Efficient manipulation of quantum states is a key step towards applications in quantum information, quantum metrology, and nonlinear optics. Recently, atomic arrays have been shown to be a promising system for exploring topological quantum optics and robust control of quantum states, where the inherent nonlinearity is included through long-range hoppings. Here we show that a one-dimensional atomic array in a periodic magnetic field exhibits characteristic properties associated with an effective two-dimensional Hofstadter-butterfly-like model. Our work points out super- and sub-radiant topological edge states localized at the boundaries of the atomic array despite featuring long-range interactions, and opens an avenue of exploring an interacting quantum optical platform with synthetic dimensions.

量子态的有效操纵是向量子信息，量子计量学和非线性光学应用的关键一步。最近，原子阵列已被证明是探索拓扑量子光学和对量子态进行鲁棒控制的有前途的系统，其中固有的非线性是通过远距离跳跃实现的。在这里，我们显示了在周期性磁场中的一维原子阵列表现出与有效的二维霍夫施塔特-蝴蝶状模型相关的特征。我们的工作指出了尽管具有长距离相互作用，但位于原子阵列边界的超辐射和次辐射拓扑边缘状态，并为探索具有合成尺寸的相互作用的量子光学平台开辟了道路。