Long-Range Correlations of Density in a Bose-Einstein Condensate Expanding in a Random Potential

We study correlations of atomic density in a weakly interacting Bose-Einstein condensate, expanding diffusively in a random potential. We show that these correlations are long range and that they are strongly enhanced at long times. The density at distant points exhibits negative correlations.

Figure 1

Drawing of a BEC expanding in a 3D random potential and confined transversally to a waveguide of typical diameter $d$. We assume $2\pi /k_{\mu }\ll d\lesssim \ell$.

Figure 2

Ensemble-averaged atomic density $\overline{n}$ of a BEC expanding in a 3D random potential inside a quasi-1D optical waveguide. The main plot shows $\overline{n}$ as a function of time for three different distances $z$. The dashed line is a $1/\sqrt{t}$ asymptote. The inset shows $\overline{n}$ as a function of $z$.

Figure 3

Diagram for the kernel $K_2$ generating the long-range correlation function of density fluctuations $C_2$ through Eq. (5). The solid lines and dashed lines represent Green’s functions $G$ and its complex conjugates $G^{*}$, respectively. The two parallel $G$ lines connected by dotted “ladders” symbolize averages of products of two Green’s functions, $\overline{G{G}^{*}}$. $H=({\ell }^5{m}^3/3\pi {\hslash }^3{ϵ}_1){\mathbf{\nabla }}_{{\mathbf{r}}_5}·{\mathbf{\nabla }}_{{\mathbf{r}}_7}$ is the Hikami box [15, 16]. Integration should be performed over ${\mathbf{r}}_5={\mathbf{r}}_6={\mathbf{r}}_7={\mathbf{r}}_8$.

Figure 4

Position dependence of the correction $c$ to the variance of atomic density in an expanding BEC for three different times $\tau =t\mu /\hslash$. The dimensionless distance is $\zeta =z\sqrt{k_{\mu }\ell }/\ell$. The dashed lines show analytic results $B{\tau }^{3/4}[1-C(\zeta /{\tau }^{1/4}{)}^2]$ (small $\zeta$) and $D\tau /\zeta$ (large $\zeta$) for $\tau =3\times {10}^4$. The numerical constants $B$, $C$, and $D$ are given in the text. The inset shows the time dependence of $c$ for a given (small) $\zeta =0.4$.

Figure 5

Long-range correlation function $c_2$ versus dimensionless distance $\Delta \zeta =2\zeta$ (solid lines) for three different times $\tau =t\mu /\hslash$. The dashed lines show asymptotes $B{\tau }^{3/4}(1-E\Delta \zeta /{\tau }^{1/4})$ and $-F\tau /\Delta \zeta$ for $\tau =3\times {10}^4$. The numerical constants $B$, $E$, and $F$ are given in the text. The inset shows the time dependence of the correlation for $\Delta \zeta =0.8$.