QND measurements and state preparation in quantum gases by light detection

We consider light scattering from ultracold atoms trapped in an optical lattice into a cavity. The measurement of photons leaking out the cavity provides a quantum nondemolition (QND) access to various atomic variables. Depending on the chosen geometry the time resolved light detection projects the atomic quantum state to various atom-number squeezed and Schroedinger cat states. Modifications of the atomic and light properties at a single quantum trajectory are demonstrated. The quantum structure of final states can be revealed by further observations of the same sample.