Localization landscape for interacting Bose gases in one-dimensional speckle potentials

While the properties and the shape of the ground state of a gas of ultracold bosons are well understood in harmonic potentials, they remain for a large part unknown in the case of random potentials. Here, we use the localization-landscape (LL) theory to study the properties of the solutions of the Gross-Pitaevskii equation (GPE) in one-dimensional (1D) speckle potentials. In the cases of intermediate and strongly attractive interactions, we find an approximate relation which allows to evaluate the disorder-averaged localization length from the nonlinear coefficient of the GPE. For weakly repulsive interactions, we illustrate that the ground state $\psi_{0}$ of the quasi-1D GPE can be understood as a superposition of a finite number of single-particle states. For intermediate repulsive interactions, we show numerically that, in the smoothing regime, $\psi_{0}$ can be predicted using a Thomas-Fermi-like approach involving the effective potential, which is given by the reciprocal of the LL. Moreover, we show that, in the Lifshitz glass phase, the particle density and the chemical potential can be well estimated by means of the LL. Our approach can be applied for any random potential endowed with finite-range correlations and can be generalized to higher-dimensional systems.