Shape of the Quantum Diffusion Front

We show that quantum diffusion has well-defined front shape. After an initial transient, the wave packet front (tails) is described by a stretched exponential $P(x,t)=A\left(t\right)\exp (-|x/w{|}^{\gamma })$, with $1<\mathrm{}\gamma <\infty$, where $w\left(t\right)$ is the spreading width which scales as $w\left(t\right)\sim t^{\beta }$, with $0<\mathrm{}\beta \le 1$. The two exponents satisfy the universal relation $\gamma =1/(1-\beta )$. We demonstrate these results through numerical work on one-dimensional quasiperiodic systems and the three-dimensional Anderson model of disorder. We provide an analytical derivation of these relations by using the memory function formalism of quantum dynamics. Furthermore, we present an application to experimental results for the quantum kicked rotor.