  We explore the use of first and second order same-time atomic spatial
correlation functions as a diagnostic for probing the small scale spatial
structure of atomic samples trapped in optical lattices. Assuming an ensemble
of equivalent atoms, properties of the local wave function at a given lattice
site can be measured using same-position first-order correlations. Statistics
of atomic distributions over the lattice can be measured via two-point
correlations, generally requiring the averaging of multiple realizations of
statistically similar but distinct realizations in order to obtain sufficient
signal to noise. Whereas two-point first order correlations are fragile due to
phase fluctuations from shot-to-shot in the ensemble, second order correlations
are robust. We perform numerical simulations to demonstrate these diagnostic
tools.

Quantum Physics