Observing quantum-classical correspondence through optical path integrals

The quantum-classical correspondence (QCC) describes how quantum systems recover classical behavior in the semiclassical limit. One of the crucial mechanisms to this correspondence is the concept of action, which governs dynamics in both classical mechanics and quantum path integrals. Despite its fundamental importance, experimental access to the action of individual quantum paths remains crucial yet challenging. Here, we experimentally demonstrate a method to measure the action of quantum paths between fixed spacetime points, based on weak-value measurement techniques. By introducing and analyzing the spectrum of the norm of the action gradient, we identify not only single but also multiple stationary-action paths in systems exhibiting complex QCC. Using an optical setup that simulates photon propagation in free space and under an impulsive potential, we reconstruct pathwise actions and observe sharp dips in the spectrum that correspond precisely to classical trajectories. This work provides a framework for directly accessing and analyzing the action, opening new avenues for experimentally exploring the QCC.