Long-time expansion of a Bose-Einstein condensate: Observability of Anderson localization

We numerically explore the long-time expansion of a one-dimensional Bose-Einstein condensate in a disorder potential employing the Gross-Pitaevskii equation. The goal is to search for unique signatures of Anderson localization in the presence of particle-particle interactions. Using typical experimental parameters, we show that the time scale for which the nonequilibrium dynamics of the interacting system begins to diverge from the noninteracting system exceeds the observation times up to now accessible in the experiment. We find evidence that the long-time evolution of the wave packet is characterized by (sub)diffusive spreading and a growing effective localization length, suggesting that interactions destroy Anderson localization.

Figure 1

Real-space correlation function for Gaussian-correlated [Eq. (7)] and speckle disorder [Eq. (9)].

Figure 2

The three scenarios discussed in this paper: scenario A (left column), scenario B (central column), and scenario C (right column). For details see text.

Figure 3

(a) Numerically obtained densities for scenario A (see Fig. 2) with the Gaussian-correlated disorder for several time steps indicated by the numbers in units of $t_0$ next to the individual curves. The densities have been calculated within a disorder average over 50 ensembles. The analytical formula, Eq. (21), is plotted as a reference (black dashed line). (b) The momentum distribution for two later times as compared to the initial momentum distribution for $t_I=20t_0$. The same ensemble as in (a) has been used. Inset: The momentum distribution in log-scale.

Figure 4

Comparison between the densities at $t=500t_0$ for the Gaussian-correlated disorder and for the speckle disorder for scenario A (see Fig. 2). Inset: Linear plot of the densities at the position of the local maxima. Equation (21) plotted as reference (black dashed).

Figure 5

(a) Numerically obtained densities for scenario B (see Fig. 2) for the Gaussian-correlated disorder at several time steps indicated by the numbers in units of $t_0$ next to the individual curves. The densities have been calculated within a disorder average over 50 ensembles. The analytical formula, Eq. (21) black dashed line, is plotted as reference. (b) The momentum distribution for two later times as compared to the initial momentum distribution at $t_I=20t_0$. The same ensemble as in (a) has been used. The vertical lines mark the position of $k_L=\pm \sqrt{\mu \left(t_s\right)}$. Inset: The momentum distribution in log-scale.

Figure 6

(a) Numerically obtained densities for scenario C (see Fig. 2) for the Gaussian-correlated disorder at several time steps indicated by the numbers in units of $t_0$ next to the individual curves. The densities have been calculated within an ensemble average over 50 ensembles. The analytical formula, Eq. (21) black dashed line, is plotted as reference. Inset: Zoom-in of the expanding density for several time steps, starting with $t=100t_0$ up to $t=500t_0$ (order indicated by the arrow). (b) The momentum distribution for two later times as compared to the initial momentum distribution for $t_I=20t_0$. The same ensemble as in (a) has been used. The vertical lines mark the position of $k_L=\pm \sqrt{\mu \left(t\right)}$, where the width of the lines corresponds to the standard deviation due to the time dependence of $\mu \left(t\right)$ and averaging over ensembles. Inset: The momentum distribution in log-scale.

Figure 7

The width $\mathrm{\Delta }x\left(t\right)$ [Eq. (15)] as a function of time for the three scenarios for (a) the Gaussian-correlated disorder and (b) the speckle disorder. The colored shaded area around each curve represents one standard deviation resulting from the ensemble of 50 disorder realizations each. The gray bar marks the time accessed experimentally [29].

Figure 8

Time-dependent second moment $\mathrm{\Delta }{x}^2\left(t\right)$ [Eq. (15)] for scenarios A and C for the Gaussian-correlated disorder. The labels “inner” and“outer” reflect the contributions to $\mathrm{\Delta }{x}^2\left(t\right)$ when only the contributions from the inner region $x\in \left[-3000:3000\right]x_0$ or only from the outer region $\left|x\right|>3000x_0$ are included. These partial second moments are additive ${〈\mathrm{\Delta }{x}^2〉}_{\mathrm{inner}}+{〈\mathrm{\Delta }{x}^2〉}_{\mathrm{outer}}={〈\mathrm{\Delta }{x}^2〉}_{\mathrm{total}}$.

Figure 9

The width $\mathrm{\Delta }k\left(t\right)$ [Eq. (17)] as a function of time for the three scenarios for the Gaussian-correlated disorder. The colored shaded area around each curve represents one standard deviation for an ensemble of 50 disorder realizations.

Figure 10

Time dependence of the exponent $\alpha \left(t\right)$ [Eq. (28)] obtained from the results in Fig. 7 for (a) the Gaussian-correlated disorder and (b) the speckle potential.

Figure 11

Localization length $L_{\mathrm{loc}}$ for scenario B with a speckle potential evaluated by fitting the densities from Sec. 3 at $t=500t_0$ to Eq. (29) for different intervals in units of $x_0$. The horizontal dashed line shows the analytic prediction [Eq. (30)].

Figure 12

Localization length $L_{\mathrm{loc}}$ for the speckle potential evaluated by fitting the densities from Sec. 3 to Eq. (29) in the interval (a) $x\in [35:100]x_0$ and (b) $x\in [100:360]x_0$. The gray shaded area indicates the time accessed in the experiment. The experimental values for $L_{\mathrm{loc}}$ are taken from Ref. [29].

Figure 13

Same as Fig. 12, but plotted as function of inverse time $1/t$ for the interval $x\in [100:360]x_0$. The horizontal dashed line shows the analytic prediction [Eq. (30)]. The dashed purple line following the points of scenario C is a fit to a function $f(1/t)=a{(1/t)}^b$ with $b\approx 0.35$.

Figure 14

(a) The width $\mathrm{\Delta }x\left(t\right)$ [Eq. (15)] as a function of time for different tolerances $\mathrm{tol}$ that result in different average time steps [Eq. (A3)] for a single realization of a Gaussian-correlated disorder for scenario C (see Fig. 2). The gray colored area represents one standard deviation around the mean value resulting from an ensemble of 50 disorder realizations. (b) The corresponding time step for different tolerances.

Figure 15

The width $\mathrm{\Delta }x\left(t\right)$ [Eq. (15)] as a function of time for different space discretizations for a single realization of scenario C (see Fig. 2). The numerical values are given up to the third significant digit. The gray colored area represents one standard deviation around the mean value resulting from an ensemble of 50 disorder realizations of the Gaussian-correlated potential.

Figure 16

The width $\mathrm{\Delta }x\left(t\right)$ [Eq. (15)] as a function of time for different box sizes for a single realization of scenario C (see Fig. 2). The box spans $[-x_{\max }:x_{\max }]$. The light-colored area represents one standard deviation around the mean value resulting from an ensemble of 50 disorder realizations