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The density-functional approach to quantum electrodynamics is extendingtraditional density-functional theory and opens the possibility to describeelectron-photon interactions in terms of effective Kohn-Sham potentials. Inthis work, we numerically construct the exact electron-photon Kohn-Shampotentials for a prototype system which consists of a trapped electron coupledto a quantized electromagnetic mode in an optical high-Q cavity. While theeffective current that acts on the photons is known explicitly, the exacteffective potential that describes the forces exerted by the photons on theelectrons is obtained from a fixed-point inversion scheme. This procedureallows us to uncover important beyond-mean-field features of the effectivepotential which mark the breakdown of classical light-matter interactions. Weobserve peak and step structures in the effective potentials, which can beattributed solely to the quantum nature of light, i.e., they are real-spacesignatures of the photons. Our findings show how the ubiquitous dipoleinteraction with a classical electromagnetic field has to be modified inreal-space in order to take the quantum nature of the electromagnetic fieldfully into account.

Kohn-Sham Approach to Quantum Electrodynamical Density Functional Theory: Exact Time-Dependent Effective Potentials in Real Space