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+{"text":"\\section{Introduction} \nControlling the photonic transport in integrated circuits is important for building future information and communication technologies. While a remarkable progress has been achieved on early implementations of such technologies \\cite{Asavanant,Larsen,Juan}, the development of an optical rectifier that can act on single or a few photon pulses remain challenging. These rectifying devices are useful for routing and isolating signals. Rectifiers based on ferromagnetic compounds have been demonstrated for increasing signal processing capabilities. However, they are inherently lossy and difficult to miniaturize and use in integrated circuits \\cite{Pozar}. Recently, using a semi-classical theory, a two atoms device has been identified as a candidate for exhibiting strong directionality \\cite{auffeves2014}. The transmission of light through this device would depend on the direction of propagation. If the input comes from one side it will pass through it, and if it comes from the other side it will get reflected. This has triggered a debate over whether such a device can perform with non-classical states of light \\cite{Alex,Auffeves2016,Vienna2016,Combes,Fang,Clemens,Gonzalez,Ordonez}.\n\nThis rectifying device, also referred to as \\textit{optical diode}, is consisting of a pair of two-level systems coupled to a one-dimensional waveguide. By changing the distance between the atoms and the detuning of their resonance frequencies, the efficiency of the diode was optimized. It has been shown that for light pulses in coherent states, the device can exhibit efficiencies of more than 60\\% \\cite{Alex,Auffeves2016,Vienna2016,Combes}. However, in the monochromatic limit, it has been found that this device cannot rectify single photon pulses \\cite{Alex}. Since significant directionality can be achieved for coherent inputs while a symmetric behaviour is predicted for single photon pulses, one might naturally ask: which components of the coherent state triggers the performance of the device? Do superpositions matter? How can we change these critical components, so we can improve the efficiency of this device for inputs with a few photon pulses? To answer this questions, we will use the same approach as in \\cite{Alex}. Using the Heisenberg equations of motion, we will show that by changing the length of the light pulse, we can reduce the number of photons necessary to activate the rectifying behaviour of the optical diode. \n\nThis article is organized as follow: in section II, we will introduce the theoretical model for this device and present how the rectification factor of the device is computed. In section III, we will discuss the cases of photon-number pulses. We will show that the main terms that will contribute to the rectification depend directly on the pulse duration. In section IV, we will present a few concluding remarks and discuss potential future directions of this research.\n\n\\section{The model}\n\n\\subsection{The rectifying device}\n\nAs illustrated in Fig.~\\ref{fig:diode}, the system of interest here is composed of a pair of atoms strongly coupled to a one dimensional waveguide. Each atom is model as a two-level system, for which $\\ket{g_j}$ and $\\ket{e_j}$ are the ground and excited states of the $j^{th}$ atom, respectively. The atom on the right is resonant with the central frequency of the light pulse $\\omega_0$. On the other hand, the atom on the left is detuned. We denote its resonance frequency by $\\omega_1$. Both frequencies are assumed to be much larger than the cutoff frequency of the waveguide \\cite{Snyder}. In this case, the dispersion relations read $\\omega = v_g \\abs{k}$, such that $k$ is the longitudinal wave-number of the field mode and $v_g$ is the group velocity of the light pulse in the waveguide. The free Hamiltonian of this system reads\n\\begin{equation} \\label{H0}\n    \\hat{\\mathcal{H}}_{0} = \\sum_{j=1}^{2}  \\hbar \\omega_j \\ketbra{e_j} + \\int_{0}^{\\infty} \\dd \\omega \\hbar \\omega  (\\hat{a}^{\\dagger}_{\\omega}\\hat{a}^{}_{\\omega} + \\hat{b}^{\\dagger}_{\\omega}\\hat{b}^{}_{\\omega}) \\;,\n\\end{equation}\nsuch that the first term corresponds to the two atoms. The second term describes the field modes. The operator $\\hat{a}^{\\dagger}_{\\omega}$ ($\\hat{b}^{\\dagger}_{\\omega}$) creates a photon with frequency $\\omega$ moving towards the right (left). The interaction between the atoms and the propagating light pulse is described by the dipole Hamiltonian within the rotating wave approximation \\cite{Snyder,Roulet}\n\\begin{equation} \\label{Hdip}\n\\begin{split}\n    \\hat{\\mathcal{H}}_{\\mathrm{dip}} =& -i \\hbar \\sum_{j=1}^{2} \\int_{0}^{\\infty} \\dd \\omega g_{\\omega}^{(j)} \\\\\n    \\times \\Big[ & \\hat{\\sigma}_{+}^{(j)} \\left( \\hat{a}_{\\omega} e^{i \\omega \\frac{ x_j}{v_g}} + \\hat{b}_{\\omega} e^{- i \\omega \\frac{ x_j}{v_g}}\\right)e^{ - i (\\omega - \\omega_j)t} - \\textbf{H.c.} \\Big] \\;.\n\\end{split}\n\\end{equation}\nHere $x_j$, $g_{\\omega}^{(j)}$ and $\\hat{\\sigma}_{+}^{(j)} = \\ketbra{e_j}{g_j} $ are the position, the coupling strength and raising operator of the $j^{th}$ atom, respectively. Similarly to the previous study \\cite{Alex}, we assume the Weisskopf-Wigner approximation \\cite{Scully}, i.e. both atoms have the same coupling to the waveguide, $g_{\\omega}^{(1)} = g_{\\omega}^{(2)} = g$. Thus, the decay rate to the waveguide for both atoms is $\\gamma = 2\\pi g^2$.\n\n\\subsection{Computing rectification}\n\nSeveral figures of merit have been used in previous papers to capture rectification: we review them in Appendix \\ref{app:defs}. Here we define the rectifying factor of the device as\n\\begin{equation}\\label{DiodeEff}\n    \\mathcal{R} = T_{\\rightarrow} - T_{\\leftarrow}\\,,\n\\end{equation} where $T_{\\rightarrow}$ is the transmittivity of the light coming from the left (i.e.~the fraction of left-incoming light scattered towards the right); analogously, $T_{\\leftarrow}$ is the fraction of right-incoming light scattered towards the left. Clearly $-1\\leq\\mathcal{R}\\leq 1$, the sign determining the direction in which the diode rectifies. Explicitly,\n\\begin{eqnarray}\\label{eqsT}\nT_{\\rightarrow}&=&\\frac{1}{F}\\lim_{t\\rightarrow\\infty}N_a^{\\rightarrow}(t)\\,=\\,\\frac{1}{F}(1-\\lim_{t\\rightarrow\\infty}N_b^{\\rightarrow}(t))\\,,\\\\\nT_{\\leftarrow}&=&\\frac{1}{F}\\lim_{t\\rightarrow\\infty}N_b^{\\leftarrow}(t)\n\\end{eqnarray}\nwhere $F$ is the mean photon flux per unit time, and where \\begin{eqnarray*}\nN_{b}^{\\textrm{input}} (t) = \\int_{0}^{\\infty} \\dd \\omega \\textrm{Tr}\\left[\\hat{b}^{\\dagger}_{\\omega}(t)\\hat{b}^{}_{\\omega}(t)\\,\\rho_{\\textrm{input}}\\right]\\;,\n\\end{eqnarray*} (and similar definitions for similar quantities).\n\nIn this paper, we shall consider monomode Fourier-limited pulses. The mode that defines a pulse coming from the left is \n\\begin{equation}\\label{hatA}\n\\hat{A}^\\dagger = \\int_{0}^{\\infty} \\dd\\omega f (\\omega) \\hat{a}_{\\omega}^{\\dagger} = \\int_{0}^{\\infty} \\dd\\tau \\xi (\\tau) \\hat{a}_{\\tau}^{\\dagger}  \\;,\n\\end{equation}\nwhere\n\\begin{equation}\n    \\begin{split}\n    \\hat{a}_{\\tau}^{} =& \\frac{1}{\\sqrt{2\\pi}} \\int \\dd\\omega \\hat{a}_{\\omega}^{} e^{- i (\\omega - \\omega_0) \\tau} \\;,\\\\\n    \\xi (\\tau) =& \\frac{1}{\\sqrt{2\\pi}} \\int \\dd\\omega f (\\omega) e^{- i (\\omega - \\omega_0) \\tau}\\,.\n    \\end{split}\n\\end{equation} For pulses coming from the right, we use the identical definition of a mode $\\hat{B}$, replacing the operators $(\\hat{a}_{.},\\hat{a}^\\dagger_{.})$ with $(\\hat{b}_{.},\\hat{b}^\\dagger_{.})$.\n\nLike in previous works, for definiteness the calculations will assume a square pulse\n\\begin{equation}\\label{xitau}\n\\xi (\\tau) = \\Bigg\\{\n    \\begin{array}{cl}\n        \\sqrt{\\Omega\/2}  &  \\text{for } 0 \\leq \\tau \\leq 2\/\\Omega,  \\\\\n        0 & \\text{otherwise}.\n    \\end{array}\n\\end{equation}\nFor an input state $\\ket{\\psi}$, the mean flux of photons coming from the left is \n\\begin{equation}\\label{flux}\n    F = \\textrm{Tr}\\left[\\hat{a}_\\tau^\\dagger(0)\\hat{a}_\\tau(0)\\rho_{\\textrm{input}}\\right]\\;,\n\\end{equation}\n\nA basis of the total states of the system is denoted $\\ket{n_a,n_b,s_1,s_2}$ where $n_{a,b}$ is a Fock state in mode $\\hat{A}$ or $\\hat{B}$, while $s_j\\in\\{g,e\\}$ is the state of the $j$-th atom. Initially, the pulse will come from either the left or the right; and both atoms are assumed to be in the ground state to ensure that the rectifying device is passive. Thus our input states will always be of the form $\\ket{\\psi_a}\\equiv\\ket{\\psi,0,g,g}$, or $\\ket{\\psi_b}\\equiv\\ket{0,\\psi,g,g}$.\n\nInstead of solving the time-dependent equations of motion and sending $t\\rightarrow\\infty$, we shall consider only long pulses and thus assume that the system is in the steady state:\n\\begin{eqnarray}\n\\lim_{t\\rightarrow\\infty}N_b^{\\textrm{input}}(t)&\\approx&\\int_{0}^{\\infty} \\dd \\omega \\expval{\\hat{b}^{\\dagger}_{\\omega}\\hat{b}^{}_{\\omega}}_{ss,\\textrm{input}}\\,.\\label{ss:assume}\n\\end{eqnarray}\n\n\n\\begin{figure}\n    \\centering\n        \\includegraphics[width=0.5\\textwidth,trim={0 9cm 0 0cm},clip]{figures\/Diode.pdf}\n    \\caption{A scheme of the rectifying device consisting of two 2-level systems coupled to a one-dimensional waveguide. The one on the right is resonant with the central frequency of the light pulse, $\\omega_0$. The atom on the left is detuned, $\\omega_1 \\neq \\omega_0$.}\n    \\label{fig:diode}\n\\end{figure}\n\n\\subsection{The case for pulses of finite length}\n\nAll previous fully-quantum studies of this rectifying device \\cite{Alex,Vienna2016,Auffeves2016} concluded that there is no rectification for single-photon Fock states. This conclusion was drawn in the limit of pulses of \\textit{infinite length}. One of the results of the next Section will be that \\textit{the device does show some rectification for single-photon Fock states} for pulses of finite length.\n\nThose same works also showed that high rectification is expected for coherent states, in qualitative (if not exact quantitative) agreement with a previous semi-classical study \\cite{auffeves2014}. Specifically, rectification was predicted \\cite{Alex} and later observed \\cite{Clemens} to have a high-value plateau when $F\/\\gamma$ ranges between $10^{-3}$ and $10^{-1}$. Let us now describe this case in more detail, to understand what may be the origin of the rectification.\n\nEvery coherent state is ultimately monomode \\cite{Alex90s,Alex90ss}: a coherent state pulse with average number of photons $\\bar{n}$, propagating towards the right, can be written as\n\\begin{equation}\\label{state:cohbis}\n    \\ket{\\alpha_a} = \\exp\\big(\\sqrt{\\bar{n}} (\\hat{A}^\\dagger - \\hat{A})\\big) \\ket{\\text{\\o{}}}\\;.\n\\end{equation} Since $\\hat{a}_\\tau(0)\\ket{\\alpha} = \\sqrt{\\bar{n}}\\, \\xi(\\tau) \\ket{\\alpha}$, the flux \\eqref{flux} is given by $F=\\frac{\\bar{n} \\Omega}{2}$. If the device performance is maximized for plateau centred around $F\/\\gamma \\approx  10^{-2}$, then the optimal average number of photons in the pulse is\n\\begin{equation}\n   \\bar{n}_{\\textrm{opt}} \\approx \\frac{2 \\gamma }{\\Omega}  \\times 10^{-2}\\,.\n\\end{equation}\nIf we were to consider infinitely long pulses, as used in the single-photon case, then we would have to take the monochromatic limit $\\Omega \\rightarrow 0$; in which case $\\bar{n}_{\\textrm{opt}}$ diverges.\n\n\n\\begin{figure}\n    \\centering\n        \\includegraphics[width=0.23\\textwidth]{figures\/f1s2.pdf}\n        \\includegraphics[width=0.23\\textwidth]{figures\/p1pow.pdf}\\\\\n        \\includegraphics[width=0.23\\textwidth]{figures\/f2s2.pdf}\n        \\includegraphics[width=0.23\\textwidth]{figures\/p2pow.pdf}\\\\\n        \\includegraphics[width=0.23\\textwidth]{figures\/f3s2.pdf}\n        \\includegraphics[width=0.23\\textwidth]{figures\/p3pow.pdf}\\\\\n        \\includegraphics[width=0.23\\textwidth]{figures\/f4s2.pdf}\n        \\includegraphics[width=0.23\\textwidth]{figures\/p4pow.pdf}\\\\\n        \\includegraphics[width=0.23\\textwidth]{figures\/f5s2.pdf}\n        \\includegraphics[width=0.23\\textwidth]{figures\/p5pow.pdf}\\\\\n       \n        \\includegraphics[width=0.23\\textwidth]{figures\/barre.pdf}\n    \\caption{The diode rectification $\\mathcal{R}$ for different Fock states ($n=1,2,3,4$ and 5 photons, respectively from top to bottom). Left panels: the rectification factors are computed as a function of the detuning $\\Delta = \\omega_0 - \\omega_1$, and the phase $\\theta = \\omega_0 d\/v_g$ while the bandwidth of the input pulse is fixed, ($\\gamma\/\\Omega = 10^{2}$. On the right panels, we plot the rectification factors for a fixed detuning, $\\Delta\/\\gamma = 0.1$ while we vary the bandwidth $\\Omega$.}\n    \\label{fig:eff}\n    \\vspace{-2cm}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n        \\includegraphics[width=0.48\\textwidth]{figures\/RectDelta.pdf}\n        \\vspace{-2mm}\n    \\caption{The diode rectification $\\mathcal{R}$ for different Fock states ($n=1,2,3,4, 5$ and 22 photons). The pulse length is fixed ($\\Omega\/\\gamma = 10^{-2}$).  The rectification factors are computed as a function of the detuning $\\Delta = \\omega_0 - \\omega_1$. For the case of $n=1$, the phase $\\theta$ is optimized in order to have the strongest negativity of the rectification factor. Ergo, we set  $\\theta \/ 2\\pi = 0.5025$.  Then we used the same value for the other cases. Here, we show the case of $n=22$ because for this Fock state we observed the strongest negative rectification.}\n    \\label{fig:negeff}\n\\end{figure}\n\n\\section{Beyond the monochromatic case}\n\nIn this section, we consider the case of a pulse with a finite length. However, for our approximation \\eqref{ss:assume} to be valid, we need the pulse to be long enough such that the transient regime remains negligible. We shall set the bandwidth to be two orders of magnitude smaller than the decay rate to the waveguide, $\\Omega\/\\gamma = 10^{-2}$. For such a value, the necessary average number of photons to be in the middle of the plateau for the coherent state should be $\\bar{n}_{\\textrm{opt}} \\approx 2$.\n\n\\subsection{The device is not sensitive to coherences between Fock states}\n\nWe first prove that coherences between Fock states don't play any role in the dynamics of the rectification device. Consider as input state a generic superposition\n\\begin{eqnarray}\n\\ket{\\psi}&=&\\sum_{n=0}^\\infty c_n\\ket{n} \\;,\n\\end{eqnarray} \nwith $\\ket{n}$ standing for the Fock state with $n$ photons. For definiteness, we look at a special case, but the argument is the same for any of the calculations of interest here.\n\nWe assume therefore that the state $\\ket{\\psi}$ is prepared in mode $\\hat{A}$, i.e.~the input state is $\\ket{\\psi_a}$ with the Fock states given by \\cite{Alex90s,Alex90ss}\n\\begin{equation}\\label{state:Fockn}\n    \\ket{n} = \\frac{(\\hat{A}^{\\dagger})^n}{\\sqrt{n!}} \\ket{\\text{\\o{}}}\\,.\n\\end{equation} Then we look at the number of reflected photons\n\\begin{equation}\\label{eq:NnmAll}\n    N_{b}^{\\psi} (t) = \n    \\sum_{m,n}c_nc_m^*\\int_{0}^{\\infty}\\dd\\omega\\bra{m_a}\\hat{b}^{\\dagger}_{\\omega}(t)\\hat{b}_{\\omega}(t)\\ket{n_a}\\,.\n\\end{equation}\nSince we are working within the rotating-wave approximation [Eq.~\\eqref{Hdip}], the number of excitations is conserved at all times; and the operator $\\hat{b}^{\\dagger}_{\\omega}(t)\\hat{b}_{\\omega}(t)$ counts the excitations in part of the system (which may have a photonic and an atomic component). If $m\\neq n$, either the number of excitations in that part, or the number of excitations in the rest, will be different between $\\ket{n_a}$ and $\\ket{m_a}$. Thus, $\\bra{m_a}\\hat{b}^{\\dagger}_{\\omega}(t)\\hat{b}_{\\omega}(t)\\ket{n_a}=\\bra{n_a}\\hat{b}^{\\dagger}_{\\omega}(t)\\hat{b}_{\\omega}(t)\\ket{n_a}\\delta_{m,n}$.\n\n\nThus, coherences across the Fock basis in the input state do not play any role. The rectifying device is fully characterised by studying its behavior on each Fock state $\\ket{n}$ separately, as we are going to do next.\n\n\n\\subsection{Photon-number pulses}\n\nLet us now study rectification for Fock states \\eqref{state:Fockn}. The corresponding flux for our chosen pulse \\eqref{xitau} can be computed from $[\\hat{a}_\\tau(0),(A^{\\dagger})^n] = n \\xi (\\tau) (A^{\\dagger})^{n-1}$, which implies $\\hat{a}_\\tau(0)\\ket{n} = \\sqrt{n} \\xi(\\tau) \\ket{n-1}$, and finally \\begin{eqnarray}\nF&=&\\expval{\\hat{a}_\\tau^\\dagger(0)\\hat{a}_\\tau(0)}{n} = \\frac{n\\Omega}{2}\\,.\n\\end{eqnarray}\n\nIn the Hamiltonian \\eqref{Hdip}, the operator $\\hat{b}_{\\omega}$ evolves as\n\\begin{equation}\\label{btomega}\n\\begin{split}\n    \\hat{b}_{\\omega} (t) =& \\;\\hat{b}_{\\omega} (0) + g_{\\omega}^{(1)} \\int_{0}^{t} \\dd t' \\hat{\\sigma}_{-}^{(1)}(t') e^{i(\\omega-\\omega_1)t'} \\\\ \n    & + g_{\\omega}^{(2)} \\int_{0}^{t} \\dd t' \\hat{\\sigma}_{-}^{(2)}(t') e^{i[(\\omega-\\omega_2)t' + \\omega d \/ v_g]}\\;.\n\\end{split}\n\\end{equation} As an example of what we have to compute, the number of reflected photons when the light comes in from the left is\n\\begin{equation}\\label{Nb:coh}\n\\begin{split}\n    N_{\\text{ref}}^{(n)} (t) =& \\int_{0}^{\\infty}\\dd\\omega\\bra{n_a}\\hat{b}^{\\dagger}_{\\omega} (t)\\hat{b}_{\\omega} (t)\\ket{n_a}\\;,\\\\\n    =&  \\frac{\\gamma}{2} \\int_{0}^{t}\\dd t' \\Big\\{ \\expval{\\hat{\\sigma}_z^{(1)}(t')}  +  \\expval{\\hat{\\sigma}_z^{(2)}(t')} + 2   \\\\\n    & + 2  \\left[ \\expval{\\hat{\\sigma}_{-}^{(1)}(t')\\hat{\\sigma}_{+}^{(2)}(t')} e^{-i (\\Delta_{12} t' - \\omega_2 \\mu)} + c.c.\\right] \n    \\Big\\} \\;,\n\\end{split}\n\\end{equation}\nwhere $\\Delta_{12} =\\omega_1 - \\omega_2$ and $\\mu =  d \/ v_g$.\n\nThe dynamics is given by the closed set of Heisenberg equations of motion presented in \\cite{Alex}, which we reproduce for completeness in Appendix \\ref{app:heis}. We solve them numerically in the steady state regime, for various input states, to obtain the desired expectation values \\eqref{ss:assume}.\n\nIn Fig.~\\ref{fig:eff}, the rectification factor for various input states is illustrated. In the case of a fixed bandwidth, $\\gamma\/\\Omega = 10^{2} $, we can see that we have a significant rectification for all the Fock states. For all these cases we observe a maximum rectification near 66\\%. However, the range of inter-atomic distances, $d$, and detunings, $\\Delta$, for which the strongest directionalities are observed, is narrower for Fock states with smaller number of photons. On the other hand, when we decrease the bandwidth, the directionality is lost for all the values of the distance between the qubits, $d$. For the case of a single photon, we observe that the rectification fades for a bandwidth between $\\gamma\/\\Omega \\sim 10^{3}$ and $10^{4}$. For an increasing number of photons, the rectification appears to be more robust against the narrowing of the bandwidth. For instance, in the case of 5 photons, the rectification of the device vanishes between $\\gamma\/\\Omega \\sim 10^{4}$ and $10^{5}$. Also, a similar behaviour can be observed for a fixed inter-atomic distance while varying the detuning. This is in agreement with the result in \\cite{Alex} that the device becomes symmetric for single photon inputs within the monochromatic limit.\n\nInterestingly, we observe that the rectification factor takes negative values (e.g. near $\\theta\/2\\pi = 0.5025$ and $\\Delta\/\\gamma = 0.04$). According to the definition \\eqref{DiodeEff}, this means that the direction of the diode became reversed. In Fig.~\\ref{fig:negeff}, we can see the maximum positive value of the rectification factor, for all the considered number states, are between 0.6 and 0.66. However, in the negative regime, the maximum efficiency of the device seems to depend on the number of photons. The strongest negativity is observed for the case of 22 photons. In other words, increasing or decreasing the number of photons results in weaker negative rectifications. Note that in this optimal case, $n=22$, the behaviour of the device is anti-symmetric. Hence, by changing the detuning between positive and negative values, one can flip the diode's preferred direction.\n\n\n\n\\section{Conclusions}\n\nIn this paper, we have clarified (and occasionally rectified, pun intended) the rectifying functionality of the simple optical diode sketched in Fig.~\\ref{fig:diode}. We have found that its behavior is completely determined by the behavior on the Fock states, and that some single-photon rectification does happen when the pulses are of finite length.\n\nThis study was done under two main assumptions: (i) that the pulses are monomode and long compared to decay time of atomic excitations, and (ii) that the rotating-wave approximation holds. To go beyond the first and consider composite and\/or short pulses, one would have to solve the time-dependent Heisenberg equations of motion. Removing the second assumption may be of interest too, given that the diode is naturally implemented with superconducting qubits \\cite{Clemens}, a platform in which ultrastrong coupling can be reached \\cite{Zueco,Niemczyk}.\n\n\n\\section*{Acknowledgment}\nThis research is supported by the National Research Foundation and the Ministry of Education, Singapore, under the Research Centres of Excellence programme. Part of this work was done when S.L.~was on exchange at the National University of Singapore in a NGNE programme. S.L.~acknowledges financial support from the School of Physics and the Qian Xuesen Honors College, Xi'an Jiaotong University.\n\n\n\\section{Introduction} \nControlling the photonic transport in integrated circuits is important for building future information and communication technologies. While a remarkable progress has been achieved on early implementations of such technologies \\cite{Asavanant,Larsen,Juan}, the development of an optical rectifier that can act on single or a few photon pulses remain challenging. These rectifying devices are useful for routing and isolating signals. Rectifiers based on ferromagnetic compounds have been demonstrated for increasing signal processing capabilities. However, they are inherently lossy and difficult to miniaturize and use in integrated circuits \\cite{Pozar}. Recently, using a semi-classical theory, a two atoms device has been identified as a candidate for exhibiting strong directionality \\cite{auffeves2014}. The transmission of light through this device would depend on the direction of propagation. If the input comes from one side it will pass through it, and if it comes from the other side it will get reflected. This has triggered a debate over whether such a device can perform with non-classical states of light \\cite{Alex,Auffeves2016,Vienna2016,Combes,Fang,Clemens,Gonzalez,Ordonez}.\n\nThis rectifying device, also referred to as \\textit{optical diode}, is consisting of a pair of two-level systems coupled to a one-dimensional waveguide. By changing the distance between the atoms and the detuning of their resonance frequencies, the efficiency of the diode was optimized. It has been shown that for light pulses in coherent states, the device can exhibit efficiencies of more than 60\\% \\cite{Alex,Auffeves2016,Vienna2016,Combes}. However, in the monochromatic limit, it has been found that this device cannot rectify single photon pulses \\cite{Alex}. Since significant directionality can be achieved for coherent inputs while a symmetric behaviour is predicted for single photon pulses, one might naturally ask: which components of the coherent state triggers the performance of the device? Do superpositions matter? How can we change these critical components, so we can improve the efficiency of this device for inputs with a few photon pulses? To answer this questions, we will use the same approach as in \\cite{Alex}. Using the Heisenberg equations of motion, we will show that by changing the length of the light pulse, we can reduce the number of photons necessary to activate the rectifying behaviour of the optical diode. \n\nThis article is organized as follow: in section II, we will introduce the theoretical model for this device and present how the rectification factor of the device is computed. In section III, we will discuss the cases of photon-number pulses. We will show that the main terms that will contribute to the rectification depend directly on the pulse duration. In section IV, we will present a few concluding remarks and discuss potential future directions of this research.\n\n\\section{The model}\n\n\\subsection{The rectifying device}\n\nAs illustrated in Fig.~\\ref{fig:diode}, the system of interest here is composed of a pair of atoms strongly coupled to a one dimensional waveguide. Each atom is model as a two-level system, for which $\\ket{g_j}$ and $\\ket{e_j}$ are the ground and excited states of the $j^{th}$ atom, respectively. The atom on the right is resonant with the central frequency of the light pulse $\\omega_0$. On the other hand, the atom on the left is detuned. We denote its resonance frequency by $\\omega_1$. Both frequencies are assumed to be much larger than the cutoff frequency of the waveguide \\cite{Snyder}. In this case, the dispersion relations read $\\omega = v_g \\abs{k}$, such that $k$ is the longitudinal wave-number of the field mode and $v_g$ is the group velocity of the light pulse in the waveguide. The free Hamiltonian of this system reads\n\\begin{equation} \\label{H0}\n    \\hat{\\mathcal{H}}_{0} = \\sum_{j=1}^{2}  \\hbar \\omega_j \\ketbra{e_j} + \\int_{0}^{\\infty} \\dd \\omega \\hbar \\omega  (\\hat{a}^{\\dagger}_{\\omega}\\hat{a}^{}_{\\omega} + \\hat{b}^{\\dagger}_{\\omega}\\hat{b}^{}_{\\omega}) \\;,\n\\end{equation}\nsuch that the first term corresponds to the two atoms. The second term describes the field modes. The operator $\\hat{a}^{\\dagger}_{\\omega}$ ($\\hat{b}^{\\dagger}_{\\omega}$) creates a photon with frequency $\\omega$ moving towards the right (left). The interaction between the atoms and the propagating light pulse is described by the dipole Hamiltonian within the rotating wave approximation \\cite{Snyder,Roulet}\n\\begin{equation} \\label{Hdip}\n\\begin{split}\n    \\hat{\\mathcal{H}}_{\\mathrm{dip}} =& -i \\hbar \\sum_{j=1}^{2} \\int_{0}^{\\infty} \\dd \\omega g_{\\omega}^{(j)} \\\\\n    \\times \\Big[ & \\hat{\\sigma}_{+}^{(j)} \\left( \\hat{a}_{\\omega} e^{i \\omega \\frac{ x_j}{v_g}} + \\hat{b}_{\\omega} e^{- i \\omega \\frac{ x_j}{v_g}}\\right)e^{ - i (\\omega - \\omega_j)t} - \\textbf{H.c.} \\Big] \\;.\n\\end{split}\n\\end{equation}\nHere $x_j$, $g_{\\omega}^{(j)}$ and $\\hat{\\sigma}_{+}^{(j)} = \\ketbra{e_j}{g_j} $ are the position, the coupling strength and raising operator of the $j^{th}$ atom, respectively. Similarly to the previous study \\cite{Alex}, we assume the Weisskopf-Wigner approximation \\cite{Scully}, i.e. both atoms have the same coupling to the waveguide, $g_{\\omega}^{(1)} = g_{\\omega}^{(2)} = g$. Thus, the decay rate to the waveguide for both atoms is $\\gamma = 2\\pi g^2$.\n\n\\subsection{Computing rectification}\n\nSeveral figures of merit have been used in previous papers to capture rectification: we review them in Appendix \\ref{app:defs}. Here we define the rectifying factor of the device as\n\\begin{equation}\\label{DiodeEff}\n    \\mathcal{R} = T_{\\rightarrow} - T_{\\leftarrow}\\,,\n\\end{equation} where $T_{\\rightarrow}$ is the transmittivity of the light coming from the left (i.e.~the fraction of left-incoming light scattered towards the right); analogously, $T_{\\leftarrow}$ is the fraction of right-incoming light scattered towards the left. Clearly $-1\\leq\\mathcal{R}\\leq 1$, the sign determining the direction in which the diode rectifies. Explicitly,\n\\begin{eqnarray}\\label{eqsT}\nT_{\\rightarrow}&=&\\frac{1}{F}\\lim_{t\\rightarrow\\infty}N_a^{\\rightarrow}(t)\\,=\\,\\frac{1}{F}(1-\\lim_{t\\rightarrow\\infty}N_b^{\\rightarrow}(t))\\,,\\\\\nT_{\\leftarrow}&=&\\frac{1}{F}\\lim_{t\\rightarrow\\infty}N_b^{\\leftarrow}(t)\n\\end{eqnarray}\nwhere $F$ is the mean photon flux per unit time, and where \\begin{eqnarray*}\nN_{b}^{\\textrm{input}} (t) = \\int_{0}^{\\infty} \\dd \\omega \\textrm{Tr}\\left[\\hat{b}^{\\dagger}_{\\omega}(t)\\hat{b}^{}_{\\omega}(t)\\,\\rho_{\\textrm{input}}\\right]\\;,\n\\end{eqnarray*} (and similar definitions for similar quantities).\n\nIn this paper, we shall consider monomode Fourier-limited pulses. The mode that defines a pulse coming from the left is \n\\begin{equation}\\label{hatA}\n\\hat{A}^\\dagger = \\int_{0}^{\\infty} \\dd\\omega f (\\omega) \\hat{a}_{\\omega}^{\\dagger} = \\int_{0}^{\\infty} \\dd\\tau \\xi (\\tau) \\hat{a}_{\\tau}^{\\dagger}  \\;,\n\\end{equation}\nwhere\n\\begin{equation}\n    \\begin{split}\n    \\hat{a}_{\\tau}^{} =& \\frac{1}{\\sqrt{2\\pi}} \\int \\dd\\omega \\hat{a}_{\\omega}^{} e^{- i (\\omega - \\omega_0) \\tau} \\;,\\\\\n    \\xi (\\tau) =& \\frac{1}{\\sqrt{2\\pi}} \\int \\dd\\omega f (\\omega) e^{- i (\\omega - \\omega_0) \\tau}\\,.\n    \\end{split}\n\\end{equation} For pulses coming from the right, we use the identical definition of a mode $\\hat{B}$, replacing the operators $(\\hat{a}_{.},\\hat{a}^\\dagger_{.})$ with $(\\hat{b}_{.},\\hat{b}^\\dagger_{.})$.\n\nLike in previous works, for definiteness the calculations will assume a square pulse\n\\begin{equation}\\label{xitau}\n\\xi (\\tau) = \\Bigg\\{\n    \\begin{array}{cl}\n        \\sqrt{\\Omega\/2}  &  \\text{for } 0 \\leq \\tau \\leq 2\/\\Omega,  \\\\\n        0 & \\text{otherwise}.\n    \\end{array}\n\\end{equation}\nFor an input state $\\ket{\\psi}$, the mean flux of photons coming from the left is \n\\begin{equation}\\label{flux}\n    F = \\textrm{Tr}\\left[\\hat{a}_\\tau^\\dagger(0)\\hat{a}_\\tau(0)\\rho_{\\textrm{input}}\\right]\\;,\n\\end{equation}\n\nA basis of the total states of the system is denoted $\\ket{n_a,n_b,s_1,s_2}$ where $n_{a,b}$ is a Fock state in mode $\\hat{A}$ or $\\hat{B}$, while $s_j\\in\\{g,e\\}$ is the state of the $j$-th atom. Initially, the pulse will come from either the left or the right; and both atoms are assumed to be in the ground state to ensure that the rectifying device is passive. Thus our input states will always be of the form $\\ket{\\psi_a}\\equiv\\ket{\\psi,0,g,g}$, or $\\ket{\\psi_b}\\equiv\\ket{0,\\psi,g,g}$.\n\nInstead of solving the time-dependent equations of motion and sending $t\\rightarrow\\infty$, we shall consider only long pulses and thus assume that the system is in the steady state:\n\\begin{eqnarray}\n\\lim_{t\\rightarrow\\infty}N_b^{\\textrm{input}}(t)&\\approx&\\int_{0}^{\\infty} \\dd \\omega \\expval{\\hat{b}^{\\dagger}_{\\omega}\\hat{b}^{}_{\\omega}}_{ss,\\textrm{input}}\\,.\\label{ss:assume}\n\\end{eqnarray}\n\n\n\\begin{figure}\n    \\centering\n        \\includegraphics[width=0.5\\textwidth,trim={0 9cm 0 0cm},clip]{figures\/Diode.pdf}\n    \\caption{A scheme of the rectifying device consisting of two 2-level systems coupled to a one-dimensional waveguide. The one on the right is resonant with the central frequency of the light pulse, $\\omega_0$. The atom on the left is detuned, $\\omega_1 \\neq \\omega_0$.}\n    \\label{fig:diode}\n\\end{figure}\n\n\\subsection{The case for pulses of finite length}\n\nAll previous fully-quantum studies of this rectifying device \\cite{Alex,Vienna2016,Auffeves2016} concluded that there is no rectification for single-photon Fock states. This conclusion was drawn in the limit of pulses of \\textit{infinite length}. One of the results of the next Section will be that \\textit{the device does show some rectification for single-photon Fock states} for pulses of finite length.\n\nThose same works also showed that high rectification is expected for coherent states, in qualitative (if not exact quantitative) agreement with a previous semi-classical study \\cite{auffeves2014}. Specifically, rectification was predicted \\cite{Alex} and later observed \\cite{Clemens} to have a high-value plateau when $F\/\\gamma$ ranges between $10^{-3}$ and $10^{-1}$. Let us now describe this case in more detail, to understand what may be the origin of the rectification.\n\nEvery coherent state is ultimately monomode \\cite{Alex90s,Alex90ss}: a coherent state pulse with average number of photons $\\bar{n}$, propagating towards the right, can be written as\n\\begin{equation}\\label{state:cohbis}\n    \\ket{\\alpha_a} = \\exp\\big(\\sqrt{\\bar{n}} (\\hat{A}^\\dagger - \\hat{A})\\big) \\ket{\\text{\\o{}}}\\;.\n\\end{equation} Since $\\hat{a}_\\tau(0)\\ket{\\alpha} = \\sqrt{\\bar{n}}\\, \\xi(\\tau) \\ket{\\alpha}$, the flux \\eqref{flux} is given by $F=\\frac{\\bar{n} \\Omega}{2}$. If the device performance is maximized for plateau centred around $F\/\\gamma \\approx  10^{-2}$, then the optimal average number of photons in the pulse is\n\\begin{equation}\n   \\bar{n}_{\\textrm{opt}} \\approx \\frac{2 \\gamma }{\\Omega}  \\times 10^{-2}\\,.\n\\end{equation}\nIf we were to consider infinitely long pulses, as used in the single-photon case, then we would have to take the monochromatic limit $\\Omega \\rightarrow 0$; in which case $\\bar{n}_{\\textrm{opt}}$ diverges.\n\n\n\\begin{figure}\n    \\centering\n        \\includegraphics[width=0.23\\textwidth]{figures\/f1s2.pdf}\n        \\includegraphics[width=0.23\\textwidth]{figures\/p1pow.pdf}\\\\\n        \\includegraphics[width=0.23\\textwidth]{figures\/f2s2.pdf}\n        \\includegraphics[width=0.23\\textwidth]{figures\/p2pow.pdf}\\\\\n        \\includegraphics[width=0.23\\textwidth]{figures\/f3s2.pdf}\n        \\includegraphics[width=0.23\\textwidth]{figures\/p3pow.pdf}\\\\\n        \\includegraphics[width=0.23\\textwidth]{figures\/f4s2.pdf}\n        \\includegraphics[width=0.23\\textwidth]{figures\/p4pow.pdf}\\\\\n        \\includegraphics[width=0.23\\textwidth]{figures\/f5s2.pdf}\n        \\includegraphics[width=0.23\\textwidth]{figures\/p5pow.pdf}\\\\\n       \n        \\includegraphics[width=0.23\\textwidth]{figures\/barre.pdf}\n    \\caption{The diode rectification $\\mathcal{R}$ for different Fock states ($n=1,2,3,4$ and 5 photons, respectively from top to bottom). Left panels: the rectification factors are computed as a function of the detuning $\\Delta = \\omega_0 - \\omega_1$, and the phase $\\theta = \\omega_0 d\/v_g$ while the bandwidth of the input pulse is fixed, ($\\gamma\/\\Omega = 10^{2}$. On the right panels, we plot the rectification factors for a fixed detuning, $\\Delta\/\\gamma = 0.1$ while we vary the bandwidth $\\Omega$.}\n    \\label{fig:eff}\n    \\vspace{-2cm}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n        \\includegraphics[width=0.48\\textwidth]{figures\/RectDelta.pdf}\n        \\vspace{-2mm}\n    \\caption{The diode rectification $\\mathcal{R}$ for different Fock states ($n=1,2,3,4, 5$ and 22 photons). The pulse length is fixed ($\\Omega\/\\gamma = 10^{-2}$).  The rectification factors are computed as a function of the detuning $\\Delta = \\omega_0 - \\omega_1$. For the case of $n=1$, the phase $\\theta$ is optimized in order to have the strongest negativity of the rectification factor. Ergo, we set  $\\theta \/ 2\\pi = 0.5025$.  Then we used the same value for the other cases. Here, we show the case of $n=22$ because for this Fock state we observed the strongest negative rectification.}\n    \\label{fig:negeff}\n\\end{figure}\n\n\\section{Beyond the monochromatic case}\n\nIn this section, we consider the case of a pulse with a finite length. However, for our approximation \\eqref{ss:assume} to be valid, we need the pulse to be long enough such that the transient regime remains negligible. We shall set the bandwidth to be two orders of magnitude smaller than the decay rate to the waveguide, $\\Omega\/\\gamma = 10^{-2}$. For such a value, the necessary average number of photons to be in the middle of the plateau for the coherent state should be $\\bar{n}_{\\textrm{opt}} \\approx 2$.\n\n\\subsection{The device is not sensitive to coherences between Fock states}\n\nWe first prove that coherences between Fock states don't play any role in the dynamics of the rectification device. Consider as input state a generic superposition\n\\begin{eqnarray}\n\\ket{\\psi}&=&\\sum_{n=0}^\\infty c_n\\ket{n} \\;,\n\\end{eqnarray} \nwith $\\ket{n}$ standing for the Fock state with $n$ photons. For definiteness, we look at a special case, but the argument is the same for any of the calculations of interest here.\n\nWe assume therefore that the state $\\ket{\\psi}$ is prepared in mode $\\hat{A}$, i.e.~the input state is $\\ket{\\psi_a}$ with the Fock states given by \\cite{Alex90s,Alex90ss}\n\\begin{equation}\\label{state:Fockn}\n    \\ket{n} = \\frac{(\\hat{A}^{\\dagger})^n}{\\sqrt{n!}} \\ket{\\text{\\o{}}}\\,.\n\\end{equation} Then we look at the number of reflected photons\n\\begin{equation}\\label{eq:NnmAll}\n    N_{b}^{\\psi} (t) = \n    \\sum_{m,n}c_nc_m^*\\int_{0}^{\\infty}\\dd\\omega\\bra{m_a}\\hat{b}^{\\dagger}_{\\omega}(t)\\hat{b}_{\\omega}(t)\\ket{n_a}\\,.\n\\end{equation}\nSince we are working within the rotating-wave approximation [Eq.~\\eqref{Hdip}], the number of excitations is conserved at all times; and the operator $\\hat{b}^{\\dagger}_{\\omega}(t)\\hat{b}_{\\omega}(t)$ counts the excitations in part of the system (which may have a photonic and an atomic component). If $m\\neq n$, either the number of excitations in that part, or the number of excitations in the rest, will be different between $\\ket{n_a}$ and $\\ket{m_a}$. Thus, $\\bra{m_a}\\hat{b}^{\\dagger}_{\\omega}(t)\\hat{b}_{\\omega}(t)\\ket{n_a}=\\bra{n_a}\\hat{b}^{\\dagger}_{\\omega}(t)\\hat{b}_{\\omega}(t)\\ket{n_a}\\delta_{m,n}$.\n\n\nThus, coherences across the Fock basis in the input state do not play any role. The rectifying device is fully characterised by studying its behavior on each Fock state $\\ket{n}$ separately, as we are going to do next.\n\n\n\\subsection{Photon-number pulses}\n\nLet us now study rectification for Fock states \\eqref{state:Fockn}. The corresponding flux for our chosen pulse \\eqref{xitau} can be computed from $[\\hat{a}_\\tau(0),(A^{\\dagger})^n] = n \\xi (\\tau) (A^{\\dagger})^{n-1}$, which implies $\\hat{a}_\\tau(0)\\ket{n} = \\sqrt{n} \\xi(\\tau) \\ket{n-1}$, and finally \\begin{eqnarray}\nF&=&\\expval{\\hat{a}_\\tau^\\dagger(0)\\hat{a}_\\tau(0)}{n} = \\frac{n\\Omega}{2}\\,.\n\\end{eqnarray}\n\nIn the Hamiltonian \\eqref{Hdip}, the operator $\\hat{b}_{\\omega}$ evolves as\n\\begin{equation}\\label{btomega}\n\\begin{split}\n    \\hat{b}_{\\omega} (t) =& \\;\\hat{b}_{\\omega} (0) + g_{\\omega}^{(1)} \\int_{0}^{t} \\dd t' \\hat{\\sigma}_{-}^{(1)}(t') e^{i(\\omega-\\omega_1)t'} \\\\ \n    & + g_{\\omega}^{(2)} \\int_{0}^{t} \\dd t' \\hat{\\sigma}_{-}^{(2)}(t') e^{i[(\\omega-\\omega_2)t' + \\omega d \/ v_g]}\\;.\n\\end{split}\n\\end{equation} As an example of what we have to compute, the number of reflected photons when the light comes in from the left is\n\\begin{equation}\\label{Nb:coh}\n\\begin{split}\n    N_{\\text{ref}}^{(n)} (t) =& \\int_{0}^{\\infty}\\dd\\omega\\bra{n_a}\\hat{b}^{\\dagger}_{\\omega} (t)\\hat{b}_{\\omega} (t)\\ket{n_a}\\;,\\\\\n    =&  \\frac{\\gamma}{2} \\int_{0}^{t}\\dd t' \\Big\\{ \\expval{\\hat{\\sigma}_z^{(1)}(t')}  +  \\expval{\\hat{\\sigma}_z^{(2)}(t')} + 2   \\\\\n    & + 2  \\left[ \\expval{\\hat{\\sigma}_{-}^{(1)}(t')\\hat{\\sigma}_{+}^{(2)}(t')} e^{-i (\\Delta_{12} t' - \\omega_2 \\mu)} + c.c.\\right] \n    \\Big\\} \\;,\n\\end{split}\n\\end{equation}\nwhere $\\Delta_{12} =\\omega_1 - \\omega_2$ and $\\mu =  d \/ v_g$.\n\nThe dynamics is given by the closed set of Heisenberg equations of motion presented in \\cite{Alex}, which we reproduce for completeness in Appendix \\ref{app:heis}. We solve them numerically in the steady state regime, for various input states, to obtain the desired expectation values \\eqref{ss:assume}.\n\nIn Fig.~\\ref{fig:eff}, the rectification factor for various input states is illustrated. In the case of a fixed bandwidth, $\\gamma\/\\Omega = 10^{2} $, we can see that we have a significant rectification for all the Fock states. For all these cases we observe a maximum rectification near 66\\%. However, the range of inter-atomic distances, $d$, and detunings, $\\Delta$, for which the strongest directionalities are observed, is narrower for Fock states with smaller number of photons. On the other hand, when we decrease the bandwidth, the directionality is lost for all the values of the distance between the qubits, $d$. For the case of a single photon, we observe that the rectification fades for a bandwidth between $\\gamma\/\\Omega \\sim 10^{3}$ and $10^{4}$. For an increasing number of photons, the rectification appears to be more robust against the narrowing of the bandwidth. For instance, in the case of 5 photons, the rectification of the device vanishes between $\\gamma\/\\Omega \\sim 10^{4}$ and $10^{5}$. Also, a similar behaviour can be observed for a fixed inter-atomic distance while varying the detuning. This is in agreement with the result in \\cite{Alex} that the device becomes symmetric for single photon inputs within the monochromatic limit.\n\nInterestingly, we observe that the rectification factor takes negative values (e.g. near $\\theta\/2\\pi = 0.5025$ and $\\Delta\/\\gamma = 0.04$). According to the definition \\eqref{DiodeEff}, this means that the direction of the diode became reversed. In Fig.~\\ref{fig:negeff}, we can see the maximum positive value of the rectification factor, for all the considered number states, are between 0.6 and 0.66. However, in the negative regime, the maximum efficiency of the device seems to depend on the number of photons. The strongest negativity is observed for the case of 22 photons. In other words, increasing or decreasing the number of photons results in weaker negative rectifications. Note that in this optimal case, $n=22$, the behaviour of the device is anti-symmetric. Hence, by changing the detuning between positive and negative values, one can flip the diode's preferred direction.\n\n\n\n\\section{Conclusions}\n\nIn this paper, we have clarified (and occasionally rectified, pun intended) the rectifying functionality of the simple optical diode sketched in Fig.~\\ref{fig:diode}. We have found that its behavior is completely determined by the behavior on the Fock states, and that some single-photon rectification does happen when the pulses are of finite length.\n\nThis study was done under two main assumptions: (i) that the pulses are monomode and long compared to decay time of atomic excitations, and (ii) that the rotating-wave approximation holds. To go beyond the first and consider composite and\/or short pulses, one would have to solve the time-dependent Heisenberg equations of motion. Removing the second assumption may be of interest too, given that the diode is naturally implemented with superconducting qubits \\cite{Clemens}, a platform in which ultrastrong coupling can be reached \\cite{Zueco,Niemczyk}.\n\n\n\\section*{Acknowledgment}\nThis research is supported by the National Research Foundation and the Ministry of Education, Singapore, under the Research Centres of Excellence programme. Part of this work was done when S.L.~was on exchange at the National University of Singapore in a NGNE programme. S.L.~acknowledges financial support from the School of Physics and the Qian Xuesen Honors College, Xi'an Jiaotong University.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\t\n\nErd\\H{o}s \\cite{Erdos} in 1946 asked the question of finding  the minimal number of distinct distances among any $N$ points in the  plane.  The breakthrough work of Guth-Katz \\cite{Guth-Katz}  gave the lower bound $\\geq C\\frac{N}{\\log N}$ for some constant $C>0$ in the Euclidean plane, which is sharp up to a factor of $\\log$. Another related and widely studied conjecture is\nthe Falconer's conjecture which asks about the lower bound of the Hausdorff dimension of the sets in $\\mathbb{R}^d$ for which the difference set has positive Lebesgue measure.  The Falconer's conjecture can be viewed as a  continuous analogue of the distinct distances problem.  Interested readers may check Falconer \\cite{Falconer}, Guth-Iosevich-Ou-Wang \\cite{Guth-Iosevich-Ou-Wang}, Iosevich \\cite{Iosevich} etc. \nThe Erd\\H{o}s-Falconer type problems have been generalized to  other spaces  and applied to certain sum-product estimates, see e.g.  Bourgain-Tao \\cite{Bourgain-Tao},   Hart-Iosevich-Koh-Rudnev \\cite{HIKR}, Roche-Newton and Rudnev \\cite{RocheNewton-Rudnev},  Rudnev-Selig \\cite{Rudnev-Selig}, Sheffer-Zahl \\cite{Sheffer-Zahl}, and blog of Tao \\cite{Tao} etc.\n  However, the distinct distances problem has not been considered in  hyperbolic surfaces until very recently by Lu and the author  in \\cite{Lu-Meng} where the modular surface and hyperbolic surfaces  with cocompact fundamental groups are studied.  But this problem is still open for  more general hyperbolic surfaces arising from  non-cocompact Fuchsian groups. \n\nIn this paper, for all cofinite Fuchsian groups $\\Gamma$,  we give complete answer to the distinct distances problem for all hyperbolic surfaces $\\Gamma\\backslash\\mathbb{H}^2$ endowed with the hyperbolic metric from $\\mathbb{H}^2$. \n\n\n\\begin{theorem}\\label{thm-cofinite}\n\tFor any cofinite Fuchsian group $\\Gamma\\subset \\mathrm{PSL}(2, \\mathbb{R})$, any set of $N$ points on the hyperbolic surface $\\Gamma\\backslash\\mathbb{H}^2$ determines $\\geq C_{\\Gamma}\\frac{N}{\\log N}$ distinct distances for some constant $C_{\\Gamma}$ depending only on  $\\Gamma$. \n\\end{theorem}\n\n\nIn particular, for finite index subgroups of the modular group $\\mathrm{PSL}(2, \\mathbb{Z})$, we extract out the dependence of the implied constants on the index. \n\n\n\n\n\\begin{theorem}\\label{thm-finite-index}\n\tFor any finite index subgroup $\\Gamma$ of $\\mathrm{PSL}(2, \\mathbb{Z})$ with $[ \\mathrm{PSL}(2,\\mathbb{Z}):\\Gamma ]=\\mu$, any set of $N$ points on the hyperbolic surface  $\\Gamma\\backslash\\mathbb{H}^2$ determines $\\geq C \\frac{N}{\\mu\\log N}$ distinct distances for some absolute constant $C>0$. \n\\end{theorem}\n\nTheorem \\ref{thm-finite-index} has application to equilateral dimension problem. The equilateral dimension of a metric space is the maximal number of points in the space with pairwise equal distance. It has been studied in various spaces, see Alon-Milman \\cite{Alon-Milman}, Guy \\cite{Guy}, Koolen \\cite{Koolen} etc. For instance, the equilateral dimension of the $n$-dimensional Euclidean space is $n+1$.  However, we are not aware of any result in literature about the equilateral dimension of general hyperbolic surfaces. We observe that the lower bound in Theorem \\ref{thm-finite-index} is not trivial for distinct distances among any set of size $N\\gg \\mu^{1+\\epsilon}$.  Thus the following corollary holds.\n\\begin{corollary}\n\tFor any subgroup $\\Gamma$ of $\\mathrm{PSL}(2, \\mathbb{Z})$ with finite index $[ \\mathrm{PSL}(2,\\mathbb{Z}):\\Gamma ]=\\mu$, the equilateral dimension of the hyperbolic surface $\\Gamma\\backslash\\mathbb{H}^2$ is $\\ll \\mu^{1+\\epsilon}$ for any $ \\epsilon>0$. \n\\end{corollary}\n\nThe isometry group of the hyperbolic plane $\\mathbb{H}^2$ is  $\\mathrm{PSL}(2, \\mathbb{R})$   which acts on $\\mathbb{H}^2$ by M\\\"{o}bius transformation:\n\\[z\\mapsto \\gamma(z):=\\frac{az+b}{cz+d}, \\text{ for }\\gamma=\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}\\in\\mathrm{PSL}(2, \\mathbb{R}), z\\in\\mathbb{H}^2.\\] \nFor any discrete  subgroup $\\Gamma$ of $\\mathrm{PSL}(2, \\mathbb{R})$, i.e. a \\textit{Fuchsian group}, the distance between any two points $p, q $ on the hyperbolic surface $ Y\\cong \\Gamma\\backslash\\mathbb{H}^2$ is \n$$d_Y(p, q):=\\min_{\\gamma_1, \\gamma_2\\in \\Gamma} d_{\\mathbb{H}^2}(\\gamma_1(p), \\gamma_2(q))=\\min_{\\gamma_1, \\gamma_2\\in\\Gamma} d_{\\mathbb{H}^2}(p, \\gamma_1^{-1}\\gamma_2(q) )=\\min_{\\gamma\\in\\Gamma}d_{\\mathbb{H}^2}(p, \\gamma(q)).$$\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.6\\textwidth]{surface-show.png}\n\t\\caption{Distances on hyperbolic surface}\n\\end{figure}\n\nInstead of calculating distances on the surface directly, we consider representatives of the points in a fundamental domain $F_{\\Gamma}$ of $\\Gamma$. In \\cite{Lu-Meng}, Lu and the author introduced the concept of a ``geodesic cover\" $\\Gamma'\\subset\\Gamma$ such that for any $p, q\\in F_{\\Gamma}$, \n$$  d_Y(p, q)= d_{\\mathbb{H}^2}(p, \\gamma' q) ~\\text{for some}~\\gamma'\\in\\Gamma'.$$\nIf there exists a finite geodesic cover,  one can derive a lower bound for the distinct distance problem on the hyperbolic surface $\\Gamma\\backslash\\mathbb{H}^2$.  For the modular surface $\\mathrm{PSL}(2, \\mathbb{Z})\\backslash\\mathbb{H}^2$, we would be able to find a finite geodesic cover by working explicitly with matrices in $\\mathrm{PSL}(2, \\mathbb{Z})$. However, it is hard to tackle general non-cocompact Fuchsian groups this way since we cannot explicitly write out all the elements. Another difficulty to find such a finite geodesic cover is, the number of representatives we need to examine would blow up if the fundamental domain has many inequivalent cusps. This is not an issue for modular surface which has only one inequivalent cusp, and that the imaginary parts of points in a fundamental domain of $\\mathrm{PSL}(2, \\mathbb{Z})$ are all bounded below (or bounded above if we choose other type of fundamental domain). Therefore,  representatives we have to examine will not have very small imaginary parts and the number of them could be bounded.  But in the general case, if a pair of points are close to two inequivalent cusps respectively, the number of representatives we have to examine might lose control. \n\n\nIn order to overcome such difficulties, we propose a more general concept of a geodesic cover defined on any subset of a fundamental domain $F_{\\Gamma}$ and also defined in different base groups, see Definition \\ref{defn-geodesic-covering-number}. By building relations between geodesic covers of different subregions in $F_{\\Gamma}$ and geodesic covers of certain regions in different groups, we  prove lower bounds for distinct distances on hyperbolic surfaces associated with any cofinite Fuchsian group. See Lemmas \\ref{lem-geodesic-number-Distinct-distance}, \\ref{lem-geodesic-cover-number-finite} and \\ref{lem-finite-index-geo-cover-number} for details. \n\n\n\n\n\n\n\\bigskip\n\\textbf{Acknowledgment.} The author is partially supported by the Humboldt Professorship of Professor Harald Helfgott. \n\n\n\\section{Preliminaries and preparations}\n\nFirst we briefly summarize the properties of Fuchsian groups (see Beardon \\cite{Beardon} or Katok \\cite{Katok} for more related materials).\nA subgroup $\\Gamma$ of $\\mathrm{PSL}(2, \\mathbb{R})$ is a \\textit{Fuchsian group} if and only if $\\Gamma$ acts \\textit{properly discontinuously} on $\\mathbb{H}^2$. Thus the $\\Gamma$-orbit of any point $z\\in\\mathbb{H}^2$ is\\textit{ locally finite}, which means any compact set $K\\subset\\mathbb{H}^2$ contains only finite number of orbit points, i.e. the set $\\Gamma z\\cap K$ is finite for any $z\\in\\mathbb{H}^2$.\n\nA \\textit{cofinite} Fuchsian group is a discrete subgroup of $\\mathrm{PSL}(2, \\mathbb{R})$ of finite covolume i.e. a fundamental domain of $\\Gamma\\backslash\\mathbb{H}^2$ has finite hyperbolic area.  A cofinite discrete subgroup is  also called a \\textit{lattice} in some other contexts. \nSiegel's theorem (see \\cite{Katok}, Theorem 4.1.1) says cofinite Fuchsian group is \\textit{geometrically finite}, i.e. there exists a convex fundamental domain with finitely many sides.\n\n\n\n\n\n\n\n\n\nThe cocompact Fuchsian groups has been considered in \\cite{Lu-Meng}. In this paper, we focus on non-cocompact case. Suppose $\\Gamma$ has parabolic elements, and thus its fundamental domain $F_{\\Gamma}$ must have a vertex on $\\hat{\\mathbb{R}}$ which is called a \\textit{cusp}. Since we assume $\\Gamma$ is cofinite, by Siegel's theorem, its fundamental domain $F_{\\Gamma}$ has finitely many cusps.\n\n\nWe use an idea of Iwaniec (see \\cite{Iwaniec}, \\S 2.2) to partition the fundamental domain of a Fuchsian group. Define the stability group as\n$$\\Gamma_z:=\\{ \\gamma\\in\\Gamma: \\gamma z=z \\}.$$\nGiven a cusp $\\mathfrak{a}\\in \\hat{\\mathbb{R}}$ for $\\Gamma$. The stability group $\\Gamma_{\\mathfrak{a}}$ is a cyclic group generated by a parabolic element, say $\\Gamma_{\\mathfrak{a}}=\\langle \\gamma_{\\mathfrak{a}}\\rangle$. There exists $\\sigma_{\\mathfrak{a}}\\in SL(2, \\mathbb{R})$ such that\n\\begin{equation}\\label{eq-conjugate-to-translation}\n\\sigma_{\\mathfrak{a}}\\infty=\\mathfrak{a}, \\quad \\sigma_{\\mathfrak{a}}^{-1}\\gamma_{\\mathfrak{a}}\\sigma_{\\mathfrak{a}}=\\begin{pmatrix}1&1\\\\ 0&1\\end{pmatrix}.\n\\end{equation}\nThen $\\sigma_{\\mathfrak{a}}^{-1}$ sends $\\mathfrak{a}$ to $\\infty$ and $\\sigma_{\\mathfrak{a}}$ maps the strip\n\\begin{equation}\\label{infinite-part-of-fundamental-domain}\nP(T):=\\{ z=x+iy: 0<x<1, y\\geq T\\}.\n\\end{equation}\ninto the cuspidal zone\n\\begin{equation}\\label{eq-F-cusp-to-F-infty}\nF_{\\mathfrak{a}, T}=\\sigma_{\\mathfrak{a}}P(T).\n\\end{equation}\nThe cuspidal zone $F_{\\mathfrak{a}, T}$ is contained in a disc (the boundary is a horocycle) tangent to $\\hat{\\mathbb{R}}$ at $\\mathfrak{a}$.\n\nWhen there are more than one cusps, we may choose $T$ large enough such that the cuspidal zones are disjoint. By doing this, we divide the fundamental domain $F_{\\Gamma}$ into cuspidal parts \\begin{equation}\\label{eq-partition-cuspidal-central}\nF_{\\bowtie,T}:=\\bigcup_{\\mathfrak{a}}F_{\\mathfrak{a}, T}\n\\end{equation} and the central part $F_T:=F_{\\Gamma}\\setminus F_{\\bowtie,T}$ (see Figure \\ref{fig-cuspidal-partition}).\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=.6\\textwidth\n\t]{cuspidal-zone.png}\n\t\\caption{Cuspidal parts and central part}\n\t\\label{fig-cuspidal-partition}\n\\end{figure}\n\n\n\nNow we give the definition of a geodesic cover and geodesic-covering number of any region in a Fuchsian group. \n\\begin{definition}\\label{defn-geodesic-covering-number}\n\tLet $F_{\\Gamma}$ be a fundamental domain of $\\Gamma$, and $Y\\cong\\Gamma\\backslash\\mathbb{H}^2$ be the hyperbolic surface associated with $\\Gamma$. For any subset $F'\\subset F_{\\Gamma}$, we say $\\Gamma'\\subset\\Gamma$ is a \\textbf{geodesic cover of $F'$ in $\\Gamma$} if \n\t\\begin{equation}\n\td_{Y}(p, q)=\\min_{\\gamma_1, \\gamma_2\\in\\Gamma'} d_{\\mathbb{H}^2}(\\gamma_1(p), \\gamma_2(q)), \\forall p, q\\in F'.\n\t\\end{equation}\n\t\n\tWe  call the smallest cardinality of $\\Gamma'\\subset\\Gamma$ the \\textbf{geodesic-covering number of $F'$  in $\\Gamma$}, denoted by $K_{\\Gamma}(F')$. \n\t\n\\end{definition}\n\n\\begin{remark}\n\tA geodesic cover  always contains identity. If we take $F'=F_{\\Gamma}$, this matches the definition of the geodesic cover in \\cite{Lu-Meng}. \n\\end{remark}\n\\begin{remark}\n\tNote that this definition depends on different regions and different base groups. We see that if $F''\\subset F'\\subset F_{\\Gamma}$, then $K_{\\Gamma}(F'')\\leq K_{\\Gamma}(F')$.  But for a subgroup $\\Gamma^*$ of $\\Gamma$, it is not clear if we have $K_{\\Gamma^*}(F')\\leq K_{\\Gamma}(F')$ or vice versa. \n\\end{remark}\n\nThen we consider the geodesic-covering numbers of  the central part $F_T$ and cuspidal parts $F_{\\mathfrak{a},T} $ for every cusp $\\mathfrak{a}$. If all of them are finite, we are able to derive a lower bound for distinct distances on hyperbolic surfaces.  \n\n\\begin{lemma}\\label{lem-geodesic-number-Distinct-distance}\nAssume $\\Gamma$ is a cofinite Fuchsian group  with a fundamental domain $F_{\\Gamma}$. If the geodesic-covering numbers of $F_T$ and $F_{\\mathfrak{a},T}$ for every cusp $\\mathfrak{a}$ are all finite for some $T=T_{\\Gamma}$, then any set of  $N$ points on the hyperbolic surface $\\Gamma\\backslash\\mathbb{H}^2$ determines $\\geq C_{\\Gamma}\\frac{N}{\\log N}$ distinct distances for some constant $C_{\\Gamma}$ depending on $\\Gamma$. \n\\end{lemma}\n\n\n\\begin{remark}\nThroughout our proof, we assume the set concerned has no points lying on the boundary of $F_{\\Gamma}$. If there are points lying on the boundary of $F_{\\mathfrak{a}, T}$ for some cusp $\\mathfrak{a}$, we may use a parabolic motion to map $F_{\\mathfrak{a}, T}$ to a translate of $P(T)$ without points on the boundary.  If there are points lying on the boundary of  $F_T$, the same proof of Lemma \\ref{lem-geodesic-cover-number-finite} also works for the closure of $F_T$. \n\\end{remark}\n\n\\begin{remark}\\label{remark-cocompact}\n\tIf $\\Gamma$ is a cocompact Fuchsian group, there is no cusp and $F_{\\Gamma}$ is bounded.  Thus we only need to assume the cuspidal part is empty and the central part $F_{T}=F_{\\Gamma}$ (for large enough $T$). In this case, the above lemma still holds. \n\\end{remark}\n\\begin{proof}[Proof of Lemma \\ref{lem-geodesic-number-Distinct-distance}]\nGiven a set $\\mathcal{S}$ of $N$ points on the hyperbolic surface $Y\\cong\\Gamma\\backslash\\mathbb{H}^2$, we consider such $N$ points on a fundamental domain $F_{\\Gamma}$. According to the partition \\eqref{eq-partition-cuspidal-central} of \t$F_{\\Gamma}$, either $F_{\\bowtie,T}$ or $F_T$ has more than $N\/2$ points on it. \n\n\\textbf{Case 1).} If $F_T$ contains more than $N\/2$ points, we only need to consider the lower bound for distinct distances among them, since this is also a lower bound for the $N$ points on the whole surface.  \n\n\n\n\n\n\n\nDenote the set of points on $F_T$ by $\\mathcal{S}_1$. \n Since we assume the geodesic-covering number of $F_T$ is finite, we choose a finite geodesic cover $\\Gamma'\\subset \\Gamma$ with cardinality $|\\Gamma'|=K_{\\Gamma}(F_T)$. Define the distance set\n$$d_{Y}(\\mathcal{S}_1):=\\{ d_{Y}(p, q): p, q\\in \\mathcal{S}_1  \\}\\subset \\{ d_{\\mathbb{H}^2}(p, q): p, q \\in \\cup_{\\gamma\\in \\Gamma'} \\gamma(\\mathcal{S}_1)\\},$$\nand the distance quadruples\n\\begin{align}\\label{eq-distance-quadruple-to-surface}\nQ_{Y}(\\mathcal{S}_1)&:=\\{  (p_1, p_2; p_3, p_3)\\in \\mathcal{S}^4_1: d_{Y}(p_1, p_2)=d_{Y}(p_3, p_4)\\neq 0 \\}  \\nonumber\\\\\n&~\\subset Q_{\\mathbb{H}^2}\\big(\\cup_{\\gamma\\in\\Gamma'}\\gamma(\\mathcal{S}_1) \\big),\n\\end{align}\nwhere \n\\begin{equation}\\label{eq-defn-QH}\nQ_{\\mathbb{H}^2}(\\mathcal{P}):=\\{ (p_1, p_2; p_3, p_4)\\in \\mathcal{P}^4: d_{\\mathbb{H}^2}(p_1, p_2)=d_{\\mathbb{H}^2}(p_3, p_4)\\neq 0 \\}.\n\\end{equation}\n\nFor any finite set of points $\\mathcal{P}$ on a hyperbolic surface $Y$, the connection between $d_Y(\\mathcal{P})$ and $Q_Y(\\mathcal{P})$ is as follows. Suppose the elements of $d_Y(\\mathcal{P})$\nare $d_1, d_2, \\cdots, d_k$ and $n_i$ is the number of pairs $(p_1, p_2)\\in\\mathcal{P}^2$ with distance $d_i$ ($1\\leq i\\leq k$). By the Cauchy-Schwarz inequality, we get\n\\begin{equation}\n{|\\mathcal{P}|\\choose 2 }^2=\\bigg( \\sum_{i=1}^k n_i \\bigg)^2\\leq \\bigg(\\sum_{i=1}^k n^2_i \\bigg)k=|Q_Y(\\mathcal{P})| |d_Y(\\mathcal{P})|,\n\\end{equation}\nthus\n\\begin{equation}\\label{Quadruple-to-distance}\n|d_Y(\\mathcal{P})|\\geq \\frac{ (|\\mathcal{P}|^2-|\\mathcal{P}| )^2  }{  |Q_Y(\\mathcal{P})|}. \n\\end{equation}\n\n\n\nFor any set of points $\\mathcal{P}$ in $\\mathbb{H}^2$, by an argument of Tao in his blog \\cite{Tao} (see also \\cite{Rudnev-Selig}), one can derive\n\\begin{equation}\\label{eq-Tao-quaruple}\n|Q_{\\mathbb{H}^2}(\\mathcal{P})|\\ll |\\mathcal{P} |^3 \\log (|\\mathcal{P}|).\n\\end{equation}\nRecently, Lu-Meng \\cite{Lu-Meng} also gave a different proof for the above estimate by modifying the framework of Guth-Katz and working explicitly with isometries of $\\mathbb{H}^2$. \nSince the geodesic-covering number $K_{\\Gamma}(F_T)$ of $F_T$ in $\\Gamma$ is finite, the cardinality of $\\cup_{\\gamma\\in\\Gamma'}\\gamma(\\mathcal{S}_1)$  is  $\\leq K_{\\Gamma}(F_T) |\\mathcal{S}_1\t| \\leq  K_{\\Gamma}(F_T) N$.  By \\eqref{eq-distance-quadruple-to-surface}, we derive that\n\\begin{equation}\n|Q_Y(\\mathcal{S}_1)|\\ll K^3_{\\Gamma}(F_T) N^3 (\\log (K_{\\Gamma}(F_T))+\\log N).\n\\end{equation}\n Thus by \\eqref{Quadruple-to-distance}, we get\n \\begin{equation}\n |d_Y(\\mathcal{S})|\\geq |d_Y(\\mathcal{S}_1)|\\gg \\frac{N}{ K^3_{\\Gamma}(F_T)  (\\log (K_{\\Gamma}(F_T))+\\log N) }\\geq C'_{\\Gamma}\\frac{N}{\\log N},\n \\end{equation}\nfor some constant $C'_{\\Gamma}>0$ depending  on $\\Gamma$. \t\n\t\n\\textbf{Case 2).} There are more than $N\/2$ points on $F_{\\bowtie,T}$. Let $n_c<\\infty$ be the number of cusps for the fundamental domain $F_{\\Gamma}$. Then there exists one cusp $\\mathfrak{b}$ such that $F_{\\mathfrak{b},T}$ contains more than $N\/2n_c$ points. We may assume all these points lie in the interior of $F_{\\mathfrak{b}, T}$. Denote the set of points on $F_{\\mathfrak{b},T}$ by $\\mathcal{S}_2$. By a similar argument as in Case 1), we deduce that\n\t\\begin{equation}\n\t|d_Y(\\mathcal{S})|\\geq |d_Y(\\mathcal{S}_2)|\\gg \\frac{N\/n_c}{ K^3_{\\Gamma}(F_{\\mathfrak{b}, T}) (\\log(K_{\\Gamma}(F_{\\mathfrak{b},T}) ) +\\log N  )  }  \\geq C''_{\\Gamma}\\frac{N}{\\log N},\n\t\\end{equation}\n\tfor some constant $C''_{\\Gamma}>0$ depending  on $\\Gamma$.\n\t\n\tCombining the cases 1) and 2), we finish the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{ Geodesic-covering numbers for cofinite Fuchsian groups }\n\n\nIn this section, we give the proof of Theorem \\ref{thm-cofinite} based on Lemma \\ref{lem-geodesic-number-Distinct-distance}. We only need to bound the geodesic-covering numbers of  $F_T$ and $F_{\\mathfrak{a}, T} $ for every cusp $ \\mathfrak{a}$. \n\\begin{lemma}\\label{lem-geodesic-cover-number-finite}\n\tAssume $\\Gamma$ is a  cofinite Fuchsian group with a fundamental domain $F_{\\Gamma}$. If we partition $F_{\\Gamma}$ as in \\eqref{eq-partition-cuspidal-central} for some large enough $T$ depending on $\\Gamma$, the geodesic-covering numbers of $F_T$ and $F_{\\mathfrak{a}, T}$ for every cusp $\\mathfrak{a}$ in $\\Gamma$ are all finite.  This is also true for the closure of $F_T$, i.e. $K_{\\Gamma}(\\overline{F}_T)<\\infty$. \n\\end{lemma}\n\n\n\\begin{proof}\nWe need to know the basic shape of a fundamental domain for any  fuchsian group. A convenient choice for us is \\textit{Ford  domain} which was first introduced by L. R. Ford \\cite{Ford}. It is known that Ford domain is a fundamental domain (see \\cite{Katok}, Theorem 3.3.5).  \nThere are concrete methods to construct fundamental domains of Fuchsian groups, interested readers  may check Voight \\cite{Voight} for an algorithmic method, and Kulkarni \\cite{Kul} for construction of special polygons (also a fundamental domain)  for subgroups of modular group using Farey symbol. \n\n\nLet $F_{\\Gamma}$ be a fundamental domain of cofinite $\\Gamma$ with finite number of sides and finite number of cusps. We partition $F_{\\Gamma}$ as in \\eqref{eq-partition-cuspidal-central} for some $T$ we choose later, \n$$F_{\\Gamma}=F_T\\bigcup_{\\mathfrak{a} ~{\\rm cusp}}F_{\\mathfrak{a}, T}.  $$\n\n\n1) First we show that, for  some $T$, the geodesic-covering number of $F_{\\mathfrak{a}, T}$ in $\\Gamma$ is finite for  every cusp $\\mathfrak{a}$. In order to do this, we make use of Ford domains. \n\nFor any cusp $\\mathfrak{a}$, by \\eqref{eq-conjugate-to-translation},  there exists $\\sigma_{\\mathfrak{a}}$ such that the stability group $\\Gamma_{\\mathfrak{a}}$ is generated by \n$$\\sigma_{\\mathfrak{a}}\\begin{pmatrix}\n1 & 1\\\\\n0 & 1\n\\end{pmatrix}\\sigma^{-1}_{\\mathfrak{a}},$$\nand the fundamental domain of $\\sigma^{-1}_{\\mathfrak{a}}\\Gamma_{\\mathfrak{a}}\\sigma_{\\mathfrak{a}}$ is \n\\begin{equation}\nP:=\\{ z\\in\\mathbb{H}^2:  0\\leq x< 1, y>0\\}.\n\\end{equation}\nDenote $\\widetilde{\\Gamma}^{\\mathfrak{a}}:=\\sigma^{-1}_{\\mathfrak{a}}\\Gamma\\sigma_{\\mathfrak{a}}$ and $$\\widetilde{\\Gamma}_{\\infty}^{\\mathfrak{a}}:=\\sigma^{-1}_{\\mathfrak{a}}\\Gamma_{\\mathfrak{a}}\\sigma_{\\mathfrak{a}}=\\left\\langle  \\begin{pmatrix}\n1 & 1\\\\\n0 & 1\n\\end{pmatrix}  \\right\\rangle.$$\n\n\nBy \\eqref{infinite-part-of-fundamental-domain} and \\eqref{eq-F-cusp-to-F-infty},   the geodesic-covering number of $F_{\\mathfrak{a}, T}$ in $\\Gamma$ is the same as the geodesic-covering number of $\\sigma^{-1}_{\\mathfrak{a}}(F_{\\mathfrak{a}, T}  )=P(T)$ in $\\sigma^{-1}_{\\mathfrak{a}}\\Gamma\\sigma_{\\mathfrak{a}}$, i.e. \n\\begin{equation}\\label{eq-geo-cover-number-to-P}\nK_{\\Gamma}(F_{\\mathfrak{a}, T})=K_{\\widetilde{\\Gamma}^{\\mathfrak{a}}}(P(T)).\n\\end{equation}\n\n\nWe define a  domain associated with cusp $\\mathfrak{a}$ as\n\\begin{align}\\label{eq-Ford-domain-cusp}\n\\mathcal{D}_{\\mathfrak{a}}:=& \\{ z\\in P:  {\\rm Im}(\\gamma z) <{\\rm Im} (z),    \\forall \\gamma\\in \\widetilde{\\Gamma}^{\\mathfrak{a}}\\setminus\\widetilde{\\Gamma}_{\\infty}^{\\mathfrak{a}} \\}\\nonumber\\\\\n=&\\{ z\\in P: |cz+d|>1, \\forall \\begin{pmatrix}\n* & *\\\\\nc &  d\n\\end{pmatrix}\\in  \\widetilde{\\Gamma}^{\\mathfrak{a}}\\setminus\\widetilde{\\Gamma}_{\\infty}^{\\mathfrak{a}}    \\}\n\\end{align}\nwhich is a \\textit{Ford domain} and thus a fundamental domain of $\\widetilde{\\Gamma}^{\\mathfrak{a}}$.  Note that $\\sigma_{\\mathfrak{a}}^{-1}(F_{\\Gamma})$ may not be the same as $\\mathcal{D}_{\\mathfrak{a}}$. \n\n\nWe want to choose large enough $T$ such that $P(T)\\subset \\mathcal{D}_{\\mathfrak{a}}$ for all cusp $\\mathfrak{a}$.  Since $\\mathcal{D}_{\\mathfrak{a}}$ is a fundamental domain of $ \\widetilde{\\Gamma}^{\\mathfrak{a}}  $, the boundary of  $\\mathcal{D}_{\\mathfrak{a}}$ consists of finite number of pieces from\\textit{ isometric circles} of the form $|z+\\frac{d}{c}|=\\frac{1}{|c|}$ for some \n$$c\\neq 0, \\begin{pmatrix}\n* & *\\\\\nc & d\n\\end{pmatrix}\\in \\widetilde{\\Gamma}^{\\mathfrak{a}}.$$\nThus there is a largest radius among these isometric circles, say $\\frac{1}{c_{\\mathfrak{a}}}$, actually (see \\cite{Iwaniec}, \\S 2.6)  \n\\begin{equation}\\label{eq-min-c}\nc_{\\mathfrak{a}}=\\min\\left\\{ c>0:  \\begin{pmatrix}\n* & *\\\\\nc & *\n\\end{pmatrix}\\in\\widetilde{\\Gamma}^{\\mathfrak{a}}\\setminus\\widetilde{\\Gamma}_{\\infty}^{\\mathfrak{a}}    \\right\\}.\n\\end{equation}\nFor the fundamental domain $F_{\\Gamma}$, there are only finite number of cusps, we choose any large enough\n$$T\\geq 100+10\\max_{\\mathfrak{a} ~{\\rm cusp}} \\tfrac{1}{c_{\\mathfrak{a}}  },$$\nthen $P(T)=\\sigma^{-1}_{\\mathfrak{a}}(F_{\\mathfrak{a}, T}  )\\subset \\mathcal{D}_{\\mathfrak{a}}$ for every cusp $\\mathfrak{a}$. \n\nFor the above choice of $T$, we are ready to estimate  $K_{\\widetilde{\\Gamma}^{\\mathfrak{a}}}(P(T))$ for any $\\mathfrak{a}$.  Consider the set\n\\begin{equation}\n\\mathcal{A}:=\\big\\{ \\gamma\\in\\widetilde{\\Gamma}^{\\mathfrak{a}}: d_{\\mathbb{H}^2}(z_1, \\gamma z_2)\\leq d_{\\mathbb{H}}(z_1, z_2), z_1, z_2\\in P(T), {\\rm Im}(z_1)\\geq {\\rm Im}(z_2)  \\big\\},\n\\end{equation}\nwhich, by Definition \\ref{defn-geodesic-covering-number},  is a geodesic cover of $P(T)$ in $\\widetilde{\\Gamma}^{\\mathfrak{a}}$.  \n\nFor any two points $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ in $P(T)$ with $y_1\\geq y_2$, the only possible isometries $\\gamma$ from\n$$ \\widetilde{\\Gamma}_{\\infty}^{\\mathfrak{a}}=\\left\\langle  \\begin{pmatrix}\n1 & 1\\\\\n0 & 1\n\\end{pmatrix}  \\right\\rangle,$$\nsuch that $d_{\\mathbb{H}^2}(z_1, \\gamma z_2)\\leq d_{\\mathbb{H}^2}(z_1, z_2)$ are \n\\begin{equation}\n\\mathcal{T}:=\\left\\{  \\begin{pmatrix}\n1 & -1\\\\\n0 & 1\n\\end{pmatrix}, \n\\begin{pmatrix}\n1 &0\\\\\n0 & 1\n\\end{pmatrix},\n\\begin{pmatrix}\n1 & 1\\\\\n0 & 1\n\\end{pmatrix}  \\right\\}.\n\\end{equation}\n\nIf $\\gamma\\in \\widetilde{\\Gamma}^{\\mathfrak{a}}\\setminus\\widetilde{\\Gamma}_{\\infty}^{\\mathfrak{a}} $, by the construction of $\\mathcal{D}_{\\mathfrak{a}}$  and \\eqref{eq-min-c}, we have\n\\begin{equation}\n{\\rm Im}(\\gamma z_2)=\\frac{y_2}{ (cx_2+d)^2+c^2 y_2^2}\\leq \\frac{1}{c^2 y_2}\\leq \\frac{1}{c^2_{\\mathfrak{a}} y_2}.\n\\end{equation}\nSince $y_2\\geq T\\geq 100+\\frac{10}{c_{\\mathfrak{a}}}$, we deduce that\n\\begin{equation}\n{\\rm Im}(\\gamma z_2)\\leq \\frac{1}{100 c^2_{\\mathfrak{a}}+10c_{\\mathfrak{a}}}<\\frac{1}{10c_{\\mathfrak{a}}}.\n\\end{equation}\nDenote $\\gamma z_2=x_0+iy_0$ ($y_0<\\frac{1}{10c_{\\mathfrak{a}}}$), then by the hyperbolic distance formula, \n\\begin{equation}\\label{eq-hyperbolic-cosh-distance}\n2\\cosh(d_{\\mathbb{H}^2}(z_1, z_2))=\\frac{(x_1-x_2)^2+y_1^2+y_2^2}{y_1 y_2},\n\\end{equation}\nand $|x_1-x_2|\\leq 1$, $y_1\\geq y_2\\geq T\\geq 100+\\frac{10}{c_{\\mathfrak{a}}}$, \nwe derive\n\\begin{align}\n&2\\cosh(d_{\\mathbb{H}^2}(z_1, \\gamma z_2))\n-2\\cosh(d_{\\mathbb{H}^2}(z_1, z_2)) \\nonumber\\\\\n&=\\frac{(x_1-x_0)^2+y_1^2+y_0^2}{y_1 y_0 }- \\frac{(x_1-x_2)^2+y_1^2+y_2^2}{ y_1 y_2}   \\nonumber\\\\\n&\\geq \\frac{y_1}{y_0}-\\frac{1}{y_1 y_2}-\\frac{y_1}{y_2}-\\frac{y_2}{y_1} \\nonumber\\\\\n&\\geq y_1\\Big(\\frac{1}{y_0}-\\frac{1}{y_2}\\Big)-\\frac{1}{100^2}-1\\nonumber\\\\\n&\\geq \\frac{10}{c_{\\mathfrak{a}}} \\Big( 10c_{\\mathfrak{a}}-\\frac{c_{\\mathfrak{a}}}{10}     \\Big)-2=99-2>0.\n\\end{align}\n\nHence we  have $\\mathcal{A}=\\mathcal{T}$. We derive that the geodesic-covering number of $P(T)$ in $\\widetilde{\\Gamma}^{\\mathfrak{a}}$ is $\\leq 3$.  Since  our choice of $T$  works for all cusps, and by \\eqref{eq-geo-cover-number-to-P}, we conclude that the geodesic-covering number of $F_{\\mathfrak{a}, T}$ in $\\Gamma$ is finite for all cusp $\\mathfrak{a}$, precisely $K_{\\Gamma}(F_{\\mathfrak{a}, T})\\leq 3$. \n\n\n2) Now we bound the geodesic-covering number of the central part $F_T$ in $\\Gamma$. \nDefine the diameter of $F_T$ as\n$$\\diam(F_T):=\\max_{p, q\\in F_T} d_{\\mathbb{H}^2}(p, q).$$\nSince $F_T$ is bounded, the diameter $\\diam(F_T)$ is finite. Pick any point $O$ inside $F_T$ which is not fixed by any element in $\\Gamma$ except identity, then the set\n$$\\mathcal{B}:=\\big\\{ \\gamma\\in\\Gamma: d_{\\mathbb{H}^2}(O, \\gamma(O))\\leq 3\\diam(F_T)   \\big\\}$$\nis a geodesic cover of $F_T$ in $\\Gamma$. Indeed, for any $\\gamma\\not\\in \\mathcal{B}$ and any two points $p, q\\in F_T$,  by triangle inequality, we get\n\\begin{align}\nd_{\\mathbb{H}^2}(p, \\gamma(q))&\\geq d_{\\mathbb{H}^2}(O, \\gamma(O))-d_{\\mathbb{H}^2}(p, O)-d_{\\mathbb{H}^2}(\\gamma(O), \\gamma(q))\\nonumber\\\\\n&\\geq 3\\diam(F_T)-\\diam(F_T)-\\diam(F_T)=\\diam(F_T)\\geq d_{\\mathbb{H}^2}(p, q).\n\\end{align}\n\nSince a Fuchsian group $\\Gamma$ acts properly discontinuously on $\\mathbb{H}^2$, the $\\Gamma$ orbit of any point is locally finite. Thus the set $\\mathcal{B}$ is finite. Therefore, the geodesic-covering number of $F_T$ in $\\Gamma$ is finite.  The same proof also works for the closure of $F_T$. \n\\end{proof}\n\n\\begin{remark}\n\tExplicitly counting the cardinality of a set of the type $\\mathcal{B}$ is the so-called \\textit{hyperbolic circle problem}, see e.g. Lax-Phillips \\cite{Lax-Phillips} and Phillips-Rudnick \\cite{Phillips-Rudnick} etc.\n\\end{remark}\n\n\n\n\\section{Finite index subgroups of the modular group}\nIn this section, we give the proof of Theorem \\ref{thm-finite-index}. \n\nLet $\\Gamma$ be a finite index subgroup of $\\mathrm{PSL}(2, \\mathbb{Z})$ with $[\\mathrm{PSL}(2,\\mathbb{Z}):\\Gamma]=\\mu$. Let $F$ be a fundamental domain of $\\mathrm{PSL}(2, \\mathbb{Z})$, we may choose \n$$F=\\Big\\{z\\in\\mathbb{H}^2: |\\Re(z)|\\leq \\frac{1}{2}, |z|\\geq 1 \\Big\\}.$$ \nIf we have the right coset decomposition $$\\mathrm{PSL}(2,\\mathbb{Z})=\\bigcup_{i=1}^{\\mu} \\Gamma \\alpha_i,$$ then \n\\begin{equation}\\label{finite-index-fundamental-domain}\nF_{\\Gamma}=\\bigcup_{i=1}^{\\mu} \\alpha_i(F)\n\\end{equation}\n is a fundamental domain of $\\Gamma$. One can choose the coset representatives properly to get a simply connected fundamental domain of $\\Gamma$ (see \\cite{Schoen}, Chapter IV, Theorem 3).\nFor example, for the principal congruence subgroup $$\\Gamma(2)=\\left\\{ \\gamma\\in\\mathrm{PSL}(2, \\mathbb{Z}): \\gamma\\equiv \\begin{pmatrix}\n1 & 0\\\\\n0 & 1\n\\end{pmatrix}  \\bmod 2  \\right\\},$$\n with index $[\\mathrm{PSL}(2, \\mathbb{Z}):\\Gamma(2)]=6$, see Figure \\ref{figure-Gamma2} (the arrows show the side parings) for a fundamental domain of $\\Gamma(2)$ and Figure \\ref{figure-surface-Gamma2} the shape of the surface $\\Gamma(2)\\backslash\\mathbb{H}^2$. \n\n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{fundamental-domain-Gamma2.jpg}\n\\caption{Fundamental domain for $\\Gamma(2) $}\n\t\\label{figure-Gamma2}\n\t\\end{figure}\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.7\\textwidth]{Gamma2-surface.png}\n\t\\caption{Shape of surface $\\Gamma(2)\\backslash\\mathbb{H}^2$}\n\t\\label{figure-surface-Gamma2}\n\\end{figure}\n\n\n\n\n\nFor a set $\\mathcal{S}$ of $N$ points on $Y\\cong\\Gamma\\backslash\\mathbb{H}^2$, we consider their representatives in a fundamental domain $F_\\Gamma$ constructed from the right coset decomposition.  Since $F_{\\Gamma}$ is a union of $\\mu$ copies of $F$, there exists an $\\alpha_j$ such that $\\alpha_j(F)$ contains $\\geq N\/\\mu$ points from $\\mathcal{S}$. Without loss of generality, we may assume $\\alpha_j$ is identity and still denote this copy as $F$.  Otherwise, we just take $\\alpha_j^{-1}(F_{\\Gamma})$ as the fundamental domain of $\\Gamma$ since $\\alpha_j$ is an isometry of $\\mathbb{H}^2$ and this transformation will not change distances and angles among the points we are considering. If we  have a lower bound for distinct distances among these $\\geq N\/\\mu$ points, this would also give us a lower bound for distinct distances among all points of $\\mathcal{S}$. \n\nWe divide $F$ into two parts $F=F_{u}\\cup F_o$ (see Figure \\ref{figure-partition-fundamental-domain}) with \n\\begin{equation}\\label{one-fundamental-domain-parition}\nF_u:=\\Big\\{ z=x+iy\\in\\mathbb{H}^2:  |x|\\leq \\frac{1}{2}, y\\geq 2  \\Big\\} ~\\text{and}~\nF_o:=F\\setminus F_u.\n\\end{equation}\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.6\\textwidth]{fundamental-domain-partition.jpg}\n\t\\caption{Partition of the fundamental domain $F=F_u+F_o$}\n\t\\label{figure-partition-fundamental-domain}\n\\end{figure}\n\n\nWe want to bound the geodesic-covering numbers of $F_u$ and $F_o$ in different base groups. We prove the following lemma. \n\\begin{lemma}\\label{lem-finite-index-geo-cover-number}\nFor any subgroup $\\Gamma$ of $\\mathrm{PSL}(2, \\mathbb{Z})$, the geodesic-covering numbers $K_{\\Gamma}(F_u)$ and $K_{\\Gamma}(F_o)$ are  both bounded by some absolute constants. Precisely\t\n\\begin{enumerate}[label=(\\roman*)]\n\t\\item\\label{finite-index-cusp-part} The geodesic-covering number of $F_u$ in $\\Gamma$ is $K_{\\Gamma}(F_u)\\leq 3$.\n\t\\item\\label{finite-index-central-part} The geodesic-covering number of $F_o$ in $\\Gamma$ is  $K_{\\Gamma}(F_o)\\leq 252$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{remark}\n\tThe estimate in \\ref{finite-index-central-part} may be improved by more careful calculations. We don't aim to optimize the constant here. The key point is that the geodesic-covering number of $F_o$ in any subgroup $\\Gamma$ is absolutely bounded and thus independent of the index of $\\Gamma$ in $\\mathrm{PSL}(2, \\mathbb{Z})$. One may also use $y\\geq U$ in the definition of $F_u$ for any large enough $U$ to optimize the estimate of $K_{\\Gamma}(F_o)$. \n\\end{remark}\n\n\n\n\nBefore giving the proof of Lemma \\ref{lem-finite-index-geo-cover-number}, we use it to prove Theorem \\ref{thm-finite-index} first. \n\\begin{proof}[Proof of Theorem \\ref{thm-finite-index}]\nSuppose $\\Gamma$ is a subgroup of $\\mathrm{PSL}(2, \\mathbb{Z})$  of finite index  $[\\mathrm{PSL}(2,\\mathbb{Z}):\\Gamma]=\\mu$. \t\nLet $\\mathcal{S}$ be a set of $N$ points on the hyperbolic surface $Y\\cong\\Gamma\\backslash\\mathbb{H}^2$, and define the distance set\n$$d_Y(\\mathcal{S}):=\\{ d_Y(p, q): p, q\\in \\mathcal{S}   \\}.$$\nIf $F$ is a fundamental domain of $\\mathrm{PSL}(2, \\mathbb{Z})$, by the fundamental domain of $\\Gamma$ in the form \\eqref{finite-index-fundamental-domain}, there exists some $j$ such that $\\alpha_j(F)$ contains more than $N\/\\mu$ points. Since $\\alpha_j$ is an isometry of $\\mathbb{H}^2$, without loss of generality, we assume $\\alpha_j(F)=F$ and let $\\mathcal{S}_F$ be these $\\geq N\/\\mu$ points on it. We observe that \n\\begin{equation}\\label{eq-distance-surface-to-one-F}\n|d_Y(\\mathcal{S})|\\geq |d_Y( \\mathcal{S}_F )|. \n\\end{equation}\nWe use Lemma \\ref{lem-finite-index-geo-cover-number} to establish a lower bound for $\t|d_Y(\\mathcal{S}_F )|$ and hence derive a lower bound for $|d_Y(\\mathcal{S})|$. \n\nWe partition the region $F=F_u\\cup F_o$ as in \\eqref{one-fundamental-domain-parition}.  Either $F_u$ or $F_o$ contains more than $\\frac{1}{2}|\\mathcal{S}_F|\\geq N\/2\\mu$ points. \n\n\\textbf{Case 1).}  The region $F_u$ contains more than $\\frac{1}{2}|\\mathcal{S}_F|$ points. Let $\\mathcal{S}_u:=\\mathcal{S}_F\\cap F_u$ be the points on $F_u$, and $\\Gamma_u$ be a geodesic-cover of $F_u$ in $\\Gamma$ with cardinality $K_{\\Gamma}(F_u)$. Then we have\n\\begin{align}\nQ_Y(\\mathcal{S}_u )&:=\\big\\{(p_1, p_2; p_3, p_4)\\in \\mathcal{S}_u^4: d_Y(p_1, p_2)=d_Y(p_3, p_4)\\neq 0 )  \\big\\}\\nonumber\\\\\n&~\\subset Q_{\\mathbb{H}^2}\\big(\\cup_{\\gamma\\in\\Gamma_u} \\gamma(\\mathcal{S}_u) \\big),\n\\end{align}\nwhere $Q_{\\mathbb{H}^2}(\\mathcal{P})$ is defined in \\eqref{eq-defn-QH}. By Lemma \\ref{lem-finite-index-geo-cover-number} \\ref{finite-index-cusp-part} and \\eqref{eq-Tao-quaruple}, we derive \n\\begin{equation}\n|Q_Y(S_u)|\\ll K_{\\Gamma}^3(F_u) |\\mathcal{S}_u|^3 \\log(K_{\\Gamma}(F_u)|\\mathcal{S}_u|)\\leq 27 |\\mathcal{S}_u|^3 \\log(3|\\mathcal{S}_u|).\n\\end{equation}\nConsequently by \\eqref{Quadruple-to-distance} and \\eqref{eq-distance-surface-to-one-F}, we get the  lower bound\n\\begin{equation}\n|d_Y(\\mathcal{S})|\\geq |d_Y(\\mathcal{S}_u)|\\gg \\frac{|\\mathcal{S}_u|}{ \\log|\\mathcal{S}_u| },\n\\end{equation}\nwhere the implied constant is absolute. Therefore, by the assumption $\\frac{N}{2\\mu}\\leq |\\mathcal{S}_u|\\leq N$,  we conclude that\n\\begin{equation}\\label{eq-distance-finite-index-cuspidal}\n|d_Y(\\mathcal{S})|\\geq C_1\\frac{N}{\\mu\\log N}\n\\end{equation}\nfor some absolute constant \t$C_1>0$. \n\t\n\t\n\\textbf{Case 2).}\tThe region $F_o$ contains more than $\\frac{1}{2}|\\mathcal{S}_F|$ points. Let $\\mathcal{S}_o=\\mathcal{S}_F\\cap F_o$ and $\\Gamma_o$ be a geodesic cover of $F_o$ in $\\Gamma$ with cardinality $K_{\\Gamma}(F_o)$. By Lemma \\ref{lem-finite-index-geo-cover-number} \\ref{finite-index-central-part} and a similar argument as in \\textbf{Case 1)}, we derive that\n\\begin{equation}\nQ_Y(\\mathcal{S}_o)\\subset Q_{\\mathbb{H}^2}\\big( \\cup_{\\gamma\\in\\Gamma_o}\\gamma(F_o)   \\big)\n\\end{equation}\nand thus\n\\begin{equation}\n|Q_Y(\\mathcal{S}_o)|\\ll K_{\\Gamma}^3(F_o) |\\mathcal{S}_o|^3 \\log(K_{\\Gamma}(F_o)|\\mathcal{S}_o|)\\leq 252^3 |\\mathcal{S}_o|^3 \\log( 252 |\\mathcal{S}_o|). \n\\end{equation}\t\nAgain by \\eqref{Quadruple-to-distance} and the assumption $\\frac{N}{2\\mu}\\leq |\\mathcal{S}_o|\\leq N$, we conclude that\n\\begin{equation}\\label{eq-distance-finite-index-central}\n|d_Y(\\mathcal{S})|\\geq |d_Y(\\mathcal{S}_o)|\\geq  C_2\\frac{N}{\\mu\\log N}\n\\end{equation}\t\nfor some absolute constant $C_2>0$. \n\nFinally, combining \\eqref{eq-distance-finite-index-cuspidal} and \\eqref{eq-distance-finite-index-central} and taking $C=\\min\\{C_1, C_2\\}$, we get the desired lower bound for distinct distances in hyperbolic surfaces associated with any finite index subgroup of $\\mathrm{PSL}(2, \\mathbb{Z})$, \n\\begin{equation}\n|d_Y(\\mathcal{S})|\\geq  C\\frac{N}{\\mu\\log N}\n\\end{equation}\nfor some absolute constant $C>0$. \n\\end{proof}\n\nIn the following we prove Lemma \\ref{lem-finite-index-geo-cover-number}.\n\n\\begin{proof}[Proof of \\ref{finite-index-cusp-part} in Lemma \\ref{lem-finite-index-geo-cover-number}]\nRecall that $F_u$ is the region \n$$\\Big\\{ z=x+iy\\in\\mathbb{H}^2:  |x|\\leq \\frac{1}{2}, y\\geq 2  \\Big\\}.$$\nWe consider the set\n\\begin{equation}\\label{eq-set-smaller-distance}\n\\mathcal{A}:=\\{\\gamma\\in \\mathrm{PSL}(2, \\mathbb{Z}): d_{\\mathbb{H}^2}(z_1, \\gamma z_2)\\leq d_{\\mathbb{H}^2}(z_1, z_2), z_1, z_2\\in F_u, {\\rm Im}(z_1)\\geq {\\rm Im}(z_2) \\},\n\\end{equation} \nwhich is a geodesic cover of $F_u$ in $\\mathrm{PSL}(2, \\mathbb{Z})$ by Definition \\ref{defn-geodesic-covering-number}.  For any subgroup $\\Gamma$ of $\\mathrm{PSL}(2, \\mathbb{Z})$, the set $\\mathcal{A}\\cap\\Gamma$ is a geodesic cover of $F_u$ in $\\Gamma$.  \n\nFor any two points $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ in $F_u$ with $y_1\\geq y_2\\geq 2$, and\n $$\\gamma=\\begin{pmatrix}\n a&b\\\\\n c&d\n \\end{pmatrix}\\in \\mathrm{PSL}(2,\\mathbb{Z}),$$\nthe imaginary part of $\\gamma(z_2)$ can be written as\n$$ \\frac{y_2}{|cz_2+d|^2}=\\frac{y_2}{(cx_2+d)^2+c^2 y_2^2 }.$$\n\nIf $c=0$, then $a=d=1$, the isometry  $\\gamma$ is actually a translation of the form\n$$\\gamma=\\begin{pmatrix}\n1&b\\\\\n0&1\n\\end{pmatrix}\\in \\mathrm{PSL}(2, \\mathbb{Z}),$$\nfor some $b\\in\\mathbb{Z}$. The only possible choices of $\\gamma$ for which $d_{\\mathbb{H}^2}(z_1, \\gamma z_2)\\leq d_{\\mathbb{H}^2}(z_1, z_2)$ are from the set\n\\begin{equation}\\label{eq-translation-cover}\n\\mathcal{T}=\\left\\{  \\begin{pmatrix}\n1 & -1\\\\\n0 & 1\n\\end{pmatrix}, \n\\begin{pmatrix}\n1 &0\\\\\n0 & 1\n\\end{pmatrix},\n\\begin{pmatrix}\n1 & 1\\\\\n0 & 1\n\\end{pmatrix}  \\right\\}.\n\\end{equation} \n\nIf $c\\neq 0$, then $|c|\\geq 1$ and thus\n$${\\rm Im}(\\gamma z_2)\\leq \\frac{1}{y_2}\\leq \\frac{1}{2}.$$\nDenote $\\gamma(z_2)=x_0+iy_0$, then $y_0\\leq \\frac{1}{2}$. By the hyperbolic distance formula  \\eqref{eq-hyperbolic-cosh-distance}\nwith the fact $y_1\\geq y_2\\geq 2$ and  $|x_1-x_2|\\leq 1$, we get\n\\begin{align}\n&2\\cosh(d_{\\mathbb{H}^2}(z_1, \\gamma z_2))\n-2\\cosh(d_{\\mathbb{H}^2}(z_1, z_2)) \\nonumber\\\\\n&=\\frac{(x_1-x_0)^2+y_1^2+y_0^2}{y_1 y_0 }- \\frac{(x_1-x_2)^2+y_1^2+y_2^2}{ y_1 y_2}   \\nonumber\\\\\n&\\geq \\frac{y_1}{y_0}-\\frac{1}{y_1 y_2}-\\frac{y_1}{y_2}-\\frac{y_2}{y_1} \\nonumber\\\\\n&\\geq 2y_1-\\frac{1}{4}-\\frac{y_1}{2}-1\\geq \\frac{7}{4}>0.\n\\end{align}\nThus for any  $\\gamma\\in \\mathrm{PSL}(2, \\mathbb{Z})$ with $c\\neq 0$, we always have $d_{\\mathbb{H}^2}(z_1, \\gamma z_2)>d_{\\mathbb{H}^2}(z_1, z_2)$.  Hence $\\mathcal{A}=\\mathcal{T}$. \n\nFor $\\Gamma$ being any subgroup  of $\\mathrm{PSL}(2, \\mathbb{Z})$, the elements of $\\gamma'\\in\\Gamma$ such that $$d_{\\mathbb{H}^2}(z_1, \\gamma' z_2)\\leq d_{\\mathbb{H}^2}(z_1, z_2) ~\\text{ with} z_1, z_2\\in F_u, {\\rm Im}(z_1)\\geq {\\rm Im}(z_2)$$ \nare also from the set $\\mathcal{A}=\\mathcal{T}$ in \\eqref{eq-set-smaller-distance} and \\eqref{eq-translation-cover}. Therefore, by Definition \\ref{defn-geodesic-covering-number}, the set $\\mathcal{T}\\cap \\Gamma$ is a geodesic cover of $F_u$ in $\\Gamma$.  (Note that $\\mathcal{T}\\cap\\Gamma$ always contains the identity.) We conclude that the geodesic-covering number of $F_u$ in any subgroup $\\Gamma$ of $\\mathrm{PSL}(2, \\mathbb{Z})$ is $K_{\\Gamma}(F_u)\\leq 3$. \n\\end{proof}\n\n\n\\begin{proof}[Proof of \\ref{finite-index-central-part} in Lemma \\ref{lem-finite-index-geo-cover-number}]\nNow we deal with the bounded part $$F_o=\\{z=x+iy\\in\\mathbb{H}^2: |z|\\geq 1, 0<y<2 \\}.$$\nWe estimate the diameter of $F_o$,\n\\begin{align}\\label{eq-Fo-diameter}\n\\cosh(\\diam(F_o))&=\\cosh\\big(\\max_{z_1, z_2\\in F_o} d_{\\mathbb{H}^2}(z_2, z_2)\\big)\\nonumber\\\\\n&\\leq \\cosh\\bigg(d_{\\mathbb{H}^2}\\Big(\\frac{-1+\\sqrt{3}i}{2}, \\frac{1}{2}+2i\\Big) \\bigg)=\\frac{23\\sqrt{3}}{24}=1.6598\\ldots\n\\end{align}\n Denote $r_0:=\\max_{z\\in F_o}d_{\\mathbb{H}^2}(2i, z) $, then\n\\begin{equation}\\label{eq-Fo-r0}\n\\cosh(r_0)=\\cosh\\bigg(d_{\\mathbb{H}^2}\\Big( 2i, \\frac{1+\\sqrt{3}i}{2} \\Big)  \\bigg)=\\frac{5\\sqrt{3}}{6}=1.4433\\ldots\n\\end{equation}\nThe point $2i$ is not fixed by any element in $\\mathrm{PSL}(2, \\mathbb{Z})$ except identity. By definition, the  set\n\\begin{equation}\n\\Gamma_o:=\\{\\gamma\\in \\mathrm{PSL}(2, \\mathbb{Z}):   d_{\\mathbb{H}^2}(2i, \\gamma(2i))\\leq \\diam(F_o)+2r_0    \\}.\n\\end{equation}\nis a geodesic cover of $F_o$ in $\\mathrm{PSL}(2, \\mathbb{Z})$. In fact, for any $\\gamma\\in\\mathrm{PSL}(2, \\mathbb{Z})$ but not in $\\Gamma_o$, we have \n\\begin{equation}\\label{eq-Fo-distance-increase}\nd_{\\mathbb{H}^2}(z_1, \\gamma z_2)\\geq \\diam(F_o)\\geq d_{\\mathbb{H}^2}(z_1, z_2), \\forall z_1, z_2\\in F_o.\n\\end{equation}\n\nNow we estimate the size of $\\Gamma_o$. The set $\\{\\gamma(F_o): \\gamma\\in \\Gamma_o \\}$ is contained in the disc $\\mathcal{D}(2i, R)$ centering at $2i$ of radius $R=\\diam(F_o)+3r_0$.  Thus,\n\\begin{equation}\n|\\Gamma_o|\\cdot {\\rm Area}(F_o)={\\rm Area}\\Big(\\bigcup_{\\gamma\\in\\Gamma_o}\\gamma(F_o)\\Big)\\leq {\\rm Area}(\\mathcal{D}(2i, R)).\n\\end{equation}\nSince the area of the fundamental domain $F$ is $\\pi\/3$ and the area of $F_u$ is $1\/2$, we derive that \n$${\\rm Area}(F_o)=\\frac{\\pi}{3}-\\frac{1}{2}=0.5471\\ldots$$\nBy the hyperbolic area formula, \\eqref{eq-Fo-diameter} and \\eqref{eq-Fo-r0}, we get\n$${\\rm Area}(\\mathcal{D}(2i, R))=2\\pi(\\cosh(R)-1)=\\frac{\\pi}{36}(848 +11\\sqrt{4381})=137.5389\\ldots\n$$\nHence,\n\\begin{equation}\n|\\Gamma_o|\\leq \\frac{{\\rm Area}(\\mathcal{D}(2i, R))}{{\\rm Area}(F_o)}   \\leq 252.\n\\end{equation}\n\nFor any subgroup $\\Gamma$ of $\\mathrm{PSL}(2, \\mathbb{Z})$, by \\eqref{eq-Fo-distance-increase}, we see that the set $\\Gamma_o\\cap\\Gamma$ is a geodesic-cover of $F_o$ in $\\Gamma$, and immediately we have $K_{\\Gamma}(F_o)\\leq |\\Gamma_o|\\leq 252$. \n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nEntanglement is a key resource for quantum information tasks, like quantum simulations \\cite{Monroe}, computation \\cite{Albash}, and communication \\cite{Pirandola}. These tasks are based on the idea of creating networks of quantum systems, where the generation of well-controlled entanglement between spatially separated nodes is essential. These networks, which may consist of distant or nearby nodes, have been thoroughly investigated for efficient process and transfer of quantum information \\cite{Awschalom}. However, due to interactions with an uncontrollable environment noisy or non-maximally entangled states are produced. To protect quantum information and to guarantee a high performance of its processing, one can use quantum error correction \\cite{Peres, Devitt} or quantum teleportation in combination with entanglement purification \\cite{Bennett1, Deutsch, Bennett2}. Quantum error correction is particularly suitable for quantum computers, however it is inferior to entanglement purification with two-way classical communication \\cite{Bennett2}, which is the subject of our investigations. In particular, we consider a so-called recurrence protocol \\cite{Dur}, an iterative approach, which operates in each purification step only on two identical copies of states.\n\nIn this paper, we discuss entanglement purification from the point of view of optimality. Recently, optimized entanglement purification has been investigated with the help of genetic algorithms \\cite{Krastanov}, where \nthe analytical and numerical studies are based on Werner states \\cite{Werner}. In fact, also the first ever proposed protocols are based on either Werner \\cite{Bennett1} or Bell diagonal states \\cite{Deutsch}. It has been \nshown that $4$ or $12$ local random $SU(2)$ transformations can bring any states into a Bell diagonal or Werner state, respectively \\cite{Bennett2}. We have already argued that these random local unitary transformations \nand states obtained in consequence are only useful\nfor grasping and consequently understanding the complex task of purification because they may waste useful entanglement \\cite{Torres}. Therefore, here, we develop a method for general states and demonstrate it for \nseveral simple examples including also the Werner state. Nonetheless, our motivation also lies in the fact that an experimental implementation may not have enough control over the generated noisy states and thus more general, adaptive, and optimal purification strategies have to be made available. In this general context, we assume that an experiment can still guarantee identical copies of states before the purification protocol takes place. \n\nOur method is based on quasi-Monte Carlo numerical sampling of the states, which undergo the purification protocol, and then the concurrence \\cite{Wooters} of the output states is integrated over an {\\it a priori} probability distribution \nfunction. This results in an average two-qubit gate-dependent cost function for the whole sample of states and then we employ a quasi-Newton algorithm \\cite{Kelley} for solving this non-linear optimization problem \non the whole $SU(4)$ group. The obtained two-qubit gate is optimal in the sense that it achieves, on average, a higher increase of entanglement of an input family of states. We discuss the performance of our method for several examples and compare with protocols based on one bilateral application of controlled-NOT gates, the paradigmatic two-qubit operation used in the seminal purification protocols \\cite{Bennett1,Deutsch}.\n\nThe paper is organized as follows. In Sec.~\\ref{Method} we introduce our\nmethod and give some elementary examples to allow further acquaintance with the concept of the introduced cost function. \nIn Sec.~\\ref{Results} we demonstrate our approach for four one-parameter and one two-parameter family of states. \nNumerical and analytical results are presented for concurrences and success probabilities.  In Sec.~\\ref{Conclusions}\nwe summarize and draw our conclusions. Some details supporting the main text are collected in Appendix~\\ref{AppendixA}.\n\n\\section{Method}\n\\label{Method}\nIn this section, we describe how the recurrence protocol is optimized with respect to an input familiy of quantum states.\n\n\\subsection{Entanglement purification protocol}\\label{EntPurif}\nLet us consider the product state of two-qubit pairs \n\\begin{equation}\n  \\boldsymbol{\\rho}=\\rho^{A_1,B_1} \\otimes\\rho^{A_2,B_2},\n\\label{eq:rhoAB4}\n\\end{equation}\nwhere qubit components of each pair are assumed to be spatially separated at distant locations. These locations are labeled by $A$ and $B$. In an \nentanglement purification protocol, one performs local quantum operations, which may not just involve two-qubit gates \\cite{Bernad1}. \nThis is followed by measurements on one of the pairs\nat both locations. A classical communication between $A$ and $B$ results in a qubit pair with a higher degree of entanglement. Both pairs are assumed to start in the same state $\\rho$ \nand the degree of the entanglement is usually measured by the \nfidelity with respect to one of the Bell states\n\\begin{align}\n  \\ket{\\Psi^\\pm}=\\tfrac{1}{\\sqrt2}\\left(\\ket{01}\\pm\\ket{10}\\right), \\,\n  \\ket{\\Phi^\\pm}=\\tfrac{1}{\\sqrt2}\\left(\\ket{00}\\pm \\ket{11}\\right).\n  \\label{eq:Bellstates}\n\\end{align}\nHowever, these states can be subject to local unitary transformations, which can cause some technical difficulties, when one uses fidelity, i.e.~using fidelity as a cost function would force us to search also for additional local unitary operations in order to align the output state of the protocol with the Bell basis. As we intend to \nanalyze the entanglement purification in a very general setup, one requires an entanglement measure that is invariant under local unitary transformations. Therefore, we turn to the concurrence as a measure \nof the the attainability of a maximally entangled state \\cite{Wooters}:\n\\begin{equation}\\label{eq:concurrence}\n \\mathcal{C}(\\rho)=\\max\\{0,\\lambda_1-\\lambda_2-\\lambda_3-\\lambda_4\\}.\n\\end{equation}\nHere $\\lambda_1, \\lambda_2, \\lambda_3, \\lambda_4$ are the square roots of the non-negative eigenvalues of the non-Hermitian matrix\n\\begin{equation}\n \\tilde \\rho = \\rho (\\sigma_{y}\\otimes\\sigma_{y})\\rho^{*}(\\sigma_{y}\\otimes\\sigma_{y}), \\nonumber\n\\end{equation}\nwith $*$ being the complex conjugation in the standard basis and $\\sigma_{y}$ is the Pauli matrix.\n\n\\begin{figure}[t!]\n    \\includegraphics[width=.4\\textwidth]{fig1.pdf}\n    \\caption{Schematic representation of bipartite entanglement purification, where the entanglement purification protocol trades two \n    entangled qubit pairs for a qubit pair with a higher degree of entanglement, which is quantified by the concurrence $\\mathcal{C}$.}\n    \\label{fig:purifscheme}\n\\end{figure}\n\nIn this manuscript, we examine and optimize a purification protocol having the following steps:\\\\\n\n(I) Two unitary transformations $\\rho \\rightarrow U(\\boldsymbol{\\alpha}) \\rho U(\\boldsymbol{\\alpha})^\\dagger$ are applied locally at $A$ and $B$ (see Fig.~\\ref{fig:purifscheme}), where $U$ is a general two-qubit unitary described by a parameter vector $\\boldsymbol{\\alpha}$.\nAfter the application of the quantum operation the four-qubit system \nattains the state  \n\\begin{equation}\n\\boldsymbol{\\rho}'=\n U_{A1,A2}(\\boldsymbol{\\alpha}) U_{B1,B2}(\\boldsymbol{\\alpha}) \\boldsymbol{\\rho} U^\\dagger_{B1,B2} (\\boldsymbol{\\alpha}) U^\\dagger_{A1,A2} (\\boldsymbol{\\alpha}). \n\\label{eq: maprho}\n\\end{equation}\n\n(II)  One of the pairs $(A_2,B_2)$ is then locally measured in the standard basis. There are four possible states, i.e., 4-dimensional vectors, in which one can find the measured pair: \n\\begin{eqnarray}\n \\ket{1}&=&\\ket{00}_{A_2,B_2}, \\quad \\ket{2}=\\ket{01}_{A_2,B_2}, \\nonumber \\\\\n \\ket{3}&=&\\ket{10}_{A_2,B_2}, \\quad \\ket{4}=\\ket{11}_{A_2,B_2}. \\nonumber\n\\end{eqnarray}\nA successful measurement of one of the states $\\ket{i}$ with $i\\in\\{1,2,3,4\\}$ results in\na state for the other qubit\n\\begin{equation}\n  \\tilde\\rho^{A_1,B_1}_i= \\frac{\\bra{i} \\boldsymbol{\\rho}' \\ket{i} }{\\mathrm{Tr}\\{\\ketbra{i}{i} \\boldsymbol{\\rho}'\\}}\n\\label{}\n\\end{equation}\nwith probability\n\\begin{equation}\n p_i=\\mathrm{Tr}\\{\\ketbra{i}{i} \\boldsymbol{\\rho}'\\}. \\nonumber\n\\end{equation}\n\n(III) Depending on value of the measurement results, which are communicated between the two parties, the state with largest concurrence and the related success probability are kept, whereas the others are discarded. The output of the protocol is the pair \n$(\\mathcal{C}',P)$, concurrence value $\\mathcal{C}'$ of the state obtained with probability $P$. If there are multiple maxima, e.g., $\\mathcal{C}(\\tilde\\rho^{A_1,B_1}_1)=\\mathcal{C}(\\tilde\\rho^{A_1,B_1}_2)$, then \n$\\mathcal{C}'=\\mathcal{C}(\\tilde\\rho^{A_1,B_1}_1)$ and $P=p_1+p_2$.\\\\\n\nGiven two copies of a state $\\rho$ with concurrence $\\mathcal{C}$, it is straightforward to see that pair $(\\mathcal{C}',P)$ depends on the vector $\\boldsymbol{\\alpha}$, which we use as the optimization parameter.\n\nRecurrence entanglement purification protocols might include symmetric and asymmetric single-qubit gates before and after the bilateral action of the two-qubit operations \\cite{Bennett1,Deutsch}. These are important in an analytical approach to maintain the form of the state after each iteration, however, entanglement is not affected by this process. In this work, we shall omit these single-qubit gates as we are focused on increasing the value of the concurrence and not on the specific form of the state after each iteration step. \n\n\\subsection{Protocol optimization}\\label{ProtOpt}\n\nIn order to optimize the protocol described in Sec.~\\ref{EntPurif}, we first define an appropriate parametrization for the two-qubit unitary operations used in Eq.~\\eqref{eq: maprho}. In principle, one should consider elements from $U(4)$, the group of $4 \\times 4$ unitary matrices, which contains the subgroup $SU(4)$. However, $U(4)$ is the semi-direct product of $U(1)$ and $SU(4)$, where elements of $U(1)$ are rotations of the unit circle \\cite{Baker}. Therefore,\nchoosing elements from $SU(4)$ in Eq.~\\eqref{eq: maprho}\nrepresents the most general unitary quantum operation involving two-qubit gates at locations $A$ and $B$.\nElements in $SU(4)$ can be parametrized in the following way \\cite{Tilma1}:\n\\begin{eqnarray} \\label{eq: eulerU}\n U(\\boldsymbol{\\alpha})=e^{i \\sigma_3 \\alpha_1} e^{i \\sigma_2 \\alpha_2} e^{i \\sigma_3 \\alpha_3} e^{i \\sigma_5 \\alpha_4} e^{i \\sigma_3 \\alpha_5} e^{i \\sigma_{10} \\alpha_6} \n e^{i \\sigma_3 \\alpha_7} e^{i \\sigma_2 \\alpha_8} \\nonumber \\\\ \n e^{i \\sigma_{3} \\alpha_9} e^{i \\sigma_5 \\alpha_{10}} e^{i \\sigma_3 \\alpha_{11}} e^{i \\sigma_{2} \\alpha_{12}} \n e^{i \\sigma_3 \\alpha_{13}} e^{i \\sigma_8 \\alpha_{14}} e^{i \\sigma_{15} \\alpha_{15}}, \\nonumber \n\\end{eqnarray}\nwith $\\boldsymbol{\\alpha}=(\\alpha_1, \\alpha_2, \\dots, \\alpha_{15})^T \\in \\mathbb{R}^{15}$ ($T$ denotes the transposition), and\n\\begin{eqnarray}\n && 0 \\leqslant \\alpha_1, \\alpha_3, \\alpha_5, \\alpha_7, \\alpha_9, \\alpha_{11}, \\alpha_{13} \\leqslant \\pi, \\nonumber \\\\\n &&  0 \\leqslant \\alpha_2, \\alpha_4, \\alpha_6, \\alpha_8, \\alpha_{10}, \\alpha_{12} \\leqslant \\frac{\\pi}{2} \\nonumber \\\\\n && 0 \\leqslant \\alpha_{14} \\leqslant \\frac{\\pi}{\\sqrt{3}}, \\quad 0 \\leqslant \\alpha_{15} \\leqslant \\frac{\\pi}{\\sqrt{6}}, \\label{eq:anglecond}\n\\end{eqnarray}\nwhere $\\sigma_i,\\ i=1, ..., 15$ form a Gell-Mann type basis of the Lie group $SU(4)$, see Appendix \\ref{AppendixA}. This is called the Euler angle parametrization of $SU(4)$, which is sufficient\nto represent every element of the Lie group. For example, the canonical parametrization $e^{i H}$, where $H$ is a $4 \\times 4$ self-adjoint matrix with trace zero, is not minimal, because after exponentiation we may have multiple wrappings around the great circles of the $7$-sphere. \n\nOur aim is to increase the concurrence, which is a non-linear function of the state $\\rho$ and the protocol's unitary matrix $U$. Furthermore, we have lower and upper bounds on $\\boldsymbol{\\alpha}$, as given in Eq.~\\eqref{eq:anglecond}. We then consider a cost function $f:\\mathbb{R}^{15} \\rightarrow \n\\mathbb{R}$ as\n\\begin{align}\n    f(\\boldsymbol{\\alpha}) = 1 - \\mathcal{C}'(\\boldsymbol{\\alpha}) \\label{eq:f}\n\\end{align}\nand our optimization problem is, to find local minimizers, e.g., a point $\\boldsymbol{\\alpha}^*$ such that\n\\begin{equation}\n f(\\boldsymbol{\\alpha}^*)\\leqslant f(\\boldsymbol{\\alpha}) \\nonumber\n\\end{equation}\nfor all $\\boldsymbol{\\alpha}$ in the Euclidean norm defined neighborhood of $\\boldsymbol{\\alpha}^*$. The gradient $\\nabla f (\\boldsymbol{\\alpha})$ can be made available to us due to an automatic differentiation \\cite{JAX}, so we \nimplement for the optimization a quasi-Newton algorithm the so-called limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method \\cite{Liu}. Here, we have a constrained optimization problem, because $\\boldsymbol{\\alpha}$ takes values in the hyperrectangle defined by Eq.~\\eqref{eq:anglecond} and thus we use the L-BFGS approach of Ref. \\cite{Ryrd}. \nAs long as the state to be purified is given, the above approach yields an optimal $\\boldsymbol{\\alpha}^*$ and a concurrence $\\mathcal{C}'$ as close as possible to one. \nThe vector $\\boldsymbol{\\alpha}^*$ gives the best two-qubit unitary gate associated with this given state. However, our main aim is to provide the best gate, which is capable to purify most effectively certain class of states and not only a fixed one. This is relevant in a scenario where the generation of distant entanglement is affected by noise, leading to slightly different types of states entering the protocol.\nIn order to formulate this quantitatively we introduce  a probability density function (PDF) $p(\\boldsymbol{x})$, where the vector $\\boldsymbol{x}$ defines uniquely the state $\\rho$ according to a parametrization. \nFor example, in the case of a  Werner state \\cite{Werner}\n\\begin{eqnarray}\n\\hat{\\rho}(x) &=& x \\ket{\\Psi^{-}}\\bra{\\Psi^{-}} + \\frac{1-x}{3} \\ket{\\Psi^{+}}\\bra{\\Psi^{+}} \\nonumber \\\\\n&+& \\frac{1-x}{3}  \\ket{\\Phi^{-}}\\bra{\\Phi^{-}}+ \\frac{1-x}{3} \\ket{\\Phi^{+}}\\bra{\\Phi^{+}},  \n\\label{eq:wernerstate}\n\\end{eqnarray}  \nwe have $x\\in[0,1]$ and the PDF satisfies\n\\begin{equation}\n \\int^1_0 p(x)=1. \\nonumber\n\\end{equation}\nIn general, a two-qubit state can be described by $15$ parameters, which have to fulfill some non-trivial conditions \\cite{Kimura,Byrd,Gamel}. The choice of the PDF is not straightforward and the only guideline \nwe have is that the support $\\operatorname{supp}(p)$ consists of all $\\boldsymbol{x}$, which define entangled states. This is motivated by \nthe fact that separable states are not purifiable. In the case of the Werner states, the support of the PDF is the interval $(0.5,1]$. Thus, the PDF may put more weight on states with a given concurrence.\n\\begin{algorithm}[H]\n\t\t\\setstretch{1.20}\n\t\t\\caption{Optimization of recurrence protocol}\n\t\t\\label{alg:ouralgo}\n\t\t\\hspace*{\\algorithmicindent} \\textbf{Input} $\\boldsymbol{\\rho}(\\boldsymbol{x})$, optimizer $\\text{OPT}$\\\\\n\t\t\\hspace*{\\algorithmicindent} \\textbf{Output} $\\boldsymbol{\\rho}(\\boldsymbol{x})$\n\t\t\\begin{algorithmic}[1]\n\t\t\\For{$j=1$ to $M$}\n\t\t\\State $\\boldsymbol{x}_j \\sim p(\\boldsymbol{x}_j)$\n\t\t\\State $\\boldsymbol{\\rho}_j = \\boldsymbol{\\rho}(\\boldsymbol{x}_j)$\n\t\t\\EndFor\n\t\t\t\\For{$i=1$ to $N$}\n\t\t\t\\State $U_{AB}(\\boldsymbol{\\alpha}) = U_{A1,A2}(\\boldsymbol{\\alpha}) U_{B1,B2}(\\boldsymbol{\\alpha})$\n\t\t\t\\For{$j=1$ to $M$}\n\t\t\t\\State $\\boldsymbol{\\sigma}^{j}(\\boldsymbol{\\alpha}) = U_{AB}(\\boldsymbol{\\alpha}) \\boldsymbol{\\rho}_j U^{\\dagger}_{AB}(\\boldsymbol{\\alpha})$\n\t\t\t\\For{$l=1$ to $4$}\n\t\t\t\\State $\\boldsymbol{\\sigma}^{jl}_A(\\boldsymbol{\\alpha}) =  \\frac{\\bra{l} \\boldsymbol{\\sigma}^{j}(\\boldsymbol{\\alpha})  \\ket{l} }{\\mathrm{Tr}\\{\\ketbra{l}{l} \\boldsymbol{\\sigma}^{j}(\\boldsymbol{\\alpha})  \\}}$\n\t\t\t\\EndFor\n\t\t\t\\EndFor\n\t\t\t\\State $\\mathcal{C}^l(\\boldsymbol{\\alpha}) = \\frac{1}{M}\\sum_{j=1}^M \\mathcal{C}(\\boldsymbol{\\sigma}^{jl}_A)$\n\t\t\t\\State $l' = \\underset{l=1,2,3,4}{\\text{argmin}}\\ \\left(1 - \\mathcal{C}^l(\\boldsymbol{\\alpha})\\right)$\n\t\t\t\\State $\\bar{f}(\\boldsymbol{\\alpha}) =  1 - \\mathcal{C}^{l'}(\\boldsymbol{\\alpha})$\n\t\t\t\\State $\\boldsymbol{\\alpha}^* = \\text{OPT}(\\bar{f}(\\boldsymbol{\\alpha}), \\nabla_{\\boldsymbol{\\alpha}} \\bar{f}(\\boldsymbol{\\alpha}))$\n\t\t\t\\State $\\boldsymbol{\\rho}_j = \\boldsymbol{\\sigma}^{jl'}_{A}(\\boldsymbol{\\alpha}^*) \\otimes \\boldsymbol{\\sigma}^{jl'}_{A}(\\boldsymbol{\\alpha}^*)$\n\t\t\t\\EndFor\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\nIn this context, we have an output concurrence $\\mathcal{C}'(\\boldsymbol{\\alpha}, \\boldsymbol{x})$ depending on both the two-qubit gate and the input state. Therefore, we are going to use an average cost function \n\\begin{equation}\n \\bar{f}(\\boldsymbol{\\alpha}) = 1 - \\int_{\\operatorname{supp}(p)} \\mathcal{C}'(\\boldsymbol{\\alpha}, \\boldsymbol{x}) p(\\boldsymbol{x})\\,  d \\boldsymbol{x}, \\label{eq:cost}\n\\end{equation}\nin the L-BFGS algorithm. The integral can be either solved numerically or approximated via a quasi-Monte Carlo approach by sampling $\\boldsymbol{x}$ over its corresponding parameter distribution $p(\\boldsymbol{x})$:\n\\begin{align}\n    \\bar{f}(\\boldsymbol{\\alpha}) = 1 - \\underset{\\boldsymbol{x} \\sim p(\\boldsymbol{x})}{\\mathbb{E}} \\left[\\mathcal{C}'(\\boldsymbol{\\alpha}, \\boldsymbol{x}) \\right].\n\\end{align}\nWe chose the second approach since it proved to be precise enough and numerically faster. A summarize of the optimization routine can be seen in Algorithm~\\ref{alg:ouralgo}. \n\nNow, we shed light on the meaning of the average cost function through the following two examples. We employ the CNOT gate\n\\begin{equation}\n U_{\\text{CNOT}}=\\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1\\\\ 0 & 0 & 1 & 0  \\end{pmatrix}=\n e^{i\\frac{3\\pi}{4}} \\times \\underbrace{U'}_{\\in SU(4)}, \\nonumber\n\\end{equation}\nwhere $U'$ in the Euler angle parametrization is given by setting\n\\begin{eqnarray}\n &&\\alpha_3= \\alpha_5=\\alpha_7= \\frac{\\pi}{4}, \\nonumber \\\\\n &&\\alpha_4= \\alpha_6=\\alpha_{10}= \\frac{\\pi}{2}, \\nonumber\n\\end{eqnarray}\nand the remaining $9$ angles to zero in Eq.~\\eqref{eq: eulerU}. By fixing the vector $\\boldsymbol{\\alpha}$, one is able to get a value for the average cost function of the CNOT gate and thus to evaluate its performance. \\\\\n\n{\\it Example 1.}\nLet us consider the state\n\\begin{equation}\n \\frac{1}{6}\\begin{pmatrix} 1+2x & 0 & 0 & 1-4x \\\\ 0 & 2-2x & 0 & 0 \\\\ 0 & 0 & 2-2x & 0\\\\ 1-4x & 0 & 0 & 1+2x  \\end{pmatrix}, \\quad \\text{with} \\quad x \\in [0,1] \\label{eq:example1}\n\\end{equation}\nsubject to the purification protocol with CNOT gates. This state is a rotated Werner state and its concurrence reads\n\\begin{equation}\n \\mathcal{C}(x)=\\begin{cases}2x-1 & x \\in (0.5,1], \\\\ 0 & x \\in [0, 0.5].\\end{cases} \\nonumber\n\\end{equation}\nIn fact, this example is based on the pioneering protocol of Ref. \\cite{Bennett}. \nThe output reads\n\\begin{equation}\n \\mathcal{C}'_{\\text{CNOT}}(x)=\\frac{3(4x^2-1)}{5-4x+8x^2}, \\quad \\text{for} \\quad x \\in (0.5,1], \\nonumber\n\\end{equation}\nwith success probability\n\\begin{equation}\n P_{\\text{CNOT}}=\\frac{5-4x+8x^2}{9}. \\nonumber\n\\end{equation}\nFor the sake of simplicity, we consider a uniform PDF $p(x)$ with $\\operatorname{supp}(p)=(0.5,1]$. Hence, the input average cost function reads\n\\begin{equation}\n \\bar{f}_{\\text{input}}=1-\\int^1_{0.5} \\mathcal{C}(x) p(x)\\, dx =0.5, \\nonumber\n\\end{equation}\nand the application of the purification protocol with CNOT gates yields\n\\begin{equation}\n \\bar{f}_{\\text{CNOT}}=1-\\int^1_{0.5} \\mathcal{C}'(x) p(x)\\, dx =0.450103. \\nonumber\n\\end{equation}\nThe result shows that the protocol with two identical copies of states allows one to increase on average the entanglement of the output states.\\\\\n\n{\\it Example 2.}\nNow, we consider the state\n\\begin{equation}\n \\begin{pmatrix} \\frac{x}{2} & 0 & 0 & -\\frac{x}{2} \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1-x & 0\\\\ -\\frac{x}{2} & 0 & 0 & \\frac{x}{2}  \\end{pmatrix}, \\quad \\text{with} \\quad x \\in [0,1]. \\label{eq:example2}\n\\end{equation}\nThis state with concurrence $\\mathcal{C}(x)=x$ is perfectly purifiable in just one iteration of the protocol \\cite{Torres}. Hence, \n\\begin{equation}\n \\mathcal{C}'_{\\text{CNOT}}(x)=1, \\nonumber\n\\end{equation}\nwith success probability\n\\begin{equation}\n P_{\\text{CNOT}}=\\frac{x^2}{2}. \\nonumber\n\\end{equation}\nIt is immediate from Eq.~\\eqref{eq:cost} that for {\\it any} PDF with $\\operatorname{supp}(p)=[0,1]$\n\\begin{equation}\n \\bar{f}_{\\text{CNOT}}=1-\\int^1_{0} \\mathcal{C}'(x) p(x)\\, dx =0. \\nonumber\n\\end{equation}\nThis means that the CNOT gate is optimal for this family of states.\n\n\n{\\it Example 3.}\nAs a last example, let us consider the initial state\n\\begin{equation}\n x\\ketbra{\\Phi^+}{\\Phi^+}+(1-x)\\ketbra{\\Phi^-}{\\Phi^-}, \\quad \\text{with} \\quad x\\in[0,1],   \n\\end{equation}\nand concurrence $\\mathcal{C}(x)=|1-2x|$. After one iteration of the purification protocol with bilaterally applied CNOT gates, it is not hard to realize that the resulting state has the concurrence \n\\begin{equation}\n\\mathcal{C}'_{\\text{CNOT}}(x)=(1-2x)^2\\nonumber \n\\end{equation}\nwith success probability $P_{\\text{CNOT}}=1$. The output\nconcurrence $\\mathcal{C}'_{\\text{CNOT}}(x)$ is less than or equal to $\\mathcal{C}(x)$ for all $x \\in [0,1]$. Thus, we conclude by using the properties of concurrence and integration that\n\\begin{equation}\n \\int^1_{0} \\mathcal{C}'_{\\text{CNOT}}(x) p(x)\\, dx \\leqslant  \\int^1_{0} \\mathcal{C}(x) p(x)\\, dx\n\\end{equation}\nfor {\\it any} PDF with $\\operatorname{supp}(p)=[0,1]$. Hence, the initial average cost function $\\bar f_{\\rm input}$ is always less than or equal to \n$\\bar f_{\\rm CNOT}$. This is an example where the purification fails with the sole implementation of the CNOT gate. It must be noted that\nfor  $x\\neq 0.5$, the state in this example can be purified with previous protocols \\cite{Bennett1,Deutsch} that work on the Bell basis, and where the implementation with the CNOT gate is now accompanied by local single-qubit gates.\n\nThese examples demonstrate the meaning of the average cost function. It is obvious that a two-qubit gate can be optimal for certain family of states and less optimal for other type of state.\nIn the subsequent section, we will investigate numerically several cases. \n\n\\begin{figure*}[t!]\n \\begin{center}\n \\includegraphics[width=.39\\textwidth]{fig2a.pdf}\n \\includegraphics[width=.39\\textwidth]{fig2b.pdf}\n \\includegraphics[width=.39\\textwidth]{fig2c.pdf}\n \\includegraphics[width=.39\\textwidth]{fig2d.pdf}\n\\caption{Optimized purification protocol for the family of states in Eq.~\\eqref{eq:example1}.\nTop left panel: average cost function $\\bar{f}$ with a uniform PDF as a function of $N$, the number of iterations. Top right panel: The concurrence $\\mathcal{C}$ as a function of $x$. \nFour curves are presented for different values of iterations $N$: $0$ or the input concurrence (dashed line), $1$ (dotted line), $2$ (dash-dotted line), and $3$ (full line).\nBottom left panel: Success probabilities as a function of $x$ for different values of iterations $N$:  $1$ (dotted line), $2$ (dash-dotted line), and $3$ (full line). Bottom right panel: The overall success \nprobability after $N=1$ (dotted line), $N=2$ (dash-dotted line), and $N=3$ (full line) iterations as a function of $x$.}\n   \\label{fig:TWerner}\n \\end{center}\n\\end{figure*}\n\\section{Results}\n\\label{Results}\n\nIn this section it is demonstrated how our proposed method can find optimal purification schemes. Our first case is the continuation of {\\it Example 1.} in Sec. \\ref{Method}. We have seen so far that \nthe CNOT gate in $N=1$ purification round can reduce the average cost function $\\bar{f}$ approximately by $0.05$. The simulation results with the input state in Eq.~\\eqref{eq:example1} show that the optimal\n$SU(4)$ gate for $N=1$ has a similar improvement on $\\bar{f}$ as the CNOT gate, see Fig.~\\ref{fig:TWerner}. The optimal gates found for $N=2$ and $3$ can further reduce $\\bar{f}$, however, it is easy to check that \nthe CNOT gate is not optimal anymore for these rounds of iterations. In Fig.~\\ref{fig:TWerner} it is also shown that a higher number of iterations result in more concave shapes of the corresponding concurrences. It is \nworth noting that the success probability of the second iteration is lower than the success probability of the first iteration. The overall success probability of $3$ iterations, \nshown also in Fig.~\\ref{fig:TWerner}, is defined as: in the first iteration $4$ qubit pairs, in the second iteration $2$ qubit pairs, and in the third iteration the final qubit pair are successfully purified. These \nresults show that states close to maximally entangled states can be produced already in the third iteration, but the overall success probability of the procedure is getting smaller with the number of \niterations. The only fixed point is  $\\mathcal{C}=1$. Thus our method seems to provide the same results to the one found in \\cite{Bennett}, which is based completely on the CNOT gate. As a result, we have investigated the circumstances where the CNOT gate is also optimal. It turns out that the local unitary transformation introduced by \\cite{Deutsch} and given by\n\\begin{equation}\n  b_{A_1}^{\\dagger}\\otimes b_{A_2}^{\\dagger}\\otimes b_{B_1}\\otimes b_{B_2}, \\label{eq:localtr}\n\\end{equation}\nwhere\n\\begin{equation}\n b=\\frac{\\mathbb{I}_2+i\\sigma_x}{\\sqrt2} \\nonumber\n\\end{equation}\nwith the Pauli matrix $\\sigma_x$ and the identity map $\\mathbb{I}_2$, is crucial for the CNOT gate. This is the transformation, which transforms a Werner state in Eq.~\\eqref{eq:wernerstate} into \nthe state in Eq.~\\eqref{eq:example1}. Now, if one applies Eq.~\\eqref{eq:localtr} before all iterations of the protocol involving only the CNOT gate, then the same curves are obtained as in Fig.~\\ref{fig:TWerner}. \nThis means that there are more optimal protocols, which yield the same results, and our approach can find them. Thus, the analytically tractable CNOT-based protocol can be used as a benchmark to test \nthe numerical accuracy of our method. Our numerics show at least a $2$ decimal accuracy until the third iteration, but up to the fourth iteration, there is a breakdown in the numerical accuracy.  \n\nThe one-parameter family of states in Eq.~\\eqref{eq:example1} has the same concurrence as the Werner state. Therefore, we consider the Werner state to be our next application. In Fig.~\\ref{fig:Werner} \nnumerical results are presented for the average cost function $\\bar{f}$ and the overall success probability which exhibit the same behavior found for the one-parameter family of states in Eq.~\\eqref{eq:example1}. \nIn this case, one can note an improvement of $0.04$ in $\\bar{f}$ after $N=3$ iterations. This is less than the previously obtained value of $0.09$ shown in Fig.~\\ref{fig:TWerner}.\nFurthermore, this is accompanied by another numerical \ninaccuracy: the overall success probability at $\\mathcal{C}=1$ is less than one. Given these results, the proposed method can find optimal entangling two-qubit gates for at least two iterations. It is also clear \nfrom these tests that the algorithm is always reducing $\\bar{f}$, but from $N=3$ iterations this might be not an optimal improvement of the concurrence. This originates from the fact that the gradient $\\nabla f (\\boldsymbol{\\alpha})$, see Eq.~\\eqref{eq:f}, is almost flat in the neighborhood of $\\mathcal{C}=1$ for $N>2$ iterations and thus the numerical search for the optimal gate, i.e, \nthe search for $\\boldsymbol{\\alpha}^* \\in \\mathbb{R}^{15}$, becomes inefficient.  \n\n\\begin{figure*}[t!]\n \\begin{center}\n \\includegraphics[width=.39\\textwidth]{fig3a.pdf}\n \\includegraphics[width=.39\\textwidth]{fig3b.pdf}\n\\caption{Optimized purification protocol for Werner states, see Eq.~\\eqref{eq:wernerstate}.\nLeft panel: average cost function $\\bar{f}$ with a uniform PDF as a function of $N$, the number of iterations.  Right panel: The overall success \nprobability after $N=1$ (dotted line), $N=2$ (dash-dotted line), and $N=3$ (full line) iterations as a function of $x$.}\n   \\label{fig:Werner}\n \\end{center}\n\\end{figure*}\n\nInspired by {\\it Example 2.}, we consider the state in Eq.~\\eqref{eq:example2} and also\n\\begin{equation}\nx\\ketbra{\\Psi^-}{\\Psi^-}+(1-x)\\ketbra{\\Upsilon}{\\Upsilon} \\label{eq:MaZs}\n\\end{equation}\nwith $x \\in [0,1]$ and \n\\begin{equation}\n \\ket{\\Upsilon}=\\frac{1}{\\sqrt2} \\left(\\ket{\\Psi^+}+i\\ket{\\Phi^-}\\right). \\nonumber\n\\end{equation}\nBy using the unitary transformation in Eq.~\\eqref{eq:localtr} on Eq.~\\eqref{eq:MaZs} one obtains Eq.~\\eqref{eq:example2} and therefore it is expected that the only optimal two-qubit gates are the ones that purify these states\nin one iteration. Both states have an input concurrence $\\mathcal{C}(x)=x$. However, it is important to note that the CNOT gate is not optimal for the state in Eq.~\\eqref{eq:MaZs}, because after one iteration round\n\\begin{equation}\n \\mathcal{C}'_{\\text{CNOT}}(x)=\\begin{cases} 0& x \\in [0,0.5], \\\\ 2x \\frac{2x-1}{1+x^2} & x \\in (0.5,1], \\end{cases} \\nonumber\n\\end{equation} \nand $\\mathcal{C}'_{\\text{CNOT}}(x) \\leqslant x$. The success probability is\n\\begin{equation}\n P_{\\text{CNOT}}=\\frac{1+x^2}{2}. \\nonumber\n\\end{equation}\nThus, the CNOT gate based purification protocol impairs the concurrence. In contrast to this analytical observation, numerical analysis with uniform PDF yields already in the first iteration for both states an average cost function $\\bar{f}\\approx 0.0002$. To demonstrate the robustness of the numerical approach we provide examples of three non-uniform PDFs for $N=1$ \niteration. First, we consider\n\\begin{equation}\np(x)=2x,\\quad \\text{with} \\quad x \\in [0,1], \\nonumber\n\\end{equation}\nwhich describes a situation, where states with higher concurrences are more likely to be subject to the purification. The resulting average cost function is $\\bar{f}\\approx 0.000004$. Secondly, we take \n\\begin{equation}\np(x)=2(1-x),\\quad \\text{with} \\quad x \\in [0,1], \\nonumber\n\\end{equation}\nwhich puts more weight on states with low concurrences and find $\\bar{f}\\approx 0.0005$. Finally, we investigate a PDF\n\\begin{equation}\np(x)=6x(1-x),\\quad \\text{with} \\quad x \\in [0,1], \\nonumber\n\\end{equation}\ni.e., the states around the concurrence $\\mathcal{C}(x)=0.5$ are more likely to participate in the purification, and obtain $\\bar{f}\\approx 0.00005$. These results demonstrate the effectiveness of our approach and up to a numerical precision these one-parameter families of states can be purified in one iteration.\n\n\\begin{figure*}[t!]\n \\begin{center}\n \\includegraphics[width=.39\\textwidth]{fig4a.pdf}\n \\includegraphics[width=.39\\textwidth]{fig4b.pdf}\n\\caption{Optimized purification protocol for the state in Eq.~ \\eqref{eq:qrstate}. Left panel: Average cost function $\\bar{f}$ with a uniform PDF as a function of $N$, the number of iterations. \nCrosses are the results of the CNOT gate, whereas dots display the numerical optimization. They have been connected by lines to guide the eye. Right panel: The concurrence $\\mathcal{C}'$\nas a function of $x$ and $y$ after one purification round. The cone-type surface is obtained with our approach. A less optimal surface with minima at $x=0$ and along the $y$ axis is the result of the purification protocol \nwith the CNOT gate.}\n   \\label{fig:XY}\n \\end{center}\n\\end{figure*}\n\nNext, we consider the following two-parameter family of states \\cite{Bernad2}\n\\begin{equation}\n\\hat{\\rho}(x,y) = \\frac{1}{4}\n\\left(\n\\begin{array}{cccc}\n 1-x & i y & -i y & x-1 \\\\\n -i y & x+1 & -x-1 & i y \\\\\n i y & -x-1 & x+1 & -i y \\\\\n x-1 & -i y & i y & 1-x \\\\\n\\end{array}\n\\right) \\label{eq:qrstate}\n\\end{equation}\nwith $x,y \\in \\mathbb{R}$ and $x^2+y^2\\leqslant1$. The concurrence of this state is $\\sqrt{x^2+y^2}$. In order to relate the performance of our approach, we apply the CNOT gate based purification \nprotocol to this family of states. One obtains\n\\begin{eqnarray}\n \\mathcal{C}'_{\\text{CNOT}}(x,y)&=&\\frac{2|x|}{1+x^2}, \\label{eq:xyCNOTc} \\\\\n P_{\\text{CNOT}}&=&\\frac{1+x^2}{2}, \\label{eq:xyCNOTp}\n\\end{eqnarray}\nafter $N=1$ and\n\\begin{eqnarray}\n \\mathcal{C}'_{\\text{CNOT}}(x,y)&=&\\frac{4 |x| (1+x^2)}{1+6x^2+x^4}, \\label{eq:xyCNOTc2} \\\\\n P_{\\text{CNOT}}&=&\\frac{1+6x^2+x^4}{2 (1+x^2)^2}, \\label{eq:xyCNOTp2}\n\\end{eqnarray}\nafter $N=2$ purification rounds, respectively. If one applies the unitary transformation in Eq.~\\eqref{eq:localtr} on the state before the bilateral CNOT gates are performed, then the above results remain \nunchanged. These results demonstrate that the CNOT gate and the additional tricks, which yielded optimal purifications for the one-parameter family of states, are not optimal in this scenario, since they are outperformed on average by our optimized protocol. \nWe assume that CNOT-based purification protocols cannot exploit all the useful entanglement, because the concurrence \nand the success probability become independent of the $y$ variable. Using these analytical results and the uniform PDF\n\\begin{equation}\n p(x,y)=\\frac{1}{\\pi},\\quad \\text{with} \\quad x^2+y^2 \\leqslant 1, \\nonumber\n\\end{equation}\nwe compare our approach with the above-presented analytical results, see Eq.~\\eqref{eq:xyCNOTc} and Eq.~\\eqref{eq:xyCNOTc2}. In Fig.~\\ref{fig:XY}, we show that our optimized protocol provides after one iteration a $x$ and $y$-dependent concurrence. Although the \nnumerically-obtained concurrence is larger at $x=0$ and $y \\in [-1,1]$ than the surface given by Eq.~\\eqref{eq:xyCNOTc}, one can also observe the opposite at $y=0$ and $x \\in [-1,1]$. However, the concurrence presents a better improvement with\nour algorithm and the average cost function with the uniform PDF stays for more iterations lower than the CNOT-based protocol, see the left figure in panel \\ref{fig:XY}.      \nIn regard to the overall success probability, we let our algorithm work until the third iteration and in Fig.~\\ref{fig:XYos} the results are compared with the CNOT-based purification protocol. The obtained \nsurfaces differ only by a $\\pi\/2$ rotation around the $z$-axis. Therefore, there is not much difference in the overall success probability and from this aspect the CNOT gate can also be considered optimal, though, \nit still can not improve the concurrence as effectively as the gate obtained with our approach. However, one might think with a proper choice of local unitary transformations, like the transformation \nin Eq.~\\eqref{eq:localtr} used for Werner and one-step purifiable states, the application of the CNOT gate may result in an optimal purification protocol. Let us consider then two general local unitary \ntransformations at locations $A$ and $B$ for the preparation of two-qubit states. It is enough to consider unitary transformations in $SU(2)$, for which the Euler angle parameterization reads \\cite{Tilma2}\n\\begin{eqnarray}\n U_A&=& e^{i \\sigma_z \\alpha_1} e^{i \\sigma_y \\alpha_2} e^{i \\sigma_z \\alpha_3}, \\\\\n U_B&=& e^{i \\sigma_z \\beta_1} e^{i \\sigma_y \\beta_2} e^{i \\sigma_z \\beta_3},\n\\end{eqnarray}\nwhere $\\sigma_y$ and $\\sigma_z$ are the Pauli matrices. Here, $\\alpha_1, \\beta_1 \\in [0, \\pi]$, $\\alpha_2,\\beta_2 \\in [0, \\pi\/2]$, and $\\alpha_3, \\beta_3 \\in [0, 2\\pi]$ are the Euler angles for  $SU(2)$. For the first\niteration, we analyze the output\nunitary of the optimized protocol with \\textit{Mathematica} \\cite{CAS} and observe that the angles $\\alpha_1$ and $\\beta_1$ do not affect the efficiency of the CNOT-based protocol. Optimal \nconcurrences for the remaining four angles yield either the result in Eq.~\\eqref{eq:xyCNOTc} or\n\\begin{equation}\n \\mathcal{C}'_{\\text{CNOT}}(x,y)=\\frac{2|y|}{1+y^2},  \n\\end{equation}\nwith success probability\n\\begin{equation} \n P_{\\text{CNOT}}=\\frac{1+y^2}{2}, \n\\end{equation}\nfor $\\alpha_2=\\frac{\\pi}{8}$, $\\beta_2=\\frac{3 \\pi}{8}$, and $\\alpha_3=\\beta_3=\\frac{3\\pi}{4}$. For these results the average cost function with uniform PDF yields $\\bar{f}\\approx 0.372$, which is still lower than\nthe average concurrence found by our approach. These results demonstrate that even with local unitary transformations the CNOT gate is globally not optimal for all values of $x$ and $y$. However, when $x$ and $y$\nare known beforehand, then one can use the following local unitary transformation\n\\begin{eqnarray}\n U^\\dagger_A \\otimes U_B, \\quad \\text{with} \\quad U=\\cos(\\theta)\\mathbb{I}_2 + \\sin(\\theta) \\sigma_x, \\label{eq:statedeptr}\n\\end{eqnarray}\nwhere the angle $\\theta$ is a function of $x$ and $y$. For example, for $x,y \\geqslant 0$, we have\n\\begin{equation}\n \\cos(\\theta)=\\sqrt{\\frac{1}{2}+\\sqrt{\\frac{x+\\sqrt{x^2+y^2}}{8 \\sqrt{x^2+y^2}}}}. \\nonumber\n\\end{equation}\nUsing this transformation before the purification protocol, one obtains the state\n\\begin{equation}\n\\frac{1+ \\sqrt{x^2+y^2}}{2} \\ketbra{\\Psi^-}{\\Psi^-}+\\frac{1- \\sqrt{x^2+y^2}}{2} \\ketbra{\\Phi^-}{\\Phi^-}. \\label{eq:Spec}\n\\end{equation}\nNow, the CNOT-based purification yields\n\\begin{equation}\n \\mathcal{C}'_{\\text{CNOT}}(x,y)=\\frac{2\\sqrt{x^2+y^2}}{1+x^2+y^2}, \n\\end{equation}\nwith success probability\n\\begin{equation}\n P_{\\text{CNOT}}=\\frac{1+x^2+y^2}{2}. \n\\end{equation}\nIt is immediate that $\\mathcal{C}'_{\\text{CNOT}}(x,y) \\geqslant \\sqrt{x^2+y^2}$, i.e., we are improving the concurrence. In this particular case, one does not need the average cost function in the numerical search, \nbecause the state is fixed. Our algorithm running with a single state with parameters $(x,y)$ as defined in Eq.~\\eqref{eq:qrstate} instead of an ensemble of states can improve the concurrence, but is unable to find the optimal solution presented above, because the transformation\nEq.~\\eqref{eq:statedeptr} together with CNOT gates is a non-symmetrical transformation at nodes $A$ and $B$.\n\nFinally, let us point out that our approach does not take into account operational or memory errors. These always depend on the implementation and if an experiment can provide us models for the errors then \nour approach can be easily extended. Another important experimental input is the PDF $p(\\boldsymbol{x})$, which is usually subject to the method of generating entangled states between locations A and B. Furthermore, \nthis PDF is assigned to the process of choosing a value $\\boldsymbol{x}$ as a random event, i.e, we have the same two two-qubit pairs parameterized by $\\boldsymbol{x}$ before the purification protocol. In effect, \nwe integrate the parameter dependence of the states away and thus our approach yields an optimal two-qubit gate on average. If the value of $\\boldsymbol{x}$ is fixed in an experimental design, then our method \nprovides an optimal two-qubit gate designed for this particular state. The optimal two-qubit gates can be can be realized by three\nCNOT gates and additional single-qubit rotations \\cite{Vidal,Vatan}, and therefore the experimental generation of this gate is possible with high fidelity \\cite{Debnath}. Also in the context of trapped ions or superconducting quantum circuits, the generation of two-qubit entangling gates can be achieved with high precision using the M\\o{}lmer-S\\o{}rensen gate \\cite{MS} or the $\\sqrt{i \\text{SWAP}}$ \\cite{Fan} gate. Since two-qubit unitaries have a straightforward implementation in many physical settings, thanks to quantum compilation \\cite{Martinez}, we argue that quantum computing platforms implementing quantum communication could actually benefit from globally optimal gates for entanglement purification.\n\n\\begin{figure*}[t!]\n \\begin{center}\n \\includegraphics[width=.39\\textwidth]{fig5.pdf}\n\\caption{The overall success probability of the state in Eq.~ \\eqref{eq:qrstate} after and $N=3$ iterations as a function of $x$ and $y$. Both surfaces are similar in appearance. The one having maxima at $x=1$ \nbelongs to the CNOT-based protocol and the other one having maxima at $y=1$ is obtained via the optimized numerical protocol.}\n   \\label{fig:XYos}\n \\end{center}\n\\end{figure*}\n\n\\section{Conclusions}\n\\label{Conclusions}\n\nIn the context of entanglement purification and recurrence protocols, we have presented a method to obtain optimal protocols. This method searches for the optimal two-qubit gate, which is applied bilaterally\nat the nodes $A$ and $B$ in order to distill from an ensemble of noisy pairs a higher fidelity state with respect to a maximally entangled state. Our strategy considers situations, where the \nexperimental apparatus does not have enough control over the entanglement generation phase and different experimental runs result in different states. Here, we assume that the same copies of the noisy state can be \ngenerated before the purification protocol takes place, but a different experimental run could result in different states. Errors originating from local operations, memory requirements, or even \nclassical communication are neglected for now.\n\nWe have numerically demonstrated the optimality of our proposal for several states. In the case of the Werner state, we have found several optimal two-qubit gates and their performances are the same as the CNOT-based Bennett protocol \\cite{Bennett}. Thus, for Werner states the optimality cannot be improved beyond the already known performance. Next, we have investigated a family of states that one can purify \nin one step, i.e., two noisy copies of states are enough to obtain a maximally entangled state, where we immediately obtain minimal average cost function $\\bar{f} \\approx 0$, as expected. Finally, we consider a two-parameter family of states which is considered in theoretical models of a quantum repeater \\cite{Bernad2}. Our numerical investigations demonstrate that a\nsingle bilaterally applied CNOT gate cannot be globally optimal for these states. On the other hand, when the state is known, then parameter-dependent, non-symmetric local transformations allow again the CNOT to be also \none of the optimal two-qubit gates. This case exemplifies the difference between protocols that are optimal for a full class of parameterized states and those that are only optimal for a single state.  \n\nIn conclusion, we have proposed a general method to optimize entanglement purification for an arbitrary family of states. Our algorithm \\cite{Francesco} can find the most optimal two-qubit gate that on average induces the highest increase of entanglement. We remark that the method is general for the set of parameters defining the states. Optimizing for a single state is also possible, as shown in one of the \npresented examples. Furthermore, we have focused on the degree of entanglement measured by the concurrence and not on the fidelity with respect to a particular Bell state. \nThis is motivated by the fact that there exist an infinite number of maximally entangled states, and therefore entangled mixed states can present a larger fidelity with respect to maximally entangled states that might not correspond to Bell states defined in the computational basis. \nA possible drawback is that one cannot know with certainty the final maximally entangled state produced by the protocol. This and other issues such as different input states, asymmetric two-qubit gates, \nand non-ideal local operations remain open questions. However, this work aims to introduce the concept of globally optimized recurrence protocols that is flexible enough to incorporate the \naforementioned points in further investigations, which we plan to carry out in the future. \n\n\\begin{acknowledgments}\nThis work is supported by the DFG under Germany's Excellence Strategy-Cluster of Excellence Matter and Light for Quantum\nComputing (ML4Q) EXC 2004\/1-390534769. The results were obtained using the libraries JAX \\cite{JAX}, Scipy \\cite{Scipy} and QuTiP \\cite{Qutip}.\n\n\\end{acknowledgments}\n\n\n\\section{Introduction}\n\nEntanglement is a key resource for quantum information tasks, like quantum simulations \\cite{Monroe}, computation \\cite{Albash}, and communication \\cite{Pirandola}. These tasks are based on the idea of creating networks of quantum systems, where the generation of well-controlled entanglement between spatially separated nodes is essential. These networks, which may consist of distant or nearby nodes, have been thoroughly investigated for efficient process and transfer of quantum information \\cite{Awschalom}. However, due to interactions with an uncontrollable environment noisy or non-maximally entangled states are produced. To protect quantum information and to guarantee a high performance of its processing, one can use quantum error correction \\cite{Peres, Devitt} or quantum teleportation in combination with entanglement purification \\cite{Bennett1, Deutsch, Bennett2}. Quantum error correction is particularly suitable for quantum computers, however it is inferior to entanglement purification with two-way classical communication \\cite{Bennett2}, which is the subject of our investigations. In particular, we consider a so-called recurrence protocol \\cite{Dur}, an iterative approach, which operates in each purification step only on two identical copies of states.\n\nIn this paper, we discuss entanglement purification from the point of view of optimality. Recently, optimized entanglement purification has been investigated with the help of genetic algorithms \\cite{Krastanov}, where \nthe analytical and numerical studies are based on Werner states \\cite{Werner}. In fact, also the first ever proposed protocols are based on either Werner \\cite{Bennett1} or Bell diagonal states \\cite{Deutsch}. It has been \nshown that $4$ or $12$ local random $SU(2)$ transformations can bring any states into a Bell diagonal or Werner state, respectively \\cite{Bennett2}. We have already argued that these random local unitary transformations \nand states obtained in consequence are only useful\nfor grasping and consequently understanding the complex task of purification because they may waste useful entanglement \\cite{Torres}. Therefore, here, we develop a method for general states and demonstrate it for \nseveral simple examples including also the Werner state. Nonetheless, our motivation also lies in the fact that an experimental implementation may not have enough control over the generated noisy states and thus more general, adaptive, and optimal purification strategies have to be made available. In this general context, we assume that an experiment can still guarantee identical copies of states before the purification protocol takes place. \n\nOur method is based on quasi-Monte Carlo numerical sampling of the states, which undergo the purification protocol, and then the concurrence \\cite{Wooters} of the output states is integrated over an {\\it a priori} probability distribution \nfunction. This results in an average two-qubit gate-dependent cost function for the whole sample of states and then we employ a quasi-Newton algorithm \\cite{Kelley} for solving this non-linear optimization problem \non the whole $SU(4)$ group. The obtained two-qubit gate is optimal in the sense that it achieves, on average, a higher increase of entanglement of an input family of states. We discuss the performance of our method for several examples and compare with protocols based on one bilateral application of controlled-NOT gates, the paradigmatic two-qubit operation used in the seminal purification protocols \\cite{Bennett1,Deutsch}.\n\nThe paper is organized as follows. In Sec.~\\ref{Method} we introduce our\nmethod and give some elementary examples to allow further acquaintance with the concept of the introduced cost function. \nIn Sec.~\\ref{Results} we demonstrate our approach for four one-parameter and one two-parameter family of states. \nNumerical and analytical results are presented for concurrences and success probabilities.  In Sec.~\\ref{Conclusions}\nwe summarize and draw our conclusions. Some details supporting the main text are collected in Appendix~\\ref{AppendixA}.\n\n\\section{Method}\n\\label{Method}\nIn this section, we describe how the recurrence protocol is optimized with respect to an input familiy of quantum states.\n\n\\subsection{Entanglement purification protocol}\\label{EntPurif}\nLet us consider the product state of two-qubit pairs \n\\begin{equation}\n  \\boldsymbol{\\rho}=\\rho^{A_1,B_1} \\otimes\\rho^{A_2,B_2},\n\\label{eq:rhoAB4}\n\\end{equation}\nwhere qubit components of each pair are assumed to be spatially separated at distant locations. These locations are labeled by $A$ and $B$. In an \nentanglement purification protocol, one performs local quantum operations, which may not just involve two-qubit gates \\cite{Bernad1}. \nThis is followed by measurements on one of the pairs\nat both locations. A classical communication between $A$ and $B$ results in a qubit pair with a higher degree of entanglement. Both pairs are assumed to start in the same state $\\rho$ \nand the degree of the entanglement is usually measured by the \nfidelity with respect to one of the Bell states\n\\begin{align}\n  \\ket{\\Psi^\\pm}=\\tfrac{1}{\\sqrt2}\\left(\\ket{01}\\pm\\ket{10}\\right), \\,\n  \\ket{\\Phi^\\pm}=\\tfrac{1}{\\sqrt2}\\left(\\ket{00}\\pm \\ket{11}\\right).\n  \\label{eq:Bellstates}\n\\end{align}\nHowever, these states can be subject to local unitary transformations, which can cause some technical difficulties, when one uses fidelity, i.e.~using fidelity as a cost function would force us to search also for additional local unitary operations in order to align the output state of the protocol with the Bell basis. As we intend to \nanalyze the entanglement purification in a very general setup, one requires an entanglement measure that is invariant under local unitary transformations. Therefore, we turn to the concurrence as a measure \nof the the attainability of a maximally entangled state \\cite{Wooters}:\n\\begin{equation}\\label{eq:concurrence}\n \\mathcal{C}(\\rho)=\\max\\{0,\\lambda_1-\\lambda_2-\\lambda_3-\\lambda_4\\}.\n\\end{equation}\nHere $\\lambda_1, \\lambda_2, \\lambda_3, \\lambda_4$ are the square roots of the non-negative eigenvalues of the non-Hermitian matrix\n\\begin{equation}\n \\tilde \\rho = \\rho (\\sigma_{y}\\otimes\\sigma_{y})\\rho^{*}(\\sigma_{y}\\otimes\\sigma_{y}), \\nonumber\n\\end{equation}\nwith $*$ being the complex conjugation in the standard basis and $\\sigma_{y}$ is the Pauli matrix.\n\n\\begin{figure}[t!]\n    \\includegraphics[width=.4\\textwidth]{fig1.pdf}\n    \\caption{Schematic representation of bipartite entanglement purification, where the entanglement purification protocol trades two \n    entangled qubit pairs for a qubit pair with a higher degree of entanglement, which is quantified by the concurrence $\\mathcal{C}$.}\n    \\label{fig:purifscheme}\n\\end{figure}\n\nIn this manuscript, we examine and optimize a purification protocol having the following steps:\\\\\n\n(I) Two unitary transformations $\\rho \\rightarrow U(\\boldsymbol{\\alpha}) \\rho U(\\boldsymbol{\\alpha})^\\dagger$ are applied locally at $A$ and $B$ (see Fig.~\\ref{fig:purifscheme}), where $U$ is a general two-qubit unitary described by a parameter vector $\\boldsymbol{\\alpha}$.\nAfter the application of the quantum operation the four-qubit system \nattains the state  \n\\begin{equation}\n\\boldsymbol{\\rho}'=\n U_{A1,A2}(\\boldsymbol{\\alpha}) U_{B1,B2}(\\boldsymbol{\\alpha}) \\boldsymbol{\\rho} U^\\dagger_{B1,B2} (\\boldsymbol{\\alpha}) U^\\dagger_{A1,A2} (\\boldsymbol{\\alpha}). \n\\label{eq: maprho}\n\\end{equation}\n\n(II)  One of the pairs $(A_2,B_2)$ is then locally measured in the standard basis. There are four possible states, i.e., 4-dimensional vectors, in which one can find the measured pair: \n\\begin{eqnarray}\n \\ket{1}&=&\\ket{00}_{A_2,B_2}, \\quad \\ket{2}=\\ket{01}_{A_2,B_2}, \\nonumber \\\\\n \\ket{3}&=&\\ket{10}_{A_2,B_2}, \\quad \\ket{4}=\\ket{11}_{A_2,B_2}. \\nonumber\n\\end{eqnarray}\nA successful measurement of one of the states $\\ket{i}$ with $i\\in\\{1,2,3,4\\}$ results in\na state for the other qubit\n\\begin{equation}\n  \\tilde\\rho^{A_1,B_1}_i= \\frac{\\bra{i} \\boldsymbol{\\rho}' \\ket{i} }{\\mathrm{Tr}\\{\\ketbra{i}{i} \\boldsymbol{\\rho}'\\}}\n\\label{}\n\\end{equation}\nwith probability\n\\begin{equation}\n p_i=\\mathrm{Tr}\\{\\ketbra{i}{i} \\boldsymbol{\\rho}'\\}. \\nonumber\n\\end{equation}\n\n(III) Depending on value of the measurement results, which are communicated between the two parties, the state with largest concurrence and the related success probability are kept, whereas the others are discarded. The output of the protocol is the pair \n$(\\mathcal{C}',P)$, concurrence value $\\mathcal{C}'$ of the state obtained with probability $P$. If there are multiple maxima, e.g., $\\mathcal{C}(\\tilde\\rho^{A_1,B_1}_1)=\\mathcal{C}(\\tilde\\rho^{A_1,B_1}_2)$, then \n$\\mathcal{C}'=\\mathcal{C}(\\tilde\\rho^{A_1,B_1}_1)$ and $P=p_1+p_2$.\\\\\n\nGiven two copies of a state $\\rho$ with concurrence $\\mathcal{C}$, it is straightforward to see that pair $(\\mathcal{C}',P)$ depends on the vector $\\boldsymbol{\\alpha}$, which we use as the optimization parameter.\n\nRecurrence entanglement purification protocols might include symmetric and asymmetric single-qubit gates before and after the bilateral action of the two-qubit operations \\cite{Bennett1,Deutsch}. These are important in an analytical approach to maintain the form of the state after each iteration, however, entanglement is not affected by this process. In this work, we shall omit these single-qubit gates as we are focused on increasing the value of the concurrence and not on the specific form of the state after each iteration step. \n\n\\subsection{Protocol optimization}\\label{ProtOpt}\n\nIn order to optimize the protocol described in Sec.~\\ref{EntPurif}, we first define an appropriate parametrization for the two-qubit unitary operations used in Eq.~\\eqref{eq: maprho}. In principle, one should consider elements from $U(4)$, the group of $4 \\times 4$ unitary matrices, which contains the subgroup $SU(4)$. However, $U(4)$ is the semi-direct product of $U(1)$ and $SU(4)$, where elements of $U(1)$ are rotations of the unit circle \\cite{Baker}. Therefore,\nchoosing elements from $SU(4)$ in Eq.~\\eqref{eq: maprho}\nrepresents the most general unitary quantum operation involving two-qubit gates at locations $A$ and $B$.\nElements in $SU(4)$ can be parametrized in the following way \\cite{Tilma1}:\n\\begin{eqnarray} \\label{eq: eulerU}\n U(\\boldsymbol{\\alpha})=e^{i \\sigma_3 \\alpha_1} e^{i \\sigma_2 \\alpha_2} e^{i \\sigma_3 \\alpha_3} e^{i \\sigma_5 \\alpha_4} e^{i \\sigma_3 \\alpha_5} e^{i \\sigma_{10} \\alpha_6} \n e^{i \\sigma_3 \\alpha_7} e^{i \\sigma_2 \\alpha_8} \\nonumber \\\\ \n e^{i \\sigma_{3} \\alpha_9} e^{i \\sigma_5 \\alpha_{10}} e^{i \\sigma_3 \\alpha_{11}} e^{i \\sigma_{2} \\alpha_{12}} \n e^{i \\sigma_3 \\alpha_{13}} e^{i \\sigma_8 \\alpha_{14}} e^{i \\sigma_{15} \\alpha_{15}}, \\nonumber \n\\end{eqnarray}\nwith $\\boldsymbol{\\alpha}=(\\alpha_1, \\alpha_2, \\dots, \\alpha_{15})^T \\in \\mathbb{R}^{15}$ ($T$ denotes the transposition), and\n\\begin{eqnarray}\n && 0 \\leqslant \\alpha_1, \\alpha_3, \\alpha_5, \\alpha_7, \\alpha_9, \\alpha_{11}, \\alpha_{13} \\leqslant \\pi, \\nonumber \\\\\n &&  0 \\leqslant \\alpha_2, \\alpha_4, \\alpha_6, \\alpha_8, \\alpha_{10}, \\alpha_{12} \\leqslant \\frac{\\pi}{2} \\nonumber \\\\\n && 0 \\leqslant \\alpha_{14} \\leqslant \\frac{\\pi}{\\sqrt{3}}, \\quad 0 \\leqslant \\alpha_{15} \\leqslant \\frac{\\pi}{\\sqrt{6}}, \\label{eq:anglecond}\n\\end{eqnarray}\nwhere $\\sigma_i,\\ i=1, ..., 15$ form a Gell-Mann type basis of the Lie group $SU(4)$, see Appendix \\ref{AppendixA}. This is called the Euler angle parametrization of $SU(4)$, which is sufficient\nto represent every element of the Lie group. For example, the canonical parametrization $e^{i H}$, where $H$ is a $4 \\times 4$ self-adjoint matrix with trace zero, is not minimal, because after exponentiation we may have multiple wrappings around the great circles of the $7$-sphere. \n\nOur aim is to increase the concurrence, which is a non-linear function of the state $\\rho$ and the protocol's unitary matrix $U$. Furthermore, we have lower and upper bounds on $\\boldsymbol{\\alpha}$, as given in Eq.~\\eqref{eq:anglecond}. We then consider a cost function $f:\\mathbb{R}^{15} \\rightarrow \n\\mathbb{R}$ as\n\\begin{align}\n    f(\\boldsymbol{\\alpha}) = 1 - \\mathcal{C}'(\\boldsymbol{\\alpha}) \\label{eq:f}\n\\end{align}\nand our optimization problem is, to find local minimizers, e.g., a point $\\boldsymbol{\\alpha}^*$ such that\n\\begin{equation}\n f(\\boldsymbol{\\alpha}^*)\\leqslant f(\\boldsymbol{\\alpha}) \\nonumber\n\\end{equation}\nfor all $\\boldsymbol{\\alpha}$ in the Euclidean norm defined neighborhood of $\\boldsymbol{\\alpha}^*$. The gradient $\\nabla f (\\boldsymbol{\\alpha})$ can be made available to us due to an automatic differentiation \\cite{JAX}, so we \nimplement for the optimization a quasi-Newton algorithm the so-called limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method \\cite{Liu}. Here, we have a constrained optimization problem, because $\\boldsymbol{\\alpha}$ takes values in the hyperrectangle defined by Eq.~\\eqref{eq:anglecond} and thus we use the L-BFGS approach of Ref. \\cite{Ryrd}. \nAs long as the state to be purified is given, the above approach yields an optimal $\\boldsymbol{\\alpha}^*$ and a concurrence $\\mathcal{C}'$ as close as possible to one. \nThe vector $\\boldsymbol{\\alpha}^*$ gives the best two-qubit unitary gate associated with this given state. However, our main aim is to provide the best gate, which is capable to purify most effectively certain class of states and not only a fixed one. This is relevant in a scenario where the generation of distant entanglement is affected by noise, leading to slightly different types of states entering the protocol.\nIn order to formulate this quantitatively we introduce  a probability density function (PDF) $p(\\boldsymbol{x})$, where the vector $\\boldsymbol{x}$ defines uniquely the state $\\rho$ according to a parametrization. \nFor example, in the case of a  Werner state \\cite{Werner}\n\\begin{eqnarray}\n\\hat{\\rho}(x) &=& x \\ket{\\Psi^{-}}\\bra{\\Psi^{-}} + \\frac{1-x}{3} \\ket{\\Psi^{+}}\\bra{\\Psi^{+}} \\nonumber \\\\\n&+& \\frac{1-x}{3}  \\ket{\\Phi^{-}}\\bra{\\Phi^{-}}+ \\frac{1-x}{3} \\ket{\\Phi^{+}}\\bra{\\Phi^{+}},  \n\\label{eq:wernerstate}\n\\end{eqnarray}  \nwe have $x\\in[0,1]$ and the PDF satisfies\n\\begin{equation}\n \\int^1_0 p(x)=1. \\nonumber\n\\end{equation}\nIn general, a two-qubit state can be described by $15$ parameters, which have to fulfill some non-trivial conditions \\cite{Kimura,Byrd,Gamel}. The choice of the PDF is not straightforward and the only guideline \nwe have is that the support $\\operatorname{supp}(p)$ consists of all $\\boldsymbol{x}$, which define entangled states. This is motivated by \nthe fact that separable states are not purifiable. In the case of the Werner states, the support of the PDF is the interval $(0.5,1]$. Thus, the PDF may put more weight on states with a given concurrence.\n\\begin{algorithm}[H]\n\t\t\\setstretch{1.20}\n\t\t\\caption{Optimization of recurrence protocol}\n\t\t\\label{alg:ouralgo}\n\t\t\\hspace*{\\algorithmicindent} \\textbf{Input} $\\boldsymbol{\\rho}(\\boldsymbol{x})$, optimizer $\\text{OPT}$\\\\\n\t\t\\hspace*{\\algorithmicindent} \\textbf{Output} $\\boldsymbol{\\rho}(\\boldsymbol{x})$\n\t\t\\begin{algorithmic}[1]\n\t\t\\For{$j=1$ to $M$}\n\t\t\\State $\\boldsymbol{x}_j \\sim p(\\boldsymbol{x}_j)$\n\t\t\\State $\\boldsymbol{\\rho}_j = \\boldsymbol{\\rho}(\\boldsymbol{x}_j)$\n\t\t\\EndFor\n\t\t\t\\For{$i=1$ to $N$}\n\t\t\t\\State $U_{AB}(\\boldsymbol{\\alpha}) = U_{A1,A2}(\\boldsymbol{\\alpha}) U_{B1,B2}(\\boldsymbol{\\alpha})$\n\t\t\t\\For{$j=1$ to $M$}\n\t\t\t\\State $\\boldsymbol{\\sigma}^{j}(\\boldsymbol{\\alpha}) = U_{AB}(\\boldsymbol{\\alpha}) \\boldsymbol{\\rho}_j U^{\\dagger}_{AB}(\\boldsymbol{\\alpha})$\n\t\t\t\\For{$l=1$ to $4$}\n\t\t\t\\State $\\boldsymbol{\\sigma}^{jl}_A(\\boldsymbol{\\alpha}) =  \\frac{\\bra{l} \\boldsymbol{\\sigma}^{j}(\\boldsymbol{\\alpha})  \\ket{l} }{\\mathrm{Tr}\\{\\ketbra{l}{l} \\boldsymbol{\\sigma}^{j}(\\boldsymbol{\\alpha})  \\}}$\n\t\t\t\\EndFor\n\t\t\t\\EndFor\n\t\t\t\\State $\\mathcal{C}^l(\\boldsymbol{\\alpha}) = \\frac{1}{M}\\sum_{j=1}^M \\mathcal{C}(\\boldsymbol{\\sigma}^{jl}_A)$\n\t\t\t\\State $l' = \\underset{l=1,2,3,4}{\\text{argmin}}\\ \\left(1 - \\mathcal{C}^l(\\boldsymbol{\\alpha})\\right)$\n\t\t\t\\State $\\bar{f}(\\boldsymbol{\\alpha}) =  1 - \\mathcal{C}^{l'}(\\boldsymbol{\\alpha})$\n\t\t\t\\State $\\boldsymbol{\\alpha}^* = \\text{OPT}(\\bar{f}(\\boldsymbol{\\alpha}), \\nabla_{\\boldsymbol{\\alpha}} \\bar{f}(\\boldsymbol{\\alpha}))$\n\t\t\t\\State $\\boldsymbol{\\rho}_j = \\boldsymbol{\\sigma}^{jl'}_{A}(\\boldsymbol{\\alpha}^*) \\otimes \\boldsymbol{\\sigma}^{jl'}_{A}(\\boldsymbol{\\alpha}^*)$\n\t\t\t\\EndFor\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\nIn this context, we have an output concurrence $\\mathcal{C}'(\\boldsymbol{\\alpha}, \\boldsymbol{x})$ depending on both the two-qubit gate and the input state. Therefore, we are going to use an average cost function \n\\begin{equation}\n \\bar{f}(\\boldsymbol{\\alpha}) = 1 - \\int_{\\operatorname{supp}(p)} \\mathcal{C}'(\\boldsymbol{\\alpha}, \\boldsymbol{x}) p(\\boldsymbol{x})\\,  d \\boldsymbol{x}, \\label{eq:cost}\n\\end{equation}\nin the L-BFGS algorithm. The integral can be either solved numerically or approximated via a quasi-Monte Carlo approach by sampling $\\boldsymbol{x}$ over its corresponding parameter distribution $p(\\boldsymbol{x})$:\n\\begin{align}\n    \\bar{f}(\\boldsymbol{\\alpha}) = 1 - \\underset{\\boldsymbol{x} \\sim p(\\boldsymbol{x})}{\\mathbb{E}} \\left[\\mathcal{C}'(\\boldsymbol{\\alpha}, \\boldsymbol{x}) \\right].\n\\end{align}\nWe chose the second approach since it proved to be precise enough and numerically faster. A summarize of the optimization routine can be seen in Algorithm~\\ref{alg:ouralgo}. \n\nNow, we shed light on the meaning of the average cost function through the following two examples. We employ the CNOT gate\n\\begin{equation}\n U_{\\text{CNOT}}=\\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1\\\\ 0 & 0 & 1 & 0  \\end{pmatrix}=\n e^{i\\frac{3\\pi}{4}} \\times \\underbrace{U'}_{\\in SU(4)}, \\nonumber\n\\end{equation}\nwhere $U'$ in the Euler angle parametrization is given by setting\n\\begin{eqnarray}\n &&\\alpha_3= \\alpha_5=\\alpha_7= \\frac{\\pi}{4}, \\nonumber \\\\\n &&\\alpha_4= \\alpha_6=\\alpha_{10}= \\frac{\\pi}{2}, \\nonumber\n\\end{eqnarray}\nand the remaining $9$ angles to zero in Eq.~\\eqref{eq: eulerU}. By fixing the vector $\\boldsymbol{\\alpha}$, one is able to get a value for the average cost function of the CNOT gate and thus to evaluate its performance. \\\\\n\n{\\it Example 1.}\nLet us consider the state\n\\begin{equation}\n \\frac{1}{6}\\begin{pmatrix} 1+2x & 0 & 0 & 1-4x \\\\ 0 & 2-2x & 0 & 0 \\\\ 0 & 0 & 2-2x & 0\\\\ 1-4x & 0 & 0 & 1+2x  \\end{pmatrix}, \\quad \\text{with} \\quad x \\in [0,1] \\label{eq:example1}\n\\end{equation}\nsubject to the purification protocol with CNOT gates. This state is a rotated Werner state and its concurrence reads\n\\begin{equation}\n \\mathcal{C}(x)=\\begin{cases}2x-1 & x \\in (0.5,1], \\\\ 0 & x \\in [0, 0.5].\\end{cases} \\nonumber\n\\end{equation}\nIn fact, this example is based on the pioneering protocol of Ref. \\cite{Bennett}. \nThe output reads\n\\begin{equation}\n \\mathcal{C}'_{\\text{CNOT}}(x)=\\frac{3(4x^2-1)}{5-4x+8x^2}, \\quad \\text{for} \\quad x \\in (0.5,1], \\nonumber\n\\end{equation}\nwith success probability\n\\begin{equation}\n P_{\\text{CNOT}}=\\frac{5-4x+8x^2}{9}. \\nonumber\n\\end{equation}\nFor the sake of simplicity, we consider a uniform PDF $p(x)$ with $\\operatorname{supp}(p)=(0.5,1]$. Hence, the input average cost function reads\n\\begin{equation}\n \\bar{f}_{\\text{input}}=1-\\int^1_{0.5} \\mathcal{C}(x) p(x)\\, dx =0.5, \\nonumber\n\\end{equation}\nand the application of the purification protocol with CNOT gates yields\n\\begin{equation}\n \\bar{f}_{\\text{CNOT}}=1-\\int^1_{0.5} \\mathcal{C}'(x) p(x)\\, dx =0.450103. \\nonumber\n\\end{equation}\nThe result shows that the protocol with two identical copies of states allows one to increase on average the entanglement of the output states.\\\\\n\n{\\it Example 2.}\nNow, we consider the state\n\\begin{equation}\n \\begin{pmatrix} \\frac{x}{2} & 0 & 0 & -\\frac{x}{2} \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1-x & 0\\\\ -\\frac{x}{2} & 0 & 0 & \\frac{x}{2}  \\end{pmatrix}, \\quad \\text{with} \\quad x \\in [0,1]. \\label{eq:example2}\n\\end{equation}\nThis state with concurrence $\\mathcal{C}(x)=x$ is perfectly purifiable in just one iteration of the protocol \\cite{Torres}. Hence, \n\\begin{equation}\n \\mathcal{C}'_{\\text{CNOT}}(x)=1, \\nonumber\n\\end{equation}\nwith success probability\n\\begin{equation}\n P_{\\text{CNOT}}=\\frac{x^2}{2}. \\nonumber\n\\end{equation}\nIt is immediate from Eq.~\\eqref{eq:cost} that for {\\it any} PDF with $\\operatorname{supp}(p)=[0,1]$\n\\begin{equation}\n \\bar{f}_{\\text{CNOT}}=1-\\int^1_{0} \\mathcal{C}'(x) p(x)\\, dx =0. \\nonumber\n\\end{equation}\nThis means that the CNOT gate is optimal for this family of states.\n\n\n{\\it Example 3.}\nAs a last example, let us consider the initial state\n\\begin{equation}\n x\\ketbra{\\Phi^+}{\\Phi^+}+(1-x)\\ketbra{\\Phi^-}{\\Phi^-}, \\quad \\text{with} \\quad x\\in[0,1],   \n\\end{equation}\nand concurrence $\\mathcal{C}(x)=|1-2x|$. After one iteration of the purification protocol with bilaterally applied CNOT gates, it is not hard to realize that the resulting state has the concurrence \n\\begin{equation}\n\\mathcal{C}'_{\\text{CNOT}}(x)=(1-2x)^2\\nonumber \n\\end{equation}\nwith success probability $P_{\\text{CNOT}}=1$. The output\nconcurrence $\\mathcal{C}'_{\\text{CNOT}}(x)$ is less than or equal to $\\mathcal{C}(x)$ for all $x \\in [0,1]$. Thus, we conclude by using the properties of concurrence and integration that\n\\begin{equation}\n \\int^1_{0} \\mathcal{C}'_{\\text{CNOT}}(x) p(x)\\, dx \\leqslant  \\int^1_{0} \\mathcal{C}(x) p(x)\\, dx\n\\end{equation}\nfor {\\it any} PDF with $\\operatorname{supp}(p)=[0,1]$. Hence, the initial average cost function $\\bar f_{\\rm input}$ is always less than or equal to \n$\\bar f_{\\rm CNOT}$. This is an example where the purification fails with the sole implementation of the CNOT gate. It must be noted that\nfor  $x\\neq 0.5$, the state in this example can be purified with previous protocols \\cite{Bennett1,Deutsch} that work on the Bell basis, and where the implementation with the CNOT gate is now accompanied by local single-qubit gates.\n\nThese examples demonstrate the meaning of the average cost function. It is obvious that a two-qubit gate can be optimal for certain family of states and less optimal for other type of state.\nIn the subsequent section, we will investigate numerically several cases. \n\n\\begin{figure*}[t!]\n \\begin{center}\n \\includegraphics[width=.39\\textwidth]{fig2a.pdf}\n \\includegraphics[width=.39\\textwidth]{fig2b.pdf}\n \\includegraphics[width=.39\\textwidth]{fig2c.pdf}\n \\includegraphics[width=.39\\textwidth]{fig2d.pdf}\n\\caption{Optimized purification protocol for the family of states in Eq.~\\eqref{eq:example1}.\nTop left panel: average cost function $\\bar{f}$ with a uniform PDF as a function of $N$, the number of iterations. Top right panel: The concurrence $\\mathcal{C}$ as a function of $x$. \nFour curves are presented for different values of iterations $N$: $0$ or the input concurrence (dashed line), $1$ (dotted line), $2$ (dash-dotted line), and $3$ (full line).\nBottom left panel: Success probabilities as a function of $x$ for different values of iterations $N$:  $1$ (dotted line), $2$ (dash-dotted line), and $3$ (full line). Bottom right panel: The overall success \nprobability after $N=1$ (dotted line), $N=2$ (dash-dotted line), and $N=3$ (full line) iterations as a function of $x$.}\n   \\label{fig:TWerner}\n \\end{center}\n\\end{figure*}\n\\section{Results}\n\\label{Results}\n\nIn this section it is demonstrated how our proposed method can find optimal purification schemes. Our first case is the continuation of {\\it Example 1.} in Sec. \\ref{Method}. We have seen so far that \nthe CNOT gate in $N=1$ purification round can reduce the average cost function $\\bar{f}$ approximately by $0.05$. The simulation results with the input state in Eq.~\\eqref{eq:example1} show that the optimal\n$SU(4)$ gate for $N=1$ has a similar improvement on $\\bar{f}$ as the CNOT gate, see Fig.~\\ref{fig:TWerner}. The optimal gates found for $N=2$ and $3$ can further reduce $\\bar{f}$, however, it is easy to check that \nthe CNOT gate is not optimal anymore for these rounds of iterations. In Fig.~\\ref{fig:TWerner} it is also shown that a higher number of iterations result in more concave shapes of the corresponding concurrences. It is \nworth noting that the success probability of the second iteration is lower than the success probability of the first iteration. The overall success probability of $3$ iterations, \nshown also in Fig.~\\ref{fig:TWerner}, is defined as: in the first iteration $4$ qubit pairs, in the second iteration $2$ qubit pairs, and in the third iteration the final qubit pair are successfully purified. These \nresults show that states close to maximally entangled states can be produced already in the third iteration, but the overall success probability of the procedure is getting smaller with the number of \niterations. The only fixed point is  $\\mathcal{C}=1$. Thus our method seems to provide the same results to the one found in \\cite{Bennett}, which is based completely on the CNOT gate. As a result, we have investigated the circumstances where the CNOT gate is also optimal. It turns out that the local unitary transformation introduced by \\cite{Deutsch} and given by\n\\begin{equation}\n  b_{A_1}^{\\dagger}\\otimes b_{A_2}^{\\dagger}\\otimes b_{B_1}\\otimes b_{B_2}, \\label{eq:localtr}\n\\end{equation}\nwhere\n\\begin{equation}\n b=\\frac{\\mathbb{I}_2+i\\sigma_x}{\\sqrt2} \\nonumber\n\\end{equation}\nwith the Pauli matrix $\\sigma_x$ and the identity map $\\mathbb{I}_2$, is crucial for the CNOT gate. This is the transformation, which transforms a Werner state in Eq.~\\eqref{eq:wernerstate} into \nthe state in Eq.~\\eqref{eq:example1}. Now, if one applies Eq.~\\eqref{eq:localtr} before all iterations of the protocol involving only the CNOT gate, then the same curves are obtained as in Fig.~\\ref{fig:TWerner}. \nThis means that there are more optimal protocols, which yield the same results, and our approach can find them. Thus, the analytically tractable CNOT-based protocol can be used as a benchmark to test \nthe numerical accuracy of our method. Our numerics show at least a $2$ decimal accuracy until the third iteration, but up to the fourth iteration, there is a breakdown in the numerical accuracy.  \n\nThe one-parameter family of states in Eq.~\\eqref{eq:example1} has the same concurrence as the Werner state. Therefore, we consider the Werner state to be our next application. In Fig.~\\ref{fig:Werner} \nnumerical results are presented for the average cost function $\\bar{f}$ and the overall success probability which exhibit the same behavior found for the one-parameter family of states in Eq.~\\eqref{eq:example1}. \nIn this case, one can note an improvement of $0.04$ in $\\bar{f}$ after $N=3$ iterations. This is less than the previously obtained value of $0.09$ shown in Fig.~\\ref{fig:TWerner}.\nFurthermore, this is accompanied by another numerical \ninaccuracy: the overall success probability at $\\mathcal{C}=1$ is less than one. Given these results, the proposed method can find optimal entangling two-qubit gates for at least two iterations. It is also clear \nfrom these tests that the algorithm is always reducing $\\bar{f}$, but from $N=3$ iterations this might be not an optimal improvement of the concurrence. This originates from the fact that the gradient $\\nabla f (\\boldsymbol{\\alpha})$, see Eq.~\\eqref{eq:f}, is almost flat in the neighborhood of $\\mathcal{C}=1$ for $N>2$ iterations and thus the numerical search for the optimal gate, i.e, \nthe search for $\\boldsymbol{\\alpha}^* \\in \\mathbb{R}^{15}$, becomes inefficient.  \n\n\\begin{figure*}[t!]\n \\begin{center}\n \\includegraphics[width=.39\\textwidth]{fig3a.pdf}\n \\includegraphics[width=.39\\textwidth]{fig3b.pdf}\n\\caption{Optimized purification protocol for Werner states, see Eq.~\\eqref{eq:wernerstate}.\nLeft panel: average cost function $\\bar{f}$ with a uniform PDF as a function of $N$, the number of iterations.  Right panel: The overall success \nprobability after $N=1$ (dotted line), $N=2$ (dash-dotted line), and $N=3$ (full line) iterations as a function of $x$.}\n   \\label{fig:Werner}\n \\end{center}\n\\end{figure*}\n\nInspired by {\\it Example 2.}, we consider the state in Eq.~\\eqref{eq:example2} and also\n\\begin{equation}\nx\\ketbra{\\Psi^-}{\\Psi^-}+(1-x)\\ketbra{\\Upsilon}{\\Upsilon} \\label{eq:MaZs}\n\\end{equation}\nwith $x \\in [0,1]$ and \n\\begin{equation}\n \\ket{\\Upsilon}=\\frac{1}{\\sqrt2} \\left(\\ket{\\Psi^+}+i\\ket{\\Phi^-}\\right). \\nonumber\n\\end{equation}\nBy using the unitary transformation in Eq.~\\eqref{eq:localtr} on Eq.~\\eqref{eq:MaZs} one obtains Eq.~\\eqref{eq:example2} and therefore it is expected that the only optimal two-qubit gates are the ones that purify these states\nin one iteration. Both states have an input concurrence $\\mathcal{C}(x)=x$. However, it is important to note that the CNOT gate is not optimal for the state in Eq.~\\eqref{eq:MaZs}, because after one iteration round\n\\begin{equation}\n \\mathcal{C}'_{\\text{CNOT}}(x)=\\begin{cases} 0& x \\in [0,0.5], \\\\ 2x \\frac{2x-1}{1+x^2} & x \\in (0.5,1], \\end{cases} \\nonumber\n\\end{equation} \nand $\\mathcal{C}'_{\\text{CNOT}}(x) \\leqslant x$. The success probability is\n\\begin{equation}\n P_{\\text{CNOT}}=\\frac{1+x^2}{2}. \\nonumber\n\\end{equation}\nThus, the CNOT gate based purification protocol impairs the concurrence. In contrast to this analytical observation, numerical analysis with uniform PDF yields already in the first iteration for both states an average cost function $\\bar{f}\\approx 0.0002$. To demonstrate the robustness of the numerical approach we provide examples of three non-uniform PDFs for $N=1$ \niteration. First, we consider\n\\begin{equation}\np(x)=2x,\\quad \\text{with} \\quad x \\in [0,1], \\nonumber\n\\end{equation}\nwhich describes a situation, where states with higher concurrences are more likely to be subject to the purification. The resulting average cost function is $\\bar{f}\\approx 0.000004$. Secondly, we take \n\\begin{equation}\np(x)=2(1-x),\\quad \\text{with} \\quad x \\in [0,1], \\nonumber\n\\end{equation}\nwhich puts more weight on states with low concurrences and find $\\bar{f}\\approx 0.0005$. Finally, we investigate a PDF\n\\begin{equation}\np(x)=6x(1-x),\\quad \\text{with} \\quad x \\in [0,1], \\nonumber\n\\end{equation}\ni.e., the states around the concurrence $\\mathcal{C}(x)=0.5$ are more likely to participate in the purification, and obtain $\\bar{f}\\approx 0.00005$. These results demonstrate the effectiveness of our approach and up to a numerical precision these one-parameter families of states can be purified in one iteration.\n\n\\begin{figure*}[t!]\n \\begin{center}\n \\includegraphics[width=.39\\textwidth]{fig4a.pdf}\n \\includegraphics[width=.39\\textwidth]{fig4b.pdf}\n\\caption{Optimized purification protocol for the state in Eq.~ \\eqref{eq:qrstate}. Left panel: Average cost function $\\bar{f}$ with a uniform PDF as a function of $N$, the number of iterations. \nCrosses are the results of the CNOT gate, whereas dots display the numerical optimization. They have been connected by lines to guide the eye. Right panel: The concurrence $\\mathcal{C}'$\nas a function of $x$ and $y$ after one purification round. The cone-type surface is obtained with our approach. A less optimal surface with minima at $x=0$ and along the $y$ axis is the result of the purification protocol \nwith the CNOT gate.}\n   \\label{fig:XY}\n \\end{center}\n\\end{figure*}\n\nNext, we consider the following two-parameter family of states \\cite{Bernad2}\n\\begin{equation}\n\\hat{\\rho}(x,y) = \\frac{1}{4}\n\\left(\n\\begin{array}{cccc}\n 1-x & i y & -i y & x-1 \\\\\n -i y & x+1 & -x-1 & i y \\\\\n i y & -x-1 & x+1 & -i y \\\\\n x-1 & -i y & i y & 1-x \\\\\n\\end{array}\n\\right) \\label{eq:qrstate}\n\\end{equation}\nwith $x,y \\in \\mathbb{R}$ and $x^2+y^2\\leqslant1$. The concurrence of this state is $\\sqrt{x^2+y^2}$. In order to relate the performance of our approach, we apply the CNOT gate based purification \nprotocol to this family of states. One obtains\n\\begin{eqnarray}\n \\mathcal{C}'_{\\text{CNOT}}(x,y)&=&\\frac{2|x|}{1+x^2}, \\label{eq:xyCNOTc} \\\\\n P_{\\text{CNOT}}&=&\\frac{1+x^2}{2}, \\label{eq:xyCNOTp}\n\\end{eqnarray}\nafter $N=1$ and\n\\begin{eqnarray}\n \\mathcal{C}'_{\\text{CNOT}}(x,y)&=&\\frac{4 |x| (1+x^2)}{1+6x^2+x^4}, \\label{eq:xyCNOTc2} \\\\\n P_{\\text{CNOT}}&=&\\frac{1+6x^2+x^4}{2 (1+x^2)^2}, \\label{eq:xyCNOTp2}\n\\end{eqnarray}\nafter $N=2$ purification rounds, respectively. If one applies the unitary transformation in Eq.~\\eqref{eq:localtr} on the state before the bilateral CNOT gates are performed, then the above results remain \nunchanged. These results demonstrate that the CNOT gate and the additional tricks, which yielded optimal purifications for the one-parameter family of states, are not optimal in this scenario, since they are outperformed on average by our optimized protocol. \nWe assume that CNOT-based purification protocols cannot exploit all the useful entanglement, because the concurrence \nand the success probability become independent of the $y$ variable. Using these analytical results and the uniform PDF\n\\begin{equation}\n p(x,y)=\\frac{1}{\\pi},\\quad \\text{with} \\quad x^2+y^2 \\leqslant 1, \\nonumber\n\\end{equation}\nwe compare our approach with the above-presented analytical results, see Eq.~\\eqref{eq:xyCNOTc} and Eq.~\\eqref{eq:xyCNOTc2}. In Fig.~\\ref{fig:XY}, we show that our optimized protocol provides after one iteration a $x$ and $y$-dependent concurrence. Although the \nnumerically-obtained concurrence is larger at $x=0$ and $y \\in [-1,1]$ than the surface given by Eq.~\\eqref{eq:xyCNOTc}, one can also observe the opposite at $y=0$ and $x \\in [-1,1]$. However, the concurrence presents a better improvement with\nour algorithm and the average cost function with the uniform PDF stays for more iterations lower than the CNOT-based protocol, see the left figure in panel \\ref{fig:XY}.      \nIn regard to the overall success probability, we let our algorithm work until the third iteration and in Fig.~\\ref{fig:XYos} the results are compared with the CNOT-based purification protocol. The obtained \nsurfaces differ only by a $\\pi\/2$ rotation around the $z$-axis. Therefore, there is not much difference in the overall success probability and from this aspect the CNOT gate can also be considered optimal, though, \nit still can not improve the concurrence as effectively as the gate obtained with our approach. However, one might think with a proper choice of local unitary transformations, like the transformation \nin Eq.~\\eqref{eq:localtr} used for Werner and one-step purifiable states, the application of the CNOT gate may result in an optimal purification protocol. Let us consider then two general local unitary \ntransformations at locations $A$ and $B$ for the preparation of two-qubit states. It is enough to consider unitary transformations in $SU(2)$, for which the Euler angle parameterization reads \\cite{Tilma2}\n\\begin{eqnarray}\n U_A&=& e^{i \\sigma_z \\alpha_1} e^{i \\sigma_y \\alpha_2} e^{i \\sigma_z \\alpha_3}, \\\\\n U_B&=& e^{i \\sigma_z \\beta_1} e^{i \\sigma_y \\beta_2} e^{i \\sigma_z \\beta_3},\n\\end{eqnarray}\nwhere $\\sigma_y$ and $\\sigma_z$ are the Pauli matrices. Here, $\\alpha_1, \\beta_1 \\in [0, \\pi]$, $\\alpha_2,\\beta_2 \\in [0, \\pi\/2]$, and $\\alpha_3, \\beta_3 \\in [0, 2\\pi]$ are the Euler angles for  $SU(2)$. For the first\niteration, we analyze the output\nunitary of the optimized protocol with \\textit{Mathematica} \\cite{CAS} and observe that the angles $\\alpha_1$ and $\\beta_1$ do not affect the efficiency of the CNOT-based protocol. Optimal \nconcurrences for the remaining four angles yield either the result in Eq.~\\eqref{eq:xyCNOTc} or\n\\begin{equation}\n \\mathcal{C}'_{\\text{CNOT}}(x,y)=\\frac{2|y|}{1+y^2},  \n\\end{equation}\nwith success probability\n\\begin{equation} \n P_{\\text{CNOT}}=\\frac{1+y^2}{2}, \n\\end{equation}\nfor $\\alpha_2=\\frac{\\pi}{8}$, $\\beta_2=\\frac{3 \\pi}{8}$, and $\\alpha_3=\\beta_3=\\frac{3\\pi}{4}$. For these results the average cost function with uniform PDF yields $\\bar{f}\\approx 0.372$, which is still lower than\nthe average concurrence found by our approach. These results demonstrate that even with local unitary transformations the CNOT gate is globally not optimal for all values of $x$ and $y$. However, when $x$ and $y$\nare known beforehand, then one can use the following local unitary transformation\n\\begin{eqnarray}\n U^\\dagger_A \\otimes U_B, \\quad \\text{with} \\quad U=\\cos(\\theta)\\mathbb{I}_2 + \\sin(\\theta) \\sigma_x, \\label{eq:statedeptr}\n\\end{eqnarray}\nwhere the angle $\\theta$ is a function of $x$ and $y$. For example, for $x,y \\geqslant 0$, we have\n\\begin{equation}\n \\cos(\\theta)=\\sqrt{\\frac{1}{2}+\\sqrt{\\frac{x+\\sqrt{x^2+y^2}}{8 \\sqrt{x^2+y^2}}}}. \\nonumber\n\\end{equation}\nUsing this transformation before the purification protocol, one obtains the state\n\\begin{equation}\n\\frac{1+ \\sqrt{x^2+y^2}}{2} \\ketbra{\\Psi^-}{\\Psi^-}+\\frac{1- \\sqrt{x^2+y^2}}{2} \\ketbra{\\Phi^-}{\\Phi^-}. \\label{eq:Spec}\n\\end{equation}\nNow, the CNOT-based purification yields\n\\begin{equation}\n \\mathcal{C}'_{\\text{CNOT}}(x,y)=\\frac{2\\sqrt{x^2+y^2}}{1+x^2+y^2}, \n\\end{equation}\nwith success probability\n\\begin{equation}\n P_{\\text{CNOT}}=\\frac{1+x^2+y^2}{2}. \n\\end{equation}\nIt is immediate that $\\mathcal{C}'_{\\text{CNOT}}(x,y) \\geqslant \\sqrt{x^2+y^2}$, i.e., we are improving the concurrence. In this particular case, one does not need the average cost function in the numerical search, \nbecause the state is fixed. Our algorithm running with a single state with parameters $(x,y)$ as defined in Eq.~\\eqref{eq:qrstate} instead of an ensemble of states can improve the concurrence, but is unable to find the optimal solution presented above, because the transformation\nEq.~\\eqref{eq:statedeptr} together with CNOT gates is a non-symmetrical transformation at nodes $A$ and $B$.\n\nFinally, let us point out that our approach does not take into account operational or memory errors. These always depend on the implementation and if an experiment can provide us models for the errors then \nour approach can be easily extended. Another important experimental input is the PDF $p(\\boldsymbol{x})$, which is usually subject to the method of generating entangled states between locations A and B. Furthermore, \nthis PDF is assigned to the process of choosing a value $\\boldsymbol{x}$ as a random event, i.e, we have the same two two-qubit pairs parameterized by $\\boldsymbol{x}$ before the purification protocol. In effect, \nwe integrate the parameter dependence of the states away and thus our approach yields an optimal two-qubit gate on average. If the value of $\\boldsymbol{x}$ is fixed in an experimental design, then our method \nprovides an optimal two-qubit gate designed for this particular state. The optimal two-qubit gates can be can be realized by three\nCNOT gates and additional single-qubit rotations \\cite{Vidal,Vatan}, and therefore the experimental generation of this gate is possible with high fidelity \\cite{Debnath}. Also in the context of trapped ions or superconducting quantum circuits, the generation of two-qubit entangling gates can be achieved with high precision using the M\\o{}lmer-S\\o{}rensen gate \\cite{MS} or the $\\sqrt{i \\text{SWAP}}$ \\cite{Fan} gate. Since two-qubit unitaries have a straightforward implementation in many physical settings, thanks to quantum compilation \\cite{Martinez}, we argue that quantum computing platforms implementing quantum communication could actually benefit from globally optimal gates for entanglement purification.\n\n\\begin{figure*}[t!]\n \\begin{center}\n \\includegraphics[width=.39\\textwidth]{fig5.pdf}\n\\caption{The overall success probability of the state in Eq.~ \\eqref{eq:qrstate} after and $N=3$ iterations as a function of $x$ and $y$. Both surfaces are similar in appearance. The one having maxima at $x=1$ \nbelongs to the CNOT-based protocol and the other one having maxima at $y=1$ is obtained via the optimized numerical protocol.}\n   \\label{fig:XYos}\n \\end{center}\n\\end{figure*}\n\n\\section{Conclusions}\n\\label{Conclusions}\n\nIn the context of entanglement purification and recurrence protocols, we have presented a method to obtain optimal protocols. This method searches for the optimal two-qubit gate, which is applied bilaterally\nat the nodes $A$ and $B$ in order to distill from an ensemble of noisy pairs a higher fidelity state with respect to a maximally entangled state. Our strategy considers situations, where the \nexperimental apparatus does not have enough control over the entanglement generation phase and different experimental runs result in different states. Here, we assume that the same copies of the noisy state can be \ngenerated before the purification protocol takes place, but a different experimental run could result in different states. Errors originating from local operations, memory requirements, or even \nclassical communication are neglected for now.\n\nWe have numerically demonstrated the optimality of our proposal for several states. In the case of the Werner state, we have found several optimal two-qubit gates and their performances are the same as the CNOT-based Bennett protocol \\cite{Bennett}. Thus, for Werner states the optimality cannot be improved beyond the already known performance. Next, we have investigated a family of states that one can purify \nin one step, i.e., two noisy copies of states are enough to obtain a maximally entangled state, where we immediately obtain minimal average cost function $\\bar{f} \\approx 0$, as expected. Finally, we consider a two-parameter family of states which is considered in theoretical models of a quantum repeater \\cite{Bernad2}. Our numerical investigations demonstrate that a\nsingle bilaterally applied CNOT gate cannot be globally optimal for these states. On the other hand, when the state is known, then parameter-dependent, non-symmetric local transformations allow again the CNOT to be also \none of the optimal two-qubit gates. This case exemplifies the difference between protocols that are optimal for a full class of parameterized states and those that are only optimal for a single state.  \n\nIn conclusion, we have proposed a general method to optimize entanglement purification for an arbitrary family of states. Our algorithm \\cite{Francesco} can find the most optimal two-qubit gate that on average induces the highest increase of entanglement. We remark that the method is general for the set of parameters defining the states. Optimizing for a single state is also possible, as shown in one of the \npresented examples. Furthermore, we have focused on the degree of entanglement measured by the concurrence and not on the fidelity with respect to a particular Bell state. \nThis is motivated by the fact that there exist an infinite number of maximally entangled states, and therefore entangled mixed states can present a larger fidelity with respect to maximally entangled states that might not correspond to Bell states defined in the computational basis. \nA possible drawback is that one cannot know with certainty the final maximally entangled state produced by the protocol. This and other issues such as different input states, asymmetric two-qubit gates, \nand non-ideal local operations remain open questions. However, this work aims to introduce the concept of globally optimized recurrence protocols that is flexible enough to incorporate the \naforementioned points in further investigations, which we plan to carry out in the future. \n\n\\begin{acknowledgments}\nThis work is supported by the DFG under Germany's Excellence Strategy-Cluster of Excellence Matter and Light for Quantum\nComputing (ML4Q) EXC 2004\/1-390534769. The results were obtained using the libraries JAX \\cite{JAX}, Scipy \\cite{Scipy} and QuTiP \\cite{Qutip}.\n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Derivation of the path integral}\n\\label{app:pathintegral}\nFor simplicity, in this section, we assume a quantum hamiltonian given by\n\\begin{equation}\n\\hat{H}=\\sul{\\alpha,\\beta}{}h_{\\alpha\\beta}\\hat{c}_{\\alpha}^{\\dagger}\\hat{c}_{\\beta}^{}+\\sul{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}{}U_{\\alpha\\beta}\\hat{c}_{\\alpha}^{\\dagger}\\hat{c}_{\\beta}^{\\dagger}\\hat{c}_{\\beta}^{}\\hat{c}_{\\alpha}^{}.\n\\label{eq:quantum_hamiltonian_app}\n\\end{equation}\nThe result for a non-diagonal interaction $U_{\\alpha\\beta\\gamma\\nu}$, however, is given in appendix \\ref{app:classical_hamiltonians}\nIn order to get from eq.~(\\ref{eq:propagator_in_grassmann_path-integral}) to the complex path integral eq.~(\\ref{eq:propagator_in_complex_path-integral}), the following two integrals with $j,j^\\prime\\in\\mathbb{N}_0$, will be inserted:\n\\begin{align}\n&\\intg{0}{2\\pi}{\\theta}\\intg{}{}{^2\\phi}\\exp\\rbr{-\\abs{\\phi}^2+\\conjg{\\phi}{\\rm e}^{{\\rm i}\\theta}-{\\rm i} j\\theta}\\phi^{j^\\prime}=2\\pi^2\\delta_{j,j^\\prime}\n\\label{eq:inserted_integral1} \\\\\n&\\intg{}{}{^2\\phi}\\intg{}{}{^2\\mu}\\exp\\rbr{-\\abs{\\phi}^2-\\abs{\\mu}^2+\\conjg{\\phi}\\mu}\\phi^j\\rbr{\\conjg{\\mu}}^{j^\\prime}=\\pi^2j!\\delta_{j,j^\\prime},\n\\label{eq:inserted_integral2}\n\\end{align}\nThereby ${\\rm d}^2\\mu={\\rm d}\\Re{\\mu}{\\rm d}\\Im{\\mu}$, \\textit{i.e.}~the integrations over $\\phi$ and, in the second case, over $\\mu$ run over the whole complex plane. One should notice, that the first of these two integrals is just the second one, but with the modulus of $\\mu$ already integrated out.\n\nThe first of these two integrals is used to decouple $\\zetakv{0}$ from $\\zetakv{1}$ by the following identity:\n\n\\begin{widetext}\n\\begin{align}\n\\int&{\\rm d}^{2J}\\zetakj{0}{}\\exp\\rbr{-\\conjg{\\zetakv{0}}\\cdot\\zetakv{0}}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\chikj{0}{j}}\\zetakj{0}{j}}}\\prodl{j=1}{J}\\rbr{\\conjg{\\zetakj{0}{j}}}^{\\nij{j}}= \\nonumber \\\\\n&\\int\\frac{{\\rm d}^{2N_i}\\phikj{0}{}}{\\pi^{N_i}}\\int\\limits_{0}^{2\\pi}\\frac{{\\rm d}^{N_i}\\thetakj{i}{}}{\\left(2\\pi\\right)^{N_i}}\\int{\\rm d}^{2J}\\zetakj{0}{}\\exp\\rbr{-\\conjg{\\zetakv{0}}\\cdot\\zetakv{0}-\\abs{\\phikv{0}}^2+\\conjg{\\phikv{0}}\\cdot\\mukv{0}}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\chikj{0}{j}}\\phikj{0}{j}}}\\cbr{\\prodl{j=0}{J-1}\\rbr{1+\\zetakj{0}{J-j}\\conjg{\\mukj{0}{J-j}}}}\\prodl{j=1}{J}\\rbr{\\conjg{\\zetakj{0}{j}}}^{\\nij{j}},\n\\label{eq:insert_integrals_0}\n\\end{align}\n\\end{widetext}\nwith $\\mukj{0}{j}=\\nij{j}\\exp\\rbr{{\\rm i}\\thetakj{i}{j}}$ for all $j\\in\\{1,\\ldots,J\\}$, where $J$ is the number of single particle states taken into account. Note that here, for the initially unoccupied single particle states, the phases $\\thetakj{i}{j}$ are arbitrary but fixed, \\textit{e.g.}~to zero, while the integration runs only over those initial phases $\\thetakj{i}{j}$, for which $\\nij{l}=1$. In this way, the integrals, that have to be performed exactly, in order to get a reasonable and correct semiclassical approximation for the propagator are already done, and do not have to be carried out later.\n\nFor the $N_i=\\sum_{j=1}^{J}\\nij{j}$ initially occupied single particle states, the identity follows directly from eq.~(\\ref{eq:inserted_integral1}), while for the unoccupied ones, it is important to notice, that the term $\\conjg{\\chikj{0}{j}}\\zetakj{0}{j}$ does vanish when integrating over $\\zetakv{0}$. This is because of the properties of the Grassmann integrals eq.~(\\ref{eq:vanishing_grassmann-integrals}) and the fact, that there is no $\\conjg{\\zetakj{0}{j}}$ for those components, for which $\\nij{j}=0$.\n\nThe thus obtained expression is the starting point for an iterative insertion of integrals of the form of eq.~(\\ref{eq:inserted_integral2}). For $1\\leqm<M$, an evaluation of the overlaps and matrix elements of eq.~(\\ref{eq:propagator_in_grassmann_path-integral}) containing $\\zetakv{m}$ yields the following expression:\n\n\\begin{widetext}\n\\begin{align}\n&\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\chikj{m}{j}}\\zetakj{m}{j}}}\\cbr{1-\\frac{{\\rm i}\\tau}{\\hbar}\\sul{\\alpha,\\beta=1}{J}\\left(h_{\\alpha\\beta}^{(m-1)}\\conjg{\\zetakj{m}{\\alpha}}\\chikj{m-1}{\\beta}+U_{\\alpha\\beta}^{(m-1)}\\conjg{\\zetakj{m}{\\alpha}}\\conjg{\\zetakj{m}{\\beta}}\\chikj{m-1}{\\beta}\\chikj{m-1}{\\alpha}\\right)}\\prodl{j=1}{J}\\rbr{1+\\conjg{\\zetakj{m}{j}}\\chikj{m-1}{j}}= \\nonumber \\\\\n&\\qquad\\left[a^{(m)}-\\frac{{\\rm i}\\tau}{\\hbar}\\sul{\\alpha}{}h_{\\alpha\\alpha}^{(m-1)}b_\\alpha^{(m)}-\\frac{{\\rm i}\\tau}{\\hbar}\\sul{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}{}h_{\\alpha\\beta}^{(m-1)}c_{\\alpha\\beta}^{(m)}-\\frac{{\\rm i}\\tau}{\\hbar}\\sul{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}{}U_{\\alpha\\beta}^{(m-1)}d_{\\alpha\\beta}^{(m)}\\right]\\prodl{j=1}{J}\\rbr{1+\\conjg{\\zetakj{m}{j}}\\chikj{m-1}{j}}.\n\\label{eq:abbreviation_mth_factor}\n\\end{align}\nWith the help of the integral eq.~(\\ref{eq:inserted_integral2}), the coefficients $a^{(m)}$, $b^{(m)}$, $c^{(m)}$ and $d^{(m)}$ can successively -- starting from $m=1$ -- be written as\n\\begin{align}\na^{(m)}=&\\int\\frac{{\\rm d}^{2J}\\mukj{m}{}}{\\pi^J}\\int\\frac{{\\rm d}^{2J}\\phikj{m}{}}{\\pi^J}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\chikj{m}{j}}\\phikj{m}{j}}}\\exp\\rbr{-\\abs{\\phikv{m}}^2-\\abs{\\mukv{m}}^2+\\conjg{\\phikv{m}}\\cdot\\mukv{m}}\\prodl{j=0}{J-1}\\cbr{1+\\zetakj{m}{J-j}\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}},\n\\label{eq:insert_integral_am} \\\\\nb_{\\alpha}^{(m)}=&\\int\\frac{{\\rm d}^{2J}\\mukj{m}{}}{\\pi^J}\\int\\frac{{\\rm d}^{2J}\\phikj{m}{}}{\\pi^J}\\exp\\rbr{-\\abs{\\phikv{m}}^2-\\abs{\\mukv{m}}^2+\\conjg{\\phikv{m}}\\cdot\\mukv{m}}\\conjg{\\zetakj{m}{\\alpha}}\\chikj{m-1}{\\alpha}\\left\\{\\prodl{j=0}{L-\\alpha-1}\\cbr{1+\\zetakj{m}{J-j}\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}}\\right\\} \\nonumber \\\\\n&\\quad\\cbr{1+\\zetakj{m}{\\alpha}\\sul{k=1}{\\infty}c_k^{(1)}\\rbr{\\phikj{m-1}{\\alpha}}\\rbr{\\conjg{\\mukj{m}{\\alpha}}}^{k}}\\left\\{\\prodl{j=J-\\alpha+1}{J-1}\\cbr{1+\\zetakj{m}{J-j}\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}}\\right\\}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\chikj{m}{j}}\\phikj{m}{j}}},\n\\label{eq:insert_integral_bm} \\\\\nc_{\\alpha\\beta}^{(m)}=&\\int\\frac{{\\rm d}^{2J}\\mukj{m}{}}{\\pi^J}\\int\\frac{{\\rm d}^{2J}\\phikj{m}{}}{\\pi^J}\\exp\\rbr{-\\abs{\\phikv{m}}^2-\\abs{\\mukv{m}}^2+\\conjg{\\phikv{m}}\\cdot\\mukv{m}}\\conjg{\\zetakj{m}{\\alpha}}\\chikj{m-1}{\\beta}\\tbr{\\prodl{j=0}{J-\\max\\rbr{\\alpha,\\beta}-1}\\cbr{1+\\zetakj{m}{J-j}\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}}} \\nonumber \\\\\n&\\quad\\cbr{1+\\zetakj{m}{\\max\\rbr{\\alpha,\\beta}}\\sul{k=1}{\\infty}c_k^{(2)}\\left(\\phikj{m-1}{\\max\\rbr{\\alpha,\\beta}}\\right)\\left(\\conjg{\\mukj{m}{\\max\\rbr{\\alpha,\\beta}}}\\right)^k}\\tbr{\\prodl{j=J-\\max\\rbr{\\alpha,\\beta}+1}{J-\\min\\rbr{\\alpha,\\beta}-1}\\cbr{1+\\zetakj{m}{J-j}\\sul{k=1}{\\infty}c_k^{(3)}\\rbr{\\phikj{m-1}{J-j}}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}}} \\nonumber \\\\\n&\\quad\\cbr{1+\\zetakj{m}{\\min\\rbr{\\alpha,\\beta}}\\sul{k=1}{\\infty}c_k^{(2)}\\left(\\phikj{m-1}{\\min\\rbr{\\alpha,\\beta}}\\right)\\left(\\conjg{\\mukj{m}{\\min\\rbr{\\alpha,\\beta}}}\\right)^k}\\tbr{\\prodl{j=J-\\min\\rbr{\\alpha,\\beta}+1}{J-1}\\cbr{1+\\zetakj{m}{J-j}\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}}}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\chikj{m}{j}}\\phikj{m}{j}}}\n\\label{eq:insert_integral_cm}\n\\end{align}\n\\begin{align}\nd_{\\alpha\\beta}^{(m)}=&\\int\\frac{{\\rm d}^{2J}\\mukj{m}{}}{\\pi^J}\\int\\frac{{\\rm d}^{2J}\\phikj{m}{}}{\\pi^J}\\exp\\rbr{-\\abs{\\phikv{m}}^2-\\abs{\\mukv{m}}^2+\\conjg{\\phikv{m}}\\cdot\\mukv{m}}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\chikj{m}{j}}\\phikj{m}{j}}}\\conjg{\\zetakj{m}{\\alpha}}\\conjg{\\zetakj{m}{\\beta}}\\chikj{m-1}{\\beta}\\chikj{m-1}{\\alpha} \\nonumber \\\\\n&\\quad\\tbr{\\prodl{j=0}{J-\\max\\rbr{\\alpha,\\beta}-1}\\cbr{1+\\zetakj{m}{J-j}\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}}}\\cbr{1+\\zetakj{m}{\\max\\rbr{\\alpha,\\beta}}\\sul{k=1}{\\infty}c_k^{(4)}\\left(\\phikj{m-1}{\\max\\rbr{\\alpha,\\beta}}\\right)\\left(\\conjg{\\mukj{m}{\\max\\rbr{\\alpha,\\beta}}}\\right)^k} \\nonumber \\\\\n&\\quad\\tbr{\\prodl{j=J-\\max\\rbr{\\alpha,\\beta}+1}{J-\\min\\rbr{\\alpha,\\beta}-1}\\cbr{1+\\zetakj{m}{J-j}\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}}}\\cbr{1+\\zetakj{m}{\\min\\rbr{\\alpha,\\beta}}\\sul{k=1}{\\infty}c_k^{(4)}\\left(\\phikj{m-1}{\\min\\rbr{\\alpha,\\beta}}\\right)\\left(\\conjg{\\mukj{m}{\\min\\rbr{\\alpha,\\beta}}}\\right)^k} \\nonumber \\\\\n&\\quad\\tbr{\\prodl{j=J-\\min\\rbr{\\alpha,\\beta}+1}{J-1}\\cbr{1+\\zetakj{m}{J-j}\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}}},\n\\label{eq:insert_integral_dm}\n\\end{align}\nwith $c_1^{(1)}=c_1^{(2)}=c_1^{(3)}=c_1^{(4)}=1$.\n\nIt is important to notice, that the integral over $\\phikv{m}$ and $\\mukv{m}$ selects only the $k=1$ terms of the occurring sums. Therefore, the terms with $k\\geq2$ can be varied, in order to modify the final path integral in the desired way.\n\nFinally, for $m=M$, a similar argument as for $m=0$ allows to restrict the integrals over $\\phikv{M}$ again to those $N_f=\\sum_{j=1}^{J}\\nfj{j}$ components with $\\nfj{j}=1$, while setting all the other components of $\\phikv{M}$ to zero.\n\nAfter this the $m$-th factor in the product over the timesteps only depends on $\\zetakv{m+1}$ and $\\chikv{m}$, such that one can easily integrate out the intermediate Grassmann variables $\\zetakv{1},\\ldots,\\zetakv{M}$ and $\\chikv{0},\\ldots,\\chikv{M-1}$ by using\n\\begin{align}\n\\int&{\\rm d}^{2J}\\zeta\\intg{}{}{^{2J}\\chi}\\exp\\rbr{-\\conjg{\\gV{\\zeta}}\\cdot\\gV{\\zeta}-\\conjg{\\gV{\\chi}}\\cdot\\gV{\\chi}}\\cbr{\\prodl{j=0}{J-1}\\rbr{1+\\zeta_{J-j}f_{J-j}^{(m)}}}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\zeta_{j}}\\chi_{j}}}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\chi_{j}}\\phikj{m}{j}}}=\\prodl{j=1}{J}\\rbr{1+f_j^{(m)}\\phikj{m}{j}},\n\\label{eq:grassman-integrated_nohamiltonian} \\\\\n\\int&{\\rm d}^{2J}\\zeta\\intg{}{}{^{2J}\\chi}\\exp\\rbr{-\\conjg{\\gV{\\zeta}}\\cdot\\gV{\\zeta}-\\conjg{\\gV{\\chi}}\\cdot\\gV{\\chi}}\\cbr{\\prodl{j=0}{J-1}\\rbr{1+\\zeta_{J-j}f_{J-j}^{(m)}}}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\zeta_{j}}\\chi_{j}}}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\chi_{j}}\\phikj{m}{j}}}\\conjg{\\zeta_{\\alpha}}\\chi_{\\beta}= \\nonumber \\\\\n&\\qquad f_\\alpha^{(m)}\\phikj{m}{\\beta}\\cbr{\\prodl{j=1}{\\min\\rbr{\\alpha,\\beta}-1}\\rbr{1+f_j^{(m)}\\phikj{m}{j}}}\\cbr{\\prodl{j=\\max\\rbr{\\alpha,\\beta}+1}{J}\\rbr{1+f_j^{(m)}\\phikj{m}{j}}}\\prodl{j=\\min\\rbr{\\alpha,\\beta}+1}{\\max\\rbr{\\alpha,\\beta}-1}\\rbr{1-f_j^{(m)}\\phikj{m}{j}},\n\\label{eq:grassman-integrated_sp-part} \\\\\n\\int&{\\rm d}^{2J}\\zeta\\intg{}{}{^{2J}\\chi}\\exp\\rbr{-\\conjg{\\gV{\\zeta}}\\cdot\\gV{\\zeta}-\\conjg{\\gV{\\chi}}\\cdot\\gV{\\chi}}\\cbr{\\prodl{j=0}{J-1}\\rbr{1+\\zeta_{J-j}f_{J-j}^{(m)}}}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\zeta_{j}}\\chi_{j}}}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\chi_{j}}\\phikj{m}{j}}}\\conjg{\\zeta_{\\alpha}}\\conjg{\\zeta_{\\beta}}\\chi_{\\beta}\\chi_{\\alpha}= \\nonumber \\\\\n&\\qquad f_\\alpha^{(m)}f_\\beta^{(m)}\\phikj{m}{\\beta}\\phikj{m}{\\alpha}\\prodl{\\substack{j=1 \\\\ j\\neq\\alpha,\\beta}}{J}\\rbr{1+f_j^{(m)}\\phikj{m}{j}}.\n\\label{eq:grassman-integrated_interaction}\n\\end{align}\n\\end{widetext}\nMoreover, the integrals over $\\zetakv{0}$ and $\\chikv{M}$ yield\n\\begin{align}\n\\int{\\rm d}^{2J}\\zetakj{0}{}&\\exp\\rbr{-\\conjg{\\zetakv{0}}\\cdot\\zetakv{0}}\\cbr{\\prodl{j=0}{J-1}\\rbr{1+\\zetakj{0}{J-j}\\conjg{\\mukj{0}{J-j}}}} \\nonumber \\\\\n&\\qquad\\prodl{j=1}{J}\\rbr{\\conjg{\\zetakj{0}{j}}}^{\\nij{j}}=\\prodl{j:\\nij{j}=1}{}\\conjg{\\mukj{0}{j}}\n\\label{eq:grasmman-integrated_initial} \\\\\n\\int{\\rm d}^{2J}\\chikj{M}{}&\\exp\\rbr{-\\conjg{\\chikv{M}}\\cdot\\chikv{M}}\\cbr{\\prodl{j=0}{J-1}\\rbr{\\chikj{M}{J-j}}^{\\nfj{J-j}}} \\nonumber \\\\\n&\\qquad\\prodl{j=1}{J}\\rbr{1+\\chikj{M}{j}\\conjg{\\phikj{M}{j}}}=\\prodl{j:\\nfj{j}=1}{}\\phikj{M}{j}\n\\label{eq:grasmman-integrated_final}\n\\end{align}\nAfter performing these integrals, one notices, that the inserted integrals have been chosen such, that the resulting sums can be performed and yield exponentials, such that the propagator is, after integrating out $\\mukv{1},\\ldots,\\mukv{M}$ as well as $\\phikv{0}$ and undo the expansion in $\\tau$, given by the path integral eq.~(\\ref{eq:propagator_in_complex_path-integral}), where the classical Hamiltonian is given by\n\n\\begin{align}\nH&_{cl}\\left(\\conjg{\\gV{\\mu}},\\gV{\\phi}\\right)= \\nonumber \\\\\n&\\sul{\\alpha}{}h_{\\alpha\\alpha}\\conjg{\\mu_\\alpha}\\phi_\\alpha f_1\\left(\\conjg{\\mu_\\alpha},\\phi_\\alpha\\right) \\nonumber \\\\\n&+\\sul{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}{}h_{\\alpha\\beta}\\conjg{\\mu_\\alpha}\\phi_\\beta f_2\\left(\\conjg{\\mu_\\alpha},\\phi_\\alpha\\right)\\exp\\left(-\\conjg{\\mu_\\beta}\\phi_\\beta\\right){\\prod\\limits_{l}}^{\\alpha,\\beta}g\\left(\\conjg{\\mu_l},\\phi_l\\right) \\nonumber \\\\\n&+\\sul{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}{}U_{\\alpha\\beta}\\conjg{\\mu_\\alpha}\\conjg{\\mu_\\beta}\\phi_\\alpha\\phi_\\beta f_3\\left(\\conjg{\\mu_\\alpha},\\phi_\\alpha\\right)f_3\\left(\\conjg{\\mu_\\beta},\\phi_\\beta\\right),\n\\label{eq:classical_hamiltonian_mu-phi}\n\\end{align}\nwhere $f_1$, $f_2$, $f_3$ and $g$ are arbitrary analytic functions satisfying the following conditions:\n\\begin{align}\nf_1\\left(0,\\phi\\right)&=f_2\\left(0,\\phi\\right)=f_3\\left(0,\\phi\\right)=1\n\\label{eq:constraint_f} \\\\\ng\\left(0,\\phi\\right)&=1\n\\label{eq:constraint_g_1} \\\\\n\\left.\\frac{\\partial}{\\partial\\mu^\\ast}g\\left(\\mu^\\ast,\\phi\\right)\\right|_{\\mu^\\ast=0}&=-2\\phi.\n\\label{eq:constraint_g_2}\n\\end{align}\nMoreover, as in section \\ref{sec:path_integral}, the product in the third line runs only over those values of $j$, which are lying between $\\alpha$ and $\\beta$, excluding $\\alpha$ and $\\beta$ themselves,\n\\begin{equation}\n{\\prod\\limits_{j}}^{\\alpha,\\beta}\\ldots=\\prod\\limits_{j=\\min\\left(\\alpha,\\beta\\right)+1}^{\\max\\left(\\alpha,\\beta\\right)-1}\\ldots\n\\end{equation}\n\\section{The semiclassical amplitude}\n\\label{app:semiclassical_amplitude}\nThe semiclassical amplitude is given by the integral over the exponential of the second variation of the path integral around the classical path wich can be written as,\n\n\\begin{widetext}\n\\begin{align}\n\\mathcal{A}_\\gamma=\\lim\\limits_{\\pisteps\\to\\infty}\\frac{1}{\\rbr{2\\pi}^{2N-1+\\rbr{M-1}J}}&\\intg{}{}{^{N-1}\\delta\\thetakj{0}{}}\\intg{}{}{^N\\delta\\Jkj{M}{}}\\intg{}{}{^N\\delta\\thetakj{M}{}}\\intg{}{}{^J\\delta\\Jkj{1}{}}\\intg{}{}{^J\\delta\\thetakj{1}{}}\\cdots\\intg{}{}{^J\\delta\\Jkj{M-1}{}}\\intg{}{}{^J\\delta\\thetakj{M-1}{}} \\nonumber \\\\\n\\exp\\Bigg\\{&-\\frac{1}{2}\\delta\\thetakv{0}\\mathbf{P}_i^\\prime\\pdiff{\\phikv{0}}{\\thetakv{i}}\\cbr{-\\exp\\cbr{-2{\\rm i}{\\rm diag}\\rbr{\\thetakv{i}}}+\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{{\\phikv{0}}^2}}\\pdiff{\\phikv{0}}{\\thetakv{i}}{\\mathbf{P}_i^\\prime}^{\\rm T}\\delta\\thetakv{0} \\nonumber \\\\\n&-\\frac{1}{2}\\rbr{\\begin{array}{c} \\delta\\thetakv{M}\\mathbf{P}_f \\\\ \\delta\\Jkv{M}\\mathbf{P}_f \\end{array}}{\\mathbf{O}^{(M)}}^{\\rm T}\\rbr{\\begin{array}{cc}\n\\exp\\cbr{-2{\\rm i}{\\rm diag}\\rbr{\\thetakv{M}}} & \\mathbf{I}_{J} \\\\ \\mathbf{I}_{J} & \\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(M-1)}}{{\\conjg{\\phikv{M}}}^2}\n\\end{array}}\\mathbf{O}^{(M)}\\rbr{\\begin{array}{c} \\mathbf{P}_f^{\\rm T}\\delta\\thetakv{M} \\\\ \\mathbf{P}_f^{\\rm T}\\delta\\Jkv{M} \\end{array}} \\nonumber \\\\\n&-\\frac{1}{2}\\sul{m=1}{M-1}\\rbr{\\begin{array}{c} \\delta\\thetakv{m} \\\\ \\delta\\Jkv{m} \\end{array}}{\\mathbf{O}^{(m)}}^{\\rm T}\\rbr{\\begin{array}{cc}\n\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{{\\phikv{m}}^2} & \\mathbf{I}_{J} \\\\ \\mathbf{I}_{J} & \\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m-1)}}{{\\conjg{\\phikv{m}}}^2}\n\\end{array}}\\mathbf{O}^{(m)}\\rbr{\\begin{array}{c} \\delta\\thetakv{m} \\\\ \\delta\\Jkv{m} \\end{array}} \\nonumber \\\\\n&+\\rbr{\\begin{array}{c} \\delta\\thetakv{1} \\\\ \\delta\\Jkv{1} \\end{array}}{\\mathbf{O}^{(1)}}^{\\rm T}\\rbr{\\begin{array}{c} 0 \\\\ \\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{\\conjg{\\phikv{1}}\\partial\\phikv{0}} \\end{array}}\\pdiff{\\phikv{0}}{\\thetakv{i}}{\\mathbf{P}_i^\\prime}^{\\rm T}\\delta\\thetakv{i} \\nonumber \\\\\n&+\\rbr{\\begin{array}{c} \\delta\\thetakv{M}\\mathbf{P}_f \\\\ \\delta\\Jkv{M}\\mathbf{P}_f \\end{array}}{\\mathbf{O}^{(M)}}^{\\rm T}\\rbr{\\begin{array}{cc} 0 & 0 \\\\ \\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(M-1)}}{\\conjg{\\phikv{M}}\\partial\\phikv{M-1}} & 0 \\end{array}}\\mathbf{O}^{(M-1)}\\rbr{\\begin{array}{c} \\delta\\thetakv{M-1} \\\\ \\delta\\Jkv{M-1} \\end{array}} \\nonumber \\\\\n&+\\sul{m=1}{M-2}\\rbr{\\begin{array}{c} \\delta\\thetakv{m} \\\\ \\delta\\Jkv{m} \\end{array}}{\\mathbf{O}^{(m)}}^{\\rm T}\\rbr{\\begin{array}{cc} 0 & 0 \\\\ \\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{\\conjg{\\phikv{m+1}}\\partial\\phikv{m}} & 0 \\end{array}}\\mathbf{O}^{(m)}\\rbr{\\begin{array}{c} \\delta\\thetakv{m} \\\\ \\delta\\Jkv{m} \\end{array}}\\Bigg\\},\n\\label{eq:reduced_propagator_phi-phistar-basis}\n\\end{align}\n\\end{widetext}\nwith\n\\begin{equation}\n\\mathbf{O}^{(m)}=\\rbr{\\begin{array}{cc}\n\\pdiff{\\phikv{m}}{\\thetakv{m}} & \\pdiff{\\phikv{m}}{\\Jkv{m}} \\\\\n\\pdiff{\\conjg{\\phikv{m}}}{\\thetakv{m}} & \\pdiff{\\conjg{\\phikv{m}}}{\\Jkv{m}}\n\\end{array}}.\n\\label{eq:jacobi-matrix}\n\\end{equation}\nMoreover, ${\\rm diag}\\rbr{\\V{v}}$ is the diagonal $d\\times d$-matrix for which the $(j,j)$-th entry is equal to $v_j$, where $d$ is the dimensionality of the vector $\\V{v}$ and $\\mathbf{P}_{i\/f}$ and $\\mathbf{P}_{i\/f}^\\prime$ are defined  as the $N\\timesJ$ and $(N-1)\\timesJ$-matrices, respectively, which project onto the subspace of initially and finally occupied single particle states, with the latter excluding the first occupied one,\n\\begin{align}\n\\rbr{\\mathbf{P}_{i\/f}}_{lj}=&\\delta_{j_{l}^{(\\prime)},j}\n\\label{eq:projection_matrices} \\\\\n\\rbr{\\mathbf{P}_{i\/f}^\\prime}_{lj}=&\\delta_{j_{l+1}^{(\\prime)},j},\n\\label{eq:restricted_projection_matrices}\n\\end{align}\nwhere $j_1<\\ldots<j_{N}\\in\\left\\{j\\in\\{1,\\ldots,J\\}:\\nij{j}=1\\right\\}$ and $j_1^\\prime<\\ldots<j_{N}^\\prime\\in\\left\\{j\\in\\{1,\\ldots,J\\}:\\nfj{j}=1\\right\\}$ are the initially, respectively finally, occupied single particle states.\n\nFor later reference, we also define $\\bar{\\mathbf{P}}_{i\/f}$ as the complement of $\\mathbf{P}_{i\/f}$ as well as\n\\begin{equation}\n\\mathbf{Q}_{i\/f}=\\rbr{\\begin{array}{c} \\bar{\\mathbf{P}}_{i\/f} \\\\ \\mathbf{P}_{i\/f} \\end{array}},\n\\label{eq:reordering_matrix}\n\\end{equation}\nwhich are the (orthogonal) matrices, which put the components corresponding to initially and finally unoccupied single particle states to the first $J-N$ positions, and those correspondig to occupied single particle states to the last $N$ positions, \\textit{i.e.}\n\\begin{equation}\n\\mathbf{Q}_{i\/f}{\\bf n}^{(i\/f)}=(\\underbrace{0,\\ldots,0}_{J-N},\\underbrace{1,\\ldots,1}_{N})^{\\rm T}.\n\\label{eq:reordering_matrix-action_on_nif}\n\\end{equation}\nThe integral over $\\delta\\thetakv{0}$ is given by\n\n\\begin{widetext}\n\\begin{align}\n&\\frac{1}{\\rbr{2\\pi}^{N-1}}\\intg{}{}{^{N-1}\\delta\\thetakj{0}{}}\\exp\\Bigg\\{-\\frac{1}{2}\\delta\\thetakv{0}\\mathbf{P}_i^\\prime\\pdiff{\\phikv{0}}{\\thetakv{i}}\\rbr{-\\exp\\cbr{-2{\\rm i}{\\rm diag}\\rbr{\\thetakv{i}}}+\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{{\\phikv{0}}^2}}\\pdiff{\\phikv{0}}{\\thetakv{i}}{\\mathbf{P}_i^\\prime}^{\\rm T}\\delta\\thetakv{0} \\nonumber \\\\\n&\\qquad\\qquad\\qquad+\\rbr{\\begin{array}{c} \\delta\\thetakv{1} \\\\ \\delta\\Jkv{1} \\end{array}}{\\mathbf{O}^{(1)}}^{\\rm T}\\rbr{\\begin{array}{c} 0 \\\\ \\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{\\conjg{\\phikv{1}}\\partial\\phikv{0}} \\end{array}}\\pdiff{\\phikv{0}}{\\thetakv{i}}{\\mathbf{P}_i^\\prime}^{\\rm T}\\delta\\thetakv{i}-\\frac{1}{2}\\rbr{\\begin{array}{c} \\delta\\thetakv{1} \\\\ \\delta\\Jkv{1} \\end{array}}{\\mathbf{O}^{(1)}}^{\\rm T}\\rbr{\\begin{array}{cc}\n\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(1)}}{{\\phikv{1}}^2} & \\mathbf{I}_{J} \\\\ \\mathbf{I}_{J} & \\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{{\\conjg{\\phikv{1}}}^2}\n\\end{array}}\\mathbf{O}^{(1)}\\rbr{\\begin{array}{c} \\delta\\thetakv{1} \\\\ \\delta\\Jkv{1} \\end{array}}\\Bigg\\}= \\nonumber \\\\\n&\\frac{1}{\\sqrt{2\\pi}^{N-1}}\\tbr{\\det\\cbr{\\mathbf{I}_{J}-\\pdiff{^2{H^{(cl)}}^{(0)}}{\\rbr{\\mathbf{P}_i\\phikv{0}}^2}\\exp\\cbr{2{\\rm i}{\\rm diag}\\rbr{\\thetakv{i}}}}}^{-1}\\exp\\tbr{-\\frac{1}{2}\\rbr{\\begin{array}{c} \\delta\\thetakv{1} \\\\ \\delta\\Jkv{1} \\end{array}}{\\mathbf{O}^{(1)}}^{\\rm T}\\rbr{\\begin{array}{cc}\n\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(1)}}{{\\phikv{1}}^2} & \\mathbf{I}_{J} \\\\ \\mathbf{I}_{J} & \\mathbf{X}^{(1)}\n\\end{array}}\\mathbf{O}^{(1)}\\rbr{\\begin{array}{c} \\delta\\thetakv{1} \\\\ \\delta\\Jkv{1} \\end{array}}},\n\\label{eq:theta0-fluctuation-integral}\n\\end{align}\nwhere $\\mathbf{X}^{(1)}$ is defined as\n\\begin{equation}\n\\mathbf{X}^{(1)}=\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{{\\conjg{\\phikv{1}}}^2}+\\rbr{\\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{\\conjg{\\phikv{1}}\\partial\\phikv{0}}}{\\mathbf{P}_i^\\prime}^{\\rm T}\\tbr{\\exp\\cbr{-2{\\rm i}{\\rm diag}\\rbr{\\mathbf{P}_i^\\prime\\thetakv{i}}}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{\\rbr{\\mathbf{P}_i^\\prime\\phikv{0}}^2}}^{-1}\\mathbf{P}_i^\\prime\\rbr{\\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{\\phikv{0}\\partial\\conjg{\\phikv{1}}}}\n\\label{eq:def_X1_fermions}\n\\end{equation}\nIt can be shown, that eq.~(\\ref{eq:def_X1_fermions}) can also be written as\n\\begin{equation}\n\\mathbf{X}^{(1)}=\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{{\\conjg{\\phikv{1}}}^2}+\\rbr{\\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{\\conjg{\\phikv{1}}\\partial\\phikv{0}}}\\mathbf{X}^{(0)}\\rbr{\\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{{\\phikv{0}}^2}\\mathbf{X}^{(0)}}^{-1}\\rbr{\\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(0)}}{\\phikv{0}\\partial\\conjg{\\phikv{1}}}},\n\\label{eq:X1_revisited}\n\\end{equation}\nwith\n\\begin{equation}\n\\mathbf{X}^{(0)}={\\mathbf{Q}_i}^{\\rm T}\\rbr{\\begin{array}{cc} 0 \\\\ & \\exp\\cbr{2{\\rm i}{\\rm diag}\\rbr{\\mathbf{P}_i^\\prime\\thetakv{i}}} \\end{array}}\\mathbf{Q}_i.\n\\label{eq:X0_def}\n\\end{equation}\nNow, consider the integral\n\\begin{align}\n&\\frac{1}{\\rbr{2\\pi}^{J}}\\intg{}{}{^J\\delta\\Jkj{m}{}}\\intg{}{}{^J\\delta\\thetakj{m}{}}\\exp\\Bigg\\{-\\frac{1}{2}\\rbr{\\begin{array}{c} \\delta\\thetakv{m+1} \\\\ \\delta\\Jkv{m+1} \\end{array}}{\\mathbf{O}^{(m+1)}}^{\\rm T}\\rbr{\\begin{array}{cc} \\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m+1)}}{{\\phikv{m+1}}^2} & \\mathbf{I}_{J} \\\\ \\mathbf{I}_{J} & \\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{{\\conjg{\\phikv{m+1}}}^2} \\end{array}}{\\mathbf{O}^{(m+1)}}\\rbr{\\begin{array}{c} \\delta\\thetakv{m+1} \\\\ \\delta\\Jkv{m+1} \\end{array}} \\nonumber \\\\\n&\\qquad-\\frac{1}{2}\\rbr{\\begin{array}{c} \\delta\\thetakv{m} \\\\ \\delta\\Jkv{m} \\end{array}}{\\mathbf{O}^{(m)}}^{\\rm T}\\rbr{\\begin{array}{cc} \\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{{\\phikv{m}}^2} & \\mathbf{I}_{J} \\\\ \\mathbf{I}_{J} & X^{(m)} \\end{array}}{\\mathbf{O}^{(m)}}\\rbr{\\begin{array}{c} \\delta\\thetakv{m} \\\\ \\delta\\Jkv{m} \\end{array}}+\\rbr{\\begin{array}{c} \\delta\\thetakv{m+1} \\\\ \\delta\\Jkv{m+1} \\end{array}}{\\mathbf{O}^{(m+1)}}^{\\rm T}\\rbr{\\begin{array}{cc} 0 & 0 \\\\ \\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{\\conjg{\\phikv{m+1}}\\partial\\phikv{m}} & 0 \\end{array}}{\\mathbf{O}^{(m)}}\\rbr{\\begin{array}{c} \\delta\\thetakv{m} \\\\ \\delta\\Jkv{m} \\end{array}} \\Bigg\\}= \\nonumber \\\\\n&\\tbr{\\det\\cbr{\\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{{\\phikv{m}}^2}\\mathbf{X}^{(m)}}}^{-1}\\exp\\tbr{-\\frac{1}{2}\\rbr{\\begin{array}{c} \\delta\\thetakv{m+1} \\\\ \\delta\\Jkv{m+1} \\end{array}}{\\mathbf{O}^{(m+1)}}^{\\rm T}\\rbr{\\begin{array}{cc} \\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m+1)}}{{\\phikv{m+1}}^2} & \\mathbf{I}_{J} \\\\ \\mathbf{I}_{J} & \\mathbf{X}^{(m+1)} \\end{array}}{\\mathbf{O}^{(m+1)}}\\rbr{\\begin{array}{c} \\delta\\thetakv{m+1} \\\\ \\delta\\Jkv{m+1} \\end{array}}}\n\\label{eq:k-th_integration_fermionic_reduced_propagator}\n\\end{align}\nwith\n\\begin{equation}\n\\mathbf{X}^{(m+1)}=\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{{\\conjg{\\phikv{m+1}}}^2}+\\rbr{\\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{\\conjg{\\phikv{m+1}}\\partial\\phikv{m}}}\\mathbf{X}^{(m)}\\rbr{\\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{{\\phikv{m}}^2}\\mathbf{X}^{(m)}}^{-1}\\rbr{\\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{\\phikv{m}\\partial\\conjg{\\phikv{m+1}}}}.\n\\label{eq:recursion_X}\n\\end{equation}\nFor $m=1$ this is exactly the integral in eq.~(\\ref{eq:reduced_propagator_phi-phistar-basis}) after integrating out $\\delta\\thetakv{0}$ and thus defines $\\mathbf{X}^{(2)}$. One then recognizes, that after the $m$-th integration, the integral is again of the form of eq.~(\\ref{eq:k-th_integration_fermionic_reduced_propagator}) up to the $(M-1)$-th integration. With this observation, the semiclassical amplitude is given by\n\\begin{align}\n\\mathcal{A}_\\gamma=&\\lim\\limits_{\\pisteps\\to\\infty}\\frac{1}{\\rbr{2\\pi}^{\\frac{3N-1}{2}}}\\intg{}{}{^N\\Jkj{M}{}}\\intg{}{}{^N\\thetakj{M}{}}\\prodl{m=0}{M-1}\\sqrt{\\det\\rbr{\\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{{\\phikv{m}}^2}\\mathbf{X}^{(m)}}}^{-1} \\nonumber \\\\\n&\\qquad\\exp\\tbr{-\\frac{1}{2}\\rbr{\\begin{array}{c} \\delta\\thetakv{M}\\mathbf{P}_f \\\\ \\delta\\Jkv{M}\\mathbf{P}_f \\end{array}}{\\mathbf{O}^{(M)}}^{\\rm T}\\rbr{\\begin{array}{cc} \\exp\\cbr{-2{\\rm i}{\\rm diag}\\rbr{\\thetakv{M}}} & \\mathbf{I}_{J} \\\\ \\mathbf{I}_{J} & X^{(M)} \\end{array}}\\mathbf{O}^{(M)}\\rbr{\\begin{array}{c} \\mathbf{P}_f^{\\rm T}\\delta\\thetakv{M} \\\\ \\mathbf{P}_f^{\\rm T}\\delta\\Jkv{M} \\end{array}}} \\nonumber \\\\\n=&\\lim\\limits_{\\pisteps\\to\\infty}\\frac{1}{\\sqrt{2\\pi}^{N-1}}\\cbr{\\prodl{m=0}{M-1}\\sqrt{\\det\\rbr{\\mathbf{I}_{J}-\\frac{{\\rm i}\\tau}{\\hbar}\\pdiff{^2{H^{(cl)}}^{(m)}}{{\\phikv{m}}^2}\\mathbf{X}^{(m)}}}^{-1}}\\sqrt{\\det\\rbr{\\mathbf{I}_{N}-\\exp\\cbr{-2{\\rm i}{\\rm diag}\\rbr{\\mathbf{P}_{f}\\thetakv{M}}}\\mathbf{P}_f\\mathbf{X}^{(M)}\\mathbf{P}_f^{\\rm T}}}^{-1}.\n\\label{eq:semiclassical_amplitude_preliminary}\n\\end{align}\n\\end{widetext}\nIn the continuous limit, the discrete set of $\\mathbf{X}^{(m)}$ turns into a function of time $\\mathbf{X}(t)$, and (by expanding it up to first order in $\\tau$) is given by eq.~(\\ref{eq:deq_X}), and the semiclassical amplitude can be written in the form given in eq.~(\\ref{eq:semiclassical_prefactor_intermediately}).\n\\section{Possible Classical Hamiltonians}\n\\label{app:classical_hamiltonians}\nIn this part, we state different possibilities for the classical hamiltonian as can be derived out of similar calculations as in appendix \\ref{app:pathintegral} without going furhter into detail.\n\\subsection{Classical Hamiltonians in the particle picture}\n\\label{subapp:particle-picture}\nFirst, we present two possibilities arising directly from the derivation presented in appendix \\ref{app:pathintegral}, but restrict ourselves to those, which contain $\\gV{\\mu}$ and $\\gV{\\phi}$ in a symmetric way and omitting the one already stated in sec.~\\ref{sec:path_integral}. These examples shall just illustrate, which kinds of classical Hamiltonians are possible:\n\\begin{align}\nH_{cl}^{(1)}&\\left(\\conjg{\\gV{\\mu}},\\gV{\\phi}\\right)= \\nonumber \\\\\n&\\sul{\\alpha}{}h_{\\alpha\\alpha}\\conjg{\\mu_\\alpha}\\phi_\\alpha\\cos\\left(\\conjg{\\mu_\\alpha}\\phi_\\alpha\\right)+\\sul{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}{}U_{\\alpha\\beta}\\conjg{\\mu_\\alpha}\\conjg{\\mu_\\beta}\\phi_\\alpha\\phi_\\beta \\nonumber \\\\\n&+\\sul{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}{}h_{\\alpha\\beta}\\conjg{\\mu_\\alpha}\\phi_\\beta\\exp\\left(-\\sul{l=\\min\\left(\\alpha,\\beta\\right)}{\\max\\left(\\alpha,\\beta\\right)}\\conjg{\\mu_l}\\phi_l\\right),\n\\label{eq:classical_hamiltonian_1}\n\\end{align}\n\\begin{align}\nH_{cl}^{(2)}&\\left(\\conjg{\\gV{\\mu}},\\gV{\\phi}\\right)= \\nonumber \\\\\n&\\sul{\\alpha}{}h_{\\alpha\\alpha}\\conjg{\\mu_\\alpha}\\phi_\\alpha\\exp\\left(\\conjg{\\mu_\\alpha}\\phi_\\alpha\\right) \\nonumber \\\\\n&+\\sul{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}{}h_{\\alpha\\beta}\\conjg{\\mu_\\alpha}\\phi_\\beta\\exp\\left(-\\conjg{\\mu_\\beta}\\phi_\\beta-\\conjg{\\mu_\\alpha}\\phi_\\alpha\\right) \\nonumber \\\\\n&\\qquad\\qquad\\times\\prodl{l=\\min\\left(\\alpha,\\beta\\right)+1}{\\max\\left(\\alpha,\\beta\\right)-1}\\left[1-\\sinh\\left(2\\conjg{\\mu_l}\\phi_l\\right)\\right] \\nonumber \\\\\n&+\\sul{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}{}U_{\\alpha\\beta}\\conjg{\\mu_\\alpha}\\conjg{\\mu_\\beta}\\phi_\\alpha\\phi_\\beta\\cosh\\left(\\conjg{\\mu_\\alpha}\\phi_\\alpha\\right)\\cosh\\left(\\conjg{\\mu_\\beta}\\phi_\\beta\\right),\n\\label{eq:classical_hamiltonian_2}\n\\end{align}\nNext, consider the more general case, that the quantum Hamiltonian is written in the form\n\\begin{equation}\n\\hat{H}=\\sul{\\alpha,\\beta}{}h_{\\alpha\\beta}\\hat{c}_\\alpha^\\dagger\\hat{c}_\\beta^{}+\\sul{\\substack{\\alpha,\\beta,\\rho,\\nu \\\\ \\alpha\\neq\\beta,\\rho\\neq\\nu}}{}U_{\\alpha\\beta\\rho\\nu}\\hat{c}_\\alpha^\\dagger\\hat{c}_\\beta^\\dagger\\hat{c}_\\rho^{}\\hat{c}_\\nu.\n\\end{equation}\nBy splitting the interaction term also into (pairwise) diagonal and non-diagonal terms, one can in a similar way as in sec.~\\ref{app:pathintegral} construct the following classical Hamiltonian\n\n\\begin{widetext}\n\\begin{align}\nH_{cl}\\left(\\conjg{\\gV{\\mu}},\\gV{\\phi}\\right)=&\\sul{\\alpha}{}h_{\\alpha\\alpha}\\conjg{\\mu_\\alpha}\\phi_\\alpha f_1\\left(\\conjg{\\mu_\\alpha},\\phi_\\alpha\\right)+\\sul{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}{}h_{\\alpha\\beta}\\conjg{\\mu_\\alpha}\\phi_\\beta f_2\\left(\\conjg{\\mu_\\alpha},\\phi_\\alpha\\right)\\exp\\left(-\\conjg{\\mu_\\beta}\\phi_\\beta\\right)\\prodl{l=\\min\\left(\\alpha,\\beta\\right)+1}{\\max\\left(\\alpha,\\beta\\right)-1}g\\left(\\conjg{\\mu_l},\\phi_l\\right) \\nonumber \\\\\n&+\\sul{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}{}U_{\\alpha\\beta\\beta\\alpha}\\conjg{\\mu_\\alpha}\\conjg{\\mu_\\beta}\\phi_\\alpha\\phi_\\beta f_3\\left(\\conjg{\\mu_\\alpha},\\phi_\\alpha\\right)f_3\\left(\\conjg{\\mu_\\beta},\\phi_\\beta\\right) \\nonumber \\\\\n&+\\sul{\\substack{\\alpha,\\beta,\\rho \\\\ \\alpha\\neq\\beta,\\rho\\neq\\alpha,\\rho\\neq\\beta}}{}\\cbr{\\Theta\\rbr{\\beta-\\alpha}\\Theta\\rbr{\\beta-\\rho}+\\Theta\\rbr{\\alpha-\\beta}\\Theta\\rbr{\\rho-\\beta}-\\Theta\\rbr{\\alpha-\\beta}\\Theta\\rbr{\\beta-\\rho}-\\Theta\\rbr{\\beta-\\alpha}\\Theta\\rbr{\\rho-\\beta}} \\nonumber \\\\\n&\\qquad\\qquad\\rbr{U_{\\alpha\\beta\\beta\\rho}-U_{\\alpha\\beta\\rho\\beta}}\\conjg{\\mu_\\alpha}\\conjg{\\mu_\\beta}\\phi_\\beta\\phi_\\rho f_1\\rbr{\\conjg{\\mu_\\alpha},\\phi_\\alpha}f_2\\rbr{\\conjg{\\mu_\\alpha},\\phi_\\alpha}\\exp\\rbr{-\\conjg{\\mu_\\rho}\\phi_\\rho}\\prodl{j=\\min\\rbr{\\alpha,\\rho}+1}{\\max\\rbr{\\alpha,\\rho}-1}g\\rbr{\\conjg{\\mu_j},\\phi_j} \\nonumber \\\\\n&+\\sul{\\substack{\\alpha,\\beta,\\rho \\\\ \\alpha\\neq\\beta,\\rho\\neq\\alpha,\\rho\\neq\\beta}}{}\\cbr{\\Theta\\rbr{\\beta-\\alpha}\\Theta\\rbr{\\rho-\\alpha}+\\Theta\\rbr{\\alpha-\\beta}\\Theta\\rbr{\\alpha-\\rho}-\\Theta\\rbr{\\alpha-\\beta}\\Theta\\rbr{\\rho-\\alpha}-\\Theta\\rbr{\\beta-\\alpha}\\Theta\\rbr{\\alpha-\\rho}} \\nonumber \\\\\n&\\qquad\\qquad\\rbr{U_{\\alpha\\beta\\rho\\alpha}-U_{\\alpha\\beta\\alpha\\rho}}\\conjg{\\mu_\\alpha}\\conjg{\\mu_\\beta}\\phi_\\alpha\\phi_\\rho f_1\\rbr{\\conjg{\\mu_\\alpha},\\phi_\\alpha}f_2\\rbr{\\conjg{\\mu_\\beta},\\phi_\\beta}\\exp\\rbr{-\\conjg{\\mu_\\rho}\\phi_\\rho}\\prodl{j=\\min\\rbr{\\beta,\\rho}+1}{\\max\\rbr{\\beta,\\rho}-1}g\\rbr{\\conjg{\\mu_j},\\phi_j} \\nonumber \\\\\n&+\\sul{\\substack{\\alpha,\\beta,\\rho,\\nu \\\\ \\alpha\\neq\\beta,\\alpha\\neq\\rho,\\alpha\\neq\\nu,\\beta\\neq\\rho,\\beta\\neq\\nu,\\rho\\neq\\nu}}{}\\cbr{\\Theta\\rbr{\\beta-\\alpha}-\\Theta\\rbr{\\alpha-\\beta}}\\cbr{\\Theta\\rbr{\\rho-\\nu}-\\Theta\\rbr{\\nu-\\rho}}U_{\\alpha\\beta\\rho\\nu}\\conjg{\\mu_\\alpha}\\conjg{\\mu_\\beta}\\phi_\\rho\\phi_\\nu f_2\\rbr{\\conjg{\\mu_\\alpha},\\phi_\\alpha}f_2\\rbr{\\conjg{\\mu_\\beta},\\phi_\\beta} \\nonumber \\\\\n&\\qquad\\qquad\\exp\\rbr{-\\conjg{\\mu_\\rho}\\phi_\\rho-\\conjg{\\mu_\\nu}\\phi_\\nu}\\cbr{\\prodl{l=\\min\\rbr{\\alpha,\\beta,\\rho,\\nu}+1}{\\min\\big\\{\\tbr{\\alpha,\\beta,\\rho,\\nu}\\setminus\\tbr{\\min\\rbr{\\alpha,\\beta,\\rho,\\nu}}\\big\\}-1}g\\rbr{\\conjg{\\mu_j},\\phi_j}}\\cbr{\\prodl{l=\\max\\big\\{\\tbr{\\alpha,\\beta,\\rho,\\nu}\\setminus\\tbr{\\max\\rbr{\\alpha,\\beta,\\rho,\\nu}}\\big\\}+1}{\\max\\rbr{\\alpha,\\beta,\\rho,\\nu}-1}g\\rbr{\\conjg{\\mu_j},\\phi_j}},\n\\end{align}\n\\end{widetext}\nwhere $f_1$, $f_2$, $f_3$ and $g$ are again arbitrary analytic functions satisfying eqs.~(\\ref{eq:constraint_f}-\\ref{eq:constraint_g_2}). Thereby, one should notice, that\n\\[\\min\\big\\{\\tbr{\\alpha,\\beta,\\rho,\\nu}\\setminus\\tbr{\\min\\rbr{\\alpha,\\beta,\\rho,\\nu}}\\big\\}\\]\nis the second smallest number out of the set $\\tbr{\\alpha,\\beta,\\rho,\\nu}$ and\n\\[\\max\\big\\{\\tbr{\\alpha,\\beta,\\rho,\\nu}\\setminus\\tbr{\\max\\rbr{\\alpha,\\beta,\\rho,\\nu}}\\big\\}\\]\nthe second largest number out of the set $\\tbr{\\alpha,\\beta,\\rho,\\nu}$.\n\\subsection{Classical Hamiltonians in the hole picture}\n\\label{subapp:hole-picture}\nThe cases considered above, we call particle picture, since the boundary conditions are such, that $\\abs{\\phi_j}^2=1$ corresponds to the $j$-th single particle state beeing occupied, while $\\abs{\\phi_j}^2=0$ corresponds to the $j$-th single particle state beeing empty. However, the role of occupied and unoccupied states can be reversed, if eqs.~(\\ref{eq:insert_integrals_0}) are replaced by\n\n\\begin{widetext}\n\\begin{align}\n&\\int{\\rm d}^{2J}\\zetakj{0}{}\\exp\\rbr{-\\conjg{\\zetakv{0}}\\cdot\\zetakv{0}}\\cbr{\\prodl{j=1}{J}\\rbr{1+\\conjg{\\chikj{0}{j}}\\zetakj{0}{j}}}\\prodl{j=1}{J}\\rbr{\\conjg{\\zetakj{0}{j}}}^{\\nij{j}}= \\nonumber \\\\\n&\\int\\frac{{\\rm d}^{2\\rbr{J-N_i}}\\phikj{0}{}}{\\pi^{J-N_i}}\\int\\limits_{0}^{2\\pi}\\frac{{\\rm d}^{J-N_i}\\thetakj{i}{}}{\\rbr{2\\pi}^{J-N_i}}\\int{\\rm d}^{2J}\\zetakj{0}{}\\exp\\rbr{-\\conjg{\\zetakv{0}}\\cdot\\zetakv{0}-\\abs{\\phikv{0}}^2+\\conjg{\\phikv{0}}\\cdot\\mukv{0}}\\cbr{\\prodl{j=1}{J}\\rbr{\\phikj{0}{j}+\\conjg{\\chikj{0}{j}}}}\\cbr{\\prodl{j=0}{J-1}\\rbr{\\conjg{\\mukj{0}{J-j}}+\\zetakj{0}{J-j}}}\\prodl{j=1}{J}\\rbr{\\conjg{\\zetakj{0}{j}}}^{\\nij{j}},\n\\label{eq:insert_integrals_0_hole}\n\\end{align}\nwhere the integrations over $\\thetakv{i}$ and $\\phikv{0}$ run over those components, which are initially empty $\\mukj{0}{j}=\\rbr{1-\\nij{j}}\\exp\\rbr{{\\rm i}\\thetakj{i}{j}}$, as well as\n\\begin{align}\na^{(m)}=&\\int\\frac{{\\rm d}^{2J}\\mukj{m}{}}{\\pi^J}\\int\\frac{{\\rm d}^{2J}\\phikj{m}{}}{\\pi^J}\\cbr{\\prodl{j=1}{J}\\rbr{\\phikj{m}{j}}+\\conjg{\\chikj{m}{j}}}\\exp\\rbr{-\\abs{\\phikv{m}}^2-\\abs{\\mukv{m}}^2+\\conjg{\\phikv{m}}\\cdot\\mukv{m}}\\prodl{j=0}{J-1}\\cbr{\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}+\\zetakj{m}{J-j}},\n\\label{eq:insert_integral_am_hole} \\\\\nb_{\\alpha}^{(m)}=&\\int\\frac{{\\rm d}^{2J}\\mukj{m}{}}{\\pi^J}\\int\\frac{{\\rm d}^{2J}\\phikj{m}{}}{\\pi^J}\\exp\\rbr{-\\abs{\\phikv{m}}^2-\\abs{\\mukv{m}}^2+\\conjg{\\phikv{m}}\\cdot\\mukv{m}}\\conjg{\\zetakj{m}{\\alpha}}\\chikj{m-1}{\\alpha}\\left\\{\\prodl{j=0}{L-\\alpha-1}\\cbr{\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}+\\zetakj{m}{J-j}}\\right\\} \\nonumber \\\\\n&\\quad\\cbr{\\sul{k=1}{\\infty}c_k^{(1)}\\rbr{\\phikj{m-1}{\\alpha}}\\rbr{\\conjg{\\mukj{m}{\\alpha}}}^{k}+\\zetakj{m}{\\alpha}}\\left\\{\\prodl{j=J-\\alpha+1}{J-1}\\cbr{\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}+\\zetakj{m}{J-j}}\\right\\}\\cbr{\\prodl{j=1}{J}\\rbr{\\phikj{m}{j}+\\conjg{\\chikj{m}{j}}}},\n\\label{eq:insert_integral_bm_hole} \\\\\nc_{\\alpha\\beta}^{(m)}=&\\int\\frac{{\\rm d}^{2J}\\mukj{m}{}}{\\pi^J}\\int\\frac{{\\rm d}^{2J}\\phikj{m}{}}{\\pi^J}\\exp\\rbr{-\\abs{\\phikv{m}}^2-\\abs{\\mukv{m}}^2+\\conjg{\\phikv{m}}\\cdot\\mukv{m}}\\conjg{\\zetakj{m}{\\alpha}}\\chikj{m-1}{\\beta}\\tbr{\\prodl{j=0}{J-\\max\\rbr{\\alpha,\\beta}-1}\\cbr{\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}+\\zetakj{m}{J-j}}} \\nonumber \\\\\n&\\quad\\cbr{\\sul{k=1}{\\infty}c_k^{(2)}\\left(\\phikj{m-1}{\\max\\rbr{\\alpha,\\beta}}\\right)\\left(\\conjg{\\mukj{m}{\\max\\rbr{\\alpha,\\beta}}}\\right)^k+\\zetakj{m}{\\max\\rbr{\\alpha,\\beta}}}\\tbr{\\prodl{j=J-\\max\\rbr{\\alpha,\\beta}+1}{J-\\min\\rbr{\\alpha,\\beta}-1}\\cbr{\\sul{k=1}{\\infty}c_k^{(3)}\\rbr{\\phikj{m-1}{J-j}}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}+\\zetakj{m}{J-j}}} \\nonumber \\\\\n&\\quad\\cbr{\\sul{k=1}{\\infty}c_k^{(2)}\\left(\\phikj{m-1}{\\min\\rbr{\\alpha,\\beta}}\\right)\\left(\\conjg{\\mukj{m}{\\min\\rbr{\\alpha,\\beta}}}\\right)^k+\\zetakj{m}{\\min\\rbr{\\alpha,\\beta}}}\\tbr{\\prodl{j=J-\\min\\rbr{\\alpha,\\beta}+1}{J-1}\\cbr{\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}+\\zetakj{m}{J-j}}}\\cbr{\\prodl{j=1}{J}\\rbr{\\phikj{m}{j}+\\conjg{\\chikj{m}{j}}}}\n\\label{eq:insert_integral_cm_hole} \\\\\nd_{\\alpha\\beta}^{(m)}=&\\int\\frac{{\\rm d}^{2J}\\mukj{m}{}}{\\pi^J}\\int\\frac{{\\rm d}^{2J}\\phikj{m}{}}{\\pi^J}\\exp\\rbr{-\\abs{\\phikv{m}}^2-\\abs{\\mukv{m}}^2+\\conjg{\\phikv{m}}\\cdot\\mukv{m}}\\cbr{\\prodl{j=1}{J}\\rbr{\\phikj{m}{j}+\\conjg{\\chikj{m}{j}}}}\\conjg{\\zetakj{m}{\\alpha}}\\conjg{\\zetakj{m}{\\beta}}\\chikj{m-1}{\\beta}\\chikj{m-1}{\\alpha} \\nonumber \\\\\n&\\quad\\tbr{\\prodl{j=0}{J-\\max\\rbr{\\alpha,\\beta}-1}\\cbr{\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}+\\zetakj{m}{J-j}}}\\cbr{\\sul{k=1}{\\infty}c_k^{(4)}\\left(\\phikj{m-1}{\\max\\rbr{\\alpha,\\beta}}\\right)\\left(\\conjg{\\mukj{m}{\\max\\rbr{\\alpha,\\beta}}}\\right)^k+\\zetakj{m}{\\max\\rbr{\\alpha,\\beta}}} \\nonumber \\\\\n&\\quad\\tbr{\\prodl{j=J-\\max\\rbr{\\alpha,\\beta}+1}{J-\\min\\rbr{\\alpha,\\beta}-1}\\cbr{\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}+\\zetakj{m}{J-j}}}\\cbr{\\sul{k=1}{\\infty}c_k^{(4)}\\left(\\phikj{m-1}{\\min\\rbr{\\alpha,\\beta}}\\right)\\left(\\conjg{\\mukj{m}{\\min\\rbr{\\alpha,\\beta}}}\\right)^k+\\zetakj{m}{\\min\\rbr{\\alpha,\\beta}}} \\nonumber \\\\\n&\\quad\\tbr{\\prodl{j=J-\\min\\rbr{\\alpha,\\beta}+1}{J-1}\\cbr{\\sul{k=1}{\\infty}\\frac{1}{k!}\\rbr{\\conjg{\\mukj{m}{J-j}}}^{k}\\rbr{\\phikj{m-1}{J-j}}^{k-1}+\\zetakj{m}{J-j}}},\n\\label{eq:insert_integral_dm_hole}\n\\end{align}\nInserting the integrals like this results in the following path integral:\n\\begin{align}\nK\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)=&\\left[\\prod\\limits_{j:n_j^{(i)}=0}^{}\\int\\limits_{0}^{2\\pi}\\frac{{\\rm d}\\theta_j^{(0)}}{2\\pi}\\exp\\left(-{\\rm i}\\theta_j^{(0)}\\right)\\right]\\left[\\prod\\limits_{m=1}^{M-1}\\prod\\limits_{j}^{}\\int\\limits_{\\mathbb{C}}^{}\\frac{{\\rm d}\\phi_j^{(m)}}{\\pi}\\exp\\left(-\\left|\\phi_j^{(m)}\\right|^2\\right)\\right]\\left[\\prod\\limits_{j:n_j^{(f)}=0}^{}\\int\\limits_{\\mathbb{C}}^{}\\frac{{\\rm d}\\phi_j^{(M)}}{\\pi}\\phi_j^{(M)}\\exp\\left(-\\left|\\phi_j^{(M)}\\right|^2\\right)\\right] \\nonumber \\\\\n&\\qquad\\exp\\left\\{\\sum\\limits_{m=1}^{M}\\left[{\\gV{\\phi}^{(m)}}^\\ast\\cdot\\gV{\\phi}^{(m-1)}-\\frac{{\\rm i}\\tau}{\\hbar}H_{cl}\\left({\\gV{\\phi}^{(m)}}^{\\ast},\\gV{\\phi}^{(m-1)}\\right)\\right]\\right\\},\n\\label{eq:propagator_in_complex_path-integral_hole}\n\\end{align}\nwith the classical hamiltonian\n\\begin{align}\n&{H^{(cl)}}^{(m)}\\rbr{\\conjg{\\gV{\\mu}},\\gV{\\phi}}= \\nonumber \\\\\n&\\quad\\sul{\\alpha=1}{J}h_{\\alpha\\alpha}^{(m)}\\exp\\rbr{-\\conjg{\\mu_\\alpha}\\phi_\\alpha}+\\sul{\\substack{\\alpha,\\beta=1 \\\\ \\alpha\\neq\\beta}}{J}U_{\\alpha\\beta}^{(m)}\\exp\\rbr{-\\conjg{\\mu_\\alpha}\\phi_\\alpha-\\conjg{\\mu_\\beta}\\phi_\\beta}+\\sul{\\substack{\\alpha,\\beta=1 \\\\ \\alpha\\neq\\beta}}{J}h_{\\alpha\\beta}^{(m)}\\conjg{\\mu_\\beta}\\phi_\\alpha\\exp\\rbr{-\\conjg{\\mu_\\alpha}\\phi_\\alpha}f\\rbr{\\conjg{\\mu_\\beta},\\phi_\\beta}\\prodl{j=\\min\\rbr{\\alpha,\\beta}+1}{\\max\\rbr{\\alpha,\\beta}-1}g\\rbr{\\conjg{\\mu_j},\\phi_j},\n\\label{eq:classical_hamiltonian_mu-phi_hole}\n\\end{align}\nwhere $f$ and $g$ are arbitrary analytical functions satisfying\n\\begin{align}\n&f(0,\\phi)=1\n\\label{eq:condition_hole_f} \\\\\n&g(0,\\phi)=-1\n\\label{eq:condition_hole_g_1} \\\\\n&\\left.\\frac{\\partial}{\\partial \\conjg{\\mu}}g\\rbr{\\conjg{\\mu},\\phi}\\right|_{\\conjg{\\mu}=0}=2\\phi.\n\\end{align}\n\\end{widetext}\n\n\\section{Grassmann coherent states}\n\\label{sec:coherent_states}\nIn order to derive the path integral representation for the fermionic propagator in Fock space, we will use Grassmann coherent states in intermediate steps. They are defined as the eigenstates of the fermionic annihilation operators \\cite{negele+orland},\n\\begin{equation}\n\\hat{c}_j\\ket{\\gV{\\zeta}}=\\zeta_j\\ket{\\gV{\\zeta}}.\n\\label{eq:definition_coherent_states}\n\\end{equation}\nHere, $\\hat{c}_j$ and $\\hat{c}_j^\\dagger$ annihilates and creates, respectively, a particle in the $j$-th single particle state, two states which coincide in the orbital degrees of freedom, but differ in the spin degree of freedom are accounted for as different single particle states, and are therefore labeled by different indexes $j$. \n\nHowever, due to the antisymmetry and the Pauli exclusion principle, the eigenvalues of the coherent states have to be (complex) anticommuting numbers, called Grassmann numbers \\cite{negele+orland,grassmann_numbers}, \\textit{i.e.}~for two of these numbers $\\zeta$ and $\\chi$\n\\begin{equation}\n\\zeta\\chi=-\\chi\\zeta.\n\\label{eq:anticommutativity_grassmann}\n\\end{equation}\nThey also anticommute with the creation and annihilation operators,\n\\begin{align}\n\\zeta\\hat{c}_j^{}&=-\\hat{c}_j^{}\\zeta, & \\zeta\\hat{c}_j^{\\dagger}&=-\\hat{c}_j^{\\dagger}\\zeta,\n\\label{eq:anticommutation_grassman_creation_annihilation}\n\\end{align}\nwhile they commute with regular complex numbers. The anticommuting property also implies $\\zeta^2=0$.\n\nIntegration over a complex Grassmann number is defined by\n\\begin{align}\n&\\int{\\rm d}\\zeta^\\ast{\\rm d}\\zeta1=\\int{\\rm d}\\zeta^\\ast{\\rm d}\\zeta\\zeta=\\int{\\rm d}\\zeta^\\ast{\\rm d}\\zeta\\zeta^\\ast=0,\n\\label{eq:vanishing_grassmann-integrals} \\\\\n&\\int{\\rm d}\\zeta^\\ast{\\rm d}\\zeta\\zeta\\zeta^\\ast=1.\n\\label{eq:nonvanishing_grassmann-integrals}\n\\end{align}\nWith the properties of the Grassmann numbers, it is possible to show that the fermionic coherent states are given by \\cite{negele+orland}\n\\begin{equation}\n\\ket{\\gV{\\zeta}}=\\exp\\left(-\\frac{1}{2}\\gV{\\zeta}^{\\ast}\\cdot\\gV{\\zeta}\\right)\\prod\\limits_{j}\\left(1-\\zeta_j^{}\\hat{c}_j^\\dagger\\right)\\ket{0},\n\\label{eq:construction_coherent_states}\n\\end{equation}\nwhere $\\ket{0}$ denotes the fermionic vacuum state. Moreover, they satisfy\n\\begin{align}\n&\\Braket{\\gV{\\zeta}|\\gV{\\chi}}=\\exp\\left[\\sum\\limits_{j}\\left(-\\frac{1}{2}\\zeta_j^\\ast\\zeta_j^{}-\\frac{1}{2}\\chi_j^\\ast\\chi_j^{}+\\zeta_j^\\ast\\chi_j^{}\\right)\\right],\n\\label{eq:overlap_coherent-coherent} \\\\\n&\\Braket{\\V{n}|\\gV{\\zeta}}=\\exp\\left(-\\frac{1}{2}\\gV{\\zeta}^{\\ast}\\cdot\\gV{\\zeta}\\right){\\prod\\limits_{j}}^\\prime\\zeta_j^{n_j}, \\\\\n&\\int{\\rm d}\\gV{\\zeta}^\\ast\\int{\\rm d}\\gV{\\zeta}\\ket{\\gV{\\zeta}}\\bra{\\gV{\\zeta}}=1,\n\\label{eq:overlap_coherent-fock}\n\\end{align}\nwith $\\ket{\\V{n}}$ being an arbitrary Fock state, such that $n_j\\in\\{0,1\\}$ is the occupation of the $j$-th single particle state. The prime at the product indicates that the order of the individual factors is reversed, \\textit{i.e.}~the factor corresponding to the largest possible value is the most left one, while the $j=1$ term is the most right one.\n\n\\section{Conclusions}\n\\label{sec:conclusion}\nWe presented a rigorous derivation of fermionic path integrals representing quantum transition amplitudes in Fock space in terms of unrestricted, commuting complex fields. In the context of semiclassical approaches we believe that this result represents an important improvement over previous approaches. First, we replace the anticommuting (Grassmann) variables, usually assumed to be the most natural representation of a fermionic path integral, by complex variables in the path integral. In this way, the propagator can be given a direct physical interpretation as a complex-valued amplitude. Second, the path integral is unrestricted (defined over the whole complex plane) and therefore avoids the complications due to the definition of path integrals in compact phase spaces.\n\nMost notably, in the approach presented here a Hamiltonian classical limit can be identified which leads to real actions and therefore explicit interference. After applying the stationary phase approximations to the path integral. In the semiclassical limit (of large particle number), we are able to derive as our major result a van Vleck-Gutzwiller type propagator for fermionic quantum fields.  \n\nIn contrast to the approaches of \\cite{miller_classical,miller_semiclassical}, here the semiclassical approximation as well as the classical limit are obtained from an exact path integral. However, there is still a freedom of choice for the classical Hamiltonian, which should be investigated further. Hence we do not exclude the possibility, that by a certain choice, the classical limits of \\cite{miller_classical,miller_semiclassical} can be recovered. Moreover it remains to be explored, which classical limit is best suited for calculations and simulations. This may actually even depend on the actual problem at hand.\n\nIn Sec.~\\ref{sec:transition_prob} we applied our results to the calculation of transition probabilities in the fermionic Fock space, and found a rich dependence of many-body interference effects on the universality class of the system. For systems with spin-orbit interaction that belong to the symplectic class, our results predict the exact cancellation of the transition probability between time-reversed many-body states, if the total number of particles is odd. This prediction that can be independently demonstrated to be a consequence of Kramer's degeneracy, is a very stringent test for the correctness of our approach. If the total particle number is even, however, the same transition is not only allowed, but its probability is enhanced by a factor of two compared to the transitions to other states. For systems without spin-flip mechanisms, we recover the coherent backscattering previously found for bosons \\cite{cbs_fock}. Upon destroying time reversal symmetry all this effects vanish, and the transition probability profile can be assumed to be more or less constant for all Fock states.\n\nFinally, we would like to note that, although the path integral eq.~(\\ref{eq:propagator_in_complex_path-integral}) is restricted to the particle picture, \\textit{i.e.}~to the case that a particle is defined through an occupied single particle state, it is also possible to construct a path integral in the hole picture (for more details see appendix \\ref{subapp:hole-picture}), where a particle is defined as an unoccupied single particle state.\n\nThe {\\it major} principle restriction of applicability of our approach is that the number of fermions $N \\gg 1$ should be large enough (our experience in the bosonic case indicates that $N \\sim 10$ is enough). Therefore, within this regime, electronic systems such as quantum dots, coupled discrete systems like spin chains modeled by Heisenberg or Ising type Hamiltonians, and molecular systems described by a discrete set of single-particle orbitals can be addressed. Still, then exist practical limitations of semiclassical approaches in concrete applications, related, \\textit{e.g.}, to the solution of the shooting problem and the correct evaluation of amplitudes and Maslov indexes. We hope that our approach is still beneficial for the Chemical Physics community.\n\nFinally, we remark that for treating emergent universal quantum fluctuations in mesoscopic systems we only need to verify that the classical limit displays chaotic behavior, a substantially easier task.\n\nFurther applications of the semiclassical methods along the lines presented here like the description of many-body spin echoes \\cite{TomS} are presently under investigation.\n\n\\section{Introduction}\n\\label{sec:intro}\nSemiclassical techniques attempt to describe quantum phenomena using only classical information as input (besides $\\hbar$), but keeping all the kinematical and interpretational aspects of quantum mechanics untouched. Semiclassical methods should therefore be distinguished from quasi-classical approaches, which are based on the quantum-classical correspondence and do not only use classical information, but also try to export classical concepts to approximate quantum mechanics. The epitome of the quasi-classical approach is the use of the Ehrenfest theorem to approximate the quantum mechanical evolution of wave packets, with systematic corrections given by the Wigner-Moyal expansion \\cite{Brack+Bhaduri}.\n\nSemiclassical methods, as understood in this contribution, attempt to link classical and quantum mechanics in a more abstract, less direct way. While for the quasi-classical program, quantum mechanics is used to construct quantities with a direct classical counterpart (like the trajectory defined by the mean position and momentum of a wavepacket), the semiclassical program employs information extracted from classical trajectories (like their actions and stabilities) to construct quantum mechanical objects. This difference becomes very explicit when we use semiclassical methods to construct quantum objects without classical analogue, such as probability amplitudes.\n\nA major goal of the semiclassical program is the construction of the semiclassical propagator $K^{{\\rm sc}}$, the asymptotic form (when $\\hbar \\to 0$) of the quantum mechanical propagator\n\\begin{equation}\nK({\\bf q},{\\bf q}^\\prime,t)=\\langle {\\bf q}|{\\rm e}^{-\\frac{i}{\\hbar}\\hat{H}t}|{\\bf q}^\\prime \\rangle,\n\\end{equation}\ndefined as the matrix element of the time-evolution operator \\cite{Sakurai}.\n\nAs reviewed in \\cite{Gutzwiller}, the challenge to construct a semiclassical propagator has a long history. Although already in 1926 it was clear for Pauli, Dirac and van Vleck that the quantum mechanical propagator can be approximated by an object of the form $K^{{\\rm sc}}\\sim {\\rm e}^{\\frac{i}{\\hbar}R}$ with the classical action $R$ appearing as a phase, it took more than forty years before Gutzwiller completed the rigorous construction of the semiclassical propagator from Feynman's path integral \\cite{Schulmann}. In its final form it reads \\cite{Gut1970}\n\\begin{equation}\nK^{\\rm sc}({\\bf q},{\\bf q}^\\prime,t)=\\sum_{\\gamma}A_{\\gamma}({\\bf q},{\\bf q}^\\prime,t){\\rm e}^{\\frac{i}{\\hbar}R_{\\gamma}({\\bf q},{\\bf q}^\\prime,t)+i\\mu_{\\gamma}\\frac{\\pi}{2}}\n\\label{eq:vvG-propagator_first_quantization}\n\\end{equation}\nwhere the sum extends over the set of solutions $\\gamma$ of the classical problem to join the classical configurations ${\\bf q}^\\prime$ and ${\\bf q}$ in time $t$. As envisioned by Dirac, $R$ is the classical action of the trajectory, while $A$ is related to its variations with respect to the initial and final configurations, and $\\mu$ is the number of focal points of the trajectory $\\gamma$. \n\nThe derivation of the  van Vleck-Gutzwiller propagator marks the starting point of modern semiclassical methods \\cite{Brack+Bhaduri,Gutzwiller,Haake}. They have been able not only to capture but also to successfully describe interference phenomena, \\textit{i.e.}~wave effects impossible to describe using quasi-classical techniques.\n\nBy Fourier-transforming $K^{{\\rm sc}}$ we get the semiclassical (Gutzwiller) Green's function, the starting point to describe stationary properties of quantum systems in the semiclassical limit, and in particular to understand the emergence of universal fluctuations in the spectra and eigenfunctions of classically chaotic quantum systems \\cite{Gutzwiller,Haake}. Also, the early semiclassical notion of the theory of molecular collisions \\cite{molecular_collisions} and related approaches in mesoscopic condensed matter to describe quantum transport \\cite{weaklocalizationBaranger,quantumchaostransportKlaus} (for reviews see \\cite{Klaus,Rodolfo,bookdaniel}) connect the van-Vleck propagator, or the semiclassical Green function, with the single-particle S-matrix in terms of transition amplitudes for transmission and reflection.\n\nThe success of the semiclassical methods has been restricted, however, predominantly to quantum systems that admit a first-quantization description. In fact, the generalization of the van Vleck-Gutzwiller propagator to describe systems of interacting particles does not pose any conceptual challenge, as the classical limit of the theory is very well understood. The semiclassical propagator is now an established tool to describe quantum dynamics of molecular systems \\cite{miller_s-matrix,cellular_dynamics,nonadiabatic,ivr} and mesoscopic electronic systems \\cite{Ullmo}.\n\nTechnical, but not conceptual, problems arise when indistinguishability comes into play. Here, the semiclassical calculation of ground and (doubly) excited states in helium by Greg Ezra et.~al.~\\cite{helium_semiclassical} marks a successful step in coping with strongly interacting two-electron dynamics. The number of classical paths we need to construct to calculate the transition amplitude between different (anti-) symmetrized configurations of a quantum system however grows extremely fast with the number of particles \\cite{Quirin}. The same vast increase of the number of classical trajectories that have to be taken into account, affects the coupled coherent state approach \\cite{ccs}, which has been developed for the treatment of fermionic many-body systems in phase space. In this approach, the wave function is expanded in a (large) set of Slater determinants of single-particle coherent states with randomly selected intitial conditions. The coherent states are then evolved along the corresponding classical trajectory.\n\nMoreover, for fermionic systems with spin orbit interactions, hybrid semiclassical approaches exist, which describe the orbital motion of non-interacting particles in phase space, while the spin is treated in a second quantized approach using spin coherent states \\cite{Paul,semiclassical_soi_strong_coupling1,*semiclassical_soi_strong_coupling2,semiclassical_soi_transport,Bolte2001,semiclassical_soi_trace-formula1,semiclassical_soi_trace-formula2,semiclassical_weak_antilocalization}\n\nImportantly, the emergence of mean-field behavior, an expected simplification of the description when the number of particles is large, cannot be rigorously included in a natural way if one sticks to the first-quantized picture where the total number of particles $N$ is not defined by the quantum many body state but is an external parameter determining the dimensionality $D=Nd$, where $d$ is the spatial dimension, of the system and thereby fixing the structure of the very space where the system lives.\n\nThese remarks indicate already a possible solution of the problem. If a second-quantized picture in Fock space is adopted instead, both quantum indistinguishability and flexibility in the number of particles are automatically included at the kinematic level: the Fock space of quantum states is by definition spanned by states which are correctly (anti-) symmetrized, and the number of particles is simply another observable represented by a hermitian operator \\cite{negele+orland}. When invoking a Fock space description, this change of perspective implies for the semiclassical program that particles appear as an emergent concept, derived from the more fundamental degree of freedom: the quantum field \\cite{Weinberg}.\n\nThe development of a semiclassical program for bosonic fields has received powerful impact from the experimental realization of their discrete version in the context of cold-atom physics \\cite{Lewenstein}. In fact, the theoretical model that describes microscopically a system of interacting bosons on a lattice, the so-called Bose-Hubbard model \\cite{coldatoms-review}, is a special realization of an interacting bosonic field. Here, again, the complementarity between quasi-classical and semiclassical approaches has been apparent. Quasi-classical methods as the ones used in \\cite{Aguiar+Barangar} work well as long as quantum interference does not come into play and eventually dominates the dynamics. However, a rigorous derivation of the van Vleck-Gutzwiller propagator in bosonic Fock space was achieved only recently \\cite{cbs_fock}.\n\nIt is fair to say that the situation in the fermionic case is more desperate. Already a quasi-classical approach faces a fundamental problem: how to define a sensible classical limit if the fermionic fields must obey the Pauli principle and therefore admit only non-commutative descriptions? The attempts and achievements to associate commuting variables to fermionic operators, that spans from the 1970's well into the 2010's, are still lacking a rigorous microscopic derivation, indicating the complexity of the problem \\cite{Klauder,miller_classical_spin-analogy,miller_classical,Grossmann}. The importance of the Chemical Physics community in this program has been obvious: electronic degrees of freedom are fundamental in the realm of molecular reactions. Moreover, chemical reactions require, in principle, simulations with anticommuting variables.\n\nIn order to avoid these anticommuting variables, in a series of important papers, Miller and collaborators proposed to use a heuristic generalization of the Heisenberg prescription \\cite{miller_classical_spin-analogy,miller_classical,miller_semiclassical} to construct the classical limit of fermionic degrees of freedom (for recent applications see \\cite{Anderson_impurity,molecular_transport}). It is a remarkable and valuable feature of this approach that it associates correct signs to expressions involving anticommuting fermionic operators $\\hat{c},\\hat{c}^{\\dagger}$ and respects the Pauli principle. In the simplest example, these key features can be seen in the mapping $F \\to F^{{\\rm cl}}$ between operators $F(\\hat{c},\\hat{c}^{\\dagger})$ and classical phase space functions $F^{{\\rm cl}}(\\sqrt{n}{\\rm e}^{i\\theta},\\sqrt{n}{\\rm e}^{-i\\theta})$, which gives for $i \\ne j$\n\\begin{eqnarray}\n\\label{eq:Miller}\n\\hat{c}_{i}^{\\dagger}\\hat{c}_{j} &\\to& \\sqrt{n_{i}n_{j}(1-n_{i})(1-n_{j})}{\\rm e}^{-i(\\theta_{i}-\\theta_{j})}, \\\\\n\\hat{c}_{j}\\hat{c}^{\\dagger}_{i} &\\to& -\\sqrt{n_{i}n_{j}(1-n_{i})(1-n_{j})}{\\rm e}^{-i(\\theta_{i}-\\theta_{j})}. \\nonumber\n\\end{eqnarray}\nThe (in general continuous) classical phase space variables $0 \\le n \\le 1$ are naturally interpreted as classical fermionic occupation numbers with the angles $\\theta$ as their corresponding canonically conjugated variables.\n\nHowever, as it is obvious from eq.~(\\ref{eq:Miller}), the thus  classical Hamiltonian obtained in this way has the physical Fock states, defined by\n\\begin{equation}\nn_{i}=0 {\\rm \\ or \\ } 1 {\\rm \\ for \\ all \\ } i,\n\\end{equation}\nas fixed points of the dynamics and the corresponding semiclassical propagator is then trivially incorrect in the relevant case where it connects physical Fock states. Moreover, as discussed at length in Sec.~\\ref{sec:path_integral}, approaching the classical limit from the quantum side by means of a formal path integral in terms of the fermionic states (introduced by Klauder \\cite{Klauder}),\n\\begin{equation}\n|b\\rangle=b|0\\rangle +\\sqrt{1-|b|^{2}}\\hat{c}^{\\dagger}|0\\rangle {\\rm \\ \\ , \\ \\ with \\ complex \\ } b,\n\\end{equation}\nshows that eq.~(\\ref{eq:Miller}) can be rigorously obtained from an exact path integral representation in terms of the commuting fields $b$. This indicates that in a representation where eq.~(\\ref{eq:Miller}) holds, the quantum mechanical propagation between Fock states is {\\it not} supported by classical trajectories and the semiclassical limit is problematic.   \n\nThis complication may be due to the fact that in Klauder's representation the path integral is {\\it restricted}, namely, the integration over the variables $b$ are defined inside the unit disk instead of over the whole complex plane. A heuristic incorporation of Langer corrections proposed in \\cite{miller_semiclassical},\n\\begin{equation}\n\\sqrt{n(1-n)} \\to \\sqrt{\\left(n+\\frac{1}{2}\\right)\\left(\\frac{3}{2}-n\\right)},\n\\end{equation}\nlifts the problem and actually leads to a classical limit that gives, for example, agreement with first-order quantum perturbation theory by using classical perturbation theory.\n\nAs this volume commemorates Greg Ezra's contributions to the description of atomic and molecular dynamics, we would like to mention that Ezra's pioneering work on the Langer correction to the semiclassical propagator \\cite{Ezra} could possibly provide the key to make rigorous the promising proposal presented in \\cite{miller_classical}. It is then tempting to check whether Ezra's insight into Langer corrections within the path integral formalism in first-quantized systems with would help to make Miller's approach justified from first-principles \\cite{us}.\n\nHere we follow a different route and present what we believe to be the first microscopic derivation of the exact propagator between $N$-particle fermionic Fock states in terms of path integrals over commuting, unrestricted classical fields. Our path integral not only incorporates and generalizes Miller's mapping $F \\to F^{{\\rm cl}}$ \"teaching\" the classsical limit of large $N$ about anticommuting operators, but it is supported in the semiclassical limit by classical paths. No extra assumptions or corrections are required.\n\nAs we will discuss in Sec.~\\ref{sec:path_integral}, the thus derived classical Hamiltonian corresponds to an approximation of the Holstein-Primakoff transformation for a single particle in a two-level system, used in \\cite{mmst-mapping3}.\n\nAfter briefly introducing Grassmann variables in Sec.~\\ref{sec:coherent_states}, in Sec.~\\ref{sec:path_integral} we present our derivation of the exact path integral for fermionic systems. Armed with this object, in Sec.~\\ref{sec:semiclassical_approximation} we follow the typical semiclassical program: we identify both the effective Planck's constant and the classical limit of the theory from the phase of the path's amplitude in the path integral, and evaluate the path integral in stationary phase approximation to obtain a van Vleck-Gutzwiller type propagator for interacting fermionic fields. The presentation will be restricted to spin-$1\/2$ systems, although a generalization to higher spins is straight forward. Finally, in Sec.~\\ref{sec:transition_prob}, we use the thus derived semiclassical propagator to calculate the transition probability from one fermionic Fock state to another one for systems without time reversal symmetry, for systems diagonal in spin space but time reversal invariant, as well as for time reversal invariant spin-$1\/2$ systems non-diagonal in spin space. \n\nTechnical details of the derivation of our main results, namely the exact complex path integral representation of the fermionic propagator in terms of commuting fields, eq.~(\\ref{eq:propagator_in_complex_path-integral}), the classical Hamiltonian eq.~(\\ref{eq:classical_hamiltonian}) and the van Vleck propagator, eqns.~(\\ref{eq:semiclassical_propagator_before_fluctuation-integration},\\ref{eq:semiclassical_prefactor}) can be found in the appendices.\n\n\\section{The path integral in complex variables}\n\\label{sec:path_integral}\n\\subsection{Derivation}\nThe aim of this part is to derive a path integral representation of the propagator in Fock space,\n\\begin{equation}\nK\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)=\\Braket{\\V{n}^{(f)}|\\exp\\left(-\\frac{\\rm i}{\\hbar}\\hat{H}t_f\\right)|\\V{n}^{(i)}},\n\\label{eq:definition_propagator_fock}\n\\end{equation}\nto which the stationary phase approximation can be applied. Note that for simplicity of presentation, the Hamiltonian has been chosen time independent, although the following calculations are also valid for the time dependent case.\n\nThe path integral representation is usually achieved by applying the Trotter Formula \\cite{trotter}, which replaces the exponential in eq.~(\\ref{eq:definition_propagator_fock}) by the product of infinitely many propagators with an infinitesimally small time step and by inserting the unit operator between two adjacent factors. Since the resolution of unity for Fock states is given by a sum, rather than an integral, they are not suitable for the construction of a path integral. This makes the coherent states the natural choice for the representation of the unit operator. However, when applying the semiclassical approximation to the coherent state path integral, one ends up with grassmannian equations of motion. On the other hand, it is desirable to have complex equations of motion leading to a real action. In order to achieve this, one has to find a way to replace the integrals over Grassmann variables by integrals over complex ones.\n\nHere, we will give a rough description of the procedure, which allows for such a transformation from Grassmann to complex integrals. However, it turns out that some of the steps contain a certain freedom of choice. The final path integral will then depend on the individual choices made during the derivation. The derivation for the specific choice presented later in this publication, is then carried out in appendix \\ref{app:pathintegral}.\n\nAfter applying Trotter's formula \\cite{trotter} the first step is to insert \\emph{two} unit operators in terms of fermionic coherent states between two adjacent exponentials,\n\\begin{align}\nK&\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)= \\nonumber \\\\\n&\\lim\\limits_{M\\to\\infty}\\left[\\prod\\limits_{m=0}^{M}\\left(\\int{\\rm d}{\\gV{\\zeta}^{(m)}}^{\\ast}\\int{\\rm d}{\\gV{\\zeta}^{(m)}}^{}\\int{\\rm d}{\\gV{\\chi}^{(m)}}^{\\ast}\\int{\\rm d}{\\gV{\\chi}^{(m)}}^{}\\right)\\right] \\nonumber \\\\\n&\\quad\\left[\\prod\\limits_{m=0}^{M-1}\\Braket{\\gV{\\zeta}^{(m+1)}|\\exp\\left(-\\frac{{\\rm i}\\tau}{\\hbar}\\hat{H}\\right)|\\gV{\\chi}^{(m)}}\\Braket{\\gV{\\chi}^{(m)}|\\gV{\\zeta}^{(m)}}\\right] \\nonumber \\\\\n&\\quad\\Braket{\\V{n}^{(f)}|\\gV{\\chi}^{(M)}}\\Braket{\\gV{\\chi}^{(M)}|\\gV{\\zeta}^{(M)}}\\Braket{\\gV{\\zeta}^{(0)}|\\V{n}^{(i)}},\n\\label{eq:propagator_in_grassmann_path-integral}\n\\end{align}\nwhere $\\tau=t_f\/M$.\n\nNext, in order to replace the Grassmann integrals by complex ones, one has to insert complex integrals such that the overlap $\\braket{\\gV{\\chi}^{(m)}|\\gV{\\zeta}^{(m)}}$ can be written as an integral over a product of two factors, with the first one depending only on $\\gV{\\chi}^{(m)}$ and the second one on $\\gV{\\zeta}^{(m)}$. Here, integrals of the form\n\\begin{equation}\n\\int\\limits_{\\mathbb{C}}^{}{\\rm d}\\phi\\int\\limits_{\\mathbb{C}}^{}{\\rm d}\\mu\\exp\\left(-\\left|\\phi\\right|^2-\\left|\\mu\\right|^2+\\phi^{\\ast}\\mu\\right)\\phi^{k^{}}\\left(\\mu^{\\ast}\\right)^{k^{\\prime}}=\\pi^2k!\\delta_{k^{}k^\\prime}\n\\label{eq:inserted_integral}\n\\end{equation}\nwill be used, since this choice allows us to construct a path integral, which for intermediate times has the same form as the one for bosons in coherent state representation \\cite{negele+orland} (see appendix \\ref{app:pathintegral}).\n\nAfter this insertion, we can decouple $\\gV{\\zeta}^{(m+1)}$ and $\\gV{\\chi}^{(m)}$ from $\\gV{\\zeta}^{(m)}$ and $\\gV{\\chi}^{(m-1)}$ in eq.~(\\ref{eq:propagator_in_grassmann_path-integral}), such that the integrand for the propagator becomes a product, in which the $m$-th factor only depends on $\\gV{\\zeta}^{(m)}$ and $\\gV{\\chi}^{(m-1)}$. Therefore the insertion of these integrals allows us to integrate out the Grassmann variables exactly after expanding the exponential up to linear order in $\\tau$.\n\nAt this point, it is important to note that not only the choice of the inserted integrals is not unique, but that, when choosing \\textit{e.g.}~integrals of the form (\\ref{eq:inserted_integral}), there is a certain freedom in choosing the combinations of $k$ and $k^\\prime$. With the choices cf.~appendix \\ref{app:pathintegral}, one arrives at\n\n\\begin{widetext}\n\\begin{align}\nK\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)=\\left[\\prod\\limits_{j:n_j^{(i)}=1}^{}\\int\\limits_{0}^{2\\pi}\\frac{{\\rm d}\\theta_j^{(0)}}{2\\pi}\\exp\\left(-{\\rm i}\\theta_j^{(0)}\\right)\\right]\\left(\\prod\\limits_{j:n_j^{(f)}=1}^{}\\int\\limits_{\\mathbb{C}}^{}\\frac{{\\rm d}\\phi_j^{(M)}}{\\pi}\\phi_j^{(M)}\\right)\\left(\\prod\\limits_{m=1}^{M-1}\\prod\\limits_{j}^{}\\int\\limits_{\\mathbb{C}}^{}\\frac{{\\rm d}\\phi_j^{(m)}}{\\pi}\\right)\\times& \\nonumber \\\\\n\\times\\exp\\left\\{\\sum\\limits_{m=1}^{M}\\left[-\\left|\\gV{\\phi}^{(m)}\\right|^2+{\\gV{\\phi}^{(m)}}^\\ast\\cdot\\gV{\\phi}^{(m-1)}-\\frac{{\\rm i}\\tau}{\\hbar}H_{cl}\\left({\\gV{\\phi}^{(m)}}^{\\ast},\\gV{\\phi}^{(m-1)}\\right)\\right]\\right\\}&,\n\\label{eq:propagator_in_complex_path-integral}\n\\end{align}\n\\end{widetext}\n\nwhere at final time the integrals over those $\\phi_j^{(M)}$ corresponding to empty single particle states, \\textit{i.e.}~for those $j$ where $n_j^{(f)}=0$, are already evaluated exactly and therefore have to be set to zero in eq.~(\\ref{eq:propagator_in_complex_path-integral}). In fact, the integrals over those components do not even have to be inserted right from the beginning, since\n\\begin{equation}\n\\int{\\rm d}{\\chi_j^{(M)}}^\\ast\\int{\\rm d}\\chi_j^{(M)}\\exp\\left(-{\\chi_j^{(M)}}^\\ast\\chi_j^{(M)}\\right)\\left(1+{\\chi_j^{(M)}}^\\ast\\zeta_j^{(M)}\\right)=1.\n\\end{equation}\nThe exact integration over the finally unoccupied states is necessary, since the stationarity conditions will not give solutions for the phases of these components and therefore, these integrals can not be performed in a stationary phase approximation. For the same reason the integrations over those $\\phi_j^{(0)}$ with $n_j^{(i)}=0$ are already performed exactly. This means that effects due to vacuum fluctuations \\cite{Nolting}, \\textit{i.e.}~the spontaneous creation and annihilation of particles out of the vacuum, are treated exactly. Furthermore, for $m=0$, the integrations over the amplitudes $J_j^{(0)}=|\\phi_j^{(0)}|^2$ for the initially occupied single particle states $j$ are performed exactly (see appendix \\ref{app:pathintegral} for details of this exact integration). As a matter of fact, these integrals could also be included in the stationary phase approximation, which would eventually result in a multiplication of our result for the semiclassical propagator with a factor $\\alpha={\\rm e}^N\/(\\sqrt{2\\pi})^N$, where $N$ is the total number of particles, which is the $N$-th power of Stirling's approximation of $n!$ for $n=1$.\n\nNow one might raise the question, why the initial amplitudes related to occupied states are integrated out, but not the final ones. Actually, the amplitudes of $\\phi_j^{(M)}$ for occupied sites could also be integrated out, which would result in dividing the result for the semiclassical approximation by the same factor $\\alpha$. However, we choose not to perform them, in order to be in accordance with the usual first quantized semiclassical approach, where the path integral, to which the stationary phase approximation is applied, consists of one integration (over the canonical variables chosen as basis) less than those over their canonical conjugate variables. For instance, the path integral for the propagator in configuration space consists (before taking the limit $M\\to\\infty$) of $M$ momentum integrals and $M-1$ position integrals. Moreover, our choice is supported by the fact that it leads to the exact result if the quantum Hamiltonian is diagonal and non-interacting.\n\nWhen comparing the path integral with the corresponding one in first quantization, eq.~(\\ref{eq:vvG-propagator_first_quantization}), the phases $\\theta_j^{(0)}$ would correspond to the initial momenta of the path. The role of $\\gV{\\phi}^{(M)}$, however, is much more sophisticated. Its phases again correspond to the final momenta, while its amplitude should somehow correspond to the final position. Yet, the value of the latter is not fixed to $n_j^{(f)}=1$, which would be the expected boundary condition for the paths. This boundary condition is hidden in the in the integration over $\\phi_j^{(M)}$ is determined by the extra factor $\\phi_j^{(M)}$ of the integrand. In a stationary phase analysis of the integrand, which will be performed below, one finally recognizes that indeed both, the stationarity condition of phase and amplitude of $\\phi_j^{(M)}$, are required in order to get the correct boundary condition. Thus, the boundary condition at final time is indeed hidden in the full integral over $\\phi_j^{(M)}$.\n\nFinally, it should be noted that the classical Hamiltonian $H_{cl}$ is not unique, but again depends on the way chosen to construct the path integral in complex variables. There remains a certain freedom to weigh individual terms in the classical Hamiltonian differently, which might help in studying effects related to particular parts of the Hamiltonian. For instance, in the Hamiltonian given in eq.~(\\ref{eq:classical_hamiltonian_2}) in appendix \\ref{subapp:particle-picture}, the interaction, single-particle energies and the antisymmetry under particle exchange are weighted exponentially, while the Pauli principle is given by an exponential suppression of hopping processes leading to occupations of one single-particle state by more than one particle. However, due to the exponential factor in the diagonal term of the single-particle part of the Hamiltonian, processes quantum mechanically forbidden by the Pauli principle are further suppressed energetically. This energetically suppression essentially corresponds to the heuristic inclusion of a Pauli potential \\cite{Grossmann,paulipotential1,paulipotential2,paulipotential3,paulipotential4}, \\textit{i.e.}~a potential, which hinders two electrons to occupy the same single-particle state.\n\nFor the quantum Hamiltonian considered here,\n\\begin{equation}\n\\hat{H}=\\sum\\limits_{\\alpha,\\beta}h_{\\alpha\\beta}\\hat{c}_{\\alpha}^\\dagger\\hat{c}_{\\beta}^{}+\\sum\\limits_{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}^{}U_{\\alpha\\beta}\\hat{c}_{\\alpha}^\\dagger\\hat{c}_{\\beta}^\\dagger\\hat{c}_{\\beta}^{}\\hat{c}_{\\alpha}^{}.\n\\label{eq:qm_hamiltonian}\n\\end{equation}\none possible classical Hamiltonian is given by\n\\begin{align}\nH_{cl}&\\left(\\gV{\\mu},\\gV{\\phi}\\right)= \\nonumber \\\\\n\\sum\\limits_{\\alpha}&h_{\\alpha\\alpha}\\mu_{\\alpha}\\phi_{\\alpha}+\\sum\\limits_{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}U_{\\alpha\\beta}\\mu_{\\alpha}\\mu_{\\beta}\\phi_{\\alpha}\\phi_{\\beta}\n\\label{eq:classical_hamiltonian} \\\\\n+&\\sum\\limits_{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}h_{\\alpha\\beta}\\mu_{\\alpha}\\phi_{\\beta}\\exp\\left(-\\mu_{\\alpha}\\phi_{\\alpha}-\\mu_{\\beta}\\phi_{\\beta}\\right){\\prod\\limits_{j}}^{\\alpha,\\beta}\\left(1-2\\mu_j\\phi_j\\right), \\nonumber\n\\end{align}\nwhere the product in the last line runs only over those values of $j$, which are lying between $\\alpha$ and $\\beta$, excluding $\\alpha$ and $\\beta$ themselves. The case $\\gV{\\mu}=\\gV{\\phi}^{*}$, \\textit{i.e.}\n\\begin{align}\nH_{cl}&\\left(\\gV{\\phi}^{*},\\gV{\\phi}\\right)= \\nonumber \\\\\n\\sum\\limits_{\\alpha}&h_{\\alpha\\alpha}|\\phi_{\\alpha}|^{2}+\\sum\\limits_{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}U_{\\alpha\\beta}|\\phi_{\\alpha}|^{2}|\\phi_{\\beta}|^{2}\n\\label{eq:classical_hamiltonian_diagonal} \\\\\n+&\\sum\\limits_{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}h_{\\alpha\\beta}\\phi_{\\alpha}^{*}\\phi_{\\beta}\\exp\\left(-|\\phi_{\\alpha}|^{2}-|\\phi_{\\beta}|^{2}\\right){\\prod\\limits_{j}}^{\\alpha,\\beta}\\left(1-2|\\phi_{j}|^{2}\\right), \\nonumber\n\\end{align}\n\nwill be of particular importance for the continuum limit. It is instructive to compare it with the classical electron analog model (CEAM) obtained from Miller's mapping which gives in this case\n\\begin{align}\nH_{cl}^{{\\rm CEAM}}&\\left(\\gV{\\phi}^{*},\\gV{\\phi}\\right)= \\nonumber \\\\\n\\sum\\limits_{\\alpha}&h_{\\alpha\\alpha}|\\phi_{\\alpha}|^{2}+\\sum\\limits_{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}U_{\\alpha\\beta}|\\phi_{\\alpha}|^{2}|\\phi_{\\beta}|^{2}\n\\label{eq:classical_hamiltonian_Miller} \\\\\n+&\\sum\\limits_{\\substack{\\alpha,\\beta \\\\ \\alpha\\neq\\beta}}h_{\\alpha\\beta}\\phi_{\\alpha}^{*}\\phi_{\\beta}\\sqrt{(1-|\\phi_{\\alpha}|^{2})(1-|\\phi_{\\beta}|^{2})}{\\prod\\limits_{j}}^{\\alpha,\\beta}\\left(1-2|\\phi_{j}|^{2}\\right), \\nonumber\n\\end{align}\nin terms of the, now {\\it restricted}, variables $\\phi_{\\alpha}$ with $|\\phi_{\\alpha}|^{2} \\le 1$.\n\nIn eq.~(\\ref{eq:classical_hamiltonian}), the factors $1-2\\mu_j\\phi_j$ are a consequence of the anticommutativity of the creation and annihilation operators (and the Grassmannians) and thus account for the antisymmetry of the fermions under particle exchange. Consider for example the following two processes for the scattering of two particles in the states $1$ and $2$ into the states $2$ and $3$: in the first process, the particle in state $1$ is scattered into state $3$, with the second particle staying in state $2$, while in the second one the particle in state $2$ is scattered into state $3$ and the particle in state $1$ is scattered into state $2$. These two processes are the same up to an exchange of the two particles. Therefore, these two processes have to yield the same contribution, but with a different sign. On the other hand, if state $2$ is empty, while a particle is scattered from state $1$ to state $3$, there is no corresponding process resulting from an odd number of exchanges of particles, and thus, the contribution has always to be the same. In general, a process where a particle is scattered from state $\\alpha$ to state $\\beta$ with $\\left|\\alpha-\\beta\\right|>1$, has to be multiplied by a factor of $-1$ for each occupied state $j$ between $\\alpha$ and $\\beta$. However, classically the occupations are not restricted to $0$ and $1$, but can be any number, such that one ends up with a factor interpolating between the two extreme values $+1$ for the case without a particle in state $j$ and $-1$ for the case where state $j$ is occupied. Furthermore, the exponential in the non-diagonal part of the single particle term accounts for the Pauli principle by the exponential suppression of processes, which lead to an enhanced number of particles within one single particle state.\n\nA (certainly not complete) list of further possible classical Hamiltonians corresponding to the quantum Hamiltonian (\\ref{eq:qm_hamiltonian}) can be found in appendix \\ref{app:classical_hamiltonians}.\n\nIt is furthermore instructive to see how our approach treats the extreme case of a {\\it single} electron, $N=1$, where the state space is spanned by two discrete states and anticommutation of the fermionic fields does not play a role. In this situation, our results can be directly compared with existing exact mappings between systems with $n=2$ discrete states and a quantum top with total angular momentum $s$ such that $n=(2s+1)\/2$. In the  Chemical Physics community these so-called Meyer-Miller-Stock-Thoss (MMST) methods \\cite{miller_classical_spin-analogy,molecular_transport,mmst-mapping2,mmst-mapping3} have been successfully used to describe non-adiabatic transitions of the nuclear dynamics between two potential surfaces corresponding to two discrete many-body states of the electrons. The MMST method maps the dynamics of a two-level system into the problem of a spinning particle, which can be in turn mapped into a set of harmonic oscillators by means of the Schwinger representation of angular momentum (see \\cite{mmst-mapping3}). In this way, a classical picture for two-level systems is obtained, as a basis for standard (continuous) semiclassical approaches.\n\nOur result for the classical limit of a single electron, included in Eq.~(\\ref{eq:classical_hamiltonian_diagonal}), appears naturally within the MMST approach as an approximate version of the Holstein-Primakoff transformation, see \\cite{mmst-mapping3} for details and \\cite{Paul} for an application to spin transport. As it is also shown there, this classical limit, however, gives unsatisfactory results when used as starting point of a semiclassical calculation of the time evolution of quantum observables. This apparent drawback is fully resolved when taking into account, as shown in detail here, that the semiclassical limit where our result holds is defined by $N \\to \\infty$. Therefore, the application of our methods to the limiting case $N=1$ is expected to poorly compare with exact quantum mechanical results. However, the main motivation of the present work is to deal semiclassically with anticommuting variables, not with few discrete degrees of freedom as in \\cite{mmst-mapping3}.\n\\subsection{Comparison with CEAM and Klauder's approach}\nMiller's heuristic approach can actually be verified by extracting the classical Hamiltonian from another path integral representation. This is by extending the $b$-fermionic states introduced by Klauder in \\cite{Klauder},\n\\begin{equation}\n\\ket{b}=\\sqrt{1-\\left|b\\right|^2}\\ket{0}+b\\ket{1},\n\\label{eq:klauder}\n\\end{equation}\nto the case of multiple single-particle states and define (see also \\cite{Levein})\n\\begin{equation}\n|{\\bf b}\\rangle=\\prod_{j}\\left(\\sqrt{1-|b_{j}|^{2}}\\hat{1}+b_{j}\\hat{c}^{\\dagger}_{j}\\right)\\ket{0}.\n\\end{equation}\nThese states define an overcomplete basis for the fermionic Hilbert space, as they form the identity\n\\begin{equation}\n\\left(\\prod\\limits_{j}^{}\\int\\limits_{\\mathbb{D}}^{}\\frac{{\\rm d}b_j^{(m)}}{\\pi}\\right)\\ket{\\bf b}\\bra{\\bf b}=\\hat{1}\n\\end{equation}\nwhere $\\mathbb{D}$ denotes the unit disc in the complex plane, and therefore can be used to construct a path integral representation of the propagator in terms of paths ${\\bf b}(t)$ in the space of commuting variables ${\\bf b}$.\n\nThe steps of the derivation of the path integral in this basis correspond to those one follows to construct the fermionic path integral using coherent states \\cite{negele+orland,Klauder}. After reaching a form where the classical Hamiltonian can be read off from an action functional giving the phase of the quantum propagator, we obtain\n\\begin{equation}\nH^{{\\rm Klauder}}_{cl}({\\bf b}^{*},{\\bf b})=\\langle {\\bf b}|\\hat{H}|{\\bf b}\\rangle.\n\\label{eq:classical_hamiltonian_Klauder}\n\\end{equation}\nA short calculation finally shows that the classical Hamiltonian (\\ref{eq:classical_hamiltonian_Klauder}) obtained using Klauder's representation  is equal to Miller's, eq.~(\\ref{eq:classical_hamiltonian_Miller}), \\textit{i.e.}\n\\begin{equation}\nH^{{\\rm Klauder}}_{cl}({\\bf b}^{*},{\\bf b})=H^{{\\rm CEAM}}_{cl}({\\bf b}^{*},{\\bf b}).\n\\end{equation}\nthus providing a rigorous construction of the classical limit of the approach by Miller and coworkers \\cite{miller_classical}. \n\nIn principle, having at hand a classical Hamiltonian as the one in eq.~(\\ref{eq:classical_hamiltonian_Miller}), a semiclassical analysis of the path integral in $b$-representation along the lines presented bellow can be carried out. The first step is to consider the classical equations of motion\n\\begin{equation}\ni\\hbar\\frac{d}{dt}{\\bf b}(t)=\\frac{\\partial}{\\partial {\\bf b}^{*}}H^{{\\rm CEAM}}_{cl}({\\bf b}^{*},{\\bf b}),\n\\end{equation}\nwhich can be canonically transformed into\n\\begin{align}\ni\\hbar\\frac{d}{dt}n_{j}(t)&=\\left.\\frac{\\partial}{\\partial \\theta_{j}}H^{{\\rm CEAM}}_{cl}({\\bf b}^{*},{\\bf b})\\right|_{b=\\sqrt{n}\\exp({i\\theta})}\\\\\ni\\hbar\\frac{d}{dt}\\theta_{j}(t)&=-\\left.\\frac{\\partial}{\\partial n_{j}}H^{{\\rm CEAM}}_{cl}({\\bf b}^{*},{\\bf b})\\right|_{b=\\sqrt{n}\\exp({i\\theta})}.\n\\end{align}\nWithout loss of generality we consider the many-body Hamiltonian (\\ref{eq:qm_hamiltonian}). Inspection of the associated equations of motion readily shows that the classical occupations $n_{j}=|b_{j}|^{2}$ evolve in time only through the terms that depend on the phases $\\theta_{j}$. Here is where the classical limit $H^{{\\rm CEAM}}_{cl}({\\bf b}^{*},{\\bf b})$ is problematic: due to the presence of the \"Pauli\" factors $\\sqrt{n(1-n)}$ in eq.~(\\ref{eq:classical_hamiltonian_Miller}) we trivially obtain\n\\begin{equation}\n\\left.\\frac{d}{dt}n_{j}(t)\\right|_{n=0 {\\rm \\ or \\ } 1}=0.\n\\end{equation}\nTherefore the classical phase-space manifolds associated with the physical Fock states, which are defined by precisely the condition $n=0 {\\rm \\ or \\ } 1$, do not evolve in time and there is no way to connect the quantum and classical dynamics, neither at the quasi-classical, nor at the semiclassical level. Remarkably, the classical limit as given for example in eq.~(\\ref{eq:classical_hamiltonian}) circumvents this problem by allowing arbitrarily high classical occupation numbers, but penalizing them in a smooth (but exponentially strong) manner. \n\nIt is important to stress that there is no reason why classical occupations must be bounded, exactly as there is no reason why they have to take only integer values. In both cases we are apparently violating what is just a classical picture of the fermionic degrees of freedom. However, fermionic fields are essentially non-classical objects and we are satisfied with being able to {\\it define} a consistent classical limit by pure formal manipulations. Adopting this pragmatical point of view of defining the classical limit formally through the exact path integral, the fields $\\phi_{\\alpha}$ in eq.~(\\ref{eq:classical_hamiltonian_diagonal}) do not need to fit our expectations on how the classical limit should look like. All that we ask them for is to correctly describe the propagation between physical Fock states.\n\n\\section{Semiclassical approximation}\n\\label{sec:semiclassical_approximation}\n\nThe reason for the semiclassical approach to any quantum system to be rooted in the path integral formulation is that it accomplishes simultaneously three major goals. First, it allows us to identify the classical limit of the theory. Second, it serves as the starting point of a systematic stationary phase analysis that eventually leads to the semiclassical propagator. Third, {\\it it is in the structure of the action functional where $\\hbar_{{\\rm eff}}$ can be identified}. The effective Planck constant is not only the dimensionless parameter that defines the classical limit $\\hbar_{{\\rm eff}}\\to0$, but also the small parameter that makes the whole semiclassical approach valid. It appears non-perturbatively, if the characteristic path integral representation of the propagator,\n\\begin{equation}\nK\\sim \\int {\\cal D}[ \\cdot ]{\\rm e~}^{R[ \\cdot ]\/\\hbar},\n\\end{equation}\nis written in terms of a dimensionless action $\\tilde{R}$,\n\\begin{equation}\nK\\sim \\int {\\cal D}[ \\cdot ]{\\rm e~}^{\\tilde{R}[ \\cdot ]\/\\hbar_{{\\rm eff}}}.\n\\end{equation}\nInspection of the exponents in eq.~(\\ref{eq:propagator_in_complex_path-integral}) shows that Planck's constant $\\hbar$ actually plays a minor role in our case. Clearly, $\\hbar$ can be absorbed simply by a redefinition of the parameters of the Hamiltonian (note that this is not the case in the usual phase-space path integral). In order to identify $\\hbar_{{\\rm eff}}$, we rescale all the fields in such a way that the exponent appearing in eq.~(\\ref{eq:propagator_in_complex_path-integral}) takes the form $\\tilde{R}\/\\hbar_{{\\rm eff}}$ with $\\tilde{R}={\\cal O}(1)$. Following this recipe, eq.~(\\ref{eq:propagator_in_complex_path-integral}) leads to\n\\begin{equation}     \n\\hbar_{{\\rm eff}}=N^{-1},\n\\end{equation}\nshowing that in the present approach {\\it the classical limit corresponds to the limit of large number of particles}. In the following, we complete the stationary analysis of the exact propagator valid in this $N \\gg 1$ limit.\n\nIn eq.~(\\ref{eq:propagator_in_complex_path-integral}) all integrals, that can and should be carried out exactly, are already performed, except for the integration over the initial phase of the first occupied single particle state. This integration has to be done exactly because of the $U(1)$ gauge symmetry, \\textit{i.e.~}the freedom to multiply the wave function by an arbitrary global phase. In order to perform this integration, one first has to substitute the integrations over the real and imaginary part of $\\phi_j^{(m)}$ by those over its modulus squared $J_j^{(m)}$ and phase $\\varphi_j^{(m)}$ and then has to substitute the latter by $\\theta_j^{(m)}-\\theta_{j_1}^{(0)}$, where $j_1$ denotes the first initially occupied single particle state,\n\\begin{equation}\nj_1=\\min\\left\\{j\\in\\{1,2,\\ldots\\}:n^{(i)}_j=1\\right\\}.\n\\label{eq:definition_j1}\n\\end{equation}\nThese substitutions can be summarized as\n\\begin{equation}\n\\phi_j^{(m)}=\\sqrt{J_j^{(m)}}\\exp\\left[{\\rm i}\\left(\\theta_j^{(m)}-\\theta_{j_1}^{(0)}\\right)\\right],\n\\label{eq:replacement_complex_amplitude_phase}\n\\end{equation}\nfor all $j$ and $m\\geq1$, while for $m=0$,\n\\begin{align}\n\\phi_j^{(0)}&=n_j^{(i)}\\exp\\left[{\\rm i}\\left(\\theta_j^{(0)}-\\theta_{j_1}^{(0)}\\right)\\right] & \\text{if } j\\neq j_1, \n\\label{eq:replacement_phase_initial} \\\\\n\\phi_{j_1}^{(0)}&=\\exp\\left({\\rm i}\\theta_{j_1}^{(0)}\\right).\n\\label{eq:phi_initial_j1}\n\\end{align}\nAfter these substitutions it is easy to see that the remaining dependence of the path integral on the global phase $\\theta_{j_1}^{(0)}$ is given by $\\exp[{\\rm i}(N_f-N_i)\\theta_{j_1}^{(0)}]$, with $N_{i\/f}=\\sum_jn_j^{(i\/f)}$ being the initial, respectively, final total number of particles. Therefore, the integration over the global phase simply yields a factor $2\\pi\\delta_{N_f,N_i}$, which accounts for the conservation of the total particle number. The remaining integrals over $J_j^{(m)}$ and $\\theta_j^{(m)}$ are then performed in stationary phase approximation, where (similar to the derivation of Stirling's approximation) for consistency and in order to include the behavior of the integrand especially for small occupations correctly, it is important to include the factors\n\\begin{equation}\n\\sqrt{J_j^{(m)}}=\\exp\\left[\\log\\left(J_j^{(m)}\\right)\/2\\right]\n\\label{eq:inclusion_prefactors}\n\\end{equation}\nin the stationarity analysis. For intermediate times, $1\\leq m<M$, the stationarity conditions for $J_j^{(m)}$ and $\\theta_j^{(m)}$ can be combined to the conditions\n\\begin{align}\n{\\rm i}\\hbar\\left(\\phi_{j}^{(m)}-\\phi_j^{(m-1)}\\right)=&\\tau\\frac{\\partial H_{cl}\\left({\\gV{\\phi}^{(m)}}^\\ast,\\gV{\\phi}^{(m-1)}\\right)}{\\partial{\\phi_j^{(m)}}^\\ast},\n\\label{eq:eom_discrete_intermediate} \\\\\n-{\\rm i}\\hbar\\left({\\phi_{j}^{(m+1)}}^\\ast-{\\phi_j^{(m)}}^\\ast\\right)=&\\tau\\frac{\\partial H_{cl}\\left({\\gV{\\phi}^{(m+1)}}^\\ast,\\gV{\\phi}^{(m)}\\right)}{\\partial{\\phi_j^{(m)}}}.\n\\label{eq:eom_discrete_intermediate_conj}\n\\end{align}\nIn the same way, the conditions for $m=M$ can be written in the form of eq.~(\\ref{eq:eom_discrete_intermediate}) with $m=M$ as well as the boundary condition\n\\begin{equation}\nJ_j^{(M)}=n_j^{(f)}.\n\\label{eq:bcf_discrete}\n\\end{equation}\nNote that a linear combination of the stationarity conditions for $\\theta_j^{(M)}$ and $J_j^{(M)}$ is required to get the stationary phase conditions in this form.\n\nSince the integration over the initial phase is performed only for occupied states, and the amplitude of $\\phi_j^{(0)}$ is equal to the initial occupation of the site $n_j^{(i)}$, the stationarity condition for $\\theta_j^{(0)}$ yields eq.~(\\ref{eq:eom_discrete_intermediate_conj}) with $m=0$. When finally taking the continuous limit $\\tau\\to0$, these conditions result in the equations of motion\n\\begin{align}\n{\\rm i}\\hbar\\dot{\\gV{\\phi}}(t)&=\\frac{\\partial H_{cl}\\left(\\gV{\\phi}^\\ast(t),\\gV{\\phi}(t)\\right)}{\\partial\\gV{\\phi}^\\ast(t)},\n\\label{eq:eom} \\\\\n-{\\rm i}\\hbar\\dot{\\gV{\\phi}}^\\ast(t)&=\\frac{\\partial H_{cl}\\left(\\gV{\\phi}^\\ast(t),\\gV{\\phi}(t)\\right)}{\\partial\\gV{\\phi}(t)},\n\\label{eq:eom_conj}\n\\end{align}\nalong with the boundary conditions\n\\begin{align}\n\\left|\\phi_j(0)\\right|^2&=n_j^{(i)}, & \\left|\\phi_j(t_f)\\right|^2&=n_j^{(f)}\n\\label{eq:bc}\n\\end{align}\nwith $\\phi_{j_1}(0)=1$. It is important to note that the equations of motion (\\ref{eq:eom}) and (\\ref{eq:eom_conj}) are complex conjugates of each other, such that for $J$ single particle states we get $J$ complex (or correspondingly $2J$ real) equations of motion with $2J$ real boundary conditions. Therefore one can always find at least one solution without the complexification necessary for the bosonic coherent state propagator \\cite{Aguiar+Barangar,Garg+Braun}. Therefore, the classical Hamiltonian and action will also be real.\n\nWe also point out the key difference in the role of the boundary conditions in eq.~(\\ref{eq:bc}) when compared with the derivation of the classical limit from the path integral in the standard first-quantized case. In the later, {\\it boundary conditions are imposed at the level of the path integral} and therefore are not subject to the stationary phase conditions. Contrary to the bosonic case where this observation remains true \\cite{cbs_fock}, here again we encounter that the classical limit of fermionic fields displays counter-intuitive features: the boundary conditions (\\ref{eq:bc}) that allow for multiple solutions of (\\ref{eq:eom}, \\ref{eq:eom_conj}) are themselves obtained from a stationary phase argument, and the corresponding quantum fluctuations must be considered at the same footing as the fluctuations around the classical solutions.\n\nEvaluating the exponent of the path integral along the stationary point (including all additional phase factors originating from the boundary terms $m=1,M$) then yields the classical action\n\\begin{align}\nR_{\\gamma}&\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)=\\int\\limits_{0}^{t_f}{\\rm d}t\\left[\\hbar\\gV{\\theta}(t)\\cdot\\dot{\\V{J}}(t)-H_{cl}\\left(\\gV{\\phi}^\\ast(t),\\gV{\\phi}(t)\\right)\\right],\n\\label{eq:action}\n\\end{align}\nof the mean field trajectories defined by the equations of motion (\\ref{eq:eom}) and the boundary conditions (\\ref{eq:bc}). In eq.~(\\ref{eq:action}) the real functions $\\gV{\\theta}(t)$ and $\\V{J}(t)$ are defined through\n\\begin{equation}\n\\phi_j(t)=\\sqrt{J_j(t)}\\exp\\left({\\rm i}\\theta_j(t)\\right).\n\\label{eq:definition_J_theta_timedependent}\n\\end{equation}\nIt is worth to note, that the equations of motion (\\ref{eq:eom},\\ref{eq:eom_conj}) in these variables can also be written as the real equations\n\\begin{align}\n\\dot{\\V{J}}(t)&=\\frac{2}{\\hbar}\\frac{\\partial H_{cl}\\left(\\gV{\\phi}^\\ast(t),\\gV{\\phi}(t)\\right)}{\\partial\\gV{\\theta}(t)},\n\\label{eq:eom_J} \\\\\n\\dot{\\gV{\\theta}}(t)&=-\\frac{2}{\\hbar}\\frac{\\partial H_{cl}\\left(\\gV{\\phi}^\\ast(t),\\gV{\\phi}(t)\\right)}{\\partial\\V{J}(t)},\n\\label{eq:eom_theta}\n\\end{align}\nwhere $\\phi_j^\\ast(t)$ and $\\phi_j(t)$ should be understood as functions of $J_j(t)$ and $\\theta_j(t)$ according to eq.~(\\ref{eq:definition_J_theta_timedependent}). Thus, the classical trajectory lives on a symplectic manifold in phase space, which is here defined as $\\{(\\V{J},\\gV{\\theta}):J_{j=1,2,\\ldots}\\in[0,\\infty),\\theta_{j=1,2,\\ldots}\\in[0,2\\pi)\\}$. Moreover, the theory of canonical transformations \\cite{Tabor} can be applied to show that the Poincar\\'e-Cartan 1-form\n\\begin{equation}\n\\gV{\\theta}\\cdot{\\rm d}\\V{J}-H{\\rm d}t\n\\label{eq:poincare-cartan}\n\\end{equation}\nis invariant under canonical transformations.\n\nThe derivatives of the action can be found by applying the equations of motion to the integrand to read\n\\begin{align}\n\\frac{\\partial R_{\\gamma}\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)}{\\partial\\V{n}^{(i)}}&=-\\hbar\\gV{\\theta}(0),\n\\label{eq:deriv_action_ni} \\\\\n\\frac{\\partial R_{\\gamma}\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)}{\\partial\\V{n}^{(f)}}&=\\hbar\\gV{\\theta}(t_f),\n\\label{eq:deriv_action_nf} \\\\\n\\frac{\\partial R_{\\gamma}\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)}{\\partial t_f}&=-E_\\gamma,\n\\label{eq:deriv_action_t}\n\\end{align}\nwhere $E_\\gamma=H_{cl}\\left(\\gV{\\phi}^\\ast(0),\\gV{\\phi}(0)\\right)$ is the energy of the trajectory.\n\nFinally, the propagator eq.~(\\ref{eq:definition_propagator_fock}) reads\n\\begin{equation}\nK^{\\rm sc}\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)=\\sum\\limits_{\\gamma}^{}\\mathcal{A}_{\\gamma}\\exp\\left[\\frac{\\rm i}{\\hbar}R_{\\gamma}\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)\\right],\n\\label{eq:semiclassical_propagator_before_fluctuation-integration}\n\\end{equation}\nwhere the sum runs over all ``classical paths'' $\\gamma$ which satisfy the equations of motion (\\ref{eq:eom}) and the boundary conditions (\\ref{eq:bc}), while $\\mathcal{A}_{\\gamma}$ is given by the still pending integrations over the second variation of the paths.\nAs is shown in appendix \\ref{app:semiclassical_amplitude}, $\\mathcal{A}_{\\gamma}$ can be written as\n\\begin{align}\n\\mathcal{A}_{\\gamma}=&\\frac{1}{\\sqrt{2\\pi}^{N-1}}\\exp\\left\\{\\frac{\\rm i}{2\\hbar}\\int\\limits_{0}^{t_f}{\\rm d}t{\\rm Tr}\\left[\\frac{\\partial^2H_{cl}}{\\partial\\gV{\\phi}(t)^2}\\mathbf{X}(t)\\right]\\right\\} \\nonumber \\\\\n&\\det\\left\\{\\mathbf{I}_{N}+\\exp\\left[-2{\\rm i}{\\rm diag}\\left(\\mathbf{P}_{f}\\gV{\\theta}(t_f)\\right)\\right]\\mathbf{P}_f\\mathbf{X}(t_f)\\mathbf{P}_f^{\\rm T}\\right\\}^{-\\frac{1}{2}},\n\\label{eq:semiclassical_prefactor_intermediately}\n\\end{align}\nwith $N=N_{i}=N_{f}$ being the total particle number and $\\mathbf{I}_N$ the $N\\times N$ unit matrix. Moreover, $\\mathbf{P}_f$ is the matrix of the projector onto the subspace of the states which are occupied at final time, such that \\textit{e.g.}\n\\begin{equation}\n\\mathbf{P}_f\\V{n}^{(f)}=(\\underbrace{1,\\ldots,1}_{N})^{\\rm T}.\n\\label{eq:example_for_action_of_Pf}\n\\end{equation}\n\nFor later reference, we also define $\\mathbf{P}_i$, which is defined in the same way as $\\mathbf{P}_f$, but selecting the initially occupied single particle states, as well as the complements $\\bar{\\mathbf{P}}_{i\/f}$ of $\\mathbf{P}_{i\/f}$. With these matrices, one can also define the (orthonormal) matrix\n\\begin{equation}\n\\mathbf{Q}_{i\/f}=\\left(\\begin{array}{c}\n\\bar{\\mathbf{P}}_{i\/f} \\\\\n\\mathbf{P}_{i\/f}\n\\end{array}\\right)\n\\label{eq:definition_Q},\n\\end{equation}\nshifting all components of a vector corresponding to an initially (finally) unoccupied single particle state in front of all the others.\n\nFinally in eq.~(\\ref{eq:semiclassical_prefactor_intermediately}) $\\mathbf{X}(t)$ satisfies the differential equation\n\\begin{align}\n\\dot{\\mathbf{X}}(t)=&\\frac{\\rm i}{\\hbar}\\frac{\\partial^2H_{cl}}{{\\partial\\gV{\\phi}^\\ast(t)}^2}-\\frac{\\rm i}{\\hbar}\\frac{\\partial^2H_{cl}}{\\partial\\gV{\\phi}^\\ast(t)\\partial\\gV{\\phi}(t)}\\mathbf{X}(t) \\nonumber \\\\\n&-\\frac{\\rm i}{\\hbar}\\mathbf{X}(t)\\frac{\\partial^2H_{cl}}{\\partial\\gV{\\phi}(t)\\partial\\gV{\\phi}^\\ast(t)}+\\frac{\\rm i}{\\hbar}\\mathbf{X}(t)\\frac{\\partial^2H_{cl}}{\\partial\\gV{\\phi}(t)^2}\\mathbf{X}(t),\n\\label{eq:deq_X}\n\\end{align}\nwith initial condition\n\\begin{equation}\n\\mathbf{X}(0)=\\mathbf{Q}_i^{\\rm T}\n\\left(\\begin{array}{cc}\n0 \\\\\n & \\exp\\left[2{\\rm i}{\\rm diag}\\left(\\mathbf{P}_i^\\prime\\gV{\\theta}(0)\\right)\\right]\n\\end{array}\\right)\\mathbf{Q}_i.\n\\label{eq:ic_X}\n\\end{equation}\nThe same differential equation, however with different initial conditions, was encountered previously in derivations of a semiclassical propagator for bosonic many body systems in coherent state representation \\cite{Aguiar+Barangar,Garg+Braun}. The solutions given there indicate, how to find $\\mathbf{X}(t)$: Consider a solution $\\boldsymbol\\psi(t)$ of the equations of motion with initial conditions ${\\bf Y}$ and ${\\bf W}$, whereby each pair $(Y_j,W_j)$ are canonically conjugate variables. Possibilities for the choice of these pairs are \\textit{e.g.}~$(\\Re\\psi_j(0),\\Im\\psi_j(0))$, where $\\Re$ and $\\Im$ denote the real and imaginary part, respectively, $(\\left|\\psi_j(0)\\right|,\\arg\\psi_j(0))$ with $\\arg\\psi$ denoting the phase of $\\psi$, or $(\\psi_j(0),{\\psi}^\\ast_j(0))$. Then, the differential equation (\\ref{eq:deq_X}) is solved by the function\n\\begin{equation}\n-\\frac{\\partial{\\boldsymbol\\psi}(t)}{\\partial\\bf W}\\left(\\frac{\\partial{\\boldsymbol\\psi}^\\ast(t)}{\\partial\\bf W}\\right)^{-1},\n\\label{eq:general_solution_X}\n\\end{equation}\nevaluated at the initial conditions corresponding to the trajectory $\\gamma$.\n\nFinally, in order to find the solution for $\\mathbf{X}(t)$, the variables ${\\bf Y}$ and ${\\bf W}$ need to be chosen such that for $t=0$, eq.~(\\ref{eq:general_solution_X}) also satisfies the initial condition (\\ref{eq:ic_X}), which yields\n\\begin{equation}\n(Y_j,W_j)=\\left\\{\\begin{array}{lc}\n({\\psi}_j(0),{\\psi}_j^\\ast(0)),\\quad & \\text{if } n_j^{(i)}=0 \\text{ or } j=j_1 \\\\\n(n_{j}^{(i)},\\theta_{j}), & \\text{else}.\n\\end{array}\\right.\n\\label{eq:choice_initial_conditions}\n\\end{equation}\nEventually, the semiclassical amplitude $\\mathcal{A}_{\\gamma}$ can be written as\n\\begin{align}\n\\mathcal{A}_{\\gamma}=&\\sqrt{\\det\\left[\\frac{1}{2\\pi{\\rm i}\\hbar}\\frac{\\partial^2R_{\\gamma}}{\\partial\\left(\\mathbf{P}_f^{\\prime}\\V{n}^{(f)}\\right)\\partial\\left(\\mathbf{P}_i^{\\prime}\\V{n}^{(i)}\\right)}\\right]} \\nonumber \\\\\n&\\sqrt{\\det\\mathbf{Q}_f\\mathbf{Q}_i}\\exp\\left(\\frac{\\rm i}{2\\hbar}\\int\\limits_{0}^{t_f}{\\rm d}t{\\rm Tr}\\frac{\\partial^2H_{cl}}{\\partial\\gV{\\phi}^\\ast\\partial\\gV{\\phi}}\\right) \\nonumber \\\\\n&\\exp\\left(\\frac{\\rm i}{2}\\sum\\limits_{j:n_j^{(f)}=1}^{}\\theta_j(t_f)-\\frac{\\rm i}{2}\\sum\\limits_{j:n_j^{(i)}=1}^{}\\theta_j(0)\\right) \\nonumber \\\\\n&\\det\\left(\\mathbf{A}-\\mathbf{B}\\mathbf{C}^{-1}\\mathbf{D}\\right)^{-\\frac{1}{2}}.\n\\label{eq:semiclassical_prefactor}\n\\end{align}\nwith $\\mathbf{P}_i^\\prime$ and $\\mathbf{P}_f^\\prime$ being the matrices resulting from $\\mathbf{P}_i$ and $\\mathbf{P}_f$, respectively, by removing the first line. The determinant consisting of the matrices\n\\begin{align}\n\\mathbf{A}&=\\frac{\\partial\\left(\\bar{\\mathbf{P}}_f\\gV{\\phi}^\\ast(t_f),J_{\\min\\left\\{j\\in\\{1,2,\\ldots\\}:n_j^{(f)}=1\\right\\}}(t_f)\\right)}{\\partial\\left(\\bar{\\mathbf{P}}_i^{\\prime}\\gV{\\phi}^\\ast(0)\\right)},\n\\label{eq:firstderivmatrix} \\\\\n\\mathbf{B}&=\\frac{\\partial\\left(\\bar{\\mathbf{P}}_f\\gV{\\phi}^\\ast(t_f),J_{\\min\\left\\{j\\in\\{1,2,\\ldots\\}:n_j^{(f)}=1\\right\\}}(t_f)\\right)}{\\partial\\left(\\mathbf{P}_i^{\\prime}\\gV{\\theta}(0)\\right)},\n\\label{eq:secondderivmatrix} \\\\\n\\mathbf{C}&=\\frac{\\partial\\left(\\mathbf{P}_f^{\\prime}\\V{J}(t_f)\\right)}{\\partial\\left(\\mathbf{P}_i^{\\prime}\\gV{\\theta}(0)\\right)},\n\\label{eq:thirdderivmatrix} \\\\\n\\mathbf{D}&=\\frac{\\partial\\left(\\mathbf{P}_f^{\\prime}\\V{J}(t_f)\\right)}{\\partial\\left(\\bar{\\mathbf{P}}_i^{\\prime}\\gV{\\phi}^\\ast(0)\\right)}.\n\\label{eq:fourthderivmatrix}\n\\end{align}\naccounts for the vacuum fluctuations that have been treated exactly.\nNote that in eq.~(\\ref{eq:semiclassical_prefactor}) the Solari-Kochetov extra-phase\n\\begin{equation}\n\\exp\\left(\\frac{\\rm i}{2\\hbar}\\int\\limits_{0}^{t_f}{\\rm d}t{\\rm Tr}\\frac{\\partial^2H_{cl}}{\\partial\\gV{\\phi}^\\ast\\partial\\gV{\\phi}}\\right)\n\\end{equation}\ntypically arises in a semiclassical approximation of the propagator in coherent state representation \\cite{Aguiar+Barangar,Solari,Kochetov,Vieira+Sacramento}, while in the standard (first quantized) van-Vleck-Gutzwiller propagator \\cite{Gut1970}, this phase is absent, due to the Weyl (symmetric) ordering of the Hamiltonian with respect to position and momentum operators. For Bosons, the Solari-Kochetov phase can be absorbed in the action by replacing the bosoinc creation and annihilation operators according to $\\hat{a}_j^\\dagger\\hat{a}_{j^\\prime}^{}\\to(\\hat{a}_j^\\dagger\\hat{a}_{j^\\prime}^{}+\\hat{a}_{j^\\prime}^{}\\hat{a}_j^\\dagger)\/2$ \\cite{Aguiar+Barangar}, which corresponds to Weyl ordering of the quantum Hamiltonian. In the same way, for the propagator in spin coherent states, this phase is absent in Weyl ordering \\cite{Pletyukhov_extra-phase}. However, this vanishing of the Solari-Kochetov phase in these cases is due to the fact that the classical Hamiltonian is obtained out of the quantum one by the simple replacements $\\hat{a}_j^\\dagger\\to\\phi_j^\\ast$ and $\\hat{a}_j^{}\\to\\phi_j^{}$, which is not valid here. Therefore, it seems that here this phase can not be eliminated by changing the chosen ordering of the fermionic creation and annihilation operators.\n\nDue to their definition eq.~(\\ref{eq:definition_Q}), the determinants $\\det\\mathbf{Q}_{i\/f}$ depend only on the choice of the initial and final occupations and accept only the values $\\pm1$. Note that this sign also depends on the definition of the Fock states, while the product of both depends only on the relative changes between the initial and final state and therefore is independent of the exact choice of ordering of the single particle states.\n\nIt is important to notice that in eq.~(\\ref{eq:semiclassical_prefactor}) the determinant $\\det(\\mathbf{A}-\\mathbf{B}\\mathbf{C}^{-1}\\mathbf{D})$ depends only on the derivatives of the values of the trajectory at final time with respect to the initial conditions and should therefore be possible to calculate in an actual application. Moreover, we expect that this determinant is just the product of the exponentials of the final and initial phases of the final unoccupied states, which can be set to zero. Thus, we assume this determinant to be equal to one. However, up to now we did not succeed in proofing this conjecture rigorously and therefore, we will keep this determinant in the following.\n\n\n\\section{Transition probability}\n\\label{sec:transition_prob}\n\\subsection{General semiclassical treatment}\nKnowing the propagator enables us, in principle, to calculate the quantum probability to measure the Fock state $\\V{n}^{(f)}$ after preparing the system of spin-$1\/2$ particles in the initial Fock state $\\V{n}^{(i)}$ and letting it evolve for some time $t$. Computing this probability is usually non-trivial, since the single particle states can on the one hand be chosen arbitrarily, and may thus not necessarily be eigenstates of the single-particle Hamiltonian and on the other hand interactions in general induce a coupling between different single particle states. This probability is given by the modulus square of the overlap between the time evolved state and $\\ket{\\V{n}^{(f)}}$,\n\\begin{equation}\nP\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)=\\left|\\Braket{\\V{n}^{(f)}|\\hat{K}\\left(t_f\\right)|\\V{n}^{(i)}}\\right|^2.\n\\label{eq:definition_transition_probability}\n\\end{equation}\nUsing the semiclassical approximation (\\ref{eq:semiclassical_propagator_before_fluctuation-integration}), it is given by a double sum over trajectories,\n\\begin{equation}\nP\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)\\approx\\sum\\limits_{\\gamma,\\gamma^\\prime}^{}\\mathcal{A}_{\\gamma^{}}\\mathcal{A}_{\\gamma^\\prime}^\\ast\\exp\\left[\\frac{\\rm i}{\\hbar}\\left(R_{\\gamma^{}}-R_{\\gamma^\\prime}\\right)\\right].\n\\label{eq:transition_probability_semiclassically}\n\\end{equation}\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{pairing.pdf}\n\\put(-235,90){${\\bf n}^{(i)}$}\n\\put(-140,78){${\\bf n}^{(f)}$}\n\\put(-109,95){${\\bf n}^{(f)}=$}\n\\put(-102,85){${\\bf n}^{(i)}$}\n\\put(-171,12){${\\bf n}^{(i)}$}\n\\put(-75,0){${\\bf n}^{(f)}=\\mathcal{T}{\\bf n}^{(i)}$}\n\\put(-193,158){GUE, GOE}\n\\put(-186,148){\\& GSE}\n\\put(-85,153){GOE}\n\\put(-127.5,145){GSE}\n\\put(-190,90){$\\textcolor{red}{\\gamma}$}\n\\put(-215,80){$\\textcolor[rgb]{0.35,0,0.35}{\\gamma^\\prime=\\gamma}$}\n\\put(-63,90){$\\textcolor{red}{\\gamma}$}\n\\put(-95,70){$\\textcolor[rgb]{0.35,0,0.35}{\\gamma^\\prime=\\mathcal{T}\\gamma}$}\n\\put(-125,12){$\\textcolor{red}{\\gamma}$}\n\\put(-158,2){$\\textcolor[rgb]{0.35,0,0.35}{\\gamma^\\prime=\\mathcal{T}\\gamma}$}\n\\caption{The quantum transition in a system of spin-$1\/2$ particles in the semiclassical limit. A trajectory $\\gamma$ is paired with a partner trajectory $\\gamma^\\prime$, where $\\gamma^\\prime$ can be either $\\gamma$ itself, or its time reverse. The annotations at the arrows indicate the symmetry class required for the corresponding pairing to be present.}\n\\label{fig:pairing}\n\\end{figure}\nUpon applying an energy or disorder average, the action difference gives rise to huge oscillations, such that most contributions to the averaged double sum will cancel, except if the paths $\\gamma$ and $\\gamma^\\prime$ are correlated. The types of trajectory pairs, which we will consider in the following are depicted in fig.~\\ref{fig:pairing}. The simplest type of correlation arises for $\\gamma=\\gamma^\\prime$. This is known as the diagonal approximation \\cite{da}. The second derivatives of the action with respect to the initial and final Fock state in the prefactor can then be used to transform the sum over trajectories into an integration over the initial phases. Then the diagonal approximation yields,\n\\begin{align}\nP&_{cl}\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)= \\nonumber \\\\\n&\\int\\limits_{0}^{2\\pi}{\\rm d}^{N-1}\\theta^{(i)}\\det\\left(\\mathbf{A}-\\mathbf{B}\\mathbf{C}^{-1}\\mathbf{D}\\right)^{-1}\\delta\\left(\\left|\\gV{\\phi}(t_f)\\right|^2-\\V{n}^{(f)}\\right),\n\\label{eq:classical_probability}\n\\end{align}\nwhich we will refer to as classical probability. Here $\\gV{\\phi}(t)$ is the solution of the equations of motion eq.~(\\ref{eq:eom}) with the initial condition $\\phi_j(0)=\\sqrt{n_j^{(i)}}\\exp\\left({\\rm i}\\theta_j^{(i)}\\right)$. It is worth to notice that the exact treatment of the vacuum fluctuations gives rise to a renormalization of the transition probability by the additional factor $\\det\\left(\\mathbf{A}-\\mathbf{B}\\mathbf{C}^{-1}\\mathbf{D}\\right)^{-1}$.\n\nFurther pairs of correlated trajectories are those given by $\\gamma$ and its time reverse, $\\gamma^\\prime=\\mathcal{T}\\gamma$. However, the time reverse of a trajectory exists only if the system is time reversal symmetric. Moreover, the initial and final occupations, respectively, of both trajectories in the double sum of eq.~(\\ref{eq:transition_probability_semiclassically}) have to be the same. On the other hand, if $\\gamma$ has initial occupations $\\V{n}^{(i)}$ and final occupations $\\V{n}^{(f)}$, the initial occupations of its time reverse are given by the time reverse of $\\V{n}^{(f)}$ and the final ones by the time reverse of $\\V{n}^{(i)}$. Therefore, in order to pair $\\gamma$ with its time reverse, we need time reversal symmetry and also the final Fock state has to be the time reverse of the initial one. To this end, one has to replace the sum over trajectories from $\\V{n}^{(i)}$ to $\\V{n}^{(f)}$ by a sum over trajectories ending at the Fock state $\\mathcal{T}\\V{n}^{(i)}$ originating from time reversing the initial one. To this end, the actions in the exponential need to be expanded in the final Fock state around $\\mathcal{T}\\V{n}^{(i)}$ up to linear order, while the prefactor is assumed to vary only very slightly with $\\V{n}^{(f)}$, such that it can be simply replaced by $\\mathcal{T}\\V{n}^{(i)}$. For pairs $\\gamma^\\prime=\\mathcal{T}\\gamma$ this procedure then gives the contribution\n\\begin{align}\n\\sum\\limits_{\\gamma}^{}&\\mathcal{A}_{\\gamma^{}}\\mathcal{A}_{\\mathcal{T}\\gamma^{}}^\\ast\\exp\\left(\\frac{\\rm i}{\\hbar}\\Delta R_{}\\right)\\times \\nonumber \\\\\n&\\times\\exp\\left[{\\rm i}\\left(\\gV{\\theta}^{(\\gamma)}(t_f)-\\gV{\\theta}^{(\\mathcal{T}\\gamma)}(t_f)\\right)\\cdot\\left(\\V{n}^{(f)}-\\mathcal{T}\\V{n}^{(i)}\\right)\\right],\n\\label{eq:cbs_contribution_before_average}\n\\end{align}\nwith the difference $\\Delta R=R_\\gamma-R_{\\mathcal{T}\\gamma}$ in the actions of $\\gamma$ and $\\mathcal{T}\\gamma$. Since for time reversal symmetric systems, the energy of a trajectory and its time-reverse is the same, we easily get\n\\begin{equation}\n\\Delta R=\\hbar\\int\\limits_{0}^{t_f}{\\rm d}t\\left(\\gV{\\theta}^{(\\gamma)}\\cdot\\dot{\\V{J}}^{(\\gamma)}-\\gV{\\theta}^{(\\mathcal{T}\\gamma)}\\cdot\\dot{\\V{J}}^{(\\mathcal{T}\\gamma)}\\right).\n\\label{eq:action_difference_time-reverse_general}\n\\end{equation}\nIn the next steps, we assume -- in accordance with the cases considered below -- that the difference $\\Delta R$, is independent of the trajectory. This is usually the case, since the second part of the integral in the action difference can be related with the first one by making use of the nature of the time reversal operation. However, as we will see later, $\\Delta R$ does not vanish in general. Moreover, we can savely assume that $\\gV{\\theta}^{(\\mathcal{T}\\gamma)}(t_f)$ depends on the initial phases of $\\gamma$, only (and through them on the initial Fock state).\n\nUpon disorder average, the phases $\\theta_j^{(\\gamma)}(t_f)$ behave, for chaotic systems, like linearly distributed random variables between $0$ and $2\\pi$. Thus, treating them as random variables and performing the average, yields a $\\delta_{\\V{n}^{(f)},\\mathcal{T}\\V{n}^{(i)}}$, such that one gets after utilizing the second derivative of the action again\n\\begin{equation}\nP_{cl}\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)\\delta_{\\V{n}^{(f)},\\mathcal{T}\\V{n}^{(i)}}\\exp\\left(\\frac{\\rm i}{\\hbar}\\Delta R\\right).\n\\label{eq:cbs_contribution_TR_not_specified}\n\\end{equation}\nThe action difference $\\Delta R$ strongly depends on whether the system is diagonal in spin space or not.\n\\subsection{Systems diagonal in spin space}\n\\label{subsec:diagonal_spin-space}\nIf the system is diagonal in spin space, \\textit{i.e.}~the Hamiltonian does not consist of terms giving rise to spin-flips, the time reversal operation amounts to a complex conjugation only, and therefore\n\\begin{equation}\n\\mathcal{T}\\V{n}^{(i)}=\\V{n}^{(i)}.\n\\label{eq:time-reversal_initial_fock-state_diagonal_spin-space}\n\\end{equation}\nIt also implies that, on the classical level, the time reverse of $\\gV{\\phi}(t)$ is given by $\\gV{\\phi}^\\ast(t_f-t)$. With this information, it is easy to prove that time reversed paths have the same action, $\\Delta R=0$. Thus in semiclassical approximation the averaged transition probability for a spin-diagonal system is given by\n\\begin{equation}\nP\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)\\approx P_{cl}\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)\\left(1+\\delta_{\\V{n}^{(f)},\\V{n}^{(i)}}\\right).\n\\label{eq:transition_probabilit_spin-diagonal}\n\\end{equation}\nThis is, apart from apart from the renormalization of the classical transition probability due to the exact treatment of the vacuum (see eq.~(\\ref{eq:classical_probability})), exactly the same result found previously for bosonic, spinless systems \\cite{cbs_fock}.\n\\subsection{Systems non-diagonal in spin space}\n\\label{non-diagonal_spin-space}\nIf the system's Hamiltonian is non-diagonal in spin space, the time reversal operation is not just complex conjugation, but also demands an exchange of the spin-up and spin-down components while at the same time introducing a relative minus sign between them,\n\\begin{equation}\n\\hat{T}=\\left[\\prod\\limits_{j}\\left(-{\\rm i}\\hat{\\sigma}_{j,y}\\right)\\right]\\hat{K}.\n\\label{eq:tr_non-diagonal_spin-space}\n\\end{equation}\nHere $\\hat{\\sigma}_{j,y}$ is the $y$-Pauli matrix for the $j$-th state and $\\hat{K}$ denotes complex conjugation. Important examples of systems with such time reversal operations are for instance systems with a Rashba spin-orbit coupling \\cite{Rashba}, whcih is of key importance in semiconductor spintronics, but more recently has also been realized using ultra-cold atoms \\cite{Spielman}.\n\nOn the classical level, this means that the time reversal of $\\gV{\\phi}=(\\gV{\\phi}_{\\uparrow},\\gV{\\phi}_{\\downarrow})^{\\rm T}$, where $\\gV{\\phi}_{\\uparrow(\\downarrow)}$ is the vector containing all spin-up (spin-down) components of $\\gV{\\phi}$, is given by\n\\begin{equation}\n\\mathcal{T}\\left(\\begin{array}{c}\n\\gV{\\phi}_{\\uparrow}(t) \\\\\n\\gV{\\phi}_{\\downarrow}(t)\n\\end{array}\\right)=\\left(\\begin{array}{c}\n-\\gV{\\phi}_{\\downarrow}^\\ast(t_f-t) \\\\\n\\gV{\\phi}_{\\uparrow}^\\ast(t_f-t)\n\\end{array}\\right),\n\\label{eq:tr-trajectory_non-diagonal_spin-space}\n\\end{equation}\nand therefore also\n\\begin{equation}\n\\mathcal{T}\\V{n}^{(i)}=\\mathcal{T}\\left(\\begin{array}{c}\n\\V{n}^{(i)}_{\\uparrow} \\\\\n\\V{n}^{(i)}_{\\downarrow}\n\\end{array}\\right)=\\left(\\begin{array}{c}\n\\V{n}^{(i)}_{\\downarrow} \\\\\n\\V{n}^{(i)}_{\\uparrow}\n\\end{array}\\right).\n\\label{eq:timereverse_ni}\n\\end{equation}\nFor the action difference, this yields\n\\begin{equation}\n\\Delta R=\\pi\\hbar\\sum\\limits_{j}^{}\\left[\\left(\\mathcal{T}\\V{n}^{(i)}\\right)_{j,\\uparrow}-n^{(i)}_{j,\\uparrow}\\right]=\\pi\\hbar\\left(N_{\\downarrow}-N_{\\uparrow}\\right),\n\\label{eq:action_difference_non-diagonal_spin-space}\n\\end{equation}\nwhere $N_{\\uparrow(\\downarrow)}$ is the total number of spin-up (spin-down) particles in the initial state.\n\nThus, invoking the widely used nomenclature of the random matrix symmetry classes and quantum chaos \\cite{Haake}, one finally finds for the averaged transition probability in semiclassical approximation\n\\begin{align}\nP&\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)= \\nonumber \\\\\n&P^{(cl)}\\left(\\V{n}^{(f)},\\V{n}^{(i)};t_f\\right)\\left\\{\\begin{array}{ll}\n1 & \\text{, GUE} \\\\\n\\left[1+\\delta_{\\V{n}^{(f)},\\V{n}^{(i)}}\\right] & \\text{, GOE} \\\\\n\\left[1+(-1)^N\\delta_{\\V{n}^{(f)}_{\\downarrow},\\V{n}^{(i)}_{\\uparrow}}\\delta_{\\V{n}^{(f)}_{\\downarrow},\\V{n}^{(i)}_{\\uparrow}}\\right] & \\text{, GSE}.\n\\end{array}\\right.\n\\label{eq:transition_probability_semiclassical}\n\\end{align}\nHere GUE (Gaussian Unitary Ensemble) means that the average runs over systems without time reversal symmetry, while for GOE (Gaussian Orthogonal Ensemble) and GSE (Gaussian Symplectic Ensemble) the average is over time reversal invariant spin-$1\/2$ systems, which are diagonal and non-diagonal in spin space, respectively. This result and in particular the origin of the deltas is illustrated in fig.~\\ref{fig:pairing}.\n\nIt is important to note that, the probability to find $\\V{n}^{(f)}=\\mathcal{T}\\V{n}^{(i)}$ is zero on average for the GSE case, if $N$ is odd. However, the transition probability is a strictly positive quantity. Therefore, in order to become zero on average, it has to be zero for each disorder realization. In other words, for a time reversal symmetric system, which is non-diagonal in spin space, the transition from an initial Fock state to its spin reversed version is semiclassically prohibited,\n\\begin{equation}\n\\Braket{\\hat{T}\\V{n}^{(i)}|\\hat{K}\\left(t_f\\right)|\\V{n}^{(i)}}=0\n\\label{eq:prohibited_transition}\n\\end{equation}\nfor an odd total number of particles. This is consistent with\n\\begin{equation}\n\\Braket{\\hat{T}\\V{n}|\\hat{H}|\\V{n}}=0.\n\\label{eq:kramers_degeneracy}\n\\end{equation}\nSimilar to the proof of Kramer's degeneracy \\cite{Sakurai}, one can show that eq.~(\\ref{eq:kramers_degeneracy}) implies that for an odd number of particles and a symplectic time reversal symmetry, the transition from a Fock state to its spin reversed version is exactly forbidden quantum mechanically.\n\nOn the other hand, if the total number of particles is even, and hence the total spin is integer, the transition probability is always enhanced by a factor of two compared to the classical one, if the final Fock state is the time reversed version of the initial one.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the main systematic errors in the neutrino experiments are those \nassociated with neutrino cross sections. \nThe high-intensity neutrino beams used in neutrino oscillation experiments are \npeaked in the 0.3-5 GeV energy domain and allow the study of neutrino-nucleus \ninteraction with unprecedented detail. In this energy \nrange the dominant contribution to neutrino-nucleus scattering comes from the \ncharged-current quasielastic (CCQE) reactions and resonance production \nprocesses.  \n\nVarious modern~\\cite{NOMAD, MiniB1, MiniB2, MiniB3, Miner1, Miner2, T2K}\nneutrino experiments reported measurements of the differential \n$d\\sigma\/d Q^2$ ($Q^2$ is squared four-momentum transfer)~\\cite{NOMAD, MiniB1, \nMiniB2, MiniB3, Miner1, Miner2}, \ndouble-differential~\\cite{MiniB1, MiniB3, T2K} and total cross \nsections~\\cite{NOMAD, MiniB1, MiniB3, SciB, T2K} of CCQE scattering. This \ninteraction represents a two-particle scattering process with lepton and \nnucleon in the final state. Since the criteria used to \nselect CCQE events are strongly influenced by both target material and  \ndetector technology, various selection techniques are applied in these \nexperiments (tracking and Cerenkov detector). In neutrino scattering \nexperiments, the neutrino energy is unknown. However, both the neutrino energy\n and $Q^2$ can be evaluated based on the kinematics of the outgoing final state \nparticles. \n\nWith the assumptions of conserved vector current and partially conserved axial \ncurrent, the only undetermined form factor in the theoretical description of \nthe CCQE neutrino scattering is the axial nucleon form factor. In most analysis \nof neutrino interaction on complex nuclei, the dipole parametrization of \n$F_A(Q^2)$ with one parameter, the axial mass $M_A$, is used. The analyses are \nmainly based on the relativistic Fermi gas model (RFGM)~\\cite{Smith} and \ninvolve some additional model dependence due to nuclear structure. While \nconstraints exist from pion electroproduction data with $M_A=1.06\\mbox{\\boldmath $p_{\\mathrm m}$} 0.016$ GeV~\n\\cite{Bernard}, neutrino experiments usually treat axial mass $M_A$ in CCQE as \n independent of that measurement. \n\nThe values of $M_A$ are obtained from a fit to observed $Q^2$ distribution of \nevents, differential and total (anti)neutrino CCQE interaction cross sections. \nThe formal averaging of $M_A$ values which are very widely spread was done in \nRef.~\\cite{Bernard}: $M_A=1.026\\mbox{\\boldmath $p_{\\mathrm m}$} 0.021$ GeV. This result is also known as \nworld-averaged value of the axial mass. Values of $M_A$ determined from the \nrecent QE (anti)neutrino-carbon scattering experiments range anywhere from \n$M_A\\approx$ 1 to 1.35 GeV. The NOMAD Collaboration~\\cite{NOMAD} reports \n$M_A=1.05\\mbox{\\boldmath $p_{\\mathrm m}$} 0.02\\mbox{\\boldmath $p_{\\mathrm m}$} 0.06$ GeV, and the most resent MINERvA~\\cite{Miner1, \nMiner2} results are consistent with $M_A \\simeq 0.99$ GeV. On the other hand, \nthe MiniBooNE Collaboration~\\cite{MiniB1, MiniB2, MiniB3} reports a large value \nof $M_A=1.35\\mbox{\\boldmath $p_{\\mathrm m}$} 0.17$ GeV. The other recent results~\\cite{K2KSciFi, K2KSciBar, \nMINOS1, MINOS2} similarly find central values higher than the above-mentioned \nworld average. The absolute values of the differential and total cross sections\n measured by MiniBooNE are about 30\\% larger as compared to the NOMAD \nresults. It is essential to obtain consistency between experiments utilizing \ndifferent beam energies, nuclear targets, and detectors.\n\nThe radius of the nucleon axial charge distribution in terms \nof the dipole mass is defined as $\\langle r^2_A\\rangle=12\/M^2_A$, and for \n$1.032 \\le M_A \\le 1.35$ GeV it can be estimated as $0.51 \\le r_A \\le 0.66$ fm.\nSo, a large value of $M_A=1.35$ GeV corresponds to a rather small axial charge \nradius of about 0.51 fm. On the other hand, the typical nucleon size as deduced \nfrom the electron scattering experiment is about 0.85 fm, whereas \nmodel-dependent analyses of proton-antiproton annihilation lead to a baryon \ncharge radius of about 0.5 fm~\\cite{Bernard}.\n\nThese results have encounraged many theoretical studies~\\cite{Benh,Sob,BAV1,\nBAV2,Meu3,Mart1,Mart2,Niev1,Niev2,Niev3,Amaro1,Amaro2,Amaro3,Bod1,\nLal} to attempt to explain the discrepancy between the data and \ntraditional nuclear model, i.e., the RFGM. Some models that are based on the  \nimpulse approximation lead to \nlarge value of $M_A$~\\cite{Benh,Sob,BAV1,BAV2} in the MiniBooNE data.\nFor instance, in Refs.~\\cite{BAV1, BAV2}, a value of $M_A = 1.37$ GeV, that \nfits the $Q^2$ shape of the measured $d\\sigma\/d Q^2$ cross section was \nobtained within the RFGM, and relativistic distorted-wave impulse approximation \n(RDWIA) approaches. \n\nOther nuclear models~\\cite{Mart1,Mart2,Niev1,Niev2,Niev3} include effects of \nmultinucleon excitations such as meson-exchange currents (MECs) and isobar \ncurrents (ICs) to describe the MiniBooNE data. The contribution of the \ntwo-particle-two-hole (2p-2h) excitations to CCQE scattering has been found \nto be sizable and allows one to reproduce the MiniBooNE cross sections with \nvalue of \n$M_A\\approx 1.03$ GeV. This result suggests that much of the cross section\n (about 30\\%) measured by the MiniBooNE experiment can be attributed to \nprocesses that are not properly QE scattering. However, fully relativistic \nmicroscopic calculations of 2p-2h contributions are extremely difficult and \nmay be bound to model-dependent assumptions. Future theoretical work is \nobviously needed to improve the present models that include effects beyond the \nimpulse approximation. The transverse enhancement (TE) effective model to \naccount for MEC effects has been proposed in Ref.~\\cite{Bod1}. In this model, \nthe magnetic form factors for nucleons bound in carbon are modified to describe\n the enhancement in the transverse electron-carbon QE cross section. The other \nnucleon form factors are the same as for free nucleons. It allows one to \ndescribe the MiniBooNE data (total cross section) by using the dipole \napproximation for $F_A(Q^2)$ with $M_A=1.014$ GeV.   \n\nThe assumption of the dipole ansatz for the axial form factor is a crucial \nelement in these studies. The dipole parametrization of $F_A(Q^2)$ has no \nstrict theoretical basis and the choice of this parametrization is made by \nanalogy with electromagnetic form factors. On the other hand the dipole ansatz \nhas been found to conflict with electron scattering data for the vector form \nfactor~\\cite{Bod1}. A model-independent description of the axial form factor \nwas presented in Ref.~\\cite{Hill}. The application of analyticity and \ndispersion relations to the axial form factor in the RFGM to find constraints \nfor the axial mass using the data from MiniBooNE produces the value of \n$M_A\\approx 0.85$ GeV which differs significantly from extractions based on the\ntraditional dipole ansatz. \n\nThe pion and nucleon form factors were discussed in Ref.~\\cite{Masjuan} within\nthe large-$N_c$ approach in the spacelike region. It was shown that, if the \nerror bars on the monopole mass are taken into account one can make the dipole \noverlap with a product of monopoles within the corresponding error bars \nprovided with the half-width rule. This construction is based on the following \nassumptions: (a) hadronic form factors in the spacelike region are dominated \nby mesonic states with the relevant quantum number, (b) the high-energy \nbehavior is given by perturbative QCD, and the number of mesons is taken to be \nminimal to satisfy these conditions, and (c) errors in the meson-dominated form \nfactors are estimated by means of the half-width rule.  \n\nIn Ref.~\\cite{Bod2} the values of $F_A(Q^2)$ as a function of $Q^2$ were \nextracted from the differential cross section of neutrino scattering on \ndeuterium (on a ``quasifree'' nucleon). A reasonable description of axial form \nfactor by dipole approximation with $M_A=1.014\\mbox{\\boldmath $p_{\\mathrm m}$} 0.016$ GeV was found. The \ncurrent data on CCQE scattering come from a variety of experiments \noperating with carbon, oxygen, and iron data. Therefore the aim of this work is\n twofold. First, we propose a method which allow one to determine $F_A(Q^2)$ as\n a function of $Q^2$ directly from the measured flux-integrated \n$d\\sigma \/d Q^2$ cross section of CCQE (anti)neutrino scattering on nuclei. \nSecond, we apply the method and extract the $Q^2$-dependence of the axial form \nfactor from MiniBooNE data~\\cite{MiniB1,MiniB3} in the RFGM, RDWIA and RFGM+TE \napproaches and show that the $Q^2$-dependence for the axial and vector form \nfactors are correlated. We also compare the extracted $Q^2$-dependence of \n$F_A(Q^2)$ with predictions of the meson-dominance ansatz and dipole \napproximation predictions with $M_A\\approx$ 1~GeV and $M_A\\approx$ 1.3~GeV\n\n\nThe outline of this paper is the following. In Sec. II we present briefly the\nRDWIA model and discuss the procedure which allows determination of the axial \nform factor from the flux-integrated $d\\sigma\/d Q^2$ (anti)neutrino cross \nsection. Section III presents results of the extraction of $F_A$ from the \nMiniBooNE data and calculations of the flux-unfolded total cross sections for \nantineutrino scattering off carbon . Our conclusions are summarized in \nSec. IV. \n\n\\section{Model and method for extraction of $F_A(Q^2)$ from flux-integrated \n$d\\sigma\/dQ^2$ differential cross sections}\n\nThe formalism of CCQE exclusive\n\n\\begin{equation}\\label{qe:excl}\n\\nu(k_i) + A(p_A)  \\rightarrow \\mu(k_f) + N(p_x) + B(p_B),     \n\\end{equation}\nand inclusive\n\\begin{equation}\\label{qe:incl}\n\\nu(k_i) + A(p_A)  \\rightarrow \\mu(k_f) + X                     \n\\end{equation}\nscattering off nuclei in the one-W-boson exchange approximation has been \nextensively described in previous works~\\cite{BAV3,BAV4,BAV5,BAV6}. Here \n$k_i=(\\varepsilon_i,\\mbox{\\boldmath $k$}_i)$ and $k_f=(\\varepsilon_f,\\mbox{\\boldmath $k$}_f)$ are the initial and \nfinal lepton momenta respectively, $p_A=(\\varepsilon_A,\\mbox{\\boldmath $p$}_A)$, and \n$p_B=(\\varepsilon_B,\\mbox{\\boldmath $p$}_B)$ are the initial and final target momenta, \nrespectively $p_x=(\\varepsilon_x,\\mbox{\\boldmath $p$}_x)$ is the ejectile nucleon momentum, \n$q=(\\omega,\\mbox{\\boldmath $q$})$ is the momentum transfer, and $Q^2=-q^2=\\mbox{\\boldmath $q$}^2-\\omega^2$. \n\n\\subsection{Model}\n\nAll the nuclear structure information and final \nstate interaction (FSI) effects are contained in the weak CC nuclear tensors \n$W_{\\mu \\nu}$, which are given by a bilinear product of the transition matrix \nelements of the nuclear CC operator $J_{\\mu}$ between the initial nucleus \nstate and the final state.\nWe describe CCQE neutrino-nuclear scattering in the impulse approximation (IA),\n assuming that the incoming neutrino interacts with only one nucleon, which is \nsubsequently emitted, while the remaining (A-1) nucleons in the target are \nspectators. The nuclear current is written as the sum of single-nucleon \ncurrents. \n\nThe single-nucleon charged current has $V{-}A$ structure $J^{\\mu} = \nJ^{\\mu}_V + J^{\\mu}_A$. For a free-nucleon vertex function \n$\\Gamma^{\\mu} = \\Gamma^{\\mu}_V + \\Gamma^{\\mu}_A$ we use the vector \ncurrent vertex function \n$\\Gamma^{\\mu}_V = F_V(Q^2)\\gamma^{\\mu} + \n{i}\\sigma^{\\mu \\nu}q_{\\nu}F_M(Q^2)\/2m$, where \n$\\sigma^{\\mu \\nu}=i[\\gamma^{\\mu}, \\gamma^{\\nu}]\/2$ and $F_V$ and $F_M$ are the weak \nvector form factors. They are \nrelated to the corresponding electromagnetic ones for protons and neutrons by \nthe hypothesis of the conserved vector current. We use the approximation \nof Ref.~\\cite{MMD} on the nucleon form factors.\nBecause the bound nucleons are off shell we employ the de Forest \nprescription~\\cite{deFor} and the Coulomb gauge for off-shell vector current \nvertex $\\Gamma^{\\mu}_V$. \n\nThe axial current vertex function can be written in terms of the axial \n$F_A(Q^2)$ and the pseudoscalar $F_P(Q^2)$ form factors:\n\\begin{eqnarray}\n\\label{Eq3}\n\\Gamma_A^{\\mu} &=& F_A(Q^2)\\gamma^{\\mu}\\gamma_5 +F_P(Q^2)q^{\\mu}\\gamma_5   \n\\end{eqnarray}\nThe pseudoscalar form factor $F_P(Q^2)$ is dominated by the pion pole and is \ngiven in term of the Godberger-Treiman relation near $Q^2\\approx 0$ if \npartially conserved axial current is assumed. We assume that the similar \nrelation is valid for high $Q^2$ as well and write\n\\begin{eqnarray}\n\\label{Eq4}\nF_p &=& \\frac{2m^2F_A(Q^2)}{m^2_{\\pi} +Q^2}=F_A(Q^2)F'_P(Q^2),   \n\\end{eqnarray}\nwhere $F'_P(Q^2)=2m^2\/(m^2_{\\pi}+Q^2)$ and $m_{\\pi}$ is the pion mass. Then the \naxial current vertex function can be written in the form\n\\begin{eqnarray}\n\\label{Eq5}\n\\Gamma_A^{\\mu} &=& F_A(Q^2)[\\gamma^{\\mu}\\gamma_5 +F'_P(Q^2)q^{\\mu}\\gamma_5].   \n\\end{eqnarray}\nThus the only undetermined form factor is the axial form factor that is \ncommonly parametrized as a dipole \n\\begin{eqnarray}\n\\label{Eq6}\nF_A &=& \\frac{F_A(0)}{(1 + Q^2\/M^2_A)^2}   \n\\end{eqnarray}\nwhere $F_A(0)$=1.267 and $M_A$ is the axial mass, which controls the $Q^2$ \ndependence of $F_A$, and ultimately, the normalization of the predicted cross \nsection. \nWe calculated the relativistic wave functions of the bound nucleon states  \nin the independent particle shell model as the self-consistent solutions of a \nDirac equation, derived within a relativistic mean field approach, from a \nLagrangian containing $\\sigma, \\omega$, and $\\rho$ mesons \n(the $\\sigma-\\omega$ model)\\cite{Serot,Horow}. \nAccording to the JLab data~\\cite{Dutta, Kelly1} the occupancy of the \nindependent particle shell-model orbitals of ${}^{12}$C equals on \naverage 89\\%. In this work we assume that the missing strength (11\\%) can be \nattributed to the short-range nucleon-nucleon ($NN$) correlations in the \nground state, leading to the appearance of the high-momentum and \nhigh-energy component in the nucleon distribution in the target.  \nThese estimates of the depletion of hole states follow from the RDWIA \nanalysis of ${}^{12}$C$({e},e^{\\prime}{p})$ for \n$Q^2 < 2$ (GeV\/c)$^2$~\\cite{Kelly1} and are consistent with a direct \nmeasurement of the spectral function~\\cite{Rohe}, which observed approximately \n0.6 protons in a region attributable to a single-nucleon knockout from a \ncorrelated cluster. \n\nIn the RDWIA, final state interaction effects for the outgoing nucleon are \ntaken into account. The distorted-wave function of the knocked out nucleon is \nevaluated as a solution of a Dirac equation containing a phenomenological \nrelativistic optical potential. \nWe use the LEA program~\\cite{LEA} for the numerical calculation of the \ndistorted wave functions with the EDAD1 parametrization~\\cite{Cooper} of the \nrelativistic optical potential for carbon. This code, initially designed for \ncomputing exclusive proton-nucleus and electron-nucleus scattering, was \nsuccessfully tested against A$(e,e'p)$ data~\\cite{Fissum, Dutta}, and we \nadopted this program for neutrino reactions.  \n\nA complex optical potential with a nonzero imaginary part generally produces \nan absorption of the flux. For the exclusive A$(l,l'N)$ channel this reflects \nthe coupling between different open reaction channels. \nHowever, for the inclusive reaction, the total flux must be conserved. \nTherefore, we calculate the inclusive and total cross sections with the EDAD1 \nrelativistic optical potential in which only the real part is included.\n\nThe inclusive cross sections with the FSI effects in the presence of the \nshort-range $NN$ correlations were calculated by using the method proposed in\n Refs.~\\cite{BAV3,BAV6}. In this approach the contribution of the $NN$ \ncorrelated pairs is evaluated in the IA, i.e., the virtual boson couples to only\n one member of the $NN$-pair. It is one-body process that leads to the emission \nof two nucleons (2p-2h excitation). The contributions of the \ntwo-body currents, such as meson-exchange currents and isobar currents,  are \nnot considered. \n\n\\subsection{Method for extraction of $F_A(Q^2)$ from flux-integrated \n$d\\sigma\/dQ^2$ cross sections}\n\nThe inclusive weak hadronic tensor $W_{\\mu\\nu}$ is given by bilinear products \nof the transition matrix elements of the nuclear weak current operators \n$J_{\\mu}$ between the initial and final nuclear states, i.e., \n$W_{\\mu\\nu} = \\langle J_{\\mu} J^{\\dagger}_{\\nu}\\rangle $, where the angle \nbrackets denote products of matrix elements, appropriately averaged over \ninitial states and summed over final states.\n\nBy using Eq.~\\eqref{Eq5}, the axial vector current can be factorized in the form\n\\begin{eqnarray}\n\\label{Eq7}\nJ_A &=& F_A(Q^2)J'_A(Q^2),   \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{Eq8}\nJ'_A &=& \\gamma^{\\mu}\\gamma_5 +F'_P(Q^2)q^{\\mu}\\gamma_5,  \n\\end{eqnarray}\nand the weak current can be expressed as $J = J_V +F_AJ'_A$. The expression for \nthe hadronic tensor is then given by\n\\begin{eqnarray}\n\\label{Eq9}\nW_{\\mu\\nu} &=& W^V_{\\mu\\nu} + F^2_A(Q^2)W^A_{\\mu\\nu} + hF_A(Q^2)W^{VA}_{\\mu\\nu},  \n\\end{eqnarray}\nwhere $W^V_{\\mu\\nu}=\\langle (J_V)_{\\mu} (J_V)^{\\dagger}_{\\nu}\\rangle$, \n$W^A_{\\mu\\nu}=\\langle (J'_A)_{\\mu} (J'_A)^{\\dagger}_{\\nu}\\rangle$,  \n$W^{VA}_{\\mu\\nu}=\\langle (J_V)_{\\mu} (J'_A)^{\\dagger}_{\\nu} + \n(J'_A)_{\\mu} (J_V)^{\\dagger}_{\\nu}\\rangle$, and $h$ is 1 for a neutrino and -1 for \nan antineutrino. \nFinally, contracting $W_{\\mu\\nu}$ with the lepton tensor we obtain the inclusive \n(anti)neutrino scattering cross section $d\\sigma\/dQ^2$ in terms of vector \n$\\sigma^V$, axial $\\sigma^A$, and vector-axial $\\sigma^{VA}$ cross sections\n\\begin{eqnarray}\n\\label{Eq10}\n\\frac{d\\sigma^{\\nu,~\\bar{\\nu}}}{dQ^2}(Q^2,\\varepsilon_i) &=& \n\\sigma^V(Q^2,\\varepsilon_i) + F^2_A(Q^2)\\sigma^A(Q^2,\\varepsilon_i) + \nhF_A(Q^2)\\sigma^{VA}(Q^2,\\varepsilon_i),\n\\end{eqnarray}\nwhere $\\sigma^V=d\\sigma\/dQ^2(F_A=0)$ and \n$\\sigma^A=d\\sigma\/dQ^2(F_V=F_M=0,F_A=1)$.\nThe vector (axial) cross section is due to the vector (axial) component of the \nweak current and it can be calculated as the $d\\sigma\/dQ^2$ cross section with \n$F_A(Q^2)=0$ $(F_V(Q^2)=F_M(Q^2)=0, F_A(Q^2)=1)$. So, the cross section \n$\\sigma^A(Q^2)$ does not depend on vector form factors, i.e., on the \nlongitudinal or transverse QE response functions. The vector-axial cross \nsection $\\sigma^{VA}$, arising from the interference between the vector and \naxial currents can be written as\n\\begin{eqnarray}\n\\label{Eq11}\n\\sigma^{VA} &=& [\\sigma(F_A=1) - \\sigma^V - \\sigma^A],  \n\\end{eqnarray}\nwhere $\\sigma(F_A=1)$ is the $d\\sigma\/dQ^2$ cross section, calculated with \n$F_A(Q^2)$=1.\n\nIn the simplest case of (anti)neutrino scattering off a free nucleon the cross \nsections $\\sigma^V$, $\\sigma^A$, and $\\sigma^{VA}$ can be expressed in terms of \nthe vector form factors $F_V$ and $F_M$. For instance~\\cite{Llew},\n\\begin{eqnarray}\n\\label{Eq12}\n\\sigma^{VA} &=& \\frac{G^2_F}{2\\pi}\\cos^2\\theta_c\\frac{Q^2}{m\\varepsilon_i}\n\\left(1-\\frac{Q^2}{4m\\varepsilon_i}\\right)[F_V(Q^2)+F_M(Q^2)],  \n\\end{eqnarray}\nwhere $G_F$ is the Fermi constant and $\\theta_c$ is the Cabibbo angle. The\ndifference \n\\begin{eqnarray}\n\\label{Eq13}\n\\frac{d\\sigma^{\\nu}}{dQ^2}(Q^2,\\varepsilon_i)-\n\\frac{d\\sigma^{\\bar{\\nu}}}{dQ^2}(Q^2,\\varepsilon_i) &=& \n2F_A(Q^2)\\sigma^{VA}(Q^2,\\varepsilon_i),\n\\end{eqnarray}\nis going to zero at $Q^2 \\to$0 and decreases with (anti)neutrino energy. \n\nOur RDWIA results for the $\\sigma^V$, $\\sigma^A$, and $\\sigma^{VA}$ cross \nsections for neutrino CCQE scattering off carbon are shown in Fig. 1 as \nfunctions of $Q^2$ for neutrino energies $\\varepsilon_{\\nu}$=0.5, 0.7, 1.2, and \n2.5 GeV. The cross section $\\sigma^V$ has a maximum at \n$Q^2\\approx 0.15$ (GeV\/c)$^2$ and depends slowly on neutrino energy. The \ncross section $\\sigma_A$ is dominant at $\\varepsilon_{\\nu} > 1$ GeV in the \nrange $Q^2 > 0.2$ (GeV\/c)$^2$\nand slowly decreases with $Q^2$. On the other hand, the cross section \n$\\sigma^{VA}$ decreases with neutrino energy as $\\sim 1\/\\varepsilon_{\\nu}$. \nAt $\\varepsilon_{\\nu} > 1$ GeV, it depends slowly on $Q^2$ in the range \n$Q^2>0.3$ (GeV\/c)$^2$.\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[height=16cm,width=16cm]{fig1_boon_FA.eps}\n  \\end{center}\n  \\caption{\\label{Fig1}(Color online) Differential cross sections $\\sigma_V$ \n(solid line), $\\sigma_A$ (dashed-dotted line), and $\\sigma_{VA}$ (dashed line) \nvs four-momentum transfer $Q^2$ for neutrino scattering off carbon calculated \nin the RDWIA approach for four values of incoming neutrino energy: \n$\\varepsilon_{\\nu}=0.5, 0.7, 1.2$ and 2.5 GeV.} \n\\end{figure*}\n \nIn neutrino experiments the differential cross sections of CCQE \nneutrino-nucleus scattering are measured within rather wide ranges of the \n(anti)neutrino energy spectrum. Therefore, flux-averaged and flux-integrated \ndifferential cross sections can be extracted.\nThe MiniBooNE $\\nu_{\\mu}$ and $\\bar{\\nu}_{\\mu}$ CCQE flux-integrated \n$\\left\\langle d\\sigma^{\\nu,\\bar{\\nu}}\/dQ^2 \\right\\rangle$ cross sections \nwere measured as functions of $Q^2$ in the range \n$0\\leq Q^2 \\leq 2$ (GeV\/c)$^2$~\\cite{MiniB1,MiniB3}. These cross sections can \nbe written as \n\\begin{eqnarray}\n\\label{Eq14}\n\\left\\langle \\frac{d\\sigma^{\\nu,\\bar{\\nu}}}{dQ^2}(Q^2)\\right\\rangle &=& \n\\int_{\\varepsilon_{min}}^{\\varepsilon_{max}}W_{\\nu,\\bar{\\nu}}(\\varepsilon_i)\n\\frac{d\\sigma^{\\nu,\\bar{\\nu}}}{dQ^2}(Q^2,\\varepsilon_i) \nd\\varepsilon_i.\n\\end{eqnarray}\nThe weight functions $W_{\\nu,\\bar{\\nu}}$ are defined as \n\\begin{eqnarray}\n\\label{Eq15}\nW_{\\nu,\\bar{\\nu}}(\\varepsilon_i) &=& I_{\\nu,\\bar{\\nu}}(\\varepsilon_i)\/\n\\Phi_{\\nu,\\bar{\\nu}},\n\\end{eqnarray}\nwhere $I_{\\nu,\\bar{\\nu}}(\\varepsilon_i)$ is the neutrino (antineutrino) spectrum \nand $\\Phi_{\\nu,\\bar{\\nu}}$ is the neutrino (antineutrino) flux in \n$\\nu(\\bar{\\nu})$-beam mode~\\cite{Flux}, integrated over \n$0\\leq \\varepsilon_i \\leq 3$ GeV. Combining Eq.~\\eqref{Eq10} with \nEq.~\\eqref{Eq14} we obtain the flux-integrated $\\langle \nd\\sigma^{\\nu,\\bar{\\nu}}\/dQ^2\\rangle$ cross section in terms of flux-integrated\n $\\langle \\sigma^V\\rangle$, $\\langle \\sigma^A\\rangle$, and \n$\\langle \\sigma^{VA}\\rangle$ cross sections \n\\begin{eqnarray}\n\\label{Eq16}\n\\left\\langle\\frac{d\\sigma^{\\nu,\\bar{\\nu}}}{dQ^2}(Q^2)\\right\\rangle &=& \n\\langle\\sigma^V(Q^2)\\rangle^{\\nu,\\bar{\\nu}} + \nF^2_A(Q^2)\\langle\\sigma^A(Q^2)\\rangle^\n{\\nu,\\bar{\\nu}} + h F_A(Q^2)\\langle\\sigma^{VA}(Q^2)\\rangle^{\\nu,\\bar{\\nu}},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{Eq17}\n{\\left\\langle \\sigma^j(Q^2)\\right\\rangle}^{\\nu,\\bar{\\nu}} &=& \n\\int_{\\varepsilon_{min}}^{\\varepsilon_{max}}W_{\\nu,\\bar{\\nu}}(\\varepsilon_i)\n[\\sigma^j(Q^2,\\varepsilon_i)]^{\\nu,\\bar{\\nu}} d\\varepsilon_i\n\\end{eqnarray}\nare the flux-integrated vector, axial, and vector-axial $(j=V,A,AV)$ cross \nsections. The values of $F_A(Q^2)$ can be extracted as the \nsolution of Eq.~\\eqref{Eq16}, by using the data for \n$\\langle d\\sigma^{\\nu,\\bar{\\nu}}\/dQ^2\\rangle$. In the case of neutrino scattering \non deuterium (quasifree nucleon) this procedure was applied in  \nRef.~\\cite{Budd} for the extraction of $F_A(Q^2)$ as a function of $Q^2$.\n\nBecause the flux-integrated cross \nsections $\\left\\langle \\sigma^j \\right \\rangle$ depend on the nuclear model, \nvector form factors, and the predicted (anti)neutrino flux, the extracted \nvalues of the axial form factor are model dependet and implicitly include the \nuncertainties in the $F_V$, $F_M$, and $(\\bar{\\nu}_{\\mu})\\nu_{\\mu}$ flux.    \n\n\n\\section{Results and analysis}\n\n\\subsection{CCQE flux-integrated $\\langle d\\sigma^i(Q^2)\\rangle$ differential \ncross sections}\n\nThe MiniBooNE $\\nu_{\\mu}(\\bar{\\nu}_{\\mu})$ CCQE flux-integrated differential \ncross sections $d\\sigma\/dQ^2$ were extracted as functions of $Q^2$ in the \nrange $0 \\leq Q^2\\leq 2$ (GeV\/c)$^2$~\\cite{MiniB1,MiniB3}. The cross sections \nare scaled with the number of neutron (proton) in the target. \n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[height=16cm,width=16cm]{fig2_boon_FA.eps}\n  \\end{center}\n  \\caption{\\label{Fig2}(Color online) Flux-integrated \n$\\langle \\sigma^V\\rangle^{\\nu,\\bar{\\nu}}$, $\\langle \\sigma^A\\rangle^{\\nu,\\bar{\\nu}}$, \nand $\\langle \\sigma^{VA}\\rangle^{\\nu,\\bar{\\nu}}$  cross sections as functions of \nfour-momentum transfer $Q^2$. As shown in the key, the cross sections were \ncalculated in the RDWIA and RFGM with BNB $\\nu_{\\mu}$ and $\\bar{\\nu}_{\\mu}$ \nfluxes.} \n\\end{figure*}\nA ``shape-only'' fit to the cross sections was performed\n to extract values for adjusted CCQE model parameters, $M_A$ and $\\kappa$, \nwithin the Fermi gas model with the dipole parametrization of $F_A(Q^2)$. \nTo tune this model to the low $Q^2$, the parameter $\\kappa$~ was introduced.\nThe ``shape-only'' fit yields the model parameters, \n$M_A=1.35\\mbox{\\boldmath $p_{\\mathrm m}$} 0.17$ (GeV\/c)$^2$ \nand $\\kappa=1.007\\mbox{\\boldmath $p_{\\mathrm m}$} 0.012$. The extracted value for $M_A$ is approximately \n30\\% higher than the world-averaged one. The prediction of the RFGM with these \nvalues of the parameters also describes well the measured (anti)neutrino \nflux-folded differential cross section $d\\sigma^{\\nu,\\bar{\\nu}}\/dQ^2$.\n\nTo extract the values of the axial form factor $F_A(Q^2)$ as a function of \n$Q^2$ from the MiniBooNE neutrino and antineutrino flux-folded \n$d\\sigma^{\\nu,\\bar{\\nu}}\/dQ^2$ cross sections we calculated the flux-integrated \n$\\langle \\sigma^V\\rangle^{\\nu,\\bar\\nu}$, $\\langle \\sigma^A\\rangle^{\\nu,\\bar\\nu}$, \nand $\\langle \\sigma^{VA}\\rangle^{\\nu,\\bar\\nu}$ cross sections with booster \nneutrino beam line (BNB) $\\nu_{\\mu}$ and $\\bar{\\nu}_{\\mu}$ fluxes. In Fig. 2 \nthese cross sections, calculated within the RDWIA \nand RFGM (with the Fermi momentum $p_F=221$ MeV\/c and a binding energy \n$\\epsilon_b=25$ MeV for carbon), are shown as functions of $Q^2$. In the region\n $Q^2 \\leq 0.25$(GeV\/c)$^2$, the Fermi gas model results are higher than those \nobtained within the RDWIA. At $Q^2 \\leq 0.05$(GeV\/c)$^2$ this discrepancy \nequals $\\sim 10$\\% for $\\langle \\sigma^V\\rangle$, $\\sim 12$\\% for \n$\\langle \\sigma^A\\rangle$, and $\\sim 8$\\% for $\\langle \\sigma^{VA}\\rangle$. \nTo extract values for the axial form factor we calculated \n$\\langle \\sigma^i \\rangle$ $(i=V,A,VA)$ cross sections with the BNB flux using \nthe $Q^2$ bins $\\Delta Q^2=Q^2_{i+1}-Q^2_i$ similar to Refs.~\\cite{MiniB1,MiniB2}\n\\begin{eqnarray}\n\\label{Eq18}\n\\langle\\sigma^i\\rangle_j=\n\\frac{1}{\\Delta Q^2}\\int_{Q^2_j}^{Q^2_{j+1}}\n\\langle\\sigma^i(Q^2)\\rangle dQ^2 \n\\end{eqnarray}\n\n\\subsection{Extraction of the axial form factor}\n\nFlux-integrated $\\langle d\\sigma^{\\nu}\/dQ^2\\rangle$ and \n$\\langle d\\sigma^{\\bar{\\nu}}\/dQ^2\\rangle$ cross sections for \n$\\nu_{\\mu}$ and $\\bar{\\nu}_{\\mu}$ CCQE scattering as functions of $Q^2$ together \nwith the MiniBooNE data~\\cite{MiniB1,MiniB3} are shown in Figs. 3 and 4 \n(upper panel) correspondingly. The neutrino (antineutrino) cross section are \nscaled with the number of neutrons (protons) in the target.\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[height=16cm,width=16cm]{fig3_boon_FA.eps}\n  \\end{center}\n  \\caption{\\label{Fig3}(Color online) Flux-integrated \n$\\langle d\\sigma\/dQ^2\\rangle^{\\nu}$ cross \nsection per neutron target for the $\\nu_{\\mu}$ CCQE scattering (upper panel) and\n the normalized axial form factor $F_A(Q^2)\/F_A(0)$ extracted from the \nMiniBooNE data (lower panel). Upper panel: Calculations from the RDWIA with \n$M_A=1.37$ GeV and RFGM with $M_A=1.36$ GeV. Lower panel: Filled \ncircles (filled squares) are the axial form factor extracted within the RDWIA \n(RFGM) and the solid line is the result of the dipole parametrization with \n$M_A=1.37$ GeV.} \n\\end{figure*}\nThey are calculated in the RDWIA\n ($M_A=1.37$ GeV) and RFGM ($M_A=1.36$ GeV) approaches. Also shown \nin Figs. 3 and 4 are the normalized axial form factor $F_A(Q^2)\/F_A(0)$ (lower \npanel) extracted within the RDWIA and Fermi gas model from the \nmeasured cross sections. \n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[height=16cm,width=16cm]{fig4_boon_FA.eps}\n  \\end{center}\n  \\caption{\\label{Fig4}(Color online) The same as Fig. 3, but for antineutrino \nscattering} \n\\end{figure*}\nThe differences of the measured cross sections \n$\\Delta [d\\sigma\/dQ^2]= \\langle d\\sigma\/dQ^2\\rangle^{\\nu} - \\langle d\\sigma\/dQ^2\n\\rangle^{\\bar{\\nu}}$ \nare shown in Fig. 5 (upper panel) as a function of $Q^2$ compared with the \nRDWIA and RFGM calculations. Also shown are the normalized axial form factors \nextracted in these approaches from $\\Delta [d\\sigma\/dQ^2]$ (lower panel). The \ntotal normalization error on the neutrino (antineutrino) cross section \nmeasurement is 10.7\\% (17.2\\%). To extract values of $F_A(Q^2)$ the measured \ncross sections with `` the shape-only'' error were used in Eq.~\\eqref{Eq16}.   \n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[height=16cm,width=16cm]{fig5_boon_FA.eps}\n  \\end{center}\n  \\caption{\\label{Fig5}(Color online) The same as Fig. 3 but for a difference \nof the measured cross sections $\\Delta [d\\sigma\/dQ^2]$.} \n\\end{figure*}\n\nThere is an overall agreement between the calculated and measured neutrino \ncross sections across the full range of $Q^2$. The values of the normalized \naxial form factors extracted within the RDWIA are similar to the RFGM result. \nA good match between the dipole parametrization with $M_A=1.37$ GeV and \nextracted form factors is observed. For antineutrino scattering there is an \nagreement between the calculated results and the data at $Q^2 \\ge 0.1$ \n(GeV\/c)$^2$ within the error of the experiment. However, the RDWIA result \nunderestimates the measured cross section at $Q^2 < 0.1$ (GeV\/c)$^2$. The \nvalues of $F_A(Q^2)$ extracted in the RDWIA and Fermi gas approaches at \n$Q^2 > 0.1$ (GeV\/c)$^2$ also agree well with the dipole parametrization, \nwhereas at $Q^2 \\le 0.1$ (GeV\/c)$^2$ the RDWIA result is higher than the RFGM \nand dipole parametrization prediction. A good match between the calculated \nand measured differences $\\Delta [d\\sigma\/dQ^2]$ is observed. The extracted \naxial form factors also agree well with the dipole approximation \nprediction with $M_A=1.37$ GeV.\n\nIn Ref.~\\cite{Bod1} it was assumed that enhancements in the transverse \n(anti)neutrino CCQE cross section are modified $F_M(Q^2)$, for a bound nucleon \nat low $Q^2\\approx 0.3$ (GeV\/c)$^2$.  The authors proposed a transverse \nenhancement function for the carbon target. If the TE   \noriginates from the MEC, then we may expect that enhancement in the \nlongitudinal or axial contributions is small. Therefore in the TE model \n$F_V(Q^2)$ and $F_A(Q^2)$ are the same as for free nucleons. This approach was \nproposed to explain the apparent discrepancy between the low-(MiniBooNE) and \nhigh-(NOMAD) energy (anti)neutrino CCQE cross sections and $M_A$ measurements.\n\nTo study the TE effects on the extracted axial form factor we compare results \nof the RFGM ($M_A=1.36$ GeV) and the Fermi gas model ($M_A=1.014$ GeV) with the \ntransverse enhancement function from Ref.~\\cite{Bod1}. We call the last \napproach the RFGM + TE model~\\cite{Sob2}. The flux-integrated cross sections \n$\\langle d\\sigma^{\\nu}\/dQ^2\\rangle$ scaled with a number of neutron per target \nas functions of $Q^2$ together with the MiniBooNE data~\\cite{MiniB1} are shown \nin Fig. 6. The upper panel shows cross sections calculated in the RFGM and \nRFGM+TE models. Also shown in Fig. 6 (lower panel) are normalized axial form \nfactors extracted in these approaches from measured cross section.\n\nIn the range $Q^2\\le 0.5$ (GeV\/c)$^2$ the differential cross section  \n$\\langle d\\sigma^{\\nu}\/dQ^2\\rangle$ calculated within the RFGM+TE model is in \nagreement with the Fermi gas model prediction. However, in the region \n$Q^2\\ge 0.7$ (GeV\/c)$^2$ the RFGM+TE result is lower than the measured cross \nsection. Therefore,\n there is a disagreement between the form factors extracted in this approach \nand predicted by the dipole parametrization with $M_A=1.014$ GeV. In the \nregion $Q^2 \\approx 0.2-0.3$ (GeV\/c)$^2$ where an enhancement in the $F_M(Q^2)$ \nwas assumed, the extracted values of $F_A$ are lower than those predicted from\n the dipole approximation. At $Q^2 > 0.6$ (GeV\/c)$^2$ the enhancement function \ndisappears with higher values of $Q^2$ and the value of $F_A$ extracted in the  \nRFGM+TE model start approaching those extracted from the RFGM.\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[height=16cm,width=16cm]{fig6_boon_FA.eps}\n  \\end{center}\n  \\caption{\\label{Fig6}(Color online) The same as Fig. 3, but calculated \n$\\langle d\\sigma^{\\nu}\/dQ^2\\rangle$ cross sections per neutron target and \nextracted form factors are from the RFGM ($M_A=1.36$ GeV) and RFGM+TE \n($M_A=1.014$ GeV)} \n\\end{figure*}\nAs shown in Fig. 6 the shape of the $Q^2$ dependence of the axial form factor \nextracted within the RFGM+TE approach cannot be well described by the dipole \nansatz. So, assuming the dipole parametrization of $F_A(Q^2)$ the \nMiniBooNE data may be described within the impulse approximation with large \naxial mass value $\\sim 1.3$ GeV as well as within the approaches that \ninclude sizable MEC and IC contributions and allow reproduction of data with \n$M_A\\sim 1$ GeV. But for the self-consistent description of the MiniBooNE \ndata with the enhancement in the transverse response function (at least as it \nwas proposed in Ref.~\\cite{Bod1}), one should use a parametrization of \n$F_A(Q^2)$ other than a dipole. \n\nAs was pointed in Ref.~\\cite{Masjuan}, the dipole approximation cannot be \njustified from a field-theoretic point of view and is in contradiction with \nthe large-$N_c$-motivated parametrization. We calculated the normalized axial \nform factor by using the minimum meson-dominance ansatz from \nRef.\\cite{Masjuan}: \n\\begin{eqnarray}\n\\label{Eq19}\nF_A(Q^2)=F_A(0)\\frac{m^2_{a_1}m^2_{a'_1}}{(m^2_{a_1} +Q^2)(m^2_{a'_1}+Q^2)}, \n\\end{eqnarray}\nwhere $m_{a_1}=1.230$ GeV, and $m_{a'_1}=1.647$ GeV. By applying the half-width \nrule $m_R \\mbox{\\boldmath $p_{\\mathrm m}$} \\Gamma_R\/2$ to this parametrization with $\\Gamma_{a_1}=0.425$ GeV \nand $\\Gamma_{a'_1}=0.254$ GeV, we get results depicted in Fig. 7. The errors in \nthe meson-dominanted form factor (the band for the form factor) are estimated \nby treating resonance masses as random variables distributed with the \ndispersion given by the width. As we can see, the agreement between the \nextracted form factor and the meson dominance ansatz is impressive. Actually, \nthe two axial mesons are incorporated as a product of monopoles, but the net \neffect is essentially a dipole form factor with an average mass which is \nlarger than $m_R=1$ GeV. \n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[height=15cm,width=15cm]{fig66_boon_FA.eps}\n  \\end{center}\n  \\caption{\\label{Fig7}(Color online) The normalized axial form factor \n$F_A(Q^2)\/F_A(0)$ extracted from MiniBooNE data. The dashed line is the result \nof the dipole parametrization with $M_A=1.36$ GeV, dashed-dotted line is the \nresult of the meson-dominance ansatz, and solid lines show the bands for the \nform factor due to the half-width of the meson masses.} \n\\end{figure*}\n\n\\subsection{CCQE total cross section}\n\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[height=17cm,width=17cm]{fig7_boon_FA.eps}\n  \\end{center}\n  \\caption{\\label{Fig8}(Color online) Total $\\nu_{\\mu} (\\bar{\\nu}_{\\mu})$ CCQE \ncross section per neutron (proton) target as a function of neutrino energy. \nData points for different targets are from Refs.~\\cite{SciB,MiniB2,Mann,Baker,\nPohl,Brunner}. Also shown are predictions of the RDWIA ($M_A=1.37$ GeV), \nPWIA ($M_A=1.37$ GeV), and RFGM ($M_A=1.36$ GeV).}\n\\end{figure*}\nWe calculated the total cross sections for CCQE antineutrino scattering on \ncarbon in the plane-wave impulse approximation (PWIA), RDWIA, and RFGM \napproaches. The cross section per proton target as a function of antineutrino \nenergy is shown in Fig. 8 (lower panel) together with the data from  \nRefs.~\\cite{MiniB1,MiniB3,Mann, Baker, Pohl, Brunner}. Also shown are (upper \npanel) the total cross sections per neutron target for neutrino scattering \nfrom Ref.~\\cite{BAV1}. The calculated cross sections, which use the values of \n$M_A$ extracted from the shape-only fit to the flux-integrated \n$d\\sigma^{\\nu,~\\bar{\\nu}}\/Q^2$ data, reproduce the MiniBooNE total cross section \nwithin the errors. At the energy of $\\varepsilon_{\\nu}\\approx 700$ MeV, the \nextracted cross section is $\\approx 30\\%$ higher than what is commonly assumed \nfor this process assuming the RFGM and world-average value of the axial mass \n$M_A=1.03$ GeV. As shown in Fig. 7 the spread in the data is much higher than \nthe difference in predictions of the RDWIA, PWIA, and RFGM approaches. \n\n\\section{Conclusions}\n\nIn this paper, we present a method which allows extraction of the axial form \nfactor as a function of $Q^2$ from flux-integrated \n$\\langle d\\sigma^{\\nu,\\bar{\\nu}}\/dQ^2\\rangle$ cross sections. This method is based\n on the fact that the $d\\sigma\/dQ^2$ can be written as the sum of the vector, \naxial and vector-axial cross sections and the contributions of the axial and \nvector-axial ones are proportional to $F_A^2(Q^2)$ and $F_A(Q^2)$ \ncorrespondingly. In our analysis we used the differential \n$\\langle d\\sigma^{\\nu,\\bar{\\nu}}\/dQ^2\\rangle$ CCQE cross sections with \n``a shape-only'' error measured in the MiniBooNE experiment. \n\nIn the RDWIA, RFGM, and RFGM+TE approaches with the BNB $\\nu_{\\mu} \n(\\bar{\\nu}_{\\mu})$ flux we calculated the flux-integrated vector, axial, and \nvector-axial cross sections which were used for extraction of the axial form \nfactor. The values of $F_A(Q^2)$ extracted in the RDWIA and Fermi gas models at\n $Q^2 > 0.1$ (GeV\/c)$^2$ agree well with the dipole parametrization with \n$M_A\\approx 1.37$ GeV. \nOn the other hand the vector meson-dominance ansatz that is simple and has good \ntheoretical base describes the extracted form factor also with a good \nagreement. We can argue that there is no need to use the dipole approximation \nfor fitting the MiniBooNE data and that the meson dominance already contains \nthe essential physical information, whereas we found that there is a \ndisagreement between the form factors extracted in the RFGM+TE approach and \npredicted by the dipole approximation with $M_A\\approx 1.04$ GeV. The RDWIA, \nPWIA and RFGM calculated with $M_A=1.37$ GeV and measured neutrino and \nantineutrino CCQE total cross sections match well within the experimental \nerror over the entire measured range. \n\nWe conclude that the MiniBooNE measured inclusive and total cross sections can \nbe described within the impulse approximation with the meson-dominance ansatz \nand\/or dipole approximation of $F_A(Q^2)$ with large axial mass value.\nIn addition, one could also describe MiniBooNE data with the approach that \nincorporates the large MEC contributions. However, \na non-dipole description of the axial from factor would have to be used in \nthat case. From the quality of the fit to the measured MiniBooNE CCQE cross \nsection, one could not discriminate between these approaches.\n\nSo, to distinguish between various possible mechanisms, it is necessary to \ndecompose the inclusive process into its constituents exclusive channels, at \nleast to study the semiexclusive $\\nu_{\\mu}+A \\to \\mu + p + B$ process. In this\n manner one hopes to disentangle the roles of $NN$ correlations in the target \nground state, MEC, IC, medium modifications properties of the bound nucleon, \nmany-body currents, FSI, relativistic corrections et cetera, \netc.~\\cite{Kelly2}  \n\n\n\\section*{Acknowledgments}\n\nThe authors gratefully acknowledge P.~Masjuan, E.~R.~Arriola, W.~Broniowsky, \nJ.~Amaro, and N.~Evans for fruitful discussion the results obtained \nwithin the meson dominance model and a critical reading of the manuscript.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}