Recent studies have shown that the nonlinear optical response of crystalline systems is fundamentally a quantum geometric property. In particular, specific second-order responses, such as shift current and second-harmonic generation, have been shown to depend on underlying geometric structures known as the quantum geometric tensor and quantum connection. In this work, we first generalize and unify these results by demonstrating how the full two-frequency second-order nonlinear conductivity decomposes into distinct quantum geometric contributions. As a corresponding probe, we propose two-dimensional coherent spectroscopy (2DCS), which naturally measures the nonlinear conductivity as a function of two independent frequencies using two time-delayed light pulses. We identify a term arising from the multiband quantum connection that does not appear in linear response, and we show that it can be measured in isolation by considering specific polarizations and enforcing time-reversal symmetry. We explore these findings via model calculations for transition-metal dichalcogenides and strontium ruthenate. Through these examples, we demonstrate how 2DCS enables study of the quantum connection, providing a way to compare the quantum geometry of different materials. We also show that one can gain rough momentum-resolved knowledge of the quantum geometry by varying the chemical potential.

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• Geometric & topological phases
• Nonlinear optics
• Optical conductivity
• Optical spectroscopy
• Two-dimensional coherent spectroscopy