Open quantum dynamics theory on the basis of periodical system-bath model for discrete Wigner function

TL;DR

This paper develops a stable numerical discretization scheme for open quantum dynamics using a periodical system-bath model, deriving hierarchical equations of motion for a discrete Wigner distribution function applicable to non-Markovian environments.

Contribution

It introduces a novel discretization approach based on a 2D periodically invariant system-bath model, enabling non-perturbative treatment of non-Markovian heat baths at finite temperatures.

Findings

Successfully derived hierarchical equations of motion for discrete Wigner functions.

Demonstrated numerical integration for 2D free rotor and harmonic systems.

Achieved stable simulations with coarse mesh and singular initial conditions.

Abstract

Discretizing a distribution function in a phase space for an efficient quantum dynamics simulation is a non-trivial challenge, in particular for a case that a system is further coupled to environmental degrees of freedom. Such open quantum dynamics is described by a reduced equation of motion (REOM) most notably by a quantum Fokker-Planck equation (QFPE) for a Wigner distribution function (WDF). To develop a discretization scheme that is stable for numerical simulations from the REOM approach, we employ a two-dimensional (2D) periodically invariant system-bath (PISB) model with two heat baths. This model is an ideal platform not only for a periodic system but also for a non-periodic system confined by a potential. We then derive the numerically ''exact'' hierarchical equations of motion (HEOM) for a discrete WDF in terms of periodically invariant operators in both coordinate and momentum spaces. The obtained equations can treat non-Markovian heat-bath in a non-perturbative manner at finite temperatures regardless of the mesh size. As demonstrations, we numerically integrate the discrete QFPE for a 2D free rotor and harmonic potential systems in a high-temperature Markovian case using a coarse mesh with initial conditions that involve singularity.