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Physical Review A

Nonadiabatic Berry phase in rotating systems

Shi-Min Cui

Published Apr 1, 1992• 33 citations
Publisher

Abstract

The time-dependent Schr\"odinger equation for a particle in general rotating systems is solved analytically, and the exact solutions are used to study the nonadiabatic Berry phase. It is shown that the nonadiabatic Berry topological phase appears in general rotating systems for purely mechanical reasons and is observable. An alternative expression for the nonadiabatic Berry phase is given and discussed.

DOI
10.1103/physreva.45.5255

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The time-dependent Schr"odinger equation for a particle in general rotating systems is solved analytically, and the exact solutions are used to study the nonadiabatic Berry phase. It is shown that the nonadiabatic Berry topological phase appears in general rotating systems for purely mechanical reasons and is observable. An alternative expression for the nonadiabatic Berry phase is given and discussed.
The time-dependent Schr\"odinger equation for a spin particle in a rotating magnetic field is solved analytically by the cranking method, and the exact solutions are employed to study the nonadiabatic Berry's phase. An alternative expression for Berry's phase is given, which shows that Berry's phase is related to the expectation value of spin along the rotating axis and gives Berry's phase a physical explanation besides its gauge geometric interpretation. This expression also presents a simple algorithm for calculating the nonadiabatic Berry's phase for Hamiltonians that are nonlinear functions of the SU(2) generators. It is shown that nonadiabaticity alters the time evolution ray and in turn changes its Berry's phase. For the SU(2) dynamical group, the nonadiabatic effect on Berry's phase manifests itself as spin alignment (a phenomenon in nuclear physics), and spin-alignment quantization (observed recently in high-spin nuclear physics) is related to Berry's-phase quantization.
Exact Schr\"odinger wave functions of a time-dependent harmonic oscillator are found in analytically closed forms for the eigenstates of the generalized invariant and the instantaneous Hamiltonian. The cyclic initial state (CIS) and corresponding nonadiabatic Berry phase are also found exactly for a $\ensuremath{\tau}$-periodic Hamiltonian. There may exist $N\ensuremath{\tau}$-periodic CISs and corresponding Berry phases, but the cases with unstable classical motions do not have CISs in which cases Berry phases do not exist.
Under non-relativistic conditions, the properties of adiabatic solutions of the Schrodinger equation in moving frame of reference and the behaviours of the corresponding Berry's Phase are analysed. Two cases of translation and rotation are discussed in detail, which show that the existence of Berry's phase depends on the choice of frame of reference. While Bitter and Dubbers's experiment is explained by the first-order approximation in our discussion, the nonadiabatic effects in this experiment are predicted by the second-order approximation when the adiabatic condition is broken.