We find topological band structures in photonic time crystals—materials in which the refractive index varies periodically and abruptly in time. When the refractive index changes abruptly, the light experiences time refraction and time reflection, analogous to refraction and reflection in photonic crystals. The interference between time-refracted and time-reflected waves gives rise to dispersion bands, which are gapped in the momentum. We show theoretically that photonic time crystals can be in a topologically nontrivial phase, and calculate the topological invariant associated with the momentum bands, which is expressed in the phase between the forward- and backward-propagating waves. When an interface is generated between two time crystals of different topologies, the Zak phase yields a localized interface state, manifested as a localized temporal peak.

中文翻译：

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光子时间晶体的拓扑方面

我们在光子时间晶体中发现了拓扑能带结构，这种材料的折射率会周期性地突然变化。当折射率突然变化时，光会经历时间折射和时间反射，类似于光子晶体中的折射和反射。时间折射波和时间反射波之间的干涉产生了色散带，该色带在动量中有缝隙。我们从理论上证明光子时间晶体可以处于拓扑非平凡相位，并计算与动量带相关的拓扑不变量，该动量带以向前传播和向后传播的波之间的相位表示。当在不同拓扑的两个时间晶体之间生成界面时，Zak相会产生局部界面状态，