Stochastic phase-space simulation of multimode cat states via the positive-P representation

Abstract

We present a comprehensive study of the transient dynamics of multimode Schrödinger cat states in dissipatively coupled resonator arrays using the positive-P phase-space method. By employing the positive-P representation, we derive the exact stochastic differential equations governing the system's dynamics, enabling the simulation of system sizes significantly larger than those accessible via direct master equation simulation. We demonstrate the utility of this method by simulating transient dynamics for networks up to N=21 sites. Furthermore, we critically examine the method's usefulness and limitations, specifically highlighting the computational instability encountered when estimating the state parity in the systems. Our results provide a pathway for scalable simulations of non-Gaussian states in large open quantum systems.

Figures (11)

• Figure 1: Schematic of the driven-dissipative resonator array. The system comprises a chain of linear cavities coupled via engineered non-local dissipation with strength $\gamma$. Each resonator is subject to a parametric two-photon drive with amplitude $\epsilon$, while experiencing local decoherence channels corresponding to single-photon loss (rate $\kappa_1$) and two-photon loss (rate $\kappa_2$).
• Figure 2: Stability plot of the positive-$P$ simulations from study of single mode systems: The orange region indicates parameter values where the trajectories of $\alpha$ and $\beta$ become unstable before the transient state is reached. In the blue region, the parity observable shows an unexpected decay while all other observables remain well-behaved. The green region represents the parameter regime in which all observables exhibit stable and reliable behavior. The yellowgreen region at the top shows where the second-order correlation function has low signal to noise ratio due to an indeterminate zero-by-zero form as the average photon number (the denominator) approaches zero in the large single-photon dissipation limit.
• Figure 3: Comparison of observable dynamics in a single mode obtained from the master equation (black dashed lines) and the positive-$P$ method (solid colored lines, expectation values in blue), including statistical errors (SE) in red, log scale. (a) $\kappa_1/\epsilon = 10^{-3}, \kappa_2/\epsilon = 1$: Stochastic trajectories become unstable before a transient state is reached. (b) $\kappa_1/\epsilon = 10^{-3}, \kappa_2/\epsilon = 0.2$: In this case the parity observable displays unexpected decay, while other observables exhibit excellent agreement with the master equation results. (c) $\kappa_1/\epsilon = 5, \kappa_2/\epsilon = 0.2$: All observables demonstrate excellent agreement for this set of parameters.
• Figure 4: Wigner functions of the density matrix obtained from (a) the master equation and (b) the positive-$P$ simulation of a single mode system. The parameters are $\kappa_1/\epsilon = 10^{-3}$ and $\kappa_2/\epsilon = 0.2$ at time $\epsilon t = 3$, corresponding to the generation of the transient cat state. An ensemble of $10^6$ trajectories was simulated.
• Figure 5: Comparison of observable dynamics calculated using the positive-$P$ representation (solid colored lines blue: mean, red: statistical error, log scale) and the quantum trajectory method in a small multimode system ($N=3$). The observables for site 1 are defined as follows: $n(1)$ is the average photon number, $\zeta(1)$ is the coherent amplitude, and $\Pi_{loc}(1)$ is the local parity. The system parameters are $\kappa_1/\epsilon=0.001$, $\kappa_2/\epsilon=0.2$, and $\gamma/\epsilon=10$.