Correlation functions of cold bosons in an optical lattice

Motivated by the experimental observation of collapses and revivals of Bose matter wave field, we investigate correlation functions of cold bosons in an optical lattice. Within a simple model we examine two kinds of states: one that employs the commonly used notion of coherent states, and one that obeys the total number of atoms conservation. We identify rare situations at which these states behave differently. Typically, however, their predictions coincide and so: As a function of "interaction time" the interference pattern in the density undergoes collapse and revival. Exactly at revival times the system mimics the ideal gas case, in which all correlation functions factorize, while in the collapsed phase of the evolution the system effectively behaves as if initially there was no long-range coherence. Even in the latter case though, an interference pattern should be seen in a single experiment. We stress the role of column averaging, which in fact corresponds to an averaged observation of an ensemble of two-dimensional realizations. We also note that, contrary to the common belief, an interference pattern should also be seen in a single observation of a Mott state.