Anderson localization of a weakly interacting one dimensional Bose gas

We consider the phase coherent transport of a quasi one-dimensional beam of Bose-Einstein condensed particles through a disordered potential of length L. Among the possible different types of flow identified in [T. Paul et al., Phys. Rev. Lett. 98, 210602 (2007)], we focus here on the supersonic stationary regime where Anderson localization exists. We generalize the diffusion formalism of Dorokhov-Mello-Pereyra-Kumar to include interaction effects. It is shown that interactions modify the localization length and also introduce a length scale L* for the disordered region, above which most of the realizations of the random potential lead to time dependent flows. A Fokker-Planck equation for the probability density of the transmission coefficient that takes this new effect into account is introduced and solved. The theoretical predictions are verified numerically for different types of disordered potentials. Experimental scenarios for observing our predictions are discussed.