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Physical Review A

Family of geometric phases

Franco Salmistraro

Published Oct 1, 1996• 1 citation
Publisher

Abstract

A family of geometric phases associated with the evolution of a quantum system is introduced. These phases share the property of depending on the curve determined by the Schr\"odinger equation as well as on the choice of an orthonormal basis in the Hilbert space of states. Their role in the expansion of the wave function is examined, and a comparison with previous results about Berry phases is made. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.2755

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•Jan 1, 1999•0 citations
A family of geometric phases associated with the evolution of a quantum system is introduced. These phases share the property of depending on the curve determined by the Schr"odinger equation as well as on the choice of an orthonormal basis in the Hilbert space of states. Their role in the expansion of the wave function is examined, and a comparison with previous results about Berry phases is made. \textcopyright{} 1996 The American Physical Society.
Recently we showed that quantum cyclic evolution occurs, provided the initial wave function was one of the eigenfunctions of the evolution operator, whatever the concrete form of the Hamiltonian [Phys. Rev. A 50, 5317 (1994)]. This paper will show that, for some specific Hamiltonians, cyclic evolution may occur, even if the initial wave function is not one of the eigenfunctions of the evolution operator. The general conditions for the occurrence of cyclic evolution is derived. Several interesting examples of quantum evolution, also including noncyclic evolution, and some specific geometric phases are given. \textcopyright{} 1996 The American Physical Society.
A group-theoretic treatment is given of the new phase found by Berry in the adiabatic evolution of a quantum-mechanical system in a finite-dimensional Hilbert space. It is shown how the Berry phases for the various eigenstates of the Hamiltonian are obtained from a set of angles associated with a group element. For the special case of a two-level system there is just one such angle which corresponds to the holonomy transformation associated with parallel transport around a closed curve on a sphere.
During the last few years, considerable interest has been focused on the phase that waves accumulate when the equations governing the waves vary slowly. The recent flurry of activity was set off by a paper by Michael Berry, where it was found that the adiabatic evolution of energy eigenfunctions in quantum mechanics contains a phase of geometric origin (now known as 'Berry's phase') in addition to the usual dynamical phase derived from Schrödinger's equation. This observation, though basically elementary, seems to be quite profound. Phases with similar mathematical origins have been identified
The geometric phase in quantum mechanics was originally elucidated in the context of the adiabatic theorem. This and other limitations were later removed. We show how this phase can be presented as the simplest functional of smooth open curves of unit vectors in Hilbert space, invariant under the two groups of local phase changes and monotonic reparametrisations. This viewpoint clarifies the role of the early Pancharatnam criterion for two Hilbert space vectors to be `in phase', in the later geometric phase context. Noncyclic evolution is handled in a very direct manner. Connections to the Bargmann invariants of quantum mechanics are immediate and can be exploited. A class of geometric phases arises in connection with unitary representations of Lie groups. In this case a surprisingly complete analysis of the structure of the geometric phase is possible, motivated by the Wigner-Eckart theorem. The interplay between geometric phase ideas and Lie group unitary representations is extraordinarily rich; the differential geometric structures of Lie groups and coset spaces play very important roles and allow classification of all possible geometric phases associated with a given Lie group. Examples involving SU(2), SU(3) and the metaplectic group Mp(2) are presented.