Driving with squeezed vacuum in circuit quantum electrodynamics

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.