  Interfacial electron-phonon coupling in ultrathin films has attracted much
interest; it can give rise to novel effects and phenomena, including enhanced
superconductivity. Here we report an observation of strong kinks in the energy
dispersions of quantum well states in ultrathin Yb films grown on graphite.
These kinks, arising from interfacial electron-phonon coupling, are most
prominent for films with a preferred ("magic") thickness of 4 monolayers, which
are strained and hole doped by the substrate. The energy position of the kinks
agrees well with the optical phonon energy of graphite, and the extracted
electron-phonon coupling strength {\lambda} shows a large subband dependence,
with a maximum value up to 0.6. The kinks decay rapidly with increasing film
thickness, consistent with its interfacial origin. The variation of {\lambda}
is correlated with evolution of the electronic wave function amplitudes at the
interface. A Lifshitz transition occurs just beyond the magic thickness where
the heavy Yb 5d bands begin to populate right below the Fermi level.
