  The hybridization of light and matter excitations in the form of polaritons
has enabled major advances in understanding and controlling optical
nonlinearities. Entering the quantum regime of strong interactions between
individual photons has however remained challenging since the strength of
achievable polariton interactions is typically limited by the available
interactions in the material. Here, we investigate collisions between
dark-state polaritons in three-level systems and discover a resonant process
that yields effective interactions, which are much larger than the underlying
interaction between their matter constituents. We systematically investigate
the underlying mechanism and demonstrate a substantial enhancement of polariton
interactions by several orders of magnitude. This suggests a promising approach
to quantum nonlinear optics in a range of physical settings, from atomic gases
to excitons in semiconductors and two-dimensional bilayer materials.
