A general treatment of geometric phases and dynamical invariants

TL;DR

This paper introduces a comprehensive method for calculating geometric phases, both Abelian and non-Abelian, for quantum systems undergoing nonadiabatic, noncyclic evolution, applicable to pure and mixed states, and unifies previous approaches.

Contribution

It provides a general framework based on parallel transport for computing geometric phases, extending to nonadiabatic and noncyclic cases, and generalizes existing non-Abelian holonomy concepts.

Findings

The method applies to pure and mixed states.

It captures nonadiabatic, noncyclic evolution.

Demonstrates non-Abelian phases in a two-level system with decay.

Abstract

Based only on the parallel transport condition, we present a general method to compute Abelian or non-Abelian geometric phases acquired by the basis states of pure or mixed density operators, which also holds for nonadiabatic and noncyclic evolution. Two interesting features of the non-Abelian geometric phase obtained by our method stand out: i) it is a generalization of Wilczek and Zee's non-Abelian holonomy, in that it describes nonadiabatic evolution where the basis states are parallelly transported between distinct degenerate subspaces, and ii) the non-Abelian character of our geometric phase relies on the transitional evolution of the basis states, even in the nondegenerate case. We apply our formalism to a two-level system evolving nonadiabatically under spontaneous decay to emphasize the non-Abelian nature of the geometric phase induced by the reservoir. We also show, through the generalized invariant theory, that our general approach encompasses previous results in the literature.