Understanding the geometric properties of quantum states and their implications in fundamental physical phenomena is a core aspect of contemporary physics. The quantum geometric tensor (QGT) is a central physical object in this regard, encoding complete information about the geometry of the quantum state. The imaginary part of the QGT is the well-known Berry curvature, which plays an integral role in the topological magnetoelectric and optoelectronic phenomena. The real part of the QGT is the quantum metric, whose importance has come to prominence recently, giving rise to a new set of quantum geometric phenomena such as anomalous Landau levels, flat band superfluidity, excitonic Lamb shifts and nonlinear Hall effect. Despite the central importance of the QGT, its experimental measurements have been restricted only to artificial two-level systems. Here, we develop a framework to measure the QGT in crystalline solids using polarization-, spin- and angle-resolved photoemission spectroscopy. Using this framework, we demonstrate the effective reconstruction of the QGT in the kagome metal CoSn, which hosts topological flat bands. Establishing this momentum- and energy-resolved spectroscopic probe of the QGT is poised to significantly advance our understanding of quantum geometric responses in a wide range of crystalline systems.

了解量子态的几何特性及其在基本物理现象中的含义是当代物理学的一个核心方面。量子几何张量 （QGT） 是这方面的中心物理对象，它编码有关量子态几何的完整信息。QGT 的虚部是众所周知的 Berry 曲率，它在拓扑磁电和光电现象中起着不可或缺的作用。QGT 的真正部分是量子度量，其重要性最近越来越突出，产生了一组新的量子几何现象，例如异常的朗道能级、平带超流体、激子兰姆位移和非线性霍尔效应。尽管 QGT 具有核心重要性，但其实验测量仅限于人工两能级系统。在这里，我们开发了一个框架，使用偏振、自旋和角度分辨的光电子电子能谱来测量结晶固体中的 QGT。使用这个框架，我们展示了 QGT 在 kagome 金属 CoSn 中的有效重建，该金属 CoSn 承载拓扑平面带。建立 QGT 的这种动量和能量分辨光谱探针有望显着推进我们对各种晶体系统中量子几何响应的理解。