Coherent quantum control is usually associated with isolated few-level systems, while transport in solids is often described by incoherent, perturbative carrier dynamics. We show that quantum geometry near a topological critical point can promote Bloch electrons into universal, quantum-coherent nonlinear responses. In a dual-gated MnBi2Te4, we achieve a topological phase transition. We observe higher-harmonic Hall (HHH) effects up to the 39th order, whose tunneling-like I-V curve differs from the conventional perturbative law. Away from the critical point, the HHH becomes nonmonotonic and develops coherent oscillations periodic in inverse current, 1/I, whose displacement field evolution shows a fan pattern, resembling the Landau fans but with current replacing magnetic field. We demonstrate topological scaling laws, including universal harmonic responses collapsed onto a single curve with a critical exponent of 3/2, as well as a decreasing harmonic onset current revealing the trend of diverging quantum metric near the singularity. Our experiments uncover a nonperturbative ``tunneling-interference'' quantum geometry regime, which is distinct from the conventional, perturbative quantum geometry multipole expansion. In this regime, quantum metric serves as coherent beamsplitters for Bloch electrons, whereas Berry curvature captures the topological criticality. Together they account for key observations, including efficient HHH, universal critical scaling as well as coherent quantum oscillation. Our results suggest that the nonperturbative quantum geometry near singularity can amplify the coherence and nonlinearity of Bloch electrons, suggesting momentum-space electron quantum optics for quantum and nonlinear applications.

This lecture was made possible by the William C. Ferguson Fund.