Speed of sound in disordered Bose-Einstein condensates

Disorder modifies the sound-wave excitation spectrum of Bose-Einstein condensates. We consider the classical hydrodynamic limit, where the disorder correlation length is much longer than the condensate healing length. By perturbation theory, we compute the phonon lifetime and the correction to the speed of sound. This correction is found to be negative in all dimensions, with universal asymptotics for smooth correlations. Considering in detail optical speckle potentials, we find a quite rich intermediate structure. This has consequences for the average density of states, particularly in one dimension, where we find a “boson dip” next to a sharp “boson peak” as function of frequency. In one dimension, our prediction is verified in detail by a numerical integration of the Gross-Pitaevskii equation.

Figure 1

(Color online) Schematic 1D representation of the system under study: an interacting Bose-Einstein condensate with original homogeneous density $n_0=\mu /g$ (dashed black line) is exposed to a weak, spatially correlated random potential $V\left(\mathit{r}\right)$ (solid blue), here a blue-detuned speckle potential with amplitude $V=0.1\mu$, centered on the mean $\overline{V}=0$ (dotted blue). We consider the Thomas-Fermi regime where the healing length $\xi$ is much shorter than the disorder correlation length $\sigma$. The resulting ground-state density (solid black) [Eq. (9)] mirrors the disorder while leaving the total average density and number of particles constant. On top of this disorder-modified ground state, an elementary plane-wave excitation (green, plotted around 1) propagates with wave vector $k$, here with $k\sigma =1$. We calculate its effective speed of sound and the corresponding average density of states.

Figure 2

(Color online) Relative correction to the speed of sound as function of reduced momentum. Black line: analytical prediction (31) for $d=1$ in the limit $\xi /\sigma =0$. Blue and red symbols: data from a numerical integration of the Gross-Pitaevskii equation (with small but finite healing length, such that $k\xi =0.05$) at $v=V/\mu =\pm 0.03$, respectively, averaged over 50 realizations of disorder. In the shaded area, the data stray far from the analytical prediction because its condition of applicability $\sigma ⪢\xi$ becomes invalid. (Inset) Theory (28) for $d=1$ (black), $d=2$ (violet), and $d=3$ (orange) together with the limiting values (2, 3).

Figure 3

(Color online) Relative correction to the speed of sound at $k\sigma =1$, divided by $v^2$, as function of disorder strength $v=V/\mu$, taken from the numerical integration with $k\xi =0.05$. (Dashed line) Analytical prediction $\Delta c/\left(c{v}^2\right)=\frac{1}{4}\text{ln}2-\frac{1}{2}\approx -0.3267$.

Figure 4

(Color online) Histograms over 50 realizations of disorder for the relative speed-of-sound correction for different values of $v=V/\mu$ obtained by numerical integration of the Gross-Pitaevskii equation with an excitation at $k\sigma =1$. Vertical lines indicate the average values plotted in Fig. 3.

Figure 5

(Color online) Correction to the density of states $g_d\left(\kappa \right)=\left(\overline{\rho }-{\rho }_0\right)/{\rho }_0$ divided by the squared disorder strength $v=V/\mu$ as function of reduced momentum $\kappa =\omega \sigma /c$ in dimension $d=1,2,3$. At $\kappa =1$, the momentum beyond which elastic backscattering becomes impossible in the Born approximation, there is a logarithmic divergence in $d=1$, a kink in $d=2$, and a curvature discontinuity in $d=3$.