Diffusion and Superdiffusion of a Particle in a Random Potential with Finite Correlation Time

We study theoretically the long time asymptotic of a quantum particle moving in a random time-dependent potential with finite correlation time, in $d=1\mathrm{}$. By applying a new unitary numerical scheme we first show the minor importance of quantum interference and then derive an effective Langevin-type equation for the corresponding clasical problem in the limit of weak potential. We find that on intermediate time scales $\overline{E_{\mathrm{kin}}\left(t\right)}\sim t^{2/5}$, while the true long time asymptotic is determined by a new friction term, which gives rise to a stationary power law velocity distribution, multifractality of the velocity moments, and a slowing down of the superdiffusive behavior.