Generating spatially separated correlated multiphoton states in nonlinear waveguide quantum electrodynamics

Strongly correlated multi-photon states are indispensable resources for advanced quantum technologies, yet their deterministic generation remains challenging due to the inherent weak nonlinearity in most optical systems. Here, we propose a scalable architecture for producing correlated few-photon entangled states via cascaded inelastic scattering in a nonlinear waveguide. When a single photon scatters off a far detuned excited two-level emitter, it coherently converts into a propagating doublon, a bound photon pair with anomalous dispersion. This doublon can subsequently scatter off a downstream excited emitter to further convert into a triplon, and so on, thereby establishing a photon-number amplification cascade $|\cdot \rangle \!\! \rightarrow \!\! |\!\!: \rangle \!\! \rightarrow \! \! |\!\!\therefore \rangle \!\! \to \!\! ...$ Central to this process is the concept of a pseudo-giant atom, which we introduce here to capture the non-local scattering potential emergent from the wave functions of bound states. By implementing this scheme using a real giant atom with multiple engineered coupling points, we achieve unidirectional and full controllable photon conversion without backscattering. The resulting output state forms a programmable superposition of spatially and temporally isolated photon-number components, automatically sorted by their distinct group velocities. This work opens a new paradigm in quantum state engineering, enabling on-demand generation of complex multi-photon resources for quantum simulation, metrology, and scalable quantum networks.