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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{Justin Yu Xiang Peh}
\author{Chang Hoong Chow}
\author{Boon Long Ng}
\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 a bright source of narrowband, correlated photon pairs based on double-$\Lambda$ four-wave mixing in a cold $^{87}\mathrm{Rb}$ atomic ensemble. By driving the medium with counter-propagating pump fields and optimizing the collection geometry to the diffraction limit, we achieve a detection rate of $10^{4}-10^{5}$ counts per second (cps), corresponding to an intrinsic generation rate on the order of $10^{6}-10^{7}$ cps. We systematically investigate the dependence of photon-pair generation on the pump-field detunings and intensities. We characterize the dependence between pump parameters and the atomic response near resonance. Crucially, we identify detuning regimes where constructive interference in the nonlinear susceptibility significantly enhances pair generation, while group delay simultaneously enables tunable temporal coherence. These results provide a blueprint for engineering efficient nonlinear interactions. With its combination of high brightness and spectrally narrow bandwidth, this source shows potential applications for high-speed quantum networking and for efficient interfacing with atom-based quantum memories.

\end{abstract}
\maketitle

\section{Introduction}\label{section:into}
Correlated photons constitute a valuable quantum resource with diverse applications, particularly in 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 produced 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, including both hot atomic gases \cite{lukinQuantumNoiseCorrelations1999,shuSubnaturallinewidthBiphotonsDopplerbroadened2016,parkSpectralTemporalBiphoton2020} and cold atomic ensembles \cite{kuzmichGenerationNonclassicalPhoton2003, kolchinGenerationNarrowBandwidthPaired2006, srivathsanNarrowBandSource2013}. In contrast to SPDC, correlated photon pairs generated from cold atomic ensembles exhibit narrow bandwidth characteristics primarily determined by the atomic natural linewidth and further shaped by the linear susceptibilities of the atomic medium. This makes them highly suitable for direct atom-photon interfaces. Moreover, two-photon coherence properties can be tunable through modifying the linear susceptibility experienced by the traveling photons \cite{duSubnaturalLinewidthBiphotons2008,duTemporallyShapingBiphoton2009,zhaoShapingBiphotonTemporal2015}, thereby offering potential advantages for long-distance quantum communication.

The SFWM processes in atomic systems can be implemented using various energy level configurations. One approach utilizes cascaded emission from ladder-shaped energy levels~\cite{srivathsanNarrowBandSource2013, gulatiPolarizationEntanglementQuantum2015, cereCharacterizationPhotonpairSource2018}. Alternatively, the double-$\Lambda$ scheme~\cite{balicGenerationPairedPhotons2005, wenBiphotonGenerationTwolevel2007, DduNarrowbandBiphotonGeneration2008} generates correlated photon pairs assisted by electromagnetically-induced transparency (EIT)~\cite{PhysRevLett.64.1107}. The EIT in one of the $\Lambda$ levels opens a spectrally transparent window, reducing photon absorption and enabling modulation of the two-photon coherence time via the slow-light effect \cite{duSubnaturalLinewidthBiphotons2008, zhaoPhotonPairsCoherence2014}. Some theoretical studies \cite{raymondooiCorrelationPhotonPairs2007,kolchinElectromagneticallyinducedtransparencybasedPairedPhoton2007, DduNarrowbandBiphotonGeneration2008,jiangQuantumLangevinTheory2023} systematically analyzed the two-photon wavefunction in EIT-assisted four-wave mixing, identifying distinct regimes characterized by damped temporal oscillations and group delay. The damped temporal oscillations are interpreted as a direct consequence of Autler–Townes splitting (ATS), which arises from the separation of dressed states induced by a strong coupling field. In contrast, the group-delay regime arises from a steep dispersion gradient within a narrow transparency window induced by quantum interference. In this regime, the group velocity of photons generated via the $\chi^{(3)}$ nonlinear process is significantly reduced, thereby dominating the phase-matching contribution to the two-photon wavefunction~\cite{zhaoNarrowbandBiphotonGeneration2016, meiEinsteinPodolskyRosenEnergyTimeEntanglement2020}.

In this work, we realize a double-$\Lambda$ SFWM parametric process in an elongated $^{87}\text{Rb}$ cold atomic ensemble. A weak \textit{pumping} field, primarily determining the nonlinear strength below the saturation regime, counter-propagates with a strong \textit{coupling} field along the optically dense axis of the atomic cloud. Both beams have a waist of approximately $w_{p}\approx 830\mathrm{\ \mu m}$, and overlap with the dense atomic cloud, which has a length of about $5 \,\text{mm}$ and a transverse size of roughly $2\,\text{mm}$ in height and $4\,\text{mm}$ in width. In such a counter-propagating configuration, correlated photon pairs, which close the double-$\Lambda$ atomic transition, are spontaneously emitted in phase-matched opposite directions. The collection mode is implemented along the opposite direction in the near-axis regime, deviating by approximately $1^\circ$ from the pump fields. This angular offset facilitates spatial separation between the pump and the generated photons, thereby reducing the collection of scattered pump light. The collected spatial mode is a focused Gaussian beam with a waist of $w_{c} \approx 175\,\mathrm{\mu m}$ centered on the atomic ensemble. Its divergence angle, $\theta = \lambda/(\pi w_c) \approx 1.4\,\text{mrad}$, is comparable to the diffraction angle of the pump field, $\theta_D = \lambda/w_p \approx 0.9\,\text{mrad}$, ensuring that only a single diffraction-limited mode in the opposite direction is coupled into the single-mode fibers. This geometry preserves the spatial coherence of the collected photons across the interaction volume.

This SFWM operates primarily in the ATS regime as the two-photon wavefunction exhibits a damped Rabi waveform, while the measured bandwidth, defined as the inverse of the single-photon correlation time, shows good agreement in trend with the slow-light prediction, though with a systematic offset. Although the counter-propagating geometry is commonly considered Doppler-free, we still observe that both the two-photon and single-photon correlation times are shorter than the natural lifetime of the excited states. This feature has been reported in various experiments involving correlated photon pairs~\cite{chaneliereQuantumTelecommunicationBased2006, srivathsanNarrowBandSource2013, gulatiPolarizationEntanglementQuantum2015}, nonlinear processes in atomic ensembles~\cite{duanLongdistanceQuantumCommunication2001, kuzmichGenerationNonclassicalPhoton2003}, and has also been theoretically predicted in cascaded emission scenarios~\cite{jenPositivePhasespacemethodSimulation2012, jenSuperradiantCascadeEmissions2015}.

This paper is structured as follows. Section~\ref{section:setup} details the experimental protocol and setup for measuring the temporally correlated photon pairs, Sec.~\ref{section:wavefunction} reviews macroscopical interpretations for photon-pair generation. In Sec.~\ref{section:result}, we investigate the dependence of photon-pair generation on the intensity (power) and detuning of the two pump fields within the experimentally accessible parameter ranges. We also present the characteristics of near-optimal photon pairs achievable under specific conditions. Finally, Sec.~\ref{section:conclusion} summarizes the main findings and conclusions of this study.

\section{Experimental details}\label{section:setup}
\begin{figure}[t]
    \centering
    \includegraphics[width=\linewidth]{bright_source_single.pdf}
    \caption{\label{fig:setup}
        (a) Double-$\Lambda$ atomic energy levels involved in the four-wave mixing process. 
        (b) Schematic representation of the experimental setup. 
        The pumping and coupling beams counter-propagate along the longest axis of a cigar-shaped atomic ensemble, 
        with a collection path angled by approximately $1^{\circ}$. (c) Directional emission and collection mode. The pump–atom interaction defines a cylindrical volume. Emission is confined within a geometrical angle of $\theta_{G}=w_{p}/L\approx 0.165 \, \mathrm{rad} \,(9.5^{\circ})$ consisting of diffraction-limited modes with $\theta_{D} =\lambda/w_{p}\approx 0.9 \,\mathrm{mrad}$. The collection Gaussian mode (waist $175\,\mu\mathrm{m}$) has a divergence of $\theta \approx \lambda/(\pi w_c) \approx 1.4\,\mathrm{mrad}$. By alignment optimization, only a single diffraction-limited mode is coupled into the paired single-mode fibers.
    }
\end{figure}

As illustrated in Fig.~\ref{fig:setup} (a), the experimental protocol involves double-$\Lambda$ atomic energy levels driven by the \textit{pumping} laser (of frequency $\omega_{p}$), which is detuned by $\Delta_{p}$ from the $|5S_{1/2}, \,F=1\rangle\leftrightarrow |5P_{3/2},\, F=2\rangle$ transition; and a \textit{coupling} laser (of frequency $\omega_{c}$), which is initially resonant to the $|5S_{1/2},\, F=2\rangle \leftrightarrow |5P_{1/2},\, F=2\rangle$. A Stokes photon, addressing the decay transition $|4\rangle\rightarrow |2\rangle$ and an anti-Stokes photon, connecting the decay transition $|3\rangle\rightarrow |1\rangle$, are spontaneously generated from the vacuum state in the SFWM nonlinear process. The cold atomic cloud is sequentially prepared using a magneto-optical trap (MOT) of $^{87}$Rb atoms. Initialization into the $|5\text{S}_{1/2}, \, F=1\rangle$ ground state is achieved through a cooling stage lasting approximately $5$–$10\,\text{ms}$, followed by a $400\,\mathrm{\mu s}$ state-transfer stage that gradually transfers the population from the $F=2$  into $F=1$ ground state, a mixture of Zeeman sublevels. After these steps, we obtain a cold atomic ensemble with an optical depth of roughly 30. The optical depth, which effectively quantifies the number of atoms participating in the interaction with the optical field, is defined as $\text{OD} = n \sigma L$, where $n$ is the atomic density, $\sigma$ the resonant cross-section, and $L$ the interaction length between the atomic cloud and the pump field. Based on the estimated interaction volume of approximately $5 \times 10^{-4}\,\mathrm{cm^3}$ and the ideal cross-section $\sigma = 3\lambda^2 / 2\pi \approx 2.9 \times 10^{-13}\,\mathrm{m^2}$, the number of atoms contributing to the collected spatial mode is on the order of $10^7$. Consequently, the atomic cloud is released from the MOT, and the pump fields are applied during a $0.5\,\text{ms}$ measurement window, within which the generated photons are collected and detected.

Before fiber coupling, photons pass through polarization optics (quarter-wave
plate, half-wave plate, and PBS). To maintain stable fiber alignment,
polarization filters are fixed: the Stokes mode transmits right-hand
circularly polarized ($\sigma^+$) photons, and the anti-Stokes mode transmits
left-hand circularly polarized ($\sigma^-$), both referenced to their
respective pump directions at the atom side. These polarization bases are
calibrated with respect to the corresponding pump field in the same
propagation direction. The pump and coupling polarizations are optimized for
maximum pair generation, yielding the highest rate when the pump is near
$\sigma^-$ and the coupling field near $\sigma^+$. Collected photons are
spectrally filtered by temperature-stabilized fused-silica etalons whose
maximal transmission is centered at $\omega_s$ and $\omega_{as}$. The Stokes
etalon (275\,MHz bandwidth, 34\,dB extinction) and anti-Stokes etalon
(345\,MHz bandwidth, 40 dB extinction) effectively suppress pump and Rayleigh
scattering. The total transmission efficiencies, including spatial coupling
efficiency, etalon transmission, and fiber transmission, are approximately
$(21\pm 2)\%$ for Stokes photons and $(23\pm 5)\%$ for anti-Stokes
photons. After filtering, photons are re-coupled into fibers and detected by
Excelitas single-photon detectors ($\approx 60\%$ efficiency). Detection events are time-tagged by a custom device with 2 ns resolution.

\section{Theory}\label{section:wavefunction}
Correlated photon pairs generated via double-$\Lambda$ SFWM can be modeled using either the Heisenberg–Langevin formalism \cite{kolchinElectromagneticallyinducedtransparencybasedPairedPhoton2007, jiangQuantumLangevinTheory2023}, which treats the coupled dynamics of atomic operators and quantized fields, or perturbation theory \cite{DduNarrowbandBiphotonGeneration2008,duSubnaturalLinewidthBiphotons2008}, which describes biphoton generation as a first-order nonlinear process. The first-order perturbative solution to the time-evolution operator acting on the vacuum state yields the two-photon component of the output field. The corresponding biphoton wavefunction, determined by the nonlinear susceptibility $\chi^{(3)}$, phase-matching, and pump–coupling field profiles, represents the joint probability amplitude for detecting a Stokes and an anti-Stokes photon at given times. In the macroscopic picture~\cite{DduNarrowbandBiphotonGeneration2008}, the temporal wavefunction $\Psi(\tau=t_{as}-t_{s})$ can be written as
\begin{equation}
\Psi (\tau) = \frac{L}{2\pi}\int d\omega_{as} \kappa(\omega_{as})\Phi (\omega_{as})e^{-i\omega_{as}\tau} . 
\end{equation}
where $\kappa(\omega_{as})$ is the nonlinear coupling spectrum and $\Phi(\omega_{as})$ the longitudinal detuning function. $\kappa(\omega_{as})$ is governed by the third-order susceptibility $\chi^{(3)}(\delta_{as})$ as
\begin{equation}
\kappa (\omega_{as}) = \frac{-i\sqrt{\omega_{as}\omega_{s}}}{2} \chi^{(3)}E_{p}E_{c}\,,
\end{equation}
\begin{equation}
\chi^{(3)} =
\frac{N\mu_{13}\mu_{32}\mu_{24}\mu_{41}}{\varepsilon_0\hbar^3(\Delta_p+i\gamma_{14})}
\frac{1}{|\Omega_{c}|^{2}-(\delta_{as}+i\gamma_{12})(\delta_{as}+i\gamma_{13})}\,,
\end{equation}
where $\mu_{ij}$ demotes the electrical dipole matrix elements $\langle i|\mathbf{\hat{d}}|j\rangle$ between states $i$ and $j$. The nonlinear spectrum is centered at the resonance $\omega_{0as} = \omega_{31}$ ($\Delta_{c}=0$) with the detuning $\delta_{as}=\omega_{as}-\omega_{0as}$. The effective coupling Rabi frequency is given by  $\Omega_{e} = \sqrt{|\Omega_{c}|^{2}-(\gamma_{13}-\gamma_{12})^{2}}$ where $\gamma_{e}= (\gamma_{13}+\gamma_{12})/2$ is the effective dephasing rate and $\gamma_{13}$ and $\gamma_{12}$ are the decay rates between $|3\rangle \to |1\rangle$ and $|2\rangle\to |1\rangle$ transitions. The poles in the denominator give rise to a spectral doublet separated by $\Omega_e$, each with a linewidth of $2\gamma_e$. This splitting originates from the Autler–Townes effect~\cite{autlerStark1955} induced by the coupling field. Additionally, the phase-matching condition is expressed by the longitudinal detuning function ${\Phi}(\omega_{as})$:
\begin{equation}
\Phi (\omega_{as}) = \text{sinc} (\frac{\Delta k L}{2})e^{i(k_{as}+k_{s})L/2}\ ,
\end{equation}
where $\Delta k ={k}_{as}+ {k}_{s} -({k}_{p} + {k}_{c})$ is the longitudinal phase mismatch, $k_{i}$ denotes the wavevector of each field along the $+z$ axis (the propagation direction of the pumping field), and $L$ is the medium length. The exponential phase factor $e^{i({k}_{as}+{k}_{s})L/2}$ accounts for the accumulated phase of the Stokes and anti-Stokes fields, while the contributions from pump fields are neglected. In the strong coupling regime, the nonlinear spectral response plays a dominant role in shaping the two-photon wavefunction. The nonlinear spectral doublet induces interference between two decay pathways, producing quantum beating in the biphoton wavefunction, with a damped oscillatory envelope given by
\begin{equation}
    |\Psi(\tau)|^2  \propto e^{-2\gamma_{e}}\sin^{2}\left(\frac{\Omega_{e}\tau}{2}\right). 
\end{equation}
which can be displayed through 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 can be evaluated through the photon pair coincidence between two coupled modes. 

\section{Results}\label{section:result}
\subsection{Pumping field dependence}\label{section:pump}
\begin{figure}
    \centering
    \includegraphics[width=1\linewidth]{bright_pair_vs780pumping.pdf}
    \caption{(a) Correlated pair rate as a function of the pumping field detuning and power. The coupling field is kept on resonance with a power of approximately $8.5\mathrm{\, mW}$. (b) Horizontal cut of (a) at $P_{p}\approx 500 \mathrm{\,\mu W}$. Left: correlated pair rate versus pumping field detuning; Right: corresponding single-photon count rates for the Stokes and anti-Stokes modes. (c) Vertical cut of (a): correlated pair rate as a function of pumping file power for different detunings. }
    \label{fig:pair_vs_pumping}
\end{figure}

When the coupling field power is sufficiently large ($P_{c}\approx 8.5 \mathrm{\, mW}$), the four-wave-mixing strength is primarily determined by the pumping field intensity and its detuning from the atomic resonance. As shown in Fig.~\ref{fig:pair_vs_pumping}~(a) and (b), the single-photon count rates in both modes follow an approximate Lorentzian dependence on the detuning. In contrast, the correlated photon pair rate is suppressed at exact resonance and exhibits an asymmetric dependence on the detuning, with a maximum occurring on the blue-detuned side of the $|1\rangle \rightarrow |4\rangle$ transition. Near resonance, the nonlinear susceptibility $\chi^{(3)}$ is strongly suppressed by destructive interference with the linear response $\mathrm{Im}[\chi^{(1)}]$~\cite{fleischhauerElectromagnetically2005}. As a result, linear absorption dominates the medium\'s response: most of the pump power is dissipated into atomic excitations and re-emitted as incoherent fluorescence, rather than contributing to the nonlinear process. This competition with the linear susceptibility markedly suppresses the effective nonlinear susceptibility $\chi^{(3)}$ and hence the correlated photon-pair generation rate at resonance. The detuning-dependent asymmetry further reflects constructive interference of the SFWM channels on the blue-detuned side of the pumping field. Tentatively, this suggests that the geometry-induced phase mismatch may be partially compensated by the linear atomic dispersion, governed by the real part of the linear susceptibility $\mathrm{Re}[\chi^{(1)}]$, under specific detuning conditions. Consistently, the heralding efficiencies show a similar asymmetric dependence on pumping detuning and exhibit a dip at resonance.

At a fixed pumping detuning, the correlated photon-pair rate exhibits a saturation behavior analogous to spontaneous emission: it increases with pumping power and then approaches a saturation value. As shown in in Fig.~\ref{fig:pair_vs_pumping}(c), the solid curves are fits to 
${P_{p}}/({1+P_{p}/P_{\mathrm{eff,sat}}})$,
where $P_{\mathrm{eff,sat}}$ denotes the effective  saturation power for each detuning. This behavior indicates that, at high pumping power, the effective nonlinear susceptibility is ultimately constrained by the finite atom number, leading to the observed saturation in the pair-generation rate.

\subsection{Coupling field dependence}
\begin{figure} % this float will be placed at top of p. 2
  \includegraphics[width=\linewidth]{bright_g2_vs_795pump_power.eps}
  \caption{\label{fig:g2_pump_dependence} The normalized cross-correlation function $g^{(2)}_{\text{as-s}}(\tau)$ under different powers of coupling field with the pumping field power $P_{p}\approx 500 \mathrm{\,\mu W}$. 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 ATS regime.  }
\end{figure}
A notable feature of the system is the modulation of the two-photon wavefunction by the coupling field intensity. Fig.~\ref{fig:g2_pump_dependence} shows the broadening of the cross-correlation function (i.e., the two-photon wavefunction) and the simultaneous decrease of its peak value as the coupling-field power is varied, at a fixed pump power~$P_{p}\approx 500\,\mu\mathrm{W}$ and a detuning~$\Delta_{p}/2\pi = -50\,\mathrm{MHz}$. The broadening originates from the slow-light effect inside the EIT transparency window: the steep dispersion within this narrow spectral region introduces a group-velocity delay to the photons generated in the SFWM process. However, within the range of coupling powers that ensures efficient photon-pair generation, the resulting group delay remains modest and does not produce a pronounced square-shaped temporal response. Instead, the measured cross-correlation functions $g^{(2)}_{\text{as-s}}(\tau)$ continue to exhibit the damped oscillatory waveform characteristic of the ATS regime.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{coherence_time_vs_power_combined.eps}
    \caption{\label{fig:bandwidth}~Intensity autocorrelation functions of (a) the Stokes field and (b) the anti-Stokes field under varying coupling field powers, with the pumping field fixed at $P_{p}\approx 800\, \mathrm{\mu W}$. (c) Reciprocal of coherence time ($1/\tau_{c}$) in both modes as a function of pumping field power, with the coupling field fixed at $8.5\,\mathrm{mW}$. The grey dashed line indicates the D2-line lifetime of $^{87}$Rb ($26.2\,  \mathrm{ns}$); while the D1-line has a similar lifetime of $27.7\,  \mathrm{ns}$. (d) The reciprocal of coherence time ($1/\tau_{c}$) with respect to the coupling field power. Solid lines are linear fits of $1/\tau_{c}$, and the black dashed line shows the theoretical group-delay prediction $1/\tau_{g}$ with $\gamma_{13}=2\pi\times 3\,\mathrm{MHz}$ and $\mathrm{OD}=30$. All measurements are taken with the pumping field detuned by $\Delta_{p}/2\pi=-50\,\mathrm{MHz}$ and the coupling field on resonance.}
\end{figure}


The group-velocity delay in the third-order susceptibility $\chi^{(3)}$ influences both the two-photon correlation time of the joint modes and the single-photon coherence time in each mode. We extract the correlation time of either Stokes or anti-Stokes field from intensity autocorrelation measurement and define it as the full width at half maximum (FWHM) of the $g^{(2)}_{\mathrm{auto}}(\tau)$ function. This function is obtained via a Hanbury Brown–Twiss (HBT) measurement using a 50:50 fiber beam splitter and coincidence detection at the two output channels. The measured autocorrelation functions for both fields (see Fig.~\ref{fig:bandwidth}(a) and (b)) exhibit thermal statistics with a maximal value $g^{(2)}_{\mathrm{auto}}(0)\approx 2$, consistent with the predication of thermal mixed state after tracing out the partner mode from the two-mode squeezed vacuum state. This justifies the use of the Siegert relation, $g^{(2)}(\tau) = 1 + |g^{(1)}(\tau)|^{2}$, for a zero-mean, stationary Gaussian random optical field, allowing us to identify the first-order coherence time directly as the intensity-correlation time. To better display the dependence of the coherence time on optical powers, Figs.\ref{fig:bandwidth}~(c) and (d) show the reciprocal of the extracted coherence time, $1/\tau_{c}$, as a function of the pump power $P_{p}$ and coupling power $P_{c}$, respectively. When $P_{c}\approx 8.5\,\mathrm{mW}$ is held fixed, varying pumping field power does not noticeably modify the coherence times of the generated photons, which remain at $\tau_{c}\approx 10\,\mathrm{ns}$, shorter than the natural lifetime of the excited state involved in the SFWM process ($26.2\,\mathrm{ns}$ for the D2 line; $27.7\,\mathrm{ns}$ for the D1 line). By contrast, decreasing the coupling power $P_{c}$ leads to a linear reduction in $1/\tau_{c}$ for both fields. The slope follows the same trend as the reciprocal of the predicted group-velocity delay time $1/\tau_{g}$, with
\begin{equation}
\tau_{g}\simeq \frac{2\gamma_{13}\,\mathrm{OD}}{|\Omega_{c}|^{2}},    
\end{equation}
where $\gamma_{13}$ is the decay rate of state $|3\rangle$, $\mathrm{OD}$ is the optical depth, and $\Omega_{c}$ is the coupling field Rabi frequency, and its mapping to optical power is obtained from the damped oscillations observed in the cross-correlation function (see Fig.~\ref{fig:g2_pump_dependence}). In addition, the measured lineshapes of $g^{(2)}_{\mathrm{auto}}(\tau)$ do not follow a purely exponential (Lorentzian thermal) or Gaussian (Gaussian thermal) profile. At higher coupling powers ($\approx 8\,\mathrm{mW}$), the Stokes field exhibits oscillatory tails, while the anti-Stokes field displays a peak sharper than a Lorentzian. At low coupling power ($\approx1\,\mathrm{mW}$), both fields show visible jagged features at comparable time delays. These deviations likely arise from a mismatch between the intrinsic short coherence time of the generated photons and the extension induced by the group delay. Since the spectral bandwidth of the generated photons is comparable to the width of the EIT transparency window, the single-photon wave packet experiences higher-order dispersion rather than the approximately linear dispersion valid only within a narrow spectral region. As a result, the photons are generally not transform-limited, and their first-order coherence time does not directly correspond to the Fourier transform of their spectral bandwidth.

\begin{figure}
    \centering
\includegraphics[width=1\linewidth]{bright_pair_vs795coupling.eps}
    \caption{(a) Correlated photon pair rate, (b) the peak value of normalized cross-correlation function $g^{(2)}_{\text{as-s}}(\tau)$, (c) heralding efficiency in the Stokes mode, and (d) heralding efficiency in the anti-Stokes mode, all as functions of the coupling-field power for different pump powers. The pump detuning is fixed at $\Delta{p}/2\pi=-50\mathrm{\,MHz}$.}
    \label{fig:rate_vs_795pumping}
\end{figure}

Besides the coherence time, we also investigate how the coupling field intensity affects photon-pair generation by varying the coupling power from $1\,\mathrm{mW}$ to $8.5\,\mathrm{mW}$ under resonance conditions, as shown in Fig.~\ref{fig:rate_vs_795pumping}. Panel (a) shows that after saturation, further increasing the field power does not enhance photon-pair generation. Consistent with Fig.~\ref{fig:g2_pump_dependence}, the cross-correlation peak increases with the coupling field power. The nonclassicality of the correlated photon pairs is quantified by the violation of the Cauchy–Schwarz inequality:
\begin{equation}\label{eq:cauchy}
\frac{(g^{(2)}_{\text{as-s}}(\tau))^{2}}{g^{(2)}_{\text{as}}(\tau)g^{(2)}_{\text{s}}(\tau)} \leq 1 \,
\end{equation}
where the denominator is the product of the autocorrelation functions $g^{(2)}_{\text{as/s}}(\tau)$, which both reach a maximal value of around 2 at zero delay for our source. Therefore, the increase in the cross-correlation peak indicates an enhancement of the nonclassicality of the photon pairs. Figure~\ref{fig:rate_vs_795pumping}~(c) shows that the heralding efficiency in the Stokes channel benefits from the increased coupling power, reflecting the suppression of incoherent scattering in the Stokes mode for all pumping conditions. In contrast, the heralding efficiency in the anti-Stokes channel exhibits a different behavior: at low pumping powers, it remains roughly unchanged, whereas at high pumping powers, it also depends on the coupling field power, indicating an increase in the incoherent scattering contribution in the anti-Stokes mode.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{bright_pair_vs795detuning.eps}
    \caption{\label{fig:vs_795detuning}~(a) Correlated photon pair rates (left) and single-photon count rates in both modes (right). (b) The peak of normalized cross-correlation $g^{(2)}_{\mathrm{as\text{-}s}}(\tau)$ as a function of the coupling-field detuning $\Delta_{c}$, with the pumping-field detuning fixed at $\Delta_{p}/2\pi = -50\,\mathrm{MHz}$. The pumping and coupling powers are $P_{p} \approx 700\,\mu\mathrm{W}$ and $P_{c} \approx 8.5\,\mathrm{mW}$, respectively. (c)~Normalized cross-correlation functions $g^{(2)}_{\mathrm{as\text{-}s}}(\tau)$ for two coupling field detunings, $\Delta_{c}/2\pi = -40\,\mathrm{MHz}$ and $+40\,\mathrm{MHz}$, together with a reference trace at resonance.}
\end{figure}

It is widely recognized that the EIT induced by the coupling field enhances the strength of the double-$\Lambda$ SFWM process and thereby increases the photon-pair generation rate. Under this condition, the nonlinear susceptibility $\chi^{(3)}$ exhibits constructive interference: its on-resonance value exceeds the simple sum of the two split Lorentzian components~\cite{fleischhauerElectromagnetically2005}, and the linear absorption term $\mathrm{Im}[\chi^{(1)}]$ no longer introduces a destructive interference with the SFWM gain. Consistently, Fig.~\ref{fig:vs_795detuning}~(a) shows the correlated photon pair rate and the single-photon count rates of both modes as functions of the coupling field detuning near resonance. Unlike the behavior observed when scanning the pump detuning, where a dip appears due to the competition between $\mathrm{Im}[\chi^{(1)}]$ and $\chi^{(3)}$, the correlated photon pair rate here does not exhibit such a suppression at resonance. It indicates that the linear susceptibility does not destructively interfere with the nonlinear susceptibility. Instead, the photon pair rate displays a broadened and asymmetric profile with a clear preference for negative detuning: it reaches a maximum near $-25\,\mathrm{MHz}$ and decreases monotonically as the detuning is scanned through resonance toward $+40\,\mathrm{MHz}$. The single-count rates follow a similar trend. This asymmetry may originate from partial phase-mismatch compensation: since the anti-Stokes propagates opposite to the Stokes field, the enhancement occurs on the red-detuned side of the coupling field detuning. Furthermore, Fig.~\ref{fig:vs_795detuning}~(b) shows that the enhancement of the photon-pair generation is accompanied by a faster reduction of the peak value of the cross-correlation function $g^{(2)}_{\text{as-s}}(\tau)$ as the detuning far from the resonance at red-detuned regime, highlighting the trade-off between pair generation efficiency and its non-classicality. Due to limitations of the experimental configuration, we did not explore a wider detuning range. The temporal cross-correlation functions at the coupling field detunings $\Delta_{c}/2\pi=\pm 40\, \mathrm{MHz}$ are shown in Fig.~\ref{fig:vs_795detuning}~(c). In both cases, the damped envelope becomes slower, corresponding to a broadened two-photon temporal coherence time compared with the on-resonance condition. The strong coupling field produces an Autler–Townes doublet in the nonlinear susceptibility that is symmetric at resonance. At finite detuning, the dressed-state component closer to the atomic resonance begins to dominate, while the other component moves further away and weakens. This imbalance in the contributions of the two dressed states reduces the quantum beating amplitude in the temporal waveform.
\subsection{Optimal photon-pair generation}
In the following, we investigate optimal photon-pair generation, which represents a trade-off between brightness and noise, rather than relying solely on the pair rate as a figure of merit. We present the characteristics under representative experimental conditions and further infer the initial photon-pair generation rate based on the evaluation of collection efficiencies.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{brighsource_best_pair.eps}
    \caption{(a) Left: measured correlated photon pair rate; Right: inferred generation rate versus pump field power $P_{p}$ at a detuning of $\Delta_{p}/2\pi = +40\,\mathrm{MHz}$, with the coupling field on resonance at $P_{c}\approx 8.5\, \mathrm{mW}$. Inset: nset: normalized cross-correlation function $\mathrm{g^{(2)}_{\text{as-s}}(\tau)}$ under the condition indicated by the blue arrow. (c) Lower: measured heralding efficiencies in both modes; upper: inferred pair ratios for photons collected in fibers. (c) The peak value of the normalized cross-correlation function $\mathrm{g^{(2)}_{\text{as-s}}(\tau)}$. (d) Correlated-to-noise ratio (CAR) calculated within a $20\, \mathrm{ns}$ coincidence window. }
    \label{fig:best_pair_rate}
\end{figure}

Figure~\ref{fig:pair_vs_pumping} shows that a blue-detuned pumping field enhances the photon-pair generation. We further characterize the system under a pump detuning of $\Delta_{p}/2\pi = +40\,\mathrm{MHz}$, with the coupling field on resonance at $P_{c}\approx 8.5\, \mathrm{mW}$. Figure~\ref{fig:best_pair_rate} presents the photon pair rate as a function of pump power up to the saturation regime. Near the saturation point at $P_p \approx 750\,\mathrm{\mu W}$, we achieve a maximal correlated photon pair rate of $7.0(1)\times 10^{4}\,\mathrm{cps}$. As shown in the inset of Fig.~\ref{fig:best_pair_rate}~(a), the photon pairs are primarily concentrated in the first peak within a $20\,\mathrm{ns}$ coincidence window, while the oscillatory tails in the cross-correlation are suppressed compared to the characterization at the red-detuned regime. Within this window, the corresponding photon pair rate is $5.8(1)\times 10^{4}\,\mathrm{cps}$, and it is associated with a heralding efficiency $\eta_{s}$ of $5.5(1)\, \%$ for Stokes mode (single count rate of $1.05(3) \times 10^{6}\,\mathrm{cps}$), and a heralding efficiency $\eta_{as}$ of $4.7(1)\, \%$ for anti-Stokes mode (single count rate of $1.25(6) \times 10^{6}\,\mathrm{cps}$). Accounting for the transmission losses and the detection efficiencies of the single-photon detectors, we infer the intrinsic photon-pair generation rate $R_{g}$ for the photons escaping the atomic ensemble and coupled into the single-mode fiber through $R_{g} = R_{p}/(\varepsilon_{s}\varepsilon_{as})$, where $\varepsilon_{s/as}$ denotes the total detection efficiency including optical transmission and the detection quantum efficiency. Consequently, the intrinsic photon-pair generation rate is inferred as $3.3(3)\times 10^{6}\, \mathrm{cps}$. The right axis of Fig.~\ref{fig:best_pair_rate}~(a) shows the inferred generation rate versus the pumping field power, highlighting the exceptionally high brightness of the source that we can achieve.   Fig.~\ref{fig:best_pair_rate}~(b) presents the pair ratio, defined as $\eta = R_{g}\varepsilon_{s/as}/(R_{s/as})$, which represents the fraction of correlated photon pairs within the total scattered photons. This quantity reflects the relative weight of the coherent nonlinear emission, and it is strongly influenced by incoherent spontaneous emission, residual absorption, and spatial phase-matching, namely, the extent to which the generated photon pairs are emitted into and subsequently collected by the single-mode fiber. At very weak pump powers, the pair ratio is limited by detector dark counts and pump leakage. After reaching its maximum before the saturation, it decreases as the pump power increases, indicating a growing contribution of incoherent scattering processes. Meanwhile, both the peak of the cross-correlation function and the CAR defined in the coincidence window (see panels (c) and (d)) decrease with increasing pump power. This behavior illustrates a general trade-off: pursuing higher brightness inevitably degrades the photon pair purity and the overall signal-to-noise ratio. In this characterization, the additional noise arises from detector dark counts ( $\approx 200\,\mathrm{cps}$ and $\approx 1300\,\mathrm{cps}$ for the two detectors) and the residual pump leakage (up to $\approx5000\,\mathrm{cps}$ in the Stokes and $\approx3000\,\mathrm{cps}$ in the anti-Stokes channel at maximum pump power). Their impact on the coincidence-based photon-pair measurement and noise-photon-sensitive quantities, such as heralding efficiencies, is negligible.


\section{Conclusion}\label{section:conclusion}
In conclusion, we have successfully developed a high-brightness correlated photon source from a cold $^{87}\text{Rb}$ atomic ensemble, based on the double-$\Lambda$ SFWM scheme. By employing a counter-propagating geometry and collecting photons in a phase-matched, near-axis direction ($1^{\circ}$ deviation), we achieved a collection with high spatial coherence by coupling a single diffraction-limited mode into a single-mode fiber. With this setup, we experimentally measured an instantaneous photon pair rate near $10^{5}\, \mathrm{cps}$, corresponding to an intrinsic generation rate on the order of  $10^{6}-10^{7}\, \mathrm{cps}$, demonstrating highly efficient photon pair generation. Furthermore, we show that the pump-field parameters allow for precise control over both nonlinear strength and temporal coherence. Specifically, within the counter-propagating geometry, we identify a regime of enhanced SFWM efficiency, characterized by a blue-detuned pumping field and a red-detuned coupling field. Simultaneously, the temporal coherence is primarily determined by the EIT-induced group delay. These diverse behaviors under different parameter spaces indicate that the atomic nonlinear response can be engineered to optimize nonlinear strength and coherence, providing a versatile platform for narrowband correlated photon pairs. 
 

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