  We model the equilibrium properties of a two-dimensional electron gas in a
square lateral superlattice of quantum dots in a GaAs heterostructure subject
to an external homogeneous perpendicular magnetic field and a far-infrared
circular cylindrical photon cavity with one quantized mode, the TE011 mode. In
a truncated linear basis constructed by a tensor product of the single-electron
states of the noninteracting system and the eigenstates of the photon number
operator, a local spin density approximation of density functional theory is
used to compute the electron-photon states of the two-dimensional electron gas
in the cavity. The common spatial symmetry of the vector fields for the
external magnetic field and the cavity photon field in the long wavelength
approximation enhances higher order magnetic single- and multi-photon processes
for both the para- and the diamagnetic electron-photon interactions. The
electron-photon coupling introduces explicit photon replicas into the
bandstructure and all subbands gain a photon content, constant for each
subband, that can deviate from an integer value as the coupling is increased or
the photon energy is varied. The subbands show a complex Rabi anticrossing
behavior when the photon energy and the coupling bring subbands into
resonances. The complicated energy subband structure leads to photon density
variations in reciprocal space when resonances occur in the spectrum. The
electron-photon coupling polarizes the charge density and tends to reduce the
Coulomb exchange effects as the coupling strength increases.
