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

Berry's phase and Hannay's angle for spin systems

Yves Brihaye, Stefan Giller, Piotr Kosiński, Paweł Maślanka

Published Jan 15, 1993• 3 citations
Publisher

Abstract

The classical limit of a quantum spin system is considered. Berry's relation between Hannay's angle and the semiclassical limit for Berry's phase is proven to be valid in this case.Received 22 June 1992DOI:https://doi.org/10.1103/PhysRevD.47.722©1993 American Physical Society

DOI
10.1103/physrevd.47.722

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The classical limit of a quantum spin system is considered. Berry's relation between Hannay's angle and the semiclassical limit for Berry's phase is proven to be valid in this case.Received 22 June 1992DOI:https://doi.org/10.1103/PhysRevD.47.722©1993 American Physical Society
We address the quantum-classical comparison of phase measurements in optomechanics in the general framework of Berry phases for composite systems. While the relation between Berry phase and Hannay angle has been proven for a large set of quadratic Hamiltonians, such correspondence has not been shown so far in the case of non-linear interactions (e.g. when three or more operators are involved). Remarkably, considering the full optomechanical interaction we recover the aforementioned mathematical relation with the Hannay angle obtained from classical equations of motion. Our results link at a fundamental level previous proposals to measure decoherence, such as the one expressed by Marshall et al., with the no-go theorem shown by Armata et al., which provides boundaries to understand the quantum-to-classical transition in optomechanics.
This paper contains an evaluation of the Berry phases associated with the following class of nonlinear spin Hamiltonians : H (B, n ) = γS S · B + γQ((S · n) 2 - S2/3), with B · n = 0. Examples of these Hamiltonians are given in Atomic and Solid State Physics. We compute exactly the Berry phases for S = 1, 3/2 and for S > 3/2, to second order with respect to the ratio γQ/(γS B ). The results involve loop integrals of an Abelian gauge field which can be considered as a generalization of the magnetic monopole field, the projective complex plane P2(C) playing the role of the sphere S2. In the case of the spin-1 the relation between the Aharonov and Anandan cyclic quantum phase and the Berry adiabatic phase is made explicit. The connection between the semiclassical limit of the Berry phase and the classical adiabatic Hannay angle is analyzed using the classical spin model recently proposed by Nielsen and Rohrlich.
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.