  The emerging field of free-electron quantum optics enables electron-photon
entanglement and holds the potential for generating nontrivial photon states
for quantum information processing. Although recent experimental studies have
entered the quantum regime, rapid theoretical developments predict that
qualitatively unique phenomena only emerge beyond a certain interaction
strength. It is thus pertinent to identify the maximal electron-photon
interaction strength and the materials, geometries, and particle energies that
enable one to approach it. We derive an upper limit to the quantum vacuum
interaction strength between free electrons and single-mode photons, which
illuminates the conditions for the strongest interaction. Crucially, we obtain
an explicit energy selection recipe for electrons and photons to achieve
maximal interaction at arbitrary separations and identify two optimal regimes
favoring either fast or slow electrons over those with intermediate velocities.
We validate the limit by analytical and numerical calculations on canonical
geometries and provide near-optimal designs indicating the feasibility of
strong quantum interactions. Our findings offer fundamental intuition for
maximizing the quantum interaction between free electrons and photons and
provide practical design rules for future experiments on electron-photon and
electron-mediated photon-photon entanglement. They should also enable the
evaluation of key metrics for applications such as the maximum power of
free-electron radiation sources and the maximum acceleration gradient of
dielectric laser accelerators.
