  Superconducting circuits are a competitive platform for quantum computation
because they offer controllability, long coherence times and strong
interactions - properties that are essential for the study of quantum materials
comprising microwave photons. However, intrinsic photon losses in these
circuits hinder the realization of quantum many-body phases. Here we use
superconducting circuits to explore strongly correlated quantum matter by
building a Bose-Hubbard lattice for photons in the strongly interacting regime.
We develop a versatile method for dissipative preparation of incompressible
many-body phases through reservoir engineering and apply it to our system to
stabilize a Mott insulator of photons against losses. Site- and time-resolved
readout of the lattice allows us to investigate the microscopic details of the
thermalization process through the dynamics of defect propagation and removal
in the Mott phase. Our experiments demonstrate the power of superconducting
circuits for studying strongly correlated matter in both coherent and
engineered dissipative settings. In conjunction with recently demonstrated
superconducting microwave Chern insulators, we expect that our approach will
enable the exploration of topologically ordered phases of matter.
