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\begin{document}
\title{Collective correlated photon pair generation from atomic ensemble in paraxial regime}
\author{Yifan~Li}
\author{Chang Hoong Chow}
\author{Boon Long Ng}
\author{Justin Yu Xiang Peh}
\author{Vindhiya Prakash}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}
\author{Christian~Kurtsiefer}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}
\affiliation{Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551}
\email[]{christian.kurtsiefer@gmail.com}

\begin{abstract}
We report an experimental realization of a bright narrowband correlated photon source employing spontaneous four-wave mixing in a double-$\Lambda$ energy level configuration associated with electromagnetic-induced transparency. We observe giant photon bunching phenomena with $g^{(2)}(\tau)>2$ for both Stokes and anti-Stokes photons in the heralding second-order auto-correlation functions by utilizing double Hanbury Brown and Twiss (HBT) measurements. \textcolor{red}{In a coincidence window of $ 76 \unit{ns}$, we record a notable four-photon event rate of 93 per hour, with a correlated pair rate of $1750 \unit{s^{-1}}$ in lossy collection channels. This implies an initial four-photon generation rate of approximately $8\times 10^{5} \unit{s^{-1}}$ and a pair generation rate of $1.8\times 10^{6} \unit{s^{-1}}$.(data will be updated based on recent measurements, the data is processing, preliminary analysis records near 6000 per second correlated photon pairs.)} Our findings indicate a propensity for correlated photons to cluster during generation, with the superradiance mechanism playing a dominant role in paraxial correlated photon generation. This investigation contributes valuable insights into the distinctive features of correlated photons from the cold atomic ensemble and their potential applications in quantum information processing.

\begin{comment}

We present a source for bright narrowband correlated photon pairs from four-wave mixing in a cold ensemble of $\mathrm{Rb}^{87}$ atoms.% double-$\Lambda$ energy level configuration associated with electromagnetic-induced transparency. 
Utilizing double Hanbury Brown and Twiss (HBT) measurements on both the Stokes and anti-Stokes photon sides, we observe significant deviations from thermal light behavior in the heralding second-order self-correlation functions. Specifically, we observe giant photon bunching phenomena with $g^{(2)}(\tau)>2$ for both Stokes and anti-Stokes photons. In a coincidence window of 76 ns, we record a notable four-photon event rate of 93 per hour, with a correlated pair rate of 1750 per second in lossy collection channels. This implies an initial four-photon generation rate of approximately $8\times 10^{5}$ per second and a pair generation rate of $1.8\times 10^{6}$ per second. Our findings indicate a propensity for correlated photons to cluster during generation, with the superradiance mechanism playing a dominant role in paraxial correlated photon generation. This investigation contributes valuable insights into the distinctive features of correlated photons from the cold atomic ensemble and their potential applications in quantum information processing.
\end{comment}
\end{abstract}

\maketitle
\section{Introduction}
Correlated and entangled photons constitute a valuable quantum resource with diverse applications, particularly within the domains of quantum communication, quantum cryptography, and quantum computation, specifically in the context of optical cluster states. Conventionally, correlated photon pairs are generated through spontaneous parametric down-conversion (SPDC) processes in nonlinear crystals \cite{PhysRevA.64.023802, doi:10.1126/science.1127676} or spontaneous four-wave mixing (SFWM) in atomic media \cite{kuzmichGenerationNonclassicalPhoton2003, kolchinGenerationNarrowBandwidthPaired2006}, encompassing both cold atomic ensembles and hot atomic gases \cite{shuSubnaturallinewidthBiphotonsDopplerbroadened2016,parkSpectralTemporalBiphoton2020}. In comparison to correlated photon pairs generated from SPDC, time-correlated photon pairs from cold atomic ensembles exhibit narrow bandwidth properties, benefiting from the atomic natural linewidth, making them suitable for direct interactions with atoms. Furthermore, it is noteworthy that the coherence time of correlated photons, and consequently the biphoton wave functions, can be modulated and controlled \cite{duSubnaturalLinewidthBiphotons2008,duTemporallyShapingBiphoton2009,zhaoShapingBiphotonTemporal2015}, offering potential advantages for long-distance quantum communication. Two prevalent protocols for biphoton generation involve cascade emission from cascade-shaped energy levels \cite{srivathsanNarrowBandSource2013, gulatiPolarizationEntanglementQuantum2015, cereCharacterizationPhotonpairSource2018} and electromagnetically-induced-transparency-based Stokes and anti-Stokes emission from double-$\Lambda$ energy levels \cite{balicGenerationPairedPhotons2005}. Following the proposition of the existence of a correlation field in the $\Lambda$ energy levels \cite{fleischhauerPulseMatchingCorrelation1995, lukinQuantumNoiseCorrelations1999, zibrovNondegenerateParametricSelfOscillation1999}, a subsequent series of theoretical interpretations \cite{raymondooiCorrelationPhotonPairs2007,kolchinElectromagneticallyinducedtransparencybasedPairedPhoton2007, DduNarrowbandBiphotonGeneration2008,jiangQuantumLangevinTheory2023} has emerged concerning the correlated photon pairs originating from the spontaneous four-wave mixing within the double-$\Lambda$ energy levels. These systematically categorize the biphoton wave function in both the oscillation regime and the group delay regime,  consistent with experimental observations \cite{balicGenerationPairedPhotons2005,zhaoPhotonPairsCoherence2014, zhaoNarrowbandBiphotonGeneration2016,meiEinsteinPodolskyRosenEnergyTimeEntanglement2020}.

Nevertheless, preceding theoretical investigations have employed a low parametric gain approximation and first-order perturbation theory\cite{DduNarrowbandBiphotonGeneration2008}, concentrating exclusively on the $\chi^{(3)}$ nonlinearity, while neglecting higher-order terms. Although the quantum Langevin equation methods do not eliminate multi-photon processes, the derived solution is confined to the biphoton wave function. Experimentally, previous studies have operated under conditions of low pump intensity and a slightly larger collection angle, thereby impeding the observation of directional emission in the paraxial direction. Furthermore, certain experimental investigations \cite{zhaoPhotonPairsCoherence2014, zhaoNarrowbandBiphotonGeneration2016} have explored the impact of optical depth (OD); however, they primarily emphasize the group delay regime, characterized by the extension of the photon coherence time due to the slow light effect. Additionally, the extensive utilization of single-photon detectors in previous experiments, lacking photon-number-resolving capabilities, has imposed constraints on the discernment of multiple-photon states. Even in scenarios where multiple photons arrive simultaneously, these detectors are limited to registering single-count signals, thereby impeding the detection of higher-order phenomena.

In this paper, we demonstrate that, in a dense atomic ensemble, higher-order nonlinearity in SFWM cannot be dismissed. Correlated photons emanating from the atomic ensemble cannot be treated as ideal correlated photon pairs or heralding single photon sources. These correlated photons manifest a significant bunching feature, surpassing the thermal light properties arising from near-resonance atomic emission. This observation indicates the presence of collective emission in correlated photons, a phenomenon modeled through the superradiance effect. Dicke \cite{dickeCoherenceSpontaneousRadiation1954,grivet1964quantum} introduced the concept that the symmetric superposition of atomic states leads to superradiant emission characterized by an enhanced collective decay rate proportional to the atom number. The exploration of this phenomenon initially assumes that the interatomic distance is smaller than the wavelength of the electromagnetic fields. However, certain studies \cite{grossSuperradianceEssayTheory1982, araujoSuperradianceLargeDilute2016, guerinSubradianceLargeCloud2016} have proposed that in a disordered atomic ensemble with $N\gg 2$, the field-induced dipole-dipole interaction induces superradiant emission, even when interatomic distances exceed the wavelength. Previously, the superradiance was investigated in the resonance, Raman scattering \cite{andreev1980collective} and theoretically proposed in the parametric cascaded four-wave mixing \cite{jenPositivePhasespacemethodSimulation2012, gulatiPolarizationEntanglementQuantum2015, jenSuperradiantCascadeEmissions2015}. The experiment realization \cite{kuzmichGenerationNonclassicalPhoton2003} of Duan-Lukin-Cirac-Zoller (DLCZ) protocol \cite{duanLongdistanceQuantumCommunication2001} have successfully observed quantum correlations among photon pairs generated through the collective emission from an atomic ensemble. In our specific parametric four-wave mixing process, correlated photons can be generated in the way of multiple folds of correlated Stokes and anti-Stokes pairs, and this kind of four-photon events can be measured using a double HBT setup for both Stokes and anti-Stokes. As the measurement deviates from the previous results of producing purely correlated photon pairs, the high four-photon event compared to the background and the giant bunching feature reveal that the correlated photons have a propensity to cluster in the coincidence window. These observations coincide with the photon statistics inherent in superradiance where the first photon enhances the emission of the second photon. From a macroscopic perspective, these correlated photon clusters can be attributed to a higher-order nonlinear process instead of $\chi^{(3)}$. Besides, the continuous pumping configuration substantiates the existence of a steady-state solution for superradiance, Notably, this differs from conventional realization involving pulsed excitation and inverted atom states. This solution offers a potential scheme for investigating superradiance within a parametric process, instead of direct measurement of spontaneous decay following pulsed excitation. Simultaneously, the employment of a continuously pumped cold atomic ensemble prepared through Magneto-Optical Trapping (MOT) offers novel insights into superradiance laser design, presenting an alternative to atomic beam-based approaches \cite{liuRuggedMHzLinewidthSuperradiant2020}. The time-correlated photon cluster states present a promising avenue for harnessing a many-body entangled quantum resource. This resource holds significant value for tasks such as creating many-body entangled photonic states in optical quantum computation and enhancing precision in phase measurements within the domains of quantum metrology and quantum sensing.

This paper is organized as follows. Firstly, we describe the experiment protocol to measure the time-correlated photons in pairs, triplets, and quadruplets in Sec. \ref{section:exp}. After that, we represent the formulation of correlated photon generation using the Heisenberg-Langevin method involving the higher-order terms in Sec.II. Then in Sec.III, we show the Glauber two-photon correlation function concerning the time delay $\tau$ between Stokes and anti-Stokes photons. The biphoton wave function can be modeled with the same scheme as the photon pair case but with an enhanced effective decay rate caused by the collective emission of superradiance. In Sec.IV, we present the heralding $g^{(2)}$ of Stokes photons and anti-Stokes photons in double-HBT experiments, leading to a giant bunching as $g^{(2)}>2$ rather than the anti-bunching reported before. In Sec.V, we report the direct four-photon event counts and residual three-photon counts due to the large loss in the collection and filter path. Moreover, we infer the initial state from the atomic ensemble. Afterward, in Sec.VI, we try to describe and model the measured results using the superradiance description. Finally, we summarize in Sec.VII. 


\section{Experimental Protocol}\label{section:exp}
\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{setup.pdf}
    \caption{Fig.1 (a) Sketch of the experimental setup. (b) Double-$\Lambda$ atomic energy levels involved in the four-wave mixing process.}
    \label{fig:setup}
\end{figure}
The correlated photon generation is based on a double-$\Lambda$ atomic system, which combines the $\mathrm{D1}$ lines and $\mathrm{D2}$ lines of $\mathrm{Rb}^{87}$ atoms. 
As shown in Fig.\ref{fig:setup} (a),  pumping laser $(\omega_{p})$ which is detuned by $\Delta_{p}$ from the transition $|5S_{1/2}, F=1\rangle\leftrightarrow |5P_{3/2}, F=2\rangle$ atomic transition and coupling laser $(\omega_{p})$ which is resonant to the $|5S_{1/2}, F=2\rangle \leftrightarrow |5P_{1/2}, F=2\rangle$ transition. These two fields associated with initial vacuum states in the Stokes mode ($\omega_{s}$) and the anti-Stokes ($\omega_{as}$) mode allow a phase-matched parametric four-wave mixing process. This process leads to the spontaneous generation of paired, phase-matched Stokes and anti-Stokes photons. The pumping laser is relatively weak, while the coupling field exhibits strength significantly greater than the spontaneous decay rate, characterized by $\Omega_{c} \gg \Gamma_{3}$. The coupling field not only participates in the FWM process to fulfill parametric requirements but also establishes an Electromagnetically Induced Transparency (EIT) window for the anti-Stokes photon near the atomic resonance, improving the transmission of anti-Stokes photons in the atomic ensemble. Simultaneously, these photons experience a slow light effect induced by a substantial change in dispersion. The phase matching condition can be satisfied in both the forward and backward geometries. In this experiment, as depicted in Fig. \ref{fig:setup} (b), we implement the backward-propagation configuration. Specifically, the coupling beam and the pump beam counter-propagate collinearly in opposite directions.

The atomic ensemble used in the experiment is an elongated elliptical $\mathrm{Rb}^{87}$ atomic cloud, characterized by a length $L \approx 5\ mm$ and a transverse width $2\times 4\ mm$. Following a cooling cycle, atoms are initially prepared in the ground state $|1\rangle$ (refer to Fig.\ref{fig:setup}). This preparation is achieved through a conventional dark-spontaneous-force optical trap \cite{ketterleHighDensitiesCold1993,townsendHighdensityTrappingCesium1996,munizCreatingSelfinducedDark2004,radwellColdatomDensitiesMore2013} both spatially and temporally. The repump beam, resonant to $|5S_{1/2}, F=1\rangle \leftrightarrow |5P_{3/2}, F=2\rangle$ transition and aligned with the cooling beams in the $\mathrm{x-y}$ plane, features a donut-shaped transverse intensity profile. In this profile, the center intensity approaches zero, while the outer shell exhibits higher intensity. This transverse mode beam is generated by using a pair of axicons. Within the intersection zone, the repump beams create a region with low intensity inside and high intensity outside. The trapped atoms are spatially confined in a hyperfine ground state that does not interact with the cooling laser, which avoids most light scattering.  Simultaneously, atoms attempting to escape from the trap must traverse the outer shell exposed to the relatively strong repump laser and cooling laser, leading to a significant cooling effect that impedes atom leakage from the trap. This spatial layout contributes to enhancing the atomic density within the confined atomic cloud. During a 10 ms cooling sequence, the repump laser gradually decreases over 300 µs, while the cooling beam remains constantly on for an additional 300 µs before being switched off. During this period, the atoms undergo a molasses cooling process, ensuring that the atomic ensemble does not experience dramatic expansion. Fluorescence detection of the cooling beam reveals that, following such a cooling sequence, the majority of atoms are prepared in the $|1\rangle$ ground state. 

% c220 65cm 
The collection path is deviated from the collinear pumping and coupling beam by an angle of $1$ degree. Two opposite single-mode fiber couplers collect the photon in a focused spatial Gaussian mode with a waist of $175\ \mu m$ at the atomic cloud position, corresponding to an interaction volume of approximate $5\times 10^{-4}\ cm^{3}$, neglecting the divergence of a spatial mode within the atomic ensemble. The collection paths can only collect the photons generated from this volume, propagating in the collection direction. Before coupling this spatial mode is single-mode fiber, the collected photon is filtered by polarization filters consisting of a half-wave plate and a polarization beam splitter. Then the collected photons are filtered via temperature-controlled fused-silica Etalon filters which help to eliminate the pump field scattering and certain Raman scattering photons whose frequency are centered at $\omega_{s}$ and $\omega_{as}$. The Etalon for the Stokes photon has a maximal transmission of $37\ \%$ at the Stokes frequency, with a bandwidth of $275\ \mathrm{MHz}$ and an extinction ratio of $34\mathrm{dB}$ at the pumping frequency. While the Etalon for the anti-Stokes photon has a maximal transmission of $30\ \%$ at the Stokes frequency, with a bandwidth of $345 \mathrm{MHz}$ and an extinction ratio of $40\mathrm{dB}$ at the pumping frequency. Following these filters, we execute double Hanbury Brown and Twiss measurements on both Stokes and anti-Stokes sides: photons pass through a 50:50 beam splitter and are detected by a pair of Silicon single-photon detectors on each side (Anti-Stokes side: Excelitas APDs, Stokes side: Home-made APDs with diodes). The single photon counts are recorded by the timestamp which has the resolution of $2\mathrm{ns}$. 

\section{Two-photon wave function}\label{section:twophoton}
\textcolor{red}{This section is not completed, the figures of the two-photon wavefunction will be updated.}
Conventionally, while the non-degenerated SPDC and four-wave mixing can be modeled as the two-mode squeezing on the initial vacuum state for both Stokes mode $s$ and anti-Stokes mode $as$. The squeezed vacuum state \cite{siskhakovSuperbunchedBrightSqueezed2012} is written in photon number basis as 
\begin{equation}
    |\Psi_{s, as}\rangle = \frac{1}{\cosh r} \sum_{n=0}^{\infty} (-e^{i\phi}\tanh r)^{n}|n_{s}, n_{as}\rangle
\end{equation}
In the low squeezing limit, the two-mode squeezed state can be expressed as 
\begin{equation}\label{eq:two_mode_squeezing_state}
    (1-\frac{p}{2})|0, 0\rangle - i\sqrt{p}|1, 1\rangle - p |2, 2\rangle +\mathcal{O}(p^{\frac{3}{2}})
\end{equation}
The probability of emitting correlated photon pair state $|1, 1\rangle$ is $p$, then there is a probability $p^{2}$ of creating multiple pairs. As discussed in Sec.\ref{section:photonevents}, we infer that the probability of multiple pairs is larger than the $p^{2}$ where $p$ is the photon pair generation rate. It implies that a self-optimal parametric amplifier (OPA) exists in Stokes and anti-Stokes mode. which collectively enhances the multiple pair generation. 
Glauber's second-order correlation function between Stokes and anti-Stokes photons is determined by the two-photon wave function $\Psi(\tau) = \langle \hat{a}_{as}\hat{a}_{s}\rangle$ as 
\begin{equation}
\begin{aligned}
G_{\mathrm{as-s}}^{(2)}(\tau) & = \langle\hat{a}_{s}^{\dagger}(\tau)\hat{a}_{as}^{\dagger}(t+\tau)\hat{a}_{as}(t+\tau)\hat{a}_{s}(t)\rangle \\
& = |\Psi(\tau)|^{2} + R_{s}R_{as} \ ,
\end{aligned}  
\end{equation}
where $R_{s}$ and $R_{as}$ denote the single photon rate in Stokes and anti-Stokes channels respectively. The normalized two-photon cross-section function is given as 
\begin{equation}
g_{\mathrm{as-s}}^{(2)} (\tau) =  \frac{G_{\mathrm{as-s}}^{(2)}(\tau)}{R_{1}R_{2}}
\end{equation}
As cited in Ref.\cite{duSubnaturalLinewidthBiphotons2008}, in the damped Rabi oscillation regime. the two-photon Glauber correlation function with  
\begin{equation}
G^{(2)}(\tau) = e^{-\Gamma_{e}\tau}(1-\cos(\Omega_{e}\tau) \Theta(\tau)
\end{equation}
where $\Omega_{e}= \sqrt{|\Omega_{c}|^{2} - (\gamma_{13} -\gamma_{12})^{2}}$. This oscillation in the temporal cross-correlation function is caused by the quantum interference between two types of biphoton generation, one pair whose anti-Stokes photon is red-detuned $\Omega_{e}/2$, the other is blued-detuned $\Omega_{e}/2$ compared to the center frequency $\omega_{as, 0}$ 



% The fit of coincidence counts in Fig. provide a fit value for $\Gamma_{e}$ as $30 \mathrm{MHz}$ which is far deviated from the expected value as  $\gamma_{12} + \gamma_{13}$ where $\gamma_{13}$ is the spontaneous decay rate from $|3\rangle$ to $|1\rangle$ while $\gamma_{12}$ is the dephasing rate between hyperfine ground state $|2\rangle$ and $|1\rangle$. Correspondingly, we model this additional decay with the $\Gamma_{N}$ which is expected to be contributed by the enhanced decay channel due to the superradiant emission. Consequently, we find the fit value for this collective decay rate as $30 \mathrm{MHz}$. With varying the control laser intensity,

\section{Characterization}\label{section:characterization}
\subsection{Field intensity dependency}
\textcolor{red}{The figure depicting the pair rate and triplet rate with respect to the pump intensity will be updated.}
\subsection{Optical depth dependency}
\textcolor{red}{The figure depicting the pair rate and triplet rate concerning the optical depth will be updated.}
The characteristics of superradiance display qualitative distinctions from conventional spontaneous emission and stimulated emission, notably featuring directional emission and fast emission time inversely proportional to the number of participating atoms. As cited in Ref.\cite{malcuitTransitionSuperfluorescenceAmplified1987}, all atoms participate in the macroscopic dipole moment in scenarios where dephasing effects are negligible. Therefore, the peak intensity scales quadratically with the number of atoms. As depicted in the Fig. the peak correlated pair detection rate within a coincidence window increases quadratically with the optical density, while the triplet photon detection rate, which is expected to stem from the initial multiple photon pairs, also increases quadratically. 

 

and the superradiant emission is characterized by a time $\tau_{\mathrm{c}}$ as
\begin{equation}
\tau_{\mathrm{c}} = \frac{8\pi \tau_{\mathrm{sp}}}{3\rho \lambda^{2}L} \simeq \frac{4}{\mathrm{OD}}\tau_{\mathrm{sp}},
\end{equation}
where $\tau_{\mathrm{sp}}$ represents the spontaneous decay time of a single atom, $\rho$ denotes the atomic number density, and $L$ signifies the length of the atomic medium. Here, we rewrite the characteristic time in terms of the optical depth $\mathrm{OD}$, defined as $\mathrm{OD} = \rho \sigma L$, where $\sigma$ denotes the cross-section. The last approximation in the above equation is made under the assumption of zero detuning and a low saturation parameter, a condition that the Stokes photons satisfy. It manifests that once the optical depth surpasses 4, the characteristic time of superradiance becomes shorter than that of spontaneous emission. 

According to Ref.\cite{srivathsanNarrowBandSource2013}, for a transform-limited spectrum, there is a match between the inferred bandwidth from coherence time and the real photon bandwidth. The collective enhanced decay can be modeled as 
\begin{equation}
\Gamma = \Gamma_{0}(1+\alpha\mathrm{OD})
\end{equation}
% It is also noteworthy that the measurement of superradiance in a disordered atomic system does not require an extremely dense atomic density. In a dilute atomic cloud, superradiance still plays a role in the emission mechanism. 

% The diagonalization of the decay matrix gives the eigenvalue 
% \begin{equation}
%     \Gamma_{N}= \frac{\overline{\Gamma}}{2}(\alpha \mathrm{OD} + 1)
% \end{equation}
% under the eigenstate which is the completely symmetric state exhibiting the collective effects. \cite{porrasCollectiveGenerationQuantum2008} states the state density of the initial state is  
% \begin{equation}
%     \rho = \sum_{n} (1-\varepsilon)^{n}|\phi_{n}\rangle \langle \phi_{n}| + \tilde{\rho}
% \end{equation}
% where $n$ is the number of the excitation, that is the pair number; $|\phi_{n}\rangle$ represents the photon pair state or multiple pair state. $\varepsilon = 1/ (1+\alpha \mathrm{OD})$ It shows that with the multiple-pair events, the initial density is not a pure state anymore 

% The existence of a photon pair enhances the conditional probability that another photon pair is present in the system. This will result in a sudden enhancement of the coherence $\langle \sigma_{13}\hat{\sigma}_{24}\rangle$ by a factor of $1+ P$ over its steady-state value \cite{changSinglephotonTransistorUsing2007}

\section{Autocorrelation function}\label{section:autocorrelation}
\subsection{Second-order correlation function}
After tracing over one of the modes in Eq. \ref{eq:two_mode_squeezing_state}, the remaining mode is in a thermal state \cite{mollowQuantumTheoryParametric1967b}. The autocorrelation function on both sides shows the thermal light property as $g^{(2)}_{a,a} (0) \approx 2$. It is in accord with the fact that in a parametric process from an ensemble of emitters, both sides without triggering obey the thermal state statistics, and the loss in the channels does not change this property. 

According to Siegert relation, under the condition of the atomic emitter number being large and the electric field obeying Gaussian statistics, the second-order correlation is related to the electric field correlation $g^{(1)}(\tau)$, further the optical spectrum $S(\omega)$. Therefore, we can extract frequency linewidth and coherence time from second-order correlation via 
\begin{equation}
g^{(2)}(\tau) = 1 +\beta|g^{(1)}(\tau)|^{2}; S(\omega) =\int g^{(1)}(\tau) e^{i\omega \tau} d\tau\ ,
\end{equation}
in which $\beta$ is the scale parameter representing the purity of the spatial mode and polarization mode. 
As Fig, the fit of $g_{\text{s-s}}^{(2)}(\omega)$ gives a full-width at half maximum (FHWM) of $\tilde{g}_{\text{s-s}}^{(2)}(\omega) = 2\pi \times $ MHz, corresponding to a coherence time of ns. While the anti-Stokes second-order correlation exhibits a FHWM of $\tilde{g}_{\text{s-s}}^{(2)}(\omega) = 2\pi \times $ MHz, corresponding to a coherence time of ns. These coherence properties show the differences compared to the scattering observation that a coherence time near to the lifetime of energy level is observed although the first pumping field is also executed in low saturation limit $s\approx8\times 10^{-3}$ and single-scattering regime $\text{OD}_{\text{eff}} = \text{OD}/(1+4\Delta^{2}/\Gamma^{2})<1$. On average, an incident photon is scattered at most once before escaping the cloud. The broadening decay is expected to be contributed by the superradiant emission. As the decay rate of atoms is enhanced, the single-scattering approximation is not valid.   

\begin{figure}
    \centering
    \includegraphics{auto_correlation.pdf}
    \caption{Caption}
    \label{fig:enter-label}
\end{figure}

\subsection{Heralding second-order correlation function}
The heralding second-order correlation function of the anti-Stokes photon is defined as the autocorrelation of the anti-Stokes photon conditioned on observing a Stokes photon as 
\begin{equation}
g^{(2)}_{\text{h-as-as}}(t_{1}, t_{2}|t_{s}) = \frac{\langle \hat{a}^{\dagger}_{as}(t_{1})\hat{a}^{\dagger}_{as}(t_{2})\hat{a}_{as}(t_{2})\hat{a}_{as}(t_{2}) \rangle_{\text{ps}}}{\langle \hat{a}^{\dagger}_{as}(t_{1}) \hat{a}_{as}(t_{1})\rangle_{\text{ps}}\langle \hat{a}^{\dagger}_{as}(t_{2}) \hat{a}_{as}(t_{2})\rangle_{\text{ps}}}
\end{equation}
where $\langle : \rangle_{\text{ps}}$ denotes the average over the post-selected anti-Stokes photon events upon detecting a Stokes photon. A similar definition applies to Stokes photons. Since Stokes photons precede anti-Stokes photons in the correlated pair, post-selection involves Stokes photons in reverse time after detecting an anti-Stokes photon.
Previously, the correlated photon pairs from double-$\Lambda$ \cite{guoTIMEFREQUENCYENTANGLEMENTNARROWBAND2018} or cascaded energy levels \cite{srivathsanHeraldedSinglePhotons2015a} show anti-bunching in the heralding second-order correlation function in one side of correlated photon pairs. In contrast to the single-emitter system, there is no physical limitation for the atomic ensemble to impede the simultaneous emission of correlated photon pairs.  To achieve this, a weak and far-detuned pumping field is necessary to facilitate the four-wave mixing process in the single scattering regime. The effectiveness of this regime is characterized by an effective optical depth given by:
\begin{equation}
    \text{OD}_{\text{eff}} = \frac{\text{OD}}{1 + s + \frac{4\Delta^{2}}{\Gamma^{2}}} \ ,
\end{equation}
where $s$ denotes the saturation parameter defined as $s=I/I_{\text{sat}}$, $\Delta$ represents the pumping detuning from the atomic transition, and $\Gamma$ is the decay rate of the energy level. In this context, $\Gamma = 2\pi \times 6.07$ MHz for the D2 line of $\text{Rb}^{87}$. If $\text{OD}_{\text{eff}}<1$, it signifies that, on average, there is only less than one scattering event in the atomic medium for this detuned pumping field. Under this condition, the atomic ensemble can be approximately treated as a heralding single-photon source.

The correlated photon pairs generated in the paraxial regime show different photon statistics, even the single scattering condition is roughly satisfied as $\text{OD}_{\text{eff}}\approx$. As illustrated in Fig. (a), the $g^{(2)}_{\text{h-as-as}}(\tau)$ show super bunching feature as $g^{(2)}_{\text{h-as-as}}(\tau = 0) \approx 3.5$, while the inverse heralding $g^{(2)}_{\text{h-s-s}}(\tau)$ exhibits a super bunching with $g^{(2)}_{\text{h-as-as}}(\tau = 0) \approx 3.5$. The super bunching properties indicate its super-Poissionan photon statistic in the reduced density matrix on one side. 

\begin{figure}
    \centering
    \includegraphics{g2_simulaiton.pdf}
    \caption{The simulation of second-order coherence $g^{(2)}$ for squeezed thermal state (a) and squeezed Fock state (b) based on the expressions in Ref. \cite{kimPropertiesSqueezedNumber1989}. Here, $n$ represents the mean photon number for the thermal state or photon number of the Fock state, while $r$ is the squeezing parameter.}
    \label{fig:g2_simulation}
\end{figure}

Conventionally, the nonclassical nature is indicated by anti-bunching ($g^{(2)}(\tau)<0.5$), while bunching and super bunching can be explained using classical theories or have classical counterparts. However, the feature of super bunching has been theoretically and experimentally associated with the collective emission \cite{temnovPhotonStatisticsCooperative2009,auffevesFewEmittersCavity2011,hoiGenerationNonclassicalMicrowave2012,grujicRepulsivelyInducedPhoton2013,bhattiSuperbunchingNonclassicalityNew2015,pagelNonclassicalLightFew2015} or single mode squeezing \cite{kimPropertiesSqueezedNumber1989, 2000LoudonQuantum}. Microscopic descriptions of collective emission predict that the presence of the first emitted photon enhances the probability of a second emitted photon occurring. Superradiance modifies the probability statistic of the emitted photon state, resulting in superbunching in second-order correlation. Predicting specific maximal bunching is challenging as it depends on numerous parameters, including the number of atoms participating in the symmetric state, dephasing rate, and relative phasing between indistinguishable photons. On the other hand, as depicted in \ref{fig:g2_simulation}, the single mode squeezing in the phase quadrature also results in super bunching.  However, super bunching is not necessary and sufficient for the quantum effect, some classical implementations such as constructive interference of multiple indistinguishable two-photon paths can generate super bunching with the thermal light \cite{jenPositivePhasespacemethodSimulation2012}. 

\section{Multiple-photon event count}\label{section:photonevents}
\textcolor{red}{The data will be updated by a higher generation rate.}
With a measurement duration of about $8.5$ hours with settings of $\Delta = - 25 \mathrm{MHz}$ detuning and $150 \mathrm{\mu W}$ power (saturation parameter $s\approx8\times 10^{-3}$) for pumping laser and $5 \mathrm{mW}$ for coupling laser, we acquire an average Stokes photon count rate of $ (9.6 \pm 0.9) \times 10^4$ per second and an average anti-Stokes photon count rate of $(1.25 \pm 0.16) \times 10^{5}$ per second. Employing a coincidence window of $76ns$, we record a photon pair generation rate of $2280 \pm 380$ per second, with a heralding efficiency of approximately $1 \%$. The three-photon event rate comprises $11.6 \pm 3.4$ per second for that with one Stokes photon, and $9.0 \pm 2.4$ per second for that with one anti-Stokes photon. The four-photon event is recorded with $196 \pm 78$ per hour. The observed low heralding efficiency is attributed to losses in the collection and filter paths \cite{blayEffectsFilteringPurity2017}, with a total transmission of $15 \%$ on the Stokes side and $14\%$ on the anti-Stokes side. To enhance these count rates, improvements can be made by increasing the collection volume in the atomic ensemble, optimizing losses in the collection path and Etalon filters, and employing high-detection efficiency single-photon detectors. The current measurements remain comfortably below the saturation regime of our homemade single-photon detector, which has a maximal photon count rate of $\sim 5 \times 10^{5}$ per second. The single-photon detector exhibits a quantum efficiency of $40 \%$ for the Stokes side and $50 \%$ for the anti-Stokes side, enabling direct inference of the photon generation rate after the atomic ensemble within the solid angle. The deduced pure photon pair generation rate reaches $3.0 \times 10^{5}$ per second, with an expected heralding efficiency of $\eta_{s} \sim 20 \%$ for Stokes photons and $\eta_{as}\sim 17 \%$ for anti-Stokes photons. Simultaneously, a four-state generation rate has an expectation value of $1.2 \times 10^{4}$ per second constituting $0.8 \%$ of Stokes heralding efficiency and $0.7\%$ for anti-Stokes. Three-photon states persist at rates of $4.9 \times 10^{4}$ per second for one anti-Stokes and $5.6 \times 10^{4}$ per second for one Stokes. Attributing the generation of three-photon states to the photon reabsorption and diffusion of four-photon states in the atomic ensemble, we estimate the four-photon state generation rate to be $1.2 \times 10^{6}$ per second, alongside a pure photon pair generation rate of $7.7 \times 10^{5}$ per second with loss coefficients of approximately $0.69$ for Stokes photons and $0.67$ for anti-Stokes photons. This inference indicates that the rate of generating multiple pairs in the coincidence window is comparable to that of generating pure photon pairs. The bright correlated photon pair source is achieved at the expense of compromising the purity of the photon density matrix. Although the occurrence of multiple photons with $n>2$ pairs is not theoretically forbidden in collective emission, their detection probability is degenerated, roughly scaled with $\eta^{n}$. Moreover, the configuration of four single-photon detectors in our experiment lack the capability to discern the photon states with $n>2$. 
\begin{figure}[b]
\includegraphics{fourphoton.pdf}
\caption{ \label{fig:fourphoton} (a)  The green histogram illustrates the four-fold photon event counts within a $76$ ns coincidence window over an observation period of $8.5$ hours. Conversely, the red histogram depicts four-fold photon counts within a $40$ ns window, predominantly within the first peak of the two-photon wave function. The dotted line represents a theoretical estimation of approximately $3.2$ accidental four-fold photon events per $2$ 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 $t_{\mathrm{coi}}=76$ ns is the duration of the coincidence window. This calculation yields an accidental detection rate of approximately $14$ per hour. The grey histogram presents an example of accidental four-fold photon event counts, obtained by averaging data from five sets of post-selected four-photon events whose selection window is distant from the coincidence window. (b) This 2D histogram illustrates the 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.}
%These counts conform to the joint probability of two individual variables.
%It appears, albeit tentatively, that the joint-probability $P_{4}(\tau_{\mathrm{s-as}}, \tau^{\prime}_{\mathrm{s-as}})$ takes the form of the product of the individual probabilities of independent photon pairs, i.e., $P_{2}(\tau_{\mathrm{s-as}}) P_{2}(\tau^{\prime}_{\mathrm{s-as}})$ where $P_{2}(\tau_{\mathrm{s-as}})$ is proportional to the biphoton second-order cross-correlation $G_{\mathrm{as-s}}^{(2)}(\tau)$.}

\end{figure}
Here, the definition of a four-photon state implies the simultaneous presence of double Stokes and anti-Stokes photon pairs within the same coincidence window. As Fig. (a) shows, the four-photon state counts in the double HBT setup exhibit an enhancement by a factor of 14 compared to the accidental four-photon counts within the same coincidence window. The occurrence of a four-fold photon event can be seen as a non-strict representation of the fourth-order autocorrelation function, denoted as $g^{(4)}(\tau)$. The higher relative coincidence counts in the $n$th order compared to the thermal background provide evidence for the existence of a nonlinear multiphoton process \cite{spasibkoMultiphotonEffectsEnhanced2017}. This observation suggests a connection with significant photon-number fluctuations, indicating that collective emission not only modulates a fast decay channel from symmetric atomic states but also enhances the probability of generating multiple photon pairs. The identification of four-photon events underscores that, aside from the single-photon superradiant state in the atomic ensemble \cite{duanLongdistanceQuantumCommunication2001,deoliveiraSinglephotonSuperradianceCold2014, piovellaSinglePhotonSuperradianceSubradiance2023} , the collective atomic state can also be mapped into a multiple bosonic excitation state \cite{porrasCollectiveGenerationQuantum2008} denoted as $|\Psi_{\mathrm{at}}\rangle$ (without normalization), 
\begin{equation}\label{eq:multiple_excitation_state}
|\Psi_{\mathrm{at}}\rangle = \frac{1}{\sqrt{n \!}}(\tilde{b}^{\dagger})^{n} |0_{\mathrm{at}}\rangle 
\end{equation}
Here, $\tilde{b}^{\dagger}$ is the creation operator of a bosonic mode, representing a non-Hermitian Holstein-Primakoff transformation of the collective atomic operator $\tilde{\sigma}_{12} =\sum{i}^{N}A_{i}\hat{\sigma}^{(i)}_{12}$. The single-atom spin operator $\hat{\sigma}^{(i)}_{12} = |1\rangle\langle2|$ corresponds to the transition between states $|1\rangle$ and $|2\rangle$ for a single atom located at position $i$ within a total of $N$ atoms participating in this symmetric atomic state. The ground state of the collective atomic operator, where all atoms remain in the ground state $|1\rangle$, is denoted as $|0_{\mathrm{at}}\rangle$. The correlation of a pair of Stokes and anti-Stokes photons is established through a single bosonic excitation, and consequently, multiple bosonic excitations correspond to the correlated emission of multiple pairs of Stokes and anti-Stokes photons. Despite the clustering of these pairs in the coincidence windows, the factorization of Eq.\ref{eq:multiple_excitation_state} in the product of two single excitation states does not guarantee the entanglement between these pairs. This is partially illustrated in Fig.\ref{fig:fourphoton} (b) and (c), where the four-fold photon counts suggest that, except for the relative delay between Stokes and anti-Stokes photon, there is no compelling evidence supporting the existence of time correlation between these multiple pairs. 

%This observation aligns with the observations in the heralding self $g^{(2)}_{\mathrm{s/as}}(\tau)$, where interference fringes are detected.


\section{Conclusion}
In conclusion, we investigated the correlated photon generation from atomic ensemble in the paraxial regime, a bright correlated photon pairs through superradiant emission are observed. We explore the properties of correlated photons using the second-order correlation function and four-fold photon events, and they exhibit distinct photon statistics when compared to both spontaneous emission and spontaneous FWM. Correlated photons tend to cluster within the coincidence window, leading to the generation of multiple photon pairs or photon clusters. In cases where the single-scattering approximation is applicable to scattering processes, it is noted that superradiant FWM does not adhere to single-excitation conditions, as evidenced by the observation of super bunching in the heralding function $g^{(2)}(\tau)$. The presence of four-fold photon events indicates a higher-order nonlinear process in which the FWM process occurs multiple times, suggesting that enhancing the nonlinearity can be achieved through the collective effect of atoms. Meanwhile, it is suggested that the single-pair generation regime may necessitate a greater detuning from atomic energy levels. However, aside from the Stokes anti-stokes time-frequency correlation, the characteristics of correlated photons in this paper cannot verify sufficiently the correlation between multiple photon pairs within the cluster. This mechanism introduces a novel approach to generating bright correlated photons, offering potential applications such as utilizing them as photonic clusters in quantum computers, enhancing sensitivity in quantum metrology, improving visibility in ghost imaging \cite{alleviSuperthermalLightImaging2017}, and implementing photon number filters \cite{changSinglephotonTransistorUsing2007} to alter the statistics of the photon state.

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