Multiband dispersion and warped vortices of strongly-interacting photons

We present a theoretical study of quantum correlations between interacting photons realized through co-propagating Rydberg polaritons. We show that the spatial evolution of the n-photon wavefunction is governed by a multiband dispersion featuring one massive mode and multiple massless modes with degenerate Dirac points and n-fold rotational symmetry. The resulting band structure is warped, departing from the single-band, parabolic approximation commonly assumed for interacting polaritons. Our analytical results are supported by rigorous numerical modeling that fully accounts for photon propagation inside the finite atomic medium. These findings advance the understanding of multi-photon interactions and support the development of future multi-photon control tools.