  We describe a scheme for the efficient generation of microwave photon pairs
by parametric downconversion in a superconducting transmission line resonator
coupled to a Cooper pair box serving as an artificial atom. By properly tuning
the first three levels with respect to the cavity modes, the down-conversion
probability may become higher than in the most efficient schemes for optical
photons. We show this by numerically simulating the dissipative quantum
dynamics of the coupled cavity-box system and discuss the effects of dephasing
and relaxation. The setup analyzed here might form the basis for a future
on-chip source of entangled microwave photons.
