Semiclassical Simulation of Homogeneous Emitter Ensembles with Local Dissipation

Abstract

Emitter ensembles constitute a fundamental component in quantum optical technologies, yet efficient and accurate simulation of large ensembles remains challenging. Here, we formulate a truncated Wigner approximation (TWA) for permutation-invariant emitter ensembles subject to local dissipation by sampling stochastic trajectories in an extended phase space encompassing the Bloch sphere. Benchmarks show that the TWA accurately captures dynamics, including nonclassical signatures, with the approximation improving with ensemble size. We demonstrate large-scale simulations of hundreds of interacting ensembles within the TWA to reveal emergent spatial coherence and selective directionality of cooperative emission in a pumped 1D chain, highlighting a practical path to studying extended light-matter systems. Our results expand the scope of scalable simulations of quantum emitter ensembles, establishing a bridge between microscopic models and emergent behavior.

Figures (4)

• Figure 1: (Bottom) An ensemble of emitters uniformly couples to a common field. (Top left) The ensemble emits collectively with rate $\Gamma$ and undergoes independent local dissipation with rate $\gamma$. (Top right) In semiclassical simulations, the ensemble spin (large arrow) is sampled by classical phase-space variables (dots). Stochastic trajectories evolve along the Bloch sphere surface under collective processes, while local dissipation drives evolution into the sphere's interior.
• Figure 2: Dynamics of an emitter ensemble under local pumping and collective decay. (a) Evolution of normalized emission rate $\langle \hat{J}^{+}\hat{J}^{-}\rangle/N^2$ for $N=10, 10^2, 10^3,$ and $10^6$ (dark to light), for scaled time $N\Gamma t$. Solid curves show exact results, with TWA shown as dashed curves. For $N=10^6$, exact simulation is not feasible here. Inset: maximum error between TWA and exact dynamics for the normalized emission rate (lower line) and normalized inversion $\langle \hat{J}^{z}\rangle/N$ (upper line), compared to a $1/N$ reference (dashed). (b) Squeezing parameter $\xi^2$ and subradiance parameter $s=\langle \hat{J}^{+}\hat{J}^{-}\rangle/\langle \hat{J}^{z}+N/2\rangle$ for $N=25$. Inset: steady-state emission rate $\langle \hat{J}^{+}\hat{J}^{-}\rangle/N$ versus pump rate for TWA (dots) and exact solution (solid line); red shaded region indicates values less than one, and the gray region marks the threshold of TWA validity where $\langle \hat{J}^{+}\hat{J}^{-}\rangle$ is of order one.
• Figure 3: (a) Illustration of composite systems described using the TWA by assigning a collective spin to each permutation-invariant constituent (indicated by color). (b) Forward emission rate $\langle \hat{I}_{\mathrm{F}}\rangle/N_{\mathrm{tot}}^2$ for $M=2$ pumped ensembles, each comprising $N=20$ emitters for a total $N_{\mathrm{tot}} = 40.$ The propagation phase $\varphi=0,\pi/4,\pi/2$ (upper to lower curves). Emitters are initially in the ground state, and the local pump $\gamma = 10\Gamma$. Inset: (identical) single-ensemble emission rates $\langle \hat{I}_{\ell}\rangle/N^2$ for $\varphi=0$ as the number of ensembles is increased with $M=1, 10, 10^2, 10^3$ (dark to light) and $\gamma = 0.25N_{\mathrm{tot}}\Gamma$. Solid (dashed) curves give the exact solution (TWA).
• Figure 4: Emission dynamics in a 1D chain of $M$ ensembles, for emitters initially in the ground state. (a) Normalized total forward emission rate $\langle \hat{I}_{F}\rangle/N^2$, (b) local emission rate $\langle \hat{I}_{\ell}\rangle/N^2$, and (c) spatial structure factor $S(p)$ as a function of time (log-axis). (d) Forward and backward intensity fractions versus normalized drive strength $\Omega/\gamma$ with $M=20,30,50,100,$ and $500$ (dark to light), fixing accumulated phase $M\varphi = 10\pi$ and effective cooperativity $\Gamma N_{\mathrm{tot}}/\gamma = 100$.