Nonlinear level-crossing models

Abstract

We examine the effect of nonlinearity at a level crossing on the probability for nonadiabatic transitions P. By using the Dykhne-Davis-Pechukas formula, we derive simple analytic estimates for P for two types of nonlinear crossings. In the first type, the nonlinearity in the detuning appears as a perturbative correction to the dominant linear time dependence. Then appreciable deviations from the Landau-Zener probability $P_{\mathrm{LZ}}$ are found to appear for large couplings only, when P is small; this explains why the Landau-Zener model is often seen to provide more accurate results than expected. In the second type of nonlinearity, called essential nonlinearity, the detuning is proportional to an odd power of time. Then the nonadiabatic probability P is qualitatively and quantitatively different from $P_{\mathrm{LZ}}$ because, on the one hand, it vanishes in an oscillatory manner as the coupling increases, and, on the other, it is much larger than $P_{\mathrm{LZ}}.$ We suggest experimental situations where the predicted deviations can be observed.