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We propose and theoretically investigate a model to realize cascaded opticalnonlinearity with few atoms and photons in one-dimension (1D). The opticalnonlinearity in our system is mediated by resonant interactions of photons withtwo-level emitters, such as atoms or quantum dots in a 1D photonic waveguide.Multi-photon transmission in the waveguide is nonreciprocal when the emittershave different transition energies. Our theory provides a clear physicalunderstanding of the origin of nonreciprocity in the presence of cascadednonlinearity. We show how various two-photon nonlinear effects includingspatial attraction and repulsion between photons, background fluorescence canbe tuned by changing the number of emitters and the coupling between emitters(controlled by the separation).

Cascaded two-photon nonlinearity in a one-dimensional waveguide with multiple two-level emitters