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International Journal of Modern Physics A

REALIZATION OF BERRY’S PHASE AND HANNAY’S ANGLE THROUGH CANONICAL TRANSFORMATIONS

Sagarika Biswas, K. DATTA, S. Rai Choudhury

Published Jun 20, 1994• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s0217751x94001047

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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.
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.
In a recent paper [Y. C. Ge and M. S. Child, Phys. Rev. Lett. 78, 2507 (1997)], by using a Gaussian wave function, Ge and Child presented a nonadiabatic relation between the quantum Berry phase and the classical Hannay angle for the time-dependent harmonic oscillator. In this paper, we present a perspective for this relation without the use of a trial wave function. In particular, an exact explicit formula for the cyclic evolution over the period T in the parameter space of action invariant is obtained; the $\ensuremath{-}(n+1/2)$ relation between the quantum geometric angle and the Hannay angle is rigorously established.