  In this paper we develop a theoretical description of the correlations
between ultra-cold bosons after free expansion from confinement in an optical
lattice. We consider the system evolution during expansion and give criteria
for a far field regime. We develop expressions for first and second order
two-point correlations based on a variety of commonly used approximations to
the many-body state of the system including Bogoliubov, meanfield decoupling,
and particle-hole perturbative solution about the perfect Mott-insulator state.
Using these approaches we examine the effects of quantum depletion and pairing
on the system correlations. Comparison with the directly calculated correlation
functions is used to justify a Gaussian form of our theory from which we
develop a general three-dimensional formalism for inhomogeneous lattice systems
suitable for numerical calculations of realistic experimental regimes.
