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We provided a theoretical framework, which described slow-light polaritons interacting via atomic Rydberg states. We used a diagrammatic method to analytically derive the scattering properties of two polaritons. We identified new parameter regimes where polariton-polariton interactions were repulsive. Furthermore, in the regime of attractive interactions, we identified multiple two-polariton bound states, calculated their dispersion, and studied the resulting scattering resonances. Finally, the twoparticle scattering properties allowed us to derive the effective low-energy many-body Hamiltonian. That theoretical platform was applicable to ongoing experiments.
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