  Multi-photon dynamics beyond linear optical materials are of significant
fundamental and technological importance in quantum information processing.
However, it remains largely unexplored in nonlinear waveguide QED. In this
work, we theoretically propose a structured nonlinear waveguide in the presence
of staggered photon-photon interactions, which supports two branches of gaped
bands for doublons (i.e., spatially bound-photon-pair states). In contrast to
linear waveguide QED systems, we identify two important contributions to its
dynamical evolution, i.e., single-photon bound states (SPBSs) and doublon bound
states (DBSs). Most remarkably, the nonlinear waveguide can mediate the
long-range four-body interactions between two emitter pairs, even in the
presence of disturbance from SPBS. By appropriately designing system's
parameters, we can achieve high-fidelity four-body Rabi oscillations mediated
only by virtual doublons in DBSs. Our findings pave the way for applying
structured nonlinear waveguide QED in multi-body quantum information processing
and quantum simulations among remote sites.
