  One of the most important issues in disordered systems is the interplay of
the disorder and repulsive interactions. Several recent experimental advances
on this topic have been made with ultracold atoms, in particular the
observation of Anderson localization, and the realization of the disordered
Bose-Hubbard model. There are however still questions as to how to
differentiate the complex insulating phases resulting from this interplay, and
how to measure the size of the superfluid fragments that these phases entail.
It has been suggested that the correlation function of such a system can give
new insights, but so far little experimental investigation has been performed.
Here, we show the first experimental analysis of the correlation function for a
weakly interacting, bosonic system in a quasiperiodic lattice. We observe an
increase in the correlation length as well as a change in shape of the
correlation function in the delocalization crossover from Anderson glass to
coherent, extended state. In between, the experiment indicates the formation of
progressively larger coherent fragments, consistent with a fragmented BEC, or
Bose glass.
