Lyapunov exponent for the laser speckle potential: a weak disorder expansion

TL;DR

This paper analytically calculates the Lyapunov exponent for matter waves in a weak laser speckle potential, revealing a sharp transition at a specific wavevector and sensitivity to disorder skewness, with implications for cold atom experiments.

Contribution

It provides the first analytical calculation of higher-order contributions to the Lyapunov exponent in a speckle potential, highlighting effects beyond the Born approximation.

Findings

Lyapunov exponent drops sharply at wavevector q_c/2

LE is sensitive to disorder skewness

Analytical results agree with numerical simulations

Abstract

Anderson localization of matter waves was recently observed with cold atoms in a weak 1D disorder realized with laser speckle potential [J. Billy et al., Nature 453, 891 (2008)]. The latter is special in that it does not have spatial frequency components above certain cutoff $q_{c}$. As a result, the Lyapunov exponent (LE), or inverse localization length, vanishes in Born approximation for particle wavevector $k>{1/2}q_{c}$, and higher orders become essential. These terms, up to the order four, are calculated analytically and compared with numerical simulations. For very weak disorder, LE exhibits a sharp drop at $k$ $={1/2}q_{c}$. For moderate disorder (a) the drop is less dramatic than expected from the fourth order approximation and (b) LE becomes very sensitive to the sign of the disorder skewness (which can be controlled in cold atom experiments). Both observations are related to the strongly non-Gaussian character of the speckle intensity.