Framework to Measure Quantum Metric from Step Response

The quantum metric is known to influence several observables, such as the dielectric constant, superfluid stiffness, and optical mass. However, a direct measurement of the metric has remained elusive. In linear response functions such as electric conductivity, only the matrix elements of the metric appear convoluted with energy prefactors, but to this date no direct observable has been identified that is directly proportional to the integrated metric over the Brillouin zone. In this Letter, we propose relaxation from constrained equilibrium to directly measure the symmetric part of the quantum geometric tensor. This response implements the integral over frequencies appearing in the Souza-Wilkens-Martin sum rule by choosing the electric field appropriately. Our work shines light on other geometric properties of insulators that are absent in the frequency expansions of conductivity in insulators but can, in principle, be revealed in step response.