  One dimensional confinement in waveguide Quantum Electrodynamics (QED) plays
a crucial role to enhance light-matter interactions and to induce a strong
quantum nonlinear optical response. In two or higher dimensional settings, this
response is reduced since photons can be emitted within a larger phase space,
opening the question whether strong photon-photon interaction can be still
achieved. In this study, we positively answer this question for the case of a
2D square array of atoms coupled to the light confined into a two-dimensional
waveguide. More specifically, we demonstrate the occurrence of long-lived
two-photon repulsive and bound states with genuine 2D features. Furthermore, we
observe signatures of these effects also in free-space atomic arrays in the
form of weakly-subradiant in-band scattering resonances. Our findings provide a
paradigmatic signature of the presence of strong photon-photon interactions in
2D waveguide QED.
