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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 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.