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Scientific Reports

Berry-Hannay relation in nonlinear optomechanics

Ludovico Latmiral, Federico Armata

Published Feb 10, 2020• 4 citations

Abstract

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.

DOI: 10.1038/s41598-020-59081-5

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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.

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.

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.