Hyperdiffusion of quantum waves in random photonic lattices

Abstract

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 \textit{et al.}, Nature Phys. \textbf{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 $\alpha =12/5$ is also obtained for arbitrary correlation properties of the random potential.

References in corpus (5)

• Observation of anomalous diffusion and fractional self-similarity in one dimension
• Theory of fractional-Lévy kinetics for cold atoms diffusing in optical lattices
• Generalised Ornstein-Uhlenbeck processes
• Super-diffusion in optical realizations of Anderson localization
• Enhanced transport when Anderson localization is destroyed

Cited by in corpus (5)

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• Fractional Schrödinger equation in gravitational optics
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• Richardson diffusion in neurons
• Evolving disorder and chaos enhances the wave speed of elastic waves