A general treatment of geometric phases and dynamical invariants

¹  Centro de Ciências Naturais e Humanas, Universidade Federal do ABC - Rua Santa Adélia 166, Santo André, São Paulo, 09210-170, Brazil
²  Departamento de Física, Universidade Federal de São Carlos - 13565-905, São Carlos, São Paulo, Brazil
³  Instituto de Física de São Carlos, Universidade de São Paulo - Caixa Postal 369, São Carlos, São Paulo, 13560-970, Brazil

Corresponding authors: duzzioni@ufabc.edu.br serra@ufabc.edu.br miled@ifsc.usp.br

Received: 24 October 2007
Accepted: 1 March 2008

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.