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Multi-photon dynamics beyond linear optical materials are of significantfundamental and technological importance in quantum information processing.However, it remains largely unexplored in nonlinear waveguide QED. In thiswork, we theoretically propose a structured nonlinear waveguide in the presenceof staggered photon-photon interactions, which supports two branches of gapedbands for doublons (i.e., spatially bound-photon-pair states). In contrast tolinear waveguide QED systems, we identify two important contributions to itsdynamical evolution, i.e., single-photon bound states (SPBSs) and doublon boundstates (DBSs). Most remarkably, the nonlinear waveguide can mediate thelong-range four-body interactions between two emitter pairs, even in thepresence of disturbance from SPBS. By appropriately designing system'sparameters, we can achieve high-fidelity four-body Rabi oscillations mediatedonly by virtual doublons in DBSs. Our findings pave the way for applyingstructured nonlinear waveguide QED in multi-body quantum information processingand quantum simulations among remote sites.

Long-Range Four-body Interactions in Structured Nonlinear Photonic Waveguides