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 counter-clockwise 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 non-magnetic 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 non-linearity. This enables robust single-mode topological lasing even in the presence of realistic levels of backscattering.