Collective dynamics and expansion of a Bose-Einstein condensate in a random potential

We investigate the dynamics of a Bose-Einstein condensate in the presence of a random potential created by optical speckles. We first consider the effect of a weak disorder on the dipole and quadrupole collective oscillations, finding uncorrelated frequency shifts of the two modes with respect to the pure harmonic case. This behavior, predicted by a sum-rules approach, is confirmed by the numerical solution of the Gross-Pitaevskii equation. Then we analyze the role of disorder on the one-dimensional expansion in an optical guide, discussing possible localization effects. Our theoretical analysis provides a useful insight into the recent experiments performed at LENS [J. E. Lye, L. Fallani, M. Modugno, D. S. Wiersma, C. Fort, and M. Inguscio, Phys. Rev. Lett. 95, 070401 (2005); C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, Phys. Rev. Lett. 95, 170410 (2005)].

Figure 1

(Color online) Typical shape of the potentials considered in this paper: (a) (blue detuned) speckles (red speckles would have the same shape but reversed sign), (b) Gaussian random, and (c) periodic. In all cases, the correlation length is $l_c=10\mathrm{\mu }\mathrm{m}$.

Figure 2

(Color online) (top) Column density of a typical ground-state configuration in the presence of a red-detuned speckle potential $(V_0=2.5\hslash {\omega }_z$). (bottom) Solid line: profile of the combined potential $V_R+V_{ho}$ (dashed line).

Figure 3

(Color online) Top: dipole (filled circles) and quadrupole (empty circles) frequency shifts for 100 different realizations of the speckle potential as obtained from the sum rules. Bottom: their probability distribution $P$ for 1000 realizations (left: dipole, right: quadrupole); the dashed lines are a Gaussian fit.

Figure 4

(Color online) Top: dipole (filled circles) and quadrupole (empty circles) frequency shifts for 100 different realizations of the Gaussian random potential as obtained from the sum rules. Bottom: their probability distribution $P$ for 1000 realizations (left: dipole, right: quadrupole); the dashed lines are a Gaussian fit.

Figure 5

(Color online) Ratios between quadrupole and dipole shifts (top) and the corresponding frequencies (bottom) for speckle (filled circles) and periodic (empty circles) potentials with $l_c=10\mathrm{\mu }\mathrm{m}$. Note that although the shifts for the periodic case are correlated in sign, the two frequencies are uncorrelated and depend on the actual position of the condensate in the periodic potential.

Figure 6

(Color online) Dipole (top) and quadrupole (bottom) frequency shifts as obtained from the GP equation (squares) and compared to the sum-rules predictions (circles), for ten different speckle realizations. The agreement is remarkable [25].

Figure 7

(Color online) Dipole oscillations in the speckle potential $(V_0=2.5\hslash {\omega }_z)$ for three different displacements $\mathrm{\Delta }z$ of the harmonic potential $(\mathrm{\Delta }z=6,15,24\mathrm{\mu }\mathrm{m})$. The insets show a density plot for $\mathrm{\Delta }z=24\mathrm{\mu }\mathrm{m}$ at $t=75\mathrm{ms}$ (left), $\mathrm{\Delta }z=15\mathrm{\mu }\mathrm{m}$ at $t=250\mathrm{ms}$ (center), and $\mathrm{\Delta }z=6\mathrm{\mu }\mathrm{m}$ at $t=500\mathrm{ms}$ (right).

Figure 8

(Color online) (a)–(d) Density profiles of the condensate (red continuous line) during the expansion in the waveguide in the presence of a red-detuned speckle potential [shown in (e)], for different times [$t=0,25,50,75\mathrm{ms}$, from (a) to (d)], compared to the free-expansion case (blue dashed line). Note that the $y$-axis scale changes from (a) to (d).

Figure 9

(Color online) (a)–(d) Density profiles of the condensate (red continuous line) during the expansion in the waveguide in the presence of a blue-detuned speckle potential [shown in (e)], for different times [$t=0,25,50,75\mathrm{ms}$, from (a) to (d)], compared to the free expansion case (blue dashed line). Note that the $y$-axis scale changes from (a) to (d).

Figure 10

(Color online) Density plots of the function $q(k,\alpha ,\beta )$ (see text) for a potential well [(a) and (b)] and a barrier [(c) and (d)], as a function of the incident momentum $k$ and the length scale $\beta$ of the potential. The maximum value of $k$ in the figures corresponds to an energy $E=U_0$. Left and right columns refer to different potential intensities: (a)–(c) $\alpha =0.2$, (b)–(d) $\alpha =1$. The plot in Figs. 8, 9 must be compared to the case $\beta =1$ in (a) and (c), respectively. Dark regions indicate complete reflection or transmission, light gray (color online) corresponds to a $50\%$ transparency.

Figure 11

(Color online) (a)–(d) Density profiles of the condensate (red continuous line) during the expansion in the waveguide in the presence of a periodic lattice [shown in (e)], for different times [$t=0,25,50,75\mathrm{ms}$, from (a) to (d)], compared to the free-expansion case (blue dashed line). Note that the $y$-axis scale changes from (a) to (d).

Figure 12

(Color online) Continuous line (red): autocorrelation function (top) and intensity distribution (bottom) for the speckle potential in Fig. 1. The (blue) dashed lines represent the expected average values over several realizations.

Figure 13

(Color online) Autocorrelation function (top) and intensity distribution (bottom) for the Gaussian random potential in Fig. 1. The blue dashed lines represent the expected average values over several realizations.

Figure 14

(Color online) Autocorrelation function (top) and intensity distribution (bottom) for the periodic potential in Fig. 1. The blue dashed lines represent the expected average values over several realizations.