Onset of dissipation in Zener dynamics: Relaxation versus dephasing

We study the onset of dissipation in an externally driven small system, coupled to an environment. More specifically, we focus on the interplay between Zener dynamics (induced by the driving source), and the interaction with the environment, which gives rise to both dephasing and relaxation. We first consider a toy model, consisting of an externally driven two-level system coupled to various tupes of environments. We demonstrate the onset of dephasing, manifested as a decay of interference terms. In addition, we derive an effective equation of motion for the density matrix of the system, in the presence of a dissipative coupling to a thermal bath characterized by a broad spectrum. We extend our analysis of dephasing and relaxation to deal with the dynamics of a multi-level system. We show that dephasing and relaxation have competing effects on the dynamics. Dephasing tends to enhance the absorption of energy into the system, due to destruction of localization in the energy space; relaxation, on the other hand, gives rise to a loss of energy, thus limiting the rate of energy absorption. We calculate the dependence of the conductance on the external bias and find that it is nonmonotonous due to the interplay between diffusion and relaxation in energy space. In the weak dephasing regime we obtain fluctuations in the conductance, due to quantum interference effects. Our results can be applied to one-dimensional conducting rings threaded by time-varying Aharonov-Bohm flux.