Effects of nonlinear sweep in the Landau-Zener-Stueckelberg effect

We study the Landau-Zener-Stueckelberg (LZS) effect for a two-level system with a time-dependent nonlinear bias field (the sweep function) W(t). Our main concern is to investigate the influence of the nonlinearity of W(t) on the probability P to remain in the initial state. The dimensionless quantity epsilon=piDelta(2)/(2hv) depends on the coupling Delta of both levels and on the sweep rate v. For fast sweep rates, i.e., epsilon much less than 1, and monotonic, analytic sweep functions linearizable in the vicinity of the resonance we find the transition probability 1-Pcongruent toepsilon(1+a), where a>0 is the correction to the LSZ result due to the nonlinearity of the sweep. Further increase of the sweep rate with nonlinearity fixed brings the system into the nonlinear-sweep regime characterized by 1-Pcongruent toepsilon(gamma) with gammanot equal1, depending on the type of sweep function. In the case of slow sweep rates, i.e., epsilon much greater than 1, an interesting interference phenomenon occurs. For analytic W(t) the probability P=P(0)e(-eta) is determined by the singularities of rootDelta(2)+W-2(t) in the upper complex plane of t. If W(t) is close to linear, there is only one singularity, which leads to the LZS result P=e(-epsilon) with important corrections to the exponent due to nonlinearity. However, for, e.g., W(t)proportional tot(3) there is a pair of singularities in the upper complex plane. Interference of their contributions leads to oscillations of the prefactor P-0 that depends on the sweep rate through epsilon and turns to zero at some epsilon. Measurements of the oscillation period and of the exponential factor would allow one to determine Delta, independently.