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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Some connections between classical and quantum anholonomy

G. Giavarini, E. Gozzi, Daniel Rohrlich, William D. Thacker

Published May 15, 1989• 18 citations
Publisher

Abstract

In this paper we study the interplay between the classical and quantum anholonomy effects (Hannay's angle and Berry's phase). When a quantum system with a finite number of energy levels has a Berry phase, it also has a nonzero Hannay angle. We show how systems with infinitely many levels can evade this correspondence, and find some necessary conditions for a system with a Berry phase to have no Hannay angle.

DOI
10.1103/physrevd.39.3007

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In this paper we study the interplay between the classical and quantum anholonomy effects (Hannay's angle and Berry's phase). When a quantum system with a finite number of energy levels has a Berry phase, it also has a nonzero Hannay angle. We show how systems with infinitely many levels can evade this correspondence, and find some necessary conditions for a system with a Berry phase to have no Hannay angle.
The Berry phase, which was discovered more than two decades ago, provides very deep insight into the geometric structure of quantum mechanics. Its classical counterpart, Hannay's angle, is defined if closed curves of action variables return to the same curves in phase space after a time evolution. In this paper we study the Berry phase and Hannay's angle in a quantum-classical hybrid system under the Born-Oppenheimer approximation. By the term quantum-classical hybrid system, we mean a composite system consists of a quantum subsystem and a classical subsystem. The effects of subsystem-subsystem couplings on the Berry phase and Hannay's angle are explored. The results show that the Berry phase has been changed sharply by the couplings, whereas the couplings have a small effect on the Hannay's angle.
In the present paper, we investigate the Berry phase and the Hannay angle of an interacting two-mode boson system and obtain their analytic expressions in explicit forms. The relation between the Berry phase and the Hannay angle is discussed. We find that, in the large-particle-number limit, the classical Hannay angle equals the particle number derivative of the quantum Berry phase except for a sign. This relationship is applicable to other many-body boson systems where the coherent-state description is available and the total particle number is conserved. The measurement of the classical Hannay angle in the many-body systems is briefly discussed as well.