We study the Bose-Hubbard model using the finite size density matrix renormalization group method. We obtain a complete phase diagram for a system in the presence of a harmonic trap and compare it with that of the homogeneous system.
The superfluid to the Mott insulator phase transition is investigated using different experimental signatures of these phases in quantities such as momentum distribution, visibility, condensate fraction and the total number of bosons at a particular density. The relationships between the various experimental signatures and the phase diagram are highlighted.