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Quantum states can be stabilized in the presence of intrinsic andenvironmental losses by either applying active feedback conditioned on anancillary system or through reservoir engineering. Reservoir engineeringmaintains a desired quantum state through a combination of drives and designedentropy evacuation. We propose and implement a quantum reservoir engineeringprotocol that stabilizes Fock states in a microwave cavity. This protocol isrealized with a circuit quantum electrodynamics platform where a Josephsonjunction provides direct, nonlinear coupling between two superconductingwaveguide cavities. The nonlinear coupling results in a single photon resolvedcross-Kerr effect between the two cavities enabling a photon number dependentcoupling to a lossy environment. The quantum state of the microwave cavity isdiscussed in terms of a net polarization and is analyzed by a measurement ofits steady state Wigner function.

Single-photon Resolved Cross-Kerr Interaction for Autonomous Stabilization of Photon-number States