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arXiv:2302.01962 · The European Physical Journal Plus

Correspondence between open bosonic systems and stochastic differential equations

Alexander Engel, Scott Parker

Published Jun 30, 2023• 3 citations
arXivHTMLPDF
DOI
10.1140/epjp/s13360-023-04205-9

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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 leads to remarkable simplification of the complicated calculations required in the conventional procedures of 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. Since our formulation is widely applicable to bosonic systems such as the Bose-Hubbard model, it facilitates studies of the open quantum many-body phenomenon.
The Monte Carlo trajectory sampling of stochastic differential equations based on the quasiprobability distribution functions, 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 Monte Carlo samplings for the initial distribution and stochastic noises incorporate quantum fluctuations, and thus, we can go beyond the mean-field approximation. However, description using stochastic differential equations is possible only when the corresponding Fokker–Planck equation has a positive-semidefinite diffusion matrix. In this work, we analytically derive the stochastic differential equations for arbitrary Hamiltonian and jump operators based on the path-integral formula, independently of the derivation of the Fokker–Planck equation. In the course of the derivation, we formulate the path-integral representation of the GKSL equation by using the -ordered quasiprobability distribution function, which systematically describes the aforementioned quasiprobability distribution functions by changing the real parameter . The essential point of this derivation is that we employ the Hubbard–Stratonovich transformation in the path integral, and its application is not always feasible. We find that the feasible condition of the Hubbard–Stratonovich transformation is identical to the positive-semidefiniteness condition of the diffusion matrix in the Fokker–Planck equation. In the benchmark calculations, we confirm that the Monte Carlo simulations of the obtained stochastic differential equations 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 stochastic differential equations to be numerically solved.