Caldeira--Leggett quantum master equation in Wigner phase space: continued-fraction solution and application to Brownian motion in periodic potentials

TL;DR

This paper develops a continued-fraction approach to solve the Caldeira--Leggett quantum master equation in Wigner phase space, enabling detailed analysis of environmental effects on nonlinear quantum systems like quantum Brownian motion in periodic potentials.

Contribution

It introduces a novel continued-fraction solution method for the quantum master equation in Wigner space, extending classical techniques to quantum systems.

Findings

Successfully applied to quantum Brownian motion in periodic potentials

Allows detailed study of environmental effects on quantum nonlinear systems

Provides full solutions of the quantum master equation using continued fractions

Abstract

The continued-fraction method to solve classical Fokker--Planck equations has been adapted to tackle quantum master equations of the Caldeira--Leggett type. This can be done taking advantage of the phase-space (Wigner) representation of the quantum density matrix. The approach differs from those in which some continued-fraction expression is found for a certain quantity, in that the full solution of the master equation is obtained by continued-fraction methods. This allows to study in detail the effects of the environment (fluctuations and dissipation) on several classes of nonlinear quantum systems. We apply the method to the canonical problem of quantum Brownian motion in periodic potentials both for cosine and ratchet potentials (lacking inversion symmetry).