Anderson localization of elementary excitations in a one-dimensional Bose-Einstein condensate

We study the elementary excitations of a transversely confined Bose-Einstein condensate in presence of a weak axial random potential. We determine the localization length (i) in the hydrodynamical low energy regime; for a domain of linear densities ranging from the Tonks-Girardeau to the transverse Thomas-Fermi regime; in the case of a white noise potential and (ii) for all the range of energies, in the one-dimensional mean field regime, in the case where the randomness is induced by a series of randomly placed point-like impurities. We discuss our results in view of recent experiments in elongated BEG systems.