Theory of quantum spin Hall topological lasers

We theoretically investigate a quantum spin Hall topological laser formed by an array of dielectric ring resonators endowed of saturable gain. The system preserves time-reversal symmetry, the clockwise and counterclockwise modes in each ring resonator acting as two pseudospin states that experience opposite synthetic magnetic fields. We consider ring resonators featuring an internal S-shaped waveguide asymmetrically coupling the two pseudospin states. In spite of the nonmagnetic nature of the configuration, we show that an effective breaking of reciprocity is induced by the interplay of spatial parity breaking with saturable gain and a Kerr optical nonlinearity. This enables robust single-mode topological lasing even in the presence of realistic levels of backscattering.