Theory of deterministic down-conversion of single photons occurring at an impedance-matched Λ system

We theoretically investigate the optical response of a Λ system interacting with a semi-infinite waveguide field. In particular, we focus on the case of an impedance-matched Λ system, in which the two decay rates from the top level are identical. We derive the analytical formulae to evaluate the amplitude and the power spectrum of the output field, and present numerical results assuming an implementation by a circuit quantum electrodynamics system. For a low input power (single-photon regime), input photons are down-converted deterministically by inducing the Raman transition in the Λ system. In contrast, for a high input power (multi-photon regime), the conversion efficiency decreases because of the saturation of the Λ system.