One-dimensional Anderson localization in certain correlated random potentials

We study Anderson localization of ultracold atoms in weak one-dimensional speckle potentials using perturbation theory beyond Born approximation. We show the existence of a series of sharp crossovers (effective mobility edges) between energy regions where localization lengths differ by orders of magnitude. We also point out that the correction to the Born term explicitly depends on the sign of the potential. Our results are in agreement with numerical calculations in a regime relevant for experiments. Finally, we analyze our findings in the light of a diagrammatic approach.

Figure 1

(Color online) Lyapunov exponent $\gamma$ calculated two orders beyond the Born approximation for particles in 1D speckle potentials created with a square diffusive plate versus the particle momentum $\hslash k$ and the strength of disorder ${ϵ}_{\text{R}}=2m{\sigma }_{\text{R}}^2{V}_{\text{R}}/{\hslash }^2$ ($V_{\text{R}}$ and ${\sigma }_{\text{R}}$ are the amplitude and correlation length of the disorder). The solid blue lines correspond to ${ϵ}_{\text{R}}=0.1$ and ${ϵ}_{\text{R}}=0.02$.

Figure 2

(Color online) Functions $f_n$ for $n=2$, 3, and 4 for a speckle potential created with a square diffusive plate [solid lines; see Eqs. (12, 13) and the Appendix] and comparison with numerical calculations (points with error bars). The inset is a magnification of function $f_4$ around $\kappa =2$.

Figure 3

(Color online) Lyapunov exponent $\gamma \left(k\right)$ versus the particle momentum $k$ as determined numerically (solid red lines) and by perturbation theory up to order 4 (dashed blue lines) for a speckle potential created with a square plate. The dotted green lines are the Born term. Inset: comparison of odd and even contributions in the Born series for ${ϵ}_{\text{R}}=0.1$.

Figure 4

Relevant fourth-order backscattering contributions. Contrary to the case of uncorrelated potentials [26, 27], the sum of diagrams (a)–(c) does not give zero for speckle potentials; only diagrams (b) and (d) contribute for $k{\sigma }_{\text{R}}∊\left[1,2\right]$.