Anderson Localization of Expanding Bose-Einstein Condensates in Random Potentials

We show that the expansion of an initially confined interacting 1D Bose-Einstein condensate can exhibit Anderson localization in a weak random potential with correlation length ${\ensuremath{\sigma}}_{R}$. For speckle potentials the Fourier transform of the correlation function vanishes for momenta $k>2/{\ensuremath{\sigma}}_{R}$ so that the Lyapunov exponent vanishes in the Born approximation for $k>1/{\ensuremath{\sigma}}_{R}$. Then, for the initial healing length of the condensate ${\ensuremath{\xi}}_{\mathrm{in}}>{\ensuremath{\sigma}}_{R}$ the localization is exponential, and for ${\ensuremath{\xi}}_{\mathrm{in}}<{\ensuremath{\sigma}}_{R}$ it changes to algebraic.

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