  We study theoretically the parametric down conversion and squeezing of
microwaves using cavity quantum electrodynamics of a superconducting Cooper
pair box (CPB) qubit located inside a transmission line resonator. The
non-linear susceptibility \chi_2 describing three-wave mixing can be tuned by
dc gate voltage applied to the CPB and vanishes by symmetry at the charge
degeneracy point. We show that the coherent coupling of different cavity modes
through the qubit can generate a squeezed state. Based on parameters realized
in recent successful circuit QED experiments, squeezing of 95% ~ 13dB below the
vacuum noise level should be readily achievable.
