Theory of the Coherence of Topological Lasers

We present a theoretical study of the temporal and spatial coherence properties of a topological laser device built by including saturable gain on the edge sites of a Harper-Hofstadter lattice for photons. For small enough lattices, the Bogoliubov analysis applies, and the coherence time is almost determined by the total number of photons in the device in agreement with the standard Schawlow-Townes phase diffusion. In larger lattices, looking at the lasing edge mode in the comoving frame of its chiral motion, the spatiotemporal correlations of long-wavelength fluctuations display a Kardar-Parisi-Zhang (KPZ) scaling. Still, at very long times, when the finite size of the device starts to matter, the functional form of the temporal decay of coherence changes from the KPZ stretched exponential to a Schawlow-Townes-like exponential, while the nonlinear dynamics of KPZ fluctuations remains visible as a broadened linewidth as compared to the Bogoliubov-Schawlow-Townes prediction. While we establish the above behaviors also for nontopological 1D laser arrays, the crucial role of topology in protecting the coherence from static disorder is finally highlighted: Our numerical calculations suggest the dramatically reinforced coherence properties of topological lasers compared to corresponding nontopological devices. These results open exciting possibilities for both fundamental studies of nonequilibrium statistical mechanics and concrete applications to laser devices.