Sampling methods to describe superradiance in large ensembles of quantum emitters

TL;DR

This work addresses the challenge of computing the two-time photon correlation $g^{(2)}(t,\tau)$ for large ensembles of quantum emitters by introducing two sampling-based approaches that bound $\mathcal{G}^{(2)}(t,0)$: pairwise sampling (an upper bound) and $m$-wise sampling (a lower bound). The authors develop the theoretical framework linking the master equation to correlation functions and then demonstrate, through benchmarks on inverted arrays and coherently driven lattices, that offset-corrected versions of both methods provide tight bounds across regimes. A key finding is an empirical crossover at $N \approx 2m$ where the optimal method switches between $m$-wise and pairwise sampling, refined by analytic offset corrections derived from the independent-emitter limit. Collectively, these tools enable accurate, bounding predictions of superradiant photon statistics in large ensembles and arbitrary EM environments, offering practical guidance for choosing sampling parameters in simulations.

Abstract

Superradiance is a quantum phenomenon in which coherence between emitters results in enhanced and directional radiative emission. Many quantum optical phenomena can be characterized by the two-time quantum correlation function $g^{(2)}(t,τ)$, which describes the photon statistics of emitted radiation. However, the critical task of determining $g^{(2)}(t,τ)$ becomes intractable for large emitter ensembles due to the exponential scaling of the Hilbert space dimension with the number of emitters. Here, we analyse and benchmark two approximate numerical sampling methods applicable to emitter arrays embedded within electromagnetic environments, which generally provide upper and lower bounds for $g^{(2)}(t,0)$. We also introduce corrections to these methods (termed offset corrections) that significantly improve the quality of the predictions. The optimal choice of method depends on the total number of emitters, such that taken together, the two approaches provide accurate descriptions across a broad range of important regimes. This work therefore provides new theoretical tools for studying the well-known yet complex phenomenon of superradiance in large ensembles of quantum emitters.

Figures (14)

• Figure 1: A square lattice of 64 emitters with lattice constant $d$ in the $x$--$y$ plane, coherently driven by a laser with wavevector $\mathbf{k}_L$ parallel to the $x$-axis, $\mathbf{k}_L = k_L \hat{\mathbf{x}}$ with $k_L=\omega_L/c$. All dipole moments are aligned parallel to the $z$-axis. Two detectors, labelled $a$ and $b$, are located in the far field of the lattice along the $x$- and $y$-directions, respectively.
• Figure 2: Schematic illustration of the approximate sampling methods. (a) Pairwise sampling method with sample size $m = 2$ and total number of samples $S_2 = 5$. (b) $m$-wise sampling method with sample size $m = 3$ and total number of samples $S_m = 4$. Dashed lines indicate inter-emitter interactions characterised by the coefficients $\gamma_{\mu\nu}$ and $\Delta_{\mu\nu}$.
• Figure 3: $\mathcal{G}^{(2)}(0,0)$ as a function of normalized separation $d/\lambda$ defining a square lattice of $64$ identical emitters with all dipole moments aligned perpendicular to the plane of the lattice. The number of samples is $S_m=1000$. The exact result is shown by the red-dashed curve. All other curves are obtained using the $m$-wise SM for different sample sizes ($m$). The inset shows the difference between the exact method and those obtained using the $m$-wise SM, denoted $\Delta\mathcal{G}^{(2)}(0,0)$.
• Figure 4: Normalized emission rate as a function of time for a 1D chain of 9 identical emitters with dipole moments aligned perpendicular to the chain and separation $d=\lambda/10$. The exact result is shown in red, and all other curves are obtained using the $m$-wise SM for different sample sizes ($m$) and $S_m = 1000$.
• Figure 5: $\mathcal{G}^{(2)}(0,0)$ as a function of the normalized emitter separation $d/\lambda$ for a fully inverted $8\times 8$ square lattice of identical emitters with dipole moments aligned perpendicular to the plane of the array. The exact result is shown by the red dashed curve. Predictions obtained using the pairwise and $m$-wise sampling methods are shown both with and without offset corrections, as indicated in the legend. Black dash-dotted lines specify the Dicke and independent emitter values. The critical separation $d_{\rm critical}$ for which $\mathcal{G}^{(2)}(0,0) = 1$ is specified by the vertical dashed line. For the pairwise method, we have chosen $S_2=10000$, and for the $m$-wise SM $m=6$ and $S_m=5000$.