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Atlantis studies in mathematical physics

Lecture 8: Properties of Free Motion, Anholonomy, Geometric Phase

Gianfausto Dell’Antonio

Published Jan 1, 2015• 0 citations
Publisher

Abstract

In this Lecture we start the analysis of the Schrodinger equation and describe the properties of some of its solutions.
DOI
10.2991/978-94-6239-118-5_8

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We describe the behavior of two coupled Bose-Einstein condensates in time-dependent (TD) trap potentials and TD Rabi (or tunneling) frequency using the two-mode approach. Starting from Bloch states, we succeed in obtaining analytical solutions for the TD Schr\"odinger equation and present a detailed analysis of the relative and geometric phases acquired by the wave function of the condensates, as well as their population imbalance. We also establish a connection between the geometric phases and constants of motion which characterize the dynamic of the system. Besides analyzing the affects of temporality on condensates that differ by hyperfine degrees of freedom (internal Josephson effect), we also do present a brief discussion of a one-species condensate in a double-well potential (external Josephson effect).
Anholonomies in the parametric dependences of the eigenvalues and the eigenvectors of Floquet operators that describe unit time evolutions of periodically driven systems, e.g., kicked rotors, are studied. First, an example of the anholonomies induced by a periodically pulsed rank-1 perturbation is given. As a function of the strength of the perturbation, the perturbed Floquet operator of the quantum map and its spectrum are shown to have a period. However, we show examples where each eigenvalue does not obey the periodicity of the perturbed Floquet operator and exhibits an anholonomy. Furthermore, this induces another anholonomy in the eigenspaces, i.e., the directions of the eigenvectors, of the Floquet operator. These two anholonomies are previously observed in a family of Hamiltonians [T. Cheon, Phys. Lett. A 248, 285 (1998)] and are different from the phase anholonomy known as geometric phases. Second, the stability of Cheon's anholonomies in periodically driven systems is established by a geometrical analysis of the family of Floquet operators. Accordingly, Cheon's anholonomies are expected to be abundant in systems whose time evolutions are described by Floquet operators. As an application, a design principle for quantum state manipulations along adiabatic passages is explained.