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Few-photon transport via waveguide-coupled local quantum systems hasattracted extensive theoretical and experimental studies. Most of the study hasfocused on atomic or atomic-like local quantum systems due to their stronglight-matter interaction useful for quantum applications. Here, we studyfew-photon transport via a waveguide-coupled multimode optical cavity withsecond-order bulk nonlinearity. We develop a Feynman diagram approach andcompute the scattering matrix of the one- and two-photon transport. Based onthe calculated scattering matrix, we show highly nonclassical photonic effects,including photon blockade and $\pi-$conditional phase shift, are achievable inthe waveguide-coupled multimode optical cavity system via quantum interferenceand linear response engineering. Our results might lead to significantapplications of quantum photonic circuits in all-optical quantum informationprocessing and quantum network protocols.

Few-photon transport via a multimode nonlinear cavity: theory and applications