  We use the Bogoliubov theory of atoms in an optical lattice to study the
approach to the Mott-insulator transition. We derive an explicit expression for
the superfluid density based on the rigidity of the system under phase
variations. This enables us to explore the connection between the quantum
depletion of the condensate and the quasi-momentum distribution on the one hand
and the superfluid fraction on the other. The approach to the insulator phase
may be characterized through the filling of the band by quantum depletion,
which should be directly observable via the matter wave interference patterns.
We complement these findings by self-consistent Hartree-Fock-Bogoliubov-Popov
calculations for one-dimensional lattices including the effects of a parabolic
trapping potential.
