We study the dynamics of a general quartic interaction Hamiltonian under the influence of dissipation and non-classical driving. We show that this scenario could be realised 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 non-linearity, therefore, does not significantly influence the highly squeezed intracavity microwave field but, for a range of realistic parameters, enables characterisation of itinerant squeezed fields.
@article{arxiv.1501.01009,
title = {Enhancement and state tomography of a squeezed vacuum with circuit quantum electrodynamics},
author = {Matthew Elliott and Eran Ginossar},
journal= {arXiv preprint arXiv:1501.01009},
year = {2015}
}