  We compute the phase diagram of the one-dimensional Bose-Hubbard model with a
quasi-periodic potential by means of the density-matrix renormalization group
technique. This model describes the physics of cold atoms loaded in an optical
lattice in the presence of a superlattice potential whose wave length is
incommensurate with the main lattice wave length. After discussing the
conditions under which the model can be realized experimentally, the study of
the density vs. the chemical potential curves for a non-trapped system unveils
the existence of gapped phases at incommensurate densities interpreted as
incommensurate charge-density wave phases. Furthermore, a localization
transition is known to occur above a critical value of the potential depth V_2
in the case of free and hard-core bosons. We extend these results to soft-core
bosons for which the phase diagrams at fixed densities display new features
compared with the phase diagrams known for random box distribution disorder. In
particular, a direct transition from the superfluid phase to the Mott
insulating phase is found at finite V_2. Evidence for reentrances of the
superfluid phase upon increasing interactions is presented. We finally comment
on different ways to probe the emergent quantum phases and most importantly,
the existence of a critical value for the localization transition. The later
feature can be investigated by looking at the expansion of the cloud after
releasing the trap.
