  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
$\pi$ 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 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.
