Conventional topological insulators support boundary states with dimension one lower than that of the bulk system that hosts them, and these states are topologically protected due to quantized bulk dipole moments. Recently, higher-order topological insulators have been proposed as a way of realizing topological states with dimensions two or more lower than that of the bulk due to the quantization of bulk quadrupole or octupole moments. However, all these proposals as well as experimental realizations have been restricted to real-space dimensions. Here, we construct photonic higher-order topological insulators (PHOTIs) in synthetic dimensions. We show the emergence of a quadrupole PHOTI supporting topologically protected corner modes in an array of modulated photonic molecules with a synthetic frequency dimension, where each photonic molecule comprises two coupled rings. By changing the phase difference of the modulation between adjacent coupled photonic molecules, we predict a dynamical topological phase transition in the PHOTI. Furthermore, we show that the concept of synthetic dimensions can be exploited to realize even higher-order multipole moments such as a fourth-order hexadecapole (16-pole) insulator supporting 0D corner modes in a 4D hypercubic synthetic lattice that cannot be realized in real-space lattices.

传统的拓扑绝缘体支持的边界态的维度比承载它们的体系统的维度低一，并且由于量子化的体偶极矩，这些状态受到拓扑保护。最近，由于体四极或八极矩的量子化，高阶拓扑绝缘体被提出作为实现比体尺寸低二维或更多维的拓扑状态的一种方式。然而，所有这些建议以及实验实现都仅限于真实空间维度。在这里，我们在合成维度构建光子高阶拓扑绝缘体（PHOTIS）。我们展示了在具有合成频率维度的调制光子分子阵列中支持拓扑保护角模式的四极 PHOTI 的出现，其中每个光子分子包含两个耦合环。通过改变相邻耦合光子分子之间调制的相位差，我们预测了 PHOTI 中的动态拓扑相变。此外，我们还表明，可以利用合成维度的概念来实现更高阶的多极矩，例如在 4D 超立方合成晶格中支持 0D 角模式的四阶十六极（16 极）绝缘体，而这在实际中无法实现-空间晶格。