Resonant phenomena in compact and extended systems

We consider the dynamics of Josephson junctions in two formulations: one where the phase is defined on a compact interval [0,2pi], and a second where it is defined on an extended interval (- infinity, infinity). We find that in general the two approaches are not equivalent: they have different sets of allowable initial conditions, and similar initial conditions can produce different values for observables. However, they do have identical predictions for the appearance of resonances. These resonances can be understood in a framework of ``resonant Zener tunneling,`` which encompasses resonant tunneling in superlattices, tunneling between Wannier-Stark states in crystals, and Landau-Zener transitions in small metal rings. This framework also reveals a type of resonance in some quantum systems that are strongly driven (i.e., in the sudden limit), wherein the system has a large amplitude to transfer out of the diabatic state.