Localization of a non interacting quantum wave packet in one-dimensional disordered potentials

We revisit the recent experiments [Billy et al., Nature 453, 891 (2008); Roati et al., Nature 453, 895 (2008)] that have lead to the observation of Anderson localization for matter waves in one-dimensional speckle and quasiperiodic potentials. We show that in presence of localization the tails of the asymptotic density distribution can be described by the diagonal term of the projection over the eigenstates of the disordered potential, and that this is equivalent of assuming a phase randomization of the off-diagonal terms. This provides an alternative and complementary viewpoint to the usual description in terms of localization of single plane wave components as a result of multiple scattering from the random potential. For the quasiperiodic case, we also discuss the implications of using continuous or discrete models.