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)]