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Journal of Physics A Mathematical and General

Berry's phase, Hannay's angle and coherent states

Mustapha Maamache, Jean-Pierre Provost, Gérard Vallée

Published Dec 21, 1990• 27 citations
Publisher

Abstract

Through the use of coherent states describing the action-angle variables the authors establish a strict parallelism between Berry's phase and Hannay's angle. In particular they define a geometrical transport for classical tori which leads to Hannay's angle. As in the case of Berry's phase, the transport can be associated with the minimization of a distance.

DOI
10.1088/0305-4470/23/24/018

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Through the use of coherent states describing the action-angle variables the authors establish a strict parallelism between Berry's phase and Hannay's angle. In particular they define a geometrical transport for classical tori which leads to Hannay's angle. As in the case of Berry's phase, the transport can be associated with the minimization of a distance.
We show how to derive the Hannay angles of Grassmannian classical mechanics from the evolution of Grassmannian action—angle quantum states. Just as in the commutative case, this evolution defines a geometric transport, which can also be obtained from a quantum canonical transformation or a variational principle. As examples, we explicitly construct the quantum states for the classical counterparts of a first-and second-quantized N-level system. In the latter case, these states reduce to standard fermionic coherent states and the classical Hannay angles coincide with the quantum Berry phases.
Ce travail de physique theorique porte sur les relations entre la phase de berry et l'angle de hannay qui sont deux notions geometriques decrivant des transports naturels en physique respectivement quantique et classique. Le but principal de ce travail a ete d'elaborer un nouveau formalisme, celui des etats quantiques action angle (generalisation des etats coherents de l'oscillateur harmonique), dans lequel le transport classique apparait comme un cas particulier du transport quantique. Cela a permis non seulement d'etablir plusieurs derivations quantiques inedites de l'angle de hannay, mais aussi d'apporter des eclairages originaux sur la relation entre la phase de berry et l'angle de hannay a travers l'introduction des notions de principe variationnel moyenne ou de minimisation de metriques sur les etats quantiques ou les trajectoires classiques. Le rapport entre la phase de berry et l'angle de hannay a egalement ete examine dans le cas des systemes grassmaniens
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.