Symmetry-protected topological lasing in a BdG system of pseudo-spin-1/2 bosons

A topological laser is an active bosonic matter where the gain is distributed preferentially along the boundaries rather than in the bulk so that the mode competition will favor lasing in the edge modes, which are guaranteed by topology to be robust against disorder. We consider a (pseudo) spin-1/2 Bogoliubov-de Gennes (BdG) system, which is engineered, through a quench process, from a spin-1 Bose-Einstein condensate (BEC) in a honeycomb optical lattice. We use it to explore a host of issues concerning the symmetry-protected topological lasers which are characterized with stable bulk bands but unstable edge modes that can be populated at an exponentially fast rate. We express the criterion for topological amplifications in terms of an unconventional commutator between the number conserving and number nonconserving parts of the BdG Hamiltonian. We show that only when this “commutator” vanishes can a BdG system be made to operate as a symmetry-protected topological laser [Ling and Kain, Phys. Rev. A 104, 013305 (2021)]. We also carry out a quantitative study of the topological properties of our system in terms of topological invariants and symmetry classes, within the 38-fold way classification of topological phases in non-Hermitian systems [Ling and Kain, Phys. Rev. A 105, 023319 (2022)].