  We study off-resonant collective light scattering from ultracold atoms
trapped in an optical lattice. Scattering from different atomic quantum states
creates different quantum states of the scattered light, which can be
distinguished by measurements of the spatial intensity distribution, quadrature
variances, photon statistics, or spectral measurements. In particular,
angle-resolved intensity measurements reflect global statistics of atoms (total
number of radiating atoms) as well as local statistical quantities (single-site
statistics even without an optical access to a single site) and pair
correlations between different sites. As a striking example we consider
scattering from transversally illuminated atoms into an optical cavity mode.
For the Mott insulator state, similar to classical diffraction, the number of
photons scattered into a cavity is zero due to destructive interference, while
for the superfluid state it is nonzero and proportional to the number of atoms.
Moreover, we demonstrate that light scattering into a standing-wave cavity has
a nontrivial angle dependence, including the appearance of narrow features at
angles, where classical diffraction predicts zero. The measurement procedure
corresponds to the quantum non-demolition (QND) measurement of various atomic
variables by observing light.
