Path-Integral Formulation of Truncated Wigner Approximation for Bosonic Markovian Open Quantum Systems

Abstract: The truncated Wigner approximation (TWA) enables us to investigate bosonic quantum many-body dynamics, including open quantum systems described by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. In the TWA, the Weyl-Wigner transformation, a way of mapping from quantum-mechanical operators to $c$-numbers, of the GKSL equation leads to the Fokker-Planck equation, which we calculate by reducing it to the corresponding stochastic differential equations. However, the Fokker-Planck equation is not always reduced to the stochastic differential equations depending on details of jump operators. In this work, we clarify the condition for obtaining the stochastic differential equations from the Fokker-Planck equation and derive analytical expressions of these equations for a system with an arbitrary Hamiltonian with jump operators that do not couple different states. This result enables us to shortcut the conventional complicated calculations in applying the TWA. In the course of the derivation, we formulate the GKSL equation by using the path-integral representation based on the Weyl-Wigner transformation, which gives us a clear interpretation of the relation between the TWA and quantum fluctuations and allows us to calculate the non-equal time correlation functions in the TWA. In the benchmark calculations, we numerically confirm that the relaxation dynamics of physical quantities including the non-equal time correlation functions obtained in our formulation agrees well with the exact ones in the numerically solvable models.