| We study the Bose-Hubbard (BH) model with a quasi-periodic modulation or on-site disorder of the Anderson type, using both exact diagonalization and a mean-field approximation. We find that in the weakly interacting regime (when the on-site interaction U is comparable to the tunneling strength J), the addition of either an incommensurate, second lattice or on-site disorder decreases the superfluid fraction but retains a large condensate fraction. These results demonstrate a localization transition to the weakly interacting Bose glass phase. In the strong interaction regime on the other hand, when U/J is larger than the critical value of the homogenous Mott transition, we find that the there is a critical strength for the incommensurate lattice/ disordered potential (Delta/J) above which a condensate and a superfluid are formed. In particular, the incommensurate lattice results indicate that there is a localization-delocalization transition already present in 1D for a critical non-zero value of Delta/J, contrary to the disordered case. This localization transition is the generalization of the Aubry-Andre model to interacting bosons. We explain the onset of a condensate (and a superfluid) by emphasizing the importance of the filling factor in the BH model. We further show how our results are affected by more realistic experimental considerations, such as a varying coupling constant J (due to the perturbations) and harmonic trapping potentials. |
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