  We investigate the coupling between a quantized electromagnetic field in a
cavity resonator and a Coulomb interacting electronic system in a nanostructure
in an external magnetic field. Effects caused by the geometry of the electronic
system and the polarization of the electromagnetic field are explicitly taken
into account. Our numerical results demonstrate that the two-level system
approximation and the Jaynes-Cummings model remain valid in the weak
electron-photon coupling regime, while the quadratic vector potential in the
diamagnetic part of the charge current leads to significant correction to the
energy spectrum in the strong coupling regime. Furthermore, we find that a
coupling to a strong cavity photon mode polarizes the charge distribution of
the system requiring a large basis of single-electron eigenstates to be
included in the model.
