DYNAMICS OF SINGLE AND MULTIPLE ZENER TRANSITIONS

The probability of remaning in a given state is evaluated for a two-level system that can cross repeatedly and periodically as a function of a control parameter. The possibility of multiple Zener transitions between the surfaces introduces the issue of the appropriate initial conditions for each successive transition. We devise a simple model, which is readily analyzed numerically, to evaluate the consequences of an initially nondiaganol density matrix on the subsequent evolution through the crossing region. The initial state strongly affects the outcome of the transition. Thus successive transitions are, in general, not independent of each other. Consequently, we find that the probability oscillates in time. This contrasts with the result of convergence at long times to a constant state occupation which would be predicted based solely on probabilities of single transitions. We also analyze the role that inelastic transitions play in modifying the nondissipative multiple-Zener-transition model. This model finds application to small tunnel junctions, where the control parameter is a current and the output is a voltage. Here, we show that the average dc voltage is nonzero when inelastic transitions are included, in contrast to the zero average voltage obtained in the absence of inelastic transitions, or when their rate is very large.