  We study properties of ultra-cold bosonic atoms in one, two and three
dimensional optical lattices by large scale quantum Monte Carlo simulations of
the Bose Hubbard model in parabolic confinement potentials. Local phase
structures of the atoms are shown to be accessible via a well defined local
compressibility, quantifying a global response of the system to a local
perturbation. An indicator for the presence of extended Mott plateaux is shown
to stem from the shape of the coherent component of the momentum distribution
function, amenable to experimental detection. Additional fine structures in the
momentum distribution are found to appear unrelated to the local phase
structure, disproving previous claims. We discuss limitations of local
potential approximations for confined Bose gases, and the absence of quantum
criticality and critical slowing down in parabolic confinement potentials, thus
accounting for the fast dynamics in establishing phase coherence in current
experiments. In contrast, we find that flat confinement potentials allow
quantum critical behavior to be observed already on moderately sized optical
lattices. Our results furthermore demonstrate, that the experimental detection
of the Mott transition would be significantly eased in flat confinement
potentials.
