Quantum Geometry in Phonon-Mediated Optical Responses

Quantum geometry is crucial for understanding intricate condensed matter systems, governing transport phenomena and optical responses. However, traditional studies of quantum geometry predominantly consider a static crystal lattice, focusing exclusively on the pure-electronic quantum geometry of the Hilbert space parametrized by electronic wave vectors, thereby overlooking the dynamic effects arising from phonons and their coupling with electrons. In this Letter, we reveal the intrinsic quantum geometry of the electron-phonon coupling (EPC), which resides in the hybrid Hilbert space parametrized by both the electronic wave vectors and phonon displacements. The EPC quantum metric, EPC Berry curvature, and EPC shift vector, as central elements, quantify the EPC-induced velocity, polarization, and anomalous charge-center shift, respectively. We further connect this geometry to phonon-mediated optical responses, particularly in-gap resonances, enabling experimental detection and characterization of EPC quantum geometry.