Exact analytical solutions to the master equation of quantum Brownian motion for a general environment

TL;DR

This paper provides exact analytical solutions to the master equation of quantum Brownian motion, addressing previous inaccuracies and offering a flexible approach for various environments and conditions.

Contribution

It introduces a compact formulation for deriving the master equation and solutions for quantum Brownian oscillators, correcting earlier mathematical subtleties and extending applicability.

Findings

Explicit solutions for ohmic, sub-ohmic, and supra-ohmic environments.

Identification of a mathematical subtlety affecting previous derivations.

Generalization to external forces and multiple oscillators.

Abstract

We revisit the model of a quantum Brownian oscillator linearly coupled to an environment of quantum oscillators at finite temperature. By introducing a compact and particularly well-suited formulation, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. The flexibility of our approach allows for an immediate generalization to cases with an external force and with an arbitrary number of Brownian oscillators. More importantly, we point out an important mathematical subtlety concerning boundary-value problems for integro-differential equations which led to incorrect master equation coefficients and impacts on the description of nonlocal dissipation effects in all earlier derivations. Furthermore, we provide explicit, exact analytical results for the master equation coefficients and its solutions in a wide variety of cases, including ohmic, sub-ohmic and supra-ohmic environments with a finite cut-off.