  For over 80 years of research, the conventional description of free-electron
radiation phenomena, such as Cherenkov radiation, has remained unchanged:
classical three-dimensional electromagnetic waves. Interestingly, in reduced
dimensionality, the properties of free-electron radiation are predicted to
fundamentally change. Here, we present the first observation of Cherenkov
surface waves, wherein free electrons emit narrow-bandwidth photonic
quasiparticles propagating in two-dimensions. The low dimensionality and narrow
bandwidth of the effect enable to identify quantized emission events through
electron energy loss spectroscopy. Our results support the recent theoretical
prediction that free electrons do not always emit classical light and can
instead become entangled with the photons they emit. The two-dimensional
Cherenkov interaction achieves quantum coupling strengths over two orders of
magnitude larger than ever reported, reaching the single-electron-single-photon
interaction regime for the first time with free electrons. Our findings pave
the way to previously unexplored phenomena in free-electron quantum optics,
facilitating bright, free-electron-based quantum emitters of heralded Fock
states.
