Phase diagram for injection locking a superradiant laser

We experimentally and theoretically study the response of a superradiant or bad-cavity laser to an applied coherent drive. We observe two forms of synchronization (injection locking) between the superradiant ensemble and the applied drive: one attractive and one repulsive in nature. We explain the region of repulsion as arising from the higher three-dimensional description of the atomic spin state that stores the laser coherence in a superradiant laser, as opposed to a two-parameter description of the electric field in a traditional good-cavity laser. We derive a phase diagram of predicted behavior and experimentally measure the response of the system across various trajectories therein.