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+{"text":"\\section{}\n\nMuch of our knowledge on quantum nature of atomic nuclei \ncomes from studies of nuclear reactions \nin which an energetic beam of one nuclear species \ncollides with a target made of another. \nAmong various collision processes, \nthe nucleon knockout reaction \nhas become recognized as one of the most sensitive tools \nfor spectroscopic studies, \nespecially for nuclei away from the stability line, \nwhich include even those beyond the drip line. \nThe knockout residue produced by removing a nucleon (or nucleons) \nfrom a fast moving beam particle, \nwhich impinges on a light target fixed in the laboratory, \nis observed in inverse kinematics \nby a detector placed in forward hemisphere efficiently. \nThe removed nucleon(s) will be selected democratically from the valence space, \nallowing states with unique, often rarely accessible configurations \nto be populated in this process. \nThe final state in the residue is identified \nby tagging de-excitation \n$\\gamma$ rays~\\cite{Navin08,Aumann00,Maddalena01} \n(see also references in Ref.~\\cite{Gade08b}) \nand by observing decay neutrons and constructing the invariant \nmass~\\cite{Simon99,Hoffman08,Hoffman09,Kondo10,Cao12,Lunderberg12,Aksyutina13,Kohley13}. \nFor one-nucleon knockout case, \nthe momentum spread of the residue reflects the Fermi motion of the nucleon \nsuddenly removed, \nand is sensitive to the orbital angular momentum (the $l$ value) \nof the struck nucleon. \nFurthermore, \nthe cross sections leading to individual final states \nrelate to the occupancy of single-particle orbits, \nproviding a link to details of the nuclear structure. \n\nThe present study aims at exploring unbound states in $^{16}{\\rm C}$ \nthrough an application of the one-neutron knockout technique \nto a $^{17}{\\rm C}$ beam \nimpinged on a proton target, \nfor which simple reaction mechanisms are expected. \nFocus is placed in a search of lowest-lying cross shell transitions, \nthe location of which reflects the shell gap between $p$ and $sd$ orbits. \nThe neutron-rich carbon (C) isotopes \nhave attracted attention as they often exhibit unique features: \nnone of the odd mass C (heavier than $^{13}{\\rm C}$) \nhas the ground-state spin parity of $J^{\\pi}_{\\rm g.s.}$=5\/2$^+$, \nthe values which are expected from a naive shell model. \nThere has been a debate about a reduced E2 transition strength \n(small proton collectivity) for the $2^+_1$ state \nin $^{16}{\\rm C}$~\\cite{Imai04,Elekes04,Ong08,Wiedeking08,Wuosmaa10,Petri12}. \nThere is evidence for neutron halo formation for \n$^{15}{\\rm C}$~\\cite{Sauvan00}, \n$^{19}{\\rm C}$~\\cite{Nakamura99}, and \n$^{22}{\\rm C}$~\\cite{Tanaka10,Kobayashi12}. \nFor some, if not all, of these features, \nnuclear deformation may play a key role, \nwhich occurs in this mass region \ndue to near degeneracy of the $\\nu d_{5\/2}$-$\\nu s_{1\/2}$ orbits: \nneutrons in these orbits can gain energy by breaking spherical symmetry \n(the Jahn-Teller effect)~\\cite{Hamamoto07}. \nThe effect of nuclear deformation \nwill further be signified by large quadrupole transition \nstrengths~\\cite{Elekes05,Ysatou08} \nand by a reduction of the major \n$p$-$sd$ shell gap~\\cite{Talmi60,Suzuki97,Wiedeking05}. \nA recent $\\beta$-delayed neutron \nemission study of $^{17}{\\rm B}$~\\cite{Ueno13} \nhas reported low-lying negative parity states in $^{17}{\\rm C}$, \namong which the lowest one was the $J^{\\pi}$=1\/2$^-$ state \nat the excitation energy of $E_x$=2.71(2) MeV. \nThe energy of this state, \nreflecting the $p$-$sd$ shell gap, \nturned out to be \nlower than those of neighboring odd mass C isotopes:  \n$E_x$=3.10 MeV for the $1\/2^-_1$ state in $^{15}{\\rm C}$ \nand $E_x$=3.09 MeV for the $1\/2^+_1$ state \nin $^{13}{\\rm C}$~\\cite{Ajzenberg91}. \nThis \nmight indicate an onset of the $p$-$sd$ shell gap quenching \ntowards heavier C isotopes. \nTo examine this picture in more detail \nit is worthwhile to accumulate data \non cross shell transitions in neighboring isotopes. \nThis Letter reports on new relevant spectroscopic information \non $^{16}{\\rm C}$ in its unbound $E_x$ region. \nBesides, \nbased on the parallel momentum distribution \nof the core fragment populated in a final state, \nit is demonstrated that the width of the distribution \nprovides a good measure of the $l$ value (and thus the parity) \nof the state populated; \nfor neutron knockout involving a proton target, \nthis has previously been shown \nbased on the transverse momentum distributions \nin the $^1{\\rm H}(^{18}{\\rm C},$$^{17}{\\rm C}^*)$~\\cite{Kondo09} \nand $^1{\\rm H}(^{14}{\\rm Be},$$^{13}{\\rm Be}^*)$~\\cite{Kondo10} reactions \nwith the aid of elaborate reaction mechanism calculations. \n\nDespite relative proximity to stability, \ninformation on energy levels of $^{16}\\textrm{C}$, \nparticularly that above the neutron threshold \n(the neutron separation energy of $^{16}{\\rm C}$ \nis $S_n$=4.250(4) MeV~\\cite{Audi03}), \nhas been limited. \nThis is partially due to ineffectiveness of $\\beta$ decay \nfor this particular nucleus, \nas recognized by the absence of a parent nucleus \n($^{16}{\\textrm B}$ is particle unstable). \nEarly spectroscopic studies on $^{16}\\textrm{C}$ \nutilized binary reactions involving transfers of neutrons. \nThe $^{14}\\textrm{C}(t,p)^{16}\\textrm{C}$ two-neutron transfer \nstudies~\\cite{Fortune77,Balamuth77,Fortune78,Sercely78} \nhave investigated levels below 7 MeV, \nincluding six bound states and an unbound state at 6.11 MeV. \nSince the ground state of $^{14}\\textrm{C}$ \nis characterized by neutron $p$-shell closure, \nthe states populated \nmostly involved configurations with two $sd$-shell neutrons, $(1s0d)^2$. \nThe $^{13}\\textrm{C}(^{12}\\textrm{C},$$^9\\textrm{C})^{16}\\textrm{C}$ \nthree-neutron transfer study~\\cite{Bohlen03} \nhas reported 14 more states up to $E_x$=17.4 MeV, \nincluding states with more complex configurations. \nDue to kinematical matching~\\cite{Bohlen03} \nstates with high angular momenta \nwere favorably populated. \nCombining information \nfrom the recent $^{15}\\textrm{C}(d,p)^{16}\\textrm{C}$ reaction study \nusing a radioactive $^{15}\\textrm{C}$ beam~\\cite{Wuosmaa10}, \nsound $J^{\\pi}$ assignments \nhave been available for bound states. \nFor unbound states \nonly the 8.92-MeV level \nhas received a firm assignment of $5^-$~\\cite{Bohlen03}. \nTwo earlier one-neutron knockout studies on $^{17}{\\rm C}$ using Be targets \nfocused on transitions leading to bound final states \nin $^{16}{\\rm C}$ by means of \nin-beam $\\gamma$-ray spectroscopy~\\cite{Maddalena01,Rodriguez-Tajes12}. \nThey provided information \nnot only on excited states of $^{16}{\\rm C}$ \nbut also on ground state properties of $^{17}{\\rm C}$, \ne.g., the spin parity, \n$J^{\\pi}_{\\rm g.s.}(^{17}{\\rm C})$=$3\/2^+$, and \nno halo formation in spite of the remarkable low neutron separation energy \nof $S_n$=0.727(18) MeV~\\cite{Audi03} \ndue to a high angular momentum of $l$=2 for the valence neutron. \n\nThe experiment was performed at the RIPS facility~\\cite{Kubo92} of RIKEN. \nDetails of the setup \nare provided in Refs.~\\cite{Ysatou08,Tshoo12}, \nand a preliminary report of this work has been presented \nin Ref.~\\cite{Hwang13}. \nThe $^{17}\\textrm{C}$ beam \nwas produced from a 110-MeV\/nucleon $^{22}\\textrm{Ne}$ beam \nwhich impinged on a 6-mm-thick Be target. \nThe typical $^{17}\\textrm{C}$ beam intensity \nwas 10.2 kcps with a momentum spread of $\\Delta P\/P$=$\\pm 0.1\\%$. \nThe beam profile \nwas monitored by a set of parallel-plate avalanche counters (PPACs) \nplaced upstream of the experimental target. \nThe target \nwas pure liquid hydrogen~\\cite{Ryuto05} contained in a cylindrical cell: \n3 cm in diameter, 120$\\pm$2 mg\/cm$^2$ in thickness, \nand having 6-$\\mu$m-thick Havar foils for the entrance and exit windows. \nThe average energy of $^{17}\\textrm{C}$ at the middle of the target \nwas 70 MeV\/nucleon. \nThe target was surrounded by a NaI(Tl) scintillator array \nused to detect $\\gamma$ rays from charged fragments. \nThe fragment was bent by a dipole magnet behind the target, \nand was detected by a plastic counter hodoscope \nplaced downstream of the magnet. \nThe $\\Delta E$ and time-of-flight (TOF) information in the hodoscope \nwas used to identify the $Z$ number of the fragment. \nThe trajectory \nwas reconstructed by a set of multi-wire drift chambers (MWDCs) \nbefore and after the magnet, \nwhich, together with TOF, \ngave mass identification. \nNeutrons were detected by two walls \nof plastic scintillator arrays placed 4.6 and 5.8 m downstream from the target. \nThe neutron detection efficiency \nwas $24.1\\pm0.8\\%$ for a threshold setting of 4 MeVee. \nThe relative energy ($E_{\\rm rel}$) of the final system \nwas calculated from momentum vectors of the charged fragment and the neutron. \nIn deducing the fragment vector, \ninformation on the impact point on target in transverse directions \n(determined by the upstream tracking detectors) \nwas taken into account \ntogether with hit information within the MWDC placed behind the target. \nNeutron coincidence events \nwere classified in terms of $E_{\\rm rel}$ \nand the Fermi momentum of the struck neutron $k_3$. \nIn the sudden approximation, \nthe latter \ncorresponds to the transferred momentum \nto the knockout residue ($^{16}{\\rm C}$). \nThe detector acceptance \nwas evaluated by a Monte Carlo simulation \nas a function of $E_{\\rm rel}$ and $k_3$. \n\nFigure~\\ref{fig:spectrum_fit} \nshows relative energy dependence of cross sections for the \n(a) $^1\\textrm{H}(^{17}\\textrm{C},$$^{15}\\textrm{C}$+$n)$ and \n(b) $^1\\textrm{H}(^{17}\\textrm{C},$$^{15}\\textrm{C}\n(5\/2^+; 0.74\\ {\\rm MeV})$+$n)$ reactions at 70 MeV\/nucleon. \nBackground contributions measured by an empty target were subtracted. \nError bars are statistical ones. \nShown in the inset of Fig.~\\ref{fig:spectrum_fit} (b) \nis the \"Doppler-corrected\" energy spectrum for $\\gamma$ rays \nemitted from the decay product nucleus $^{15}{\\rm C}$. \nA peak around $E_{\\gamma}$=0.8 MeV \narises from the decay of the first $5\/2^+$ level at 0.74 MeV \n(the only bound excited state) \nto $^{15}{\\rm C}_{\\rm g.s.}$. \nThe $5\/2^+$ state \nis an isomeric state having a half-life of 2.61$\\pm$0.07 ns~\\cite{Ajzenberg91}. \nThis \nlong life time \ncaused the emission point \nof the de-excitation $\\gamma$ ray \nto be distributed along the path of the fast-moving decay product. \nThe Doppler correction for the $\\gamma$ ray energy \nwas made by assuming that the decay occurs at 40.7 cm downstream of the target \n(an average decay point expected from the average beam energy \nand the known mean life for the isomeric state) \nin both data reduction and simulation by the \\textsc{geant} code~\\cite{GEANT}. \nThe latter \nwas done fully taking into account realistic geometry of the experiment. \nA higher energy tail for the photo-electric peak \nis due to this incomplete Doppler correction procedure. \nThe simulated response, however, \nreproduced the data well \nas shown by the green solid line. \nThe photo-peak efficiency over $E_{\\gamma}$=0.60--1.12 MeV \n($\\gamma$-ray window) \nwas estimated to be 5.1(3)\\% by the \\textsc{geant} simulation. \nFigure~\\ref{fig:spectrum_fit} (b) \nwas obtained by gating on the $5\/2^+$ $\\gamma$ peak \nwith the above window \nand by correcting for the detection efficiency. \nThe background component \nwas subtracted \nby assuming (i) \nthat the background portion is the same \nas that in the $\\gamma$-ray spectrum \nin the inset of Fig.~\\ref{fig:spectrum_fit} (b): \nthe portion of the area beneath the dotted line over the $\\gamma$-ray window, \nwhich amounts to 46\\%, \nand (ii) \nthat the background shape is \ncharacterized by \nthe inclusive spectrum in Fig.~\\ref{fig:spectrum_fit} (a). \nA peak \nis clearly seen at $E_{\\rm rel}$=0.46 MeV \nin Fig.~\\ref{fig:spectrum_fit} (a). \nThis is also evident in Fig.~\\ref{fig:spectrum_fit} (b), \nindicating that \nthis peak feeds the $5\/2^+$ state in $^{15}{\\rm C}$ after emitting a neutron. \nBesides, another resonance \nis visible at $E_{\\rm rel}$$\\approx$1.3 MeV \nin Fig.~\\ref{fig:spectrum_fit} (b). \nThese were observed for the first time. \n\n\\begin{figure}[t]\n\\resizebox{0.90\\columnwidth}{!}{%\n\\includegraphics[angle=-90]{spectrum_sub.eps}%\n}\n\\caption{(Color online.) \nRelative energy spectra \nfor the \n(a) $^1\\textrm{H}(^{17}\\textrm{C},$$^{15}\\textrm{C}$+$n)$ and \n(b) $^1\\textrm{H}(^{17}\\textrm{C},$$^{15}\\textrm{C}\n(5\/2^+; 0.74\\ {\\rm MeV})$+$n)$ \none-neutron knockout reactions at 70 MeV\/nucleon. \nShown in the inset of panel (b) \nis the Doppler-corrected energy spectrum of $\\gamma$ rays \nemitted from $^{15}{\\rm C}$. \nNeutron coincidence is required for this spectrum. \nGreen solid lines represent the results of the fit; \ndotted lines assumed background; \nred dashed lines extracted individual resonances. \nA decay scheme for states populated is shown in panel (a). \n\\label{fig:spectrum_fit}}\n\\end{figure}\n\nThe relative energy spectrum of Fig.~\\ref{fig:spectrum_fit} (a) \nwas used in a fitting analysis \nto extract the parameters for the resonances: \n$E_{\\rm rel}$ and the populating cross section $\\sigma^{\\rm exp}_{-1n}$. \nTheir responses (dashed lines) \nwere generated by a Monte Carlo simulation \nwhich takes into account the detector resolution, beam profile, \nCoulomb multiple scattering, \nand range difference inside the target. \nThe relative energy resolution \nwas estimated to scale as $\\Delta E_{\\rm rel}$=$0.17\\sqrt{E_{\\rm rel}}$ MeV \n(in rms). \nFor the $E_{\\rm rel}$=0.46-MeV state, \na finite line width ${\\mathit \\Gamma}$ for an $l$=1 neutron emission \nwas considered by adopting \na single Breit-Wigner function~\\cite{Lane-Thomas58,Ysatou08} \n(for other states, \nnot as well isolated in the present data as this state, \nonly the instrumental resolution was taken into account). \nIn the analysis an excess strength \nwas recognized at $E_{\\rm rel}$$\\approx$1.9 MeV. \nThis may correspond to the nearest known state \nat $E_x$=6.11 MeV~\\cite{Fortune77,Sercely78} \nif the decay product nucleus $^{15}{\\rm C}$ is populated in the ground state \n(in Fig.~\\ref{fig:spectrum_fit} (a) the response of this strength \nwas created by assuming $E_{\\rm rel}$=1.86 MeV: \nthe difference between $E_x$=6.11 MeV and $S_n$ for $^{16}{\\rm C}$). \nA similar fitting analysis using the $\\gamma$-ray coincidence spectrum \nin Fig.~\\ref{fig:spectrum_fit} (b), however, \ndid not exclude the possibility that this component \nis absent from this spectrum. \nIt is quoted that the upper limit on the fraction of $^{15}{\\rm C}$ fragments \nfrom the decay of the $E_{\\rm rel}$=1.86-MeV resonance, \nthat were in the 0.74-MeV state \nis 20\\% of the strength found in the spectrum \nin Fig.~\\ref{fig:spectrum_fit} (a). \nThe solid line in Fig.~\\ref{fig:spectrum_fit} (a) \nshows the result of the fit to the total inclusive spectrum. \nThe background (dotted line) coming from transitions to the continuum and \nfrom detecting neutrons \nnot associated with the decay of excited states in $^{16}{\\rm C}$, \ne.g., neutrons emitted from excited $^{17}{\\rm C}$ nuclei \nthat are created by inelastic scattering processes, \nwas simulated by a function \n$a(E_{\\rm rel})^b\\exp(-cE_{\\rm rel})$ \nwith $a$, $b$, and $c$ free parameters. \nThe results of the fit \nare summarized in Table~\\ref{tbl:sigma_1n}. \n$E_x$ \nwas calculated by $E_x$=$E_{\\rm rel}$+$S_n$+$E^*$, \nwith $E^*$ the excitation energy of $^{15}{\\rm C}$. \nThe errors include statistical ones \nand those due to the choice of the background shape \n(the latter, \nestimated by further assuming thermal emission of a neutron ($b$=1\/2) \nand neutron evaporation ($b$=1) for the background shape, \nturned out to be a dominant source of the error: \nthey were 80, 90, and 70\\% of the errors quoted in Table~\\ref{tbl:sigma_1n} \nfor $E_{\\rm rel}$, ${\\mathit \\Gamma}$, and $\\sigma^{\\rm exp}_{-1n}$, respectively, \nfor the $E_{\\rm rel}$=0.46-MeV state, \nwhile they \nwere 90, 20, and 70\\% in the (upper bounds of) errors \nof $\\sigma^{\\rm exp}_{-1n}$ for the $E_{\\rm rel}$=1.86-MeV state \nand of $E_{\\rm rel}$ and $\\sigma^{\\rm exp}_{-1n}$ \nfor the $E_{\\rm rel}$=1.29-MeV state, respectively). \nFor $\\sigma^{\\rm exp}_{-1n}$, \nuncertainties originating from the target thickness and \nneutron detection efficiency \nare included. \nThe same fit was repeated \nfor the $\\gamma$-ray coincidence spectrum of Fig.~\\ref{fig:spectrum_fit} (b) \nusing the responses \nfor the $E_{\\rm rel}$=0.46 and 1.29-MeV states, \nobtained from the fit to the inclusive spectrum \nin Fig.~\\ref{fig:spectrum_fit} (a). \nTo quantify the character of the latter state, \nas a state built on the $^{15}{\\rm C}^*$(0.74 MeV) excited core, \nthe fit was repeated by changing the strength from the original one. \nBy finding the fractional value at which the $\\chi^2$ of the fit \nalters from the minimum by one unit, \na lower limit of 32\\% was deduced. \nThe extraction of the $l$ value of the knocked-out neutron \nfrom a differential quantity \nfor the $E_{\\rm rel}$=0.46-MeV state \nis explained later. \n\n\nTo allow discussion in terms of nuclear structure, \nreaction model calculations based on the Glauber \napproximation~\\cite{Hansen03} \nwere performed. \nThe one-neutron removal cross section $\\sigma^{\\rm th}_{-1n}$ \nis expressed for a given final state with $J^{\\pi}$ as \n\\begin{equation}\n\\sigma^{\\rm th}_{-1n}=\\sum_{nlj}\\left( \\frac{A}{A-1}\\right)^{N}\nC^2S(J^{\\pi},nlj)\n\\sigma_{\\rm sp}(nlj,S_n^{\\rm eff}), \n\\end{equation}\nwhere $A$ is the projectile mass, $N$ the major oscillator quantum number, \n$C^2S$ the spectroscopic factor, \nand $\\sigma_{\\rm sp}$ the single-particle cross section. \nThe quantum numbers of the removed neutron are denoted by $nlj$. \n$S_n^{\\rm eff}$ is the effective separation energy given by \nthe sum of $S_n$ of the projectile \nand $E_x$ of the residue. \n$\\sigma_{\\rm sp}$ \nwas calculated by the code \\textsc{csc\\_gm}~\\cite{Abu-Ibrahim03} \ntaking into account \nboth stripping and diffractive processes \n(effective nature of the nucleon-nucleon ($NN$) profile function used \nresulted in small non-zero stripping cross sections)~\\cite{Hansen03}. \nThe elastic $S$ matrix \nfor the collision \nof the residue (core) with the proton target \nwas calculated by folding the finite-range Gaussian $NN$ \nprofile function~\\cite{Abu-Ibrahim08} \nwith the point proton and neutron densities of the core \nobtained from \nthe Hartree-Fock (HF) calculation using the SkX interaction~\\cite{Brown98}. \nThe $S$ matrix for describing the scattering of \nthe valence neutron with the target proton \nwas given by $S(b)$=1$-{\\mathit \\Gamma}_{pn}(b)$, \nhere $b$ is the impact parameter of the colliding nucleons, \nand ${\\mathit \\Gamma}_{pn}$ \nthe profile function for proton-neutron scattering. \nThe parameters chosen for the profile function \nare those \ndescribing the $NN$ total and elastic cross sections \nconsistently~\\cite{Abu-Ibrahim08}.\nThe neutron-residue relative motion \nwas calculated in a Woods-Saxon potential. \nThe depth was adjusted \nso as to reproduce $S_n^{\\rm eff}$, \nfor a diffuseness $a_0$=$0.7$ fm and a reduced radius $r_0$ specifically chosen \nto be consistent with the HF calculation~\\cite{Terry04,Gade08}: \n$r_0$ generates a single-particle \nwave function with a rms neutron-core separation \nof $r_{\\rm sp}$=$[A\/(A-1)]^{1\/2}r_{\\rm HF}$ at the HF-predicted binding energy, \nwhere $r_{\\rm HF}$ is the HF rms radius of each orbit. \nThe spin-orbit potential \nhad the same $a_0$ and $r_0$ as the central one with a strength of $-12$ MeV \nin the notation of Ref.~\\cite{Bertulani06}. \nThe HF radius for the $p_{1\/2}$ ($p_{3\/2}$) orbit of $^{17}\\textrm{C}$, \nfor example, \nis \n2.966 (2.779) fm; \nthis translates into $r_{\\rm sp}$=3.057 (2.865) fm, \nwhich is reproduced by taking $r_0$=$1.263$ (1.234) fm. \nThe $C^2S$ values \nwere obtained by the shell-model code \\textsc{nushell}~\\cite{nushell} \nusing the WBT interaction~\\cite{Warburton92} in the $spsdpf$ model space. \nThe calculated results \nfor relevant states \nare given in Table~\\ref{tbl:sigma_1n}. \n$\\sigma^{\\rm th}_{-1n}$ \nincludes contributions from both stripping ($\\sigma_{\\rm str}$) and \ndiffractive ($\\sigma_{\\rm diff}$) mechanisms. \nDue to inert nature of the proton, \nthe latter dominates the knockout processes. \n\n\\begin{table}[t]\n\\caption{States populated by \nthe $^1{\\rm H}$($^{17}{\\rm C}$,$^{16}{\\rm C}$) reaction. \nTheoretical cross sections were obtained by using the Glauber-model code \n\\textsc{csc\\_gm}~\\cite{Abu-Ibrahim03} and the shell-model spectroscopic factors \ncalculated with the WBT interaction~\\cite{Warburton92}. \nCalculations \nused $S_n^{\\rm eff}$ involving experimental $E_x$ values. \n\\label{tbl:sigma_1n}} \n\\begin{ruledtabular}\n\\begin{tabular}{ccccccccccc}\n\\multicolumn{4}{l}{Experiment}  & &\n\\multicolumn{3}{l}{Theory} \\\\\n\\cline{1-5} \n\\cline{6-10}\n$E_{\\rm rel}$   & \n$E_x$   & \n${\\mathit \\Gamma}$ & \n$l$ &\n$\\sigma^{\\rm exp}_{-1n}$ & &\n$E_x$   & \n$\\sigma_{\\rm str}$  & \n$\\sigma_{\\rm diff}$  & \n$\\sigma^{\\rm th}_{-1n}$  & \n$J^{\\pi}$     \\\\ \n(MeV)  & (MeV) & (MeV) & ($\\hbar$) & (mb)  &  & \n(MeV)  & (mb)  & (mb)  & (mb)  & \n    \\\\ \\hline\n0.463(3)\\protect\\footnotemark[1] & 5.45(1) & 0.03(1) & 1 & 10.6(6) & & \n5.57 & 1.38 & 14.23 & 15.61 & \\multicolumn{1}{c}{$2^-_1$} \\\\\n1.86\\protect\\footnotemark[2]  & \n6.11 & \n--- & --- & \n$2.0^{+0.4}_{-0.8}$ \n& &  \n5.75  & 0.05 & 0.53 &  0.58 & ($3^-_1$) \\\\\n      &                              &     &     &           & &  \n7.60  & 0.09 & 1.37 &  1.46 & ($2^+_3$) \\\\\n      &                              &     &     &           & &  \n8.81  & 0.04 & 0.31 &  0.35 & ($4^+_2$) \\\\\n1.29(2)\\protect\\footnotemark[1]  & 6.28(2) & --- & --- & \n$2.5^{+0.2}_{-1.9}$\n& &  \n6.55  &  0.61  & 5.43  & 6.04 & ($1^-_2$) \\\\\n                                 &         &     &     &          & &  \n6.63  &  0.28  & 2.57  & 2.85 & ($2^-_2$)\n\\end{tabular}\n\\footnotetext[1]{Observed in coincidence with the 0.74-MeV \n$\\gamma$ ray from $^{15}{\\rm C}$. }\n\\footnotetext[2]{Derived from the energy $E_x$=6.11 MeV \nin Ref.~\\cite{Fortune77} \nby assuming the $^{15}{\\rm C}$ core is in the ground state. }\n\\end{ruledtabular}\n\\end{table}\n\nThe state observed at $E_x$=5.45 MeV \nwas found to be well explained by the $2^-_1$ shell-model state \nin both position and cross section, \nmaking an assignment of $2^-$ appropriate. \nThe $2^-_1$ state \nexhibited the highest cross section of unidentified shell-model states \nin the energy region of interest. \nThe summed $\\sigma^{\\rm th}_{-1n}$ for predicted $2^-$ and $1^-$ states \nbelow 8 MeV, \nwhere major strengths are concentrated, \nare 20.9 \n(15.5, 2.8, and 2.6 mb at $E_x$=5.57, 6.63, and 7.23 MeV, respectively) \nand 11.5 mb (see below for composition), respectively. \nTheir ratio is near to the statistical ratio of 5:3 \nexpected for a doublet with spins $J$=2 and 1, \nallowing an interpretation that the $2^-$ and $1^-$ states \nare formed by coupling a hole in the $\\nu p_{1\/2}$ orbit \nto three neutrons with $J$=3\/2 in the $sd$ orbits \n(note that $J^{\\pi}_{\\rm g.s.}(^{17}{\\rm C})$=$3\/2^+$). \nThe predicted $1^-$ strength is distributed \namong states at $E_x$=5.79, 6.55, and 6.98 MeV \nwith $\\sigma^{\\rm th}_{-1n}$=0.6, 6.0, and 4.9 mb, respectively. \nThe fragmentation of the strength \nand the general trend in the Glauber model \nto overestimate the cross section~\\cite{Gade08b,Gade08} \nwould exclude \nan assignment of $1^-$ for the 5.45-MeV state \nwith $\\sigma^{\\rm exp}_{-1n}$=10.6(6) mb. \n\nFigure~\\ref{fig:pl} \nshows the laboratory parallel momentum ($p_{||}$) distribution \nleading to the 5.45-MeV state. \nThis was obtained by subdividing, in terms of $p_{||}$, \nthe inclusive spectrum \nand repeating the fitting procedure described above. \nThe errors are statistical ones. \nAlso plotted in Fig.~\\ref{fig:pl} \nare the $p_{||}$ distributions calculated with \\textsc{csc\\_gm} \nfor varying $l$ values. \nAn experimental resolution of 43(1) MeV\/$c$ in rms is convoluted. \nFactors relevant to stripping mechanisms are dropped, \nand the curves \nrepresent the Fourier transform of the single-particle wave functions. \nThe full width at half maximum (FWHM) of the experimental distribution \nfor the 5.45-MeV state \nwas determined by a fit using a Gaussian \nto be 210(11) MeV\/$c$ after unfolding the resolution. \nIn the fit, \na low-energy tail ($p_{||} <$ 5.72 GeV\/$c$), \nwhich often suffers from higher-order effects~\\cite{Tostevin02}, \nwas \neliminated. \nThe fit curve (not shown) is similar to the $l$=1 curve (solid line). \nThe width \nagrees well with 233 MeV\/$c$ FWHM \ncalculated \nfor $p$-wave knockout, \nwhereas for $s$- (dotted line) and $d$-wave (dashed line) knockout, \nwidths of 121 and 377 MeV\/$c$ FWHM were respectively predicted, \nincompatible with the measurement. \nThis observation \nagrees to the expected character \nof the 5.45-MeV, $2^-$ state as having a neutron hole in the $p$ orbit, \nillustrating the robust feature of the $p_{||}$ distribution \nas an $l$ identifier. \n\nThe large populating cross sections observed for the 6.11-MeV state \nin the $^{14}{\\rm C}(t,p)^{16}{\\rm C}$ reactions~\\cite{Fortune77,Sercely78} \nhave suggested that this state is either of the natural parity \n$2^+$, $3^-$, or $4^+$ states. \nThe knockout cross sections calculated for the relevant \n$2_3^+$, $3_1^-$, and $4_2^+$ shell-model states, \ntogether with their shell-model energies, \nare compared to the data in Table~\\ref{tbl:sigma_1n}. \nThe present data turned out not to provide a strong constraint \non the $J^{\\pi}$ values for this state, \nalthough in terms of comparisons in both $E_x$ and $\\sigma_{-1n}$ \nthey seem to prefer the assignment of $2^+$ or $3^-$. \nThe 6.28-MeV state exhibited the same decay pattern as the strongest \n5.45-MeV $2^-$ state \nwith a sizable cross section. \nThe $1^-_2$ and $2^-_2$ states \npredicted at 6.55 and 6.63 MeV, respectively, \nhad large populating cross sections and are candidates for this state. \n\n\n\\begin{figure}[t]\n\\resizebox{0.90\\columnwidth}{!}{%\n\\includegraphics[angle=0]{pl_lab.eps}%\n}\n\\caption{Laboratory $p_{||}$ distribution of $^{16}{\\rm C}$ populated \nin the 5.45-MeV state after knockout from $^{17}{\\rm C}$ (open circles). \nThe dotted, solid, and dashed lines \nare the Fourier transform of single-particle wave functions \nof orbitals with $l$=0, 1, and 2, respectively. \n\\label{fig:pl}}\n\\end{figure}\n\nThe presently observed $2^-$, 5.45-MeV state in $^{16}{\\rm C}$ \nbelongs to a member of the lowest-lying states \nhaving \nan opposite parity to the ground state. \nThe location of such states provides a measure \nof the $p$-$sd$ shell gap and \nit is well explained by the shell model using the WBT interaction \nacross the C isotopes, $^{11-15}{\\rm C}$. \nTo illustrate the latter, \ntheir energies are compared \nto the shell-model values \nin Fig.~\\ref{fig:wbt_cross_shell_states}. \nIn a recent study of $\\beta$-delayed neutron emission \nof $^{17}{\\rm B}$~\\cite{Ueno13}, \nthree low-lying negative parity states \nwere newly identified in just one-neutron heavier nucleus $^{17}{\\rm C}$. \nThe WBT interaction turned out to fail \nin predicting their location by about 1 MeV \n(theory predicts lower values, see also Fig.~\\ref{fig:wbt_cross_shell_states}), \nand several possible mechanisms, \nsuch as reduction in pairing energy for neutrons in the $sd$ orbits, \nwere discussed. \nThe present study adds a case \nin which the shell model with the WBT interaction \npredicts the location of the lowest-lying cross shell transition properly \n(see also Fig.~\\ref{fig:wbt_cross_shell_states}), \nshowing \nthat this interaction \ndescribes the $p$-$sd$ shell gap in $^{16}{\\rm C}$ adequately. \nTo pin down the source of the discrepancy \nbetween theory and experiment \non the position of the cross shell transition in $^{17}{\\rm C}$, \nas discussed in Ref.~\\cite{Ueno13}, \nand to better understand the dynamical evolution of single-particle orbits \nand relevant residual interactions \naway from stability, \nfurther spectroscopic studies on such states \nin heavier C as well as neighboring isotopes are of help. \n\n\\begin{figure}[t]\n\\resizebox{0.90\\columnwidth}{!}{%\n\\includegraphics[angle=0]{wbt_cross_shell_states.eps}%\n}\n\\caption{(Color online.) \nThe migration of energies of (known) lowest-lying states in C isotopes, \nwhose parities are opposite to those of their respective ground states, \nin comparison to shell-model values \nobtained by using the WBT interaction~\\cite{Warburton92}. \nData for $^{11-15}{\\rm C}$ (filled circles) \nare from Refs.~\\cite{Ajzenberg91,Ajzenberg90}. \nThe data point for $^{16}{\\rm C}$ (red filled square) \nis from the present study, \nwhile that for $^{17}{\\rm C}$ (open diamond) is from Ref.~\\cite{Ueno13}. \nThe shell-model calculations \nwere performed within the 2$\\hbar \\omega$ and 0$\\hbar \\omega$ bases \n(for both positive and negative parity states) \nfor $^{11-15}{\\rm C}$ and $^{16,17}{\\rm C}$, respectively. \n\\label{fig:wbt_cross_shell_states}}\n\\end{figure}\n\n\nIn summary, \none-neutron knockout from $^{17}{\\rm C}$ on a proton target \nwas exploited in populating two new states at 5.45(1) and 6.28(2) MeV, \nand a previously known state at 6.11 MeV in $^{16}{\\rm C}$. \nThe energy spectrum \nwas constructed utilizing the invariant mass method \ninvolving a decay neutron and a $^{15}{\\rm C}$ fragment. \nDe-excitation $\\gamma$ rays from the latter \nwere measured to correctly locate the resonances. \nFor the 5.45-MeV state, \nan attempt was made to deduce the orbital angular momentum \nof the knocked-out neutron \nfrom the parallel momentum distribution \nassociated with the unbound knockout residue. \nThis, together with a comparison in terms of the measured and calculated \nknockout cross sections, \nhas led to a spin-parity assignment of $2^-$ for this state. \nPossible spins and parities have been suggested for the other states, \nbringing about an advanced understanding \nof the level scheme of $^{16}{\\rm C}$. \nThe energy of the first $2^-$ state was adequately reproduced by the \nstandard shell-model calculation using the WBT interaction \nwithout invoking modifications to the residual interaction. \n\n\n\n\n\n\n\n\\begin{acknowledgments}\nThis work was supported in part by \nthe Grant-in-Aid for Scientific Research (15740145) of MEXT Japan \nand the NRF grant (R32-2008-000-10155-0 (WCU), 2010-0027136) of MEST Korea. \n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAveraging techniques have been widely used in the construction and the stability analysis of solutions to time varying differential equations \\cite{bogoliubov1961, sanders2007averaging, sarychev2001lie, bullo2002averaging, vela2003ageneral,maggia2020higher}. The rigorous application of the method of averaging begins by writing the system on the form:\n\\begin{align}\n    \\label{eq:voc_form}\\frac{d\\bm{\\zeta}}{d\\tau} & = \\textbf{f}_0(\\bm{\\zeta},\\tau) + \\varepsilon\\,\\textbf{f}_1(\\bm{\\zeta},\\tau,\\varepsilon)\n\\end{align}\npossibly via coordinate changes and time scaling, where the vector field $\\textbf{f}_1$ is periodic in $\\tau$ and $\\varepsilon$ is a small parameter. When $\\textbf{f}_0=0$, the system is said to be on the averaging canonical form and the averaging theorem can be directly applied \\cite[Chapter 2]{sanders2007averaging}. If $\\textbf{f}_0\\neq 0$, the Variation of Constants (VOC) formula may be used to force the system on the averaging canonical form \\cite[Section 1.6]{sanders2007averaging},\\cite[Section 9.1]{bullo2004geometric}. However, even when the vector field $\\textbf{f}_0$ is linear and time invariant, i.e. $\\textbf{f}_0(\\bm{\\zeta},\\tau) = \\textbf{A}\\bm{\\zeta}$ for some matrix $\\textbf{A}$, the VOC formula is practically useful only when the eigenvalues of the matrix $\\textbf{A}$ are purely imaginary. This is due to the fact that the pullback of the vector field $\\textbf{f}_1$ under the flow of $\\textbf{f}_0$ will contain exponentially growing terms (see the discussion in \\cite[Section 1.7]{sanders2007averaging}). \n\nIn this manuscript, we analyze a class of singularly perturbed high-frequency, high-amplitude oscillatory systems described by equation (\\ref{eq:orig_sys}) which naturally arises in extremum seeking applications \\cite{krstic2000stability, durr2013lie}, and can be put on the form (\\ref{eq:voc_form}). Yet, the VOC formula is not useful in analyzing this class of systems for the reasons outlined in the previous paragraph. Moreover, recent results in the literature such as the singularly perturbed Lie Bracket Approximation framework \\cite{durr2015singularly, durr2017extremum} do not capture the stability properties as we illustrate below.\n\nTo analyze the behavior of this class of systems, we combine the higher order averaging theorem \\cite{sanders2007averaging} with singular perturbation techniques \\cite{khalil2002nonlinear} in a way that accounts for the interaction between the fast periodic time scale and the singularly perturbed part of the system. Furthermore, we propose a novel 3D source seeking algorithm for rigid bodies with a non-collocated sensor. The proposed algorithm is inspired by the chemotactic strategy of sea urchins sperm cells for seeking the egg in 3D \\cite{friedrich2007chemotaxis,abdelgalil2021sperm}, and it utilizes the special structure of the matrix group SO(3). We prove the practical stability of the proposed algorithm using the singularly perturbed averaging results we state here. \n\\iffalse 0\\section{Motivational Example}\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figs\/motivational_example_numerical_results.pdf}\n    \\caption{Numerical results for the motivation example. The initial condition is taken as $(-2,2,2,2)$, and the parameters are $\\beta=1\/2$, $\\omega=4\\pi$ and $\\mu=1\/(4\\pi)$.}\n    \\label{fig:motivation_exmp}\n\\end{figure}\nWe begin the discussion with a simple example to illustrate the motivation behind our results. Consider the system:\n\\begin{gather}\n    \\label{eq:mot_ex_1}\\frac{dx_1}{d\\tau}= \\mu \\beta x_1 + \\mu \\sqrt{2\\omega}\\sin(x_1^2+ x_4-x_3 +\\omega \\mu \\tau)\\\\\n    \\label{eq:mot_ex_2}\\frac{dx_2}{d\\tau}= \\mu \\beta x_2 + \\mu \\sqrt{2\\omega}\\cos(x_1^2+ x_4-x_3 +\\omega \\mu \\tau)\\\\\n    \\label{eq:mot_ex_3}\\frac{dx_3}{d\\tau}= x_4 - x_3, \\qquad     \\frac{dx_4}{d\\tau}= x_2^2 -x_4\n\\end{gather}\nwith an initial condition $(x_{1,0},x_{2,0},x_{3,0},x_{4,0})\\in\\mathbb{R}^4$, where $\\beta$ is a fixed positive parameter, and $\\mu$ and $\\omega$ are positive control parameters. Observe that the system (\\ref{eq:mot_ex_1})-(\\ref{eq:mot_ex_3}) has the same form as the class of systems considered in \\cite{durr2015singularly}. If we attempt to apply the results in \\cite{durr2015singularly}, we obtain the quasi-steady state $(x_3,x_4)=\\bm{\\varphi}_0(x_1,x_2)=(x_2^2,x_2^2)$ for the singularly perturbed part of the system. Hence, the reduced order system according to \\cite{durr2015singularly} becomes:\n\\begin{gather}\n    \\label{eq:mot_ex_reduced_1}\\frac{dx_1}{d\\tau}= \\beta x_1+\\sqrt{2\\omega} \\sin(x_1^2+\\omega\\tau)\\\\\n    \\label{eq:mot_ex_reduced_2}\\frac{dx_2}{d\\tau}= \\beta x_2+\\sqrt{2\\omega} \\cos(x_1^2+\\omega\\tau)\\\\\n    \\label{eq:mot_ex_reduced_3}x_3 = x_2^2, \\qquad    x_4= x_2^2\n\\end{gather}\nIt is clear that the system (\\ref{eq:mot_ex_reduced_1})-(\\ref{eq:mot_ex_reduced_3}) has no practically asymptotically stable subsets for any value of $\\omega$, and so the results in \\cite{durr2015singularly} are not useful here. However, numerical simulations shown in Fig.\\ref{fig:motivation_exmp} demonstrate that the system exhibits practical stability-like behavior, which is not captured by the results in \\cite{durr2015singularly}.\n\\fi\n\\section{Singularly Perturbed Second Order Averaging}\nIn this section, we state the main theorem of our work. Consider the interconnection of systems on the form:\n\\begin{align}\n    \\label{eq:orig_sys}\n    \\begin{aligned}\n        \\dot{\\textbf{x}}&= \\textbf{f}(\\textbf{x},\\textbf{y},t,\\omega) + \\frac{1}{\\sqrt{\\omega}}\\textbf{f}_3(\\textbf{x},\\textbf{y},t,\\omega),  & \\textbf{x}(t_0) &= \\textbf{x}_0\\\\ \\dot{\\textbf{y}}&=\\textbf{g}(\\textbf{x},\\textbf{y},t,\\omega)+\\frac{1}{\\sqrt{\\omega}}\\textbf{g}_3(\\textbf{x},\\textbf{y},t,\\omega),  & \\textbf{y}(t_0) &= \\textbf{y}_0\n    \\end{aligned}\n\\end{align}\nwhere $\\textbf{x},\\textbf{x}_0\\in\\mathbb{R}^n,\\,\\textbf{y},\\textbf{y}_0\\in\\mathbb{R}^m,\\,t,t_0\\in\\mathbb{R}$, $\\omega\\in(0,\\infty)$, and the maps $\\textbf{f},\\,\\textbf{g}$ are given by:\n\\begin{gather*}\n    \\textbf{f}(\\textbf{x},\\textbf{y},t,\\omega) = \\sum_{i\\in\\{1,2\\}}\\omega^{1-\\frac{i}{2}}\\, \\textbf{f}_i(\\textbf{x},\\textbf{y},\\omega t)\\\\\n    \\textbf{g}(\\textbf{x},\\textbf{y},t,\\omega)= \\omega \\textbf{A}\\,(\\textbf{y}-\\bm{\\varphi}_0(\\textbf{x})) +\\sum_{i\\in\\{1,2\\}}\\omega^{1-\\frac{i}{2}}\\, \\textbf{g}_i(\\textbf{x},\\textbf{y},\\omega t)\n\\end{gather*}\nWe adopt the following assumptions on the regularity of the right-hand side of equation (\\ref{eq:orig_sys}):\n\\begin{asmp}\\thlabel{asmp:A}\nSuppose that for $i\\in\\{1,2\\}$:\n    \\begin{enumerate}\n        \\item \\label{asmp:item_A1}  ${ \\textbf{f}_i(\\cdot,\\cdot,\\tau)\\in\\mathcal{C}^{3-i}(\\mathbb{R}^{n+m};\\mathbb{R}^n)}$, ${ \\textbf{f}_i\\in\\mathcal{C}^0(\\mathbb{R}^{n+m+1};\\mathbb{R}^n)}$,\n        \\item $\\textbf{g}_i(\\cdot,\\cdot,\\tau)\\in\\mathcal{C}^{3-i}(\\mathbb{R}^{n+m};\\mathbb{R}^n)$, ${ \\textbf{g}_i\\in\\mathcal{C}^0(\\mathbb{R}^{n+m+1};\\mathbb{R}^n)}$,\n        \\item \\label{asmp:item_A2} $\\exists T>0$ s.t. ${ \\textbf{f}_i(\\cdot,\\cdot,\\tau+T) = \\textbf{f}_i(\\cdot,\\cdot,,\\tau)}$, and $ \\textbf{g}_i(\\cdot,\\cdot,\\tau+T)$ $= \\textbf{g}_i(\\cdot,\\cdot,,\\tau)$, $\\,\\forall \\tau \\in\\mathbb{R}$,\n        \\item $\\textbf{f}_3$ and $\\textbf{g}_3$ are locally Lipschitz continuous in $\\textbf{x},\\textbf{y}$ and jointly continuous in all of their arguments,\n        \\item ${ \\int_{0}^{T}\\textbf{f}_1(\\cdot,\\cdot,s)ds} = 0$, $\\bm{\\varphi}_0(\\cdot)\\in\\mathcal{C}^{3}(\\mathbb{R}^n;\\mathbb{R}^m)$, and the matrix $ \\textbf{A}$ is Hurwitz.\n    \\end{enumerate}\n\\end{asmp}\n\\begin{rem}\nWe restrict our treatment here to the periodic case for simplicity. The extension to the case when the vector fields are quasi-periodic is straightforward but the computations are more involved.\n\\end{rem} \nNext, consider the reduced order system:\n\\begin{gather} \\label{eq:reduced_system}\n    \\dot{\\tilde{\\textbf{x}}}= \\sqrt{\\omega}\\,\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}},\\omega t )+ \\tilde{\\textbf{f}}_2(\\tilde{\\textbf{x}},\\omega t),\\qquad\\qquad\n    \\tilde{\\textbf{y}}= \\bm{\\varphi}_0(\\tilde{\\textbf{x}})\n\\end{gather}\nwhere the time-varying vector fields $\\tilde{\\textbf{f}}_i$ are defined by:\n\\begin{gather}\n    \\label{eq:red_avg_sys_dets_1}\\tilde{\\textbf{f}}_1(\\textbf{x},\\tau)= \\textbf{f}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau) \\\\\n    \\tilde{\\textbf{f}}_2(\\textbf{x},\\tau)= \\textbf{f}_2(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau) + \\textbf{C}(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\bm{\\varphi}_1(\\textbf{x},\\tau)\\\\\n    \\textbf{C}(\\textbf{x},\\textbf{y},\\tau)= \\partial_\\textbf{w} \\textbf{f}_1(\\textbf{x},\\textbf{w},\\tau)|_{\\textbf{w}=\\textbf{y}} \\\\\n    \\bm{\\varphi}_1(\\textbf{x},\\tau)= \\left(\\text{Id}-\\text{e}^{T \\textbf{A}}\\right)^{-1}\\textstyle\\int_{0}^T \\text{e}^{ (T-s)\\textbf{A}}\\,\\textbf{b}_1(\\textbf{x},s+\\tau)ds \\\\\n    \\label{eq:red_avg_sys_dets_5}\\textbf{b}_1(\\textbf{x},\\tau)= \\textbf{g}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)-\\partial_{\\textbf{x}}\\bm{\\varphi}_0(\\textbf{x})\\, \\textbf{f}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)\n\\end{gather}\nand $\\text{Id}$ is the identity matrix. In a companion paper \\cite{abdelgalil2022recursive}, see also \\cite{murdock1983some}, we showed that the higher order averaging theorem may be applied to the reduced order system (\\ref{eq:reduced_system}) to obtain the reduced order averaged system:\n\\begin{gather}\\label{eq:reduced_avg_system}\n    \\dot{\\bar{\\textbf{x}}}= \\bar{\\textbf{f}}(\\bar{\\textbf{x}}), \\qquad\\qquad \\bar{\\textbf{y}} = \\bm{\\varphi}_0(\\textbf{x})\n\\end{gather}\nwhere the vector field $\\bar{\\textbf{f}}(\\cdot)$ is given by:\n\\begin{equation}\n\\begin{aligned}\n    \\bar{\\textbf{f}}(\\textbf{x})= \\frac{1}{T}\\int_0^{T}\\bigg(\\tilde{\\textbf{f}}_2(\\textbf{x},\\tau_1)+\\frac{1}{2}\\Big[ \\int_0^{\\tau_1}&\\tilde{\\textbf{f}}_1(\\textbf{x}, \\tau_2) d\\tau_2,\\\\\n    &\\tilde{\\textbf{f}}_1(\\textbf{x},\\tau_1)\\Big]\\bigg)\\,d\\tau_1\n\\end{aligned}\n\\end{equation}\nUnder \\thref{asmp:A}, we have the following theorem concerning the relation between the stability of the system (\\ref{eq:orig_sys}) and the reduced order averaged system (\\ref{eq:reduced_avg_system}):\n\\begin{thm}\\thlabel{thm:A}\n    Let \\thref{asmp:A} be satisfied, and suppose that a compact subset $\\mathcal{S}\\subset\\mathbb{R}^n$ is globally uniformly asymptotically stable for the reduced order averaged system (\\ref{eq:reduced_avg_system}). Then, $\\mathcal{S}$ is singularly semi-globally practically uniformly asymptotically stable for the original system (\\ref{eq:orig_sys}).\n\\end{thm}\n\\begin{rem}\n    We note that our definition of singular semi-global practical uniform asymptotic stability, which can be found in the appendix A, is slightly different from that in \\cite{durr2015singularly}. The proof of \\thref{thm:A} proceeds by establishing each part of \\thref{defn:A} similar to \\cite{moreau2000practical,durr2015singularly}, relying on \\thref{prop:A} which we can be found along with a proof sketch in the appendix B. We omit the proof of the theorem from this manuscript to be included in the extended version. \n\\end{rem}\n\n\\iffalse 0\nGoing back to the motivational example, we see that when $\\mu=1\/\\omega$, the system can be written on the form:\n\\begin{gather}\n    \\label{eq:mot_ex_sol_1}\\dot{x}_1= \\beta x_1 + \\sqrt{2\\omega}\\sin(x_1^2+ x_4-x_3 +\\omega t)\\\\\n    \\label{eq:mot_ex_sol_2}\\dot{x}_2= \\beta x_2 + \\sqrt{2\\omega}\\cos(x_1^2+ x_4-x_3 +\\omega t)\\\\\n    \\label{eq:mot_ex_sol_3}\\dot{x}_3=\\omega\\,(x_4 - x_3), \\qquad     \\dot{x}_4= \\omega\\,(x_2^2 -x_4)\n\\end{gather}\nIf we apply our results, leaving the details of the computations as an easy exercise for the reader, we obtain the reduced order system:\n\\begin{align}\n\\dot{\\tilde{x}}_1&= \\beta \\tilde{x}_1  + \\sqrt{2\\omega} \\sin(\\tilde{x}_1^2+\\omega t)+ \\tilde{x}_2 \\sin(2\\tilde{x}_1^2+2\\omega t)\\\\\n\\dot{\\tilde{x}}_2&=\\alpha \\tilde{x}_2  + \\sqrt{2\\omega} \\cos(\\tilde{x}_1^2+\\omega t)+ \\tilde{x}_2 \\cos(2\\tilde{x}_1^2+2\\omega t)\n\\end{align}\nwhere $\\alpha = \\beta-1$. By applying the averaging procedure, we obtain the reduced order averaged system:\n\\begin{align}\n\\dot{\\bar{x}}_1&= (\\beta-2) \\bar{x}_1 \\\\\n\\dot{\\bar{x}}_2&= (\\beta-1) \\bar{x}_2 \n\\end{align}\nIt is clear that the reduced order averaged system has a globally uniformly asymptotically stable equilibrium point at the origin when $\\beta<1$. Hence, we conclude from \\thref{thm:A} that the origin $x_1=x_2=0$ is a singularly semi-globally practically uniformly asymptotically stable subset for the original system (\\ref{eq:mot_ex_1})-(\\ref{eq:mot_ex_3}). \n\\fi\n\\section{3D Source Seeking}\nSource seeking is the problem of locating a target that emits a scalar measurable signal, typically without global positioning information \\cite{cochran20093,krstic2008extremum}. Interestingly, microorganisms are routinely faced with the source seeking problem. In particular, sea urchin sperm cells seek the egg by swimming up the gradient of the concentration field  of a chemical secreted by the egg \\cite{abdelgalil2021sperm,friedrich2007chemotaxis}. The sperm cells do so by swimming in helical paths that dynamically align with the gradient. In this section, we propose a bio-inspired 3D source seeking algorithm for rigid bodies with a non-collocated signal strength sensor that partially mimics the strategy of sperm cells for seeking the egg.\n\nThe kinematics of a rigid body in 3D space are given by:\n\\begin{align}\n    \\label{eq:kin}\\dot{\\textbf{p}} &= \\textbf{R}\\textbf{v}, & \\dot{\\textbf{R}} &= \\textbf{R}\\widehat{\\bm{\\Omega}}\n\\end{align}\nwhere $\\textbf{p}$ denotes the position of a designated point on the body with respect to a fixed frame of reference, $\\textbf{R}$ relates the body frame to the fixed frame, and $\\textbf{v}$ and $\\bm{\\Omega}$ are the linear and angular velocities in body coordinates, respectively. The map $\\widehat{\\bullet}:\\mathbb{R}^3\\rightarrow\\mathbb{R}^{3\\times 3}$ takes a vector $\\bm{\\Omega}=\\left[\\Omega_1,\\Omega_2,\\Omega_3\\right]^\\intercal \\in \\mathbb{R}^3$ to the corresponding skew symmetric matrix. \n\nWe assume a vehicle model in which the linear and angular velocity vectors are given in body coordinates by:\n\\begin{align}\n    \\textbf{v}&= v\\textbf{e}_1, & \\bm{\\Omega} &= \\Omega_\\parallel\\textbf{e}_1 + \\Omega_\\perp \\textbf{e}_3\n\\end{align} \nwhere $\\textbf{e}_i$ for $i\\in\\{1,2,3\\}$ are the standard unit vectors. \\begin{rem}\nThis model is a natural extension of the unicycle model to the 3D setting. It is well known that this system is controllable using depth one Lie brackets \\cite{leonard1993averaging}.\n\\end{rem}\n\nLet $c:\\mathbb{R}^3\\rightarrow\\mathbb{R}$ be the signal strength field emitted by the source, and consider the case of a non-collocated signal strength sensor that is mounted at $\\textbf{p}_s$, where:\n\\begin{align}\\label{eq:sens_loc}\n    \\textbf{p}_s&= \\textbf{p} + r \\textbf{R}\\textbf{e}_2\n\\end{align}\n\\begin{asmp}\\thlabel{asmp:B}\nSuppose that the signal strength field $c\\in \\mathcal{C}^3(\\mathbb{R}^3;\\mathbb{R})$ is radially unbounded, $\\exists!\\textbf{p}^*\\in \\mathbb{R}^3$ such that $\\nabla c(\\textbf{p})= 0 \\iff \\textbf{p}=\\textbf{p}^*$, and it satisfies $c(\\textbf{p}^*)-c(\\textbf{p})\\leq \\kappa \\lVert \\nabla c(\\textbf{p})\\lVert^2, \\, \\forall \\textbf{p}\\in\\mathbb{R}^n$ and $\\kappa>0$.\\\\\n\\end{asmp}\n\nNow, consider the following control law:\n\\begin{gather}\n\\label{eq:ctrl_law_1}v = \\sqrt{4\\omega}\\cos(2\\omega t - c(\\textbf{p}_s))\\\\\n    \\label{eq:ctrl_law_2}\\dot{\\textbf{y}} = \\omega\\,\\textbf{A}\\,\\textbf{y} + \\omega\\,\\textbf{B}\\, c(\\textbf{p}_s) \\\\\n    \\Omega_\\perp  = \\sqrt{\\omega}\\,\\textbf{C}\\, \\textbf{y}, \\qquad\n    \\label{eq:ctrl_law_3}\\Omega_\\parallel = \\omega\n   \n\\end{gather}\nwhere $\\textbf{y}\\in\\mathbb{R}^2$, and $\\textbf{A},\\textbf{B},\\textbf{C}$ are given by:\n\\begin{align}\n    \\label{eq:flter_mats}\\textbf{A} &= \\left[\\begin{array}{cc} -1 & 1\\\\ 0 & -1 \\end{array}\\right], & \\textbf{B}&= \\left[\\begin{array}{c} 0\\\\ 1\\end{array}\\right], & \\textbf{C}&= \\left[\\begin{array}{cc} -1 & 1\\end{array}\\right]\n\\end{align}\nThe static part of this controller, i.e. equation (\\ref{eq:ctrl_law_1}) is a 1D extremum seeking control law \\cite{scheinker2014extremum}. Note that other choices of this control law are possible \\cite{grushkovskaya2018class}. \nThe dynamic part of this controller, i.e. the equations (\\ref{eq:ctrl_law_2})-(\\ref{eq:ctrl_law_3}), is a narrow band-pass filter centered around the frequency $\\omega$. The motivation to consider this setup is that in the presence of noise, a narrow band-pass filter centered around the dither frequency $\\omega$ optimally extracts the gradient information in the measured output while attenuating noise. In addition, assume that the distance $r$ specifying the offset of the sensor from the center of the frame is such that $r=1\/\\sqrt{\\omega}$. This assumption may seem artificial at first glance, though its implication is clear; we require that as the frequency of oscillation $\\omega$ tends to $\\infty$, the distance $r$ from the center of the vehicle is small enough so as not to amplify unwanted nonlinearities in the signal strength field. Alternatively, one may consider this assumption as a ``distinguished limit'' \\cite{kevorkian2012multiple} for the perturbation calculation in the presence of the two parameters $\\omega$ and $r$. Under these assumptions, we have the following proposition:\n\\begin{prop}\nLet \\thref{asmp:B} be satisfied, and let $r=1\/\\sqrt{\\omega}$. Then, the compact subset $\\mathcal{S}=\\{\\textbf{p}^*\\}\\times \\text{SO}(3)$ is singularly semi-globally practically uniformly asymptotically stable for the system defined by equations (\\ref{eq:kin})-(\\ref{eq:flter_mats}).\n\\end{prop}\n\\begin{proof}\nLet $\\textbf{R}_0 =\\text{exp}\\left(\\omega t \\,\\widehat{\\textbf{e}}_1 \\right),\\,\\textbf{Q} = \\textbf{R}\\textbf{R}_0^\\intercal$, and compute:\n\\begin{align}\n    \\dot{\\textbf{Q}} &= \\dot{\\textbf{R}}\\textbf{R}_0^\\intercal +  \\textbf{R}\\dot{\\textbf{R}}_0^\\intercal = \\Omega_\\perp\\textbf{R}\\widehat{\\textbf{e}}_3\\textbf{R}_0^\\intercal \\\\\n    &=  \\Omega_\\perp\\textbf{Q}\\textbf{R}_0 \\widehat{\\textbf{e}}_3\\textbf{R}_0^\\intercal = \\Omega_\\perp\\textbf{Q}\\widehat{\\textbf{R}_0 \\textbf{e}_3}\n\\end{align}\nLet $\\bm{\\Lambda}(\\textbf{y},\\omega t)= \\textbf{C}\\, \\textbf{y}\\,\\textbf{R}_0\\textbf{e}_3$ and observe that:\n\\begin{gather}\n    \\dot{\\textbf{p}}= \\textbf{R}\\textbf{v} = \\textbf{R}\\textbf{R}_0^\\intercal \\textbf{R}_0 \\textbf{v} = v \\,\\textbf{Q} \\textbf{R}_0 \\textbf{e}_1 = v\\,\\textbf{Q} \\textbf{e}_1 \\\\\n    \\dot{\\textbf{Q}}= \\sqrt{\\omega}\\,\\textbf{Q} \\widehat{\\bm{\\Lambda}}(\\textbf{y},\\omega t)\n\\end{gather}\nTo simplify the presentation, we embed $\\text{SO}(3)$ into $\\mathbb{R}^9$ by partitioning the matrix $\\textbf{Q}=[\\textbf{q}_1,\\, \\textbf{q}_2,\\, \\textbf{q}_3 ]$, and defining the state vector $\\textbf{q}=[\\textbf{q}_1^\\intercal,\\, \\textbf{q}_2^\\intercal,\\, \\textbf{q}_3^\\intercal ]^\\intercal$. Restrict the initial conditions for $\\textbf{q}$ to lie on the compact submanifold $\\mathcal{M} = \\left\\{\\textbf{q}_i\\in\\mathbb{R}^{3}:\\, \\textbf{q}_i^\\intercal\\textbf{q}_j=\\delta_{ij},~\\textbf{q}_i\\times\\textbf{q}_j = \\epsilon_{ijk}\\textbf{q}_k  \\right\\}$, \nwhere $\\delta_{ij}$ is the Kronecker symbol and $\\epsilon_{ijk}$ is the Levi-Civita symbol.\nOn $\\mathbb{R}^3\\times\\mathcal{M}\\times\\mathbb{R}^2$, the system is governed by:\n\\begin{align}\n    \\label{eq:src_seek_sys_1} \\dot{\\textbf{p}} &= \\sqrt{4\\omega}\\cos(2\\omega t - c(\\textbf{p}_s))\\,\\textbf{q}_1\\\\\n    \\label{eq:src_seek_sys_2}\\dot{\\textbf{q}}_i &= \\sqrt{\\omega}\\sum\\limits_{j,k=1}^3\\Lambda_j(\\textbf{y},\\omega t)\\epsilon_{ijk}\\textbf{q}_k \\\\\n    \\label{eq:src_seek_sys_3}\\dot{\\textbf{y}} &=\\omega \\left(\\textbf{A}\\,\\textbf{y} + \\,\\textbf{B}\\, c(\\textbf{p}_s) \\right)\n\\end{align}\nFor more details on this embedding, see the companion paper \\cite{abdelgalil2022recursive}. The signal strength field can be expanded as a series in $1\/\\sqrt{\\omega}$ using Taylor's theorem:\n\\begin{gather*}\n    c(\\textbf{p}_s)=\\,c(\\textbf{p})+\\frac{1}{\\sqrt{\\omega}} \\nabla c(\\textbf{p})^\\intercal (\\cos(\\omega t) \\textbf{q}_2 + \\sin(\\omega t) \\textbf{q}_3) \\\\\n    +\\frac{1}{2\\omega }\\nabla^2 c(\\textbf{p})[(\\cos(\\omega t) \\textbf{q}_2 + \\sin(\\omega t) \\textbf{q}_3)] \\\\\n    + \\frac{1}{\\omega \\sqrt{\\omega}}\\bm{\\rho}(\\textbf{p},\\textbf{p}_s,\\omega t,1\/\\sqrt{\\omega})\n\\end{gather*}\nwhere the remainder $\\bm{\\rho}$ is Lipschitz continuous in all of its arguments, and $\\nabla^2c(\\textbf{p})[\\textbf{w}]=\\textbf{w}^\\intercal \\nabla^2c(\\textbf{p}) \\textbf{w}$. \nNow, observe that the system governed by the equations (\\ref{eq:src_seek_sys_1})-(\\ref{eq:src_seek_sys_3}) belongs to the class of systems described by (\\ref{eq:orig_sys}). Hence, we may employ \\thref{thm:A} in analyzing the stability of the system. In order to proceed, the reduced order averaged system needs to be computed. Due to space constraints, we leave the computations as an exercise for the interested reader in the light of equations (\\ref{eq:red_avg_sys_dets_1})-(\\ref{eq:red_avg_sys_dets_5}), and we provide only the end result of the computation:\n\\noindent\\begin{minipage}{.5\\linewidth}\n\\begin{gather*}\n    \\dot{\\bar{\\textbf{p}}} = \\vphantom{\\frac{1}{4}}\\bar{\\textbf{q}}_1\\bar{\\textbf{q}}_1^\\intercal \\nabla c(\\bar{\\textbf{p}}) \\\\\n    \\dot{\\bar{\\textbf{q}}}_2 = -\\frac{1}{4}\\bar{\\textbf{q}}_1\\bar{\\textbf{q}}_3^\\intercal \\nabla c(\\bar{\\textbf{p}}),\n\\end{gather*}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\begin{gather*}\n    \\dot{\\bar{\\textbf{q}}}_1= \\frac{1}{4}(\\bar{\\textbf{q}}_2\\bar{\\textbf{q}}_2^\\intercal + \\bar{\\textbf{q}}_3\\bar{\\textbf{q}}_3^\\intercal)\\nabla c(\\bar{\\textbf{p}}) \\\\\n    \\dot{\\bar{\\textbf{q}}}_3 = -\\frac{1}{4}\\bar{\\textbf{q}}_1\\bar{\\textbf{q}}_2^\\intercal \\nabla c(\\bar{\\textbf{p}})\n\\end{gather*}\n\\end{minipage}\\vspace*{0.1in} \\\\\nEquivalently, this system can be written as:\n\\begin{gather}\\label{eq:exmp_red_avg_sys_1}\n    \\dot{\\bar{\\textbf{p}}} = \\bar{\\textbf{Q}}\\textbf{e}_1\\textbf{e}_1^\\intercal \\bar{\\textbf{Q}}^\\intercal \\nabla c(\\bar{\\textbf{p}}) \\\\\n    \\label{eq:exmp_red_avg_sys_2}\\dot{\\bar{\\textbf{Q}}} = \\bar{\\textbf{Q}} \\widehat{\\bar{\\bm{\\Lambda}}}(\\textbf{p},\\textbf{Q})\n\\end{gather}\nwhere the average angular velocity vector $\\bar{\\bm{\\Lambda}}$ is given by:\n\\begin{align}\n    \\label{eq:avg_rot_vel} \\bar{\\bm{\\Lambda}}(\\textbf{p},\\textbf{Q})&= \\frac{1}{4}\\textbf{Q}^\\intercal \\nabla c(\\textbf{p})\\times\\textbf{e}_1\n\\end{align}\nWe claim that the compact subset $\\mathcal{S}$ is globally uniformly asymptotically stable for the reduced order averaged system (\\ref{eq:exmp_red_avg_sys_1})-(\\ref{eq:exmp_red_avg_sys_2}). To prove this claim, we use the negative of the signal strength field as a Lyapunov function $V_c(\\textbf{p})=c(\\textbf{p}^*)-c(\\textbf{p})$. Observe that the system (\\ref{eq:exmp_red_avg_sys_1})-(\\ref{eq:exmp_red_avg_sys_2}) is autonomous, and so the function $V_c$ is indeed a Lyapunov function for the compact subset $\\mathcal{S}$ due to \\thref{asmp:B} \\cite{khalil2002nonlinear}. We proceed to compute the derivative of $V_c$:\n\\begin{gather}\n    \\dot{V}_c = -\\nabla c(\\bar{\\textbf{p}})^\\intercal \\bar{\\textbf{Q}}\\textbf{e}_1\\textbf{e}_1^\\intercal \\bar{\\textbf{Q}}^\\intercal \\nabla c(\\bar{\\textbf{p}})\\leq 0\n\\end{gather}\nNow, consider the subset $\\mathcal{N}=\\{(\\textbf{p},\\textbf{Q})\\in\\mathbb{R}^3\\times \\text{SO}(3): \\dot{V}_c=0\\}$, and observe that $\\mathcal{S}\\subset \\mathcal{N}$, and that $\\mathcal{S}$ is an invariant subset of the reduced order averaged system (\\ref{eq:exmp_red_avg_sys_1})-(\\ref{eq:exmp_red_avg_sys_2}). Suppose that a trajectory $(\\bar{\\textbf{p}}(t),\\bar{\\textbf{Q}}(t))$ of the system (\\ref{eq:exmp_red_avg_sys_1})-(\\ref{eq:exmp_red_avg_sys_2}) exists such that $(\\bar{\\textbf{p}}(t),\\bar{\\textbf{Q}}(t))\\in \\mathcal{N}\\backslash\\mathcal{S},\\, \\forall t\\in I$, where $I$ is the maximal interval of existence and uniqueness of the trajectory. Such a trajectory must satisfy:\n\\begin{align}\n    \\nabla c(\\bar{\\textbf{p}}(t))^\\intercal \\bar{\\textbf{Q}}(t)\\textbf{e}_1 = 0,\\quad \\forall t\\in I\n\\end{align}\nThe differentiability of the trajectories allows us to compute the derivative of this identity and obtain that:\n\\begin{align}\n    \\frac{d}{dt}\\big(\\nabla c(\\bar{\\textbf{p}}(t))^\\intercal \\bar{\\textbf{Q}}(t)\\textbf{e}_1\\big) = 0,\\quad \\forall t\\in I\n\\end{align}\nwhich simplifies to:\n\\begin{align}\n    \\label{eq:second_identity}\\nabla c(\\bar{\\textbf{p}}(t))^\\intercal \\bar{\\textbf{Q}}(t)(\\bar{\\bm{\\Lambda}}(\\bar{\\textbf{p}}(t),\\bar{\\textbf{Q}}(t))\\times \\textbf{e}_1) = 0\n\\end{align}\nRecalling equation (\\ref{eq:avg_rot_vel}), we see that:\n\\begin{gather}\n    \\bar{\\bm{\\Lambda}}(\\bar{\\textbf{p}}(t),\\bar{\\textbf{Q}}(t))\\times \\textbf{e}_1 = \\frac{1}{4}(\\text{Id}-\\textbf{e}_1\\textbf{e}_1^\\intercal)\\bar{\\textbf{Q}}(t)^\\intercal \\nabla c(\\bar{\\textbf{p}}(t))\\\\\n    = \\frac{1}{4}\\bar{\\textbf{Q}}(t)^\\intercal \\nabla c(\\bar{\\textbf{p}}(t))\n\\end{gather}\nHence, the equation (\\ref{eq:second_identity}) necessitates that:\n\\begin{align}\n    \\lVert \\nabla c(\\bar{\\textbf{p}}(t)) \\lVert^2 = 0,\\quad \\forall t\\in I\n\\end{align}\nwhich is clearly in contradiction with \\thref{asmp:B}. Accordingly, it follows from LaSalle's Invariance principle \\cite[Corollary 4.2 to Theorem 4.4]{khalil2002nonlinear} that the compact subset $\\mathcal{S}$ is globally uniformly asymptotically stable for the system (\\ref{eq:exmp_red_avg_sys_1})-(\\ref{eq:exmp_red_avg_sys_2}). Hence, we conclude by \\thref{thm:A} that the subset $\\mathcal{S}$ is singularly semi-globally practically uniformly asymptotically stable for the original system defined by (\\ref{eq:kin})-(\\ref{eq:flter_mats}). \n\\end{proof}\n\\begin{figure*}[t]\n    \\centering \\includegraphics[width=1\\textwidth]{figs\/signal_strength_and_pos.pdf}\n    \\caption{Numerical results: the history of the signal strength at the vehicle center and the position coordinates (left), and the 3D spatial trajectory (right)}\n    \\label{fig:num_res}\n\\end{figure*}\n\\begin{rem}\nIf we attempt to apply the framework of singularly pertubed Lie Bracket Approximation introduced in \\cite{durr2015singularly} to the system (\\ref{eq:src_seek_sys_1})-(\\ref{eq:src_seek_sys_3}), then the quasi-steady state of the system will be $\\textbf{y} = [c(\\textbf{p}_s),c(\\textbf{p}_s)]^\\intercal$. Hence, according to \\cite{durr2015singularly}, the reduced order system is:\n\\begin{gather}\n    \\dot{\\textbf{p}} = \\sqrt{4\\omega}\\cos(2\\omega t - c(\\textbf{p}_s))\\,\\textbf{q}_1,\\qquad \n    \\dot{\\textbf{q}}_i =0\n\\end{gather}\nwhich yields the Lie Bracket system:\n\\begin{gather}\n    \\dot{\\bar{\\textbf{p}}} = \\bar{\\textbf{Q}}\\textbf{e}_1\\textbf{e}_1^\\intercal \\bar{\\textbf{Q}}^\\intercal \\nabla c(\\bar{\\textbf{p}}), \\qquad \n    \\dot{\\bar{\\textbf{Q}}} =0\n\\end{gather}\nIt is clear that the compact subset $\\mathcal{S}$ is not asymptotically stable for the Lie Bracket system, and so the framework in \\cite{durr2015singularly} does not capture the stability of this system. \\end{rem}\n\\section{Numerical Simulations}\nWe demonstrate our results by providing a numerical example. Consider the signal strength field given by $c(\\textbf{p})= -\\log\\left(1+\\textbf{p}^\\intercal\\textbf{p}\/2\\right)$, which represents a stationary source located at the origin. We take the initial conditions as $\\textbf{p}(0)=[6,~2,~-2]^\\intercal$ and $\\textbf{R}(0)=\\textbf{I}_{3\\times3}$, and the frequency as $\\omega=4\\pi$. The numerical simulations are shown in Fig.(\\ref{fig:num_res}). Observe that the behavior near the source is nontrivial, i.e. there is a limit cycle. However, in the limit $\\omega\\rightarrow \\infty$ and $r=O(1\/\\sqrt{\\omega})$, this complex behavior does not appear in the reduced order averaged system.\n\n\\section{Conclusion}\nIn this manuscript, we analyzed a class of singularly perturbed high-amplitude, high-frequency oscillatory systems that naturally arises in extremum seeking applications and stabilization by oscillatory controls. We combined singular perturbation with the higher order averaging theorem in order to capture the stability properties of this class of systems. As an application, we proposed a novel 3D source seeking algorithm for rigid bodies with a non-collocated sensor inspired by the chemotaxis of sea urchin sperm cells.\n\\section*{Acknowledgement} The authors like to acknowledge the support of the NSF Grant CMMI-1846308. \\\\\n\\appendices\n\\renewcommand{\\thelem}{\\thesection.\\arabic{lem}} \n\\renewcommand{\\thecor}{\\thesection.\\arabic{cor}}\n\\renewcommand{\\thedefn}{\\thesection.\\arabic{defn}} \n\\renewcommand{\\theprop}{\\thesection.\\arabic{prop}} \n\\section{Definitions}\n\\begin{defn}\\thlabel{defn:A}\n The set $\\mathcal{S}$ is said to be \\textit{singularly semi-globally practically uniformly asymptotically stable} for system (\\ref{eq:orig_sys}) if the following is satisfied:\n\\begin{enumerate}\n    \\item $\\forall \\epsilon_x,\\epsilon_z\\in(0,\\infty)$ there exists $\\delta_x,\\delta_z\\in(0,\\infty)$ and $\\omega^*\\in(0,\\infty)$ such that $\\forall \\omega\\in(\\omega^*,\\infty)$, $\\forall t_0\\in\\mathbb{R}$, and $\\forall t\\in[0,\\infty)$ we have: \n\\begin{align*}\n\\begin{drcases}\n     \\hfill \\textbf{x}_0\\in \\mathcal{U}_{\\delta_x}^{\\mathcal{S}}  \\hfill \\\\\n     \\textbf{y}_0-\\bm{\\varphi}_0(\\textbf{x}_0)\\in \\mathcal{U}_{\\delta_z}^{0}\n\\end{drcases} &\\implies \\begin{cases} \n     \\hfill\\textbf{x}(t)\\in \\mathcal{U}_{\\epsilon_x}^{\\mathcal{S}}\\hfill \\\\\n     \\textbf{y}(t)-\\bm{\\varphi}_0(\\textbf{x}(t))\\in \\mathcal{U}_{\\epsilon_z}^{0} \n\\end{cases}\n\\end{align*}\n\\item $\\forall \\epsilon_x,\\epsilon_z\\in(0,\\infty)$ and all $\\delta_x,\\delta_z\\in(0,\\infty)$, there exists a time $T_f\\in(0,\\infty)$ and $\\omega^*\\in(0,\\infty)$ such that $\\forall \\omega\\in(\\omega^*,\\infty)$, $\\forall t_0\\in\\mathbb{R}$, $\\forall t_1\\in[T_f,\\infty)$, and $\\forall t_2\\in[T_f\/\\omega,\\infty)$ we have: \n\\begin{align*}\n\\begin{drcases}\n     \\hfill \\textbf{x}_0\\in \\mathcal{U}_{\\delta_x}^{\\mathcal{S}}  \\hfill\\\\\n     \\textbf{y}_0-\\bm{\\varphi}_0(\\textbf{x}_0)\\in \\mathcal{U}_{\\delta_z}^{0}\n\\end{drcases} &\\implies \\begin{cases} \n     \\hfill \\textbf{x}(t_1)\\in \\mathcal{U}_{\\epsilon_x}^{\\mathcal{S}} \\hfill \\\\\n     \\textbf{y}(t_2)-\\bm{\\varphi}_0(\\textbf{x}(t_2))\\in \\mathcal{U}_{\\epsilon_z}^{0}\n\\end{cases}\n\\end{align*}\n\\item $\\forall \\delta_x,\\delta_z\\in(0,\\infty)$ there exists $\\epsilon_x,\\epsilon_z\\in(0,\\infty)$ and $\\omega^*\\in(0,\\infty)$ such that $\\forall \\omega\\in(\\omega^*,\\infty)$, $\\forall t_0\\in\\mathbb{R}$, and $\\forall t\\in[0,\\infty)$ we have: \n\\begin{align*}\n\\begin{drcases}\n     \\hfill \\textbf{x}_0\\in \\mathcal{U}_{\\delta_x}^{\\mathcal{S}}  \\hfill\\\\\n     \\textbf{y}_0-\\bm{\\varphi}_0(\\textbf{x}_0)\\in \\mathcal{U}_{\\delta_z}^{0}\n\\end{drcases} &\\implies \\begin{cases} \n     \\hfill \\textbf{x}(t)\\in \\mathcal{U}_{\\epsilon_x}^{\\mathcal{S}}\\hfill \\\\\n     \\textbf{y}(t)-\\bm{\\varphi}_0(\\textbf{x}(t))\\in \\mathcal{U}_{\\epsilon_z}^{0}\n\\end{cases}\\\\\n\\end{align*}\n\\end{enumerate}\n\\end{defn}\nObserve that our definition of singular semi-global practical uniform asymptotic stability is different from the definitions introduced in \\cite{durr2015singularly} due to the absence of a second parameter. Intuitively, when two or more parameters are involved, the so called `distinguished limit' is a standard technique in perturbation theory that simplifies the interaction between the limiting behavior of the two parameters on the trajectories of differential equations \\cite[Chapter 2]{kevorkian2012multiple}. Our definitions are motivated by this concept of distinguished limits. \n\\section{Trajectory Approximation}\nFor the purpose of brevity, we state here some notations that may enhance the readability of the proof. Whenever a Lipschitz property of a map $\\textbf{f}$ over a subset $\\mathcal{K}$ is employed, the corresponding Lipschitz constant is labelled as $L_{\\textbf{f},\\mathcal{K}}$. Similarly, when a uniform bound is employed, it is labelled as $B_{\\textbf{f},\\mathcal{K}}$. Sometimes we use $M_{\\textbf{f},\\mathcal{K}}$ as a generic constant when a mix of the two properties is used. Finally, we may omit mentioning the map in the constant label when it is too long or when it is clear from the context. \n\nUnder \\thref{asmp:A}, we have a trajectory approximation result between the original system (\\ref{eq:orig_sys}) and the reduced order averaged system (\\ref{eq:reduced_avg_system}): \n\\begin{prop}\\thlabel{prop:A}\nLet \\thref{asmp:A} be satisfied, and suppose that a compact subset $\\mathcal{S}\\subset\\mathbb{R}^n$ is globally uniformly asymptotically stable for the averaged reduced order system (\\ref{eq:reduced_avg_system}). Then, there exist constants $\\lambda >0 $ and $\\gamma > 0$ such that for every bounded subset $ \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times\\mathbb{R}^m$, $\\forall t_f\\in(0,\\infty)$, and $\\forall D\\in(0,\\infty)$, there exists $\\omega^*\\in(0,\\infty)$ such that $\\forall \\omega\\in(\\omega^*,\\infty)$, $\\forall t_0\\in\\mathbb{R}$, $ \\forall (\\textbf{x}_0,\\textbf{y}_0-\\bm{\\varphi}_0(\\textbf{x}_0))\\in \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}$, and $\\forall t\\in[t_0,t_0+t_f]$, unique trajectories of the system (\\ref{eq:orig_sys}) exist and satisfy:\n\\begin{gather}\n    \\lVert \\textbf{x}(t)-\\bar{\\textbf{x}}(t) \\lVert < D \\\\\n     \\lVert \\textbf{y}(t)-\\bm{\\varphi}_0(\\textbf{x}(t))\\lVert < \\gamma \\lVert \\textbf{y}_0-\\bm{\\varphi}_0(\\textbf{x}_0)\\lVert \\,\\text{e}^{-\\omega(t-t_0)\\lambda } + D\n\\end{gather}\n\\end{prop}\n\\begin{proof}\nWe apply the time scaling $\\tau=\\omega (t-t_0)$, and we let $\\varepsilon=1\/\\sqrt{\\omega}$. In contrast to the standard singular perturbation analysis which starts with a coordinate shift for the singularly perturbed part of the system from $\\textbf{y}$ to $\\textbf{y}-\\bm{\\varphi}_0(\\textbf{x})$ (e.g. \\cite[Chapter 11]{khalil2002nonlinear}, \\cite[Section I]{durr2015singularly}), we augment the standard coordinate shift with a \\textit{near-identity} part:\n\\begin{equation}\\label{eq:nearid_shift}\n\\textbf{z}=\\textbf{y}-\\bm{\\varphi}_0(\\textbf{x})-\\varepsilon\\, \\bm{\\varphi}_1(\\textbf{x},\\tau)-\\varepsilon^2\\bm{\\varphi}_2(\\textbf{x},\\tau),\n\\end{equation}\nwhere the maps $\\bm{\\varphi}_i(\\textbf{x},\\tau)$ for $i\\in\\{1,2\\}$ are yet to be determined. This is coordinate shift is inspired by the standard near identity transform common in the higher order averaging literature \\cite[Section 2.8]{sanders2007averaging}. Observe that under this coordinate and time scale change, we have:\n\\begin{equation}\n\\begin{aligned}\\label{eq:shifted_orig_sys}\n    \\frac{d\\textbf{x}}{d\\tau}&=\\sum_{i=1}^2 \\varepsilon^i \\textbf{v}_i(\\textbf{x},\\textbf{z},\\tau) +\\varepsilon^3 \\textbf{v}_3(\\textbf{x},\\textbf{z},\\tau,\\varepsilon)\\\\\n    \\frac{d\\textbf{z}}{d\\tau}&= \\textbf{A}\\,\\textbf{z} + \\sum_{i=1}^2 \\varepsilon^i\\textbf{h}_i(\\textbf{x},\\textbf{z},\\tau) + \\varepsilon^3 \\textbf{h}_3(\\textbf{x},\\textbf{z},\\tau,\\varepsilon)\n\\end{aligned}\n\\end{equation}\nwhere the vector fields $\\textbf{v}_i$ and $\\textbf{h}_i$ for $i\\in\\{1,2\\}$ are given by:\n\\begin{align}\n    &\\begin{aligned}\n    \\textbf{v}_1(\\textbf{x},\\textbf{z},\\tau)&= \\textbf{f}_1(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\n    \\end{aligned}\\\\\n    &\\begin{aligned}\n    \\textbf{v}_2(\\textbf{x},\\textbf{z},\\tau)&= \\textbf{f}_2(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\\\\n     &+\\textbf{C}(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau) \\bm{\\varphi}_1(\\textbf{x},\\tau)\n    \\end{aligned}\\\\\n    &\\begin{aligned}\n    \\textbf{h}_1(\\textbf{x},\\textbf{z},\\tau)&= \\textbf{A}\\,\\bm{\\varphi}_1(\\textbf{x},\\tau)-\\partial_\\tau\\bm{\\varphi}_1(\\textbf{x},\\tau) \\\\\n    &- \\partial_{\\textbf{x}} \\bm{\\varphi}_0(\\textbf{x}) \\textbf{f}_1(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\\\\n    &+\\textbf{g}_1(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\n    \\end{aligned}\\\\\n    &\\begin{aligned}\n    \\textbf{h}_2(\\textbf{x},\\textbf{z},\\tau)&=\\textbf{A}\\, \\bm{\\varphi}_2(\\textbf{x},\\tau)-\\partial_\\tau\\bm{\\varphi}_2(\\textbf{x},\\tau)\\\\\n    &-\\partial_{\\textbf{x}} \\bm{\\varphi}_1(\\textbf{x},\\tau) \\textbf{f}_1(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau) \\\\\n    &- \\partial_{\\textbf{x}} \\bm{\\varphi}_0(\\textbf{x}) \\textbf{f}_2(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\\\\n    &+\\partial_{\\textbf{w}} \\textbf{g}_1(\\textbf{x},\\textbf{w},\\tau)|_{\\textbf{w}=\\bm{\\varphi}_0(\\textbf{x})}\\bm{\\varphi}_1(\\textbf{x},\\tau) \\\\\n    &+\\textbf{g}_2(\\textbf{x},\\textbf{z}+\\bm{\\varphi}_0(\\textbf{x}),\\tau)\n    \\end{aligned}\n\\end{align}\nNow, we let $\\bm{\\varphi}_1(\\textbf{x},\\tau)$ and $\\bm{\\varphi}_2(\\textbf{x},\\tau)$ be the solutions of the linear non-homogeneous two point boundary value problems:\n\\begin{align}\\label{eq:BVPi_1}\n    \\partial_\\tau\\bm{\\varphi}_i(\\textbf{x},\\tau)&= \\textbf{A}\\, \\bm{\\varphi}_i(\\textbf{x},\\tau) + \\textbf{b}_i(\\textbf{x},\\tau)\\\\ \\label{eq:BVPi_2}\n    \\bm{\\varphi}_i(\\textbf{x},\\tau) &= \\bm{\\varphi}_i(\\textbf{x},\\tau+T)\n\\end{align}\nfor $i\\in\\{1,2\\}$, where:\n\\begin{align}\n     &\\textbf{b}_1(\\textbf{x},\\tau)= \\textbf{g}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)-\\partial_{\\textbf{x}} \\bm{\\varphi}_0(\\textbf{x}) \\textbf{f}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\\\\n     &\\begin{aligned}\n     \\textbf{b}_2(\\textbf{x},\\tau)&= \\textbf{g}_2(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)-\\partial_{\\textbf{x}} \\bm{\\varphi}_0(\\textbf{x}) \\textbf{f}_2(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau)\\\\\n     &-\\partial_{\\textbf{x}} \\bm{\\varphi}_1(\\textbf{x},\\tau) \\textbf{f}_1(\\textbf{x},\\bm{\\varphi}_0(\\textbf{x}),\\tau) \\\\\n     &+\\partial_{\\textbf{w}} \\textbf{g}_1(\\textbf{x},\\textbf{w},\\tau)|_{\\textbf{w}=\\bm{\\varphi}_0(\\textbf{x})}\\bm{\\varphi}_1(\\textbf{x},\\tau)\n     \\end{aligned}\n\\end{align}\nThe following lemma is a simple consequence of \\thref{asmp:A} and standard linear systems theory:\n\\begin{lem}\\thlabel{lem:B}\nLet \\thref{asmp:A} be satisfied. Then, the non-homogeneous BVPs (\\ref{eq:BVPi_1})-(\\ref{eq:BVPi_2}) have unique solutions $\\bm{\\varphi}_i\\in\\mathcal{C}^{3-i}(\\mathbb{R}^{n};\\mathbb{R}^m)$ defined by:\n\\begin{align}\n    \\bm{\\varphi}_i(\\textbf{x},\\tau)&= \\left(\\text{Id}-\\text{e}^{T \\textbf{A}}\\right)^{-1}\\int_{0}^T \\text{e}^{ (T-s)\\textbf{A}}\\,\\textbf{b}_i(\\textbf{x},s+\\tau)ds\n\\end{align}\nwhere $\\text{Id}$ is the identity map on $\\mathbb{R}^m$. \n\\end{lem}\n\\begin{proof}\n    The result can be verified by direct substitution, and the regularity of the solutions follows from \\thref{asmp:A}.\n\\end{proof}\n\nWith this choice of the maps $\\bm{\\varphi}_i(\\textbf{x},\\tau)$ for $i\\in\\{1,2\\}$, observe that $\\textbf{v}_1(\\textbf{x},0,\\tau)=\\tilde{\\textbf{f}}_1(\\textbf{x},\\tau),\\,\\textbf{v}_2(\\textbf{x},0,\\tau)=\\tilde{\\textbf{f}}_2(\\textbf{x},\\tau)$, and that $\\textbf{h}_1(\\textbf{x},0,\\tau)=\\textbf{h}_2(\\textbf{x},0,\\tau)=0$, $\\forall \\textbf{x}\\in\\mathbb{R}^n,\\,\\forall \\tau\\in\\mathbb{R}$. That is, the origin $\\textbf{z}=0$ is an equilibrium point for the boundary layer model:\n\\begin{align}\n    \\label{eq:bl_model_z}\\frac{d\\textbf{z}}{d\\tau} &= \\textbf{A}\\,\\textbf{z} + \\sum_{i=1}^2 \\varepsilon^i\\textbf{h}_i(\\textbf{x},\\textbf{z},\\tau), & \\textbf{z}(0)&= \\textbf{z}_0\n\\end{align}\nMoreover, it can be shown that the vector fields $\\textbf{h}_i$ for $i\\in\\{1,2\\}$ are Lipschitz continuous and bounded on every compact subset $\\mathcal{K}\\subset\\mathbb{R}^n\\times\\mathbb{R}^m$, uniformly in $\\tau$, for some Lipschitz constants $L_{\\textbf{h}_i,\\mathcal{K}}>0$ and bounds $B_{\\textbf{h}_i,\\mathcal{K}}>0$, and that the remainder terms $\\textbf{h}_3$ and $\\textbf{v}_3$ are continuous and bounded on any compact subset $\\mathcal{K}\\subset\\mathbb{R}^n\\times\\mathbb{R}^m$ uniformly in $\\tau\\in\\mathbb{R}$ and $\\varepsilon\\in[0,\\varepsilon_0]$ for some $\\varepsilon_0>0$. Next, we have the following lemma:\n\\begin{lem}\\thlabel{lem:A}\nLet \\thref{asmp:A} be satisfied, and suppose that a compact subset $\\mathcal{S}\\subset\\mathbb{R}^n$ is globally uniformly asymptotically stable for the averaged reduced order system (\\ref{eq:reduced_avg_system}). Then, there exist constants $\\lambda >0 $ and $\\gamma > 0$ such that for every bounded subset $ \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times\\mathbb{R}^m$, $\\forall t_f\\in(0,\\infty)$, and $\\forall D\\in(0,\\infty)$, there exists $\\varepsilon^*\\in(0,\\varepsilon_0)$ such that $\\forall \\varepsilon\\in(0,\\varepsilon^*)$,  $ \\forall (\\textbf{x}_0,\\textbf{z}_0)\\in \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}$, and $\\forall \\tau\\in[0,t_f\/\\varepsilon^{2}]$, unique trajectories of the system (\\ref{eq:shifted_orig_sys}) exist and satisfy:\n\\begin{gather}\n    \\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau) \\lVert < D \\\\\n     \\lVert \\textbf{z}(\\tau)\\lVert < \\gamma \\lVert \\textbf{z}_0\\lVert \\,\\text{e}^{-\\lambda\\,\\tau} + D\n\\end{gather}\n\\end{lem}\n\\begin{proof}\n\\iffalse 0    \nThis lemma borrows ideas from \\cite[Lemma 1]{durr2015singularly} and \\cite[Lemma 2.8.2]{sanders2007averaging} but is rather long, so we refer the reader to the preprint of the journal version of the current manuscript for the full proof. Instead, we provide a sketch of the proof here. We emphasize that the lemma does not follow directly from the arguments in \\cite[Lemma 1]{durr2015singularly}.\n    \n    First, observe that the behavior of the singularly perturbed part of the system $\\textbf{z}$ is dominated by the linear vector field $\\textbf{A}\\,\\textbf{z}$ in the limit $\\varepsilon\\rightarrow 0$. Since the origin is an equilibrium point for the boundary layer model (\\ref{eq:bl_model_z}), only $O(\\varepsilon^3)$ terms affect the behavior of $\\textbf{z}$. Using an adapted version of ultimate boundedness \\cite[Theorem 4.18]{khalil2002nonlinear}, it can be shown that solutions to (\\ref{eq:shifted_orig_sys}) are such that $\\textbf{z}(t)$ satisfies:\n    \\begin{align}\n        \\lVert \\textbf{z}(\\tau)\\lVert < \\gamma\\,\\lVert \\textbf{z}_0\\lVert \\text{e}^{-\\lambda \\tau} + \\alpha\\,\\varepsilon^{\\frac{3}{2}}\n    \\end{align}\n    for some constants $\\gamma, \\,\\lambda, \\,\\alpha>0$ and small enough $\\varepsilon$. Next, one argues that the transient behavior of $\\textbf{z}(t)$ dies fast enough so as not to affect the behavior of $\\textbf{x}(t)$. This can be established by estimating the difference $\\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert$. Intuitively, $\\textbf{x}(\\tau)$ and $\\tilde{\\textbf{x}}(\\tau)$ move with an $O(\\varepsilon^2)$ velocity on average because the average of the $O(\\varepsilon)$ terms vanish. Hence, we expect the difference $\\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert$ to remain small on the time scale $O(1\/\\varepsilon^2)$, provided that $\\textbf{z}(\\tau)$ converges fast enough to a small enough neighborhood of the origin. Indeed, utilizing the fact that the time average of $\\textbf{f}_1$ is zero, one can obtain that for small enough $\\varepsilon$ the following estimate holds:\n    $$\\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert < \\delta(\\varepsilon) + M \\varepsilon^2\\int_{0}^{\\tau} \\lVert \\textbf{x}(s) - \\tilde{\\textbf{x}}(s)\\lVert ds$$\n    where $\\delta(\\varepsilon)$ is $o(1)$, i.e. $\\lim_{\\varepsilon\\rightarrow 0}\\delta(\\varepsilon) = 0$. An application of Gr\\\"onwall's inequality leads to:\n    \\begin{align}\n        \\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert<\\delta(\\varepsilon) \\text{e}^{ M\\varepsilon^2\\tau}\n    \\end{align}\n    Now it is clear that for any fixed $t_f>0$, and $\\forall\\tau\\in[0,t_f\/\\varepsilon^2]$, we have:\n    \\begin{align}\n        \\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert< o(1)\n    \\end{align}\n    In other words, the difference $\\lVert \\textbf{x}(\\tau) - \\tilde{\\textbf{x}}(\\tau)\\lVert$ can be made arbitrarily small $\\forall \\tau\\in[0,t_f\/\\varepsilon^2]$, which is what was to be proven. \n\\fi\nThe full proof of this Lemma is rather long. So we include it here for review purposes, but it will be replaced by a sketch in the final manuscript to conform with the page limit. The full proof will appear in a journal version of the current manuscript. The proof combines ideas from \\cite[Lemma 1]{durr2015singularly} and \\cite[Lemma 2.8.2]{sanders2007averaging}. \n\nFix an arbitrary bounded subset $\\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times\\mathbb{R}^m$, an arbitrary $D\\in(0,\\infty)$, and an arbitrary $t_f\\in (0,\\infty)$. Due to \\thref{asmp:A}, we know that  $\\forall (\\textbf{x}_0,\\textbf{y}_0)\\in \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}$, and $\\forall \\varepsilon\\in(0,\\varepsilon_0]$, unique trajectories of the system (\\ref{eq:shifted_orig_sys}) exist.  Let $[0,\\tau_e)$ with $\\tau_e\\in(0,\\infty)$ be the maximal interval of existence and uniqueness of a given solution $(\\textbf{x}(\\tau),\\textbf{z}(\\tau))$, where the dependence of the solution on the initial condition is suppressed for brevity. By assumption, we know that the compact subset $\\mathcal{S}$ is globally uniformly asymptotically stable for the reduced order averaged system (\\ref{eq:reduced_avg_system}). Hence, according to \\cite{abdelgalil2022recursive,durr2013lie}, we know that the compact subset $\\mathcal{S}$ is semi-globally practically uniformly asymptotically stable for the reduced order system (\\ref{eq:reduced_system}). That is, we know that $\\exists \\varepsilon_1\\in (0,\\varepsilon_0)$ and a compact subset $\\mathcal{N}_{\\textbf{x}}\\subset \\mathbb{R}^n$ such that $\\forall \\textbf{x}_0\\in \\mathcal{B}_{\\textbf{x}}$, and $\\forall \\varepsilon\\in(0,\\varepsilon_1)$, solutions $\\tilde{\\textbf{x}}(\\tau)$ to system (\\ref{eq:reduced_system}) exist on the interval $[0,\\infty)$, and $\\tilde{\\textbf{x}}(\\tau)\\in \\mathcal{N}_{\\textbf{x}},\\,\\forall \\tau\\in [0,\\infty)$. Define an open tubular neighborhood $\\mathcal{O}_\\textbf{x}(\\tau)$ around $\\tilde{\\textbf{x}}(\\tau)$ by $\\mathcal{O}_\\textbf{x}(\\tau)=\\{\\textbf{x}\\in\\mathbb{R}^n\\,:\\,\\lVert \\textbf{x} - \\tilde{\\textbf{x}}(\\tau)\\lVert < D\\}$, and observe that the $\\textbf{x}$-component of the solution to (\\ref{eq:shifted_orig_sys}) is initially inside $\\mathcal{O}_\\textbf{x}(0)$, i.e. $\\textbf{x}(0)=\\textbf{x}_0\\in \\mathcal{O}_\\textbf{x}(0)$. Moreover, define the compact subset $\\mathcal{M}_\\textbf{x}=\\overline{\\{x\\in\\mathbb{R}^n:\\,\\inf_{\\textbf{x}'\\in \\mathcal{N}_\\textbf{x}}\\lVert \\textbf{x}-\\textbf{x}'\\lVert <D\\}}$, where the overline indicates the closure of a set. The continuity of the solutions $(\\textbf{x}(\\tau),\\textbf{z}(\\tau))$ implies that one of the following cases holds: C1) $\\exists \\tau_D\\in (0,\\tau_e)$ s.t. $\\textbf{x}(\\tau)\\in \\mathcal{O}_\\textbf{x}(\\tau),\\,\\forall \\tau\\in[0,\\tau_D)$, and $\\lVert \\textbf{x}(\\tau_D)-\\tilde{\\textbf{x}}(\\tau_D)\\lVert = D$, or C2) $\\textbf{x}(\\tau)\\in \\mathcal{O}_\\textbf{x}(\\tau),\\,\\forall \\tau\\in [0,\\tau_e)$. Suppose that case C1) holds and observe that $\\tilde{\\textbf{x}}(\\tau)\\in \\mathcal{M}_\\textbf{x},\\,\\forall \\tau\\in[0,\\tau_D]$. From \\thref{asmp:A}, we know that the matrix $\\textbf{A}$ is Hurwitz, which implies \\cite{khalil2002nonlinear} that there exists a function $ V\\in\\mathcal{C}^1(\\mathbb{R}^m;\\mathbb{R})$, and positive constants $\\alpha_i,\\,i\\in\\{1,2,3,4\\}$ such that:\n        \\begin{gather}\n            \\label{eq:Lyapunov_1}\\alpha_1\\lVert \\textbf{z} \\lVert^2 \\leq V(\\textbf{z}) \\leq \\alpha_2 \\lVert \\textbf{z} \\lVert ^2 \\\\\n            \\label{eq:Lyapunov_2}\\partial_{\\textbf{z}}V(\\textbf{z}) \\textbf{A}\\,\\textbf{z} \\leq -\\alpha_3 \\lVert \\textbf{z} \\lVert ^2, \\quad \\left\\lVert \\partial_{\\textbf{z}}V(\\textbf{z})  \\right\\lVert \\leq \\alpha_4 \\lVert \\textbf{z} \\lVert\n        \\end{gather}\n    \n    Let $c\\in (0,\\infty)$ be such that the compact subset $\\mathcal{N}_\\textbf{z}=\\{\\textbf{z}\\in\\mathbb{R}^m :\\, \\lVert \\textbf{z} \\lVert \\leq \\sqrt{c\/\\alpha_2} \\}$ contains the bounded set $\\mathcal{B}_{\\textbf{z}}$, and define the compact subset $\\mathcal{M}_\\textbf{z}=\\{z\\in\\mathbb{R}^m :\\, \\lVert z \\lVert \\leq \\sqrt{c\/\\alpha_1} \\}$. Furthermore, define the compact subset $\\mathcal{K}=\\mathcal{M}_\\textbf{x}\\times \\mathcal{M}_\\textbf{z}$, and compute the derivative of $V$ along the trajectories of (\\ref{eq:shifted_orig_sys}):\n    \\begin{align}\n        \\frac{dV}{d\\tau} &= \\partial_{\\textbf{z}}V(\\textbf{z}) \\textbf{A}\\,\\textbf{z} +\\sum_{i=1}^2 \\varepsilon^i\\partial_{\\textbf{z}}V(\\textbf{z})  \\textbf{h}_i(\\textbf{x},\\textbf{z},\\tau)+O(\\varepsilon^3)\\\\\n        &\\leq -\\left(\\alpha_3- \\alpha_4\\sum_{i=1}^2 \\varepsilon^i L_{\\textbf{h}_i,\\mathcal{K}}\\right) \\lVert \\textbf{z} \\lVert ^2 + B_{\\textbf{h}_3,\\mathcal{K}}\\,\\varepsilon^3\n    \\end{align}\n    where we used the fact that the $O(\\varepsilon^3)$ terms are uniformly bounded on the compact subset $\\mathcal{K}$, and the vector fields $\\textbf{h}_1$ and $\\textbf{h}_2$ have an equilibrium point at $\\textbf{z}=0$ and are Lipschitz continuous. Let $\\varepsilon_2=\\min\\{\\varepsilon_1,1,\\alpha_3\/(2\\alpha_4(L_{h_1,\\mathcal{K}}+L_{h^2,\\mathcal{K}}))\\}$, and observe that $\\forall \\varepsilon\\in(0,\\varepsilon_2)$, $\\forall (\\textbf{x},\\textbf{z})\\in \\mathcal{K}$, $\\forall \\tau\\in \\mathbb{R}$, we have:\n    \\begin{align}\n        \\frac{d V(\\textbf{z})}{d\\tau} \\leq -\\frac{\\alpha_3}{2}\\lVert \\textbf{z} \\lVert ^2 + B_{\\textbf{h}_3,\\mathcal{K}}\\, \\varepsilon^3\n    \\end{align}\n    Let $\\varepsilon_3=\\min\\{\\varepsilon_2,\\sqrt{\\alpha_3 c \/(4\\alpha_2 B_{\\textbf{h}_3,\\mathcal{K}})}\\}$, then observe that $\\forall\\varepsilon\\in(0,\\varepsilon_3),\\,\\forall \\tau\\in\\mathbb{R}$, and $\\forall (\\textbf{x},\\textbf{z})\\in \\mathcal{M}_\\textbf{x}\\times \\mathcal{M}_\\textbf{z}\\backslash \\mathcal{N}_\\textbf{z}$, we have that $\\dot{V}\\leq 0$, and so the solution $(\\textbf{x}(\\tau),\\textbf{z}(\\tau))$ stays inside $\\mathcal{M}_\\textbf{x}\\times \\mathcal{M}_\\textbf{z}$, $\\forall \\tau\\in[0,\\tau_D]$. In addition, similar to \\cite[Theorem 4.18]{khalil2002nonlinear}, we have that there exists constants $\\gamma,\\lambda,\\alpha >0$ such that the estimate $\\lVert \\textbf{z}(\\tau)\\lVert < \\gamma\\,\\lVert \\textbf{z}_0\\lVert \\text{e}^{-\\lambda \\tau} + \\alpha\\,\\varepsilon^{\\frac{3}{2}}$, holds on the time interval $\\tau\\in[0,\\tau_D]$, $\\forall (\\textbf{x}_0,\\textbf{z}_0)\\in\\mathcal{B}_\\textbf{x}\\times \\mathcal{B}_\\textbf{z}$, $\\forall \\varepsilon\\in(0,\\varepsilon_3)$. We emphasize that the constants $\\gamma, \\lambda, \\alpha$ depend on the constants $\\alpha_j,L_{\\textbf{h}_i,\\mathcal{K}},B_{\\textbf{h}_i,\\mathcal{K}}$, but do not depend on the choice of $\\varepsilon\\in(0,\\varepsilon_3)$. Similar arguments can be used to establish that in C2), we have that $(\\textbf{x}(\\tau),\\textbf{z}(\\tau))\\in \\mathcal{M}_\\textbf{x}\\times \\mathcal{M}_\\textbf{z},\\,\\forall \\tau\\in [0,\\tau_e)$, which implies that $[0,\\infty)\\subset [0,\\tau_e)$. Now observe that we may choose $\\varepsilon<(D\/\\alpha)^{\\frac23}$ in to ensure that the result of the lemma holds in case C2) (see also the proof of Lemma 1 in \\cite{durr2015singularly}). \n    \n    Next, we define an $\\varepsilon$-dependent time $\\tau_\\varepsilon$ by requiring that the following inequality is satisfied:\n    \\begin{align}\n        \\label{eq:t_epsilon} \\gamma\\,\\lVert \\textbf{z}_0\\lVert \\text{e}^{-\\lambda\\,\\tau} &< \\alpha\\, \\varepsilon^{\\frac32}, & \\forall \\textbf{z}_0&\\in \\mathcal{M}_\\textbf{z},\\,\\forall \\tau>\\tau_\\varepsilon\n    \\end{align}\n    and observe that this is always possible for $\\varepsilon >0$. In fact, it can be shown that $\\tau_\\varepsilon=$ $\\max\\{(3\/(2\\lambda)) $ $ \\log((\\gamma \\sqrt{c\/\\alpha_2})\/(\\alpha\\,\\varepsilon)),$ $0\\}$ satisfies the inequality (\\ref{eq:t_epsilon}).  Now, we show that $\\exists \\varepsilon_4\\in(0,\\varepsilon_3)$ such that $\\tau_\\varepsilon<\\tau_D,\\, \\forall \\varepsilon\\in (0,\\varepsilon_4)$. To obtain a contradiction, suppose that there exists a bounded subset $\\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times \\mathbb{R}^m$, and a $D\\in(0,\\infty)$, such that $\\forall \\varepsilon_4\\in(0,\\varepsilon_3)$, $\\exists \\varepsilon\\in(0,\\varepsilon_4)$ such that $\\tau_\\varepsilon \\geq \\tau_D$. We estimate the difference $\\lVert \\textbf{x}(\\tau_D)-\\tilde{\\textbf{x}}(\\tau_D)\\lVert = \\lVert \\int_{0}^{\\tau_D}\\big(\\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(\\tau),\\textbf{z}(\\tau),\\tau))-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(\\tau),\\tau) + O(\\varepsilon^3)\\big) d\\tau\\lVert \\leq \\varepsilon \\int_{0}^{\\tau_D} B_{\\textbf{v}+\\textbf{f},\\mathcal{K}} d\\tau \\leq B_{\\textbf{v}+\\textbf{f},\\mathcal{K}} \\tau_D\\varepsilon \\leq B_{\\textbf{v}+\\textbf{f},\\mathcal{K}} \\tau_\\varepsilon\\,\\varepsilon$, where $B_{\\textbf{v}+\\textbf{f},\\mathcal{K}}$ is a uniform upper bound on the norm of the integrand inside the compact subset $\\mathcal{K}$ whose existence is guaranteed by \\thref{asmp:A}. Now, observe that $\\lim_{\\varepsilon\\rightarrow 0} \\tau_\\varepsilon\\,\\varepsilon = 0$, and so $\\forall D\\in(0,\\infty)$, $\\exists \\varepsilon_4\\in(0,\\varepsilon_3)$ such that $B_{\\textbf{v}+\\textbf{f},\\mathcal{K}} \\tau_\\varepsilon\\,\\varepsilon \\leq D\/2,\\,\\forall \\varepsilon\\in(0,\\varepsilon_4)$. Hence, we have that $\\forall \\varepsilon\\in(0,\\varepsilon_4)$, $\\lVert x(\\tau_D)-\\tilde{\\textbf{x}}(\\tau_D)\\lVert \\leq D\/2$ which contradicts the definition of $\\tau_D$. Accordingly, we have that for all bounded subsets $\\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times \\mathbb{R}^m$, $\\forall D\\in(0,\\infty)$, $\\exists \\varepsilon_4\\in(0,\\varepsilon_3)$, such that $\\forall \\varepsilon\\in(0,\\varepsilon_4)$, $\\forall (\\textbf{x}_0,\\textbf{z}_0)\\in \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}$, we have that $\\tau_\\varepsilon < \\tau_D$.\n    \n    Next, we show that $\\exists\\varepsilon_5\\in(0,\\varepsilon_4)$ such that $t_f\/\\varepsilon^2< \\tau_D$ $\\forall \\varepsilon\\in(0,\\varepsilon_5)$. To obtain a contradiction, suppose that there exists a bounded subset $\\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times \\mathbb{R}^m$, a $t_f\\in(0,\\infty)$, and a $D\\in(0,\\infty)$, such that $\\forall \\varepsilon_5\\in(0,\\varepsilon_4)$, $\\exists \\varepsilon\\in(0,\\varepsilon_5)$ such that $t_f\/\\varepsilon^2 \\geq \\tau_D$. Once again, we estimate the difference $\\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert$ on the interval $[0,\\tau_D]$. First, for $\\tau\\leq\\tau_\\varepsilon$, observe that $\\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert= \\lVert \\int_{0}^{\\tau}\\big(\\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(s),\\textbf{z}(s),s)-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(s),s)) + O(\\varepsilon^3)\\big) ds\\lVert\\leq B_{\\textbf{v}+\\textbf{f},\\mathcal{K}}\\tau_{\\varepsilon}\\varepsilon$ for some constant $B_{\\textbf{v}+\\textbf{f},\\mathcal{K}}$. Since $\\lim_{\\varepsilon\\rightarrow 0}\\tau_\\varepsilon\\varepsilon = 0$, we conclude that when $\\tau<\\tau\\varepsilon$, the difference $\\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert$ can be made arbitrarily small by choosing $\\varepsilon$ small enough. Second, for $\\tau>t_\\varepsilon$, we have that:\n    $\\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert= \\lVert \\int_{0}^{\\tau}\\big(\\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(s),\\textbf{z}(s),s)-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(s),s)) + O(\\varepsilon^3)\\big) ds\\lVert \\leq \\lVert \\int_{0}^{\\tau_\\varepsilon}\\big(\\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(s),\\textbf{z}(s),s)-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(s),s))\\big) ds\\lVert+\\lVert \\int_{\\tau_\\varepsilon}^{\\tau}\\big(\\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(s),\\textbf{z}(s),s)-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(s),s))\\big) ds\\lVert$ $+ B_{\\textbf{v}+\\textbf{f},\\mathcal{K}}\\tau_D\\varepsilon^3$, which leads to the estimate: \n    \\begin{gather}\\label{eq:I_estimate}\n    \\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert \\leq B_{\\textbf{v}+\\textbf{f},\\mathcal{K}}(\\tau_\\varepsilon+\\tau_D\\varepsilon^2)\\varepsilon+ \\lVert \\textbf{I}_1\\lVert\\\\\n    \\textbf{I}_1 = \\int_{\\tau_\\varepsilon}^{\\tau} \\sum_{i=1}^2\\varepsilon^i (\\textbf{v}_i(\\textbf{x}(s),\\textbf{z}(s),s))-\\tilde{\\textbf{f}}_i(\\tilde{\\textbf{x}}(s),s)) ds\n    \\end{gather}\n    on the interval $[0,\\tau_D]$. We proceed to estimate $\\lVert \\textbf{I}_1 \\lVert $ as follows:\n    \\begin{align}\\label{eq:I1_estimate}\n        \\lVert \\textbf{I}_1\\lVert &\\leq \\varepsilon (\\lVert \\textbf{I}_2\\lVert + \\lVert \\textbf{I}_3\\lVert ) + \\varepsilon^2 (\\lVert \\textbf{I}_4\\lVert +\\lVert \\textbf{I}_5\\lVert )\n    \\end{align}\n    where $\\textbf{I}_i$ for $i\\in\\{2,3,4,5\\}$ are given by:\n    \\begin{align}\n        \\textbf{I}_2&= \\int_{\\tau_\\varepsilon}^{\\tau}(\\textbf{v}_1(\\textbf{x}(s),\\textbf{z}(s),s))-\\textbf{v}_1(\\textbf{x}(s),0,s)) ds\\\\\n        \\textbf{I}_3&= \\int_{\\tau_\\varepsilon}^{\\tau} (\\textbf{v}_1(\\textbf{x}(s),0,s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s)) ds\\\\\n        \\textbf{I}_4&= \\int_{\\tau_\\varepsilon}^{\\tau}(\\textbf{v}_2(\\textbf{x}(s),\\textbf{z}(s),s))-\\textbf{v}_2(\\textbf{x}(s),0,s)) ds\\\\\n        \\textbf{I}_5&= \\int_{\\tau_\\varepsilon}^{\\tau} (\\textbf{v}_2(\\textbf{x}(s),0,s)-\\tilde{\\textbf{f}}_2(\\tilde{\\textbf{x}}(s),s)) ds\n    \\end{align}\n    Observe that $\\textbf{v}_i(\\textbf{x},0,s)=\\tilde{\\textbf{f}}_i(\\textbf{x},s)$, and so we have that:\n    \\begin{align}\n        \\textbf{I}_3&= \\int_{\\tau_\\varepsilon}^{\\tau} (\\tilde{\\textbf{f}}_1(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s)) ds \\\\\n        \\textbf{I}_5&= \\int_{\\tau_\\varepsilon}^{\\tau} (\\tilde{\\textbf{f}}_2(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_2(\\tilde{\\textbf{x}}(s),s)) ds\n    \\end{align}\n    We estimate each of the integrals above, starting by $\\textbf{I}_2$, $\\textbf{I}_4$ and $\\textbf{I}_5$, which can be estimated as:\n    \\begin{align}\\label{eq:I2_estimate}\n        \\lVert \\textbf{I}_2\\lVert &\\leq \\int_{\\tau_\\varepsilon}^{\\tau} L_{\\textbf{v}_1,\\mathcal{K}} \\lVert \\textbf{z}(s) \\lVert d\\tau \\\\\\label{eq:I4_estimate}\n        \\lVert \\textbf{I}_4\\lVert &\\leq \\int_{\\tau_\\varepsilon}^{\\tau} L_{\\textbf{v}_2,\\mathcal{K}} \\lVert \\textbf{z}(s) \\lVert ds \\\\\\label{eq:I5_estimate}\n        \\lVert \\textbf{I}_5\\lVert &\\leq \\int_{\\tau_\\varepsilon}^{\\tau} L_{\\textbf{f}_2,\\mathcal{K}} \\lVert \\textbf{x}(s)-\\tilde{\\textbf{x}}(s) \\lVert ds\n    \\end{align}\n    where $L_{\\textbf{v}_1,\\mathcal{K}},L_{\\textbf{v}_2,\\mathcal{K}},L_{\\textbf{f}_2,\\mathcal{K}}>0$ are Lipschitz constants. Next, we estimate $\\lVert \\textbf{I}_3\\lVert $. We proceed by dividing the interval $\\mathcal{I}=[\\tau_\\varepsilon,\\tau]$ into sub-intervals of length $T$ and a left over piece:\n    $$\\mathcal{I}=\\left(\\bigcup_{i=1}^{k(\\varepsilon)} [T_\n    {i-1},T_i]\\right) \\bigcup \\,[k(\\varepsilon)T,\\tau], $$ where $T_i=\\tau_\\varepsilon+i\\,T$, and $k(\\varepsilon)$ is the unique integer such that $k(\\varepsilon) T \\leq \\tau < k(\\varepsilon) T + T$. Then, we split $\\textbf{I}_3$ into a sum of sub-integrals:\n\\begin{align}\n      \\textbf{I}_3\n      &=\\sum_{i=1}^{k(\\varepsilon)} \\textbf{I}_{3,i}+ \\int_{k(\\varepsilon)T}^{\\tau}\\left(\\tilde{\\textbf{f}}_1(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s) \\right) ds \\\\\n      \\textbf{I}_{3,i}&= \\int_{T_{i-1}}^{T_i}\\left(\\tilde{\\textbf{f}}_1(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s) \\right)ds\n\\end{align}\nThe part of the integral on the leftover piece can be bounded independently from $\\varepsilon$ as follows:\n\\begin{align}\n    \\left\\lVert\\int_{k(\\varepsilon)T}^{\\tau}\\left(\\tilde{\\textbf{f}}_1(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s) \\right) ds\\right\\lVert \\leq 2 B_{\\textbf{f}_1,\\mathcal{K}} T\n\\end{align}\nNext, we employ Hadamard's lemma to obtain:\n\\begin{equation}\n\\begin{aligned} \n      \\textbf{I}_{3,i}&=\\int_{T_{i-1}}^{T_i}\\textbf{F}_1(\\textbf{x}(s),\\tilde{\\textbf{x}}(s),s) (\\textbf{x}(s)-\\tilde{\\textbf{x}}(s)) ds\n\\end{aligned}\n\\end{equation}\nwhere the matrix valued map $\\textbf{F}_1$ is given by:\n\\begin{align}\n&\\textbf{F}_1(\\textbf{x},\\tilde{\\textbf{x}},s)  =\\int_{0}^{1}\\partial_{\\textbf{w}}\\tilde{\\textbf{f}}_1(\\textbf{w},s)|_{\\textbf{w}=\\tilde{\\textbf{x}}+\\lambda(\\textbf{x}-\\tilde{\\textbf{x}})} d\\lambda\n\\end{align}\nThrough adding and subtracting a term, we may write:\n\\begin{equation}\n\\begin{aligned} \n     \\textbf{I}_{3,i}=&\\int_{T_{i-1}}^{T_i} \\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),s)\\,(\\textbf{x}(s)-\\tilde{\\textbf{x}}(s)) ds \\\\\n     +&\\int_{T_{i-1}}^{T_i}\\Delta_i\\big[\\textbf{F}_1\\big](s)\\,(\\textbf{x}(s)-\\tilde{\\textbf{x}}(s)) ds\n\\end{aligned}\n\\end{equation}\nwhere the term $\\Delta_i[\\textbf{F}_1]$ is given by:\n\\begin{equation*}\n\\begin{aligned} \n    \\Delta_i\\big[\\textbf{F}_1\\big](s) &= \\textbf{F}_1(\\textbf{x}(s),\\tilde{\\textbf{x}}(s),s)-\\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),s)\n\\end{aligned}\n\\end{equation*}\nNext, since the matrix-valued map $\\textbf{F}_1$ is periodic with zero average over its third argument when the other arguments are fixed, we have that\n\\begin{align}\n    \\int_{T_{i-1}}^{T_i} \\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),s)\\,\\textbf{w}\\,ds = 0 \n\\end{align}\nfor any fixed $\\textbf{w}$. Thus, we may write:\n\\begin{equation}\n\\begin{aligned} \n    &\\textbf{I}_{3,i}=\\int_{T_{i-1}}^{T_i}\\Delta_i\\big[\\textbf{F}_1\\big]\\,(\\textbf{x}(s)-\\tilde{\\textbf{x}}(s)) ds \\\\\n     &+\\int_{T_{i-1}}^{T_i} \\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),s)\\Delta_i[\\textbf{x}-\\tilde{\\textbf{x}}] ds\n\\end{aligned}\n\\end{equation}\nwhere $\\Delta_i[\\textbf{x}-\\tilde{\\textbf{x}}] = (\\textbf{x}(s)-\\textbf{x}(T_{i-1}))-(\\tilde{\\textbf{x}}(s)-\\tilde{\\textbf{x}}(T_{i-1}))$. The fundamental theorem of calculus yields: \n\\begin{align}\n    &(\\textbf{x}(s)-\\textbf{x}(T_{i-1}))-(\\tilde{\\textbf{x}}(s)-\\tilde{\\textbf{x}}(T_{i-1}))\\\\\n    =&\\varepsilon \\int_{T_{i-1}}^{s}(\\textbf{v}_1(\\textbf{x}(\\nu),\\textbf{z}(\\nu),\\nu)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(\\nu),\\nu))d\\nu + O(\\varepsilon^2) \\\\\n    =&\\varepsilon \\int_{T_{i-1}}^{s}(\\textbf{v}_1(\\textbf{x}(\\nu),\\textbf{z}(\\nu),\\nu)-\\textbf{v}_1(\\textbf{x}(\\nu),0,\\nu))d\\nu \\\\\n    +&\\varepsilon \\int_{T_{i-1}}^{s}(\\tilde{\\textbf{f}}_1(\\textbf{x}(\\nu),\\nu)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(\\nu),\\nu))d\\nu+ O(\\varepsilon^2)\n\\end{align}\nThrough integration by parts, we obtain:\n\\begin{equation}\n\\begin{aligned} \n    &\\int_{T_{i-1}}^{T_i}\\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),s)\\Delta_i[\\textbf{x}-\\tilde{\\textbf{x}}] ds \\\\\n    =&\\textbf{I}_{\\textbf{F},i}(s)\\,\\Delta_i[\\textbf{x}-\\tilde{\\textbf{x}}]\\bigg|_{s=T_{i-1}}^{s=T_i}-\\varepsilon\\int_{T_{i-1}}^{T_i}\\textbf{I}_{\\textbf{F},i}(s)\\,   \\Delta[\\tilde{\\textbf{f}}_1] ds \\\\\n    -&\\varepsilon\\int_{T_{i-1}}^{T_i}\\textbf{I}_{\\textbf{F},i}(s)\\, \\Delta[\\textbf{v}_1]ds + O(\\varepsilon^2)\n\\end{aligned}\n\\end{equation}\nwhere we have that:\n\\begin{align}\n    \\Delta[\\textbf{v}_1] &= \\textbf{v}_1(\\textbf{x}(s),\\textbf{z}(s),s)-\\textbf{v}_1(\\textbf{x}(s),0,s) \\\\\n    \\Delta[\\tilde{\\textbf{f}}_1]&= \\tilde{\\textbf{f}}_1(\\textbf{x}(s),s)-\\tilde{\\textbf{f}}_1(\\tilde{\\textbf{x}}(s),s) \\\\\n    \\textbf{I}_{\\textbf{F},i}(s)&=\\int_{T_{i-1}}^s\\textbf{F}_1(\\textbf{x}(T_{i-1}),\\tilde{\\textbf{x}}(T_{i-1}),\\nu)d\\nu\\,\n\\end{align}\nThe boundary term coming out of the integration by parts vanishes because the right factor vanishes at $s=T_{i-1}$ and the left factor vanishes at $s=T_{i}$, leaving only the integral terms. Using Lipschitz continuity and boundedness on compact subsets, it is not hard to see that:\n\\begin{align}\n    &\\left\\lVert \\int_{T_{i-1}}^{T_i}\\textbf{I}_{\\textbf{F},i}(s)\\,\\Delta[\\textbf{v}_1]ds \\right\\lVert\\leq \\int_{T_{i-1}}^{T_i} M_{\\textbf{I}_\\textbf{F},\\textbf{v}_1,\\mathcal{K}}\\,\\lVert \\textbf{z}(s) \\lVert ds \\\\\n    &\\left\\lVert \\int_{T_{i-1}}^{T_i}\\textbf{I}_{\\textbf{F},i}(s)\\,\\Delta[\\tilde{\\textbf{f}}_1] ds\\right\\lVert \\leq \\int_{T_{i-1}}^{T_i} M_{\\textbf{I}_\\textbf{F},\\textbf{f}_1,\\mathcal{K}}\\,\\lVert \\Delta[\\textbf{x}] \\lVert ds\\\\\n    &\\left\\lVert\\int_{T_{i-1}}^{T_i}\\Delta_i\\big[\\textbf{F}_1\\big]\\,\\Delta[\\textbf{x}] ds\\right\\lVert \\leq \\varepsilon\\int_{T_{i-1}}^{T_i} L_{\\textbf{F}_1,\\mathcal{K}}\\,\\,\\lVert \\Delta[\\textbf{x}] \\lVert ds\n\\end{align}\nwhere $\\Delta[\\textbf{x}]=\\textbf{x}(s)-\\tilde{\\textbf{x}}(s)$. By utilizing the above estimates, the integral on the sub-intervals can be shown to satisfy the bound:\n\\begin{align}\n    \\left\\lVert\\textbf{I}_{3,i}\\right\\lVert \\leq L_{\\mathcal{K}} \\,\\varepsilon \\,\\int_{T_{i-1}}^{T_i} \\left(\\lVert \\Delta[\\textbf{x}] \\lVert + \\lVert \\textbf{z}(s) \\lVert\\right)ds\n\\end{align}\nfor some Lipschitz constant $L_{\\mathcal{K}}$ and consequently the integral term $\\textbf{I}_3$ satisfies the bound:\n\\begin{align}\\label{eq:I3_estimate}\n    \\lVert \\textbf{I}_3\\lVert &\\leq L_{\\mathcal{K}} \\,\\varepsilon \\,\\int_{\\tau_\\varepsilon}^{\\tau} \\left(\\lVert \\Delta[\\textbf{x}] \\lVert + \\lVert \\textbf{z}(s) \\lVert\\right)ds+ 2B_{\\textbf{f}_1,\\mathcal{K}}T\n\\end{align}\nCombining (\\ref{eq:I_estimate}), (\\ref{eq:I1_estimate}), (\\ref{eq:I2_estimate}), (\\ref{eq:I5_estimate}), (\\ref{eq:I4_estimate}), and  (\\ref{eq:I3_estimate}), in addition to the fact that $\\tau_\\varepsilon<\\tau_D,\\,\\forall \\varepsilon\\in(0,\\varepsilon_4)$, we can show that the following estimate holds:\n\\begin{equation}\n    \\begin{aligned}\n    &\\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert \\leq (M_{\\mathcal{K},1}+M_{\\mathcal{K},2} \\tau_\\varepsilon  + M_{\\mathcal{K},3}\\tau_D\\varepsilon^2)\\varepsilon \\\\\n    &+ M_{\\mathcal{K},4}\\varepsilon \\int_{\\tau_\\varepsilon}^{\\tau}\\lVert \\textbf{z}(s)\\lVert ds +M_{\\mathcal{K},5}\\varepsilon^2\\int_{\\tau_\\varepsilon}^{\\tau}\\lVert \\textbf{x}(s)-\\tilde{\\textbf{x}}(s)\\lVert ds\n    \\end{aligned}\n\\end{equation}\nUsing the fact that $\\lVert \\textbf{z}(\\tau)\\lVert<2 \\alpha\\,\\varepsilon^{\\frac32},\\,\\forall \\tau>\\tau_\\varepsilon$ by definition, we obtain that:\n\\begin{align}\n    \\int_{\\tau_\\varepsilon}^{\\tau}\\lVert \\textbf{z}(s)\\lVert ds \\leq \\alpha\\, \\tau\\,\\varepsilon^{\\frac32} \\leq \\alpha\\, \\tau_D\\,\\varepsilon^{\\frac32}\n\\end{align}\nNow, remember that in order to obtain a contradiction we assumed that $\\tau_D\\leq t_f\/\\varepsilon^2$, and so we will have:\n\\begin{align}\n\\lVert \\textbf{x}&(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert \\leq \\delta(\\varepsilon) +\\int_{0}^{\\tau} M_{\\mathcal{K},5}\\varepsilon^2 \\lVert \\textbf{x}(s)-\\tilde{\\textbf{x}}(s)\\lVert ds\n\\end{align}\nwhere the function $\\delta(\\varepsilon)$ is given by:\n\\begin{align}\n\\delta(\\varepsilon)&= M_{\\mathcal{K},1}\\varepsilon + M_{\\mathcal{K},2} \\tau_\\varepsilon\\varepsilon + M_{\\mathcal{K},3}t_f\\varepsilon + M_{\\mathcal{K},4}t_f \\varepsilon^{\\frac12}\n\\end{align}\nAn application of Gr\\\"onwall's inequality yields:\n\\begin{align}\n    \\lVert \\textbf{x}&(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert \\leq \\delta(\\varepsilon) \\text{e}^{M_{\\mathcal{K},5}\\varepsilon^2\\tau} \\leq \\delta(\\varepsilon) \\text{e}^{M_{\\mathcal{K},5}\\varepsilon^2\\tau_D}\n\\end{align}\non the interval $\\tau\\in[0,\\tau_D]$. Once again, recall that we assumed that $\\tau_D\\leq t_f\/\\varepsilon^2$, and so we have: \n\\begin{align}\n    \\lVert \\textbf{x}(\\tau)-\\tilde{\\textbf{x}}(\\tau)\\lVert \\leq \\delta(\\varepsilon) \\text{e}^{M_{\\mathcal{K},5}t_f}\n\\end{align}\nNow, observe that $\\lim_{\\varepsilon\\rightarrow 0} \\delta(\\varepsilon) = 0$, and so we are guaranteed the existence of an $\\varepsilon_5\\in(0,\\varepsilon_4)$ such that $\\forall\\varepsilon\\in(0,\\varepsilon_5)$ we will have:\n\\begin{align}\n     \\lVert \\textbf{x}&(\\tau_D)-\\tilde{\\textbf{x}}(\\tau_D)\\lVert \\leq D\/2\n\\end{align}\nwhich contradicts the definition of $\\tau_D$. Hence, the assumption that $\\tau_D\\leq t_f\/\\varepsilon^2$ is wrong, and we have that for all bounded subsets $\\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}\\subset\\mathbb{R}^n\\times \\mathbb{R}^m$, $\\forall D\\in(0,\\infty)$, and $\\forall t_f\\in(0,\\infty)$, $\\exists \\varepsilon_5\\in(0,\\varepsilon_4)$, such that $\\forall \\varepsilon\\in(0,\\varepsilon_5)$, $\\forall (\\textbf{x}_0,\\textbf{z}_0)\\in \\mathcal{B}_{\\textbf{x}}\\times \\mathcal{B}_{\\textbf{z}}$, we have that $t_f\/\\varepsilon^2 < \\tau_D$, which shows that the lemma also holds in case C1).\n\\end{proof}\n\n\\thref{prop:A} follows from this lemma, after reversing the time scaling $\\tau=\\omega (t-t_0)$ and the near identity part of the coordinate shift (\\ref{eq:nearid_shift}), coupled with \\cite[Theorem 2.1]{abdelgalil2022recursive} and the fact that for $\\textbf{x}\\in\\mathcal{K}$ where $\\mathcal{K}$ is any compact subset, the maps $\\bm{\\varphi}_i$ for $i\\in\\{1,2\\}$ are uniformly bounded in time due to continuity and periodicity.\n\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn 2015 the Laser Interferometer Gravitational-Wave Observatory (LIGO) Scientific Collaboration and the Virgo Collaboration \\cite{Abbott:2016blz,TheLIGOScientific:2016agk} directly observed the first gravitational wave (GW) event GW150914 coming from the coalescence of binary black holes (BHs), \nand the discovery opened us a new window to understand the property of BHs and gravity in the nonlinear and strong field regimes.\nUntil now tens of GW events in the frequency range of tens to hundreds Hertz have been confirmed \\cite{LIGOScientific:2018mvr,LIGOScientific:2020ibl,LIGOScientific:2021usb,LIGOScientific:2021djp,LIGOScientific:2017vwq,LIGOScientific:2020aai,LIGOScientific:2021qlt}.\nHowever, due to the seismic noise and gravity gradient noise, the ground-based GW observatories can only measure transient GWs in the frequency range $10-10^3$ Hz, radiated by the coalescences of stellar-mass compact binaries.\nApart from transient GW sources, the future space-based GW detectors like the Laser Interferometer Space Antenna (LISA) \\cite{Danzmann:1997hm,Audley:2017drz}, TianQin \\cite{Luo:2015ght} and Taiji \\cite{Hu:2017mde} will help us uncover unprecedented information about GW sources and fundamental physics \\cite{LISA:2017pwj,TianQin:2020hid,Ruan:2018tsw,Amaro-Seoane:2022rxf,LISA:2022kgy}. \nOne of the most conspicuous sources for the future space-based GW detectors is the extreme mass-ratio inspirals (EMRIs) \\cite{Amaro-Seoane:2007osp,Babak:2017tow}.\nEMRIs, which consist of a stellar-mass compact object (secondary object) with mass $m_p\\sim1-100~M_{\\odot}$ such as BHs, neutron stars, white dwarfs, etc. orbiting around a supermassive black hole (SMBH) (primary object) with mass $M\\sim10^5-10^7~M_{\\odot}$, \nwith the mass ratio $m_p\/M$ in the range of $10^{-7}-10^{-4}$, \nradiate millihertz GWs expected to be observed by the future space-based GW detectors.\n\nFuture detections of EMRIs with space-based detectors can provide highly precise measurements on source parameters such as the BH masses, spins, etc.\nIn \\cite{Barack:2003fp}, the authors introduced a family of approximate waveforms for EMRIs to make parameter estimation with LISA.\nFor a typical source of $m_p=10~M_{\\odot}$ and $M=10^6~M_{\\odot}$ with a signal-to-noise ratio (SNR) of $30$, \nLISA can determine the masses of both primary and secondary objects to within a fractional error of $\\sim 10^{-4}$, measure the spin of the primary object to within $\\sim 10^{-4}$, and localize the source on the sky within $\\sim 10^{-3}$ steradians.\nThe improved augmented analytic kludge model \\cite{Chua:2017ujo} provides more accurate and efficient GW waveforms to improve the errors of parameters by one order of magnitude.\nThus, EMRIs can be used to precisely measure the slope of the black-hole mass function  \\cite{Gair:2010yu} or as standard sirens \\cite{Holz:2005df} to  constrain cosmological parameters and investigate the expansion history of the Universe \\cite{MacLeod:2007jd,Laghi:2021pqk,LISACosmologyWorkingGroup:2022jok}.\nThe observations of EMRIs can also help us explore gravitational physics.\nFor example, they can be used to figure out the spacetime structure around the central SMBH to high precision, \nallowing us to test if the spacetime geometry is described by general relativity or an alternative theory \nand analyze the environments such as dark matter surrounding the central SMBH \\cite{Amaro-Seoane:2007osp,eLISA:2013xep,Eda:2013gg,Eda:2014kra,Barausse:2014tra,Yue:2017iwc,Yue:2018vtk,Babak:2017tow,Berry:2019wgg,Hannuksela:2019vip,Destounis:2020kss,Burton:2020wnj,Torres:2020fye,Barausse:2020rsu,Cardoso:2019rou,Maselli:2020zgv,Maselli:2021men,Guo:2022euk,Zhang:2022rfr,Barsanti:2022ana,Cardoso:2021wlq,Dai:2021olt,Jiang:2021htl,Zhang:2022hbt,Gao:2022hho,Gao:2022hsn,Destounis:2022obl,Barsanti:2022vvl,Liang:2022gdk}.\nIn \\cite{Cardoso:2020iji,Liu:2020vsy,Liu:2020bag}, \nthe authors investigated the eccentricity and orbital evolution \n of BH binaries under the influence of accretion in addition to the scalar\/vector and gravitational radiations, and discussed the competition between radiative mechanisms and accretion effects on eccentricity evolution.\nHowever, these discussions were mainly based on the Newtonian orbit and dipole emission.\nA generic, fully-relativistic formalism to study EMRIs in spherically symmetric and non-vacuum BH spacetime was established in \\cite{Cardoso:2022whc}.\nConsidering the secondary object of mass $m_p$ orbiting the galactic BHs (GBHs) immersed in an astrophysical environment, \nlike an accretion disk or a dark matter halo, the authors found that the relative flux difference at $\\omega M=0.02$ between a vacuum and a GBH with the halo mass $M_{\\text{halo}}=0.1 M$ and the lengthscale $a_0=10^2 M_{\\text{halo}}$ and $10^3 M_{\\text{halo}}$ is $\\sim 10\\%$ and $1\\%$, respectively.\nThe results clearly indicate that EMRIs can constrain smaller-scale matter distributions around GBHs \\cite{Cardoso:2022whc}.\n\nAccording to the no-hair theorem, any BH can be described by three parameters: the mass, angular momentum, and electric charge.\nCurrent observations have not yet been able to confirm the  no-hair theorem or the existence of extra fields besides the gravitational fields in modified gravity theories.\nThe coupling between scalar fields and higher-order curvature invariants can invalidate the no-hair theorem so that BHs can carry scalar charge which depends on the mass of BH \\cite{Sotiriou:2013qea,Silva:2017uqg,Doneva:2017bvd,Antoniou:2017acq}.\nThe possible detection of scalar fields with EMRIs was discussed in \\cite{Maselli:2020zgv,Maselli:2021men,Barsanti:2022ana,Guo:2022euk,Zhang:2022rfr,Barsanti:2022vvl}.\nSMBHs are usually neutral because of long-time charge dissipation through the presence of the plasma around them or the spontaneous production of electron-positron pairs \\cite{Gibbons:1975kk,Eardley:1975kp,1982PhRvD..25.2509H,1982Prama..18..385J,Gong:2019aqa}.\nHowever, the existence of stable charged astrophysical stellar-mass compact objects such as BHs, neutron stars, white dwarfs, etc. in nature remains controversial.\nIn Refs. \\cite{Bekenstein:1971ej,de1995relativistic,deFelice:1999qp,Ivanov:2002jy,Majumdar:1947eu,zhang_influence_1982,Anninos:2001yb,Bonnor:1975gba}, \nit was shown that there can be a maximal huge amount of charge with the charge-to-mass ratio of the order one for highly compact stars, whose radius is on the verge of forming an event horizon.\nThe balance between the attractive gravitational force from the matter part and the repulsive force from the electrostatic part is unstable and charged compact stars will collapse to a charged BH due to a decrease in the electric field \\cite{Ray:2003gt}.\nAlso, BHs can be charged through the Wald mechanism by selectively accreting charge in a magnetic field \\cite{Wald:1974np}, or by accreting minicharged dark matter beyond the standard model \\cite{Cardoso:2016olt}.\nThe charge carried by compact objects can affect the parameter estimation of  the chirp mass and BH merger \\cite{Christiansen:2020pnv,Jai-akson:2017ldo}\nand the merger rate distribution of primordial BH binaries \\cite{Liu:2022wtq,Liu:2020cds}.\nThe electromagnetic self-force acting on a charged particle in an equatorial circular orbit of Kerr BH was calculated in \\cite{Torres:2020fye}. \nIt showed the dissipative self-force balances with the sum of the electromagnetic flux radiated to infinity and down the BH horizon,\nand prograde orbits can stimulate BH superradiance although the superradiance is not sufficient to support floating orbits even at the innermost stable circular orbit (ISCO) \\cite{Torres:2020fye}.\nThe observations of GW150914 can constrain the charge-to-mass ratio of charged BHs to be as high as 0.3 \\cite{Bozzola:2020mjx}.\n\nIn \\cite{Zhang:2022hbt}, the energy fluxes for both gravitational and electromagnetic waves induced by a charged particle orbiting around a Schwarzschild BH were studied.\nIt was demonstrated that the electric charge leaves a significant imprint on the phase of GWs and is observable with space-based GW detectors.\nIn this paper, based on the Teukolsky formalism for BH perturbations \\cite{Teukolsky:1973ha,Press:1973zz,Teukolsky:1974yv}, \nwe numerically calculate the energy fluxes for both tensor and vector perturbations induced by a charged particle moving in an equatorial circular orbit around a Kerr BH, the orbital evolution of EMRIs up to the ISCO, \nand investigate the capabilities to detect vector charge carried by the secondary compact object with space-based GW detectors such as LISA, TianQin, and Taiji. \nWe apply the methods of faithfulness and Fisher information matrix (FIM) to assess the capability of space-based GW detectors to detect the vector charge carried by the secondary compact object.\nThe paper is organized as follows.\nIn Sec. \\ref{sec2}, we introduce the model with vector charge and the Teukolsky perturbation formalism.\nIn Sec. \\ref{sec3}, we give the source terms as well as procedures for solving the inhomogeneous Teukolsky equations. \nThen we numerically calculate the energy fluxes for gravitational and vector fields using the Teukolsky and generalized Sasaki-Nakamura (SN) formalisms in the background of a Kerr BH.\nIn Sec. \\ref{sec4}, we give the numerical results of energy fluxes falling onto the horizon and radiated to infinity for gravitational and vector fields, \nthen we use the dephasing of GWs to constrain the charge.\nIn Sec. \\ref{sec5}, we calculate the faithfulness between GWs with and without vector charge and perform the FIM to estimate the errors of detecting vector charge with LISA, TianQin, and Taiji.\nSec. \\ref{sec5} is devoted to conclusions and discussions.\nIn this paper, we set $c=G=M=1$.\n\n\\section{Einstein-Maxwell field equations}\n\\label{sec2}\nThe simplest model including vector charges is Einstein-Maxwell theory, which is modeled via the massless vector field\n\\begin{equation}\nS=\\int d^4x\\frac{\\sqrt{-g}}{16\\pi}\\left[R-\\frac{1}{4}F^{\\mu\\nu}F_{\\mu\\nu}-A_{\\mu}J^{\\mu}\\right]-S_{\\text{matter}}(g_{\\mu\\nu},\\Phi),\n\\end{equation}\nwhere $R$ is the Ricci scalar, $A_\\mu$ is a massless vector field, $F_{\\mu\\nu}=\\nabla_{\\mu}A_{\\nu}-\\nabla_{\\nu}A_{\\mu}$ is the field strength and $\\Phi$ is the matter field, $J^\\mu$ is the electric current density.\nVarying the action with respect to the metric tensor and the vector field yields the Einstein-Maxwell field equations\n\\begin{equation}\nG^{\\mu\\nu}=8\\pi T^{\\mu\\nu}_p+8\\pi T^{\\mu\\nu}_e,\n\\end{equation}\n\\begin{equation}\n\\nabla_\\nu F^{\\mu\\nu}=4\\pi J^\\mu,\n\\end{equation}\nwhere $G_{\\mu\\nu}$ is the Einstein tensor, $T_p^{\\mu\\nu}$ and $T_e^{\\mu\\nu}$ are the particle's material stress-energy tensor and vector stress-energy tensor, respectively.\nSince the amplitude of the vector stress-energy $T_e^{\\mu\\nu}$ is quadratic in the vector field, the contribution to the background metric from the vector field is second order.\nFor an EMRI system $(m_p\\ll M)$ composing of a small compact object with mass $m_p$ and  charge-to-mass ratio $q$ orbiting around a Kerr BH with mass $M$ and spin $Ma$, we can ignore the contribution to the background metric from the vector field.\nThe perturbed Einstein and Maxwell equations for EMRIs are\n\\begin{equation}\nG^{\\mu\\nu}=8\\pi T^{\\mu\\nu}_p,\n\\end{equation}\n\\begin{equation}\n\\nabla_\\nu F^{\\mu\\nu}=4\\pi J^\\mu,\n\\end{equation}\nwhere\n\\begin{equation}\nT^{\\mu\\nu}_p(x)=m_p\\int d\\tau~u^{\\mu}u^\\nu\\frac{\\delta^{(4)}\\left[x-z(\\tau)\\right]}{\\sqrt{-g}},\n\\end{equation}\n\\begin{equation}\nJ^{\\mu}(x)=q m_p\\int d\\tau~ u^{\\mu}\\frac{\\delta^{(4)}\\left[x-z(\\tau)\\right]}{\\sqrt{-g}},\n\\end{equation}\nand $u^{\\mu}$ is the velocity of the particle.\nWe use the Newman-Penrose formalism \\cite{Newman:1966ub} to study perturbations around a Kerr BH induced by a charged particle with mass $m_p$ and charge $q$.\nIn Boyer-Lindquist coordinate,  the metric of Kerr BHs is\n\\begin{equation}\\label{SBH}\n\\begin{split}\nds^2=&(1-2r\/\\varSigma)dt^2+(4ar\\sin(\\theta)\/\\varSigma)dtd\\varphi-(\\varSigma\/\\Delta)dr^2-\\varSigma d\\theta^2\\\\\n&-\\sin^2{\\theta}(r^2+a^2+2a^2r\\sin^2{\\theta}\/\\varSigma)d\\varphi^2.\n\\end{split}\\end{equation}\nwhere  $\\varSigma=r^2+a^2\\cos^2{\\theta}$, and $\\Delta=r^2-2r+a^2$. When $a=0$, the metric reduces to the Schwarzschild metric.\nBased on the metric \\eqref{SBH}, we construct the null tetrad,\n\\begin{equation}\n\\begin{split}\n\\begin{split}\nl^\\mu&=[(r^2+a^2)\/\\Delta,1,0,a\/\\Delta],\\\\\nn^\\mu&=[r^2+a^2,-\\Delta,0,a]\/(2\\varSigma),\\\\\nm^\\mu&=[ia\\sin{\\theta},0,1,i\/\\sin{\\theta}]\/(2^{1\/2}(r+ia\\cos{\\theta})),\\\\\n\\bar m^\\mu&=[-ia\\sin{\\theta},0,1,-i\/\\sin{\\theta}]\/(2^{1\/2}(r-ia\\cos{\\theta})).\n\\end{split}\n\\end{split}\n\\end{equation}\nThe propagating vector field is described by the two complex quantities,\n\\begin{equation}\n\\phi_0=F_{\\mu\\nu}l^\\mu m^\\nu,\\qquad \\phi_2=F_{\\mu\\nu}\\bar{m}^\\mu n^\\nu.\n\\end{equation}\nThe propagating gravitational field is described by the two complex Newman-Penrose variables\n\\begin{equation}\n\\begin{split}\n\\psi_0&=-C_{\\alpha\\beta\\gamma\\delta}l^\\alpha m^\\beta l^\\gamma m^\\delta    ,\\\\\n\\psi_4&=-C_{\\alpha\\beta\\gamma\\delta}n^\\alpha \\bar{m}^\\beta n^\\gamma \\bar{m}^\\delta,\n\\end{split}\n\\end{equation}\nwhere $C_{\\alpha\\beta\\gamma\\delta}$ is the Weyl tensor.\nA single master equation for tensor ($s=-2$) and vector ($s=-1$) perturbations was derived as \\cite{Teukolsky:1973ha},\n\n\\begin{equation}\n\\label{TB}\n\\begin{split}\n&\\left[\\frac{(r^2+a^2)^2}{\\Delta}-a^2\\sin^2{\\theta}\\right]\\frac{\\partial^2\\psi}{\\partial t^2}+\\frac{4ar}{\\Delta}\\frac{\\partial^2\\psi}{\\partial t\\partial\\varphi}+\\left[\\frac{a^2}{\\Delta}-\\frac{1}{\\sin^2{\\theta}}\\right]\\frac{\\partial^2\\psi}{\\partial \\varphi^2}\\\\\n&\\qquad-\\Delta^{-s}\\frac{\\partial}{\\partial r}\\left(\\Delta^{s+1}\\frac{\\partial\\psi}{\\partial r}\\right)-\\frac{1}{\\sin\\theta}\\frac{\\partial}{\\partial \\theta}\\left(\\sin\\theta\\frac{\\partial\\psi}{\\partial \\theta}\\right)-2s\\left[\\frac{a(r-1)}{\\Delta}+\\frac{i\\cos\\theta}{\\sin^2{\\theta}}\\right]\\frac{\\partial\\psi}{\\partial \\varphi}\\\\\n&\\qquad\\qquad\\qquad\\qquad-2s\\left[\\frac{(r^2-a^2)}{\\Delta}-r-ia\\cos\\theta\\right]\\frac{\\partial\\psi}{\\partial t}+(s^2\\cot^2\\theta-s)\\psi=4\\pi\\varSigma T,\n\\end{split}\n\\end{equation}\nthe explicit field $\\psi$ and the corresponding source $T$ are given in  Table \\ref{source} \\cite{Teukolsky:1973ha}.\n \\begin{table}[h]\n  \\centering\n  \t\\begin{tabular}{|p{0.5cm}<{\\centering}|p{3.4cm}<{\\centering}|p{3.4cm}<{\\centering}|}\n\t\t\\hline\n$s$ & $\\psi$ &  $T$\\\\ \\hline\n-1   & $(r-i a\\cos\\theta)^{2}\\phi_2$ & $(r-i a\\cos\\theta)^{2}J_2$  \\\\ \\hline\n -2   & $(r-i a\\cos\\theta)^{4}\\psi_4$ &$2(r-i a\\cos\\theta)^{4}T_4$  \\\\ \\hline\n\t\\end{tabular}\n \\caption{The explicit expressions for the the field $\\psi$ and the corresponding source $T$.}\n    \\label{source}\n\\end{table}\nIn terms of the eigenfunctions ${_{s}}S_{lm}(\\theta)$ \\cite{Teukolsky:1973ha,Goldberg:1966uu}, the field $\\psi$ can be written as   %\n\\begin{equation}\n\\psi=\\int d\\omega \\sum_{l,m}R_{\\omega lm}(r)~{_{s}}S_{lm}(\\theta)e^{-i\\omega t+im\\varphi},\n\\end{equation}\nwhere the radial function $R_{\\omega lm}(r)$ satisfies the inhomogeneous Teukolsky equation\n\\begin{equation}\n\\label{Teukolsky}\n\\Delta^{-s}\\frac{d}{d r}\\left(\\Delta^{s+1}\\frac{d R_{\\omega lm}}{d r}\\right)-V_{T}(r)R_{\\omega lm}=T_{\\omega lm},\n\\end{equation}\nthe function\n\\begin{equation}\nV_{T}=-\\frac{K^2-2is(r-1)K}{\\Delta}-4is\\omega r+\\lambda_{lm\\omega},\n\\end{equation}\n $K=(r^2+a^2)\\omega-am$, $\\lambda_{lm\\omega}$ is the corresponding eigenvalue which can be computed by the BH Perturbation Toolkit \\cite{BHPToolkit}, and the source $T_{\\omega lm}(r)$ is\n\\begin{equation}\nT_{\\omega lm}(r)=\\frac{1}{2\\pi}\\int dt d\\Omega ~4\\pi \\Sigma T ~{_s}S_{lm}(\\theta)e^{i\\omega t-im\\varphi}.\n\\end{equation}\nFor the equatorial circular trajectory at $r_0$ under consideration,\nthe sources are\n\\begin{equation}\n\\begin{split}\nT^{\\mu\\nu}_p(x)&=\\frac{m_p}{r_0^2}\\frac{u^{\\mu}u^{\\nu}}{u^t}\\delta(r-r_0)\\delta(\\cos\\theta)\\delta(\\varphi-\\hat{\\omega} t),\\\\\nJ^{\\mu}(x)&=q\\frac{m_p}{r_0^2}\\frac{u^{\\mu}}{u^t}\\delta(r-r_0)\\delta(\\cos\\theta)\\delta(\\varphi-\\hat{\\omega} t),\n\\end{split}\n\\end{equation}\nwhere $\\hat{\\omega}$ is the orbital angular frequency.\nGeodesic motion in Kerr spacetime \nadmits three constants of motion: the specific energy $\\hat{E}$, the angular momentum $\\hat{L}$, and the Carter constant $\\hat{Q}$,\nand the geodesic equations are \n\\begin{eqnarray}\nm_p\\Sigma \\frac{d t}{d \\tau} &=&\\hat{E} \\frac{\\varpi^{4}}{\\Delta}+a \\hat{L}\\left(1-\\frac{\\varpi^{2}}{\\Delta}\\right)-a^{2} \\hat{E}\\sin^{2}\\theta, \\label{timequa}\\\\\nm_p\\Sigma \\frac{d r}{d \\tau} &=&\\pm \\sqrt{V_{r}\\left(r_{0}\\right)}, \\\\\nm_p\\Sigma \\frac{d \\theta}{d \\tau} &=&\\pm \\sqrt{V_{\\theta}\\left(\\theta\\right)}, \\\\\nm_p\\Sigma \\frac{d \\varphi}{d \\tau} &=&a \\hat{E}\\left(\\frac{\\varpi^{2}}{\\Delta}-1\\right)-\\frac{a^{2} \\hat{L}}{\\Delta}+ \\hat{L} \\csc ^{2} \\theta,\\label{anglequa}\n\\end{eqnarray}\nwhere $\\varpi\\equiv\\sqrt{r^2+a^2}$, the radial and polar potentials are\n\\begin{eqnarray}\n&V_{r}(r)& =\\left(\\hat{E} \\varpi^{2}-a \\hat{L}\\right)^{2}-\\Delta\\left(r^{2}+\\left(\\hat{L}-a \\hat{E}\\right)^{2}+\\hat{Q}\\right), \\\\\n&V_{\\theta}(\\theta)& = \\hat{Q}-\\hat{L}^{2} \\cot ^{2} \\theta-a^{2}\\left(1-\\hat{E}^{2}\\right) \\cos ^{2} \\theta.\n\\end{eqnarray}\n\nIn the adiabatic approximation, for a quasi-circular orbit on the equatorial plane,\nthe coordinates $r$ and $\\theta$ are considered as constants,\nthen Eqs. \\eqref{timequa}  and \\eqref{anglequa} are the remaining equations.\nThe conserved constants are \\cite{Detweiler:1978ge}\n\\begin{eqnarray}\\label{orbitE}\n\t\\hat{E}&=& m_p\\frac{r_0^{3 \/ 2}-2 r_{0}^{1 \/ 2} \\pm a }{r_{0}^{3 \/ 4}\\left(r_{0}^{3 \/ 2}-3  r_{0}^{1 \/ 2} \\pm 2 a \\right)^{1 \/ 2}},    \\\\\n\t\\hat{L}&=&m_p \\frac{\\pm (r_{0}^{2}\\mp 2a r_{0}^{1 \/ 2} + a^2)}{r_{0}^{3 \/ 4}\\left(r_{0}^{3 \/ 2}-3  r_{0}^{1 \/ 2} \\pm 2 a \\right)^{1 \/ 2}},\\\\\n\t\\hat{Q}&=&0.\n\\end{eqnarray}\nThe orbital angular frequency is\n\\begin{equation}\\label{orbitF}\n\\hat{\\omega} \\equiv \\frac{d\\varphi}{dt}=\\frac{\\pm 1}{r_{0}^{3 \/ 2} \\pm a},\n\\end{equation}\nwhere $\\pm$ corresponds to  co-rotating and counter-rotating, respectively.\nIn the following discussions, we use positive $a$ for co-rotating cases and negative $a$ for counter-rotating cases.\n\n\n\\section{Numerical calculation for the energy flux}\n\\label{sec3}\nThe homogeneous Teukolsky equation \\eqref{Teukolsky} admits two linearly independent solutions $R^{\\text{in}}_{\\omega lm}$ and $R^{\\text{up}}_{\\omega lm}$, with the following asymptotic values at the horizon $r_+$ and at infinity,\n\\begin{equation}\nR^{\\text{in}}_{\\omega lm}=\n\\begin{cases}\nB^{\\text{tran}}\\Delta^{-s}e^{-i\\kappa r^*},&\\quad (r\\to r_+)\\\\\nB^{\\text{out}}\\frac{e^{i\\omega r^*}}{r^{2s+1}}+B^{\\text{in}}\\frac{e^{-i\\omega r^*}}{r}, &\\quad (r\\to+\\infty)\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nR^{\\text{up}}_{\\omega lm}=\n\\begin{cases}\nD^{\\text{out}}e^{i\\kappa r^*}+\\frac{D^{\\text{in}}}{\\Delta^{s}}e^{-i\\kappa r^*},&\\quad (r\\to r_+)\\\\\nD^{\\text{tran}}\\frac{e^{i\\omega r^*}}{r^{2s+1}},&\\quad (r\\to+\\infty)\\\\\n\\end{cases}\n\\end{equation}\nwhere $\\kappa=\\omega-m a\/(2r_+)$, $r_\\pm=1\\pm\\sqrt{1-a^2}$, and the tortoise radius of the Kerr metric\n\\begin{equation}\nr^*=r+\\frac{2r_+}{r_+-r_-}\\ln \\frac{r-r_+}{2}-\\frac{2r_-}{r_+-r_-}\\ln \\frac{r-r_-}{2}.\n\\end{equation}\nThe solutions $R^{\\text{in}}_{\\omega lm}$ and $R^{\\text{up}}_{\\omega lm}$  are purely outgoing at infinity and purely ingoing at the horizon.\nWith the help of these homogeneous solutions, the solution to Eq.~\\eqref{Teukolsky} is\n\n\\begin{equation}\n\\begin{split}\nR_{\\omega lm}(r)=\\frac{1}{W}\n\\left(R^{\\text{in}}_{\\omega lm}\\int_{r}^{+\\infty}\\Delta^{s}R^{\\text{up}}_{\\omega lm}T_{\\omega lm}dr+R^{\\text{up}}_{\\omega lm}\\int_{r_+}^{r}\\Delta^{s}R^{\\text{in}}_{\\omega lm}T_{\\omega lm}dr\\right).\n\\end{split}\n\\end{equation}\nwith the constant Wronskian given by\n\\begin{equation}\nW= \\Delta^{s+1} \\left(R^{\\text{in}}_{\\omega lm}\\frac{d R^{\\text{up}}_{\\omega lm}}{dr}-R^{\\text{up}}_{\\omega lm}\\frac{d R^{\\text{in}}_{\\omega lm}}{dr}\\right)=2i\\omega B^{\\text{in}} D^{\\text{tran}}.\n\\end{equation}\nThe solution is purely outgoing at infinity and purely ingoing at the horizon,\n\\begin{equation}\n\\begin{split}\nR_{\\omega lm}(r\\to r_+)=Z^{\\infty}_{\\omega lm}\\Delta^{-s}e^{-i\\kappa r^*},\\\\\nR_{\\omega lm}(r\\to \\infty)=Z^{H}_{\\omega lm}r^{-2s-1}e^{i\\omega r^*},\n\\end{split}\n\\end{equation}\nwith\n\\begin{equation}\n\\begin{split}\nZ^{\\infty}_{\\omega lm}&=\\frac{B^{\\text{tran}}}{W}\\int_{r_+}^{+\\infty}\\Delta^{s}R^{\\text{up}}_{\\omega lm}T_{\\omega lm}dr,\\\\\nZ^{H}_{\\omega lm}&=\\frac{D^{\\text{tran}}}{W}\\int_{r_+}^{+\\infty}\\Delta^{s}R^{\\text{in}}_{\\omega lm}T_{\\omega lm}dr.\n\\label{amplitudes}\n\\end{split}\n\\end{equation}\nFor a circular equatorial orbit with orbital angular frequency $\\hat{\\omega} $, we get\n\\begin{equation}\nZ^{H,\\infty}_{\\omega lm}=\\delta(\\omega-m \\hat{\\omega})\\mathcal{A}^{H,\\infty}_{\\omega lm}.\n\\end{equation}\nFor $s=-1$, the energy fluxes at infinity and the horizon read\n\\begin{equation}\n\\begin{split}\n\\dot{E}_q^{\\infty}=\\left(\\frac{d E}{dt}\\right)_{EM}^\\infty&=\\sum_{l=1}^{\\infty}\\sum_{m=1}^{l}\\frac{|\\mathcal{A}^{H}_{\\omega lm}|^2}{\\pi}, \\\\\n\\dot{E}_q^H=\\left(\\frac{d E}{dt}\\right)_{EM}^H&=\\sum_{l=1}^{\\infty}\\sum_{m=1}^{l}\\alpha^E_{lm}\\frac{|\\mathcal{A}^{\\infty}_{\\omega lm}|^2}{\\pi},\n\\end{split}\n\\end{equation}\nwhere the coefficient $\\alpha^E_{lm}$ is \\cite{Teukolsky:1974yv}\n\\begin{equation}\\label{energyformula}\n\\alpha^E_{lm}=\\frac{128\\omega\\kappa r_+^3(\\kappa^2+4\\epsilon^2)}{|B_E|^2}\n\\end{equation}\nwith $\\epsilon=\\sqrt{1-a^2}\/(4r_+)$ and\n\\begin{equation}\n|B_E|^2=\\lambda_{lm\\omega}^2+4ma\\omega-4a^2\\omega^2.\n\\end{equation}\n\nFor $s=-2$, the gravitational energy fluxes at infinity and the horizon are given by\n\\begin{equation}\n\\begin{split}\n\\dot{E}_{\\text{grav}}^{\\infty}=\\left(\\frac{d E}{dt}\\right)_{GW}^\\infty&=\\sum_{l=2}^{\\infty}\\sum_{m=1}^{l}\\frac{|\\mathcal{A}^{H}_{\\omega lm}|^2}{2\\pi\\omega^2}, \\\\\n\\dot{E}_{\\text{grav}}^H=\\left(\\frac{d E}{dt}\\right)_{GW}^H&=\\sum_{l=2}^{\\infty}\\sum_{m=1}^{l}\\alpha^G_{lm}\\frac{|\\mathcal{A}^{\\infty}_{\\omega lm}|^2}{2\\pi\\omega^2},\n\\end{split}\n\\end{equation}\nwhere the coefficient $\\alpha^G_{l m}$ is \\cite{Hughes:1999bq}\n\\begin{equation}\\label{energyformula2}\n\\alpha^G_{l m}=\\frac{256\\left(2 r_{+}\\right)^5 \\kappa\\left(\\kappa^2+4 \\epsilon^2\\right)\\left(\\kappa^2+16 \\epsilon^2\\right)\\omega^3}{\\left|B_G\\right|^2},\n\\end{equation}\nand\n\\begin{equation}\n\\begin{aligned}\n\\left|B_G\\right|^2 &=\\left[\\left(\\lambda_{l m \\omega}+2\\right)^2+4 a\\omega-4 a^2\\omega^2\\right]\\times\\left[\\lambda_{l m \\omega}^2+36 m a\\omega-36 a^2\\omega^2\\right] \\\\\n&+\\left(2 \\lambda_{l m \\omega}+3\\right)\\left[96 a^2\\omega^2-48 m a\\omega\\right]+144\\omega^2\\left(1-a^2\\right) .\n\\end{aligned}\n\\end{equation}\n\nTherefore, the total energy fluxes emitted from the EMRIs read\n\\begin{equation}\n\\dot{E}=    \\dot{E}_q+\\dot{E}_{\\text{grav}},\n\\end{equation}\nwhere\n\\begin{equation}\n    \\dot{E}_q=\\dot{E}_q^\\infty+\\dot{E}_q^H,\\ \\ \\dot{E}_{\\text{grav}}=\\dot{E}_{\\text{grav}}^\\infty+\\dot{E}_{\\text{grav}}^H.\n\\end{equation}\nThe detailed derivation of the above results is given in Appendix \\ref{gsne}.\nThe energy flux emitted by tensor fields can be computed with the BH Perturbation Toolkit \\cite{BHPToolkit}.\n\n\\section{Results}\\label{sec4}\nThe top  panel of Fig. \\ref{energyd} shows the normalized vector energy flux $m_p^{-2}M^2\\dot{E}_q$ for a charged particle with different charge values of $q$ on a circular orbit about a Kerr BH with the spin $a=0.9$, as a function of orbital radius.\nThe vector energy flux is proportional to the square of the vector charge $q^2$.\nThe vector energy flux increases as the charged particle inspirals into the central Kerr BH.\nThe ratio between the vector and gravitational energy flux is shown in the bottom panel of Fig. \\ref{energyd}.\nBoth the vector and gravitational fluxes are in the same order of $(m_p\/M)^2$,\nthe ratio of fluxes is independent of the mass ratio and increases as the orbital radius because the gravitational contribution falls off faster than the vector energy flux at a large orbital radius.\nFigure \\ref{energya} shows the normalized vector energy flux $m_p^{-2}M^2\\dot{E}_q$ and the ratio of energy fluxes $\\dot{E}_q\/\\dot{E}_{\\textrm{grav}}$ as a function of the orbital radius for different values of $a$.\nFor larger $a$, the ISCO is smaller, resulting in higher GW frequency at the coalescence.\nFor the same orbital radius, the vector energy flux is slightly larger for smaller $a$.\nHowever, the total energy flux increases with $a$ for one-year observations before the merger due to the smaller ISCO.\nFigure \\ref{energyHI} shows the ratio of energy flux falling onto the horizon to the energy flux radiated away to infinity, \nas a function of the orbital radius, for various spin  $a$, and for the vector and gravitational fields.\nIt is interesting to note that the sign of ratio becomes negative for Kerr BH with positive $a$ (co-rotating orbit) at  a small orbital radius.\nIn these cases, the vector and gravitational fields generate superradiance, leading to extraction of energy from the horizon.\nThe superradiance only happens when the coefficient $\\kappa$ in Eqs. \\eqref{energyformula} and \\eqref{energyformula2} becomes negative, which means that the orbital frequency slows down the rotation of the Kerr BH.\nOur results are consistent with those found in Ref. \\cite{Torres:2020fye}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\columnwidth]{energyd.pdf}\n    \\caption{The energy fluxes versus the orbital distance. \n    The top panel shows the vector flux normalized with the mass ratio from a charged particle orbiting around a Kerr BH with the spin of $a\/M=0.9$ for different values of the vector charge $q$.\n    The bottom panel shows the ratio between vector and gravitational energy fluxes for different values of the vector charge $q$.}\n    \\label{energyd}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\columnwidth]{energya.pdf}\n    \\caption{Same as Fig. \\ref{energyd}, but for different values of the primary spin $a$.\n    The vector charge $q$ is set to 1.}\n    \\label{energya}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\columnwidth]{energyHI.pdf}\n    \\caption{The ratio of the energy flux falling onto the horizon to the energy flux radiated to infinity, as a function of the orbital radius, for various spin $a$, and for the vector (the top panel) and gravitational cases (the bottom panel).}\n    \\label{energyHI}\n\\end{figure}\n\nThe extra energy leakage due to the vector field accelerates the coalescence of binaries.\nTherefore, we expect the vector charge to leave a significant imprint on the GW phase over one-year evolution before the merger for EMRIs.\nTo detect the vector charge carried by the small compact object in EMRIs, \nwe study the dephasing of GWs caused by the additional energy loss during inspirals.\nThe observation time is one year before the merger,\n\\begin{equation}\\label{sol-fre}\n\tT_{\\text{obs}}=\\int^{f_{\\text{max}}}_{f_{\\text{min}}}\\frac{1}{\\dot{f}}df=1\\ \\text{year},\n\\end{equation}\nwhere\n\\begin{equation}\n    f_{\\text{max}}=\\text{min}(f_{\\text{ISCO}},f_{\\text{up}}),~~~~~~f_{\\text{min}}=\\text{max}(f_{\\text{low}},f_{\\text{start}}),\n\\end{equation}\n$f=\\hat{\\omega}\/\\pi$ is the GW frequency, $f_{\\text{ISCO}}$ is the frequency at the ISCO \\cite{Jefremov:2015gza},\n$f_{\\text{start}}$ is the initial frequency at $t=0$,\nthe cutoff frequencies $f_{\\text{low}}=10^{-4}$ Hz and $f_{\\text{up}}=1$ Hz.\nThe orbit evolution is determined by\n \\begin{equation}\\label{orbittime}\n \\frac{d r}{dt}=-\\dot{E}\\left(\\frac{d \\hat{E}}{dr}\\right)^{-1},\\qquad \\frac{d \\varphi_{\\text{orb}}}{d t}=\\pi f,\n \\end{equation}\n where $\\dot{E}=\\dot{E}_q+\\dot{E}_{\\text{grav}}$.\nThe total number of GW cycles accumulated over one year before the merger is \\cite{Berti:2004bd}\n\\begin{equation}\\label{phase-end}\n\\mathcal{N}=\\int_{f_{\\text{min}}}^{f_{\\max }} \\frac{f}{\\dot{f}} d f.\n\\end{equation}\nConsidering EMRIs with the mass of the second compact object being fixed to be $m_p=10~M_{\\odot}$, \nwe calculate the dephasing $\\Delta\\mathcal{N}=\\mathcal{N}(q=0)-\\mathcal{N}(q)$ for different vector charge $q$, spin $a$ and mass $M$,\nand the results are shown in Fig. \\ref{phase}.\nFor one-year observations before the merger,\nthe charged particle starts further away from ISCO due to extra radiation of the vector field and the difference $\\Delta\\mathcal{N}$ is always positive.\nAs shown in Fig. \\ref{phase}, $\\Delta \\mathcal{N}$ increases monotonically with the spin $a$ and the charge-to-mass ratio $q$, and it strongly depends on the mass of the central BH such that lighter BHs have larger $\\Delta \\mathcal{N}$.\nThis means that the observations of EMRIs with a lighter and larger-spin Kerr BH can detect the vector charge easier.\nFor the same EMRI configuration in the Kerr background, the co-rotating orbit can detect the vector charge easier than the counter-rotating orbit.\nFollowing Refs. \\cite{Berti:2004bd,Maselli:2020zgv},\nwe take the threshold for a detectable dephasing that two signals are distinguishable by space-based GW detectors as $\\Delta \\mathcal{N}=1$.\nObservations of EMRIs over one year before the merger may be able to reveal the presence of a vector charge as small as $q\\sim 0.007$ for Kerr BHs with $a=0.9$ and $M=10^6~M_\\odot$, and $q\\sim 0.01$ for Schwarzschild BHs with $M=10^6~M_\\odot$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\columnwidth]{dephasing.pdf}\n    \\caption{The difference between the number of GW cycles accumulated by EMRIs with and without the vector charge in circular orbits. The left panel shows the dephasing as a function of the mass of the Kerr BH in the range $M\\in\\left[2\\times 10^5,2\\times 10^7\\right]M_\\odot$ for different spin values, the charge $q=0.01$.\n    The right panel shows the dephasing as a function of the charge-to-mass ratio $q$ for different $M$, the spin $a=0.9$. The red dashed line corresponds to the threshold above which two signals are distinguishable with space-based GW detectors. All observational time is one year before the merger.}\n    \\label{phase}\n\\end{figure}\n\n\\section{Parameter Estimation}\nTo make the analysis more accurate and account for the degeneracy among parameters, \nwe calculate the faithfulness between two GW waveforms and carry out parameter estimation with the FIM method.\n\\label{sec5}\n\\subsection{Signals}\nWe can obtain the inspiral trajectory from adiabatic evolution in Eq. \\eqref{orbittime}, then compute GWs in the quadrupole approximation.\nThe metric perturbation in the transverse-traceless (TT) gauge is\n\\begin{equation}\nh_{i j}^{\\mathrm{TT}}=\\frac{2}{d_L}\\left(P_{i l} P_{j m}-\\frac{1}{2} P_{i j} P_{l m}\\right) \\ddot{I}_{l m},\n\\end{equation}\nwhere $d_L$ is the luminosity distance of the source,\n$P_{ij}=\\delta_{ij}-n_i n_j$ is the projection operator acting onto GWs with the unit propagating direction $n_j$,\n$\\delta_{ij}$ is the Kronecker delta function,\nand $\\ddot{I}_{ij}$ is the second time derivative of the mass quadrupole moment.\nThe GW strain measured by the detector is\n\\begin{equation}\\label{signal}\nh(t)=h_{+}(t) F^{+}(t)+h_{\\times}(t) F^{\\times}(t),\n\\end{equation}\nwhere $h_+(t)=\\mathcal{A}\\cos\\left[2\\varphi_{\\rm orb}+2\\varphi_0\\right]\\left(1+\\cos^2\\iota\\right)$, $h_\\times(t)=-2\\mathcal{A}\\sin\\left[2\\varphi_{\\rm orb}+2\\varphi_0\\right]\\cos\\iota$,  $\\iota$ is the inclination angle between the binary orbital angular momentum and the line of sight,\nthe GW amplitude  $\\mathcal{A}=2m_{\\rm p}\\left[M\\hat{\\omega}(t)\\right]^{2\/3}\/d_L$ and $\\varphi_0$ is the initial phase.\nThe interferometer pattern functions $F^{+,\\times}(t)$ and $\\iota$ can be expressed in terms of four angles which specify the source orientation, $(\\theta_s,\\phi_s)$,\nand the orbital angular direction $(\\theta_1,\\phi_1)$.\nThe faithfulness between two signals is defined as\n\\begin{equation}\\label{eq:def_F}\n\\mathcal{F}_n[h_1,h_2]=\\max_{\\{t_c,\\phi_c\\}}\\frac{\\langle h_1\\vert\n\th_2\\rangle}{\\sqrt{\\langle h_1\\vert h_1\\rangle\\langle h_2\\vert h_2\\rangle}}\\ ,\n\\end{equation}\nwhere $(t_c,\\phi_c)$ are time and phase offsets \\cite{Lindblom:2008cm},\nthe noise-weighted inner product between two templates $h_1$ and $h_2$ is\n\\begin{equation}\\label{product}\n\\left\\langle h_{1} \\mid h_{2}\\right\\rangle=4 \\Re \\int_{f_{\\min }}^{f_{\\max }} \\frac{\\tilde{h}_{1}(f) \\tilde{h}_{2}^{*}(f)}{S_{n}(f)} df,\n\\end{equation}\n$\\tilde{h}_{1}(f)$ is the Fourier transform of the time-domain signal $h(t)$,\nits complex conjugate is $\\tilde{h}_{1}^{*}(f)$,\nand $S_n(f)$ is the noise spectral density  for space-based GW detectors.\nThe signal-to-noise ratio (SNR) can be obtained by calculating $\\rho=\\left\\langle h|h \\right\\rangle^{1\/2}$.\nThe sensitivity curves of LISA, TianQin, and Taiji are shown in Fig. \\ref{sensitivity}.\nAs pointed out in \\cite{Chatziioannou:2017tdw},\ntwo signals can be distinguished by LISA if $\\mathcal{F}_n\\leq0.988$.\nHere we choose the source masses $m_p=10~M_{\\odot}$, $M=10^6~M_{\\odot}$, \nthe source angles $\\theta_s=\\pi\/3,~\\phi_s=\\pi\/2$ and $\\theta_1=\\phi_1=\\pi\/4$, \nthe luminosity distance is scaled to ensure SNR $\\rho=30$, \nthe initial phase is set as $\\varphi_0=0$ and the initial orbital separation is adjusted to experience one-year adiabatic evolution before the plunge $r_{\\text{end}}=r_{\\text{ISCO}}+0.1~M$.\nIn Fig. \\ref{faithfulness}, we show\nthe faithfulness between GW signals with and without the vector charge for LISA as a function of the vector charge.\nThe results show that one-year observations of EMRIs with LISA may be able to reveal the presence of a vector charge as small as $q\\sim 0.002$ for Kerr BHs with $a=0.9$ and $M=10^6~M_\\odot$ (co-rotating orbit), \n$q\\sim 0.003$ for Schwarzschild BHs with $a=0$ and $M=10^6~M_\\odot$, and $q\\sim 0.004$ for Kerr BHs with $a=-0.9$ and $M=10^6~M_\\odot$ (counter-rotating orbit).\nLarger positive spin of the Kerr BH (co-rotating orbit) can help us detect the vector charge easier, which is consistent with the results obtained from the dephasing in the previous section.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\columnwidth]{sensitivity.pdf}\n\t\\caption{The sensitivity curves for LISA, TianQin, and Taiji.\nThe horizontal solid lines represent the frequency band $f_{\\textrm{start}}$ to $f_{\\textrm{ISCO}}$ for EMRIs with $a=0.9$, and $M=2\\times10^5~M_{\\odot}$, $M=10^6~M_{\\odot}$ and $M=10^7~M_{\\odot}$ over one-year evolution before the merger.\n}\n\\label{sensitivity}\n\\end{figure}\n\n\\begin{figure}[thbp]\n\\center{\n\\includegraphics[scale=0.74]{Faithfulness.pdf}\n\\caption{Faithfulness between GW signals with and without the vector charge for LISA as a function of the charge $q$. The spin of the Kerr BH is $a=0$, $a=0.9$, and $a=-0.9$. The horizontal dashed line represents the detection limit with LISA, $\\mathcal{F}_n= 0.988$.\n}\\label{faithfulness}}\n\\end{figure}\n\n\\subsection{Fisher information matrix}\nThe signals \\eqref{signal} measured by the detector are determined by the following eleven parameters\n\\begin{equation}\n\\xi=(\\ln M, \\ln m_p, a, q, r_0, \\varphi_0, \\theta_s, \\phi_s, \\theta_1, \\phi_1, d_L).\n\\end{equation}\n In the large SNR limit,\nthe posterior probability distribution of the source parameters $\\xi$ can be approximated by a multivariate Gaussian distribution centered around the true values $\\hat{\\xi}$.\nAssuming flat or Gaussian priors on the source parameters $\\xi$,\ntheir covariances are given by the inverse of the FIM\n\\begin{equation}\n\\Gamma_{i j}=\\left\\langle\\left.\\frac{\\partial h}{\\partial \\xi_{i}}\\right| \\frac{\\partial h}{\\partial \\xi_{j}}\\right\\rangle_{\\xi=\\hat{\\xi}}.\n\\end{equation}\nThe statistical error on $\\xi$ and the correlation coefficients between the parameters are provided by the diagonal and non-diagonal parts of ${\\bf \\Sigma}={\\bf \\Gamma}^{-1}$, i.e.\n\\begin{equation}\n\\sigma_{i}=\\Sigma_{i i}^{1 \/ 2} \\quad, \\quad c_{\\xi_{i} \\xi_{j}}=\\Sigma_{i j} \/\\left(\\sigma_{\\xi_{i}} \\sigma_{\\xi_{j}}\\right).\n\\end{equation}\nBecause of the triangle configuration of the space-based GW detector, the total SNR is defined by $\\rho=\\sqrt{\\rho_1^2+\\rho_2^2}$, so the total covariance matrix of the binary parameters is obtained by inverting the sum of the Fisher matrices $\\sigma_{\\xi_i}^2=(\\Gamma_1+\\Gamma_2)^{-1}_{ii}$.\nHere we fix the source angles $\\theta_s=\\pi\/3,~\\phi_s=\\pi\/2$ and $\\theta_1=\\phi_1=\\pi\/4$, \nthe initial phase is set as $\\varphi_0=0$ and the initial orbital separation is adjusted to experience one-year adiabatic evolution before the plunge $r_{\\text{end}}=r_{\\text{ISCO}}+0.1~M$.\nThe luminosity distance $d_L$ is set to be $1$ Gpc.\nWe apply the FIM method for LISA, TianQin, and Taiji to estimate the errors of the vector charge.\n\nThe relative errors of the vector charge $q$ as a function of the vector charge with LISA, TianQin and Taiji are shown in Fig. \\ref{sigmaq}.\nFor one-year observations before the merger,\nthe charged particle starts further away from ISCO due to extra radiation of the vector field,\nso the $1\\sigma$ error for the charge decreases with the charge $q$.\nFor EMRIs with $M=2\\times 10^5~M_{\\odot}$ and $a=0.9$, as shown in the top panel,\nthe relative errors of the charge $q$ with TianQin are better than LISA and Taiji.\nFor $M=10^6~M_{\\odot}$ and $a=0.9$ as shown in the middle panel, \nthe relative errors of the charge $q$ with TianQin and LISA are almost the same.\nFor $M=10^7~M_{\\odot}$ and $a=0.9$ as shown in the bottom panel, \nthe relative errors of the charge $q$ with TianQin are worse than LISA and Taiji.\nIn all the cases, the relative errors of the charge $q$ with Taiji are better than LISA, for the reason that the sensitivity of Taiji is always better than LISA.\nFor $M=2\\times 10^5~M_{\\odot}$ and $a=0.9$, \nthe relative errors with TianQin are better than LISA and Taiji since the sensitivity of TianQin is better than LISA and Taiji in the high-frequency band, but worse than LISA and Taiji in the low-frequency band as shown in Fig. \\ref{sensitivity}.\nFor EMRIs with $m_p=10~M_{\\odot}$, $M=10^6~M_{\\odot}$ and $a=0.9$, \nthe vector charge can be constrained for LISA as small as $q\\sim0.021$, \nfor TianQin as small as $q\\sim0.028$ and for Taiji as small as $q\\sim0.016$.\nFor EMRIs with $m_p=10~M_{\\odot}$, $M=2\\times10^5~M_{\\odot}$ and $a=0.9$, \nthe vector charge can be constrained for LISA as small as $q\\sim0.0082$, \nfor TianQin as small as $q\\sim0.0049$ and for Taiji as small as $q\\sim0.0057$.\n\nFigure \\ref{sigmaa} shows the relative errors of the vector charge $q$ versus the spin $a$ with LISA, TianQin, and Taiji.\nIn general, the relative errors of the charge for the co-rotating orbit are better than those for the counter-rotating orbit.\nWe only consider the co-rotating orbit for simplicity.\nThe $1\\sigma$ error for the charge decreases with the spin $a$.\nComparing the relative errors for Kerr BHs with spin $a=0.9$ and spin $a=0$, \nwe find that the spin of Kerr BHs can decrease the charge uncertainty by about one or two orders of magnitude, depending on the mass of the Kerr BH.\nFor EMRIs with $M=10^6~M_\\odot$, $m_p=10~M_\\odot$, $q=0.05$, and different $a$,\nthe corner plots for source parameters with LISA are shown in Figs. \\ref{corner09}, \\ref{corner0} and \\ref{cornerm09}.\nFor comparison, we also show the corner plot for charged EMRIs in the Schwarzschild BH background in Fig. \\ref{corner0s}.\nFor $a=(0.9,0,-0.9)$, the corresponding errors of charge $\\sigma_q$ are $(0.0031,0.086,0.65)$, respectively.\nAs expected, $\\sigma_q$ is smaller for co-rotating orbits and bigger for counter-rotating orbits.\nFor EMRIs in the Kerr background, the co-rotating orbit can better detect the vector charge.\nIt is interesting to note that the charge $q$ are anti-correlated with the mass $M$  and the spin $a$ of Kerr BHs,\nand the correlations between $q$ and $M$, and $q$ and $m_p$ in the Kerr BH background\nare opposite to those in the Schwarzschild BH background.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{M2e5q.pdf}\\\\\n\\includegraphics[width=0.6\\columnwidth]{M1e6q.pdf}\\\\\n\\includegraphics[width=0.6\\columnwidth]{M1e7q.pdf}\\\\\n\\caption{\nThe $1\\sigma$ interval for the charge $q$ as a function of the charge $q$, inferred after one-year observations of EMRI with $a=0.9$ and different $M$ with LISA, Taiji, and TianQin. The horizontal dashed lines represent the $3\\sigma$ limit $33.3\\%$.\n}\n\\label{sigmaq}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{M2e5a.pdf}\\\\\n\\includegraphics[width=0.6\\columnwidth]{M1e6a.pdf}\\\\\n\\includegraphics[width=0.58\\columnwidth]{M1e7a.pdf}\\\\\n\\caption{\nThe $1\\sigma$ interval for the charge $q$ as a function of the spin $a$, inferred after one-year observations of EMRI with $q=0.05$ and different $M$ with LISA, Taiji, and TianQin.\n}\n\\label{sigmaa}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{LISA1e6A0.9.pdf}\n\\caption{Corner plot for the probability distribution of the source parameters $(\\ln M,\\ln m_p, a, q)$ with LISA, inferred after one-year observations of EMRIs with $q=0.05$ and $a=0.9$.\nVertical lines show the $1\\sigma$ interval for the source parameter.\nThe contours correspond to the $68\\%$, $95\\%$, and $99\\%$ probability confidence intervals.}\n\\label{corner09}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{LISA1e6A0.0.pdf}\n\\caption{Corner plot for the probability distribution of the source parameters $(\\ln M,\\ln m_p, a, q)$ with LISA, inferred after one-year observations of EMRIs with $q=0.05$ and $a=0$.\nVertical lines show the $1\\sigma$ interval for the source parameter.\nThe contours correspond to the $68\\%$, $95\\%$, and $99\\%$ probability confidence intervals.}\n\\label{corner0}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{LISA1e6A-0.9.pdf}\n\\caption{Corner plot for the probability distribution of the source parameters $(\\ln M,\\ln m_p, a, q)$ with LISA, inferred after one-year observations of EMRIs with $q=0.05$ and $a=-0.9$.\nVertical lines show the $1\\sigma$ interval for the source parameter.\nThe contours correspond to the $68\\%$, $95\\%$, and $99\\%$ probability confidence intervals.}\n\\label{cornerm09}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{LISA1e6A0.0s.pdf}\n\\caption{Corner plot for the probability distribution of the source parameters $(\\ln M,\\ln m_p, q)$ with LISA, inferred after one-year observations of EMRIs with $q=0.05$ in the Schwarzschild BH background.\nVertical lines show the $1\\sigma$ interval for the source parameter.\nThe contours correspond to the $68\\%$, $95\\%$, and $99\\%$ probability confidence intervals.}\n\\label{corner0s}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec6}\nWe study the energy emissions and GWs from EMRIs consisting of a small charged compact object with mass $m_p$ and the charge to mass ratio $q$ inspiraling into a Kerr BH with spin $a$.\nWe derive the formula for solving the inhomogeneous Teukolsky equation with a vector field and calculate the power emission due to the vector field in the Kerr background.\nBy using the difference between the number of GW cycles $\\Delta\\mathcal{N}$ accumulated by EMRIs with and without the vector charge in circular orbits over one year before the merger, \nwe may reveal the presence of a vector charge as small as $q\\sim 0.007$ for Kerr BHs with $a=0.9$ and $M=10^6~M_\\odot$, and $q\\sim 0.01$ for Schwarzschild BHs with $M=10^6~M_\\odot$.\nThe dephasing increases monotonically with the charge-to-mass ratio $q$, and it strongly depends on the mass of the Kerr BH such that lighter BHs have larger dephasing.\n\nWe also apply the faithfulness between GW signals with and without the vector charge \nto discuss the detection of the vector charge $q$.\nWe find that positive larger spin of the Kerr BH can help us detect the vector charge easier.\nWe show that one-year observations of EMRIs with LISA may be able to reveal the presence of a vector charge as small as $q\\sim 0.002$ for Kerr BHs with $a=0.9$ and $M=10^6~M_\\odot$, \n$q\\sim 0.003$ for Schwarzschild BHs with $a=0$ and $M=10^6~M_\\odot$, and $q\\sim 0.004$ for Kerr BHs with $a=-0.9$ and $M=10^6~M_\\odot$.\n\nTo determine vector charge more accurately and account for the degeneracy among parameters, \nwe calculate the FIM to estimate the errors of the vector charge $q$.\nFor EMRIs with $M=2\\times10^5~M_{\\odot}$ and $a=0.9$, \nthe vector charge $q$ can be constrained as small as $q\\sim0.0049$ with TianQin, $q\\sim0.0057$ with Taiji, and $q\\sim 0.0082$ with LISA.\nFor EMRIs with $M=10^6~M_{\\odot}$ and $a=0.9$, \nthe vector charge can be constrained as small as $q\\sim 0.016$ with Taiji, $q\\sim 0.021$ with LISA, and $q\\sim 0.028$ with TianQin.\nSince the sensitivity of TianQin is better than LISA and Taiji in the high-frequency bands, \nthe ability to detect the vector charge for TianQin is better than LISA and Taiji when the mass $M$ of the Kerr BH with $a=0.9$ is lighter than $\\sim 2\\times 10^5~M_{\\odot}$.\nFor the mass $M$ of the Kerr BH with $a=0.9$ above $10^6~M_{\\odot}$, LISA and Taiji are more likely to detect smaller vector charges.\nThe relative errors of the charge $q$ with Taiji are always smaller than LISA  because the sensitivity of Taiji is always better than LISA.\nDue to the extra radiation of the vector field, \nthe charged particle starts further away from ISCO,\nso the $1\\sigma$ error for the charge decreases with the charge $q$.\nAs the spin $a$ of Kerr BHs increases, the ISCO becomes smaller, \nthe positive spin of Kerr BHs (co-rotating) can decrease the charge uncertainty by about one or two orders of magnitude, depending on the mass of the Kerr BH.\nFor EMRIs with $M=10^6~M_\\odot$, $m_p=10~M_\\odot$, $q=0.05$, and different $a=(0.9,0,-0.9)$, the corresponding errors of charge $\\sigma_q$ with LISA are $(0.0031,0.086,0.65)$, respectively,\nso co-rotating orbits can better detect the vector charge.\nIt is interesting to note that the charge $q$ are anti-correlated with the mass $M$  and the spin $a$ of Kerr BHs,\nand the correlations between $q$ and $M$, and $q$ and $m_p$ in the Kerr BH background\nare opposite to those in the Schwarzschild BH background.\nIn summary, the observations of EMRIs with a lighter and larger-spin Kerr BH can detect the vector charge easier.\n\n \n\n\\begin{acknowledgments}\nThis work makes use of the Black Hole Perturbation Toolkit package.\nThis research is supported in part by the National Key Research and Development Program of China under Grant No. 2020YFC2201504\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\tThe certification of quantum correlations is essential for the ever-accelerating development of quantum technologies.\n\tBeyond this practical demand, a fundamental understanding of quantum correlations, including their characterization and quantification, plays a key role when exploring the boundary between classical theories and the unique features of quantum physics.\n\tStill, the question remains which kinds of correlation are genuinely quantum.\n\tThat is, which of the many contenders---be it an established or recently proposed concept (e.g., quantum coherence and resource theory \\cite{SAP17,CG19}, entanglement \\cite{HHHH09}, discord \\cite{MBCPV12}, etc.)---does describe the concept of a nonclassical correlation best?\n\tIn this work, we show that the notion of nonclassicality in quantum optics \\cite{TG65,M86} can supersede contemporary forms of quantumness in its ability to unveil quantum correlations.\n\n\tIn the context of quantum optics, any observations which cannot be fully described in terms of Maxwell's wave theory of light are called nonclassical \\cite{TG65,M86}.\n\tThis well-established concept of nonclassicality is based on the impossibility of describing field correlations as done in classical electrodynamics, and it is commonly defined in terms of the Glauber-Sudarshan $P$ representation \\cite{G63,S63}.\n\tThe latter describes nonclassical light through phase-space distributions that are, in the case of quantum light, incompatible with a classical concept of a non-negative probability distribution.\n\n\tThe more recently developed concept of quantum coherence adapts some ideas of the notion of nonclassicality to quantify resources required for quantum information processing  \\cite{SAP17,CG19}.\n\tIn this framework, quantum superpositions equally serve as the origin of quantumness in a system.\n\tHowever, in most cases, the classical reference is defined through incoherent mixtures of orthonormal basis states, contrasting the notion of quantum-optical nonclassicality in terms of nonorthogonal eigenstates of the non-Hermitian annihilation operator.\n\n\tEntanglement, which can be embedded into the concept of coherence \\cite{SSDBA15,KSP16}, is by far the most frequently studied form of quantum correlation among the many contenders \\cite{HHHH09}.\n\tThis is due to its fundamental role as well as its many applications, e.g., in quantum metrology, cryptography, computing, and teleportation.\n\tThe phenomenon of entanglement was discovered in early seminal discussions about the implications of quantum physics \\cite{S35,EPR35}, long before the conception of the relatively young field of quantum information processing.\n\n\tMany other notions and measures of quantum correlations have been proposed too \\cite{MBCPV12}.\n\tFor instance, discord is a feature which includes correlations caused by entangled but also by nonentangled states \\cite{HV01,OZ01}, and it can be connected to quantum coherence as well \\cite{MYGVG16}.\n\tIn this context, it is worth mentioning that the label quantum for this sort of correlation is a topic of ongoing debates \\cite{GOS15}.\n\tNonetheless, it has been demonstrated that discord is maximally inequivalent to the notion of quantum-optical nonclassicality \\cite{FP12}.\n\tTo date, it remains an open problem to decide---not only in theory, but also experimentally---which of the candidates is best suited for characterizing quantum correlations.\n\n\tIn this Letter, we address this issue experimentally by realizing and analyzing a fully phase-randomized two-mode squeezed vacuum (TMSV) state \\cite{ASV13}.\n\tThis state of quantum light has the following properties:\n        It is nonentangled;\n        it has zero discord;\n        it does not exhibit quantum coherence in the photon-number basis;\n        its reduced single-mode states are classical;\n        and it has a non-negative, two-mode Wigner function.\n\tDespite these strong signatures of classicality, we demonstrate the presence of quantum correlations as defined through the notion of nonclassicality in quantum optics with a statistical significance next to certainty.\n\tBecause of the phase independence of the generated state, this nonclassical feature is robust against dephasing.\n\tFurthermore, the activation of entanglement using this kind of state is developed to show its resourcefulness for quantum information processing applications.\n\n\n\\section{Quantum correlations}\n\tFor analyzing quantum correlations, we consider a class of two-mode states that are phase insensitive \\cite{FP12,ASV13}.\n\tStill, intensity-intensity (likewise, photon-photon) correlations are present in such states.\n\tFor instance, this can be achieved by a full phase randomization of a TMSV state, resulting in\n\t\\begin{align}\n        \\label{eq:theo}\n\t\t\\hat\\rho=\\sum_{n\\in\\mathbb N} (1-p)p^n |n\\rangle\\langle n|\\otimes|n\\rangle\\langle n|,\n\t\\end{align}\n\twhere $p=\\tanh|\\xi|$ is a value between zero and one and $\\xi$ is the complex squeezing parameter.\n\tSuch states are, for example, a relevant resource for boson sampling tasks \\cite{SLR17}.\n\n\tIn terms of quantum correlations, it can be directly observed that the state in Eq. \\eqref{eq:theo} is an incoherent mixture of photon-number states, thus exhibiting no quantum coherence in the form of quantum superpositions of photon-number states;\n\tit is a classical mixture of tensor-product states, thus exhibiting no entanglement;\n\tand it has zero discord because $\\hat\\rho=\\sum_{n\\in\\mathbb N}\\hat\\rho_{A|n}\\otimes |n\\rangle\\langle n|$ holds true, where the photon-number states form the eigenbasis to $\\hat\\rho_{A|n}=(\\hat 1\\otimes \\langle n|)\\hat\\rho(\\hat 1\\otimes|n\\rangle)$ and $\\mathrm{tr}_A\\hat\\rho$ \\cite{D11}.\n\tFor those criteria of quantum correlations, it suffices in our scenario to consider the contribution of off-diagonal elements,\n\t\\begin{align}\n\t\t\\label{eq:CohMeasure}\n\t\t\\mathcal C(\\hat\\rho)\\stackrel{\\text{def.}}{=}\\sum_{\\substack{m,n,k,l\\in\\mathbb N: \\\\ m\\neq n, k\\neq l}}\n\t\t\\big|(\\langle m| \\otimes \\langle k|)\\hat\\rho(|n\\rangle\\otimes |l\\rangle)\\big|,\n\t\\end{align}\n\twhich quantifies the coherent contributions \\cite{BCP14} and is $\\mathcal C(\\hat\\rho)=0$ for the state in Eq. \\eqref{eq:theo}.\n\tIt is worth emphasizing that the nullity of coherence in the two-mode photon-number basis implies the nullity of discord which further implies no entanglement.\n\tIn addition, the incoherent mixture of photon-number states under study further implies a classical interpretation in a particle picture \\cite{SDNTBBS19}.\n\tFurthermore, the marginal states $\\mathrm{tr}_A\\hat\\rho$ and $\\mathrm{tr}_B\\hat\\rho$ are thermal states and, thus, classical, too.\n\tAlso, the two-mode state under study is a mixture of Gaussian TMSV states, implying a non-negative Wigner function.\n\n\tAt this point, one sees no indication of quantum correlations.\n\tYet, we have not considered the notion of nonclassicality in quantum optics so far.\n\tThis concept is defined through the Glauber-Sudarshan $P$ representation \\cite{G63,S63}\n\t\\begin{align}\n\t    \\label{eq:GS}\n\t\t\\hat\\rho=\\int d^2\\alpha \\int d^2\\beta\\, P(\\alpha,\\beta) |\\alpha\\rangle\\langle\\alpha|\\otimes |\\beta\\rangle\\langle\\beta |,\n\t\\end{align}\n\twhere $|\\alpha\\rangle$ and $|\\beta\\rangle$ denote classically coherent states of the harmonic oscillator \\cite{S26}.\n\tWhenever $P$ cannot be interpreted as a classical probability density, the state of light $\\hat\\rho$ refers to a nonclassical one \\cite{TG65,M86}.\n\tSince the $P$ distribution is in many cases highly singular \\cite{S16}, thus experimentally inaccessible, regularization and direct sampling procedures have been proposed and implemented to reconstruct a function $P_\\Omega$ which is always regular and non-negative for any classical states of light \\cite{KV10,KVHS11}.\n\tThis is achieved by a convolution of the Glauber-Sudarshan $P$ function with a suitable, non-Gaussian kernel $\\Omega$, resulting in $P_\\Omega$.\n\tOur previous theoretical studies suggest that the state in Eq. \\eqref{eq:theo} indeed demonstrates nonclassical correlations \\cite{ASV13}, i.e.,\n\t\\begin{align}\n\t    \\label{eq:Ncl}\n\t\tP_\\Omega(\\alpha,\\beta)\\stackrel{\\text{ncl.}}{<}0\n\t\\end{align}\n\tfor some complex phase-space amplitudes $\\alpha$ and $\\beta$.\n\n\\section{Experimental implementation}\n\tFigure \\ref{fig:setup} shows our experimental setup for the preparation and detection of the quantum state in Eq. \\eqref{eq:theo}.\n\tThe challenge here is that a full two-mode state tomography with long-term stability, phase control, and phase readout is paramount for our coherence analysis.\n\tSee Supplemental Material \\cite{SuppMat} for technical details.\n\t\n\\begin{figure}[b]\n\t\\includegraphics*[width=8cm]{setup.eps}\n\t\\caption{\n\t\tSetup outline.\n\t\tTwo single-mode squeezed states are prepared by two OPAs.\n\t\tCombining these beams on a $50{:}50$ beam splitter with a $\\pi\/2$ phase shift yields a TMSV state.\n\t\tIntroducing a phase randomization (indicated by the phase fluctuation $\\delta\\varphi$) in one of the output arms approximates the desired target state in Eq. \\eqref{eq:theo} \\cite{ASV13}.\n\t\tBoth final beams are probed by balanced homodyne detectors with controllable phases $\\varphi_{\\mathrm{LO},A}$ and $\\varphi_{\\mathrm{LO},B}$.\n\t}\\label{fig:setup}\n\\end{figure}\n\n\tTwo amplitude-squeezed fields at $1064\\,\\mathrm{nm}$ are produced by optical parametric amplifiers (OPAs).\n\tOne OPA consists of a type-I hemilithic, standing wave, nonlinear cavity with a $7\\%$ MgO:LiNbO\\textsubscript{3} crystal;\n\tthe other OPA uses a periodically poled potassium titanyl phosphate crystal instead.\n\tTo combine distinct sources of squeezed light in one setup, the seed and pump powers are adjusted such that the squeezed output fields of the two crystals are of equal intensity and squeezing.\n\tFor this purpose, the two pump powers of the second harmonic are chosen as $242$ and $50~\\mathrm{mW}$, respectively.\n\tMoreover, we have to minimize power fluctuations of the pump light since the optical parametric gain of each OPA responds differently \\cite{SuppMat}.\n\n\n\tThe two squeezed fields are superposed with a visibility of $96.5\\%$ and a relative phase of $\\pi\/2$ on a $50{:}50$ beam splitter.\n\tThe high visibility demonstrates a successful combination of the two sources that results in a TMSV state:\n\t\\begin{align}\n\t    |\\mathrm{TMSV}\\rangle=\\frac{1}{\\cosh|\\xi|}\\sum_{n=0}^\\infty \\left(\n\t        e^{i\\arg\\xi}\\tanh|\\xi| \\right)^n |n\\rangle\\otimes|n\\rangle.\n\t\\end{align}\n\n\tBoth output modes $A$ and $B$ of the state are probed by balanced homodyne detectors.\n\tWe observed $(96.7\\pm0.7)\\%$ visibility between the fields and their corresponding local oscillators.\n    For one OPA, we measured a single-mode squeezing variance of $-1.3~\\mathrm{dB}$ and $+3.7~\\mathrm{dB}$ antisqueezing.\n\tThis yields an initial squeezing of $-7.3~\\mathrm{dB}$ and an overall efficiency of $(63\\pm2)\\%$.\n\tThe latter figure was multiplied by two to compensate for the vacuum input, because blocking the second OPA effectively introduces additional $50\\%$ loss at the first (i.e., leftmost) beam splitter in Fig. \\ref{fig:setup}.\n\n    We use piezoelectric transducers to control the optical phases and realize a random phase shift $\\delta\\varphi$ in one of the arms.\n    To achieve a uniform dephasing over the full $ 2\\pi$ interval, white noise is applied with sufficiently high amplitude.\n    Because of the bandwidth limitations of the transducers, a uniformly distributed phase---as required to exactly obtain the state in Eq. \\eqref{eq:theo}--- can be approximated only via long measurement times, requiring to maintain a high stability of our setup.\n    Further technical details are provided in the Supplemental Material \\cite{SuppMat}.\n\n\\section{Results}\n\tFigure \\ref{fig:dment} depicts the first $625$ density-matrix elements of the reconstructed state, without (top panel) and with (bottom panel) phase randomization, which requires the full two-mode state tomography.\n\n\tThe initially generated state shows strong contributions of off-diagonal elements, relating to the dominance of quantum coherence [Eq. \\eqref{eq:CohMeasure}], $\\mathcal C(|\\mathrm{TMSV}\\rangle\\langle \\mathrm{TMSV}|)=1.789\\pm0.021$.\n\tThe implemented phase randomization then leads to a 45-fold suppression of the initial coherence, resulting in almost no subsisting coherence, $\\mathcal C(\\hat\\rho)=0.041\\pm0.005$.\n\t(Errors have been obtained through a Monte Carlo approach; see Supplemental Material \\cite{SuppMat} for details on data processing and a discussion of the residual amount of quantum coherence, caused by experimental imperfections.)\n\tThis loss of coherence implies that entanglement and discord do not contribute to the phase-randomized state that is almost completely characterized by its diagonal elements in the photon-number basis; see Eq. \\eqref{eq:theo} and the bottom plot in Fig. \\ref{fig:dment}.\n\n\\begin{figure}[t]\n\t\\includegraphics*[width=8cm]{dm-abs-rand0-noise27-63-nexp4-nc.eps}\n\t\\includegraphics*[width=8cm]{dm-abs-rand1-noise27-63-nexp4-nc.eps}\n\t\\caption{\n\t\tReconstructed density matrix elements of the TMSV state in photon-number bases, before (top) and after (bottom) phase randomization.\n\t\tEach entry provides the absolute values of the density matrix elements $|\\rho_{(k,m),(l,n)}|$, where $\\hat\\rho=\\sum_{k,l,m,n} \\rho_{(k,m),(l,n)} |k\\rangle\\langle l|\\otimes |m\\rangle\\langle n|$.\n\t\tPlease note the logarithmic scale.\n\t\tIn both plots, $k$ (bottom axis) and $l$ (right axis) denote photon numbers for $A$, and $m$ (left axis) and $n$ (top axis) indicate photon numbers for $B$.\n\t\tLarge off-diagonal contributions certify the presence of quantum coherence (top).\n\t\tStrongly diminished off-diagonal elements indicate the absence of quantum coherence (bottom).\n\t}\\label{fig:dment}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics*[width=82mm]{regPw13bars.eps}\n\t\\caption{\n\t\tRegularized $P$ function sampled from experimental data for the phase-randomized TMSV state, including negativities.\n\t\n\t}\\label{fig:regP}\n\\end{figure}\n\n\tWe showed that quantum correlations in terms of quantum coherence are negligible for the produced state.\n\tThat is, the phase-averaged state is mostly consistent with a classical statistical mixture of orthonormal two-mode, tensor-product, photon-number states, thus also ruling out entanglement and discord as a source of quantum correlations as discussed earlier.\n    However, from our balanced homodyne detection data, we can further directly sample the regularized phase-space function $P_\\Omega$ \\cite{KVHS11}.\n    Here, the reconstruction is based on phase-insensitive pattern functions \\cite{SuppMat}.\n    Thus, no amount of residual coherence can contribute to nonclassicality in the form of Eq. \\eqref{eq:Ncl}.\n\n\tThe resulting distribution is shown in Fig. \\ref{fig:regP}.\n    We found a maximal statistical significance of more than 150 standard deviations for the negativity of the reconstructed quasiprobability distribution, $P_\\Omega(\\alpha=0,\\beta=1.5)=(-1.570\\pm0.010)\\times 10^{-3}$.\n\tBecause of these highly significant negativities, quantum correlations in the generated state are confirmed beyond the previously considered notions.\n\n\tAgain, we emphasize that quantum coherence, entanglement, and discord cannot contribute to the negativity as our approach is, by construction, insensitive to such phase-sensitive phenomena.\n\tFurthermore, our method is, to our knowledge, the only existing approach to uncover this form of quantumness.\n\tAs mentioned before, the Wigner function, for instance, is completely positive for our state.\n\tThus, we experimentally generated quantum correlations which are inaccessible via other means.\n\tMoreover, phase stability is not required for the kind of quantum effect.\n\tIn fact, we artificially introduced phase noise---which is often omnipresent in realistic quantum channels---to produce the sought-after state.\n\n\\section{Entanglement activation}\n\tBecause of the ever-growing importance for quantum information processing \\cite{HHHH09,NC00}, the question arises if entanglement can be activated, like for coherence and discord \\cite{SSDBA15,KSP16,MYGVG16}, from the nonclassically correlated state under study.\n\tAchieving such an activation renders this state a useful resource for quantum technologies.\n\n\tFor this purpose, let us recall that single-mode nonclassicality can be converted into entanglement via simple beam splitters \\cite{X02,KSBK02,VS14}.\n\tSimilarly, we consider combining each of our two modes separately on a 50:50 beam splitter with vacuum, where annihilation operators for the nonvacuum input map as $\\hat a\\mapsto(\\hat a+\\hat a')\/\\sqrt 2$ and $\\hat b\\mapsto(\\hat b+\\hat b')\/\\sqrt 2$;\n\tadditional modes obtained from the splitting are indicated by prime superscripts.\n\tSuch operations are free (i.e., classical) ones with respect to the reference $|\\alpha\\rangle\\otimes|\\alpha'\\rangle$ of the Glauber-Sudarshan representation [Eq. \\eqref{eq:GS}] since the beam-splitter output for mode $A$ remains in this family of states, $|(\\alpha+\\alpha')\/\\sqrt{2}\\rangle\\otimes|(\\alpha-\\alpha')\/\\sqrt 2\\rangle$, and likewise for $B$.\n\tFurthermore, applied to a photon-number input state $|n\\rangle\\otimes |0\\rangle$, the map yields the output state $|\\Psi_n\\rangle=2^{-n\/2}\\sum_{j=0}^n \\binom{n}{j}^{1\/2} (-1)^{n-j} |j\\rangle\\otimes|n-j\\rangle'$.\n\tTherefore, the state in Eq. \\eqref{eq:theo} results in the final four-mode state\n\t\\begin{align}\n\t\t\\label{eq:4mode}\n\t\t\\hat\\rho_{AA'BB'}=\\sum_{n\\in\\mathbb N} (1-p)p^n |\\Psi_n\\rangle\\langle\\Psi_n|\\otimes|\\Psi_n\\rangle\\langle\\Psi_n|.\n\t\\end{align}\n\n\tClearly, this state is still nonentangled when separating the joint subsystems $AA'$ and $BB'$ from each other.\n\tHowever, multimode entanglement is much richer since entanglement in various mode decompositions can be considered;\n\tsee, e.g., Refs. \\cite{GSVCRTF15,GSVCRTF16}.\n\tHere, we address the question whether there is entanglement between the primed and unprimed modes, i.e., with respect to the separation of $AB$ and $A'B'$.\n\n\tTo answer this question, we consider the partial transposition criterion \\cite{P96,HHH96}.\n\tIn one form \\cite{HHH96}, this criterion states that a state is entangled if the expectation value of a so-called entanglement witness $\\hat W=(|\\Phi\\rangle\\langle\\Phi|)^\\mathrm{PT'}$ is negative, where $\\mathrm{PT'}$ denotes the partial transposition of the primed modes.\n\tFor example, we can choose $|\\Phi\\rangle=|0\\rangle\\otimes|0\\rangle'\\otimes|1\\rangle\\otimes|1\\rangle'-|1\\rangle\\otimes|1\\rangle'\\otimes|0\\rangle\\otimes|0\\rangle'$, which results in\n\t\\begin{align}\n\t\t\\mathrm{tr}(\\hat W\\hat\\rho_{AA'BB'})=-\\frac{(1-p)p}{2}<0,\n\t\\end{align}\n\tfor all nontrivial parameters $0<p<1$ that define our states in Eq. \\eqref{eq:theo}.\n\tTherefore, the two-mode nonclassical correlations of the generated state can be successfully activated to produce four-mode entanglement, which is a useful mixed-state quantum resource, such as for teleportation \\cite{P94,G96,HH96,HHH01}, and which can be further enhanced via entanglement distillation \\cite{BBPSSW96,BBPS96,PSBZ01,KBSG01,HSDFFS08}.\n\n\\section{Conclusion}\n\tWe experimentally realized a quantum state with quantum correlations that are inaccessible by means of two-mode coherence, thus entanglement and discord, but can be intuitively visualized by negative quasiprobabilities.\n\tQuantum coherence---a recently explored resource for quantum information processing---exists in terms of superpositions of orthogonal photon-number states of the initial TMSV state.\n\tBut a phase averaging destroys this and related kinds of quantum correlations.\n\tThus, contemporary quantum-information-based concepts of correlation fail to certify the quantumness of the state under the challenging, but common scenario of dephasing.\n\tHowever, the Glauber-Sudarshan-based concept of nonclassiality of light---frequently considered to be a dated notion, or not considered at all--- is still capable of uncovering the quantum nature of the generated state.\n\n\tSince the Glauber-Sudarshan distribution often displays a highly singular behavior, we employ a technique which enables us to directly sample a regularized version of such a two-mode phase-space function, allowing us to certify nonclassical negativities with a statistical significance of more than 150 standard deviations.\n\tThus, the generated phase-independent state exhibits nonclassical correlations beyond entanglement and discord.\n\tFurthermore, we theoretically devised a method to activate multimode entanglement, utilizing only the produced state and simple beam splitter operations.\n\n\n\tIn conclusion, we realized a type of quantum correlation that can be accessed via phase-space approaches but not through more recent notions of quantumness.\n\tThis kind of correlation is intrinsically robust against dephasing and can be easily converted into multimode entanglement.\n\tTherefore, this finding offers useful quantum correlations that can be realized without experimentally costly phase stabilization and is resourceful for modern quantum information protocols.\n\n\\section*{Acknowledgements}\n\tE. A. acknowledges funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk\\l{}odowska-Curie IF InDiQE (EU Project No. 845486).\n\tM. S. acknowledges financial support by the Deutsche Forschungsgemeinschaft through STA-543\/9-1.\n\tJ. S. thanks Torsten Meier and Tim Bartley for discussions.\n    This work was supported by the Deutsche Forschungsgemeinschaft through SFB 652, Projects No. B12 and No. B13.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\textit{Mesoscopic} electronic systems, those with physical dimensions comparable to the Fermi wavelength $\\lambda_F$, can display a variety of striking effects such as size quantisation, ballistic electron transport and quantum interference. Quantum interference effects such as Aharonov-Bohm oscillations, weak localisation and universal conductance fluctuations (UCFs) are all direct manifestations of phase coherent electron transport. Hence these are dependent on an additional lengthscale, namely the phase coherence length $l_\\phi$ over which the electron loses memory of its phase through inelastic processes. UCFs are aperiodic but reproducible features in the electrical conductivity $\\sigma$ as a function of an external parameter such as the magnetic field $B$ and are a consequence of the non-self-averaging nature of electron interference over lengthscales comparable to $l_\\phi$. The resulting fluctuations are not limited to $\\sigma$ but will tell on other transport parameters such as the thermopower as well. The thermopower or Seebeck coefficient $S$ of a system is defined as $V_{\\rm{th}}\/\\Delta T$ where $V_{\\rm{th}}$ is the voltage that develops between its end due to an imposed temperature difference $\\Delta T$. $S$ and $\\sigma$ are related by the Mott formula \\cite{1}: \n\n\\begin{equation}\n\\label{MottFormula}\nS_{\\rm{MOTT}} = \\frac{\\pi^2 k_B^2 T}{3e}\\frac{\\mbox{d}\\ln\\sigma}{\\mbox{d}E}\\vline_{E = \\mu}\n\\end{equation}\n\nHere $k_B$ is the Boltzmann constant, $T$ is the temperature, $e$ the electronic charge, $E$ the energy of the system and $\\mu$ the chemical potential. Indeed, 'universal thermopower fluctuations' have been experimentally demonstrated in quantum wires \\cite{2}, quantum dots \\cite{3} and, more recently, in other mesoscopic structures \\cite{4}. While mesoscopic fluctuations have been reported most often as a function of $B$, they are expected \\cite{4, 5, 6} and seen \\cite{7, 8, 9} even as a function of $\\mu$. The relative magnitude of the $S$-fluctuations can be much larger than those in $\\sigma$ by virtue of the derivate relation in equation \\ref{MottFormula}. Lesovik and Khmelnitskii (henceforth LK) show \\cite{10} that precisely for this reason, the mesoscopic contribution to the thermopower can be exceedingly large and either positive or negative. Consequently, they can potentially swamp the regular, 'bulk' contribution even causing sign reversals in $S$. The important difference between such fluctuations and thermopower sign-reversals resulting from Coulomb Blockade \\cite{11, 12, 13} is that the former occur in the high-conductivity regime $\\sigma >> h\/e^2$, while the latter become significant at $\\sigma \\sim h\/e^2$.\n\nRecently some of us reported $S$ measurements in low-density two-dimensional electron gases (2DEGs) of mesoscopic dimensions \\cite{14, 15} as a function of density $n_s$ and temperature $T$. Specifically, $S$ displayed large oscillations and even sign changes as the 2DEG density $n_s$ was varied whereas the electrical resistivity $\\rho \\equiv 1\/\\sigma$ was completely monotonic. However, these measurements focused on the low-conductivity regime of the samples (300$h\/e^2 > \\rho > 5h\/e^2$), where many-body effects suppressed the strong localisation (see also refs~\\cite{16} and \\cite{17}) and it is not clear whether weak localisation, and therefore, mesoscopic fluctuations are to be expected. In this work we measure $S$ in similar samples while concentrating specifically on the medium-$\\rho$ range around the onset of $S$-oscillations where, in principle, UCFs are expected. We perform a statistical analysis of these oscillations and examine whether they are consistent with the mesoscopic fluctuations described by LK.\n\nThe remainder of the article is structured as follows: Section 2 describes the experimental system and measurement setup, section 3 presents the experimental data and statistical analysis and Section 4 presents a discussion and concluding remarks.\n\n\\section{EXPERIMENTAL DETAILS}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=4in]{Figure1.jpg}\n\t\\caption{Schematic representation of the device and measurement scheme. The blue area represents the conducting mesa and ohmic contacts are represented in green. The yellow rectangles represent top-gates.}\n\t\\label{fig1}\n\\end{figure}\n\nOur experiments are all performed at $T$ = 0.3~K in GaAs-based 2DEGs with an as-grown mobility of 220~m$^2$\/Vs at $n_s$ = 2.1$\\times$10$^{15}$~m$^{-2}$. We use a wet etch to define a conducting mesa and deposit ohmic contacts to perform electrical and thermoelectric measurements. The mesoscopic device D (see figure 1) is defined by a metallic gate of lithographical dimensions L$\\times$W = $2 - 3~\\mu \\rm{m} \\times 8~\\mu \\rm{m}$. Figure \\ref{fig1} shows a schematic representation of the device. In addition to D, there are four gate-defined bar-gates (BGs) that serve as a pair of thermometers to measure the local electron temperature $T_{\\rm{e}}$on either side of D~\\cite{14}. To measure $S$ we heat one end of the device with an AC heating current $I_{\\rm{h}}$ = 4~$\\mu$A at frequency $f_{\\rm{h}}$ = 11~Hz. $V_{\\rm{th}}$ is measured using a lock-in amplifier at 2$f_{\\rm{h}}$ as shown in figure~\\ref{fig1}. The local electron temperatures on either side of the device $T_{\\rm{e1}}$ and $T_{\\rm{e2}}$ are measured using bar-gate pairs BG$_{\\rm{1a,b}}$ and BG$_{\\rm{2a,b}}$, respectively. Details of this measurement procedure can be found in Ref.~\\cite{14}. $S$ is then calculated as $V_{\\rm{th}}\/(T_{\\rm{e1}} - T_{\\rm{e2}})$. We measure the resistance $R$ using a standard four-probe lock-in technique with a constant current $I_{\\rm{ex}}$ = 1~nA at a frequency $f$ = 7~Hz. However, the results are unaffected upon increasing $I_{\\rm{ex}}$ by an order of magnitude to 10~nA. In all our measurements $V_{\\rm{th}}$ and $R$ are measured simultaneously. $\\Delta T$ is 20~mK in all the measurements reported here.\n\n\\section{Experimental Data}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=6in]{Figure2.jpg}\n\t\\caption{(a) $S$ as a function of $n_s$ (bottom axis) and $r_s$ (top axis). The inset shows the same data on a linear scale where the sign-reversals of $S$ are clearly visible. Figure 2b shows $R$ measured simultaneously with $S$ over the same $n_s$. Blue and red traces correspond to devices with L = 2~$\\mu$m and 3~$\\mu$m, respectively and the arrow marks the density below which thermopower oscillations are present.}\n\t\\label{fig2}\n\\end{figure}\n\nFigures \\ref{fig2}a and \\ref{fig2}b show $S$ and $R$ as functions of $n_s$ for two different samples labelled D1 and D2. The top horizontal axis in figure \\ref{fig2}a shows the interaction parameter $r_s \\equiv 1\/a_B\\sqrt{\\pi n_s}$, where $a_B$ is the effective Bohr radius in GaAs $\\approx$ 11~nm. D1 has L = 2~$\\mu$m and D2 has L = 3~$\\mu$m. Figure \\ref{fig2}a also shows $S_{\\rm{MOTT}}$ for $\\sigma = n_se^2\\tau\/m$\n\n\\begin{equation}\n\\label{DrudeMott}\nS_{\\rm{MOTT}} = \\frac{\\pi^2 k_B^2 T}{3e}\\frac{\\mbox{d}\\ln\\sigma}{\\mbox{d}E}\\vline_{E = \\mu} = \\frac{\\pi k_B^2 T m}{3e \\hbar^2} \\frac{1 + \\alpha}{n_s}\n\\end{equation}\n\nHere $\\tau$ is the inelastic scattering time and $\\alpha = (n_s\/\\tau)(\\mbox{d}\\tau\/\\mbox{d}n_s) \\approx$~1. It is seen that $S$, while agreeing closely with $S_{\\rm{MOTT}}$at high $n_s$, shows a sudden departure from Mott-like behaviour at $n_s \\approx$~2.5$\\times$10$^{14}$~m$^{-2}$ with strong oscillations and even sign reversals. The sign reversals are seen most clearly in the inset to figure \\ref{fig2}a which also serves to distinguish the smooth $S$ oscillations from the measurement noise ($\\sim$10~$\\mu$V\/K which corresponds to a voltage uncertainty $V_{err}$ of 200~nV), the latter being most prominent in the range $n_s >$ 2$\\times$10$^{14}$~m$^{-2}$. It is worth emphasising that the former are large oscillations and completely reproducible between $n_s$ sweeps (see figure \\ref{fig3}), while the latter correspond to measurement noise. In the same $n_s$-range, $R$ grows monotonically in both devices except for some broad features accentuated by the log scale which, importantly, bear no obvious correlation to the oscillations in $S$. Hence, this very unusual behaviour cannot simply be reconciled with $S$ oscillations in a disordered 2DEG that arise, say, due to Coulomb blockade resulting from the 2DEG fragmenting into charge puddles.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=6in]{Figure3.jpg}\n\t\\caption{(a) shows five separate $S$ traces vertically offset to each other. The inset shows the corresponding $R$ traces. All data are taken at $T$ = 300~mK. Figure 3(b) shows five $R(n_s)$ traces, again, with a relative vertical offset, at five different $T$ and concentrating on high-$n_s$ (boxed region in the inset to figure 3(a)). The inset shows the scale of fluctuations $\\delta_R$ as a function of $T$.}\n\t\\label{fig3}\n\\end{figure}\n\nIn figure~\\ref{fig3}a we plot five consecutive $S$ traces taken at $T$ = 0.3~K vertically offset relative to each other. In the inset we plot the simultaneously measured $R(n_s)$ corresponding to each $S(n_s)$ trace, again, with a relative vertical offset. These figures firmly establish both the reproducible nature of the $S$ oscillations as well as the smooth character of $R(n_s)$. In figure~\\ref{fig3}b we investigate whether the $R$ data at high-$n_s$ contain any signatures of UCFs. We see that in the five $R(n_s)$ traces at 0.3~K $< T <$ 1.5~K, vertically offset relative to each other by 100~$\\Omega$, there are no reproducible features but a fluctuation level that has no obvious $T$-dependence. This becomes clearer in the inset where we plot $\\delta R$, the magnitude of fluctuations in $R$ as a function of $T$. $\\delta R$ is defined as the standard deviation in $R$ after subtracting a smooth background. The dependence of $\\delta R$ on $T$ is not consistent with weak localisation and $\\delta R$ simply represents the experimental noise levels.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=6in]{Figure4.jpg}\n\t\\caption{The correlation function $C_{SS}(\\Delta n_s)$. The grey arrows indicate the minimum in each instance. The inset shows the same data without normalisation.}\n\t\\label{fig4}\n\\end{figure}\n\nWe now return to the $S$ oscillations and to analyse their statistical nature, we look at the autocorrelation of $S$. In figure~\\ref{fig4} we plot $C_{SS}(\\Delta n_s) \\equiv \\langle\\delta S(n_s) \\delta S(n_s + \\Delta n_s)\\rangle\/R^2(n_s)$ normalised to $C_{SS}(0)$ = 1. Here the angular brackets represent an average over $n_s$ and $\\delta S(\\Delta n_s)$ is defined as $S$ - $S_{\\rm{MOTT}}$.  The correlator is defined with $R^2(n_s)$ in the denominator rather than $S^2(n_s$) to avoid divergences at the zero-crossings of $S(n_s)$ and also to make a quantitative comparison with the LK theory (see next section). The inset shows the data without normalising and error bars as estimated from the standard deviation in the data. We find that $C_{SS}(\\Delta n_s)$ in both devices agree quantitatively and are consistent with an initial decay $C_{SS}(\\Delta n_s) \\sim \\Delta n_s^{-1\/2}$. Furthermore, there is a minimum and subsequent maximum that occur in the vicinity of $\\Delta n_s \\approx$ 2.5$\\times$10$^{13}$~m$^{-2}$. These features suggest a degree of periodicity in the $S$ oscillations which are further reflected in the fourier transforms (FTs) of the data shown in figure~\\ref{fig5}. The grey arrows mark the FT peaks which correspond to periodicities of $\\Delta n_s \\approx$~4.2$\\times$10$^{-14}$~m$^{-2}$ and 2.7$\\times$10$^{-14}$~m$^{-2}$ for D1 and D2, respectively. We note that figure~\\ref{fig4} shows several features at $\\Delta n_s$-values larger than 5$\\times$10$^{-13}$~m$^{-2}$ which can be correlated to peaks in the FT in figure~\\ref{fig5}a. However, these must be taken with caution due to the lack of sufficient statistics. In figure~\\ref{fig5}b we plot the FT of $S$ as a function of $r_s$. Interestingly we note that the periodicities correspond to $\\Delta r_s \\approx$ 0.4 \u2013- 0.5, reminiscent of conductance oscillations seen in disordered silicon inversion layers~\\cite{18}.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=6in]{Figure5.jpg}\n\t\\caption{Figures 5a and 5b show the fourier transform of the data in figure 2(a).}\n\t\\label{fig5}\n\\end{figure}\n\n\\section{Discussion}\n\nThe strong oscillations in $S$ without corresponding features in $R$ is extremely surprising and signifies a breakdown of the Mott formula (equation \\ref{MottFormula}). While there are instances where this is expected (see section 5), we first ascertain whether the mesoscopic nature of the 2DEG has a role to play in the observed experimental data. As mentioned earlier, mesoscopic fluctuations are necessarily present in all transport parameters of mesoscopic samples. However, oscillations in $S$ can be significantly amplified compared to those in $R$ due to the derivative relation between them. Conversely, therefore, one can expect situations where $S$ oscillations are visible but immeasurably small in $R$ and such situations would be wrongly perceived as a breakdown of the Mott relation. Motivated by this we now investigate whether the observed oscillations are consistent with the LK-type oscillations~\\cite{10} mentioned earlier.\n\nLK calculated the mesoscopic contribution to $S$ of a sample of small size as a function of $\\mu$ and the magnetic field $B$. Using the Mott formula (equation~\\ref{MottFormula}) they then showed that this contribution can be  positive or negative and even exceed the regular term in magnitude, potentially resulting in sign reversals of $S$. Furthermore, they constructed and evaluated the correlation function: $C_{SS}^{LK}(\\Delta \\mu, \\Delta B) \\equiv \\langle\\delta S(0,0) \\delta S(\\Delta \\mu, \\Delta B)\\rangle = (R^2\/\\hbar^2)F_S(\\Delta \\mu \\tau_F\/\\hbar, \\Delta B LW\/\\phi_0)$. Here $F_S$ is a scaling function, $\\tau_F$ is the time taken for an electron to diffuse across the sample of length $L$ and is given by $\\tau_F \\approx L^2\/D$, with $D$ being the electron diffusivity and $\\phi_0 = hc\/e$. At $\\Delta B$ = 0 and low $T$ this correlation functions is essentially identical to the correlation function evaluated in figure~\\ref{fig4}. The low-$T$ requirement is in order to approximate $\\mu$ by the Fermi energy $E_F$ and this is met in our experimental results since $k_BT\/E_F \\leq$ 0.25 in the $n_s$-range being considered. LK predict a universal form for the autocorrelation function which they numerically show to initially decay and then go through a minimum. This is in striking agreement with the result in figure~\\ref{fig4} in that the data from both devices has the same initial $\\Delta n_s^{-1\/2}$ decay and a clear minimum.\n\nAdditionally, LK consider the cross-correlation between $S$ and $R$: $C_{SR}^{LK}(\\Delta \\mu, \\Delta B) \\equiv \\langle\\delta S(0,0) \\delta R(\\Delta \\mu, \\Delta B)\\rangle$, where $\\delta R$ are fluctuations about the mean $R$. LK show $C_{SR}^{LK}(\\Delta \\mu, \\Delta B)$ to be 0 at $\\Delta \\mu$ = 0, i.e., oscillations in $S$ and $R$ at the same value of chemical potential are uncorrelated, but gradually grow as a function of $\\Delta \\mu$ and ultimately go through a maximum. While the statement that $S$ and $R$ oscillations at a given $\\mu$ are uncorrelated seems consistent with a visual inspection of the data in figure~\\ref{fig2}, a quantitative comparison reveals discrepancies with the LK picture. First, at the point where $S$ oscillations set in, there are absolutely no oscillations in $R$ (see figures~\\ref{fig2} and \\ref{fig3}), thereby trivially rendering $C_{SR}$ zero. In this region, $R \\approx$ 3~K$\\Omega$ and therefore the expected magnitude of $\\delta R = e^2R^2\/h \\approx$ 350~$\\Omega$. This is well within the limit of accuracy even if, considering the wide aspect ratio of the sample, we divide the expected $\\delta R$ by a factor $\\sqrt{W\/L} \\approx$~2. Given the above estimate for $\\delta R$, it is then reasonable to ask why UCFs are absent in the first place. While this is unclear at the moment, a possible explanation lies in the nature of the disorder being sampled by the device. The earlier experiments on mesoscopic 2DEGs~\\cite{16} that motivated the present studies were performed especially to circumvent the long-range disorder due to remote ionised dopants in the host GaAs wafer. The residual short-scale 'white' disorder would then restore the self-averaging nature of the electron paths, leading to the absence of UCFs. We hope to verify this argument by examining whether UCFs appear in larger mesoscopic samples.\n\nIt is worth mentioning here that preliminary results of $S$ and $R$ as a function of $B$ also indicate the absence of UCFs. Not only is this consistent with with the arguments in the previous paragraph, it is another discrepancy between the experimental data and the LK picture. The LK analysis predicts similar cross-correlation and autocorrelation functions as $B$ is tuned, but we observe no reproducible oscillations in $S$ or $R$ as functions of $B$, or at least no oscillations (other than the $B^{-1}$-periodic Shubnikov-de Haas oscillations) comparable in magnitude to those observed by tuning $n_s$. This data will be presented in a separate report.\n\n\\section{Conclusions}\n\nTo summarise, we find some aspects of the data to be consistent with oscillations due to the mesoscopic nature of the device, but several others that are at odds with it. Furthermore, the $S$ oscillations persist to much lower $n_s$ even up to $R \\approx$~100~$h\/e^2$~\\cite{14,15}, a regime in which there is no reason to expect LK-type oscillations. While the oscillations may indeed be related to those seen in Ref.~\\cite{18}, there have recently been several theoretical reports describing situations where the Mott relation (equation~\\ref{MottFormula}) breaks down. These include the vicinity of a quantum critical point~\\cite{19}, proximity to a Lifshitz transition~\\cite{20}, and, remarkably, even far away from a critical point~\\cite{21}.\n\nIn conclusion, we have observed intriguing oscillations in the thermopower $S$ of mesoscopic 2DEGs and analysed their statistical nature. We find that the experimental data from the 2 devices measured suggests several common aspects in the nature of oscillations despite the devices being of dissimilar sizes: 1. the decay of the autocorrelation function of $S$ is consistent with a $\\Delta n_s^{-1\/2}$ dependence; 2. the minima and maxima of the autocorrelation in the two devices occur at approximately similar locations; and 3. the oscillations have a degree of periodicity as revealed by the fourier transform. We have compared these results to the oscillations predicted by Lesovik and Khmelnitskii in Ref.~\\cite{10} and though there are suggestive similarities between the two, it seems unlikely that they are related.\n\n\\section{Acknowledgements}\n\nWe acknowledge funding from the Leverhulme Trust and EPSRC, UK. VN acknowledges a Fellowship from the Herchel Smith Fund, University of Cambridge. VN also acknowledges useful discussions with Sriram Shastry and Charles Smith.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}