  We show that light scattering from an ultracold gas reveals not only density
correlations, but also matter-field interference at its shortest possible
distance in an optical lattice, which defines key properties such as tunneling
and matter-field phase gradients. This signal can be enhanced by concentrating
probe light between lattice sites rather than at density maxima. As addressing
between two single sites is challenging, we focus on global nondestructive
scattering, allowing probing order parameters, matter-field quadratures and
their squeezing. The scattering angular distribution displays peaks even if
classical diffraction is forbidden and we derive generalized Bragg conditions.
Light scattering distinguishes all phases in the Mott insulator - superfluid -
Bose glass phase transition.
