Subdiffusive spreading of a Bose-Einstein condensate in random potentials

We study numerically the long-time dynamics of a one-dimensional Bose-Einstein condensate expanding in a speckle or impurity disorder potential. Using the mean-field Gross-Pitaevskii equation, we demonstrate subdiffusive spreading of the condensate for long times. We find that interaction-assisted hopping between normal modes leads to this subdiffusion. A possible (partial) reason why the root-mean-square (rms) width saturates in the experiment [Nature (London) 453, 891 (2008)] is provided. We suggest that observing both the participation length and the rms width of a condensate, rather than only the rms width, could provide a more complete description of the long-time behavior of ultracold atoms in disorder potentials. Our study confirms subdiffusive spreading in spatially continuous disordered interacting models and highlights new features which spatially discrete models do not possess.

Figure 1

Averaged BEC density profiles after $t=150/\omega$ in a speckle potential. The densities are averages of ${\left|\psi \right|}^2$ over 10 realizations in the interval with width $0.2a$, where $a={(\hslash /m\omega )}^{1/2}$. The curves for $t=150/\omega$ were computed with the CNFD (red solid) and TSSP algorithms (blue dot), respectively. The parameters are $V_R=0.12\mu ,\hslash \omega =0.024\mu ,{\sigma }_R=0.65{\xi }_{\mathrm{in}}$. The space grid is $\Delta z=1.2{\sigma }_R$ and time grids are $\Delta t=0.00005/\omega$ (CNFD algorithm) and $\Delta t=0.002/\omega$ (TSSP).

Figure 2

Averaged BEC density profiles after various expansion times in a speckle potential. The densities are averages of ${\left|\psi \right|}^2$ over 7 realizations in the interval with width $0.4a$, where $a={(\hslash /m\omega )}^{1/2}$. Top panel: Density profiles at $t={10}^2/\omega ,{10}^3/\omega ,{10}^4/\omega ,{10}^5/\omega$ (from bottom to top at $z/a=500$). The magenta line (bottom line at $z/a=250$): The contact interaction was set to 0 after $t_0=10/\omega$ for total evolution time $t={10}^5/\omega$. The parameters are $V_R=0.12\mu ,\hslash \omega =0.024\mu$ (i.e., $g=243ah\omega /\pi N$), ${\sigma }_R=0.65{\xi }_{\mathrm{in}}$. Bottom panel: Development of a central plateau in the density profiles at $t=3\times {10}^4/\omega$ with small coupling constant. Other parameters are $V_R=4.85\hslash \omega ,{\sigma }_R=0.051a$.

Figure 3

Top panel: Evolution of the rms width $L$ and participation length $P$ [39]. At long times, $L$ and $P$ follow the power laws Eq. (4) with exponents ${\alpha }_1$ and ${\alpha }_2$ about $0.2\sim 0.3$ (fit for ${10}^2/\omega \le t\le {10}^5/\omega$). Inset of top panel: $L$ integrated over a space domain $[-4000a,4000a]$ (red solid) and $L$ integrated up to typical resolution limit of experiments [22] (black dashed; see text). All plots are averaged over 7 realizations. Bottom panel: Evolution of $P$ for the noninteracting case (black dash-dotted), attractive interactions (red solid), and repulsive interactions (blue dashed) with the same NM as initial state. All other parameters are as in Fig. 2.

Figure 4

Top panel: Dependence of $p(R_{\nu ,{\rho }_0}<1)$ on the interaction energy $gn$ averaged over 50 realizations in a domain with width $123a$ (solid lines). The dashed line indicates decay proportional to $n$. Parameters are $V_R=0.24\mu$ (black with $\times$), $0.34\mu$ (blue with $+$), $0.48\mu$ [red (indistinguishable)], $\hslash \omega =0.024\mu ,{\sigma }_R=0.65{\xi }_{\mathrm{in}}$. Bottom panel: Probability densities (PDF) of the distance between strongest coupling pairs which satisfies $R_{\nu ,{\rho }_0}<0.1$ for $gn/\hslash \omega =14$ (colors indicate the same parameters as in the top panel). Inset: Multipeak structure of a typical strongest coupling NM pair (black dashed and magenta lines).

Figure 5

Evolution of the participation length $P$ with generalized contact interaction for $\sigma =0.5$ for one realization. At long time, $P$ follows a power law with exponent roughly about $0.2\sim 0.3$ (fit for ${10}^2/\omega \le t\le {10}^4/\omega$). Other parameters are $V_h=4.86$, ${\sigma }_h=0.05$, $\beta =100$.

Figure 6

Top panel: Evolution of the participation length $P$ for impurity disorder for one realization. At long times, $P$ follows a power law with exponent ${\alpha }_2$ about $0.2\sim 0.3$ (fit for ${10}^2/\omega \le t\le {10}^5/\omega$). Other parameters are $V_R=0.12\mu ,\hslash \omega =0.024\mu ,{\sigma }_R=0.65{\xi }_{\mathrm{in}},d_0=a$. Bottom panel: Evolution of $P$ for the noninteracting case (black dashed-dotted), attractive interactions (red solid), and repulsive interactions (blue dashed) with the same initial state (some NM). Other parameters are as the top panel.