  We investigate numerically the zero-temperature physics of the one-dimensional Bose-Hubbard model in an incommensurate cosine potential, recently realized in experiments with cold bosons in optical superlattices L. Fallani et al., Phys. Rev. Lett. 98, 130404, (2007)]. An incommensurate cosine potential has intermediate properties between a truly periodic and a fully random potential, displaying a characteristic length scale (the quasi-period) which is shown to set a finite lower bound to the excitation energy of the system at special incommensurate fillings. This leads to the emergence of gapped incommensurate band-insulator (IBI) phases along with gapless Bose-glass (BG) phases for strong quasi-periodic potential, both for hardcore and softcore bosons. Enriching the spatial features of the potential by the addition of a second incommensurate component appears to remove the IBI regions, stabilizing a continuous BG phase over an extended parameter range. Moreover we discuss the validity of the local-density approximation in presence of a parabolic trap, clarifying the notion of a local BG phase in a trapped system; we investigate the behavior of first- and second-order coherence upon increasing the strength of the quasi-periodic potential; and we discuss the ab-initio derivation of the Bose-Hubbard Hamiltonian with quasi-periodic potential starting from the microscopic Hamiltonian of bosons in an incommensurate superlattice. 
Bosons in one-dimensional incommensurate superlattices