Finding new phases of matter and understanding their physical properties are primary goals of condensed matter physics. Over the past decade, topological phases of matter have taken center stage in condensed matter research, culminating in the 2016 Nobel Prize. Many topological phases have been experimentally discovered in quantum materials. Fundamentally, the topological physics arises from the geometric properties of quantum wavefunctions (e.g., Berry curvature, Berry connection, quantum metric, etc.), which have remained difficult to detect experimentally.

In this talk, I will show that the quantum geometry can strongly modify electrons’ response to external electromagnetic waves and lead to a wide range of previously unexplored nonlinear responses. First, I will demonstrate how we use infrared nonlinear photogalvanic effect to probe the chirality of Weyl fermions in 3D topological Weyl semimetals. Second, I will discuss our observation of the nonlinear Hall effect in the DC limit, in the 2D quantum ferroelectric metal bilayer WTe_{2}. Remarkably, this is a new electrical Hall effect realized in a nonmagnetic material and without external magnetic field. Third, I will show our optoelectronic detection and manipulation of a novel electronic instability, the gyrotropic order, in the correlated semimetal TiSe_{2}. Looking forward, the combination of advanced nonlinear optoelectronic techniques and new material platforms both in 2D and 3D can enable ample exciting possibilities to discover topology and geometry beyond the current paradigm.