Stability of dissipatively-prepared Mott insulators of photons

Reservoir engineering is a powerful approach for using controlled driven-dissipative dynamics to prepare target quantum states and phases. In this work, we study a paradigmatic model that can realize a Mott insulator of photons in its steady state. We show that, while in some regimes its steady state approximates a Mott-insulating ground state, this phase can become unstable through a nonequilibrium transition towards a coherent yet nonclassical limit-cycle phase, driven by doublon excitations. This instability is completely distinct from the ground-state Mott-insulator to superfluid transition. This difference has dramatic observable consequences and leads to an intrinsic fragility of the steady-state Mott phase: a fast pump compared to losses is required to sustain the phase, but also determines a small critical hopping. We identify unique features of the steady-state Mott phase and its instability that distinguish them from their ground-state counterpart and can be measured in experiments.