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