Coherent control via periodic modulation, also known as Floquet engineering, has emerged as a powerful experimental method for the realization of novel quantum systems with exotic properties. In particular, it has been employed to study topological phenomena in a variety of different platforms. In driven systems, the topological properties of the quasienergy bands can often be determined by standard topological invariants, such as Chern numbers, which are commonly used in static systems. However, due to the periodic nature of the quasienergy spectrum, this topological description is incomplete and new invariants are required to fully capture the topological properties of these driven settings. Most prominently, there are two-dimensional anomalous Floquet systems that exhibit robust chiral edge modes, despite all Chern numbers being equal to zero. Here we realize such a system with bosonic atoms in a periodically driven honeycomb lattice and infer the complete set of topological invariants from energy gap measurements and local Hall deflections.

通过周期调制的相干控制（也称为Floquet工程）已经成为一种强大的实验方法，用于实现具有奇异特性的新型量子系统。特别是，它已被用来研究各种不同平台中的拓扑现象。在驱动系统中，准能带的拓扑特性通常可以由静态系统中常用的标准拓扑不变量（例如Chern数）确定。但是，由于准能谱的周期性，此拓扑描述是不完整的，需要新的不变量来完全捕获这些驱动设置的拓扑特性。最显着的是，尽管所有Chern数均等于零，但二维的Floquet异常系统显示出强健的手性边缘模式。