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\begin{document}
\title{Bright correlated photon pairs based on double-lambda scheme in a cold atomic ensemble}
\author{Yifan~Li}
\author{Chang Hoong Chow}
\author{Boon Long Ng}
\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 demonstrate a bright narrowband correlated photon pair source in a cold atomic $^{87}\text{Rb}$ ensemble using the double-$\Lambda$ four-wave mixing. The time-frequency correlated photon pairs are generated in the twin Stokes and anti-Stokes modes in the presence of two pump fields. We have determined that the initial correlated photon pair rate can reach up to approximately $1.7\times 10^{6}$ per second with heralding efficiencies around $37\%$ and $27\%$ for twin modes within around $6 \times 10^{-6}$ solid angle in the paraxial regime. We have examined the characteristics of this photon pair source in relation to the intensity and detuning of the pump field and the optical depth of the atomic cloud. These correlated photon pairs display distinct properties that existing theoretical models cannot fully interpret. 

We demonstrate a bright narrowband correlated photon-pair source using the double-$\Lambda$ four-wave mixing in a cold atomic $^{87}\text{Rb}$ ensemble. The time-frequency-correlated photon pairs are generated in the twin Stokes and anti-Stokes modes in the presence of two pump fields. We have determined that the correlated photon pair generation rate can reach $10^{6}$ per second within a spatial mode with a waist of $175 \mathrm{\ \mu m}$ in the paraxial region. By investigating the influence of pump field intensity, detuning, and atomic cloud optical depth, we have characterized the properties of this photon pair source. These findings highlight the need for further theoretical development to fully comprehend the distinct properties exhibited by these correlated photon pairs.

% within a around $6 \times 10^{-6}$ solid angle in the paraxial regime

% The correlated photon pairs exhibit unique properties, indicating that they are influenced by the collective emission effects within the atomic ensemble.

\end{abstract}
\maketitle
\section{Introduction}\label{section:into}
Correlated photons constitute a valuable quantum resource with diverse applications, particularly within the domains of quantum communication \cite{bennettQuantumCryptographyPublic2014, xuSecureQuantumKey2020}, quantum computation \cite{RevModPhys.79.135} and quantum metrology \cite{RevModPhys.90.035005}. Correlated photon pairs are conventionally generated through spontaneous parametric downconversion (SPDC) processes in nonlinear crystals \cite{PhysRevA.64.023802, doi:10.1126/science.1127676} or spontaneous four-wave mixing (FWM) in atomic media comprising both hot atomic gases \cite{lukinQuantumNoiseCorrelations1999,shuSubnaturallinewidthBiphotonsDopplerbroadened2016,parkSpectralTemporalBiphoton2020} and cold atomic ensembles \cite{kuzmichGenerationNonclassicalPhoton2003, kolchinGenerationNarrowBandwidthPaired2006, srivathsanNarrowBandSource2013}. In contrast to SPDC, temporally correlated photon pairs derived from cold atomic ensembles exhibit narrow bandwidth characteristics inherited from the intrinsic atomic natural linewidth. This property makes them favorable for direct interactions with atoms. Moreover, these photon pairs possess an adjustable coherence time \cite{duSubnaturalLinewidthBiphotons2008,duTemporallyShapingBiphoton2009,zhaoShapingBiphotonTemporal2015}, thereby offering potential advantages for long-distance quantum communication.
The parametric FWM process can resort to various energy level configurations, an approach relies on cascaded emission originating from cascade-shaped energy levels \cite{srivathsanNarrowBandSource2013, gulatiPolarizationEntanglementQuantum2015, cereCharacterizationPhotonpairSource2018}. The alternative protocols employ the double-$\Lambda$ scheme \cite{balicGenerationPairedPhotons2005, wenBiphotonGenerationTwolevel2007, DduNarrowbandBiphotonGeneration2008} to generate correlated photon pairs. The existence of electromagnetically-induced transparency (EIT) \cite{PhysRevLett.64.1107}  within one of the $\Lambda$ energy levels benefits the photon pair generation, as it can establish a spectrally transparent window that significantly reduces the absorption loss of resonant photons traveling through the atomic ensemble. Furthermore, the slow light effect induced by EIT allows for the modulation of the coherence time of correlated photon pairs \cite{duSubnaturalLinewidthBiphotons2008, zhaoPhotonPairsCoherence2014}. A series of theoretical investigations \cite{raymondooiCorrelationPhotonPairs2007,kolchinElectromagneticallyinducedtransparencybasedPairedPhoton2007, DduNarrowbandBiphotonGeneration2008,jiangQuantumLangevinTheory2023} has emerged to describe the two-photon wave function of correlated photon pairs from such SFWM. These studies systematically categorize two-photon wave functions in both the oscillation regime and the group delay regime, which are consistent with experimental observations \cite{balicGenerationPairedPhotons2005,zhaoPhotonPairsCoherence2014, zhaoNarrowbandBiphotonGeneration2016,meiEinsteinPodolskyRosenEnergyTimeEntanglement2020}.

In this work, we present a bright correlated photon source from a $^{87}\text{Rb}$ cold atomic ensemble based on the double-$\Lambda$ four-wave mixing parametric process. This source is implemented within the paraxial regime ($\sim 1^{\circ}$) of continuous-wave pump fields within a small collective spatial mode of $175 \mathrm{\ \mu m}$ waist. We recorded a high pair generation rate, which is compatible with the previous correlated photon pair source in the cascaded FWM protocol \cite{cereCharacterizationPhotonpairSource2018} and in a similar double-$\Lambda$ scheme \cite{shiuObservationHighlyCorrelated2023}. Considering the geometric symmetry of our counterpropagating pump configurations, small collection solid angle, and the subpar performance of the spectral filter systems used in our experimental setup, it is reasonable to anticipate a higher photon pair generation rate by optimizing the collection setup and improving filter efficiencies. In our experiment, we observed distinct properties in the characteristics of the correlated photon pair source. These features, such as a fast decay envelope in the two-photon wave function and dependencies on optical depth (OD), can be tentatively attributed to collective emission effects, also known as superradiance \cite{dickeCoherenceSpontaneousRadiation1954, grivet1964quantum}. Some analogous collective behaviors have also been reported in the nonlinear process in the atomic ensemble, such as the ``read" process of the Duan-Lukin-Cirac-Zoller (DLCZ) protocol \cite{duanLongdistanceQuantumCommunication2001, kuzmichGenerationNonclassicalPhoton2003} and the parametric process within cascaded four-wave mixing \cite{chaneliereQuantumTelecommunicationBased2006, jenPositivePhasespacemethodSimulation2012, srivathsanNarrowBandSource2013, gulatiPolarizationEntanglementQuantum2015, jenSuperradiantCascadeEmissions2015}. 

This paper is structured as follows. Sec.\ref{section:setup} details the experimental protocol and setup for measuring the time-correlated photon pairs. Sec.\ref{section:wavefunction} reviews theoretical interpretations for the two-photon wave function based on existing theoretical results and represents an example of photon pair characteristics under specific parameters. Then in Sec.\ref{section:pump}, Sec.\ref{section:od}, and Sec.\ref{section:detuning}, we characterize the correlated photon pair source with respect to pump field intensity, optical depth (OD), and pump field detuning, respectively, within certain experimental parameter ranges. Finally, we summarize in Sec.\ref{section:conclusion}.

\section{Experimental Setup}\label{section:setup}
\begin{figure}
    \centering
     \includegraphics[width=\linewidth]{pair_source_setup.pdf}
    \caption{\label{fig:setup}(a) The double-$\Lambda$ atomic energy levels involved in the four-wave mixing process. (b) Schematic representation of the experimental setup. The pumping beam and coupling beam counterpropagate along the longest axis of a cigar-shaped atomic ensemble, with a collection path angled by approximately $1$ degrees.}
\end{figure}

The experimental protocol involves double$\Lambda$ atomic energy levels driven by the ``pumping'' field and the ``coupling'' field, denoted as $\omega_{p}$ and $\omega_{c}$, respectively. As shown in Fig.\ref{fig:setup} (a), the pumping laser is detuned by $\Delta_{p}$ from the $|5S_{1/2}, F=1\rangle\leftrightarrow |5P_{3/2}, F=2\rangle$ transition and the coupling laser is initially resonant to the $|5S_{1/2}, F=2\rangle \leftrightarrow |5P_{1/2}, F=2\rangle$ transition with $\Delta_{c}=0$. The pumping field is right circularly polarized ($\sigma^{+}$) while the coupling field is left circularly polarized ($\sigma^{-}$). In the experimental setup, depicted in Fig.\ref{fig:setup} (b), a backward-propagation configuration is implemented, wherein the coupling beam and pumping beam with a waist of roughly $850 \mathrm{\ \mu m}$ counter-propagate collinearly. This arrangement minimizes residual Doppler effects and facilitates spatial discrimination of the generated photons from the pump fields. The experimental setup employs an elongated cloud of $^{87}\mathrm{Rb}$ atoms, with a length of approximately $ 5\mathrm{\ mm}$ and a transverse cross section of $2\times 4\mathrm{\ mm}$. The control sequence consists of a $10 \ \mathrm{ms}$ cooling stage followed by a $400 \ \mathrm{\mu s}$ state transfer stage and a $1\ \mathrm{ms}$ measurement stage. After the state preparation, the majority of atoms are prepared in the $|5\text{S}_{1/2}, F=1\rangle$ state.

The collection path diverges from the pumping and coupling beam by a small angle of roughly $1$ degree with a focused Gaussian mode with an estimated waist of around $175\ \mathrm{\mu m}$, which spatially overlaps with the pump area in the atomic ensemble. This configuration corresponds to a collection volume of approximately $5\times 10^{-4}\mathrm{\ cm^{3}}$. Before coupling into the single-mode fibers, the collected photons are filtered by polarization filters consisting of a quatre-wave plate and a polarizing beam splitter. The angles of quatre-wave plates are chosen to maximize the photon pair rate. Subsequently, the collected photons undergo spectral filterings via temperature-controlled fused-silica Etalons. These filters serve to eliminate scattering photons from the pump fields and transmit photons centered at target frequencies $\omega_{s}$ and $\omega_{as}$. The Etalon for Stokes photon has a bandwidth of $275\ \mathrm{\ MHz}$ and an extinction ratio of $34\mathrm{\ dB}$ at the pumping frequency. Conversely, the Etalon for the anti-Stokes photon displays a bandwidth of $345 \mathrm{\ MHz}$ and an extinction ratio of $40\mathrm{\ dB}$ at the control field frequency. The total collection efficiencies (total transmission ratio) are characterized as around $17 \%$ for Stokes photons and $16 \%$ for anti-Stokes photons. 
In our measurement setup, we use two Excelitas single-photon detectors. These detectors have expected detection efficiencies of approximately $70 \%$ for the Stokes channel and $50 \%$ for the anti-Stokes channel. To measure auto-correlation functions, we modify the setup to include a 50:50 fiber beam splitter. This splitter directs the light into two paths, each leading to a single-photon detector. The photon event signals from these detectors are then recorded using a timestamp card with a timing resolution of 2 ns.

\section{Two-photon wave function}\label{section:wavefunction}
The correlated photon pairs from this FWM process have been theoretically modeled using the Heisenberg-Langevin method \cite{kolchinElectromagneticallyinducedtransparencybasedPairedPhoton2007, jiangQuantumLangevinTheory2023} and perturbation theory \cite{DduNarrowbandBiphotonGeneration2008,duSubnaturalLinewidthBiphotons2008}. The perturbation theory method provides a macroscopic interpretation of the correlated pair generation mechanism in the nonlinear process, neglecting the effect of the higher terms of the Hamiltonian. For the sake of simplicity, here we present the theoretical result using perturbation theory \cite{DduNarrowbandBiphotonGeneration2008}. The biphoton function $\Psi(\tau = t_{as}-t_{s})$ between correlated Stokes and anti-Stokes photons is a convolution of the nonlinear response $\tilde{\kappa}(\tau)$ and the phase matching $\tilde{\Phi}(\tau)$ in the time domain as 
\begin{equation}\label{eq:wave_function_time}
\Psi(\tau) = \tilde{\kappa}(\tau) \star \tilde{\Phi}(\tau) \ .
\end{equation}
These two functions are the Fourier transform of nonlinear coupling coefficient $\kappa(\omega_{as})$ and longitudinal detuning function $\Phi(\omega_{as})$ in the frequency domain as 
\begin{equation}
\begin{aligned}
\Psi (\tau) &= \frac{L}{2\pi}\int d\omega_{as} \kappa(\omega_{as})\Phi (\omega_{as})e^{-\omega_{as}\tau} \ ,  \\
\Phi (\omega_{as}) &= \text{sinc} (\frac{\Delta k L}{2})e^{i(k_{as}+k_{s})L/2}\ , \\
\kappa (\omega_{as})& = \frac{-i\sqrt{\omega_{as}\omega_{s}}}{2} \chi^{(3)}E_{p}E_{c} \ ,
\end{aligned}
\end{equation}
% \kappa(\omega_{as}) &= \frac{-i\sqrt{\omega_{as}\omega_{s}}E_{p}E_{c} \cdot N \mu_{13}\mu_{32}\mu_{24}\mu_{41}/(\varepsilon_{0}\hbar^{3})}{2(\Delta_{p}+i\gamma_{14})[|\Omega_{c}|^{2}-4(\omega+i\gamma_{13})(\omega+i\gamma_{12})]}
where $\Delta k =(\vec{k}_{as}+ \vec{k}_{s}) -(\vec{k}_{p} + \vec{k}_{c})$ is the phase mismatching. $\kappa(\omega_{as})$ represents the effect of third-order nonlinear susceptibility $\chi^{(3)}$ in the double-$\Lambda$ atomic energy levels, while $\Phi (\omega_{as})$ denotes the phase-matching bandwidth, which will be affected by the linear susceptibility in atomic media. The nonlinear coupling coefficient predicts two resonances separated by effective coupling Rabi frequency $\Omega_{e}$ in the atomic spectrum. The effective coupling Rabi frequency is defined as $\Omega_{e} = \sqrt{|\Omega_{c}|^{2}-(\gamma_{13}-\gamma_{12})^{2}}$, where $\gamma_{12}$ is the dephasing rate of $|1\rangle \leftrightarrow |2\rangle$ and $\gamma_{13}$ is the decay rate of $|1\rangle \leftrightarrow |3\rangle$. These resonances have a linewidth of $2\gamma_{e}=(\gamma_{12}+\gamma_{13})/2$. Additionally, the linear susceptibilities $\chi_{s}$ and $\chi_{as}$ vary the dispersion relations of the generated photons in the atomic media. Resonant coupling field $\Omega_{c}$ generates a transparency EIT window for the anti-Stokes photons and modulates its group velocity due to the slow-light effect. Therefore, depending on which spectrum dominates, such correlated photon pair source is classified into the damped Rabi oscillation regime and the group delay regime. When the group delay time $\tau_{g} = L/v_{g} \approx (2\gamma_{13}/|\Omega_{c}|^{2})\text{OD}$ is smaller than the coherence time $\tau_{\text{e}} = 1/\gamma_{e}$, the phase-matching bandwidth is constant as $\Phi(\omega)=1$. The two-photon wave function is determined by the nonlinear coupling function as 
\begin{equation}\label{eq:nonlinear_wavefunction}
\begin{aligned}
\tilde{\kappa}(\tau) &= B e^{-\gamma_{e}\tau}\sin (\frac{\Omega_{e}\tau}{2})\Theta(\tau)e^{-i\omega_{as}\tau}\ , \\
B &=-i\frac{N\mu_{13}\mu_{32}\mu_{24}\mu_{41}\sqrt{\omega_{as}\omega_{s}}}{4c\varepsilon_{0}\hbar^{3}\Omega_{e}(\Delta_{p}+i\gamma_{14})} \ .
\end{aligned}
\end{equation} 
Here, $\Theta(\tau)$ is the step function starting from $\tau=0$, $B$ represents the nonlinear strength, $N$ denotes the number of atoms, $\mu_{jk}$ denotes the dipole moment for the atomic transition from states $|j\rangle$ to $|k\rangle$, and $\gamma_{jk}$ represents the decay rate for corresponding atomic transition. When the group delay time is larger than the coherence time $\tau_{e}$, the group-delay bandwidth dominates in the biphoton spectrum as 
\begin{equation}\label{eq:phase_matching}
\Phi(\omega)\simeq \text{sinc}\left(\frac{\omega L}{2 V_{g}} +i\frac{\alpha L}{2}\right)\exp(i\frac{\omega L}{2V_{g}}-\frac{\alpha L}{2}) \ , 
\end{equation}
where $\alpha$ represents the EIT loss and $V_{g}$ denotes the group velocity for anti-Stokes photon. This phase-matching bandwidth leads to a rectangle function delayed by $\tau_{g}$ group delay function in $\tilde{\Phi}(\tau)$ in the time domain. 

\section{Cross-correlation}
\begin{figure*} % this float will be placed at top of p. 2
  \includegraphics[width=\textwidth]{g2_pump_dependence}
  \caption{\label{fig:g2_pump_dependence} (a)-(d) illustrate the corrected normalized cross-correlation function $g^{(2)}_{\text{as-s}}(\tau)$ under different power of coupling beams with the same power of pumping beams $P_{p}\simeq 500 \mathrm{\ \mu W}$ and $\mathrm{OD}=30$. The dashed lines are corresponding fits using the waveform model as $\exp(-\Gamma_{e}t)(1-\cos(\Omega_{e}\tau))\Theta(\tau)$ from the two-photon wave function in the damped oscillation regime. The fitted Rabi frequencies $\Omega_{e}$ and effective decay rate $\Gamma_{e}$ are denoted in terms of the natural linewidth of $|5P_{1/2},  F=2\rangle$ as $\Gamma = 2\pi \times 5.746 \text{\ MHz}$. The correction is performed using the transmission values  $\eta_{s} =0.17, \eta_{as}=0.16$, and detection efficiencies $\eta_{\text{d-s}}=70 \%, \eta_{\text{d-as}}=50 \%$.}
\end{figure*}

\subsection{Normalized cross-correlation function}
The two-photon wave function $\Psi(\tau)$ determines the Glauber's second-order correlation function between Stokes and anti-Stokes photons as
\begin{equation}\label{eq:cross_correlation}
\begin{aligned}
G_{\text{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. This cross-correlation function can be obtained from the photon coincidence counting between twin modes. Furthermore, the normalized cross-section function is given as 
\begin{equation}
g_{\text{as-s}}^{(2)} (\tau) =  \frac{G_{\text{as-s}}^{(2)}(\tau)}{R_{s}R_{as}} \ . 
\end{equation}
This function signifies the strong time correlation for twin photons as a Stokes photon precedes the presence of an anti-Stokes photon within the coherence time. Fig.\ref{fig:g2_pump_dependence} (a)-(d) illustrates the normalized cross-correlation functions $g^{(2)}_{\text{as-s}}(\tau)$ corrected by the accidental background caused by the imperfections in the collection and detection setup (refer to Appendix.\ref{app:correction}). In the damped Rabi oscillation regime, the theories reviewed in Sec.\ref{section:wavefunction} predict the cross-correlation function $g^{(2)}_{\text{as-s}}(\tau)$ has a decay envelope as  $e^{-\Gamma_{e}\tau}\simeq e^{-2\gamma_{13}\tau}$. However, in an example shown in Fig.\ref{fig:g2_pump_dependence} (a), the waveform of $g^{(2)}_{\text{as-s}}(\tau)$ is characterized by a distinct decay envelope feature with an enhanced decay rate as $\Gamma_{e}\simeq 3.1 \Gamma$. Conversely, the experimental parameters corresponding to Fig.\ref{fig:g2_pump_dependence} (d) predict a group delay about $\tau_{g} \simeq 91 \mathrm{\ ns}$ for the propagation of anti-Stokes photon in a $\mathrm{OD}=30$ atomic ensemble. While it is anticipated that the waveform would be delayed, but the observed waveform does not match the expected time convolution of this group delay time. 

\subsection{Coincidence count and pair detection}
In this section, we exemplify the characteristics of correlated photon pairs measured under specific experimental conditions, These conditions were chosen to balance the trade-off between photon pair rate and noise, rather than simply maximizing the pair rate. Furthermore, the initial state of the photon state is inferred by accounting for the effects of transmission ratios and detection efficiencies within our experimental setup.

% Under the experimental parameters $P_{p}\simeq 1 \mathrm{\ mW}$, $P_{c}\simeq 8.5 \mathrm{\  mW}$, $\Delta_{p}= -50\mathrm{\ MHz}$ and $\Delta_{c}=0$, we record Stokes photons at an instantaneous rate of $ (5.1 \pm 0.1) \times 10^{5} \mathrm{\ s^{-1}}$, and anti-Stokes photons at the rate of $ (5.0 \pm 0.1) \times 10^{5} \mathrm{\ s^{-1}}$. Within a coincidence window of $22 \mathrm{ns}$, the correlated photon pair rate $R_{p}^{m}$, defined as the average detection rate of correlated photons during the measurement time, is $(1.65  \pm 0.05) \times 10^{4} \mathrm{\ s^{-1}}$, with heralding efficiencies $2.6 \%$ for Stokes channel and $2.7 \%$ for anti-Stokes. The instantaneous coincidence rate $C(\tau)$, represents the coincidence probability in one of the twin modes and is linked to the correlated photon counts $N_{c}(\tau)$ via the relation $N_{c}(\tau) = C^{2}(\tau)\delta t T_{m}$, where $\delta t$ is the time bin and $T_{m}$ is the total measurement time.  With the minimal $2 \mathrm{\ ns}$ time resolution, the measured instantaneous coincidence rate $C^{m}(\tau)$ reaches a maximal value of $ 1.2 \times 10^{6} \mathrm{\ s^{-1}}$. The correlated-to-accidental ratio (CAR) is measured at around $2.4$ within the coincidence window. The quantum properties of correlated photon pairs are typically confirmed by demonstrating the violation of the Cauchy–Schwarz inequality:
% \begin{equation}
% {(g^{(2)}_{\text{as-s}}(\tau))^{2}}/{g^{(2)}_{\text{as}(\tau)}g^{(2)}_{\text{s}(\tau)}} \leq 1 \ .
% \end{equation}
% For Stokes and anti-Stokes photons, the auto-correlation functions $g^{(2)}_{\text{as/s}(\tau)}$ are approximately equal to 2 due to their thermal statistics after tracing out the corresponding twin mode. Therefore, the violation of the Cauchy–Schwarz inequality is related to the normalized cross-correlation peaks $g^{(2)}_{\text{as-s}}$. Here we observe $g^{(2)}_{\text{as-s}}$ has a measured maximum of $(6.5 \pm 0.1)$, indicating that it violates the Cauchy–Schwarz inequality by a factor of $10$. 

% Given the transmission ratio $\eta_{t}^{s} \simeq 0.17$ for Stokes channels and $\eta_{t}^{as} \simeq 0.16$ for the anti-Stokes channel, and the corresponding detection efficiencies $70 \%$ and $50 \%$, respectively, we can infer the initial photon state for the current collection setup. Using the relation $R_{p} = R_{p}^{m}/(\eta_{s}\eta_{as})$, the initial photon state has a pair rate of around $R_{p} \simeq 1.7 \times 10^{6} \mathrm{\ s^{-1}}$. The heralding efficiencies are about $37\%$ for the Stokes channel and $27\%$ for anti-Stokes channels. Based on the corrected normalized cross-correlation, we predict that the Cauchy–Schwarz inequality is violated by a factor of $26$. This correction method accounts for contributions from dark counts, laser leakage, and unpaired photons due to optical losses in the collection channels. The accidental photon counts, attributed to the dark counts of detectors and laser leakages are evaluated as $R_{s}^{d}=3000 \mathrm{\ s^{-1}}$ for Stokes channel, and $ R_{as}^{d}= 7000 \mathrm{\ s^{-1}}$ for anti-Stokes channel. 

Under the experimental pumping field powers as $P_{p}\simeq 300 \mathrm{\ \mu W}$, $P_{c}\simeq 8.5 \mathrm{\  mW}$, and field detuning settings as $\Delta_{p}=+25\mathrm{\ MHz}$ and $\Delta_{c}=0$, we record Stokes photons at an instantaneous rate of $ (5.1 \pm 0.2) \times 10^{5} \mathrm{\ s^{-1}}$, and anti-Stokes photons at the rate of $ (6.1 \pm 0.2) \times 10^{5} \mathrm{\ s^{-1}}$. Within a coincidence window of $22 \mathrm{\ ns}$, the measured correlated photon pair rate $R_{p}^{m}$, defined as the average detection rate of correlated photons during the measurement time, is $(2.05  \pm 0.07) \times 10^{4} \mathrm{\ s^{-1}}$, with heralding efficiency $4.0 \%$ for Stokes channel and $3.3 \%$ for anti-Stokes. 
%The instantaneous coincidence rate $C(\tau)$, represents the coincidence probability in one of the twin modes and is linked to the correlated photon counts $N_{c}(\tau)$ via the relation $N_{c}(\tau) = C^{2}(\tau)\delta t T_{m}$, where $\delta t$ is the time bin and $T_{m}$ is the total measurement time.  With the minimal $2 \mathrm{\ ns}$ time bin, the measured instantaneous coincidence rate $C^{m}(\tau)$ reaches a maximal value of $ 1.5 \times 10^{6} \mathrm{\ s^{-1}}$. 
The correlated-to-accidental ratio (CAR) is measured at around $3$ within this coincidence window. Typically, the quantum properties of correlated photon pairs are confirmed by demonstrating the violation of the Cauchy–Schwarz inequality:
\begin{equation}
{(g^{(2)}_{\text{as-s}}(\tau))^{2}}/{g^{(2)}_{\text{as}}(\tau)g^{(2)}_{\text{s}}(\tau)} \leq 1 \ .
\end{equation}
For Stokes and anti-Stokes photons, the auto-correlation functions at zero-delay are given as $g^{(2)}_{\text{s-s}}(0) = 1.95 \pm 0.12$ and $g^{(2)}_{\text{as-as}}(0) = 1.99 \pm 0.11$, indicating residual thermal statistics in one mode after tracing out the twinned mode. Therefore, the violation of the Cauchy–Schwarz inequality is related to the normalized cross-correlation peak as $g^{(2)}_{\text{as-s}}$. $g^{(2)}_{\text{as-s}}$ has a measured maximum of $(7.5 \pm 0.1)$, indicating that it violates the Cauchy–Schwarz inequality by a factor of $14$. 

One of the noise sources in photon coincidence counting measurements arises from optical losses in the subpar collection and filtering system, resulting in expected total transmission ratios of $\eta_{t}^{s} \simeq 0.17$ for Stokes channels and $\eta_{t}^{as} \simeq 0.16$ for the anti-Stokes channel. By inclusively accounting for these losses and the detection efficiencies of single-photon detectors, we can correct the noise caused by these imperfections in the collection and measurement devices. This allows us to infer the initial photon state within the same spatial modes. Using the relation $R_{p} = R_{p}^{m}/(\eta_{s}\eta_{as})$, the initial photon state has a pair rate of around $2.5 \times 10^{6} \mathrm{\ s^{-1}}$. The heralding efficiencies are about $58\%$ for the Stokes channel and $32\%$ for anti-Stokes channels. Based on the corrected normalized cross-correlation function, we anticipate that the Cauchy–Schwarz inequality is violated by a factor of 65. This correction for cross-correlation refers to the method detailed in Appendix.\ref{app:correction} which corrects the contributions from dark counts, laser leakage, and unpaired photons in accidental coincidence counting background. Among these, the accidental photon counts, attributed to the dark counts of detectors and laser leakages are evaluated as $R_{s}^{d}\simeq 5300 \mathrm{\ s^{-1}}$ for Stokes channel, and $ R_{as}^{d}\simeq 3100 \mathrm{\ s^{-1}}$ for anti-Stokes channel. 

\section{Pump intensity dependence}\label{section:pump}
Conventionally, we use laser power to represent the corresponding field intensity levels. The pump field intensities affect the two-photon wave function and the strength of the nonlinear process, as indicated by Eq.\ref{eq:nonlinear_wavefunction}. Additionally, the pump field intensities also influence the phase-matching condition and the propagation of correlated photon pairs through linear susceptibility, whereby the group velocity is modulated in the atomic medium as described in Eq.\ref{eq:phase_matching}.
%\subsection{Pumping field intensity}
Fig.\ref{fig:vs_780pumping_detuning} (a) illustrates that the correlated photon pair mechanism operates at both negative and positive detuning sides of the energy level $|5P_{3/2}, F=2\rangle$. At a specific detuning, as the pumping field power increases, the correlated photon pair rate initially rises and then saturates. Further increases in the pumping field power cannot significantly enhance the photon pair rate; instead, they lead to an increase in the single-photon rates $R_{s/as}$, due to the incoherent scattering photons. Consequently, the increased noise photons reduce the heralding efficiencies in both modes, as shown in Fig.\ref{fig:vs_780pumping_detuning} (c) and (d), and increase the accidental photon pair rate $R_{s}R_{as}t_{\text{coi}}$, defined within a coincidence window  $t_{\text{coi}}$. The $g^{(2)}_{\text{as-s}}$ maximum value and CAR in Fig.\ref{fig:g2_vs_780pumping_detuning} also suggest that the increases of the pumping field power undermine the quantum features by reducing the violation factor of Cauchy–Schwarz inequality and minimize the purity of the obtained correlated photon pairs. 

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{v_780pumping_detuning_largerange.pdf}
    \caption{\label{fig:vs_780pumping_detuning}Panels (a)-(b) show the characteristics as joint functions of the pumping field power and its detuning, using a resonant coupling field with a power of $P_{c}\simeq 8.5 \mathrm{\ mW}$. The characteristics displayed are (a) the correlated photon pair rate, (b) the single photon detection rate in the Stokes channel, (c) the heralding efficiency for Stokes photons, and (d) the heralding efficiency for anti-Stokes photons.}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{g2_v_780pumping_detuning_large_detuning.pdf}
    \caption{\label{fig:g2_vs_780pumping_detuning}(a) The $g^{(2)}_{\text{as-s}}$ maximum value, only corrected by dark counts. (b) CAR within a coincidence window of $22 \mathrm{\ ns}$, corresponding to the dominant peak in the cross-correlation function.}
\end{figure}

When varying the power of the coupling beam within the range of $1 \mathrm{\ mW}$ to $8.5  \mathrm{\ mW}$ for different settings of pumping field power. Fig.\ref{fig:rate_vs_795pumping} suggests that correlated photon pair rate saturates once the coupling field power reaches a certain threshold. The heralding efficiency in the Stokes channel benefits from this increase, while the anti-Stokes heralding efficiency suffers due to increased incoherent scattering photon. Fig.\ref{fig:g2_pump_dependence} (a)-(d) also depict the evolution of cross-correlation functions with varying coupling field power. As the coupling field power increases, the total correlated photon pair rate remains roughly unchanged, while the coherence time becomes narrow, corresponding to the increase of $g^{(2)}_{\text{as-s}}$ maximum in Fig.\ref{fig:rate_vs_795pumping} (d). 
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{rate_vs_795pumping.pdf}
    \caption{\label{fig:rate_vs_795pumping}Figures (a)-(d) illustrate the correlated photon pair rate, heralding efficiency in Stokes channels, heralding efficiency in anti-Stokes channel, and the maximum value of $g^{(2)}_{\text{as-s}}$ as functions of coupling field power under varying pumping field powers.}
\end{figure}

% In summary, Fig.\ref{fig:sweep_pumping} depicts the scanning in a specific power range both for the pumping and coupling fields. It demonstrates that achieving the optimal correlated photon pair source necessitates power matching for both pump fields. 
% \begin{figure}
%     \centering
%     \includegraphics[width=\linewidth]{sweep_pumping.pdf}
%     \caption{\label{fig:sweep_pumping} Figure (a) shows a 2D histogram of the correlated photon pair rate as a joint function of the powers of both pump fields. Figures (b) and (c) display the corresponding heralding efficiencies in the Stokes and anti-Stokes channels, respectively.}
% \end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{bandwidth.pdf}
    \caption{\label{fig:bandwidth}Figure (a) illustrates the bandwidth as a function of pumping field power, while maintaining the coupling field power at a constant level of $8.5\mathrm{\ mW}$. Figure (b) displays the bandwidths with respect to the coupling field power. The solid lines are linear fits for the data while the dashed black line represents the theoretical group-delay bandwidth $\Delta\omega_{g}$ with parameters $\gamma_{13} = 2\pi\times 3 \mathrm{\ MHz}$, $\text{OD} = 30$. These bandwidths are fitted from the respective auto-correlation coincidence measurements.}
\end{figure}

The bandwidth observations depicted in Fig.\ref{fig:bandwidth} demonstrate a consistent linear relationship between the spectral bandwidth and the coupling field power. As the coupling field power increases, the bandwidths vary from sub-natural linewidth to an enhanced bandwidth that is broader than the natural linewidth. When the power of the coupling beam is kept constant, the spectral bandwidths remain stable, irrespective of the pumping field power. This dependence on the coupling field power follows a similar trend to the theoretical prediction of the group delay bandwidth, given by $\Delta\omega_{g}\simeq 0.88/ \tau_{g}$, where the group delay time $\tau_{g}$ is characterized by $\tau_{g} \simeq 2\gamma_{13}\text{OD}/|\Omega_{c}|^2$. 


\section{Optical depth dependence}\label{section:od}
Optical depth (OD) provides an alternative representation for the number of atoms involved, as it is directly proportional to the atom number within the interaction volume. It is defined as $\text{OD} = n\sigma L$, where $n$ is the atomic density, $\sigma$ represents the cross-section and $L$ indicates the length of the atomic ensemble (volume length). Some theoretical investigations \cite{DduNarrowbandBiphotonGeneration2008} predict that the correlated photon pair rate and the atom number follow a quadratic relation in the damped Rabi oscillation regime of a double-$\Lambda$ FWM process. Fig.\ref{fig:vs_od} illustrates the characteristics of correlated photon pair rate within a $22\mathrm{\ ns}$ coincidence window with respect to OD, showing that the photon pair rate scales faster than the quadratically. This scaling can be interpreted as a result of superradiance, where the superradiant emission intensity scales quadratically with the number of atoms. Considering the contribution of this effect, the correlated photon pair rate is proportional to the fourth power of the optical depth (OD). The fit with $\text{OD}^{4}$ in Fig.\ref{fig:vs_od} (a) shows good agreement with the measured data, thereby supporting this superradiance interpretation to a certain degree.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{vs_od.pdf}
    \caption{\label{fig:vs_od}Figure (a) illustrates the relationship between correlated photon pairs and optical depth (OD), with a dashed line depicting a fitted curve proportional to $\text{OD}^{4}$. In Figure (b), the decay rate is derived from two-photon cross-correlation functions. For oscillations excluding the initial peak, the decay rate (green plot) closely aligns with the natural linewidth.  while when considering the first oscillation, an OD dependence in the decay rate occurs. The red dashed line is fitted to the data using the model $\Gamma_{0} (1+\alpha \text{OD})$. The OD is measured by fitting the transmission profile at various detuning frequencies of the weak probe light. Following a cooling stage, the initial optical depth for the $|F=2\rangle$ hyperfine state can reach approximately 80. Subsequently, after the transfer stage to the $|F=1\rangle$ hyperfine state, the remaining maximal OD for this ground state is measured at around 30.}
\end{figure}
We have observed that the two-photon wave function exhibits a decay envelope with an enhanced decay rate compared to the natural linewidth $\Gamma_{0}$ I the damped Rabi oscillation regime, as illustrated in Fig.\ref{fig:g2_pump_dependence}. We investigate these decay rates, fitted from the normalized cross-correlation function $g^{(2)}_{\text{as-s}}(\tau)$ as a function of the OD in Fig.\ref{fig:vs_od} (b).  The fitted decay rates derived from the residual oscillations, excluding the first peak, remain constant around the $\Gamma_{0}$ even as the OD varies. In contrast, the fitted decay rates from the first oscillations increase with the rise of OD, displaying a linear relationship with respect to the OD. This dependence aligns well with the prediction of an enhanced decay rate in the superradiance emission interpretation as given as \begin{equation}\label{eq:gamma_vs_od}
\Gamma = \Gamma_{0}(1+\alpha\mathrm{OD}) \ . 
\end{equation}
The slope parameter $\alpha$ is fitted to be $0.15$. The observation of two decay rates in different regimes indicates the existence of two distinct decay channels in the correlated photon pair mechanism: one decay rate close to the natural linewidth of the energy level and another enhanced decay rate, which  not accounted for in the two-photon wave function as described in Eq.\ref{eq:nonlinear_wavefunction}. 


\section{Detuning dependence}\label{section:detuning}
The detuning of the pump fields is a critical factor influencing the generation of correlated photon pairs from a nonlinear process. As shown in Fig.\ref{fig:vs_780pumping_detuning} (a), approaching the atomic resonance of the pumping field does not intrinsically enhance the nonlinear process. Instead, the photon pair generation rates exhibit maximum at specific detunings on both positive and negative sides of the resonance. Specifically, the correlated photon pair rate is higher for positive detuning of the pumping field than negative detuning. This asymmetry results in a higher heralding efficiency in the Stokes channel for positive detuning. In contrast, the incoherent scattering photon rate follows a Lorentizan profile centered at the resonance. 

% \begin{figure}
%     \centering
%     \includegraphics[width=\linewidth]{vs_780detuning.pdf}
%     \caption{\label{fig:vs_780detuning}(a) The correlated photon pair rate (black, left axis) and the total photon rate in the Stokes and anti-Stokes channels (right axis) as functions of the pumping field detuning, $\Delta_{p}$. Figure (b)-(d) exhibit the heralding efficiency in each channels, normalized $g^{(2)}{\text{as-s}}$ peak and the coincidence-to-accidental ratio (CAR), respectively. The experiment was conducted with a pumping power of $P_{p} \simeq 1.2\ \mathrm{mW}$, a coupling power of $P_{c} \simeq 9\ \mathrm{mW}$, and a resonant coupling field ($\Delta_{c} = 0$). The CAR was determined within a fixed coincidence window of $22\ \mathrm{ns}$, corresponding to the dominant first peak in the cross-correlation function.}
% \end{figure}


\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{vs_795detuning.pdf}
    \caption{\label{fig:vs_795detuning}(a)-(c) investigate the correlated photon pair rate, single count rate, and the peak of corrected $g^{(2)}_{\text{as-s}}$ as functions of the coupling field detuning $\Delta_{c}$, with the constant pump field powers as $P_{p} \approx  700 \mathrm{\ \mu W}$ and $P_{c} \approx 8.5 \mathrm{\ m W}$ and a fixed pumping detuning $\Delta_{p} = -50 \text{\ MHz}$. The photon pair rate is determined by measuring the coincidence counts above the accidental background counts, without imposing a fixed coherence time. The similar trends in photon counts observed in both Stokes and anti-Stokes channels suggest that the frequency-dependent attenuation due to deviation from the center frequency of the anti-Stokes etalon can be considered negligible within this frequency range. The cross-correlation functions $g^{(2)}_{\text{as-s}}$ are only corrected by accounting for dark count rates of about $1300 \mathrm{\  s^{-1}}$ for the Stokes channel and approximately $100 \mathrm{\  s^{-1}}$ for the anti-Stokes channel.}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{g2_with_coupling_detuning.pdf}
    \caption{\label{fig:g2_with_coupling_detuning}Panel (a) depicts the waveform of the cross-correlation functions $g^{(2)}_{\text{as-s}}(\tau)$ at the coupling field detuning of $\Delta_{c}= -40 \mathrm{\ MHz}$, while (b) shows the waveform under $\Delta_{c}= 40 \mathrm{\ MHz}$. The experimental parameters remain consistent with the data in  Fig. \ref{fig:vs_795detuning}. }
\end{figure}
Exploring the dependence on the coupling field detuning, Fig. \ref{fig:vs_795detuning} examines how the detuning of the coupling field $\Delta_{c}$ affects the correlated photon pair generation, while the pumping field is fixed to $\Delta_{p} = -50 \text{\ MHz}$. Within a detuning range from $-40 \mathrm{\ MHz}$ to $40 \mathrm{\ MHz}$, the correlated photon pair generation mechanism remains active, with the photon pair rate reaching its maximum at a specific negative detuning (around $-25  \mathrm{\ MHz}$) rather than at the resonance, Excluding the Dopper broadening, which can be neglected in the counterpropagating pump configuration, the observed redshift and broadening in the photon pair rate may be attributed to the Lorentz-Lorenz shift \cite{RevModPhys.64.619, Javanainen:16} or the collective Lamb shift in a dense medium \cite{scullyCollectiveLambShift2009, keaveneyCooperativeLambShift2012, roofObservationSinglePhotonSuperradiance2016}. Meanwhile, the waveform of $g^{(2)}_{\text{as-s}}(\tau)$ with large coupling field detuning exhibits distinct features compared to the scenario with a resonant coupling field. As illustrated in Fig. \ref{fig:g2_with_coupling_detuning} (a), when the coupling field detuning $\Delta_{c}$ is set to $- 40 \mathrm{\ MHz}$, the heights of successive peaks become more pronounced, accompanied by a slower damped envelope. Furthermore, it is noted that the oscillations in the waveform do not extend to the background level of $g^{(2)}_{\text{as-s}} = 1$. This indicates that one frequency component in the Rabi splitting of the anti-Stokes mode dominates the four-wave mixing process rather than both frequencies exhibiting equal intensities in the case of the resonant coupling field. Our investigations into the detunings of the pumping field and coupling field reveal that correlated photon pairs can be generated efficiently over a relatively large detuning range in this double-$\Lambda$ FWM nonlinear process. 


\section{Conclusion}\label{section:conclusion}
In conclusion, we have successfully developed a bright correlated photon source in the paraxial regime of counter-propagating pump fields from a cold $^{87}\text{Rb}$ atomic ensemble. The correlated photon pairs are generated from a four-wave mixing parametric process based on the double-$\Lambda$ energy scheme. This quantum source offers narrowband spectral characteristics and long coherence times, making it highly suitable for applications in atom-repeater quantum communication. The observed spectral properties, influenced predominantly by the coupling field intensity rather than the pump field intensity, highlight the role of linear susceptibility in determining bandwidth constraints. Although the dependence of spectral properties on the coupling field intensity shows a similar trend to the theoretical group-delay bandwidth, the predicted large group delay is not observed in the two-photon wave function, indicating experimental deviations from the previous theoretical models. Moreover, we also observe some features such as enhanced decay rate in the two-photon wave function and the dependence on OD, these properties suggest the involvement of the superradiance effects within the atomic ensemble for the generated photon pairs. To further validate the attributions of the
superradiance in the nonlinear process and fully understand our observations, a comprehensive theoretical interpretation involving the collective effects of atomic emitters is necessary. This may help fully explain the behavior of this bright correlated photon pair source.



\appendix
\section{Instantaneous coincidence rate}
The average correlated photon pair generation rate $R_{p}$ is calculated from the total correlated photon pair number $N_{p}$ detected over a measurement time $T_{m}$ as follows: 
\begin{equation}
    R_{p} = \frac{N_{p}}{T_{m}}
\end{equation}
From practical measurement counting, the measured pair detection rate is obtained from the coincidence counting between twin channels
as 
\begin{equation}
    R_{p}^{m} = \frac{\sum_{n=0}^{n_{c}}N_{c}^{m}n \delta t}{T_{m}} 
\end{equation}
where $n_{c}$ represents the number of coherence time bins, $\delta t$ is the bin time, $N_{c}$ denotes the number of coincidence counts above the background level within the time bin $n\delta t$ over a total measurement duration of $T_{m}$. Considering the total detection efficiencies $\eta_{1}$ for channel 1 and $\eta_{2}$ (including the transmission in the system and detection efficiencies of single-photon detectors), the measured pair detection rate can be expressed as
\begin{equation}
    R_{p}^{m} = \eta_{1}\eta_{2} R_{p}
\end{equation}

Using the model for correlated photon pair generation from Ref.\cite{}, the instantaneous coincidence rate $C(\tau)$ in one channel can also characterize the correlated photon source \cite{}. The quantity is linked to the measured correlated photon number as 
\begin{equation}
    N_{c}(\tau) = C(\tau)^{2}\delta T_{m} \ . 
\end{equation}
Therefore,  the normalized cross-correlation function $g^{(2)}(\tau)$ is expressed as  \cite{}
\begin{equation}
    g^{(2)}(\tau) = 1 + \frac{C^{2}(\tau)}{R_{1}R_{2}} \ ,
\end{equation}
where $R_{1}$ and $R_{2}$ represent the single photon rate in twin channels, respectively. This indicates that the normalized cross-correlation function can be obtained by normalizing the coincidence counts between the twin channels against the average background counts in the far-coherence time regime. Accordingly, the measured instantaneous coincidence rate, $C^{m}(\tau)$ is given by
\begin{equation}
    C^{(m)}(\tau) = \sqrt{\frac{M_{c}^{m}(\tau)}{\delta t T_{m}}} \ .
\end{equation}

\section{Coincidence histogram correction}\label{app:correction}
In this four-wave mixing process, the rate of correlated photon pairs, denoted as $R_{2}$, is proportional to the third-order susceptibility $\chi^{(3)}$. Concurrently, the linear susceptibility generates uncorrelated scattering photons in both Stokes and anti-Stokes modes, resulting in noise photon rates $R_{s}^{n}$ and $R_{as}^{n}$ respectively. In the experimental setup, these noise photon rates also account for the uncorrelated photons due to an imperfect collection system. Therefore the total initial photon rate as 
\begin{equation}
\begin{aligned}
R_{1} &= R_{p} + R_{1}^{n} \\ 
R_{2} &= R_{p} + R_{2}^{n} \ . \\ 
\end{aligned}
\end{equation}
These photons are detected with probabilities (total detection efficiencies) $\eta_{1}$ and $\eta_{2}$ in the twin channels. These probabilities account for the total transmission ratio and detection efficiencies of the single-photon detectors. Additionally, accidental counts arise from laser leakage and dark counts of detectors, independent of the detection efficiencies, denoted as $R_{1}^{d}$ for channel 1 and $R_{2}^{d}$ for channel 2. Hence, the measured count rates $R_{1}^{m}$ and $R_{2}^{m}$ are given by 
\begin{equation}
\begin{aligned}
R_{1}^{m} &= R_{p}\eta_{1}\eta_{2} + R_{p}\eta_{1}(1-\eta_{2}) + R_{1}^{n}\eta_{1} + R_{1}^{d} \\ 
R_{2}^{m} &= R_{p}\eta_{1}\eta_{2} + R_{p}\eta_{2}(1-\eta_{1}) + R_{2}^{n}\eta_{2} + R_{2}^{d} \ . \\ 
\end{aligned}
\end{equation}
In the measurement of coincidence counting histogram, only terms  $R_{p}\eta_{1}\eta_{2}$ in two channels result in the coincidence peak above the background counts, corresponding to the measured correlated photon pairs. The mutual counts between other terms contribute to the background counts. Given that the normalized cross-correlation $g^{(2)}(\tau)$ is linked to the practical photon counting number via 
\begin{equation}
\begin{aligned}
g^{(2)}(\tau) & =  1 + \frac{C^{2}(\tau)\delta t T_{m}}{R_{1}R_{2}\delta t T_{m}} =  1 + \frac{\eta_{1}\eta_{2}C^{2}(\tau)\delta t T_{m}}{\eta_{1}R_{1}^{n}\eta_{2}R_{2}^{n}\delta t T_{m}} \\
& = 1 + \frac{N_{p}^{m}(\tau)}{N_{bg}} 
\end{aligned}
\end{equation}
The valid background count per bin $N_{bg} =\eta_{1}R_{1}^{n}\eta_{2}R_{2}^{n}\delta t T_{m}$ forms a portion of the measured histogram background ($R_{1}^{m}R_{2}^{m}\delta t T_{m}$). Therefore, the coincidence histogram should be corrected by subtracting the following terms 
\begin{itemize}
    \item $R_{1}^{d}R_{2}^{m}\delta t T_{m}$ 
    \item $R_{p}\eta_{1}(1-\eta_{2}) R_{2}^{m}\delta t T_{m}$
    \item $R_{1}^{n}\eta_{1}R_{p}\eta_{2} + R_{1}^{n}\eta_{1}R_{2}^{d}$
    \item $R_{p}^{m} (R_{2}\eta_{2}(1-\eta_{1})+R_{2}^{n}\eta_{2} + R_{2}^{d}) $
\end{itemize}
After subtracting these terms from the coincidence histogram background, the residual background per bin corresponds to $R_{1}^{n}R_{2}^{n}\eta_{1}\eta_{2}\delta t T_{m}$. 


\bibliography{reference}

\end{document}


