Non-adiabatic Non-cyclic Generalization of the Berry Phase for a Spin-1/2 Particle in a Rotating Magnetic Field

TL;DR

This paper introduces a generalized Berry phase for a spin-1/2 particle in a rotating magnetic field that applies beyond adiabatic and cyclic conditions, unifying various limits and exploring its properties.

Contribution

It defines a non-adiabatic, non-cyclic Berry phase for spin-1/2 systems and analyzes its behavior across different dynamical regimes, extending the concept of geometric phase.

Findings

The generalized Berry phase reduces to the standard Berry phase in the adiabatic limit.

The phase vanishes in the extreme non-adiabatic limit, consistent with expectations.

The phase becomes real at specific points, confirming its geometric nature.

Abstract

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 $2\pi$, 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.