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 {spatio-temporal} 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 have established the above behaviors also for {non-topological 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 {non-topological} devices. {These results} open exciting possibilities for {both} fundamental studies of non-equilibrium statistical mechanics and concrete applications to laser devices.

中文翻译：

---

拓扑激光器相干理论

我们提出了一种拓扑激光设备的时空相干特性的理论研究，该设备通过在光子的Harper-Hofstadter晶格的边缘位置包括饱和增益而构建。{对于足够小的晶格，将应用Bogoliubov分析，并且相干时间几乎取决于设备中光子的总数{与标准的Schawlow-Townes相扩散一致。在较大的晶格中，在手征运动的共同运动帧中观察其激射边缘模式，长波波动的时空相关性显示出Kardar-Parisi-Zhang（KPZ）缩放比例。{仍然，在很长的时间}，{当}设备的有限尺寸开始变得重要时，{相干性的时间衰减}的功能形式从KPZ指数扩展到{Schawlow-Townes样}指数，尽管与{Bogoliubov Schawlow-Townes预测相比}，KPZ涨落的非线性动力学{仍显示为更宽的线宽。}尽管我们已经为{非拓扑一维激光}阵列建立了上述行为，但拓扑的关键作用最后，我们着重介绍了保护相干免受静电干扰的作用{：我们的数值计算表明，{与}相应的{非拓扑}设备相比，拓扑激光器的相干特性得到了显着增强。{这些结果}为非平衡统计力学的基础研究和激光设备的具体应用都提供了令人兴奋的可能性。}尽管我们已经为{非拓扑一维激光}阵列建立了上述行为，但拓扑结构在保护相干免受静电干扰方面的关键作用终于得到了{{：我们的数值计算表明，拓扑激光器的相干特性得到了显着增强}与}相应的{非拓扑}设备进行比较。{这些结果}为非平衡统计力学的基础研究和激光设备的具体应用都提供了令人兴奋的可能性。}尽管我们已经为{非拓扑一维激光}阵列建立了上述行为，但拓扑结构在保护相干免受静电干扰方面的关键作用终于得到了{{：我们的数值计算表明，拓扑激光器的相干特性得到了显着增强}与}相应的{非拓扑}设备进行比较。{这些结果}为非平衡统计力学的基础研究和激光设备的具体应用都提供了令人兴奋的可能性。