For many materials, a precise knowledge of their dispersion spectra is insufficient to predict their ordered phases and physical responses. Instead, these materials are classified by the geometrical and topological properties of their wavefunctions. A key challenge is to identify and implement experiments that probe or control these quantum properties. In this Review, we describe recent progress in this direction, focusing on nonlinear electromagnetic responses that arise directly from quantum geometry and topology. We give an overview of the field by discussing theoretical ideas, experiments and the materials that drive them. We conclude by discussing how these techniques can be combined with device architectures to uncover, probe and ultimately control quantum phases with emergent topological and correlated properties.

对于许多材料，对其色散光谱的精确了解不足以预测它们的有序相和物理响应。相反，这些材料根据其波函数的几何和拓扑特性进行分类。一个关键的挑战是确定和实施探测或控制这些量子特性的实验。在这篇综述中，我们描述了这方面的最新进展，重点关注直接来自量子几何和拓扑的非线性电磁响应。我们通过讨论理论思想、实验和驱动它们的材料来概述该领域。最后，我们讨论了如何将这些技术与设备架构相结合，以发现、探测并最终控制具有新兴拓扑和相关属性的量子相位。