  For certain correlated electron-photon systems we construct the exact
density-to-potential maps, which are the basic ingredients of a
density-functional reformulation of coupled matter-photon problems. We do so
for numerically exactly solvable models consisting of up to four fermionic
sites coupled to a single photon mode. We show that the recently introduced
concept of the intra-system steepening (T.Dimitrov et al., 18, 083004 NJP
(2016)) can be generalized to coupled fermion-boson systems and that the
intra-system steepening indicates strong exchange-correlation (xc) effects due
to the coupling between electrons and photons. The reliability of the
mean-field approximation to the electron-photon interaction is investigated and
its failure in the strong coupling regime analyzed. We highlight how the
intra-system steepening of the exact density-to-potential maps becomes apparent
also in observables such as the photon number or the polarizability of the
electronic subsystem. We finally show that a change in functional variables can
make these observables behave more smoothly and exemplify that the
density-to-potential maps can give us physical insights into the behavior of
coupled electron-photon systems by identifying a very large polarizability due
to ultra-strong electron-photon coupling.
