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arXiv:solv-int/9501009 · Physical Review Letters

Geometric Phase Effects for Wave-Packet Revivals

Christopher Jarzynski

Published Feb 20, 1995• 16 citations
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Abstract

The study of wave-packet revivals is extended to the case of Hamiltonians which are made time dependent through the adiabatic cycling of some parameters. It is shown that the quantal geometric phase (Berry's phase) causes the revived packet to be displaced along the classical trajectory by an amount equal to the classical geometric phase (Hannay's angle), in one degree of freedom. A physical example illustrating this effect in three degrees of freedom is mentioned.

DOI
10.1103/physrevlett.74.1264

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The study of wave-packet revivals is extended to the case of Hamiltonians which are made time dependent through the adiabatic cycling of some parameters. It is shown that the quantal geometric phase (Berry's phase) causes the revived packet to be displaced along the classical trajectory by an amount equal to the classical geometric phase (Hannay's angle), in one degree of freedom. A physical example illustrating this effect in three degrees of freedom is mentioned.
The theory of quantal revivals is extended to Hamiltonians that are time dependent through the slow noncyclic change of some parameters. It is shown by generalizing the cyclic geometric phase (Berry's phase) to noncyclic Hamiltonians that there is a noncyclic geometric displacement for all quantum revivals. The treatment of the classical mechanical geometric angle is also extended to open paths in parameter space and its semiclassical relation with the noncyclic geometric phase derived. The displacement of the revivals is then shown to be given by the total angle shift on the final torus sum of the dynamical angle and the noncyclic geometric angle.
The Gaussian wave packet solution to the Schr\"odinger equation is studied for time-dependent Hamiltonians. The geometrical phase is obtained for a cyclic wave packet solution of the generalized harmonic oscillator with a nonadiabatic time-periodic Hamiltonian. It is found that the geometrical phase is independent of $\ensuremath{\Elzxh}$, and is equal to one-half of the classical nonadiabatic Hannay angle. The Hannay angle is shown to be independent of the classical action and does not involve averaging.
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 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.