\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\citation{teichSqueezedStateLight1989}
\citation{teichSqueezedStateLight1989,dengGeneralizedTwomodeSqueezed1993}
\citation{PhysRevLett.75.4337}
\citation{greenbergerGoingBellTheorem1989,pan2000experimental}
\citation{RevModPhys.79.135}
\citation{PhysRevLett.47.460,PhysRevLett.111.130406}
\citation{bennettQuantumCryptographyPublic2014,xuSecureQuantumKey2020}
\citation{RevModPhys.90.035005}
\citation{PhysRevA.64.023802,doi:10.1126/science.1127676}
\citation{lukinQuantumNoiseCorrelations1999,shuSubnaturallinewidthBiphotonsDopplerbroadened2016,parkSpectralTemporalBiphoton2020,kuzmichGenerationNonclassicalPhoton2003,kolchinGenerationNarrowBandwidthPaired2006,srivathsanNarrowBandSource2013,cereCharacterizationPhotonpairSource2018}
\citation{sangouardQuantumRepeatersBased2011,gutierrezExperimentalFockStateSuperradiance2018,spasibkoSpectralStatisticalProperties2020}
\citation{zhangExperimentalGenerationHighfidelity2016,dongExperimentalRealizationNarrowband2017,parkGenerationBrightFourphoton2022}
\citation{hubelDirectGenerationPhoton2010}
\citation{PhysRevA.57.2076}
\citation{spasibkoSpectralStatisticalProperties2020,PhysRevA.93.043849}
\citation{}
\citation{PhysRevA.97.053835}
\citation{fox2006quantum,spasibkoSpectralStatisticalProperties2020}
\citation{spasibkoSpectralStatisticalProperties2020}
\citation{dickeCoherenceSpontaneousRadiation1954,jenSuperradiantCascadeEmissions2015}
\citation{kuzmichGenerationNonclassicalPhoton2003,yoshimura2012dynamics,jenSuperradiantCascadeEmissions2015}
\citation{jahnke2016giant,temnovPhotonStatisticsCooperative2009,bhattiSuperbunchingNonclassicalityNew2015,cygorekSignaturesCooperativeEmission2023}
\newlabel{FirstPage}{{}{1}{}{section*.1}{}}
\@writefile{toc}{\contentsline {title}{Multiphoton state from four-wave mixing}{1}{section*.2}\protected@file@percent }
\@writefile{toc}{\contentsline {abstract}{Abstract}{1}{section*.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {I}Introduction}{1}{section*.3}\protected@file@percent }
\citation{sangouardQuantumRepeatersBased2011}
\citation{PhysRev.131.2766}
\citation{mollowQuantumTheoryParametric1967b}
\citation{zielnickiJointSpectralCharacterization2018}
\citation{TEICH19881}
\citation{razaviCharacterizingHeraldedSinglephoton2009,bocquillonCoherenceMeasuresHeralded2009}
\citation{bettelliCommentCoherenceMeasures2010,bashkanskySignificanceHeraldingSpontaneous2014}
\citation{srivathsanHeraldedSinglePhotons2015a,guoTIMEFREQUENCYENTANGLEMENTNARROWBAND2018}
\newlabel{eq:effective_interaction_hamiltonian}{{1}{2}{}{equation.1.1}{}}
\newlabel{eq:two_mode_squeezing_state}{{2}{2}{}{equation.1.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {II}Auto-correlation}{2}{section*.4}\protected@file@percent }
\newlabel{section:auto_correlation}{{II}{2}{}{section*.4}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces (a) The double-$\Lambda $ atomic energy levels involved in the four-wave mixing process. In this experiment, the pumping field detuning is set at $\Delta _{p} = -50 \text  {\ MHz}$ while the coupling field is resonant. (b) A schematic representation of the experimental setup. The pumping beam and coupling beam counter-propagate along the longest axis of a cigar-shaped atomic ensemble, with a collection path angled by approximately $1$ degree. The collected Stokes and anti-Stokes photons are split by a 50:50 fiber-based beam splitter and detected by two single-photon detectors, respectively.}}{2}{figure.1}\protected@file@percent }
\newlabel{fig:setup}{{1}{2}{(a) The double-$\Lambda $ atomic energy levels involved in the four-wave mixing process. In this experiment, the pumping field detuning is set at $\Delta _{p} = -50 \text {\ MHz}$ while the coupling field is resonant. (b) A schematic representation of the experimental setup. The pumping beam and coupling beam counter-propagate along the longest axis of a cigar-shaped atomic ensemble, with a collection path angled by approximately $1$ degree. The collected Stokes and anti-Stokes photons are split by a 50:50 fiber-based beam splitter and detected by two single-photon detectors, respectively}{figure.1}{}}
\newlabel{eq:redefined_heralded_auto}{{3}{2}{}{equation.2.3}{}}
\citation{chaneliereQuantumTelecommunicationBased2006,jenPositivePhasespacemethodSimulation2012,srivathsanNarrowBandSource2013,gulatiPolarizationEntanglementQuantum2015,jenSuperradiantCascadeEmissions2015,PhysRevLett.122.143601}
\citation{temnovPhotonStatisticsCooperative2009,auffevesFewEmittersCavity2011,hoiGenerationNonclassicalMicrowave2012,grujicRepulsivelyInducedPhoton2013,bhattiSuperbunchingNonclassicalityNew2015,pagelNonclassicalLightFew2015,PhysRevApplied.4.044018}
\citation{kimPropertiesSqueezedNumber1989,2000LoudonQuantum}
\citation{PhysRevA.87.053819}
\citation{jenPositivePhasespacemethodSimulation2012}
\citation{cygorekSignaturesCooperativeEmission2023}
\citation{}
\citation{gutierrezExperimentalFockStateSuperradiance2018}
\@writefile{toc}{\contentsline {section}{\numberline {III}Photon statistics}{3}{section*.5}\protected@file@percent }
\newlabel{section:photon_statistics}{{III}{3}{}{section*.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Figures (a) and (b) illustrate the auto-correlation functions associated with the unheralded Stokes photon and anti-Stokes photon, respectively. This data originates from the identical dataset presented in Fig.\ref  {fig:heralded_auto}. The mechanism responsible for the oscillation feature in (a) remains unidentified.}}{3}{figure.2}\protected@file@percent }
\newlabel{fig:auto_correlation}{{2}{3}{Figures (a) and (b) illustrate the auto-correlation functions associated with the unheralded Stokes photon and anti-Stokes photon, respectively. This data originates from the identical dataset presented in Fig.\ref {fig:heralded_auto}. The mechanism responsible for the oscillation feature in (a) remains unidentified}{figure.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The heralded auto-correlation functions defined as Eq. \ref  {eq:redefined_heralded_auto} for Stokes photons and Anti-Stokes photons are evaluated. In panels (a) and (b), we observe the heralded auto-correlation for Stokes and anti-Stokes photons, respectively, with selected pairs falling within a $20\mathrm  {\ ns}$ coincidence window, corresponding to the first peak in the cross-correlation. In panels (c) and (d), the auto-correlation functions for Stokes and anti-Stokes photons are depicted for pairs occurring within a $20\text  {-}40 \mathrm  {\ ns}$ coincidence window, corresponding to the second peak in the cross-correlation. The delayed peak observed in Fig.(c) suggests a higher probability of detecting another Stokes photon after a certain delay relative to the Stokes photon paired in a $20\text  {-}40 \mathrm  {\ ns}$ coincidence window. Conversely, the early peak observed in Fig.(d) indicates an increased probability of detecting another anti-Stokes photon preceding the anti-Stokes photons paired within a $20\text  {-}40 \mathrm  {\ ns}$ coincidence window.}}{4}{figure.3}\protected@file@percent }
\newlabel{fig:heralded_auto}{{3}{4}{The heralded auto-correlation functions defined as Eq. \ref {eq:redefined_heralded_auto} for Stokes photons and Anti-Stokes photons are evaluated. In panels (a) and (b), we observe the heralded auto-correlation for Stokes and anti-Stokes photons, respectively, with selected pairs falling within a $20\mathrm {\ ns}$ coincidence window, corresponding to the first peak in the cross-correlation. In panels (c) and (d), the auto-correlation functions for Stokes and anti-Stokes photons are depicted for pairs occurring within a $20\text {-}40 \mathrm {\ ns}$ coincidence window, corresponding to the second peak in the cross-correlation. The delayed peak observed in Fig.(c) suggests a higher probability of detecting another Stokes photon after a certain delay relative to the Stokes photon paired in a $20\text {-}40 \mathrm {\ ns}$ coincidence window. Conversely, the early peak observed in Fig.(d) indicates an increased probability of detecting another anti-Stokes photon preceding the anti-Stokes photons paired within a $20\text {-}40 \mathrm {\ ns}$ coincidence window}{figure.3}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces  (a) illustrates the ratios of photon pair rate, photon triplet rate, and photon quadruplet rate relative to the Stokes photon count rate. In contrast, (b) depicts the same parameters relative to the anti-Stokes photon count rate. Both axes are plotted on a logarithmic scale.}}{4}{figure.4}\protected@file@percent }
\newlabel{fig:probability_vs_count_rate}{{4}{4}{(a) illustrates the ratios of photon pair rate, photon triplet rate, and photon quadruplet rate relative to the Stokes photon count rate. In contrast, (b) depicts the same parameters relative to the anti-Stokes photon count rate. Both axes are plotted on a logarithmic scale}{figure.4}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces  Triplet photon coincidence within a coincidence window $0\text  {-}80 \mathrm  {\ ns}$. (a) The histogram shows the detected triplet events consisting of one Stokes photon and two anti-Stokes photons in Ch3 and Ch4 respectively. The x-axis denotes the time delay of photon events between Ch3 and Stokes mode, while the y-axis represents the time delay between CH4 and Stokes mode. The theoretical accidental count per bin is $R_{s}R_{\text  {ch3}}R_{\text  {ch4}}t_{\text  {coi}}^{2}t_{\text  {meas}}/ n^{2}\simeq 418$ (b) The histogram exhibits the triplet events consisting of two Stokes photons and one anti-Stokes photon in Ch3 and Ch4 respectively. The theoretical accidental count per bin is $R_{as}R_{\text  {ch1}}R_{\text  {ch2}}t_{\text  {coi}}^{2}t_{\text  {meas}}/n^{2}\simeq 623$.}}{4}{figure.5}\protected@file@percent }
\newlabel{fig:triplet_photon_distribution}{{5}{4}{Triplet photon coincidence within a coincidence window $0\text {-}80 \mathrm {\ ns}$. (a) The histogram shows the detected triplet events consisting of one Stokes photon and two anti-Stokes photons in Ch3 and Ch4 respectively. The x-axis denotes the time delay of photon events between Ch3 and Stokes mode, while the y-axis represents the time delay between CH4 and Stokes mode. The theoretical accidental count per bin is $R_{s}R_{\text {ch3}}R_{\text {ch4}}t_{\text {coi}}^{2}t_{\text {meas}}/ n^{2}\simeq 418$ (b) The histogram exhibits the triplet events consisting of two Stokes photons and one anti-Stokes photon in Ch3 and Ch4 respectively. The theoretical accidental count per bin is $R_{as}R_{\text {ch1}}R_{\text {ch2}}t_{\text {coi}}^{2}t_{\text {meas}}/n^{2}\simeq 623$}{figure.5}{}}
\citation{PhysRevA.100.04180}
\citation{doi:10.1080/09500340008244039}
\citation{HashemiRafsanjani:17}
\citation{zhai2014direct}
\citation{alleviSuperthermalLightImaging2017}
\bibdata{mainNotes,reference}
\bibcite{teichSqueezedStateLight1989}{{1}{1989}{{Teich\ and\ Saleh}}{{}}}
\bibcite{dengGeneralizedTwomodeSqueezed1993}{{2}{1993}{{Deng\ \emph  {et~al.}}}{{Deng, Zhang,\ and\ Sun}}}
\bibcite{PhysRevLett.75.4337}{{3}{1995}{{Kwiat\ \emph  {et~al.}}}{{Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko,\ and\ Shih}}}
\bibcite{greenbergerGoingBellTheorem1989}{{4}{1989}{{Greenberger\ \emph  {et~al.}}}{{Greenberger, Horne,\ and\ Zeilinger}}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces (a) The green histogram illustrates the four-fold photon event counts within an $80\mathrm  {\ ns}$ coincidence window over an observation period of $7$ hours. The dashed line represents a theoretical estimation of approximately $110$ accidental four-fold photon events per $2\mathrm  {\ ns}$ bin during the same measurement duration. The accidental four-fold photon detection ratio ($R_{\mathrm  {acc4}}$) is determined by $R_{1}R_{2}R_{3}R_{4} t_{\mathrm  {coi}}^{3}$, where $R_{i}$ denotes the detection rate in each single-photon detector, and $t_{\mathrm  {coi}}$ is the coincidence window. The grey histogram presents an example of accidental four-fold photon event counts from an uncorrelated coincidence window. (b) This 2D histogram illustrates the spatial distribution of four-photon event counts against the relative delay between two Stokes photons and the delay between two anti-Stokes photons in a single four-fold photon event. (c) This displays the four-fold event counts versus the delay time of the anti-Stokes photon after the Stokes photon. Ch1 and Ch2 are two detector channels for Stokes photons, while Ch3 and Ch4 are two channels for anti-Stokes photons.}}{5}{figure.6}\protected@file@percent }
\newlabel{fig:fourphoton_dis}{{6}{5}{(a) The green histogram illustrates the four-fold photon event counts within an $80\mathrm {\ ns}$ coincidence window over an observation period of $7$ hours. The dashed line represents a theoretical estimation of approximately $110$ accidental four-fold photon events per $2\mathrm {\ ns}$ bin during the same measurement duration. The accidental four-fold photon detection ratio ($R_{\mathrm {acc4}}$) is determined by $R_{1}R_{2}R_{3}R_{4} t_{\mathrm {coi}}^{3}$, where $R_{i}$ denotes the detection rate in each single-photon detector, and $t_{\mathrm {coi}}$ is the coincidence window. The grey histogram presents an example of accidental four-fold photon event counts from an uncorrelated coincidence window. (b) This 2D histogram illustrates the spatial distribution of four-photon event counts against the relative delay between two Stokes photons and the delay between two anti-Stokes photons in a single four-fold photon event. (c) This displays the four-fold event counts versus the delay time of the anti-Stokes photon after the Stokes photon. Ch1 and Ch2 are two detector channels for Stokes photons, while Ch3 and Ch4 are two channels for anti-Stokes photons}{figure.6}{}}
\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{5}{section*.6}\protected@file@percent }
\newlabel{section:conclusion}{{IV}{5}{}{section*.6}{}}
\@writefile{toc}{\contentsline {section}{\numberline {}References}{5}{section*.7}\protected@file@percent }
\bibcite{pan2000experimental}{{5}{2000}{{Pan\ \emph  {et~al.}}}{{Pan, Bouwmeester, Daniell, Weinfurter,\ and\ Zeilinger}}}
\bibcite{RevModPhys.79.135}{{6}{2007}{{Kok\ \emph  {et~al.}}}{{Kok, Munro, Nemoto, Ralph, Dowling,\ and\ Milburn}}}
\bibcite{PhysRevLett.47.460}{{7}{1981}{{Aspect\ \emph  {et~al.}}}{{Aspect, Grangier,\ and\ Roger}}}
\bibcite{PhysRevLett.111.130406}{{8}{2013}{{Christensen\ \emph  {et~al.}}}{{Christensen, McCusker, Altepeter, Calkins, Gerrits, Lita, Miller, Shalm, Zhang, Nam, Brunner, Lim, Gisin,\ and\ Kwiat}}}
\bibcite{bennettQuantumCryptographyPublic2014}{{9}{2014}{{Bennett\ and\ Brassard}}{{}}}
\bibcite{xuSecureQuantumKey2020}{{10}{2020}{{Xu\ \emph  {et~al.}}}{{Xu, Ma, Zhang, Lo,\ and\ Pan}}}
\bibcite{RevModPhys.90.035005}{{11}{2018}{{Pezz\`e\ \emph  {et~al.}}}{{Pezz\`e, Smerzi, Oberthaler, Schmied,\ and\ Treutlein}}}
\bibcite{PhysRevA.64.023802}{{12}{2001}{{Kurtsiefer\ \emph  {et~al.}}}{{Kurtsiefer, Oberparleiter,\ and\ Weinfurter}}}
\bibcite{doi:10.1126/science.1127676}{{13}{2006}{{Thompson\ \emph  {et~al.}}}{{Thompson, Simon, Loh,\ and\ Vuletić}}}
\bibcite{lukinQuantumNoiseCorrelations1999}{{14}{1999}{{Lukin\ \emph  {et~al.}}}{{Lukin, Matsko, Fleischhauer,\ and\ Scully}}}
\bibcite{shuSubnaturallinewidthBiphotonsDopplerbroadened2016}{{15}{2016}{{Shu\ \emph  {et~al.}}}{{Shu, Chen, Chow, Zhu, Xiao, Loy,\ and\ Du}}}
\bibcite{parkSpectralTemporalBiphoton2020}{{16}{2020}{{Park\ \emph  {et~al.}}}{{Park, Jeong,\ and\ Moon}}}
\bibcite{kuzmichGenerationNonclassicalPhoton2003}{{17}{2003}{{Kuzmich\ \emph  {et~al.}}}{{Kuzmich, Bowen, Boozer, Boca, Chou, Duan,\ and\ Kimble}}}
\bibcite{kolchinGenerationNarrowBandwidthPaired2006}{{18}{2006}{{Kolchin\ \emph  {et~al.}}}{{Kolchin, Du, Belthangady, Yin,\ and\ Harris}}}
\bibcite{srivathsanNarrowBandSource2013}{{19}{2013}{{Srivathsan\ \emph  {et~al.}}}{{Srivathsan, Gulati, Brenda, Maslennikov, Matsukevich,\ and\ Kurtsiefer}}}
\bibcite{cereCharacterizationPhotonpairSource2018}{{20}{2018}{{Cer{\`e}\ \emph  {et~al.}}}{{Cer{\`e}, Srivathsan, Gulati, Chng,\ and\ Kurtsiefer}}}
\bibcite{sangouardQuantumRepeatersBased2011}{{21}{2011}{{Sangouard\ \emph  {et~al.}}}{{Sangouard, Simon, De~Riedmatten,\ and\ Gisin}}}
\bibcite{gutierrezExperimentalFockStateSuperradiance2018}{{22}{2018}{{{Ortiz-Guti{\'e}rrez}\ \emph  {et~al.}}}{{{Ortiz-Guti{\'e}rrez}, {Mu{\~n}oz-Mart{\'i}nez}, Barros, Morales, Moreira, Alves, Tieco, Saldanha,\ and\ Felinto}}}
\bibcite{spasibkoSpectralStatisticalProperties2020}{{23}{2020}{{Spasibko}}{{}}}
\bibcite{zhangExperimentalGenerationHighfidelity2016}{{24}{2016}{{Zhang\ \emph  {et~al.}}}{{Zhang, Huang, Liu, Li,\ and\ Guo}}}
\bibcite{dongExperimentalRealizationNarrowband2017}{{25}{2017}{{Dong\ \emph  {et~al.}}}{{Dong, Zhang, Hou, Yu, Shi, Ding,\ and\ Shi}}}
\bibcite{parkGenerationBrightFourphoton2022}{{26}{2022}{{Park\ and\ Moon}}{{}}}
\bibcite{hubelDirectGenerationPhoton2010}{{27}{2010}{{H{\"u}bel\ \emph  {et~al.}}}{{H{\"u}bel, Hamel, Fedrizzi, Ramelow, Resch,\ and\ Jennewein}}}
\bibcite{PhysRevA.57.2076}{{28}{1998}{{Keller\ \emph  {et~al.}}}{{Keller, Rubin, Shih,\ and\ Wu}}}
\bibcite{PhysRevA.93.043849}{{29}{2016}{{Iskhakov\ \emph  {et~al.}}}{{Iskhakov, Usenko, Filip, Chekhova,\ and\ Leuchs}}}
\bibcite{PhysRevA.97.053835}{{30}{2018}{{Lettau\ \emph  {et~al.}}}{{Lettau, Leymann, Melcher,\ and\ Wiersig}}}
\bibcite{fox2006quantum}{{31}{2006}{{Fox}}{{}}}
\bibcite{dickeCoherenceSpontaneousRadiation1954}{{32}{1954}{{Dicke}}{{}}}
\bibcite{jenSuperradiantCascadeEmissions2015}{{33}{2015}{{Jen}}{{}}}
\bibcite{yoshimura2012dynamics}{{34}{2012}{{Yoshimura\ \emph  {et~al.}}}{{Yoshimura, Sasao,\ and\ Tanaka}}}
\bibcite{jahnke2016giant}{{35}{2016}{{Jahnke\ \emph  {et~al.}}}{{Jahnke, Gies, A{\ss }mann, Bayer, Leymann, Foerster, Wiersig, Schneider, Kamp,\ and\ H{\"o}fling}}}
\bibcite{temnovPhotonStatisticsCooperative2009}{{36}{2009}{{Temnov\ and\ Woggon}}{{}}}
\bibcite{bhattiSuperbunchingNonclassicalityNew2015}{{37}{2015}{{Bhatti\ \emph  {et~al.}}}{{Bhatti, Von~Zanthier,\ and\ Agarwal}}}
\bibcite{cygorekSignaturesCooperativeEmission2023}{{38}{2023}{{Cygorek\ \emph  {et~al.}}}{{Cygorek, Scerri, Santana, Koong, Gerardot,\ and\ Gauger}}}
\bibcite{PhysRev.131.2766}{{39}{1963}{{Glauber}}{{}}}
\bibcite{mollowQuantumTheoryParametric1967b}{{40}{1967}{{Mollow\ and\ Glauber}}{{}}}
\bibcite{zielnickiJointSpectralCharacterization2018}{{41}{2018}{{Zielnicki\ \emph  {et~al.}}}{{Zielnicki, {Garay-Palmett}, {Cruz-Delgado}, {Cruz-Ramirez}, O'Boyle, Fang, Lorenz, U'Ren,\ and\ Kwiat}}}
\bibcite{TEICH19881}{{42}{1988}{{Teich\ and\ Saleh}}{{}}}
\bibcite{razaviCharacterizingHeraldedSinglephoton2009}{{43}{2009}{{Razavi\ \emph  {et~al.}}}{{Razavi, S{\"o}llner, Bocquillon, Couteau, Laflamme,\ and\ Weihs}}}
\bibcite{bocquillonCoherenceMeasuresHeralded2009}{{44}{2009}{{Bocquillon\ \emph  {et~al.}}}{{Bocquillon, Couteau, Razavi, Laflamme,\ and\ Weihs}}}
\bibcite{bettelliCommentCoherenceMeasures2010}{{45}{2010}{{Bettelli}}{{}}}
\bibcite{bashkanskySignificanceHeraldingSpontaneous2014}{{46}{2014}{{Bashkansky\ \emph  {et~al.}}}{{Bashkansky, Vurgaftman, Pipino,\ and\ Reintjes}}}
\bibcite{srivathsanHeraldedSinglePhotons2015a}{{47}{2015}{{Srivathsan}}{{}}}
\bibcite{guoTIMEFREQUENCYENTANGLEMENTNARROWBAND2018}{{48}{2018}{{Guo}}{{}}}
\bibcite{chaneliereQuantumTelecommunicationBased2006}{{49}{2006}{{Chaneli{\`e}re\ \emph  {et~al.}}}{{Chaneli{\`e}re, Matsukevich, Jenkins, Kennedy, Chapman,\ and\ Kuzmich}}}
\bibcite{jenPositivePhasespacemethodSimulation2012}{{50}{2012}{{Jen}}{{}}}
\bibcite{gulatiPolarizationEntanglementQuantum2015}{{51}{2015}{{Gulati\ \emph  {et~al.}}}{{Gulati, Srivathsan, Chng, Cer{\`e},\ and\ Kurtsiefer}}}
\bibcite{PhysRevLett.122.143601}{{52}{2019}{{Park\ \emph  {et~al.}}}{{Park, Kim,\ and\ Moon}}}
\bibcite{auffevesFewEmittersCavity2011}{{53}{2011}{{Auff{\`e}ves\ \emph  {et~al.}}}{{Auff{\`e}ves, Gerace, Portolan, Drezet,\ and\ Fran{\c c}a~Santos}}}
\bibcite{hoiGenerationNonclassicalMicrowave2012}{{54}{2012}{{Hoi\ \emph  {et~al.}}}{{Hoi, Palomaki, Lindkvist, Johansson, Delsing,\ and\ Wilson}}}
\bibcite{grujicRepulsivelyInducedPhoton2013}{{55}{2013}{{Grujic\ \emph  {et~al.}}}{{Grujic, Clark, Jaksch,\ and\ Angelakis}}}
\bibcite{pagelNonclassicalLightFew2015}{{56}{2015}{{Pagel\ \emph  {et~al.}}}{{Pagel, Alvermann,\ and\ Fehske}}}
\bibcite{PhysRevApplied.4.044018}{{57}{2015}{{Leymann\ \emph  {et~al.}}}{{Leymann, Foerster, Jahnke, Wiersig,\ and\ Gies}}}
\bibcite{kimPropertiesSqueezedNumber1989}{{58}{1989}{{Kim\ \emph  {et~al.}}}{{Kim, De~Oliveira,\ and\ Knight}}}
\bibcite{2000LoudonQuantum}{{59}{2000}{{Loudon}}{{}}}
\bibcite{PhysRevA.87.053819}{{60}{2013}{{Leymann\ \emph  {et~al.}}}{{Leymann, Hopfmann, Albert, Foerster, Khanbekyan, Schneider, H\"ofling, Forchel, Kamp, Wiersig,\ and\ Reitzenstein}}}
\bibcite{doi:10.1080/09500340008244039}{{61}{2000}{{Dür\ and\ Cirac}}{{}}}
\bibcite{HashemiRafsanjani:17}{{62}{2017}{{Rafsanjani\ \emph  {et~al.}}}{{Rafsanjani, Mirhosseini, {n}a Loaiza, Gard, Birrittella, Koltenbah, Parazzoli, Capron, Gerry, Dowling,\ and\ Boyd}}}
\bibcite{zhai2014direct}{{63}{2014}{{Zhai\ \emph  {et~al.}}}{{Zhai, Becerra, Fan,\ and\ Migdall}}}
\bibcite{alleviSuperthermalLightImaging2017}{{64}{2017}{{Allevi\ \emph  {et~al.}}}{{Allevi, Cassina,\ and\ Bondani}}}
\bibcite{hockelDirectMeasurementHeralded2011}{{65}{2011}{{H{\"o}ckel\ \emph  {et~al.}}}{{H{\"o}ckel, Koch,\ and\ Benson}}}
\bibcite{Panyukov:22}{{66}{2022}{{Panyukov\ \emph  {et~al.}}}{{Panyukov, Shishkov,\ and\ Andrianov}}}
\newlabel{LastBibItem}{{66}{7}{}{section*.7}{}}
\@writefile{toc}{\appendix }
\@writefile{toc}{\contentsline {section}{\numberline {A}Coincidence histogram}{7}{section*.8}\protected@file@percent }
\citation{razaviCharacterizingHeraldedSinglephoton2009,bocquillonCoherenceMeasuresHeralded2009,hockelDirectMeasurementHeralded2011}
\citation{Panyukov:22}
\@writefile{toc}{\contentsline {section}{\numberline {B}Heralded coincidence histogram}{8}{section*.9}\protected@file@percent }
\newlabel{app:heralded_coincidence}{{B}{8}{}{section*.9}{}}
\@writefile{toc}{\contentsline {section}{\numberline {C}Former definition of heralded auto-correlation function}{8}{section*.10}\protected@file@percent }
\newlabel{app:former_heralded}{{C}{8}{}{section*.10}{}}
\citation{bettelliCommentCoherenceMeasures2010,hockelDirectMeasurementHeralded2011,bashkanskySignificanceHeraldingSpontaneous2014}
\@writefile{toc}{\contentsline {section}{\numberline {D}Multiple photon detection model}{9}{section*.11}\protected@file@percent }
\newlabel{app:detection_model}{{D}{9}{}{section*.11}{}}
\newlabel{eq:event_rate}{{D1}{9}{}{equation.D.1}{}}
\newlabel{eq:approx_event_rate}{{D2}{9}{}{equation.D.2}{}}
\newlabel{eq:theo_pair_rate_vs_single_rate}{{D3}{9}{}{equation.D.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {E}Collection transmission and quantum efficiencies}{9}{section*.12}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {F}Initial state deduction}{10}{section*.13}\protected@file@percent }
\newlabel{app:initial_state}{{F}{10}{}{section*.13}{}}
\bibstyle{apsrev4-2}
\citation{REVTEX42Control}
\citation{apsrev42Control}
\newlabel{LastPage}{{}{11}{}{}{}}
\gdef \@abspage@last{11}