We analyze the topological and dynamical properties of a system formed by two chains of identical emitters coupled to a waveguide, whose guided modes induce all-to-all excitation hopping. We find that, in the single excitation limit, the bulk topological properties of the Hamiltonian that describes the coherent dynamics of the system are identical to the ones of a one-dimensional Su-Schrieffer-Heeger (SSH) model. However, due to the long-range character of the exchange interactions, we find weakening of the bulk-boundary correspondence. This is illustrated by the variation of the localization length and mass gap of the edge states encountered as we vary the lattice constant and offset between the chains. Most interestingly, we analytically identify parameter regimes where edge states arise which are fully localized to the boundaries of the chain, independently of the system size. These edge states are shown to be not only robust against positional disorder of the atoms in the chain, but also subradiant, i.e., dynamically stable even in the presence of inevitable dissipation processes, establishing the capacity of waveguide QED systems for the realization of symmetry protected topological phases.

中文翻译：

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具有波导介导跳跃的原子链中的亚辐射边缘状态

我们分析了由耦合到波导的两条相同发射器链形成的系统的拓扑和动力学特性，其导模引起全对全激发跳跃。我们发现，在单激发极限下，描述系统相干动力学的哈密顿量的体拓扑性质与一维 Su-Schrieffer-Heeger (SSH) 模型的拓扑性质相同。然而，由于交换相互作用的长期特征，我们发现体边界对应的弱化。当我们改变晶格常数和链之间的偏移时，遇到的边缘状态的局部化长度和质量间隙的变化说明了这一点。最有趣的是，我们通过分析识别出完全定位于链边界的边缘状态出现的参数区域，与系统大小无关。这些边缘状态不仅对链中原子的位置无序具有鲁棒性，而且还具有亚辐射性，即即使存在不可避免的耗散过程也具有动态稳定性，从而确立了波导 QED 系统实现对称保护的能力拓扑相。