  In a pioneering experiment, Hanbury Brown and Twiss (HBT) demonstrated that
noise correlations could be used to probe the properties of a (bosonic)
particle source through quantum statistics; the effect relies on quantum
interference between possible detection paths for two indistinguishable
particles. HBT correlations -- together with their fermionic counterparts --
find numerous applications, ranging from quantum optics to nuclear and
elementary particle physics. Spatial HBT interferometry has been suggested as a
means to probe hidden order in strongly correlated phases of ultracold atoms.
Here we report such a measurement on the Mott insulator phase of a rubidium
Bose gas as it is released from an optical lattice trap. We show that strong
periodic quantum correlations exist between density fluctuations in the
expanding atom cloud. These spatial correlations reflect the underlying
ordering in the lattice, and find a natural interpretation in terms of a
multiple-wave HBT interference effect. The method should provide a useful tool
for identifying complex quantum phases of ultracold bosonic and fermionic
atoms.
