Volume 82, Number 2, April 2008
Article Number 20007
Number of page(s) 5
Section General
Published online 15 April 2008
EPL, 82 (2008) 20007
DOI: 10.1209/0295-5075/82/20007

A general treatment of geometric phases and dynamical invariants

E. I. Duzzioni1, 2, R. M. Serra1 and M. H. Y. Moussa3

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

received 24 October 2007; accepted in final form 1 March 2008; published April 2008
published online 15 April 2008

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.

03.65.Ca - Formalism.
03.65.Vf - Phases: geometric; dynamic or topological.
03.65.Yz - Decoherence; open systems; quantum statistical methods.

© EPLA 2008