Quantum linear Boltzmann equation

TL;DR

This paper reviews the quantum linear Boltzmann equation, detailing its derivation, properties, numerical solutions, and various limiting cases, with applications to quantum Brownian motion, decoherence, and matter wave optics.

Contribution

It provides a comprehensive overview of the quantum linear Boltzmann equation, including derivation, symmetry analysis, numerical methods, and extensions to complex systems.

Findings

Derivation of the quantum linear Boltzmann equation as a Lindblad master equation

Analysis of its symmetry properties and relaxation dynamics

Discussion of limiting cases like quantum Brownian motion and decoherence

Abstract

We review the quantum version of the linear Boltzmann equation, which describes in a non-perturbative fashion, by means of scattering theory, how the quantum motion of a single test particle is affected by collisions with an ideal background gas. A heuristic derivation of this Lindblad master equation is presented, based on the requirement of translation-covariance and on the relation to the classical linear Boltzmann equation. After analyzing its general symmetry properties and the associated relaxation dynamics, we discuss a quantum Monte Carlo method for its numerical solution. We then review important limiting forms of the quantum linear Boltzmann equation, such as the case of quantum Brownian motion and pure collisional decoherence, as well as the application to matter wave optics. Finally, we point to the incorporation of quantum degeneracies and self-interactions in the gas by relating the equation to the dynamic structure factor of the ambient medium, and we provide an extension of the equation to include internal degrees of freedom.