  Few-photon transport via waveguide-coupled local quantum systems has
attracted extensive theoretical and experimental studies. Most of the study has
focused on atomic or atomic-like local quantum systems due to their strong
light-matter interaction useful for quantum applications. Here, we study
few-photon transport via a waveguide-coupled multimode optical cavity with
second-order bulk nonlinearity. We develop a Feynman diagram approach and
compute the scattering matrix of the one- and two-photon transport. Based on
the calculated scattering matrix, we show highly nonclassical photonic effects,
including photon blockade and $\pi-$conditional phase shift, are achievable in
the waveguide-coupled multimode optical cavity system via quantum interference
and linear response engineering. Our results might lead to significant
applications of quantum photonic circuits in all-optical quantum information
processing and quantum network protocols.
