Hyperdiffusion of quantum waves in random photonic lattices

TL;DR

This paper provides a quantum-mechanical analysis of hyper-fast wave packet diffusion in random photonic lattices, revealing power-law spreading with exponents up to 3, depending on the potential's correlation properties.

Contribution

It introduces a rigorous quantum-mechanical calculation of wave spreading in random lattices, demonstrating hyper-diffusive regimes with specific power-law exponents based on potential correlations.

Findings

Mean squared displacement scales as t^α with 2<α≤3

For delta-correlated potentials, MSD scales as t^3 (Richardson diffusion)

Hyper-diffusion with α=12/5 occurs for arbitrary potential correlations

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 $2<\alpha\leq 3$. 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.