Theory of implementation of an impedance-matched $\Lambda$ system in circuit QED*

In one-dimensional optical setups, light-matter interaction is drastically enhanced by the interference between the incident and scattered fields. Particularly, in an impedance-matched $\Lambda$-type three-level system, which has two identical radiative decay rates from the top level and interacts with a semi-infinite one-dimensional field in reflection geometry, a single photon deterministically induces the Raman transition and switches the electronic state of the system. Here we theoretically investigate a circuit QED system composed of a driven superconducting qubit and a resonator in the dispersive regime. We show that the dressed states of this system constitute an impedance-matched $\Lambda$ system under a proper choice of the frequency and power of the qubit drive. When we apply a resonant probe field to this system, it is down-converted nearly perfectly after a single reflection as long as the probe power is sufficiently weak. This indicates a deterministic quantum dynamics induced by single photons, which is applicable, for example, to the detection of single microwave photons and the bidirectional quantum memory (swapping) between a microwave photon and a superconducting qubit.