq \\omega_k \\leq 1,\\,\\forall\\,k,\n\\end{split}\n\\label{eq:opt_problem_UL_SOPA}\n\\end{equation}\nwhere ${\\sigma_q^2}_{um}^{(i)}=F_u\\left(\\bs{\\Omega}^{(i-1)}, \\left\\{\\bs{R}_{mk}^{RF}\\right\\}_{k=1}^K, P_u, {C_F}_u\\right)$. Note that, again, this is a convergent quasi-linear optimization problem that can be solved using conventional convex optimization methods \\cite{Ngo17,Nayebi17}.\n\n\\section{Numerical results}\n\\label{sec:numerical_results}\n\nIn this section, simulation results are obtained in order to quantitatively study the performance of the proposed cell-free \\gls{mmWave} massive \\gls{MIMO} network with constrained-capacity fronthaul links. In particular, we demonstrate the impact of using different pilot allocation strategies, the effects of modifying the capacity of the fronthaul links and the \\gls{RF} infrastructure at the \\glspl{AP}, and the repercussion of changing the density of \\glspl{AP} per area unit. For simplicity of exposition, and without loss of essential generality, a cell-free scenario is considered where the $M$ \\glspl{AP} and $K$ \\glspl{MS} are uniformly distributed at random within a square coverage area of size $D \\times D$~$m^2$. As described in subsection \\ref{subsec:Channel_model}, a modified version of the discrete-time narrowband clustered channel model proposed by Akdeniz \\emph{et al.} in \\cite{Akdeniz14} is used in the performance evaluation. The parameters necessary to implement this channel model can be found in \\cite[Table I]{Akdeniz14}. Furthermore, similar to what was done by Ngo \\emph{et al.} in \\cite{Ngo17}, a shadow fading spatial correlation model with two components is also considered (see \\cite[eqs. (54) and (55)]{Ngo17}) where the decorrelation distance is set to $d_{decorr}=50$~m and the parameter $\\delta$ is set to 0.5. Default parameters used to set-up the simulation scenarios under evaluation in the following subsections are summarized in Table \\ref{tab:parameters}.\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.2}\n\\caption{\\small Summary of default simulation parameters}\n\\label{tab:parameters}\n\\centering\n\\begin{tabular}{l|c}\n\\hline\n\\bfseries Parameters & \\bfseries Value\\\\\n\\hline\n\\footnotesize {Carrier frequency: $f_0$} & \\footnotesize {28 GHz}\\\\\n\\footnotesize {Bandwidth: $B$} & \\footnotesize {20 MHz}\\\\\n\\footnotesize {Side of the square coverage area: $D$} & \\footnotesize {200 m}\\\\\n\\footnotesize {AP antenna height: $h_{AP}$} & \\footnotesize {15 m}\\\\\n\\footnotesize {MS antenna height: $h_{MS}$} & \\footnotesize {1.65 m}\\\\\n\\footnotesize {Noise figure at the MS: ${NF}_{MS}$} & \\footnotesize {9 dB} \\\\\n\\footnotesize {Noise figure of the LNA at the AP: ${NF}_{LNA}$} & \\footnotesize {1.6 dB} \\\\\n\\footnotesize {Gain of the LNA at the AP: $G_{LNA}$} & \\footnotesize {22 dB} \\\\\n\\footnotesize {Attenuation of the phase splitters at the AP: $L_{PS}$} & \\footnotesize {3 dB}\\\\\n\\footnotesize {Attenuation of the power combiner at the AP: ${L_{PC}}_{in}$} & \\footnotesize {3 dB}\\\\\n\\footnotesize {Noise figure of the RF chain at the AP: $NF_{RF}$} & \\footnotesize {7 dB}\\\\\n\\footnotesize {Available average power at the AP: $\\overline{P}_m$} & \\footnotesize {200 mW}\\\\\n\\footnotesize {Available average power at the MS: $P_u=P_p$} & \\footnotesize {100 mW}\\\\\n\\footnotesize {Coherence interval length: $\\tau_c$} & \\footnotesize {200 samples}\\\\\n\\footnotesize {Training phase length: $\\tau_p$} & \\footnotesize {15 samples}\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[t!]\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_var_Pilot_Alloc_min_rate}\n        \\label{fig:DL_var_Pilot_Alloc_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_var_Pilot_Alloc_min_rate}\n        \\label{fig:UL_var_Pilot_Alloc_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of active \\glspl{MS} for different pilot allocation strategies ($N=64$ antennas, $L=8$ \\gls{RF} chains, ${C_F}_d={C_F}_u=64$ bit\/s\/Hz).}\n    \\label{fig:var_Pilot_Alloc_min_rate}\n\\end{figure}\n\n\\subsection{Impact of the pilot allocation process}\n\nOur aim in this subsection is to benchmark the performance of the proposed large-scale \\gls{CSI}-aware \\gls{DCPA} strategy against both the pure \\gls{RPA} and the \\gls{BRPA} schemes. Accordingly, the average max-min rate per user \\emph{versus} the number of active \\glspl{MS} is presented in Fig. \\ref{fig:var_Pilot_Alloc_min_rate} for each of these pilot allocation strategies and for both the \\gls{DL} and the \\gls{UL}. All results have been obtained assuming the default system parameters described in Table \\ref{tab:parameters}, the use of $L=8$ \\gls{RF} chains fully connected to uniform linear antenna arrays with $N=64$ antenna elements, and fronthaul links with a capacity of ${C_F}_d={C_F}_u=64$ bit\/s\/Hz. The first important result to note from Fig. \\ref{fig:var_Pilot_Alloc_min_rate} is that the pure \\gls{RPA} scheme is clearly outperformed by both the \\gls{BRPA} and the \\gls{DCPA} strategies irrespective of the of active \\glspl{MS} in the network. In fact, the \\gls{RPA} scheme cannot guarantee neither the absence of pilot reuse, even for those cases in which $K\\leq\\tau_p$ (in this setup, $\\tau_p=15$ time\/frequency samples), nor the possibility of having pilots that are allocated to a high number of \\glspl{MS} and\/or to \\glspl{MS} exhibiting very similar large-scale propagation patterns to the \\glspl{AP}. Therefore, the higher the number of active \\glspl{MS}, the higher the probability of having one or more users suffering from high levels of pilot contamination, with the consequent reduction of the achievable max-min user rate. If we turn our attention to results provided by the \\gls{BRPA} and \\gls{DCPA} strategies, two disjoint operation regions can be distinguished. In the first one, comprising the scenarios in which $K\\leq\\tau_p$, both approaches allocate orthogonal pilots to the users (absence of pilot contamination) and thus naturally provide the same performance. In the second one, however, comprising the scenarios in which $K>\\tau_p$, pilots have to be reused and, as a consequence, pilot contamination appears (note the rather abrupt performance drop when going from $K\\leq\\tau_p$ to $K>\\tau_p$). In these scenarios, based on a smart exploitation of the available large-scale \\gls{CSI}, the proposed \\gls{DCPA} approach reduces the amount of pilot contamination experienced by the worst users in the network and it clearly improves the achievable max-min user rates provided by the channel-unaware \\gls{BRPA} scheme.\n\nAnother result that is worth emphasizing, since it will repeatedly appear in the following subsections, is that, although in scenarios with high-capacity fronthaul links the achievable max-min \\gls{DL} user rate is higher than that provided in the \\gls{UL}, as the number of active users in the network increases, the performance obtained in both the \\gls{DL} and the \\gls{UL} tend to become increasingly similar. This behavior can be easily deduced from the analysis of the \\gls{SINR} expressions in \\eqref{eq:SINRdk} and \\eqref{eq:SINRuk}. As the number of active \\glspl{MS} in the cell-free network increases, provided that it is greater than $\\tau_p$, the term in the denominator corresponding to the residual interuser interference due to pilot contamination becomes increasingly dominant in comparison to the quantification and thermal noise terms, eventually reaching the point where they can be considered virtually negligible. Under these conditions, and since the pre-coding filters used on both links are identical, the \\gls{DL} and the \\gls{UL} experience similar \\gls{SINR} values and, therefore, tend to provide the same achievable max-min rate per user, except for small differences that can be attributed to, on the one and, the dissimilar amount of quantified information that has to be conveyed through the corresponding fronthaul links and, on the other hand, disparities among the thermal noise powers experienced at both the \\glspl{AP} and the \\glspl{MS}.\n\n\\begin{figure}[t!]\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_varCFd_min_rate}\n        \\label{fig:DL_varCFd_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_varCFu_min_rate}\n        \\label{fig:DL_varCFu_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of active \\glspl{MS} for different values of the fronthaul capacities ($N=64$ antennas, $L=8$ \\gls{RF} chains, \\gls{DCPA}).}\n    \\label{fig:varCF_min_rate}\n\\end{figure}\n\n\\subsection{Modifying the capacity of the fronthaul links and the \\gls{RF} infrastructure at the \\glspl{AP}}\n\nThe max-min achievable rate per user is plotted in Fig. \\ref{fig:varCF_min_rate} against the number of active \\glspl{MS} in the network, assuming the use of fronthaul links with different constraining capacities equal to 16, 32, 64 and 256 bit\/s\/Hz (for the network setups under consideration, using fronthaul links with a capacity of 256 bit\/s\/Hz is virtually equivalent to using infinite-capacity fronthauls). As expected, results show that increasing the fronthaul capacity is always beneficial if the main aim is to increase the achievable max-min user rate. Nevertheless, it is worth stressing that, keeping all the other parameters constant, the marginal increment of performance produced by each new increment of the fronthaul capacity suffers from the law of diminishing returns, especially for network setups with a high number of active \\glspl{MS}. That is, although the performance increase produced by doubling the fronthaul capacity from 16 bit\/s\/Hz to 32 bit\/s\/Hz, or even from 32 bit\/s\/Hz to 64 bit\/s\/Hz, can be justifiable, increasing the fronthaul capacity beyond 64 bit\/s\/Hz does not seem to be reasonable from the point of view of increasing the achievable performance of the system under the considered network setups. As observed in the previous subsection, in cell-free \\gls{mmWave} massive \\gls{MIMO} networks using high-capacity fronthaul links, the achievable max-min \\gls{DL} user rate is always slightly higher than that achieved in the \\gls{UL} irrespective of the number of active \\glspl{MS}. In scenarios with low-capacity fronthaul links and a large number of active \\glspl{MS}, however, the quantization noise experienced in the \\gls{DL} is higher than its \\gls{UL} counterpart and thus, the achievable per-user rate in the \\gls{UL} is slightly higher that than supplied in the \\gls{DL}.\n\n\\begin{figure}[t!]\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_varN_min_rate}\n        \\label{fig:DL_varN_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_varN_min_rate}\n        \\label{fig:DL_varN_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of active \\glspl{MS} for different values of the number of antennas at the \\glspl{AP} ($L=8$ \\gls{RF} chains, ${C_F}_d={C_F}_u=64$ bit\/s\/Hz, \\gls{DCPA}).}\n    \\label{fig:varN_min_rate}\n\\end{figure}\n\n\\begin{figure}[t!]\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_varL_min_rate}\n        \\label{fig:DL_varL_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_varL_min_rate}\n        \\label{fig:DL_varL_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of active \\glspl{MS} for different values of the number of \\gls{RF} chains at the \\glspl{AP} ($N=64$ antennas, ${C_F}_d={C_F}_u=64$ bit\/s\/Hz, \\gls{DCPA}).}\n    \\label{fig:varL_min_rate}\n\\end{figure}\n\nTo understand how the \\gls{RF} infrastructure used at the \\glspl{AP} influences the performance of the proposed cell-free \\gls{mmWave} massive \\gls{MIMO} system under constrained-capacity fronthaul links, Figs. \\ref{fig:varN_min_rate} and \\ref{fig:varL_min_rate} show the achievable max-min user rate against the number of active \\glspl{MS} assuming the use of uniform linear antenna arrays with different number of elements and fully-connected analog \\gls{RF} precoders with different number of \\gls{RF} chains, respectively. In particular, results presented in Fig. \\ref{fig:varN_min_rate} have been obtained assuming the use of an analog precoder with $L=8$ \\gls{RF} chains fully-connected to a linear uniform antenna array with $N=8$, 16, 32, 64 or 128 antenna elements, whereas results presented in Fig. \\ref{fig:varL_min_rate} have been obtained assuming the use of $L=2$, 4, 8 or 16 \\gls{RF} chains fully-connected to a linear uniform antenna array with $N=64$ antenna elements. The first conclusion we may draw when looking at the results presented in Fig. \\ref{fig:varN_min_rate} is that, irrespective of the number of active \\glspl{MS} in the cell-free network, increasing the number of antenna elements at the \\glspl{AP} in scenarios with high capacity fronthaul links (${C_F}_d={C_F}_u=64$ bit\/s\/Hz), although moderate and subject to the law of diminishing returns, always produces an increase in the achievable max-min user rate. As shown in Fig. \\ref{fig:varL_min_rate}, in contrast, the impact produced by an increase in the number of \\gls{RF} chains at the \\glspl{AP} depends on the number of active \\glspl{MS} in the network. In particular, when the number of active users is high, the interuser interference term due to pilot contamination (imperfect \\gls{CSI}) dominates the factors in the denominator of the \\gls{SINR} (i.e., makes the quantization and thermal noises negligible) and thus, increasing the number of \\gls{RF} chains is always beneficial when trying to increase the achievable max-min user rate. When the number of active users in the network is low, however, the quantization noise, which is an increasing function of $L$, is not negligible anymore when compared to the interuser interference term (recall that this term is null when the number of active \\glspl{MS} is less than or equal to $\\tau_p$) and thus, increasing the number of \\gls{RF} chains at the \\glspl{AP} can be clearly disadvantageous.\n\n\\begin{figure}\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_FixK_varN_varCFd_min_rate}\n        \\label{fig:DL_FixK_varN_varCFd_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_FixK_varN_varCFu_min_rate}\n        \\label{fig:UL_FixK_varN_varCFu_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of antennas at the \\glspl{AP} for different values of the fronthaul capacities ($K=20$ users, $L=8$ \\gls{RF} chains, \\gls{DCPA}).}\n    \\label{fig:FixK_varN_varCF_min_rate}\n\\end{figure}\n\n\\begin{figure}\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_FixK_varL_varCFd_min_rate}\n        \\label{fig:DL_FixK_varL_varCFd_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_FixK_varL_varCFu_min_rate}\n        \\label{fig:UL_FixK_varL_varCFu_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of \\gls{RF} chains at the \\glspl{AP} for different values of the fronthaul capacities ($K=20$ users, $N=64$ antennas, \\gls{DCPA}).}\n    \\label{fig:FixK_varL_varCF_min_rate}\n\\end{figure}\n\n\\begin{figure*}\n\n    \\centering\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[height=5.9cm]{DL_CFR_VarM}\n        \\label{fig:DL_CFR_VarM}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[height=5.9cm]{UL_CFR_VarM}\n        \\label{fig:UL_CFR_VarM}\n    \\end{subfigure}\n    \\caption{\\small CDF of the \\gls{DL} and \\gls{UL} achievable max-min rate per user for different values of the number of \\glspl{AP} and active \\glspl{MS} in the cell-free network ($N=64$ antennas, $L=8$ \\gls{RF} chains, ${C_F}_d={C_F}_u=64$ bit\/s\/Hz, \\gls{DCPA}).}\n    \\label{fig:CFR_VarM}\n\\end{figure*}\n\nResults presented in Figs. \\ref{fig:varCF_min_rate}, \\ref{fig:varN_min_rate} and \\ref{fig:varL_min_rate} were obtained assuming high-capacity fronthaul links with ${C_F}_d={C_F}_u=64$ bit\/s\/Hz. However, the amount of quantized data that has to be conveyed from (to) the \\gls{CPU} to (from) the \\glspl{AP} in the \\gls{DL} (\\gls{UL}) depends on the number of antennas and \\gls{RF} chains at the \\glspl{AP} (see Section \\ref{sec:fronthaul_capacity}). Thus, in order to deepen in the study of the impact the \\gls{RF} infrastructure may have on the achievable performance of the proposed cell-free \\gls{mmWave} massive \\gls{MIMO} system under constrained-capacity fronthaul links, the average max-min user rate is plotted in Figs. \\ref{fig:FixK_varN_varCF_min_rate} and \\ref{fig:FixK_varL_varCF_min_rate} against the number of antenna elements and \\gls{RF} chains, respectively, for different values of the fronthaul capacities and assuming a fixed number of $K=20$ active \\glspl{MS} in the network. In network setups using very high capacity fronthaul links (i.e., ${C_F}_d={C_F}_u=256$ bit\/s\/Hz), increasing the number of antenna elements $N$ and\/or the number of \\gls{RF} chains $L$ (up to $L=K$) is always beneficial as, in this case, the noise introduced by the quantization process is negligible and the system can take full advantage of the increased \\gls{RF} resources. As the capacity of the fronthaul links decreases, however, the amount of noise introduced by the quantization process increases with both $N$ and $L$ and, therefore, a situation arises where the potential performance improvement provided by the increase of $N$ and\/or $L$ is compromised by the performance reduction due to fronthaul capacity constraints. On the one hand, it can be observed in Fig. \\ref{fig:FixK_varN_varCF_min_rate} that, for fixed numbers of users and \\gls{RF} chains, there is a certain fronthaul capacity constraint value (near 24 bit\/s\/Hz in the setup used in this experiment) under which increasing the number of antenna elements at the array is counterproductive. On the other hand, results presented in Fig. \\ref{fig:FixK_varL_varCF_min_rate} show that, for fixed numbers of users and antenna elements at the arrays, there is always an optimal number of \\gls{RF} chains to be deployed (or activated) at the \\glspl{AP} that is dependent on the capacity of the fronthaul links. In particular, for the network setups under consideration, the optimal number of \\gls{RF} chains is equal to $L=10$, 4, and 1 when using fronthaul links with a capacity of 64 bit\/s\/Hz, 32 bit\/s\/Hz and less than 24 bit\/s\/Hz, respectively.\n\n\\subsection{Impact of the density of \\glspl{AP}}\n\nWith the aim of evaluating the impact the density of \\glspl{AP} per area unit may have on the performance of the proposed cell-free \\gls{mmWave} massive \\gls{MIMO} system, Fig. \\ref{fig:CFR_VarM} represents the \\gls{CDF} of the \\gls{DL} and \\gls{UL} achievable max-min user rate for different values of the number of \\glspl{AP} in the network. It has been assumed in these experiments a fixed number of active \\glspl{MS} equal to either $K=25$ or $K=8$ \\glspl{MS}, the use of $L=8$ \\gls{RF} chains fully-connected to a linear uniform antenna array with $N=64$ antenna elements, and the use of \\gls{DL} and \\gls{UL} fronthaul links with a capacity ${C_F}_d={C_F}_u=64$ bit\/s\/Hz. As expected, cell-free massive \\gls{MIMO} scenarios with a high density of \\glspl{AP} per area unit significantly outperform those with a low density of \\glspl{AP} per area unit in both median and 95\\%-likely achievable per-user rate performance. However, the achievable max-min user rate increase due to increasing the number of \\glspl{AP} in the network is, again, subject to the law of diminishing returns. For instance, in scenarios with $K=25$ \\glspl{MS}, the 95\\%-likely achievable user rate is equal to 2.55, 4.33, 6.11 and 6.50 bit\/s\/Hz for cell-frre massive \\gls{MIMO} networks with $M=25$, 50, 100 and 200 \\glspl{AP}, respectively. That is, doubling the number of \\glspl{AP} per area unit does not result in doubling the 95\\%-likely achievable user rate. Similar conclusions can be drawn when looking at either the median or the average achievable user rates.\n\nAs was observed in results presented in previous subsections for high-capacity fronthaul setups, when the number of active users in the system is low, the achievable max-min rate values in the \\gls{DL} are slightly higher than those achievable in the \\gls{UL}. Instead, when the number of active users increases, the achievable max-min user rates are virtually identical in both the \\gls{DL} and the \\gls{UL}. Also, note that the dispersion of the achievable max-min user rates around the median tends to diminish as the density of \\glspl{AP} increases. That is, cell-free massive \\gls{MIMO} networks with a high density of \\glspl{AP} per area unit tend to offer max-min achievable rates that suffer little variations irrespective of the location of the \\glspl{AP} (i.e, irrespective of the scenario under evaluation).\n\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nA novel analytical framework for the performance analysis of cell-free \\gls{mmWave} massive \\gls{MIMO} networks has been introduced in this paper. The proposed framework considers the use of low-complexity hybrid precoders\/decoders where the \\gls{RF} high-dimensionality phase shifter-based precoding\/decoding stage is based on large-scale second-order channel statistics, while the low-dimensionality baseband multiuser \\gls{MIMO} precoding\/decoding stage can be easily implemented by standard ZF signal processing schemes using small-scale estimated \\gls{CSI}. Furthermore, it also takes into account the impact of using capacity-constrained fronthaul links that assume the use of large-block lattice quantization codes able to approximate a Gaussian quantization noise distribution, which constitutes an upper bound to the performance attained under any practical quantization scheme. Max-min power allocation and fronthaul quantization optimization problems have been posed thanks to the development of mathematically tractable expressions for both the per-user achievable rates and the fronthaul capacity consumption. These optimization problems have been solved by combining the use of block coordinate descent methods with sequential linear optimization programs. Results have shown that the proposed \\gls{DCPA} suboptimal pilot allocation strategy, which is based on the idea of clustering by dissimilarity, overcomes the computational burden of the optimal small-scale CSI-based pilot allocation scheme while clearly outperforming the pure random and balanced random schemes. It has also been shown that, although increasing the fronthaul capacity and\/or the density of \\glspl{AP} per area unit is always beneficial from the point of view of the achievable max-min user rate, the marginal increment of performance produced by each new increment of these parameters suffers from the law of diminishing returns, especially for network setups with a high number of active \\glspl{MS}. Moreover, simulation results indicate that, as the capacity of the fronthaul links decreases, the potential performance improvement provided by the increase of the number of antenna elements $N$ and\/or the number of \\gls{RF} chains $L$ is compromised by the performance reduction due to the corresponding increase of the fronthaul quantization noise. In particular, for fixed numbers of users and \\gls{RF} chains, there is a certain fronthaul capacity constraint value (near 24 bit\/s\/Hz in the setups under consideration) under which increasing the number of antenna elements at the array is counterproductive. Similarly, for fixed numbers of users and antenna elements at the arrays, there is always an optimal number of \\gls{RF} chains to be deployed (or activated) at the \\glspl{AP} that is dependent on the capacity of the fronthaul links. For future work, it would be interesting to develop low-complexity pilot- and power-allocation techniques specifically designed to maximize the energy efficiency of cell-free \\gls{mmWave} massive \\gls{MIMO} networks considering both the fronthaul capacity constraints and the fronthaul power consumption. It would also be interesting to explore the use of partially-connected \\gls{RF} precoding\/decoding architectures and the implementation of baseband \\gls{MU-MIMO} precoding\/decoding other than the \\gls{ZF} scheme.\n\n\\appendices\n\n\\section{Proof of Theorem \\ref{theo:theorem_DL}}\n\\label{app:Appendix_1}\n\nFollowing an approach similar to that proposed by Nayebi \\emph{et al.} in \\cite{Nayebi17}, the signal received by the $k$th \\gls{MS} in \\eqref{eq:ydk} can be rewritten as ${y_d}_k={y_d}_{k\\,0} + {y_d}_{k\\,1} + {y_d}_{k\\,2} + {n_d}_k$, where the useful, interuser interference, and quantization noise terms can be expressed as ${y_d}_{k\\,0}=\\sqrt{\\upsilon_k} {s_d}_k$, ${y_d}_{k\\,1}=\\tilde{\\bs{g}}_k^T \\bs{W}_d^{BB} \\bs{\\Upsilon}^{1\/2}\\bs{s}_d$, and ${y_d}_{k\\,2} = \\bs{g}_k^T \\bs{q}_d=\\sum_{m=1}^M \\bs{g}_{km}^T {\\bs{q}_d}_m$, respectively. Now, considering that data symbols, quantization noise, thermal noise, and channel-related coefficients are mutually independent, the terms ${y_d}_{k\\,0}$, ${y_d}_{k\\,1}$, ${y_d}_{k\\,2}$ and ${n_d}_k$ are mutually uncorrelated and thus, based on the worst-case uncorrelated additive noise \\cite{Hassibi03}, the achievable \\gls{DL} rate for user $k$ is lower bounded by ${R_d}_k=\\log_2\\left(1+{\\SINR_d}_k\\right)$, with\n\\begin{equation*}\n   {\\SINR_d}_k=\\frac{\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,0}\\right|^2\\right\\}}{\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,1}\\right|^2\\right\\}+\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,2}\\right|^2\\right\\}+\\sigma_d^2},\n\\end{equation*}\nwhere $\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,0}\\right|^2\\right\\}=\\upsilon_k$,\n\\begin{equation*}\n\\begin{split}\n   &\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,1}\\right|^2\\right\\}=\\mathbb{E}\\left\\{\\bs{s}_d^H\\bs{\\Upsilon}^{1\/2} {\\bs{W}_d^{BB}}^H \\tilde{\\bs{g}}_k^* \\tilde{\\bs{g}}_k^T \\bs{W}_d^{BB} \\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right\\} \\\\\n   &\\qquad=\\tr\\left(\\bs{\\Upsilon} \\mathbb{E}\\left\\{{\\bs{W}_d^{BB}}^H \\tilde{\\bs{g}}_k^* \\tilde{\\bs{g}}_k^T \\bs{W}_d^{BB}\\right\\}\\right) \\\\\n   &\\qquad=\\sum_{k'=1}^K \\upsilon_{k'} \\left[\\diag\\left(\\mathbb{E}\\left\\{{\\bs{W}_d^{BB}}^H \\tilde{\\bs{g}}_k^* \\tilde{\\bs{g}}_k^T \\bs{W}_d^{BB}\\right\\}\\right)\\right]_{k'},\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n   \\mathbb{E}\\left\\{\\left|{y_d}_{k\\,2}\\right|^2\\right\\}&=\\sum_{m=1}^M \\mathbb{E}\\left\\{{\\bs{q}_d}_m^H \\bs{g}_{km}^* \\bs{g}_{km}^T {\\bs{q}_d}_m\\right\\} \\\\\n                                                       &=\\sum_{m=1}^M {\\sigma_q^2}_{dm} \\tr\\left(\\bs{R}_{mk}^{RF}\\right).\n\\end{split}\n\\end{equation*}\n\n\n\\section{Proof of Theorem \\ref{theo:theorem_UL}}\n\\label{app:Appendix_2}\n\nThe detected signal at the \\gls{CPU} corresponding to the symbol transmitted by the $k$th \\gls{MS} in \\eqref{eq:yuk} can be rewritten as ${y_u}_k={y_u}_{k\\,0} + {y_u}_{k\\,1} + {y_u}_{k\\,2} + {y_u}_{k\\,3}$, where the useful, interuser interference, quantization noise and thermal noise terms can be expressed as ${y_u}_{k\\,0}=\\sqrt{P_u} \\sqrt{\\omega_k} {s_u}_k$, ${y_u}_{k\\,1}=\\sqrt{P_u}\\left[\\bs{W}_u^{BB} \\tilde{\\bs{G}} \\bs{\\Omega}^{1\/2} \\bs{s}_u\\right]_k$, ${y_u}_{k\\,2} = \\left[\\bs{W}_u^{BB} \\bs{q}_u \\right]_k$, and ${y_u}_{k\\,3} = \\left[\\bs{W}_u^{BB} \\bs{n}_u\\right]_k$, respectively. As in the \\gls{DL}, since data symbols, quantization noise, thermal noise, and channel-related coefficients are mutually independent, the terms ${y_u}_{k\\,0}$, ${y_u}_{k\\,1}$, ${y_d}_{k\\,2}$ and ${y_d}_{k\\,3}$ are mutually uncorrelated and thus, based on the worst-case uncorrelated additive noise \\cite{Hassibi03}, the achievable \\gls{UL} rate for user $k$ is lower bounded by ${R_u}_k=\\log_2\\left(1+{\\SINR_u}_k\\right)$, with\n\\begin{equation*}\n   {\\SINR_u}_k=\\frac{\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,0}\\right|^2\\right\\}}{\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,1}\\right|^2\\right\\}+\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,2}\\right|^2\\right\\}+\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,3}\\right|^2\\right\\}},\n\\end{equation*}\nwhere $\\mathbb{E}\\left\\{\\left|{y_u}_{k\\,0}\\right|^2\\right\\}=P_u \\omega_k$,\n\\begin{equation*}\n\\begin{split}\n   &\\mathbb{E}\\left\\{\\left|{y_u}_{k\\,1}\\right|^2\\right\\}=P_u\\mathbb{E}\\left\\{\\bs{s}_u^H\\bs{\\Omega}^{1\/2} \\tilde{\\bs{G}}^H {\\bs{w}_{uk}^{BB}}^H \\bs{w}_{uk}^{BB} \\tilde{\\bs{G}}\\bs{\\Omega}^{1\/2} \\bs{s}_u\\right\\} \\\\\n   &\\qquad=P_u \\tr\\left(\\bs{\\Omega} \\mathbb{E}\\left\\{\\tilde{\\bs{G}}^H {\\bs{w}_{uk}^{BB}}^H \\bs{w}_{uk}^{BB} \\tilde{\\bs{G}}\\right\\}\\right) \\\\\n   &\\qquad=P_u \\sum_{k'=1}^K \\omega_{k'} \\left[\\diag\\left(\\mathbb{E}\\left\\{\\tilde{\\bs{G}}^H {\\bs{w}_{uk}^{BB}}^H \\bs{w}_{uk}^{BB} \\tilde{\\bs{G}}\\right\\}\\right)\\right]_{k'},\n\\end{split}\n\\end{equation*}\nwith $\\bs{w}_{uk}^{BB}$ denoting the $k$th row of $\\bs{W}_u^{BB}$, or, equivalently,\n\\begin{equation*}\n\\begin{split}\n   &\\mathbb{E}\\left\\{\\left|{y_u}_{k\\,1}\\right|^2\\right\\} = P_u\\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_u^{BB} \\tilde{\\bs{G}} \\bs{\\Omega} \\tilde{\\bs{G}}^H {\\bs{W}_u^{BB}}^H \\right\\}\\right)\\right]_k \\\\\n   &\\qquad=P_u \\sum_{k'=1}^K \\omega_{k'} \\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_u^{BB} \\tilde{\\bs{g}}_{k'} \\tilde{\\bs{g}}_{k'}^H {\\bs{W}_u^{BB}}^H \\right\\}\\right)\\right]_k,\n\\end{split}\n\\end{equation*}\nand, finally,\n\\begin{equation*}\n\\begin{split}\n   &\\mathbb{E}\\left\\{\\left|{y_u}_{k\\,2}\\right|^2\\right\\}=\\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_u^{BB} \\bs{q}_u \\bs{q}_u^H {\\bs{W}_u^{BB}}^H\\right\\}\\right)\\right]_k \\\\\n   &\\qquad=\\sum_{m=1}^M \\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_u^{BB} \\bs{q}_{um} \\bs{q}_{um}^H {\\bs{W}_u^{BB}}^H\\right\\}\\right)\\right]_k \\\\\n   &\\qquad=\\sum_{m=1}^M {\\sigma_q^2}_{um}\\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_{u\\,m}^{BB} {\\bs{W}_{u\\,m}^{BB}}^H\\right\\}\\right)\\right]_k,\n\\end{split}\n\\end{equation*}\nand, analogously,\n\\begin{equation*}\n\\begin{split}\n   \\mathbb{E}\\left\\{\\left|{y_d}_{k\\,3}\\right|^2\\right\\}=\\sigma_u^2(N)\\sum_{m=1}^M \\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_{u\\,m}^{BB} {\\bs{W}_{u\\,m}^{BB}}^H\\right\\}\\right)\\right]_k.\n\\end{split}\n\\end{equation*}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\noindent Speech enhancement aims at separating speech from background interference signals. Mainstream deep learning-based methods learn to predict the speech signal in a supervised manner, as shown in Figure \\ref{fig0} (a). Most prior works operate in the time-frequency (T-F) domain by predicting a mask between noisy and clean spectra \\cite{wang2014IRM, williamson2015complex} or directly predicting the clean spectrum \\cite{xu2013experimental, tan2018convolutional}. Some methods operate in the time domain by estimating speech signals from raw-waveform noisy signals in an end-to-end way \\cite{fu2017raw, pascual2017segan, pandey2019tcnn}. These methods have considerably improved the quality of enhanced speech compared with traditional signal processing based schemes. However, speech distortion or residual noise can often be observed in the enhanced speech, showing that there are still correlations between predicted speech and the residual signal by subtracting enhanced speech from noisy signal.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{illustration_1}\n    \\caption{Illustration of different methods. (a) Most existing deep-learning-based methods directly model speech. (b) Most traditional methods predict speech with noise estimate. (c) Our method simultaneously models speech and noise with information interaction.}\n    \\label{fig0}\n\\end{figure}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.95\\textwidth]{overview_network_structure_v2}\n    \\caption{Overall network structure of SN-Net.}\n    \\label{fig1}\n\\end{figure*}\n\nInstead of only predicting speech and ignoring the characteristics of background noises, traditional signal processing and modeling based methods mostly take the other way (see Figure \\ref{fig0} (b)), i.e. estimating noise or building noise models for speech enhancement \\cite{boll1979suppression, hendriks2010noisePSD, wang2017model, wilson2008NMF1, moham2013NMF2}. Some model-based methods instead model both speech and noise \\cite{srinivasan2006Codebook, srinivasan2005Codebook}, possibly with alternate model update. However, they typically cannot generalize well when prior noise assumption cannot be met or the interference signal is not structured. In deep-learning-based methods, two recent attempts \\cite{odelowo2017noise, odelowo2018study} focus on directly predicting noise considering that noise is dominant in low-SNR conditions. However, the benefit is limited.\n\n\nThe remaining correlation between predicted speech and noise motivates us to explore the information flow between speech and noise estimations, as shown in Figure \\ref{fig0} (c). Since speech-related information exists in predicted noise, and vice versa, adding information communication between them may help to recover some missing components and remove undesired information from each other. In this paper, we propose a two-branch convolutional neural network, namely SN-Net, to simultaneously predict speech and noise signals. Between them are information interaction modules, by which noise or speech related information are extracted from the noise branch and added back to speech features to counteract the undesired noise part or recover the missing speech, and vice versa. In this way, the discrimination capability is largely enhanced. The two branches share the same network structure, which is an encoder-decoder-based model with several residual-convolution-and-attention (RA) blocks in between for separation. Motivated by the success of self-attention technique in machine translation and computer vision tasks \\cite{vaswani2017attention, wang2018nonlocal}, we propose to combine temporal self-attention and frequency-wise self-attention parallelly inside each RA block for capturing global dependency along temporal and frequency dimensions in a separable way.\n\nOur main contributions are summarized as follows.\n\n\\begin{itemize}\n    \\item We propose to simultaneously model speech and noise in a two-branch deep neural network and introduce information flow between them. In this way, speech part is enhanced while residual noise is suppressed for speech estimation, and vice versa.\n    \\item We propose a RA block for feature extraction. Separable self-attention is utilized in this block to globally capture the temporal and frequency dependencies.\n    \\item We validate the superiority of proposed scheme in an ablation study and comparison with state-of-the-art algorithms on two public datasets. Moreover, we extend our method to speaker separation, which also shows great performance. These results demonstrate the superiority and potential of the proposed method.\n\\end{itemize}\n\n\\section{Related Work}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{overview_Encoder-Decoder}\n    \\includegraphics[width=0.4\\textwidth]{attention_structure_in_Decoder}\n    \\caption{(a) Encoder-decoder structure. The dashed arrow denotes the separation module using RA blocks. (b) Detailed structure of the gated block inside the decoder.}\n    \\label{fig2}\n\\end{figure*}\n\n\\subsection{Deep Learning-based Speech Enhancement}\n\n\\noindent Deep learning-based methods mainly study how to build a speech model. According to the adopted signal domain, these methods can be classified into two categories. Time-Frequency (T-F) domain methods take T-F representation, either complex or log power spectrum of the magnitude, as input. They typically estimate a real or complex ratio mask for each T-F bin to map noisy spectra to speech spectra \\cite{williamson2015complex, wang2014IRM, choi2019phase} or directly predict the speech representation \\cite{xu2013experimental, tan2018convolutional}. Time-domain methods take waveform as input and typically extract a hidden representation of the raw waveform through an encoder and reconstruct an enhanced version from that \\cite{fu2017raw, pascual2017segan, pandey2019tcnn}. Although these methods have shown great improvements over traditional methods, they only focus on modeling speech and neglect the importance of understanding noise characteristics.\n\n\\subsection{Noise-Aware Speech Enhancement}\n\n\\noindent Noise information is often considered in traditional signal processing based methods \\cite{boll1979suppression, hendriks2010noisePSD, wang2017model} with prior distribution assumptions for speech and noise. However, it is a challenging task to estimate the noise power spectral density for non-stationary noises and thus mostly stationary noise is assumed. They are unsuitable in generalization to low SNR and non-stationary noise conditions. Instead, some model-based methods build models for speech and noise and show more promising results, e.g., codebook \\cite{srinivasan2006Codebook, srinivasan2005Codebook} and nonnegative matrix factorization (NMF) \\cite{wilson2008NMF1, moham2013NMF2} based methods. However, they either need prior knowledge of the noise type \\cite{srinivasan2006Codebook, srinivasan2005Codebook} or are only effective for structured noise \\cite{wilson2008NMF1, moham2013NMF2}; therefore their generalization capability is limited.\n\nDeep learning-based methods can better generalize to various noise conditions. There are also some attempts on incorporating noise information, for example, by adding constraints to loss functions \\cite{fan2019noise, xu2020using, xia2020weighted} or by directly predicting noise instead of speech \\cite{odelowo2017noise, odelowo2018study}. The former does not model noise at all and the characteristics of noise are not exploited. The latter loses the speech information and show even worse quality than corresponding speech prediction method in low SNR and unseen noise conditions. A more relevant work utilizes two deep auto encoders (DAEs) to estimate speech and noise \\cite{sun2015unseen} . It first trains a DAE for speech spectrum reconstruction and then introduces another DAE to model noise with the constraint that the sum of outputs of the two DAEs is equal to the noisy spectrum.\n\nDifferent from aforementioned approaches, we proposed a two-branch CNN to predict speech and noise simultaneously and introduce interaction modules at several intermediate layers to make them benefit from each other. Such a paradigm makes it suitable for speaker separation as well. \n\n\\subsection{Two-Branch Neural Networks}\n\n\\noindent Two-branch neural networks have been explored in various tasks for capturing cross-modality information \\cite{nam2017dual, image-text} or different levels of information \\cite{simonyan2014two, wang2020dual}. For speech enhancement, a two-branch modeling is proposed to predict the amplitude and phase of the enhanced signal, respectively \\cite{yin2020phasen}. In this paper, we aim to exploit the two correlated tasks, i.e. speech and noise estimations and explicitly modeling them in an interactive two-branch framework for better discrimination.\n\n\\subsection{Self-Attention Model}\n\n\\noindent Self-attention mechanism has been widely used in many tasks, e.g., machine translation \\cite{vaswani2017attention}, image generation \\cite{zhang2019self} and video question answering \\cite{li2019beyond}. For video, spatio-temporal attention is also considered to exploit long-term dependency along both spatial and temporal dimensions \\cite{wu2019video}. Recently, speech-related tasks have also benefited from self-attention, e.g., speech recognition \\cite{salazar2019ctc} and speech enhancement \\cite{kim2020tgsa, koizumi2020speech}. In these works, self-attention is applied along the temporal dimension only, neglecting the global dependency inside each frame. Motivated by the spatio-temporal attention in video-related tasks, we propose to employ both frequency-wise and temporal self-attention to better capture dependencies along different dimensions. Such an attention is employed in both speech and noise branches for simultaneous modeling the two signals.\n\n\\section{Proposed Method}\n\n\\subsection{Overview}\n\n\\noindent Figure \\ref{fig1} shows the overall network structure of SN-Net. The input is the complex T-F spectrum computed by short-time Fourier transform (STFT), denoted as $ X^I\\in R^{T\\times F\\times 2} $, where $ T $ is the number of frames and $ F $ is the number of frequency bins. There are two branches in SN-Net, one of which predicts speech and the other predicts noise. They share the same network structure but have separate network parameters. Each branch is an encoder-decoder based structure, with several RA blocks inserted inbetween them. In this way, it is capable of simultaneously mining the potential of different components of the noisy signal. Between the two branches are interaction modules designed to transform and share information. After each branch gets its output, a merge branch is employed to adaptively combine the two outputs to generate the final enhanced speech.\n\n\\subsection{Encoder and Decoder}\n\n\\noindent As shown in Figure \\ref{fig2} (a), the encoder has three 2-D convolutional layers, each with a kernel size of (3, 5). The stride is (1, 1) for the first layer and (1, 2) for the following two. The channel numbers are 16, 32, 64, respectively. As a result, the output feature of the encoder is $ \\mathcal{F}^E_k\\in \\mathbb{R}^{T\\times F'\\times C} $, where $ F'=\\frac{F}{4}$, $C=64 $ and $ k\\in \\left\\{S,N\\right\\} $. $ S $ and $ N $ denote speech and noise branches, respectively. For simplicity, the subscript $ k $ will be ignored in the following.\n\nThe decoder consists of three gated blocks followed by one 2-D convolutional layer, which reconstructs the output $ \\mathcal{F}^D\\in \\mathbb{R}^{T\\times F\\times 2} $. As shown in Figure \\ref{fig2} (b), the gated block learns a multiplicative mask on corresponding feature from the encoder, aiming to suppress its undesired part. The masked encoder feature is then concatenated with the deconvolutional feature and fed into another 2-D convolutional layer to generate the residual representation. After three gated blocks, the final convolutional layer learns the amplitude gain and the phase for reconstruction, similar to that in \\cite{choi2019phase}. The kernel size for all 2-D deconvolutional layers is (3,5). The stride is (1,2) for the first two gated blocks and (1,1) for the last one. The channel numbers are 64, 32, 16, respectively. All the 2-D convolutional layers in the decoder have a kernel size of (1,1), a stride of (1,1) and a channel number the same as that of their inputs.\n\nAll the convolutional layers in the encoder and the decoder are followed by a batch normalization (BN) and a parametric ReLU (PReLU). No down-sampling is performed along the temporal dimension to preserve the temporal resolution.\n\n\\subsection{RA Block}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{RA_block}\n    \\caption{Structure of the RA block.}\n    \\label{fig3}\n\\end{figure}\n\n\\noindent The RA block is designed to extract features and perform separation for both speech and noise branches. It is challenging because of the diversities of noise types and the difference between speech and noises. We employ the separable self-attention (SSA) technique to capture the global dependencies along temporal and frequency dimensions, respectively. It is intuitive to use attention for these two dimensions as humans tend to put more attention to some parts of an audio signal (e.g., speech) while less to the surrounding part (e.g., noise) and they perceive differently on different frequencies. When it comes to the speech-noise network in SN-Net, the SSA modules in speech and noise branches perceive signals differently, which will be demonstrated in the ablation study section afterwards. \n\nIn SN-Net, there are four RA blocks between the encoder and the decoder. Each block consists of two residual blocks and a SSA module, as shown in Figure \\ref{fig3}, capturing both local and global dependencies inside the signal. Each residual block has two 2-D convolutional layers with a kernel size of (5,7), a stride of (1,1) and the same number of channels as their inputs. The output feature of two residual blocks $ \\mathcal{F}^{Res}_i\\in \\mathbb{R}^{T\\times F'\\times C} $ ($ i\\in \\left\\{1,2,3,4\\right\\} $ represents the $ i^{th} $ RA block and will be ignored in the following) is fed parallelly into temporal self-attention and frequency-wise self-attention blocks. These two attention blocks produce the outputs $ \\mathcal{F}^{Temp}\\in \\mathbb{R}^{T\\times F'\\times C} $ and $ \\mathcal{F}^{Freq}\\in \\mathbb{R}^{T\\times F'\\times C} $. The three features $\\mathcal{F}^{Res} $, $ \\mathcal{F}^{Temp} $ and $ \\mathcal{F}^{Freq} $ are then concatenated and fed into a 2-D convolutional layer to generate the block output $ \\mathcal{F}^{RA}\\in \\mathbb{R}^{T\\times F'\\times C} $, used in the interaction module.\n\nFor self-attention, we employ the scaled dot-product self-attention here. Considering the computational complexity, channels are reduced by half inside SSA. The temporal self-attention can be represented as\n\\begin{equation}\n\\begin{aligned} \n    \\mathcal{F}_t^{k}&=Reshape^t(Conv(\\mathcal{F}^{Res})), k\\in \\left\\{K, Q, V\\right\\},\\\\\n    SA^{t}&=Softmax(\\mathcal{F}_t^{Q}\\cdot (\\mathcal{F}_t^{K})^T\/\\sqrt{\\frac{C}{2}\\times F'})\\cdot \\mathcal{F}_t^{V},\\\\\n    \\mathcal{F}^{Temp}&=\\mathcal{F}^{Res}+Conv(Reshape^{t*}(SA^{t})),\n\\end{aligned}\n\\end{equation}\nwhere $\\mathcal{F}_t^{k} \\in \\mathbb{R}^{T\\times (\\frac{C}{2}\\times F')}$, $SA^{t}\\in \\mathbb{R}^{T\\times (\\frac{C}{2}\\times F')}$ and $\\mathcal{F}^{Temp} \\in \\mathbb{R}^{T\\times F'\\times C}$, respectively. $(\\cdot)$ denotes matrix multiplication. $Reshape^t(\\cdot)$ denotes a tensor reshape from $\\mathbb{R}^{T\\times F'\\times \\frac{C}{2}}$ to $\\mathbb{R}^{T\\times (\\frac{C}{2}\\times F')}$ and $Reshape^{t*}(\\cdot)$ is the opposite. The frequency-wise self-attention is given by\n\\begin{equation}\n\\begin{aligned} \n    \\mathcal{F}_f^{k}&=Reshape^f(Conv(\\mathcal{F}^{Res})), k\\in \\left\\{K, Q, V\\right\\},\\\\\n    SA^{f}&=Softmax(\\mathcal{F}_f^{Q}\\cdot (\\mathcal{F}_f^{K})^T\/\\sqrt{\\frac{C}{2}\\times T})\\cdot \\mathcal{F}_f^{V},\\\\\n    \\mathcal{F}^{Freq}&=\\mathcal{F}^{Res}+Conv(Reshape^{f*}(SA^{f})),\n\\end{aligned}\n\\end{equation}\nwhere $\\mathcal{F}_f^{k}\\in \\mathbb{R}^{F'\\times (\\frac{C}{2}\\times T)}$, $SA^{f}\\in \\mathbb{R}^{F'\\times(\\frac{C}{2}\\times T)}$ and $\\mathcal{F}^{Freq}\\in \\mathbb{R}^{T\\times F'\\times C}$, respectively. $Reshape^f(\\cdot)$ reshapes a tensor from $\\mathbb{R}^{T\\times F'\\times \\frac{C}{2}}$ to $\\mathbb{R}^{F'\\times (\\frac{C}{2}\\times T)}$.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.33\\textwidth]{Interaction_block}\n    \\caption{Structure of the interaction module.}\n    \\label{fig4}\n\\end{figure}\n\nIn the above equations, $ Conv $ denotes a convolutional layer followed by BN and PReLU. All the convolutional layers have a kernel size of (1,1) and a stride of (1,1). \n\n\\subsection{Interaction Module}\n\n\\noindent In SN-Net, the speech and noise branches share the same input signal, which suggests that the internal features of two branches are correlated. In light of this, we propose an interaction module to exchange information between the branches. With this block, information transformed from the noise branch is expected to enhance the speech part and counteract the noise features inside the speech branch, and vice versa. We will show in ablation study afterwards that this module plays a key role in simultaneously modeling the speech and noises. \n\nThe structure of the interaction module is shown in Figure \\ref{fig4}. Taking speech branch as an example, feature from the noise branch $ \\mathcal{F}_N^{RA} $ is first concatenated with that from the speech branch $ \\mathcal{F}_S^{RA}$. They are then fed into a 2-D convolutional layer to generate a multiplicative mask $ \\mathcal{M}^N $, predicting the suppressed and preserved areas of $ \\mathcal{F}_N^{RA} $. A residual representation $ \\mathcal{H}^{N2S} $ is then obtained by multiplying $ \\mathcal{M}^N $ with $ \\mathcal{F}_N^{RA} $ elementally. Finally, the block adds $ \\mathcal{F}_S^{RA} $ and $ \\mathcal{H}^{N2S} $ to get a \"filtered\" version of the speech feature, which will be fed into the next RA block. The process is given by\n\\begin{equation}\n\\begin{aligned} \n    \\mathcal{F}_{S_{out}}^{RA}&=\\mathcal{F}_S^{RA}+\\mathcal{F}_N^{RA}*Mask(\\mathcal{F}_N^{RA},\\mathcal{F}_S^{RA}),\\\\\n    \\mathcal{F}_{N_{out}}^{RA}&=\\mathcal{F}_N^{RA}+\\mathcal{F}_S^{RA}*Mask(\\mathcal{F}_S^{RA},\\mathcal{F}_N^{RA}),\n\\end{aligned}\n\\end{equation}\nwhere $Mask(\\cdot)$ is short for concatenation, convolution and sigmoid operations. $(*)$ denotes element-wise multiplication.\n\n\\subsection{Merge Branch}\n\n\\noindent After reconstructing the speech and noise signals in two branches, a merge module is further employed to combine the two outputs. This is done in the time domain to achieve the cross-domain benefit \\cite{Kim2018MDPhD}. The two decoder outputs are transformed to time-domain and overlapped framed representation using the same window length as the STFT we use, resulting in $ \\tilde{s}\\in \\mathbb{R}^{T\\times K} $ and $ \\tilde{n}\\in \\mathbb{R}^{T\\times K} $, where $K$ is the frame size. These two representations are stacked with the noisy waveform $ x $ and fed into the merge branch. The merge network uses a 2-D convolutional layer, followed by an temporal self-attention block to capture global temporal dependency and two other convolutional layers to learn an element-wise mask $ m\\in \\mathbb{R}^{T\\times K} $. The kernel size of all three convolutional layers is (3,7) and the channel number is 3, 3, 1, respectively. BN and PReLU are used after each convolutional layer except the last one. Sigmoid activation is used in the last layer. Finally, the 2D enhanced signal is obtained by\n\\begin{equation}\n    \\hat{s} = m\\times \\tilde{s} + (1-m)\\times (x-\\tilde{n}).\n\\end{equation}\nThe 1D signal is reconstructed from $\\hat{s}$ after overlap and add.\n\n\\section{Experiments}\n\n\\subsection{Datasets}\n\n\\noindent Three public datasets are used in our experiments.\n\n\\textbf{DNS Challenge} The DNS challenge \\cite{reddy2020interspeech} at Interspeech 2020 provides a large dataset for training. It includes 500 hours clean speech across 2150 speakers collected from Librivox and 60000 noise clips from Audioset \\cite{gemmeke2017audio} and Freesound with 150 classes. For training, we synthesized 500 hours noisy samples with SNR levels of -5dB, 0dB, 5dB, 10dB and 15dB. For evaluation, we use 150 synthetic noisy samples without reverberation inside the test set, whose SNR levels are randomly distributed between 0 dB and 20 dB.\n\n\\textbf{Voice Bank + DEMAND} This is a small dataset created by Valentini-Botinhao et al. \\cite{valentini2016investigating}. Clean speech clips are collected from the Voice Bank corpus \\cite{veaux2013voice} with 28 speakers for training and another 2 unseen speakers for test. Ten noise types with two artificially generated and eight real recordings from DEMAND \\cite{thiemann2013diverse} are used for training. Five other noise types from DEMAND are chosen for the test, without overlapping with the training set. The SNR values are 0dB, 5dB, 15dB and 20dB for training and 2.5dB, 7.5dB, 12.5dB and 17.5dB for test. \n\n\\textbf{TIMIT Corpus} This dataset is used for our speaker separation experiment. It contains recordings of 630 speakers, each reading 10 sentences and there are 462 speakers in the training set and 168 speakers in the test set. Two sentences from different speakers are mixed with random SNRs to generate mixture utterances. Shorter sentences are zero padded to match the size of longer ones. In total, the training set includes 4620 sentences and the test set 1680 sentences.\n\n\\begin{table}[t]\n    \\centering\n    \\caption{Ablation study on DNS Challenge dataset}\n    \\label{table1}\n    \\begin{tabular}{lrcc}\n        \\hline\n        \\multicolumn{2}{l}{\\textbf{Models}} & \\textbf{SDR(dB)} & \\textbf{PESQ}\\\\\n        \\hline\n        \\multicolumn{2}{l}{Noisy} & 9.09 & 1.58\\\\\n        \\hline\n        \\multicolumn{2}{l}{Speech branch w\/o SSA (baseline)} & 18.06 & 3.05\\\\\n        \\multicolumn{2}{l}{Speech branch} & 18.75 & 3.28\\\\\n        \\hline\n        \\multicolumn{2}{l}{SN-Net w\/o interaction} & 19.04 & 3.29 \\\\\n        \\multicolumn{2}{l}{SN-Net} & \\textbf{19.52} & \\textbf{3.39} \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\\subsection{Evaluation Metrics}\n\n\\noindent To evaluate the quality of the enhanced speech, the following objective measures are used. Higher scores indicate better quality.\n\n\\begin{itemize}\n    \\item SSNR: Segmental SNR.\n    \\item SDR \\cite{vincent2006performance}: Signal-to-distortion ratio.\n    \\item PESQ \\cite{rec2005p}: Perceptual evaluation of speech quality, using the wide-band version recommended in ITU-T P.862.2 (from -0.5 to 4.5).\n    \\item CSIG \\cite{hu2007evaluation}: Mean opinion score (MOS) prediction of the signal distortion (from 1 to 5).\n    \\item CBAK \\cite{hu2007evaluation}: MOS prediction of the intrusiveness of background noises (from 1 to 5).\n    \\item COVL \\cite{hu2007evaluation}: MOS prediction of the overall effect (from 1 to 5).\n\\end{itemize}\n\n\\subsection{Implementation Details}\n\\subsubsection{Input}\nAll signals are resampled to 16kHz and clipped to 2 seconds long. We take the STFT complex spectrum as input, with a Hann window of length 20ms, a hop length of 10ms and a DFT length of 320. \n\n\\subsubsection{Loss Function}\nThe loss function includes three terms, i.e. $ \\mathcal{L}= \\mathcal{L}_{Speech}+\\alpha\\mathcal{L}_{Noise}+\\beta\\mathcal{L}_{Merge} $, where $ \\mathcal{L}_{Speech} $, $ \\mathcal{L}_{Noise} $ and $ \\mathcal{L}_{Merge}$ represent the loss of three branches, respectively. $ \\alpha$ and $\\beta $ are weighting factors balancing among the three. All terms use a mean-squre-error (MSE) loss on the power-law compressed STFT spectrum \\cite{ephrat2018looking}. An inverse STFT and forward STFT are conducted on speech and noise branches before calculating the loss to ensure STFT consistency as that in \\cite{wisdom2019differentiable}.\n\n\\subsubsection{Training}\nThe proposed algorithm is implemented in TensorFlow. We use adam optimizer with a learning rate of 0.0002. All the layers are initialized with Xavier initialization. The training is conducted in two stages. The speech and noise branches are jointly trained first with the loss weight $ \\alpha=1 $ and $ \\beta=0 $. Then the merge branch is trained with the parameters of previous two fixed, using only the loss $\\mathcal{L}_{Merge}$. We train both stages for 60 epochs for DNS Challenge and 400 epochs for Voice Bank + DEMAND dataset. The batch size for all experiments is set to 32, unless otherwise specified. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{Visualization_1_v2}\n    \\caption{Log-scale feature visualization for the fourth interaction module. (a) Input feature of speech branch. (b) Transformed feature from noise to speech branch. (c) Output feature of speech branch. (d) Input feature of noise branch. (e) Transformed feature from speech to noise branch. (f) Output feature of noise branch. Three channels with the highest activities are visualized here.}\n    \\label{fig7}\n\\end{figure}\n\n\\subsection{Ablation Study}\n\\subsubsection{Objective Quality}\nWe first evaluate the effectiveness of different parts of the proposed SN-Net based on the DNS Challenge dataset. As shown in Table \\ref{table1}, we take the speech branch without SSA as the baseline. After adding SSA to the single-branch model, we observe a 0.69 dB gain on SDR and 0.23 on PESQ. By comparing \"Speech branch\" with \"SN-Net w\/o interaction\", we can see that when no interaction is employed, adding another branch with merge module at the output only marginally improves the SDR by 0.29 dB and no improvement on PESQ. After introducing the information flow, it evidently improves the SDR by 0.77 dB and PESQ by 0.11 compared to single branch. These results verify the effectiveness of the proposed RA and interaction modules for simultaneously modeling speech and noises.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{SSA_visualization1_v5}\n    \\caption{Visualization of temporal self-attention matrices from different RA blocks. (a) Speech branch. (b) Noise branch. Each matrix is linearly scaled to $[0, 1]$.}.\n    \\label{fig5}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{SSA_visualization2_v5}\n    \\caption{Visualization of frequency-wise self-attention matrices from different RA blocks. (a) Speech branch. (b) Noise branch. Each matrix is linearly scaled to $[0, 1]$.}\n    \\label{fig6}\n\\end{figure}\n\n\\subsubsection{Visualization of Information Flow}\nIn order to further understand how the interaction module works, we visualize the input feature, the output and the feature transformed from the other branch of this module in Figure \\ref{fig7}. An audio signal corrupted by white noises is used for illustration, whose spectrum is shown in the first column.\n\nThe transformed feature shown in Figure \\ref{fig7} (b) is learned from the feature in (d) and added to the feature in (a), resulting in the output feature of speech branch in (c) and vice versa. Comparing (a) and (c), we can see that the speech area is better separated with noise after interaction. For noise branch, the speech part is mostly removed in (f) compared with (d). These results show that the interaction module indeed helps the simultaneous speech and noise modeling with better separation capabilities. In terms of interchanged information, the undesired speech part in (d) is counteracted by features learned from the speech branch (e.g., the second channel of the noise branch) and the undesired noise part in (a) is suppressed by features learned from the noise branch (e.g., the third channel of the speech branch). These observations comply with our previous analysis.\n\n\\subsubsection{Visualization of Separable Self-Attention}\nWe further visualize the attention matrix to explore what it has learned. Figure \\ref{fig5} shows the temporal self-attention matrix inside different RA blocks for the same audio signal as that in Figure \\ref{fig7}. From (a) and (b), we can see that besides the diagonal line, each frame shows strong attentiveness to other frames and speech and noise branches behave differently for each RA module. This is reasonable as the two branches model different signals and their focus differs. For noise branch, the attention goes from local to global as the network goes deeper. The noise branch shows wider attentiveness than the speech branch as white noises spread in all frames while speech signal occurs only at some time.\n\nFigure \\ref{fig6} shows the frequency-wise self-attention matrix for the same audio signal. For speech branch, the focus goes from low-frequency area to full frequencies and from local to global, showing that as the network goes deeper, the frequency-wise self-attention tends to capture global dependency along the frequency dimension. For noise branch, all four RA blocks show a local attention as white noises have a constant power spectral density.\n\n\\subsection{Comparison with the State-of-the-Art}\n\n\\subsubsection{Speech Enhancement}\n\\begin{table}[t]\n    \\centering\n    \\small\n    \\caption{Quality comparisons on Voice Bank + DEMAND}\n    \\label{table2}\n    \\begin{tabular}{lccccc}\n        \\hline\n        \\textbf{Methods} & \\textbf{SSNR} & \\textbf{PESQ} & \\textbf{CSIG} & \\textbf{CBAK} & \\textbf{COVL}\\\\\n        \\hline\n        Noisy & 1.68 & 1.97 & 3.35 & 2.44 & 2.63\\\\\n        \\hline\n        SEGAN & 7.73 & 2.16 & 3.48 & 2.94 & 2.80\\\\\n        MMSE-GAN & - & 2.53 & 3.80 & 3.12 & 3.14\\\\\n        \\hline\n        PHASEN & \\textbf{10.18} & 2.99 & 4.21 & 3.55 & 3.62\\\\\n        \\hline\n        Koizumi et al. & - & 2.99 & 4.15 & 3.42 & 3.57\\\\\n        \\hline\n        Ours & 9.83 & \\textbf{3.12} & \\textbf{4.39} & \\textbf{3.60} & \\textbf{3.77} \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\n\\begin{table}[t]\n    \\centering\n    \\caption{Quality comparisons on DNS Challenge}\n    \\label{table3}\n    \\begin{tabular}{lccc}\n        \\hline\n        \\textbf{Methods} & \\textbf{SDR(dB)} & \\textbf{PESQ} \\\\\n        \\hline\n        Noisy & 9.09 & 1.58 \\\\\n        \\hline\n        TCNN & 16.86 & 2.34  \\\\\n        TCNN-L & 16.58 & 2.78 \\\\\n        \\hline\n        Conv-TasNet-SNR & - & 2.73 \\\\\n        \\hline\n        DTLN & 16.54 & 2.34 \\\\\n        \\hline\n        MultiScale+ & - & 2.71 \\\\\n        \\hline\n        PoCoNet & - & 2.75 \\\\\n        \\hline\n        Ours & \\textbf{19.52} & \\textbf{3.39} \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\\noindent Table \\ref{table2} shows the comparisons with state-of-the-art methods on Voice Bank + DEMAND. SEGAN \\cite{pascual2017segan} and MMSE-GAN \\cite{soni2018time} are two GAN-based methods. PHASEN \\cite{yin2020phasen} is a two-branch T-F domain approach where one branch predicts the amplitude and the other predicts the phase. Koizumi et al. \\cite{koizumi2020speech} is a multi-head self-attention based method. Our method outperforms all of them in almost all metrics. The large improvements on PESQ, CSIG and COVL indicate that our method preserves better speech quality. \n\nTable \\ref{table3} shows the comparison with state-of-the-art methods on DNS Challenge dataset. TCNN \\cite{pandey2019tcnn} is a time-domain low-latency approach. We implemented two versions of it.  \"TCNN\" is exactly the same as described in the paper and \"TCNN-L\" is the long-latency version using the same T-F domain loss function as ours. Conv-TasNet-SNR \\cite{koyama2020exploring} and DTLN \\cite{westhausen2020dual} are real-time approaches. MultiScale+ \\cite{choi2020phase} and PoCoNet \\cite{isik2020poconet} are non-real-time methods, among which the PoCoNet took 1st place in the 2020 DNS challenge's Non-Real-Time track. Since narrow-band PESQ number was reported in the DTLN paper, we used the released model\\footnote{https:\/\/github.com\/breizhn\/DTLN} to generate the enhanced speech and compute the metrics. For other methods, we use the numbers reported in their papers. Our method outperforms all of them by a large margin.\n\n\\begin{table}[t]\n    \\centering\n    \\small\n    \\caption{Two-speaker speech separation on TIMIT}\n    \\label{table4}\n    \\begin{tabular}{lcc}\n        \\hline\n        \\textbf{Methods} & \\textbf{SDRi(dB)} & \\textbf{PESQ}\\tablefootnote{Note that Conv-TasNet outputs 8khz audios. We use narrow-band PESQ here instead of wide-band. Accordingly, we downsample audios to 8khz for our method to match this evaluation.}\\\\\n        \\hline\n        Conv-TasNet & 7.57 & 2.14\\\\\n        Ours & \\textbf{8.39} & \\textbf{2.50} \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\\subsubsection{Extension to Speaker Separation}\n\n\\noindent As SN-Net can simultaneously model two signals, it is natural to extend it for speaker separation task. The merge branch is removed as two outputs are needed. Permutation invariant training \\cite{yu2017permutation} is employed during training to avoid the permutation problem. We conduct the two-speaker separation experiment based on the TIMIT corpus. The batch size is set to 16. For comparison, we train a non-causal version of Conv-TasNet \\cite{luo2019conv}, the state-of-the-art method, using the released code\\footnote{https:\/\/github.com\/kaituoxu\/Conv-TasNet}.\n\nThe results are shown in Table \\ref{table4}. We use SDR improvement (SDRi) and PESQ for evaluation. Our method achieves a considerable gain on PESQ by 0.36 and SDRi by 0.82 dB, compared with Conv-TasNet. This suggests that our method is not limited to specific tasks and has the potential to extract different additive parts from a mixture signal.\n\n\\section{Conclusion}\n\n\\noindent We propose a novel two-branch convolutional neural network to interactively modeling speech and noises for speech enhancement. Particularly, an interaction between two branches is proposed to leverage information learned from the other branch to enhance the target signal modeling. This interaction makes the simultaneous modeling of two signals feasible and effective. Moreover, we design a sophisticated RA block for feature extraction of both branches, which can accommodate the diversities across speech and various noise signals. Evaluations verify the effectiveness of these modules and our method significantly outperforms the state-of-the-art. The two-signal simultaneous modeling paradigm makes it applicable to speaker separation as well. \n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSuperconductivity in V$_3$Si has been studied since 1953\\cite{Har53a} and was considered to be of  conventional s-wave, single-band nature for most of the time. It was only recently that, in the wake of MgB$_2$, V$_3$Si was placed on the list of potential two-band superconductors to explain unconventional experimental results. In particular, its superfluid density was reported  \\cite{Nef05a,Kog09a} not only to deviate strongly from the single-band BCS behavior but to match a two-band model, whose interband coupling strength is all but negligible. These conclusions were backed by infrared spectroscopy results and by calculations of the Fermi surface, which was reported to be crossed by several electronic bands \\cite{Per10a}. In contrast, a single-band description worked well for the field dependence of the specific-heat and the thermal conductivity \\cite{Boa03a}. Furthermore, the field dependence of the magnetic penetration depth and of the vortex core size, determined by muon-spin rotation, were regarded as single-band behavior \\cite{Son04b}, and in \\cite{Gur04a} the temperature dependence of the specific heat was reported to be conventional. We conclude that we face a confusing situation, which we wished to clarify by further experiments.\n\nV$_3$Si is a member of the A15 superconductors, whose crystal structure changes from cubic at room temperature to tetragonal in the superconducting state \\cite{Bat64a}. It is a type II superconductor with a Ginzburg-Landau parameter of about 20, a transition temperature of 16 - 17\\,K, and an upper critical field of around 20\\,T at 0\\,K \\cite{Orl79a}; the anisotropy of the superconducting properties is marginal \\cite{Khl99a}. The vortex distribution may change from a hexagonal to a cubic lattice as the magnetic field increases \\cite{Yet99a}.\n\nIn this article we will present new experimental results capable of probing a possible two-band state of V$_3$Si. In the next section, the experimental details and the evaluation methods will be introduced. We will start with the basic characterization of the sample, go on with the measurements of the magnetic moment, and will then show how the reversible magnetization was obtained and fitted using the Ginzburg-Landau model. Finally, the direct determination of the lower critical field will be explained. In the third section, we will present the results. First, we will summarize how two-band effects show up in MgB$_2$ and will then compare this with our findings on V$_3$Si. Potential modifications \\cite{Zeh13a} of the field dependencies will be discussed by analyzing the reversible magnetization, those of the temperature dependencies by analyzing the superfluid density and the lower critical field. We will report differences between literature data and our results and present possible reasons. In the final section, we will summarize and once again explain why we believe that V$_3$Si is a single-band superconductor.  \n\n\\section{Experiment and evaluation \\label{sec:experiment}}\n\nA V$_3$Si single crystal was cut into two pieces of equal sizes of about $3 \\times 0.55 \\times 0.55$ mm$^3$. Both samples, named VA and VB, became superconducting below 16.7\\,K with a transition width of 0.2\\,K and had a residual resistance ratio of 33.\n\nUsing a SQUID magnetometer, we measured the magnetic moment of sample VB at temperatures from 2 to 16\\,K in 1\\,K steps as a function of applied magnetic field from 0 to 7\\,T. Sample VA was analyzed with a non-commercial rotating sample magnetometer, where the sample is glued on the rim of a circular plate, which rotates at a frequency of 15\\,Hz. During one rotation the sample passes four pick-up coils, where it induces electrical voltages proportional to its magnetic moment. For details about the instrument and how the magnetic moment is determined, the reader is referred to \\cite{Eis11a}. The main advantage of the rotating sample magnetometer is its fitting into a cryostat with a 15\\,T magnet. Accordingly, magnetization loops up to 15\\,T were recorded at temperatures of 5.2, 7, 9, 11, 13, and 15\\,K.\n\nThe resulting magnetization loops, either measured in the SQUID or in the rotating sample magnetometer, revealed reversible behavior over most of the field range. Irreversible effects, caused by flux-pinning, emerged merely at low fields and became dominant near 0\\,T. For instance, the critical current density at 5.2\\,K, evaluated using the methods presented in \\cite{Zeh09a}, decreased with field from some 10$^9$\\,Am$^{-2}$ at 0.1\\,T to $5 \\times 10^7$\\,Am$^{-2}$ at 1\\,T and to a negligible value at and above 2\\,T. Accordingly, more than 85 per cent of the loop fell into the reversible regime. The irreversible effects became slightly larger at lower temperatures but considerably smaller at higher temperatures. The critical currents, which are proportional to the hysteresis width of the magnetization loops, were found to be somewhat larger in the SQUID than in the rotating sample magnetometer, yet the reversible parts agreed well.        \n\nAs already mentioned, irreversibility appeared merely at low fields, but even there, knowing the magnetization as a function of increasing, $M(H_{\\rm a}^+$), and decreasing applied field, $M(H_{\\rm a}^-$), allowed us to determine the reversible part via $M_{\\rm r}(H_{\\rm a})$ $\\simeq$ 0.5[$M(H_{\\rm a}^+)$ + $M(H_{\\rm a}^-)$], where $H_{\\rm a}^+$ and $H_{\\rm a}^-$ refer to the same applied field $H_{\\rm a}$. This procedure gives reliable results as long as the hysteresis width is not much larger than the corresponding reversible signal. To get $M_{\\rm r}(B)$ from $M_{\\rm r}(H_{\\rm a})$, we calculated the magnetic induction via $B = \\mu_0(H - D M_{\\rm r} + M_{\\rm r})$, with $\\mu_0$ = $4 \\pi \\times 10^{-7}$\\,NA$^{-2}$, $H$ the applied field corrected by the field induced by the macroscopic currents \\cite{Zeh09a}, and $D$ the numerically calculated demagnetization factor of the sample in the Meissner state. \n\nThe next step was to compare the reversible magnetization, acquired from experiment, with theory. The theoretical magnetization curves of a single-band superconductor were taken from approximate equations of the Ginzburg-Landau theory, provided by Brandt \\cite{Bra03a}. According to this paper \\cite{Bra03a} the approximation errors should be less than two per cent, for the Ginzburg-Landau parameter of V$_3$Si is large. The Ginzburg-Landau curves depend on just two parameters, namely the upper critical field, at which the magnetization becomes zero, and the Ginzburg-Landau parameter, which determines the shape of the curve. The well-known Ginzburg-Landau relations \\cite{Tin75a} allowed us to calculate further quantities, such as the lower and the thermodynamic critical field, the coherence length, the magnetic penetration depth, and the superfluid density.  \n\nThe lower critical field was additionally determined by measuring the field at which vortices start to penetrate the sample. After having minimized the stray fields in the SQUID cryostat, we cooled the sample below its transition temperature in zero field. Then we enhanced the applied field stepwise but interrupted each step by measuring the corresponding remanent magnetic moment in zero field. The remanent moment should vanish at low fields and start to increase above the lower critical field, where vortices are created and pinned in the sample.\n\n\n\n\\section{Results and discussion}\n\nIn this section we will analyze whether superconductivity in V$_3$Si is more reliably described by a single or a two-band model. If V$_3$Si is a two-band superconductor, we expect some properties to deviate from the single-band BCS behavior \\cite{Zeh13a}. Possible effects on the field dependence will be discussed via the reversible magnetization and possible effects on the temperature dependence via the superfluid density. The results will be compared with the behavior of MgB$_2$, a well-known two-band material. We will see that the single-band models provide a fair description of our results in V$_3$Si.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[clip, width = 8.5cm]{Fig1.eps}\n   \\caption{\\label{MgB2GLFit} The reversible magnetization of MgB$_2$ as a function of the magnetic induction. The open circles show the experimental data for the applied field oriented parallel ($H \\parallel c$) or perpendicular ($H \\parallel ab$) to the uniaxial sample axis at 10, 20, and 30\\,K (the reduced temperatures are 0.26, 0.51, and 0.77). The solid lines are fits of the single-band Ginzburg-Landau model to the experimental data over the whole field range, while the broken lines are such fits either to the low or to the high field part of the experimental data, reflecting  the two-band nature of MgB$_2$. For details see \\cite{Zeh04b}.}\n\\end{figure}\n\nWe start by summarizing some two-band effects of MgB$_2$ \\cite{Eis07a}. This material consists of the near-isotropic $\\pi$-band and the anisotropic $\\sigma$-band, having similar electronic densities of states. Due to interband coupling, the gaps are expected to close at the same field. Yet the superconducting properties of the $\\pi$-band are heavily suppressed above a particular field, which is commonly called the upper critical field of the $\\pi$-band and whose value is about a third of that of the $\\sigma$-band. Accordingly, the field dependence of several superconducting properties deviates significantly from the single-band behavior. This is illustrated in figure~\\ref{MgB2GLFit}, where the reversible magnetization of an MgB$_2$ single crystal, indicated by open circles, is shown as a function of the magnetic induction. The solid lines present the single-band Ginzburg-Landau fits. Barring the 30\\,K results, the agreement between theory and experiment is poor and the differences are not merely of quantitative but also of qualitative nature. In contrast, the curves can be nicely fitted by {\\it two} single-band Ginzburg-Landau curves, as shown by the broken lines in the diagrams. One is adjusted to the low and the other to the high field region of the experiment, thus reflecting the two bands with their different upper critical fields \\cite{Zeh04b}. Applying the methods to different field orientations reveals the different anisotropies of the two bands. Similar effects were observed in NbSe$_2$ \\cite{Zeh10a}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[clip, width = 8.5cm]{Fig2.eps}\n   \\caption{\\label{GLFit} The magnetization of V$_3$Si as a function of magnetic induction. The open circles  show the reversible data at 15, 13, 11, 9, 7, and 5.2\\,K (the reduced temperatures are 0.90, 0.78, 0.66, 0.54, 0.42, and 0.31) and the solid lines the corresponding Ginzburg-Landau model fits. The solid lines of the insets show the irreversible data.}\n\\end{figure}\n\nWe are now prepared to shift our focus to V$_3$Si. Figure~\\ref{GLFit} shows the magnetization as a function of the magnetic induction at temperatures of about 15, 13, 11, 9, 7, and 5\\,K. The open circles indicate the reversible magnetization acquired from the rotating sample magnetometer measurements of a V$_3$Si single crystal and the solid lines the single-band Ginzburg-Landau behavior adjusted to the experimental data. In the insets, the solid lines indicate the irreversible magnetization, though only  at low fields, where a significant hysteresis shows up. At not too low temperatures, say about 9 - 15\\,K, we consider the agreement between experiment and single-band theory very good. As the temperature is reduced, the differences between theory and experiment get larger,  becoming apparent at low fields, for we adjusted the fits mainly to the high field regions, where the experiments reveal the reversible data directly. \n\nNext, we will analyze the quality of the fits in more detail. First, we determined the areas under the reversible magnetization curves, which are proportional to the condensation energies. The ratio of the condensation energy obtained from the Ginzburg-Landau fit to that from the experimental curves is considered a sensible measure for the fit quality. In MgB$_2$, the differences in the condensation energies of the high-field fit and the experimental data were some 18 per cent at 30\\,K and 30 per cent at 10 and 20\\,K. In V$_3$Si, we found much smaller deviations, namely 3 - 8 per cent at 9 - 15\\,K and 10 - 12 per cent at 7 and 5.2\\,K. So, even the low-temperature fits of V$_3$Si agree with the experimental data better than any single-band fit of MgB$_2$.\n\nThere are several possible reasons for the larger deviations between theory and experiment in V$_3$Si at low temperatures. To begin with, as shown in figure~\\ref{GLFit}, the low-field irreversible magnetization becomes large at low temperatures, thus enhancing possible errors in the corresponding reversible data. The insets of figure~\\ref{GLFit}, however, illustrate that the irreversible magnetization does not become much larger than the reversible data even at low temperatures and small fields, and hence the errors in calculating the reversible magnetization are not serious. On the other hand, we are faced with the shortcomings of the Ginzburg-Landau theory. As this theory is derived from BCS theory in the vicinity of the transition temperature, the potential errors grow when we go to lower temperatures. Yet the theory has been successfully applied to evaluating and describing experimental data at high and low temperatures, as a function of temperature and as a function of field in a large number of publications. In particular, using adjustable parameters instead of the microscopic BCS ones apparently extends the applicability to much lower temperatures (e.g. \\cite{Sil12a}). Watanabe et al.~\\cite{Wat05a} calculated the field dependence of the reversible magnetization of an s-wave system using the Eilenberger equations, which hold at arbitrary temperatures. We verified that the Ginzburg-Landau model with a Ginzburg-Landau parameter of 49 reproduces the Eilenberger curve of \\cite{Wat05a} close to the transition temperature (see also \\cite{Bel12a}). Reducing the temperature makes the Eilenberger curves slightly steeper at low fields and slightly flatter at high fields. This is basically what we find for the experimental curves in figure~\\ref{GLFit}, namely reducing the temperature makes the experimental curves slightly steeper at low fields and slightly flatter at high fields compared with the Ginzburg-Landau fit. Accordingly, the qualitative deviations between our experiments and the fits are just as expected when we assume that the Eilenberger model describes experiment at all temperatures. The deviations from the Ginzburg-Landau model, assessed via the condensation energies, agree even quantitatively, i.e., we found some 12 per cent for the experimental data and 14 per cent for the Eilenberger curves at a reduced temperature of about 0.3. Granted, the Eilenberger calculations and our experimental data refer to systems with different Ginzburg-Landau parameters ($\\kappa$), but both systems belong to the high-$\\kappa$ regime and hence should behave similarly. To conclude, the changes of the Eilenberger curves are not substantial when the temperature is lowered, which affirms the usefulness of our approach. Still, we do not expect an exact description at the lowest temperatures of figure~\\ref{GLFit}.\n\nIn contrast to V$_3$Si, presented in figure~\\ref{GLFit}, the experimental data of MgB$_2$, presented in figure~\\ref{MgB2GLFit}, also qualitatively disagree with the single-band models. In particular, we found the high-field fit to lead to a much smaller lower critical field, i.e. the magnetization value at $B = 0$\\,T, than obtained from experiment via extrapolation of the low-field data, while in V$_3$Si the two curves result in almost the same lower critical fields.\n\nSummarizing, we consider the agreement between single-band theory and experiment in V$_3$Si as good as can be expected in view of the uncertainties within both the theoretical and the experimental data. Accordingly, we conclude that the reversible magnetization of V$_3$Si supports the single-band scenario.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[clip, width = 8.5cm]{Fig3.eps}\n   \\caption{\\label{Bc2} The upper critical field, presented in the left panel, and the Ginzburg-Landau parameter, presented in the right panel, of V$_3$Si as a function of temperature. The open circles show experimental data acquired from the rotating sample magnetometer and the full circles experimental data from the SQUID measurements. The upper critical field follows the clean-limit BCS behavior, depicted by the solid line, leading to 22.7\\,T at 0\\,K. The Ginzburg-Landau parameter decreases roughly linearly from about 24 to 19 as the temperature increases from 0\\,K to the transition temperature.}\n\\end{figure}\n\nFigure~\\ref{Bc2} presents the upper critical field and the Ginzburg-Landau parameter of V$_3$Si. The open circles indicate results from the rotating sample and the full circles those from the SQUID magnetometry. The upper critical field has been evaluated by adjusting the Ginzburg-Landau model merely to the high-field regime of the reversible magnetization and may thus slightly deviate from the results obtained from full range fits. The solid line presents the clean-limit single-band BCS behavior \\cite{Hel66a}, is in excellent agreement with our experimental data, and leads to 22.7\\,T at 0\\,K, which matches literature data well \\cite{Orl79a}. In contrast to many two-band materials, no clear upward curvature near the transition temperature appears \\cite{Shu98a,Zeh02a,Mun12a}. The Ginzburg-Landau parameter, taken from fits over the whole field range and presented in the right panel, decreases quite linearly from about 24 at 0\\,K to 19 at the transition temperature, a behavior that is also close to the single-band BCS prediction \\cite{Hel66a}. Calculating further properties employing the Ginzburg-Landau relations \\cite{Tin75a} resulted in about 90\\,nm for the magnetic penetration depth, 4\\,nm for the coherence length, 0.6\\,T for the thermodynamic critical field, and 0.07\\,T for the lower critical field at 0\\,K. The critical lengths are close to literature data \\cite{Son04b,Yet05a}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[clip, width = 8.5cm]{Fig4.eps}\n   \\caption{\\label{SFLD} The superfluid density as a function of reduced temperature (left panel) and the reduced lower critical field as a function of temperature (right panel). In the left panel, the open circles show our experimental result on V$_3$Si, obtained from the reversible magnetization, which is in good agreement with the single-band BCS behavior, indicated by the solid line. The broken and the dot-dashed lines show the two-band-like superfluid density of V$_3$Si reported in Refs.~\\cite{Nef05a} and \\cite{Kog09a} schematically; the full diamonds illustrate the two-band behavior of MgB$_2$ \\cite{Zeh04b}. In the right panel, the open circles show the reduced lower critical field of our V$_3$Si single crystal, evaluated from the reversible magnetization, which is all but identical to the superfluid density of the left panel. The open diamonds and the full squares are obtained from direct measurements of the first vortex-penetration field as indicated by the arrows in the inset. The symbols in the inset present the square root of the remanent magnetic moment as a function of applied field at 7\\,K; the solid line is a linear fit to the high-field data.}\n\\end{figure}\n\nWe proceed by analyzing the temperature dependent effects by means of the superfluid density. Having evaluated the upper critical field, $B_{\\rm c2}$, and the Ginzburg-Landau parameter, $\\kappa$, via the above fit procedure, we acquired the magnetic penetration depth, $\\lambda$, by using the Ginzburg-Landau relations, and the superfluid density, $\\rho_{\\rm s}$, via $\\rho_{\\rm s}$(T) = [$\\lambda$(0\\,K) \/ $\\lambda$(T)]$^2$, with $\\lambda = \\kappa \\xi$, $\\xi^2 = \\Phi_0 \/ (2 \\pi B_{\\rm c2})$, and $\\Phi_0 \\simeq 2.07$ $\\times 10^{-15}$\\,Vs. To assess the penetration depth at 0\\,K, we used the SQUID measurements, where the temperature could be reduced to 2\\,K, but the magnetic fields were limited to 7\\,T. We therefore took the inter- and extrapolated upper critical fields from the rotating sample magnetometer results for fitting the SQUID data, so that only the Ginzburg-Landau parameter remained to be determined. To justify the use of these data, we verified that the reversible curves from SQUID and rotating sample magnetometer agree in the overlapping field and temperature range.\n\nThe left panel of figure~\\ref{SFLD} presents the superfluid density of V$_3$Si, indicated by open circles, as a function of reduced temperature. The solid line shows the expected behavior of a single-band BCS superconductor, which is close to our experimental data. Figure~\\ref{SFLD} shows also the superfluid density of MgB$_2$, indicated by full symbols, as an example for two-band superconductivity. In comparison with V$_3$Si, the MgB$_2$ curve decreases much faster at low temperatures and then becomes almost linear as the temperature increases. To explain this behavior, we need to consider that a two-band superconductor has two distinct energy gaps. The smaller gap reduces the excitation energy on the corresponding part of the Fermi surface and hence makes the superfluid density decrease more rapidly at small temperatures. This may lead to a near-linear behavior at intermediate temperatures, as found in several two-band materials, such as MgB$_2$, NbSe$_2$, and the iron-based superconductors \\cite{Man02a,Fle07a,Mar09b}. We conclude that also the superfluid density of V$_3$Si supports the single-band scenario.\n\nWe now come back to the two-band scenario of V$_3$Si proposed in Refs.~\\cite{Nef05a} and \\cite{Kog09a}, based on measurements of the superfluid density. The broken line in figure~\\ref{SFLD} shows the result presented in \\cite{Nef05a}, which was obtained by measuring the microwave surface impedance of a single crystal. There is no doubt that this curve is totally different from our result, that is, it decreases more rapidly at low temperatures and has a sharp kink at about 0.6\\,$T_{\\rm c}$. Analyzing their data with a two-band model, the authors found distinctly different energy gaps, though similar intraband coupling strengths for the two bands and, as indicated by the sharp kink at intermediate temperatures, almost negligible interband coupling. The superfluid density published by Kogan et al.~\\cite{Kog09a}, shown schematically in figure~\\ref{SFLD} by the dot-dashed line, was measured by a tunnel diode resonator technique and is quite different from that of \\cite{Nef05a}, yet it also reflects a two-band scenario with similar intraband and near-negligible interband coupling strengths. \n\nWhat are the possible reasons for the differences between those literature \\cite{Nef05a,Kog09a} and our data? Two points are obvious. First, while we evaluated the penetration depth from fits to the reversible magnetization over the whole field range, save for very small fields, where the experimental data are not available or not reliable, the authors of Refs.~\\cite{Nef05a} and \\cite{Kog09a} evaluated their data solely at very low fields. Thus, a second band, but one with a very small upper critical field, would resolve the contradictions. The second point is that while our method probes the bulk, the methods of Refs.~\\cite{Nef05a} and \\cite{Kog09a} probe the sample surface. Thus, surface irregularities or inhomogeneities that may change the properties on the surface would also resolve the contradictions. Such a statement, however, remains a speculation, for we know nothing about the surfaces of the samples used in the studies. Diener et al.~\\cite{Die09a} faced that problem in MgCNi$_3$. Acquiring data with the same method as used in \\cite{Kog09a} resulted in a penetration depth behavior similar to the broken curves of figure~\\ref{SFLD}, while acquiring the results from measurements of the lower critical field resulted in a BCS-like behavior. The authors suggested that those differences might be caused by inhomogeneities at the sample surface and hence considered the BCS-like behavior correct. \n\nWe also determined the superfluid density at very low magnetic fields,  namely by measuring the lower critical field directly. But measuring the lower critical field directly is anything but trivial. To begin with, we can merely determine the field at which flux lines start to penetrate into the sample. This is accomplished by recording the remanent magnetic moment of the sample in zero field as a function of the maximum applied field, as described at the end of Sec.~\\ref{sec:experiment}. Unfortunately, this first penetration field is usually not simply connected with the lower critical field via a single-valued demagnetization factor. First, sample edges give rise to very high stray fields, which may surpass the lower critical field and hence enforce the creation of vortices at very low applied fields, but we do not know well how the local induction is related to the applied field in such a configuration. Second, surface irregularities would affect the creation of vortices, while clean surfaces may induce additional barriers. \n\nSo, how can we acquire useful results from that procedure? To begin with, we are mainly interested in the relative temperature dependence of the lower critical field and not so much in its absolute value. Second, we aligned the longest length of our sample, the size of which is $3 \\times 0.55 \\times 0.55$ mm$^3$, parallel to the applied field, so that the geometry effects were as small as possible and assumed those geometry effects to be temperature independent. As expected, when we increased the applied field, the remanent magnetic moment remained zero at the beginning, then started to rise at a slope that became gradually steeper, indicating that first vortices had been formed and pinned in the sample, and eventually increased quadratically with field (see inset of figure~\\ref{SFLD}). Assuming a constant current density parallel to the sample borders and ignoring the stray fields revealed this quadratic behavior also in calculations (cf. with \\cite{Mos91a}). We thus consider the extrapolated onset of the quadratic behavior (cf. with inset of figure~\\ref{SFLD}) as a reliable assessment of the lowest field where vortices parallel to the applied field were in the sample. \n\nWe determined both the smallest field at which we observed a slope in the remanent magnetic moment and the field obtained from extrapolating the quadratic part to zero (see inset of figure~\\ref{SFLD}). The temperature dependence of both was found in good agreement with the superfluid density acquired from the Ginzburg-Landau fits, as shown in the right panel of figure~\\ref{SFLD}. Note that Ginzburg-Landau theory predicts the same temperature dependence for the superfluid density and the lower critical field when the changes in the logarithm of the Ginzburg-Landau parameter can be ignored, as is the case in our sample. To assess the absolute lower critical field values, we multiplied the penetration fields with the factor $(1-D)^{-1}$ $\\simeq$ 1.15, where $D \\simeq 0.13$ is the averaged demagnetization factor. In comparison to the lower critical fields from the Ginzburg-Landau fits, leading to 68\\,mT at 0\\,K, this led to lower values by some 10 per cent, namely 60\\,mT at 0\\,K, from evaluating the onset fields, and to larger values by 40 per cent, namely 95\\,mT at 0\\,K, from evaluating the fields where the quadratic behavior started. We consider these differences to be reliable. Finally, we conclude that also at very low magnetic fields, the temperature dependence shows no indication of a two-band behavior.\n\nHaving presented clear support for the single-band behavior, we may ask if our results rule out the two-band scenario completely. This is not the case, for our experiments would not detect a second band if its contribution to the measured quantities were marginal or if its superconducting properties were all but identical to the first band. These scenarios, however, would be different from those proposed in literature \\cite{Nef05a,Kog09a}. Which properties would a hypothetical second band have in our sample? Above, we have discussed the fit quality of the reversible magnetization data in terms of the ratio of the condensation energies from the fit and from experiment. In MgB$_2$ the differences, which are some 30 per cent at low temperatures, are ascribed to the two-band scenario. Accordingly, 70 per cent of the condensation energy are induced by the $\\sigma$-band, which is the band probed by the high-field fits \\cite{Zeh04b}, and 30 per cent by the second band, the $\\pi$-band, {\\it and} interband coupling effects, in rough agreement with \\cite{Eis05a}. In V$_3$Si, the deviations are much smaller and we have shown that they can qualitatively and quantitatively be explained by the imperfections of the Ginzburg-Landau model. Nevertheless, we analyzed the results assuming the deviations to be caused by a hypothetical second band. For example, at 9\\,K the fit error was 5 per cent and hence the second band would contribute to the condensation energy ($E_{\\rm c}$) by {\\it less} than 5 per cent, for these 5 per cent include the interband contributions. Employing BCS theory, where $E_{\\rm c}$ = $0.5 N \\Delta^2$, and assuming both bands to have the same density of states ($N$) would result in a gap ($\\Delta$) ratio larger than 4.4 : 1 (1 : $\\sqrt{5\/95}$), which would be quite large and much larger than the ratios proposed in Refs.~\\cite{Nef05a,Kog09a}, namely 1.4 - 1.8 : 1. \n\nStrong interband effects would also mask two-band effects \\cite{Nic05b}, though it should be noticed that the two-band scenario for V$_3$Si \\cite{Nef05a, Kog09a} came along with a negligible interband coupling strength and the high purity of our single crystals makes strong interband impurity-scattering unlikely. \n\n\\section{Summary and conclusion}\n\nLet us sum up what we have learned from testing V$_3$Si for two-band superconductivity. We have shown that the field dependence of the reversible magnetization matches the single-band Ginzburg-Landau theory reliably well. The differences between the experimental data and the theoretical fits grow as the temperature is reduced but remain small and can qualitatively and quantitatively be explained by the imperfections of the Ginzburg-Landau model. We have also found that the temperature dependence of the superfluid density, determined with different methods at different magnetic fields, follows single-band BCS behavior. Accordingly, all our results support a single-band scenario for V$_3$Si. \n\n\\section*{Acknowledgments}\nThis work was supported by the Austrian Science Fund under Contract No. 21194\nand 23996.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}