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Evolution and Berry Phase of Spin-1 Particles in a Rotating Magnetic Field

Yan Yu-zhen

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

The evolutions of spin particles in a rotating magnetic field are discussed. We obtained the wavefunctions of the spin 1/2 and spin 3/2 particles by the trial function method. The Berry phase of the system are also discussed after an evolution period.

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Using the method of dynamical algebras, the solution of a homotriparticle linear spin cluster (each particle with S=1∕2S=1∕2S=1∕2) in a rotating magnetic field is obtained. We derive degeneracy energy levels and each level's Berry phase of this system. The Berry phase as a function of $\ensuremath{\omega}$ and $\ensuremath{\theta}$ has been determined by using the relation between the Berry phase and the angular velocity $\ensuremath{\omega}$ of the rotating magnetic field as well as the angle $\ensuremath{\theta}$ between the magnetic field and ZZZ axis. We obtain the changing diagram of the Berry phase of the basis state.
In this paper we define a non-dynamical phase for a spin-1/2 particle in a rotating magnetic field in the non-adiabatic non-cyclic case, and this phase can be considered as a generalized Berry phase. We show that this phase reduces to the geometric Berry phase, in the adiabatic limit, up to a factor independent of the parameters of the system. We could add an arbitrary phase to the eigenstates of the Hamiltonian due to the gauge freedom. Then, we fix this arbitrary phase by comparing our Berry phase in the adiabatic limit with the Berry's result for the same system. Also, in the extreme non-adiabatic limit our Berry phase vanishes, modulo 2π2π2π, as expected. Although, our Berry phase is in general complex, it becomes real in the expected cases: the adiabatic limit, the extreme non-adiabatic limit, and the points at which the state of the system returns to its initial form, up to a phase factor. Therefore, this phase can be considered as a generalization of the Berry phase. Moreover, we investigate the relation between the value of the generalized Berry phase, the period of the states and the period of the Hamiltonian.
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.