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American Journal of Physics

Measuring the Hannay geometric phase

Huner Fanchiotti A, Carlos A. García Canal, Miguel Angel Mayosky, A. Veiga, Vicente Vento

Published May 23, 2022• 4 citations
Publisher

Abstract

The Hannay geometric phase is the classical analog of the well-known Berry phase. Its most familiar example is the effect of the latitude λ on the motion of a Foucault pendulum. We describe an electronic network whose behavior is exactly equivalent to that of the pendulum. The circuit can be constructed from off-the-shelf components using two matched transconductance amplifiers that comprise a gyrator to introduce the non-reciprocal behavior needed to mimic the pendulum. One may precisely measure the dependence of the Hannay phase on λ by circuit simulation and by laboratory measurements on a constructed circuit.

DOI
10.1119/5.0081149

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The Hannay geometric phase is the classical analog of the well-known Berry phase. Its most familiar example is the effect of the latitude λ on the motion of a Foucault pendulum. We describe an electronic network whose behavior is exactly equivalent to that of the pendulum. The circuit can be constructed from off-the-shelf components using two matched transconductance amplifiers that comprise a gyrator to introduce the non-reciprocal behavior needed to mimic the pendulum. One may precisely measure the dependence of the Hannay phase on λ by circuit simulation and by laboratory measurements on a constructed circuit.
The well-known geometric phase present in the quantum adiabatic evolution discovered by Berry many years ago has its analogue, the Hannay phase, in the classical domain.We calculate the Berry phase with examples for quantum hermitian and non-hermitian PTPTPT-symmetric Hamiltonians and compare with the Hannay phase in their classical equivalents.We use the analogy to propose resonant electric circuits which reproduce the theoretical solutions in simulated laboratory experiments.
Abstract The decomplexification procedure allows one to show mathematically (stricto sensu) the equivalence (isomorphism) between the quantum dynamics of a system with a finite number of basis states and a classical dynamics system. This unique way of connecting different dynamics was used in the past to analyze the relationship between the well-known geometric phase present in the quantum evolution discovered by Berry and its generalizations, with their analogs, the Hannay phases, in the classical domain. In here, this analysis is carried out for several quantum hermitian and non-hermitian PT-symmetric Hamiltonians and compared with the Hannay phase analysis in their classical isomorphic equivalent systems. As the equivalence ends in the classical domain with oscillator dynamics, we exploit the analogy to propose resonant electric circuits coupled with a gyrator, to reproduce the geometric phase coming from the theoretical solutions, in simulated laboratory experiments.
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 are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.