Heisenberg-Scaling Quantum Metrology of Electron-Photon Coupling via Multi-Electron PINEM

Free-electron quantum optics has recently emerged as a promising platform for coherent electron-light interactions, yet its potential for quantum-enhanced metrology remains largely unexplored. Here, we investigate the fundamental precision limit of coupling-strength estimation in multi-electron photon-induced near-field electron microscopy (PINEM), demonstrating that the quantum Fisher information scales quadratically with electron number. We then propose an experimentally accessible scheme combining squeezed-light probing and optical homodyne detection that attains Heisenberg scaling with respect to electron number, provided the electrons exhibit nonzero inter-electron coherence, while surpassing conventional PINEM sensing in photon-number enhancement as well. We further analyze the effect of imperfect electron preparation, showing that electron phase noise is converted into a fluctuating optical displacement and eventually limits the precision. To overcome this limitation, an auxiliary optical mode coupled to the same electron ensemble measures the displacement noise in the signal mode, allowing its conditional suppression and recovery of the Heisenberg scaling. Our work establishes multi-electron coherence as a resource for free-electron quantum metrology and opens the route to precision measurements of nanoscale electron-light interactions.