The open path phase for degenerate and non-degenerate systems and its relation to the wave-function modulus

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 att ains pi or 2 pi depending on whether there is or is not a degeneracy in the part of the parameter space-enclosed by the trajectory. Oscillations are f ound in the phase. As adiabaticity is approached, these become both more fr equent and less pronounced and the phase jump becomes increasingly more ste ep. Integral relations between the phase and the amplitude modulus (having the form of Kramers-Kronig relations, but in the time domain) are used as a n alternative way to calculate open path phases. These relations attest to the observable nature of the open path phase.