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We analyze the single-photon transport in a single-mode waveguide coupled toa whispering-gallery-type resonator interacting with a two-level atom. Thesingle-photon transport properties such as the transmission and reflectionamplitudes, as well as the resonator and the atom responses, are solved exactlyvia a real-space approach. The treatment includes the inter-mode backscatteringbetween the two degenerate whispering gallery modes of the resonator, and thedissipations of the resonator and the atom. We also show that a generalizedcritical coupling condition, that the single-photon transmission at the outputof the waveguide goes to zero on resonance for a matched system, holds for thefull coupled waveguide-ring resonator-atom system.

Fully quantum-mechanical analytic results for single-photon transport in  a single-mode waveguide coupled to a whispering-gallery resonator interacting  with a two-level atom