High-order topological phases, such as those with nontrivial quadrupole moments [1,2], protect edge states that are themselves topological insulators in lower dimensions. So far, most quadrupole phases of light are explored in linear optical systems, which are protected by spatial symmetries [3] or synthetic symmetries [1,2,4–7]. Here we present Floquet quadrupole phases in driven nonlinear photonic crystals that are protected by space-time screw symmetries [8]. We start by illustrating space-time symmetries by tracking the trajectory of instantaneous optical axes of the driven media. Our Floquet quadrupole phase is then confirmed in two independent ways: symmetry indices at high-symmetry momentum points and calculations of the nested Wannier bands. Our Letter presents a general framework to analyze symmetries in driven optical materials and paves the way to further exploring symmetry-protected topological phases in Floquet systems and their optoelectronic applications.

中文翻译：

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时空对称保护的 Floquet 四极光子晶体

高阶拓扑相位，例如具有非平凡四极矩的那些 [1,2]，保护边缘状态，这些边缘状态本身就是低维拓扑绝缘体。到目前为止，在线性光学系统中探索了大多数光的四极相位，这些系统受到空间对称性 [3] 或合成对称性 [1,2,4-7] 的保护。在这里，我们展示了受时空螺旋对称性保护的驱动非线性光子晶体中的 Floquet 四极相 [8]。我们首先通过跟踪驱动介质的瞬时光轴轨迹来说明时空对称性。然后以两种独立的方式确认我们的 Floquet 四极相：高对称动量点的对称指数和嵌套 Wannier 带的计算。