Microscopic Origin of Superradiant Biphoton Emission in Atomic Ensembles

Abstract

Superradiant biphoton emission from atomic ensembles provides a powerful route to generating highly correlated quantum light, yet its microscopic physical origin has remained incompletely understood. In particular, it is often unclear how collective enhancement, spontaneous emission, and vacuum fluctuations jointly give rise to both paired biphoton generation and unavoidable unpaired background within a single, self-consistent framework. Here we present a fully quantum microscopic theory within a unified Heisenberg--Langevin--Maxwell framework that explicitly incorporates dissipation and quantum noise, thereby revealing the microscopic origin of superradiant biphoton emission in atomic ensembles. The theory provides a consistent description of parametric gain and unpaired noise within the same open-quantum-system framework and applies to both Doppler-free cold atomic ensembles and Doppler-broadened warm vapors. In the high-optical-depth regime, the coupled propagation equations admit analytical solutions, under which the biphoton dynamics rigorously reduce to an effective collective two-level emission process. Within this limit, the biphoton correlation time and spectral properties are shown to obey closed-form scaling relations governed by optical depth and excited-state decoherence. Our results establish a unified microscopic picture of superradiant biphoton generation and clarify the fundamental role of vacuum fluctuations and dissipation in setting the brightness, pairing efficiency, and temporal structure of atomic biphoton sources, with direct relevance to quantum networking and atomic quantum interfaces.

Figures (9)

• Figure 1: Schematic illustration of biphoton generation and superradiant emission in a diamond-type atomic ensemble. (a) Four-level diamond-type configuration for biphoton generation. Co-propagating coupling ($\Omega_c$) and driving ($\Omega_d$) fields sequentially excite atoms via the $|1\rangle \rightarrow |2\rangle \rightarrow |4\rangle$ transitions. Vacuum-fluctuation–seeded spontaneous emission through the $|4\rangle \rightarrow |3\rangle \rightarrow |1\rangle$ cascade is isotropic, while directional temporal correlations between the signal ($\hat{a}^{\dagger}_s$) and idler ($\hat{a}^{\dagger}_i$) photons emerge from collectively enhanced, phase-matched four-wave mixing in the ensemble, as characterized by the coincidence rate $R_\mathrm{C}$. (b) Illustration of $R_\mathrm{C}$ as a function of the idler-photon delay time. In the low-OD limit, the $1/e$ decay time approaches the single-atom spontaneous lifetime $\tau_{31}$ on $|3\rangle \rightarrow |1\rangle$ transition. As OD increases, cooperative emission produces a brighter and temporally narrower $R_C$, indicating superradiant shortening of the decay time $\tau_d$.
• Figure 2: Spectral characteristics of signal and idler photons and their dependence on OD. (a) Pairing spectra $R_{s,p}$ and $R_{i,p}$, (b) unpairing spectra $R_{s,u}$ and $R_{i,u}$, and (c) total spectra $R_s$ and $R_i$ of the signal and idler fields for $\mathrm{OD}=10$. In (a), the pairing spectra coincide, $R_{s,p}=R_{i,p}$. (d) Total generation rates $R_s$ and $R_i$ as functions of OD. The inset shows an alternative biphoton-generation configuration with 1367 nm (signal) and 780 nm (idler). Other parameters are $\Omega_c=\Omega_d=1\Gamma$, $\Delta_1=-50\Gamma$, and $\Delta_2=0$.
• Figure 3: Spectral characteristics of paired photons and their dependence on OD. (a), (b) Spectral profiles of the pairing rate $R_p$ (black solid) together with its absorption (green dash-dotted) and emission (yellow dashed) components for $\mathrm{OD}=0.1$ and $\mathrm{OD}=10$, respectively. The absorption and emission gain spectra are normalized independently, and each $R_p$ spectrum is normalized to the maximum of its corresponding emission spectrum. The inset highlights the OD-induced spectral broadening of $R_p$. (c) Pairing rate as a function of OD obtained from the general ($R_p$) and analytical-solution ($R_{p,\mathrm{A}}$) results. The two curves nearly overlap for $\mathrm{OD}>10$. The inset shows the pairing rate for biphoton generation at 1367 nm (signal) and 780 nm (idler), which yields a higher generation rate under otherwise identical driving conditions. (d) Pairing ratios of the signal ($r_{p,s}$) and idler ($r_{p,i}$) photons as functions of OD. All panels use $\Omega_c=\Omega_d=1\Gamma$, $\Delta_1=-50\Gamma$, and $\Delta_2=0$.
• Figure 4: Superradiant behavior of biphoton temporal correlations and quantitative comparison with experimental data. (a), (b) Signal-side coincidence rate $R_{\mathrm{C},s}$ as a function of the idler-photon delay time for $\mathrm{OD}=0.1$ and $\mathrm{OD}=10$, respectively, with $\Omega_c=\Omega_d=1\Gamma$, $\Delta_1=-50\Gamma$, and $\Delta_2=0$. Exponential fits (yellow dashed curves) yield $1/e$ decay times of approximately $26~\mathrm{ns}$ and $7~\mathrm{ns}$. (c) Theoretical prediction of $R_{\mathrm{C},s}$ at $\mathrm{OD}=25$. (d) Experimental trace (red dots) reproduced from Ref. Chaneliere for direct quantitative comparison. The fitted decay times in (c) and (d) are 3.3 ns and 3.2 ns, respectively, demonstrating close agreement between theory and experiment. (e) Theoretical decay time (black curve) as a function of $1/\mathrm{OD}$, with red diamonds indicating experimental values extracted from the same reference. Panels (c) and (e) are calculated using the backward-field scheme and the corresponding experimental energy-level configuration. Experimental data are reproduced with permission.
• Figure 5: Comparison between the superradiant scaling time and the biphoton correlation decay time, and validation of the analytical approximation. (a) Decay time extracted from the full correlation function $\tau_d$ (black solid, from $G^{(2)}_{s\textrm{-}i}(\tau)$) compared with the superradiant scaling time $\tau_s$ (red dots). The inset highlights the small deviation between the two curves. (b) $1/e$ decay time as a function of OD obtained from the general ($\tau_d$) and analytical ($\tau_{d,\mathrm{A}}$) expressions. The discrepancy remains below $25\%$ for $10\lesssim\alpha\lesssim30$, and the two results converge for $\alpha\gtrsim30$, where $\tau_d\approx\tau_{d,\mathrm{A}}\approx4/(\alpha\Gamma_{31})$. (c), (d) Signal-side coincidence rate as a function of idler delay $\tau$ obtained from the full solution [$G^{(2)}_{s\textrm{-}i}(\tau)$, black dashed] and the analytical approximation [$G^{(2)}_{s\textrm{-}i,\mathrm{A}}(\tau)$, yellow solid] for $\mathrm{OD}=30$ and $\mathrm{OD}=100$. The insets highlight oscillatory structures that become more pronounced at higher OD. All panels use $\Omega_c=\Omega_d=1\Gamma$, $\Delta_1=-50\Gamma$, and $\Delta_2=0$.