  With the assistance of a single cyclic three-level system, which can be
realized by a superconducting flux qubit, we study theoretically the degenerate
microwave parametric down-conversion (PDC) in a superconducting transmission
line resonator with the fundamental and second harmonic modes involved. By
adiabatically eliminating the excited states of the three-level system, we
obtain an effective microwave PDC Hamiltonian for the two resonator modes in
such a circuit QED system. The corresponding PDC efficiency in our model can be
much larger than that in the similar circuit QED system based on a single
two-level superconducting qubit [K. Moon and S. M. Girvin, Phys. Rev. Lett.
{\bf 95}, 140504 (2005)]. Furthermore, we consider the squeezing and bunching
behavior of the fundamental mode resulting from the coherent drive to the
second harmonic one.
