The Open Path Phase for Degenerate and Non-degenerate Systems and its Relation to the Wave-function Modulus

Abstract

We calculate the open path phase in a two state model with a slowly (nearly adiabatically) varying time-periodic Hamiltonian and trace its continuous development during a period. We show that the topological (Berry) phase attains $π$ or $2 π$ depending on whether there is or is not a degeneracy in the part of the parameter space enclosed by the trajectory. Oscillations are found in the phase. As adiabaticity is approached, these become both more frequent and less pronounced and the phase jump becomes increasingly more steep. Integral relations between the phase and the amplitude modulus (having the form of Kramers-Kronig relations, but in the time domain) are used as an alternative way to calculate open path phases. These relations attest to the observable nature of the open path phase.