  Light enables manipulating many-body states of matter, and atoms trapped in
optical lattices is a prominent example. However, quantum properties of light
are completely neglected in all quantum gas experiments. Extending methods of
quantum optics to many-body physics will enable phenomena unobtainable in
classical optical setups. We show how using the quantum optical feedback
creates strong correlations in bosonic and fermionic systems. It balances two
competing processes, originating from different fields: quantum backaction of
weak optical measurement and many-body dynamics, resulting in stabilized
density waves, antiferromagnetic and NOON states. Our approach is extendable to
other systems promising for quantum technologies.
