  We show that the effect of measurement back-action results in the generation
of multiple many-body spatial modes of ultracold atoms trapped in an optical
lattice, when scattered light is detected. The multipartite mode entanglement
properties and their nontrivial spatial overlap can be varied by tuning the
optical geometry in a single setup. This can be used to engineer quantum states
and dynamics of matter fields. We provide examples of multimode generalizations
of parametric down-conversion, Dicke, and other states, investigate the
entanglement properties of such states, and show how they can be transformed
into a class of generalized squeezed states. Further, we propose how these
modes can be used to detect and measure entanglement in quantum gases.
