  The density-functional approach to quantum electrodynamics is extending
traditional density-functional theory and opens the possibility to describe
electron-photon interactions in terms of effective Kohn-Sham potentials. In
this work, we numerically construct the exact electron-photon Kohn-Sham
potentials for a prototype system which consists of a trapped electron coupled
to a quantized electromagnetic mode in an optical high-Q cavity. While the
effective current that acts on the photons is known explicitly, the exact
effective potential that describes the forces exerted by the photons on the
electrons is obtained from a fixed-point inversion scheme. This procedure
allows us to uncover important beyond-mean-field features of the effective
potential which mark the breakdown of classical light-matter interactions. We
observe peak and step structures in the effective potentials, which can be
attributed solely to the quantum nature of light, i.e., they are real-space
signatures of the photons. Our findings show how the ubiquitous dipole
interaction with a classical electromagnetic field has to be modified in
real-space in order to take the quantum nature of the electromagnetic field
fully into account.
