Competition between relaxation and external driving in the dissipative Landau-Zener problem

We study Landau-Zener transitions in a dissipative environment by means of the quasiadiabatic propagator path-integral scheme. It allows to obtain numerically exact results for the full range of the involved parameters. We discover a non-monotonic dependence of the Landau-Zener transition probability on the sweep velocity which is explained in terms of a simple physical picture. This feature results from a nontrivial competition between relaxation processes and the external sweep and is not captured by perturbative approaches. In addition to the Landau-Zener transition probability, we study the excitation survival probability and also provide a qualitative understanding of the involved competition of time scales.