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 $\ifmmode\pm\else\textpm\fi{}\ensuremath{\mu}$. 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 $\ensuremath{\mu}$, while with an additional static field $\ensuremath{\epsilon}$ we find oscillatory motion with a superposition of frequencies $(\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}\ensuremath{\mu})$. 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 $\ensuremath{\delta}p$ 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 $\ensuremath{\mu}\ensuremath{\delta}p$. This is an example of incoherent destruction of Wannier-Stark localization.