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

Anderson localization of matter waves was recently observed with cold atoms in a weak one-dimensional disorder realized with laser speckle potential [Billy et al., Nature (London) 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 wave vector $k>\frac{1}{2}{q}_c$, and higher orders become essential. These terms, up to the fourth order, are calculated analytically and compared with numerical simulations. For very weak disorder, LE exhibits a sharp drop at $k=\frac{1}{2}{q}_c$. 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.

Figure 1

(Color online) Lyapunov exponent $\lambda$ for ${\eta }_0=0.08$ (main panel) and ${\eta }_0=0.15$ (inset) as a function of the dimensionless disorder correlation parameter $\gamma =2k{R}_c$. Circles and squares show numerical data for positive $\left(\epsilon =+1\right)$ and negative $\left(\epsilon =-1\right)$ skewness of disorder, while solid and dashed lines give corresponding fourth order expansion. Curve’s spike at $\gamma =2$ reflects the logarithmic divergence of ${\lambda }_4^G$.

Figure 2

(Color online) LE calculated at fixed $\gamma =2.2$ as a function of disorder strength and “sign” $\epsilon \beta$. Circles represent numerical LE divided by $k{\beta }^4{\lambda }_4$. Line and equation show the fourth degree fit over all the points except the leftmost and the rightmost ones. Inset: expansion coefficients ${\lambda }_i\left(\gamma \right)$.