Focused Sampling for Low-Cost and Accurate Ehrenfest Modeling of Cavity Quantum Electrodynamics

TL;DR

This paper addresses critical limitations of mean-field cQED modeling by merging decoupled mean-field (DC-MF) dynamics with a focused sampling scheme that enforces zero-point energy at the single-trajectory level, yielding accurate results across both short- and long-cavity regimes. A renormalization of transition dipole moments further aligns mixed quantum-classical results with full quantum benchmarks, enabling near-quantitative agreement with CISD for atomic occupancies and optical-field intensities while dramatically reducing trajectory counts. In the short-cavity regime, MF and DC-MF with focused sampling and TDM renormalization reproduce the correct Rabi frequency (with $g=\mu_{12}\lambda_1\sqrt{\omega_1/(2\hbar)}$ and $\,\omega\approx 2g$) and can converge with as few as a single trajectory; in the long-cavity regime, DC-MF+$f1.33$ achieves near-CISD accuracy for key observables, with substantial reductions in required samples. The approach promises a practical route to accurate, low-cost cQED simulations and suggests avenues for integrating DC-MF dynamics with classical-optics techniques, though questions remain about the exact decoupling functional form and the generality of renormalization factors.

Abstract

An economic modeling approach for cavity quantum electrodynamics is provided by mean-field dynamics, wherein the optical field is described classically while a self-consistent interaction with quantum emitters is incorporated through the Ehrenfest theorem. However, conventional implementations of mean-field dynamics are known to suffer from a catastrophic leakage of zero-point energy, to lose accuracy in the short-cavity limit, and to require large numbers of trajectories to be sampled. Here, we address these three shortcomings within a single integrated approach. This approach builds on our recently-proposed modification of the Ehrenfest theorem, referred to as decoupled mean-field (DC-MF) dynamics, in combination with a focused sampling scheme that enforces zero-point energy at the single-trajectory level. The approach is shown to yield high accuracy in both short and long-cavity limits while reaching convergence within a minimal amount of trajectories.

Figures (6)

• Figure 1: Schematic illustration of the probability distributions for a single cavity mode (denoted $\alpha$) under Wigner sampling (a,c) and under focused sampling (b,d). Shown as a heat map are the distributions as a function of position $Q_\alpha$ and mode energy $E_\alpha = \frac{1}{2}(P_\alpha^2 + \omega_\alpha^2 Q_\alpha^2)$ (a,b) and as a function of position $Q_\alpha$ and momentum $P_\alpha$ (c,d).
• Figure 2: Transient excited-state occupancy of a two-level atom embedded in a half-wavelength cavity, resulting from MF(W) (red) and DC-MF(W) (blue) dynamics. Results from CISD are also shown (gray).
• Figure 3: Transient excited-state occupancy of the two-level atom embedded in a half-wavelength cavity resulting from MF dynamics (a) and DC-MF dynamics (b) as a function of the initial optical occupancy, $n^{(0)}$. Also shown are the corresponding Fourier amplitudes (c,d). Indicated with dashes are the expected Rabi oscillation frequency at $\omega = 2g$ (red) and the optical occupancy corresponding to the ZPE at $n^{(0)}=1/2$ (white).
• Figure 4: Same as Fig. \ref{['fig_Rho_small']}, but for results from MF(f1.06) (orange) and DC-MF(f1.33) (purple) dynamics.
• Figure 5: Transient excited-state occupancy of a two-level atom embedded in a long cavity ($L=12.5~\mathrm{\mu}\mathrm{m}$). Shown are results from MF(f1.06) (orange), DC-MF(W) (blue), and DC-MF(f1.33) (purple) dynamics, compared against those from CISD (gray). The inset shows early-time detail.