Microwave degenerate parametric down-conversion with a single cyclic three-level system in circuit QED

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 the 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 modes in the resonator. In our system, the PDC efficiency can be much larger than that in the case of two-level system interacting with two-mode transmission line resonator [K. Moon and S. M. Girvin, Phys. Rev. Lett. {\bf 95}, 140504 (2005)]. With the effective coupling between those two resonator modes, a coherent driving of the second harmonic mode can lead to the squeezing and bunching effect of the fundamental one.