Enhancement and state tomography of a squeezed vacuum with circuit quantum electrodynamics

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.