Adiabatic theorems for generators of contracting evolutions

TL;DR

This paper develops a comprehensive adiabatic theory for generators of contracting evolutions on Banach spaces, unifying various quantum and classical systems under a single framework.

Contribution

It introduces a unified adiabatic framework applicable to quantum, open, and classical systems, extending existing theories to more general Banach space settings.

Findings

Provides a uniform adiabatic approximation for diverse systems.

Analyzes both gapped and gapless cases.

Discusses applications to quantum and classical dynamics.

Abstract

We develop an adiabatic theory for generators of contracting evolution on Banach spaces. This provides a uniform framework for a host of adiabatic theorems ranging from unitary quantum evolutions through quantum evolutions of open systems generated by Lindbladians all the way to classically driven stochastic systems. In all these cases the adiabatic evolution approximates, to lowest order, the natural notion of parallel transport in the manifold of instantaneous stationary states. The dynamics in the manifold of instantaneous stationary states and transversal to it have distinct characteristics: The former is irreversible and the latter is transient in a sense that we explain. Both the gapped and gapless cases are considered. Some applications are discussed.