Modelling many-body quantum dynamics with stochastic trajectories: a critical test on the Tavis-Cummings model

TL;DR

This study tests a stochastic-trajectory framework based on the positive-$P$ representation for many-body quantum dynamics in the Tavis-Cummings model with dissipation. It demonstrates intrinsic divergences in stochastic trajectories (movable singularities) that skew ensemble averages beyond a finite time, and shows that damping or drift-gauge regularization only partially mitigates these issues and depends strongly on system size and initial energy. The results indicate that the method reliably reproduces short-time or semi-classical dynamics in highly dissipative, large-$N$ regimes but fails to capture long-time quantum features like collapses and revivals, highlighting fundamental limitations of stochastic phase-space sampling for strongly quantum light–matter interactions. Consequently, the paper clarifies the practical limits of this approach and motivates the search for improved regularization techniques or alternative methods for accurately simulating long-time quantum dynamics in open many-body systems.

Abstract

We critically explore the applicability of a recently proposed framework to sample the quantum dynamics of a many-body quantum system interacting with light by stochastic trajectories, applying it to the closed and open Tavis-Cummings model (TCM). The stochastic differential equations (SDEs) sample the positive P phase-space representation by analog complex-valued dynamical variables that are linked to the quantum operators. Statistical average over the stochastic trajectories yields the evolution of the quantum mechanical expectation values. However, numerical implementation of these SDEs for the TCM indicates divergent solutions, also known from other phase-space methods. This limits the applicability of the framework to finite propagation times, that are strongly dependent on the physical parameters and initial conditions of the system. We outline the underlying mathematical reason for these divergences and show that their contribution to the averages are, however, essential. To attempt to regularize the divergences, we transform the SDEs to an equivalent set of SDEs with different noise realisations, thereby pushing the valid time boundary. Quantum collapse and revival of the TCM, however, cannot be recovered by the stochastic trajectory approach, pointing to the general difficulty of the applicability of stochastic phase-space sampling methods to systems with strong quantum features.

Figures (5)

• Figure 1: (Color online) Individual trajectories for the stochastic variable $\rho_{ee}$ and exact solution for the excited state probability of the closed TCM as a function of dimensionless time. Simulation parameters: $n_{ph} = 10N$, $\gamma/f =0$, atoms start from ground state; (a) $N = 10$ atoms; (b) $N = 100$ atoms.
• Figure 2: (Color online) (Top row) Amount of remaining trajectories and (Middle row) average of the stochastic variable $\rho_{ee}$ as a function of dimensionless time for the closed TCM ($\gamma = 0$). (Bottom row) Average of the stochastic variable $\rho_{ee}$ as a function of dimensionless time for the open TCM (values of $\gamma$ are given in the legend of each plot: bottom value corresponds to the minimal $\gamma$ leading to the converged results, top value is one half of the bottom value). Simulation parameters: $N=10$, all atoms start from the ground state and (a), (d), (g) $n_\text{ph} = 0.1N$; (b), (e), (h) $n_\text{ph} = N$; (c), (f), (i) $n_\text{ph} = 10N$. The vertical dashed line in (a)--(f) and dotted line in (g)--(i) show the time instant, at which the threshold of $0.5\%$ runaway trajectories has been reached. Solid lines correspond to the exact solution. Values in (g) are magnified x8 times.
• Figure 4: (Color online) (Top row) Amount of remaining trajectories and (Middle row) average of the stochastic variable $\rho_{ee}$ as a function of dimensionless time for the closed TCM ($\gamma = 0$). (Bottom row) Average of the stochastic variable $\rho_{ee}$ as a function of dimensionless time for the open TCM (values of $\gamma$ are given in the legend of each plot: bottom value corresponds to the minimal $\gamma$ leading to the converged results, top value is one half of the bottom value). Simulation parameters: $N=100$, all atoms start from the ground state and (a), (d), (g) $n_\text{ph} = 0.1N$; (b), (e), (h) $n_\text{ph} = N$; (c), (f), (i) $n_\text{ph} = 10N$. The vertical dashed in (a)--(f) and dotted in (h)--(i) lines show the time instant, at which the threshold of $0.5\%$ runaway trajectories has been reached. Solid lines correspond to the exact solution. Values in (g) are magnified x6 times and included as an example of open TCM.
• Figure 6: (Color online) Effective potential and effective Hamiltonian (\ref{['eqn:eff_potential']}) as a function of the population inversion $w$.
• Figure 7: (Color online) (Top row) The weighted and non-weighted averages of the stochastic variable $\rho_{ee}$ following the eqns. (\ref{['eqn:final']}), (\ref{['eqn:g-transform']}) and (Bottom row) real value of $\rho_{ee}$, $C_0$ and switch function (\ref{['eqn:switch']}) of a single trajectory as a function of the dimensionless time. Simulation parameters: $N=10$, $n_{\text{ph}} = 10N$ and (a), (c) $\gamma/f=0$; (b), (d) $\gamma=2.6f$. Parameters of switch function (\ref{['eqn:switch']}): $x_1 = -1$, $x_2 = 2$, $k = 1$.