Effect of noise on Bloch oscillations and Wannier-Stark localization

We calculate an exact expression for the probability propagator for a noisy electric field driven tight-binding lattice. The noise considered is a two level jump process or a telegraph process (TP) which jumps randomly between two values ±μ. In the absence of a static field, and in the limit of zero jump rate of the noisy field, we find that the dynamics yields Bloch oscillations with frequency μ, while with an additional static field ϵ we find oscillatory motion with a superposition of frequencies (ϵ±μ). On the other hand, when the jump rate is rapid, and in the absence of a static field, the stochastic field averages to zero if the two states of the TP are equally probable a priori. In that case we see a delocalization effect. The intimate relationship between the rapid relaxation case and the zero field case seems to be a generic effect found in a wide variety of systems. It is interesting to note that even for zero static field and rapid relaxation, Bloch oscillations ensue if there is a bias δp in the probabilities of the two levels. Remarkably, the Wannier-Stark localization caused by an additional static field is destroyed if the latter is tuned to be exactly equal and opposite to the average stochastic field μδp. This is an example of incoherent destruction of Wannier-Stark localization.