On geometric phases and dynamical invariants

Abstran A formalism is developed for calculahg Beny phases for non-adiabatic time-periodic quantum systems when a dynamical invariant is known. It is found that, when the invariant is periodic and has a non-degenerate spectrum, this method allows a convenient way to obtain generalired Beny phases and the pmper cyclic initial states. The method is applied to Lhe generalized harmonic oscillator and the two-level system, where the invariant operaton are explicilly constructed. Formulae for the mnventional Beny’s phases are readily obtained by taking the adiabatic limit of the exact results. 1