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Acta Physica Sinica

Berry phase and Hannay's angle of an electromagnetic mode system driven by harmonic field with Born-Oppenheimer approximation

Liu Hao-Di

Published Jan 1, 2013• 0 citations
Publisher

Abstract

In this paper, we investigate the Berry phase and Hannay's angle of an electromagnetic mode system driven by harmonic field with Born-Oppenheimer approximation and obtain their algebraic expressions by theoretical calculation. The semiclassical relation between Berry phase and Hannay's angle is discussed. We find that besides the usual connection term, the Berry phase of BO hybrid system contains a novel term brought forth by the coupling induced effective gauge potential. This quantum modification can be viewed as an effective Aharonov-Bohm effect. Moreover, a similar phenomenon is founded in the Hannay's angle of classical BO hybrid system, which indicates that the Berry phase and Hannay's angle possess the same relation as the usual one. Besides, our theory can also be used to generate Artificial gauge potentials for neutral atoms.

DOI
10.7498/aps.62.100302

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In this paper, we investigate the Berry phase and Hannay's angle of an electromagnetic mode system driven by harmonic field with Born-Oppenheimer approximation and obtain their algebraic expressions by theoretical calculation. The semiclassical relation between Berry phase and Hannay's angle is discussed. We find that besides the usual connection term, the Berry phase of BO hybrid system contains a novel term brought forth by the coupling induced effective gauge potential. This quantum modification can be viewed as an effective Aharonov-Bohm effect. Moreover, a similar phenomenon is founded in the Hannay's angle of classical BO hybrid system, which indicates that the Berry phase and Hannay's angle possess the same relation as the usual one. Besides, our theory can also be used to generate Artificial gauge potentials for neutral atoms.
The Berry phase, which was discovered more than two decades ago, provides very deep insight into the geometric structure of quantum mechanics. Its classical counterpart, Hannay's angle, is defined if closed curves of action variables return to the same curves in phase space after a time evolution. In this paper we study the Berry phase and Hannay's angle in a quantum-classical hybrid system under the Born-Oppenheimer approximation. By the term quantum-classical hybrid system, we mean a composite system consists of a quantum subsystem and a classical subsystem. The effects of subsystem-subsystem couplings on the Berry phase and Hannay's angle are explored. The results show that the Berry phase has been changed sharply by the couplings, whereas the couplings have a small effect on the Hannay's angle.
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.
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.
In the Coulomb gauge, the solution to the Maxwell equations for slowly time-varying homogeneous nondispersive conducting linear media without charge source is obtained. We show that the electromagnetic field acquires the extra geometrical Hannay angle associated to the time-dependent generalized harmonic oscillator. The quantization scheme for the electromagnetic field provides a connection between the classical Hannay angle and the quantum adiabatic Berry phase shifts. We confirm that the interference pattern in the optical system is more or less altered by the existence of the geometric phase.
The Lewis' invariant and exact solution for the driven generalized time-dependent harmonic oscillator is found by making use of the Lewis-Riesenfeld quantum theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase for this system is obtained. We then proceed to use the exact solution to construct the coherent state and calculate the corresponding exact classical phase angle. This phase angle can give the Hannay's angle in the adjabatic limit. The relation between the exact Lewis' phase and the corresponding classical phase angle is discussed.
On the basis of quantum-classical correspondence for two-dimensional anisotropic oscillator, we study quantum-classical correspondence for two-dimensional rotation and translation harmonic oscillator system from both quantum-classical orbits and geometric phases. Here, the two one-dimensional oscillators refer to a common harmonic oscillator and a rotation and translation harmonic oscillator. In terms of the generalized gauge transformation, we obtain the stationary Lissajous orbits and Hannay's angle. On the other hand, the eigenfunctions and Berry phases are derived analytically with the help of time-dependent gauge transformation. We may draw the conclusion that the nonadiabatic Berry phase in the original gauge is-n times the classical Hannay's angle, here n is the eigenfunction index. As a matter of fact, the quantum geometric phase and the classical Hannay's angle have the same nature according to Berry. Finally, by using the SU(2) coherent superposition of degenerate two-dimensional eigenfunctions for a fixed energy value, we construct the stationary wave functions and show that the spatial distribution of wave-function probability clouds is in excellent accordance with the classical orbits, indicating the exact quantum-classical correspondence. We also demonstrate the quantum-classical correspondences for the geometric phase-angle and the quantum-classical orbits in a unified form.