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We consider the evolution of a quantum particle hopping on a cubic lattice inany dimension and subject to a potential consisting of a periodic part and arandom part that fluctuates stochastically in time. If the random potentialevolves according to a stationary Markov process, we obtain diffusive scalingfor moments of the position displacement, with a diffusion constant that growsas the inverse square of the disorder strength at weak coupling. Moregenerally, we show that a central limit theorem holds such that the squareamplitude of the wave packet converges, after diffusive rescaling, to asolution of a heat equation.

Diffusion in the mean for a periodic Schr\"{o}dinger equation perturbed by a fluctuating potential