  Flatbands emerge from a myriad of structures such as Landau levels, Lieb and
Kagome lattices, linegraphs, and more recently moire superlattices. They enable
unique properties including slow light in photonics, correlated phases in
electronics, and supercollimation in both systems. Despite these intense
parallel efforts, flatbands have never been shown to affect the core
light-matter interaction between electrons and photons, which is limited by a
dimensionality mismatch. Here, we reveal that a photonic flatband can overcome
this mismatch between localized electrons and extended photons and thus
remarkably boost their light-matter interaction. We design flatband resonances
in a silicon-on-insulator photonic crystal slab to control and enhance the
radiation emission from free electrons by tuning their trajectory and velocity.
In particular, we record a 100-fold radiation enhancement from the conventional
diffraction-enabled Smith-Purcell radiation, and show the potential of our
approach to achieve $10^6$-fold enhancements and beyond. The enhancement also
enables us to perform polarization shaping of free electron radiation from
multiple flatbands and demonstrate an approach to measure photonic bands via
angle-resolved electron-beam measurements. Our results suggest flatbands as
ideal test beds for strong light-electron interaction in various systems, with
particular relevance for efficient and compact free-electron light sources and
accelerators.
