Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths

TL;DR

This paper reviews an approach to modeling quantum systems embedded in classical-like baths using operator-valued quasi-probability functions and quasi-Lie brackets, highlighting the breaking of time-translation symmetry.

Contribution

It introduces a framework for describing quantum systems in non-canonical or non-Hamiltonian baths via quasi-Lie brackets and Quantum-Classical Liouville Equations.

Findings

Operator-valued quasi-probability functions effectively describe quantum-classical dynamics.

Quasi-Lie brackets facilitate the evolution of quantum subsystems in complex baths.

The approach captures symmetry breaking in time-translation for embedded quantum systems.

Abstract

Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations.