Berry's phase in the two-level model

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].Comment: 11 pages, no figur

Higher order geometric phase for qubits in a bichromatic field

“…The higher order corrections to the geometric phase [15] have been used to explain phase fluctuations observed in the superconducting circuit system [17]. Even for simple quantum systems such as spin qubits in a slowly changing electromagnetic field, the Berry phase, in its standard definition, exists only when the field is circularly polarized [18]. For the linearly polarized field, the first-order term in ε in equation ( 1) that describes the standard Berry phase becomes zero, but the even higher order terms in the adiabaticity parameter differ from zero.…”

Phases of the electronic two-level model under rotating wave approximation

“…Recently, we have studied the adiabatic evolution of a two-level model [6] (i) under a (real) classical monochromatic electric field and (ii) when only the contribution of the positive frequency of the electric field is taken into account (its RWA). In the RWA of the two-level model, we recover the geometric phases acquired by the instantaneous eigenstates of energy already known in the literature [7].…”

“…The matter-field coupling can be approximated by the two-level model only in the quasi-resonant regime [1], in which ω ≈ ε (in natural units, c = h = 1). Certainly, this condition does not fulfill the adiabatic condition [6] ω ε. In 2002, Li et al [8] proposed a scheme to perform quantum geometric computation in the non-adiabatic regime with trapped ions.…”