Optical parametric free-electron--photon quantum interaction

Optical parametric processes underpin quantum photonics, while free-electron--photon interactions offer agile pathways to generate nontrivial quantum photonic states. These threads have so far largely progressed independently, whereas placing free electrons in a driven nonlinear system can potentially activate coherent parametric interaction channels for joint state engineering of both types of particles. Here we unify these paradigms by developing a general theoretical framework for parametric free-electron--photon interactions in a nonlinear optical system driven by degenerate parametric down-conversion. Unlike free electrons in a linear bath, here they can couple to Bogoliubov quasiparticles through two detuned phase-matching channels, where the parametric process and free-electron interactions can quantum amplify each other. Seeding the interaction with squeezed vacuum yields gain-only or loss-only electron energy spectra, and enables electron-heralded squeezed Fock states; with bare vacuum, postselecting electron energy sidebands generates high-fidelity Schrödinger cat states. Our results show how optical parametric interactions can quantum shape free electrons and photons, potentially enabling a quantum parametric dielectric laser accelerator that mitigates the need for temporal phase synchronization, thereby allowing acceleration probabilities to approach unity even for phase-random electrons.