Saturated transitions in exactly soluble models of two-state curve crossing with time-dependent potentials

Two exactly soluble two-state curve crossing models, with potential matrices formed by a linear combination of time-independent terms and (I) inversely proportional [Formula Presented] or (II) exponential time-dependent terms, are considered here. It is shown that the two models are related to each other by a simple transformation of the time variable. These models can be further transformed to a simpler one (the “reduced” model), in which the two crossing potentials are a constant (horizontal) one and an inversely proportional time-dependent one, and the interaction coupling them is time-independent. Analysis of the exact solution of the reduced model discloses that the nonadiabatic transition probability saturates at increasingly large coupling strengths without vanishing (in contradiction of the linear Landau-Zener theory and various adiabatic theories). Similar saturation can also occur in more general models, within certain ranges of the potential parameters involved.