  Clarifying the interplay of interactions and disorder is fundamental to the
understanding of many quantum systems, including superfluid helium in porous
media, granular and thin-film superconductors, and light propagating in
disordered media. One central aspect for bosonic systems is the competition
between disorder, which tends to localize particles, and weak repulsive
interactions, which instead have a delocalizing effect. Since the required
degree of independent control of the disorder and of the interactions is not
easily achievable in most available physical systems, a systematic experimental
investigation of this competition has so far not been possible. Here we employ
an ultracold atomic Bose-Einstein condensate with tunable repulsive
interactions in a quasi-periodic lattice potential to study this interplay in
detail. We characterize the entire delocalization crossover through the study
of the average local shape of the wavefunction, the spatial correlations, and
the phase coherence. Three different regimes are identified and compared with
theoretical expectations: an exponentially localized Anderson glass, the
formation of locally coherent fragments, as well as a coherent, extended state.
Our results illuminate the role of weak repulsive interactions on disordered
bosonic systems and show that the system and the techniques we employ are
promising for further investigations of disordered systems with interactions,
also in the strongly correlated regime.
