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The Berry Phase as an Appropriate Correspondence Limit of the Aharonov-Anandan Phase in a Simple Model

Joy Christian, Abner Shimony

Published Mar 1, 1991• 3 citations
Publisher
DOI
10.1142/9789814439251_0011

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arXiv:1004.4008•Apr 22, 2010•1 citation
The authors obtain general formulae to calculate exactly the orbital Aharonov-Anandan phase for a bounded system interacting with a precessing magnetic field in the case where the binding potential is rotationally invariant around the axis of rotation of the magnetic field. The particular case of a quadratic potential is considered in full detail, and the connection with the stability problem is established. The adiabatic limit of the Aharonov-Anandan phase to obtain the corresponding Berry phase is also considered.
We show the existence of a nonadiabatic geometric phase, i.e., an Aharonov-Anandan (AA) phase, in the Aharonov-Casher (AC) topological interference effect in one-dimensional mesoscopic rings. We find the AC phase is the phase accumulated by the spin wave function during a cyclic evolution, and show it is the sum of a geometric AA phase and a dynamical phase. In the adiabatic limit, the AA phase becomes the spin-orbit Berry phase introduced by Aronov and Lyanda-Geller. By solving exactly the model of a quasi-one-dimensional ring formed by the 2DEG on a semiconductor heterostructure, we discuss the observability of the AA phase in the AC effect.
We present the time evolution of the electronic two-level model in the rotating wave approximation (RWA). We calculate the Aharonov-Anandan phase of the vector states with cyclic evolution either within the period of an external monochromatic electric field or the period corresponding to Rabi's frequency. The Aharonov-Anandan phase is shown to be dependent on the initial vector state, unless the system evolves in the adiabatic regime; in the latter case, the Aharonov-Anandan phase recovers Berry's phase. Our results are also discussed in the quasi-resonant regime.