  When atoms are loaded into an optical lattice, the process of gradually
turning on the lattice is almost adiabatic. In this paper we investigate how
the temperature changes when going from the gapless superfluid phase to the
gapped Mott phase along isentropic lines. To do so we calculate the entropy in
the single-band Bose-Hubbard model for various densities, interaction strengths
and temperatures in one and two dimensions for homogeneous and trapped systems.
Our theory is able to reproduce the experimentally observed visibilities and
therefore strongly supports that current experiments remain in the quantum
regime for all considered lattice depths with low temperatures and minimal
heating.
