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Journal of Physics A Mathematical and General

Berry's phase and Hannay's angle from quantum canonical transformations

Mustapha Maamache, Jean-Pierre Provost, Gérard Vallée

Published Feb 7, 1991• 6 citations
Publisher

Abstract

Using an 'action-angle' coherent states formalism, introduced by the authors in a preceding paper, it is shown that Berry's phase and Hannay's angle can both be derived from the same quantum unitary transformation; their relationship is easily established in this framework.
DOI
10.1088/0305-4470/24/3/027

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Motivated by some recent remarks on the ``removability'' of Berry's phase [Ph. De Sousa Gerbert, Ann. Phys. (N.Y.) 189, 155 (1989)] or its classical counterpart, the Hannay angle, we present here a fresh analysis of Berry's-phase calculations for some coherent states of general theoretical interest. Our reasoning, though semiclassical, lends itself easily for comparison with the exact results. Unlike spin-coherent states, both the (ordinary) harmonic-oscillator and squeezed coherent states are contained in Hamiltonians for which Berry's phase can be removed globally by a canonical transformation.
A new relationship between Berry’s phase and Hannay’s angle (which is its classical counterpart) is established. This relationship is exact and does not depend on semiclassical arguments, though in this note the aforementioned relationship is proved only for a restricted class of Hamiltonians. It is shown that if a Hamiltonian, for which Berry’s phase and Hannay’s angle are nonvanishing, can, in classical theory, be canonically transformed into one for which both are zero, the generating function of this canonical transformation yields, through appropriate averages, both the quantal geometrical phase and its classical analog.
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.
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.