Hyperdiffusion of quantum waves in random photonic lattices

A quantum-mechanical analysis of hyper-fast (faster than ballistic) diffusion of a quantum wave packet in random optical lattices is presented. The main motivation of the presented analysis is experimental demonstrations of hyper-diffusive spreading of a wave packet in random photonic lattices [L. Levi $\mathit{\text{et}}\mathit{\text{al.}}$, Nature Phys. $\mathbf{\text{8}}$, 912 (2012)]. A rigorous quantum-mechanical calculation of the mean probability amplitude is suggested, and it is shown that the power law spreading of the mean squared displacement (MSD) is $<x^2(t)>\sim t^{\alpha }$, where $2<\alpha \le 3$. The values of the transport exponent $\alpha$ depend on the correlation properties of the random potential $V(x,t)$, which describes random inhomogeneities of the medium. In particular, when the random potential is $\delta$ correlated in time, the quantum wave packet spreads according Richardson turbulent diffusion with the MSD $\sim t^3$. Hyper-diffusion with $\alpha =12\mathrm{/}5$ is also obtained for arbitrary correlation properties of the random potential.