Path-Integral Formulation of Bosonic Markovian Open Quantum Dynamics with Monte Carlo stochastic trajectories using the Glauber-Sudarshan P, Wigner, and Husimi Q Functions and Hybrids

TL;DR

This paper develops a path-integral approach to derive stochastic differential equations for bosonic open quantum systems using quasiprobability distributions, enabling more accurate simulations beyond mean-field approximations.

Contribution

It introduces a systematic derivation of SDEs for arbitrary Hamiltonians and jump operators based on path-integral formulas, clarifying the conditions for their applicability.

Findings

MC simulations accurately reproduce exact dynamics

Method applies to Bose-Hubbard model

Hubbard-Stratonovich transformation feasibility linked to diffusion matrix positivity

Abstract

The Monte Carlo (MC) trajectory sampling of stochastic differential equations (SDEs) based on the quasiprobabilities, such as the Glauber-Sudarshan P, Wigner, and Husimi Q functions, enables us to investigate bosonic open quantum many-body dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. In this method, the MC samplings for the initial distribution and stochastic noises incorporate quantum fluctuations, and thus, we can go beyond the mean-field approximation. However, description using SDEs is possible only when the corresponding Fokker-Planck equation has a positive-semidefinite diffusion matrix. In this work, we analytically derive the SDEs for arbitrary Hamiltonian and jump operators based on the path-integral formula, independently of the derivation of the Fokker-Planck equation (FPE). In the course of the derivation, we formulate the path-integral representation of the GKSL equation by using the $s$-ordered quasiprobability, which systematically describes the aforementioned quasiprobabilities by changing the real parameter $s$. The essential point of this derivation is that we employ the Hubbard-Stratonovich (HS) transformation in the path integral, and its application is not always feasible. We find that the feasible condition of the HS transformation is identical to the positive-semidefiniteness condition of the diffusion matrix in the FPE. In the benchmark calculations, we confirm that the MC simulations of the obtained SDEs well reproduce the exact dynamics of physical quantities and non-equal time correlation functions of numerically solvable models, including the Bose-Hubbard model. This work clarifies the applicability of the approximation and gives systematic and simplified procedures to obtain the SDEs to be numerically solved.