Bose-Einstein-condensate superfluid-Mott-insulator transition in an optical lattice

We present an analytical model for a cold bosonic gas on an optical lattice (with densities of the order of 1 particle per site), targeting the critical regime of the Bose-Einstein-condensate superfluid-Mott-insulator transition. We focus on the computation of the one-body density matrix and its Fourier transform, the momentum distribution which is directly obtainable from "time-of-flight" measurements. The expected number of particles with zero momentum may be identified with the condensate population if it is close to the total number of particles. Our main result is an analytic expression for this observable, interpolating between the known results valid for the two regimes separately: the standard Bogoliubov approximation valid in the superfluid regime and the strong-coupling perturbation theory valid in the Mott regime.