While optical lattices are well-established systems, the quantum nature of light is neglected in all setups so far. We show theoretically that the light quantization significantly broadens the physical picture enabling to go beyond effects expected in many-body dissipative systems.
We prove that the quantum backaction of weak global measurement constitutes a novel source of competitions in many-body systems (in addition to atom tunnelling and interactions) [1,2], leading to novel effects: multimode oscillations of macroscopic superposition states, protection and break-up of fermion pairs , as well as generation of antiferromagnetic states . Novel processes beyond the standard Hubbard models can be designed by the measurement, entering the field of non-Hermitian many-body physics: long-range correlated pair tunnelling and Raman-like second-order transitions beyond the typical quantum Zeno dynamics . We demonstrate the generation of multipartite mode entanglement and feedback control of many-body states .
The quantization of optical lattice potentials enables quantum simulations of various long-range interacting systems unobtainable using classical optical lattices . This leads to new quantum phases (dimers, trimers, etc. of matter waves similar to valence bond solids) beyond density orders (e.g. supersolids and density waves) directly benefiting from the collective light-matter interaction.
 W. Kozlowski, S. F. Caballero-Benitez, and I. B. Mekhov, Scientific Rep. 7, 42597 (2017);
 G. Mazzucchi, W. Kozlowski, S. F. Caballero-Benitez, T. J. Elliott, and I. B. Mekhov, Phys. Rev. A 93, 023632 (2016);
W. Kozlowski and I. B. Mekhov, Phys. Rev. A 94, 012123 (2016);
 G. Mazzucchi, S. F. Caballero-Benitez, and I. B. Mekhov, Scientific Rep. 6, 31196 (2016);
 G. Mazzucchi, S. F. Caballero-Benitez, D. A. Ivanov, and I. B. Mekhov, Optica (OSA) 3, 1213 (2016);
 S. F. Caballero-Benitez and I. B. Mekhov, Phys. Rev. Lett. 115, 243604 (2015);
S. F. Caballero-Benitez, G. Mazzucchi, and I. B. Mekhov, Phys. Rev. A 93, 063632 (2016).