Relation between the Berry phase in quantum hermitian and non-hermitian systems and the Hannay phase in the equivalent classical systems

TL;DR

This paper explores the relationship between the Berry phase in quantum hermitian and non-hermitian systems and the Hannay phase in classical systems, proposing experimental circuits to demonstrate these geometric phases.

Contribution

It provides a comparative analysis of Berry and Hannay phases in quantum and classical systems, including non-hermitian Hamiltonians and experimental circuit proposals.

Findings

Berry phase calculated for hermitian and non-hermitian Hamiltonians

Comparison with Hannay phase in classical analogs

Proposal of resonant electric circuits for experimental validation

Abstract

The well-known geometric phase present in the quantum adiabatic evolution discovered by Berry many years ago has its analogue, the Hannay phase, in the classical domain.We calculate the Berry phase with examples for quantum hermitian and non-hermitian $PT$-symmetric Hamiltonians and compare with the Hannay phase in their classical equivalents.We use the analogy to propose resonant electric circuits which reproduce the theoretical solutions in simulated laboratory experiments.