Criteria for Feasible Monte Carlo Stochastic Simulations of Bosonic Markovian Open Quantum Dynamics

Abstract

The Monte Carlo sampling of the stochastic differential equations (SDEs) based on the quasiprobability distribution function, such as the Glauber--Sudarshan P, Wigner, and Husimi Q functions provides a powerful framework for investigating bosonic open quantum many-body dynamics described by the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation, while considering the effects of quantum fluctuations beyond the mean-field approximation. However, the stochastic Monte Carlo simulation is possible only when the corresponding Fokker--Planck equation has a positive-semidefinite diffusion matrix, and the general conditions for the diffusion matrix to be positive semidefinite have remained unclear. In this work, starting from the path integral formulation, we first derive the sufficient conditions under which the diffusion matrix is positive semidefinite for an arbitrary Hamiltonian, jump operators, and choice of quasiprobability distribution functions. We also analytically derive the corresponding SDEs to be solved. We then investigate the dynamics of the GKSL equation in the thermodynamic limit and show that, depending on the form of the jump operators, the mean-field approximation may fail to describe the dynamics accurately, making stochastic Monte Carlo simulations indispensable. Furthermore, we derive the sufficient conditions under which the higher-order quantum fluctuation terms beyond the Fokker--Planck description vanish identically, even when the jump operators contain quadratic terms. Under these conditions, whenever the corresponding SDEs can be derived, the stochastic Monte Carlo simulation reproduces the exact dynamics. These results clarify the conditions under which the stochastic Monte Carlo simulations are both feasible and necessary for accurately describing the dynamics governed by the GKSL equation in phase space.