We study the dynamics of a general quartic interaction Hamiltonian under the influence of dissipation and non-classical driving. In this scenario, we show that an effective Hartree-type decoupling yields a good approximation to the dynamics of the system. We find that the stationary states are squeezed vacuum states of the non-interacting system which are enhanced by the Q-factor of the cavity. We show that this effective interaction could be realised with a cascaded superconducting cavity-qubit system in the strong dispersive regime in a setup similar to recent experiments. The qubit non-linearity, therefore, does not significantly influence the highly squeezed intracavity microwave field but, for a range of parameters, enables quantum state tomography of the cavity.
We study the dynamics of a general quartic interaction Hamiltonian under the influence of dissipation and nonclassical driving. We show that this scenario could be realized with a cascaded superconducting cavity-qubit system in the strong dispersive regime in a setup similar to recent experiments. In the presence of dissipation, we find that an effective Hartree-type decoupling with a Fokker-Planck equation yields a good approximation. We find that the stationary state is approximately a squeezed vacuum, which is enhanced by the
Q factor of the cavity but conserved by the interaction. The qubit nonlinearity, therefore, does not significantly influence the highly squeezed intracavity microwave field but, for a range of realistic parameters, enables characterization of itinerant squeezed fields.