Multi-excitation scattering in atomic arrays

In free space, photons only weakly couple to single atoms. However, recent research has shown that arranging atoms in arrays with subwavelength lattice spacings can enable efficient and controlled interactions between free-space photons and atomic media. These systems exhibit collective atomic excitations with extraordinarily long lifetimes (i.e., subradiant states) and are promising for quantum information applications. To fully leverage their potential, it is crucial to understand how to manipulate multiple excitations (both photonic and atomic) in the presence of atomic nonlinearities. Although the theory of single excitation is well-established, studies of multiple excitations often rely on numerical methods, and a comprehensive theoretical framework for general interactions is still lacking. In this work, we develop a scattering formalism to investigate few-excitation interactions between photons and atomic excitations that applies to general atomic nonlinearities. Our method can be used to explore the generation of multi-excitation subradiance and non-classical light in subwavelength atomic arrays.