Shape of the quantum diffusion front

We show that quantum diffusion has well-defined front shape. After an initi al transient, the wave packet front (tails) is described by a stretched exp onential P(chi, t) = A(t) exp(- \ chi /w \ (gamma)), with 1 < gamma < oa, w here w(t) is the spreading width which scales as w(t) similar to t(beta), w ith 0 < beta less than or equal to 1. The two exponents satisfy the univers al relation gamma = 1/(1 - beta). We demonstrate these results through nume rical work on one-dimensional quasiperiodic systems and the three-dimension al Anderson model of disorder. We provide an analytical derivation of these relations by using the memory function formalism of quantum dynamics. Furt hermore, we present an application to experimental results for the quantum kicked rotor.