Microscopic derivation of a completely positive master equation for the description of Open Quantum Brownian Motion of a particle in a potential

TL;DR

This paper derives a completely positive master equation for Open Quantum Brownian Motion (OQBM) of a particle in a potential, using the rotating wave approximation and adiabatic elimination, and analyzes its dynamics and non-Gaussian features.

Contribution

The authors derive a new completely positive hybrid quantum-classical master equation for OQBM, incorporating internal and external degrees of freedom, and analyze its properties and dynamics.

Findings

Gaussian distributions emerge in the limiting behavior of OQBM.

The third-order cumulant is nonzero, indicating non-Gaussian dynamics.

The derived master equation accurately describes the evolution of the position distribution.

Abstract

Open Quantum Brownian Motion (OQBM) was introduced as a scaling limit of discrete-time open quantum walks. This limit defines a new class of quantum Brownian motion, which incorporates both the external and internal degrees of freedom of the Brownian particle. We consider a weakly driven Brownian particle confined in a harmonic potential and dissipatively coupled to a thermal bath. Applying the rotating wave approximation (RWA) to the system-bath interaction Hamiltonian, we derive a completely positive Born-Markov master equation for the reduced dynamics. We express the resulting master equation in the coordinate representation and, utilizing the adiabatic elimination of fast variables, derive a completely positive hybrid quantum-classical master equation that defines OQBM. We illustrate the resulting dynamics using examples of initial Gaussian and non-Gaussian distributions of the OQBM walker. Both examples reveal the emergence of Gaussian distributions in the limiting behavior of the OQBM dynamics, which closely matches that of the standard OQBM. With the help of the obtained OQBM master equation, we derive the equations for the $n$-th moments and the cumulants of the position distribution of the open Brownian walker. We subsequently solve these equations numerically for Gaussian initial distributions across various parameter regimes. Notably, we find that the third-order cumulant is nonzero, indicating that the dynamics' intrinsic generator is non-Gaussian.