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We calculate entropy-temperature curves for interacting bosons in unit filledoptical lattices for both homogeneous and harmonically trapped situations, anduse them to understand how adiabatic changes in the lattice depth affect thetemperature of the system. In a translationally invariant lattice, the zerotunneling limit facilitates a rather detailed analytic description. Unlike thenon-interacting bosonic system which is always cooled upon adiabatic loadingfor low enough initial temperature, the change in the excitation spectruminduced by interactions can lead to heating. Finite tunneling helps to reducethis heating. Finally, we study the spatially inhomogeneous system confined ina parabolic potential and show that the presence of the trap can significantlyreduce the final available temperature, due to the non-vanishing superfluidcomponent at the edge of the cloud which is present in trapped systems.Comment: 9 pages and 6 figures. Two typos in Sec.IIIA were corrected and some  references were update

The role of interactions, tunneling, and harmonic confinement on the adiabatic loading of bosons in an optical lattice