Elastic Scattering Time of Matter Waves in Disordered Potentials

We report on an extensive study of the elastic scattering time ${\tau }_s$ of matter waves in optical disordered potentials. Using direct experimental measurements, numerical simulations, and comparison with the first-order Born approximation based on the knowledge of the disorder properties, we explore the behavior of ${\tau }_s$ over more than 3 orders of magnitude, ranging from the weak to the strong scattering regime. We study in detail the location of the crossover and, as a main result, we reveal the strong influence of the disorder statistics, especially on the relevance of the widely used Ioffe-Regel-like criterion $k{l}_s\sim 1$. While it is found to be relevant for Gaussian-distributed disordered potentials, we observe significant deviations for laser speckle disorders that are commonly used with ultracold atoms. Our results are crucial for connecting experimental investigation of complex transport phenomena, such as Anderson localization, to microscopic theories.

Figure 1

Elastic scattering and Born approximation. (a) Scattering of a matter wave by a laser speckle disordered potential of typical correlation length $\sigma$. During a scattering event, which happens on the characteristic time ${\tau }_s$, a momentum ${\mathbf{k}}_{\mathrm{dis}}$ is transferred to the initial momentum ${\mathbf{k}}_i$. In the Born approximation, the final momentum ${\mathbf{k}}^{\prime }={\mathbf{k}}_i+{\mathbf{k}}_{\mathrm{dis}}$ lies on the elastic scattering ring (dotted circle). For positive atom-light detuning $\mathrm{\Delta }>0$, the laser speckle potential is repulsive. Inset: for $\mathrm{\Delta }<0$, it is attractive, having identical spatial properties but opposite amplitude distribution. (b) Illustrations of the 2D-momentum distributions $n(\mathbf{k},t)$ after a typical time ${\tau }_s$ (1st row: side view, 2nd row: top view) for the isotropic ($k_i\ll {\sigma }^{-1}$) and forward ($k_i\gg {\sigma }^{-1}$) scattering regimes.

Figure 2

Measurements of the elastic scattering time ${\tau }_{\mathbf{s}}$. (a) Measurement procedure. The momentum distributions $n(\mathbf{k},t)$ are observed for different propagation times $t$ in the disorder, here shown for the parameters $V_R/h=-104\mathrm{Hz}$ (attractive case) and $k_i=2.31{\sigma }^{-1}$. The normalized height ${\tilde{n}}_i(t)$ is determined from $n(\mathbf{k},t)$ by a Gaussian fit of the radially integrated angular profile [32]. When plotted as a function of the time $t$, it shows an exponential decay from which we extract ${\tau }_s$, as illustrated for two different initial momenta $k_i=0.76{\sigma }^{-1}$ and $k_i=2.31{\sigma }^{-1}$, still at $V_R/h=-104\mathrm{Hz}$. (b) Experimental (points) and numerical (solid lines) values of ${\tau }_s$ as a function of the initial momentum $k_i$ for different values of the disorder amplitude $|V_R|$, for attractive disorder (left panel) and repulsive disorder (right panel). The initial momenta are shown in units of the characteristic frequency ${\sigma }^{-1}$ of the disorder. Born predictions ${\tau }_s^{\text{Born}}$ are indicated by dashed lines. Note that the Born curves are simply shifted down for the various disorder amplitudes due to the scaling ${\tau }_s^{\text{Born}}\propto 1/|V_R{|}^2$ (see text).

Figure 3

Deviations from the Born predictions for different disorder amplitude distributions. 2D representation (logarithmic color scale) of the ratio ${\tau }_s/{\tau }_s^{\text{Born}}$ as a function of $|V_R|$ and $k_i$ for attractive (1st row) and repulsive (2nd row) disordered potentials. Both experimental (left column) and numerical (right column) data are shown. 3rd row: same representation for a Gaussian-distributed disorder (numerical study). The amplitude probability distributions $P(V)$ for the three types of disorders are plotted in the inset.