Schrödinger cat states and steady states in subharmonic generation with Kerr nonlinearities

We discuss general properties of the equilibrium state of parametric down-conversion in superconducting quantum circuits with detunings and Kerr anharmonicities, in the strongly nonlinear regime. By comparing moments of the steady state and those of a Schrodinger cat, we show that true Schrodinger cats cannot survive in the steady state if there is any single-photon loss. A delta-function "catlike" steady-state distribution can be formed, but this only exists in the limit of an extremely large nonlinearity. The steady state is a mixed state, which is more complex than a mixture or linear combination of delta functions, and the purity of which is reduced by driving. We expect this general behavior to occur in other driven, dissipative quantum subharmonic nonequilibrium open systems.