Berry's phase in the two-level model

TL;DR

This paper investigates the presence of Berry's phase in a two-level quantum system under classical electric fields, showing no geometric phase with real fields but recovering it with complex fields and under the rotating wave approximation.

Contribution

It demonstrates the absence of geometric phases in the two-level model with real electric fields and clarifies conditions under which Berry's phase appears, including complex fields and RWA.

Findings

No geometric phase with real periodic electric fields.

Geometric phases appear with complex electric fields.

Berry's phase is present only under the rotating wave approximation.

Abstract

We study the adiabatic evolution of a two-level model in the presence of an external classical electric field. The coupling between the quantum model and the classical field is taken in the electric dipole approximation. In this regime, we show the absence of geometric phases in the interacting two-level model in the presence of any periodic real time-dependent classical electric field. We obtain a conservative scalar potential in the calculation of Berry's phases of the instantaneous eigenstates of the model. For complex electric fields, we recover the existence of geometric phases. In particular, the geometric phases of the instantaneous eigenstates of the model in the presence of a positive or of a negative frequency component of the monochromatic electric field differ by an overall sign. As a check on our results, we map this interacting two-level model onto a spin-1/2 model under the action of a classical magnetic field. We confirm that the first one acquires Berry's phase only in the rotating wave approximation [RWA].