Density correlations in cold atomic gases: Atomic speckles in the presence of disorder

The phenomenon of random intensity patterns, for waves propagating in the presence of disorder, is well known in optics and in mesoscopic physics. We study this phenomenon for cold atomic gases expanding, by a diffusion process, in a weak random potential. We show that the density-density correlation function of the expanding gas is strongly affected by disorder and we estimate the typical size of a speckle spot, i.e., a region of enhanced or depleted density. Both a Fermi gas and a Bose-Einstein condensate (in a mean-field approach) are considered.

Figure 1

The normalized density-density correlation function $I\left(\Delta r,t\right)$ of a Fermi gas with $g_s=2$, for $k_{\text{F}}{l}_{\text{F}}=10$, $r=50l_{\text{F}}$, and $t=3r^2/\left(2d{D}_{k_{\text{F}}}\right)$. (a) corresponds to $d=3$ and (b) to $d=2$. The dashed curves in (a) and (b) correspond to Eqs. (23, 24), respectively. The solid curves are the result of an exact numerical integration, as described in the text. The insets show the oscillating decay of $I\left(\Delta r,t\right)$ with increasing $\Delta r$.

Figure 2

The normalized density-density correlation function $I\left(\Delta r,t\right)$ of a BEC for (a) $d=3$ and (b) $d=2$. As in Fig. 1 the parameters were set to $k_0{l}_0=10$, $r=50l_0$, $t=3r^2/\left(2d{D}_{k_0}\right)$, $D_k=D_{k_0}{\left(k/k_0\right)}^{4-d}$, $l_k=l_0{\left(k/k_0\right)}^{3-d}$.