  We explore higher order dynamical effects in the transport through a
two-dimensional nanoscale electron system embedded in a three-dimensional
far-infrared photon cavity. The nanoscale system is considered to be a short
quantum wire with a single circular quantum dot defined in a GaAs
heterostructure. The whole system, the external leads and the central system
are placed in a constant perpendicular magnetic field. The Coulomb interaction
of the electrons, the para- and diamagnetic electron-photon interactions are
all treated by a numerically exact diagonalization using step-wise truncations
of the appropriate many-body Fock spaces. We focus on the difference in
transport properties between a description within an electric dipole
approximation and a description including all higher order terms in a single
photon mode model. We find small effects mostly caused by an electrical
quadrupole and a magnetic dipole terms that depend strongly on the polarization
of the cavity field with respect to the transport direction and the photon
energy. When the polarization is aligned along the transport direction we find
indications of a weak self-induction that we analyze and compare to the
classical counterpart, and the self-energy contribution of high-order
interaction terms to the states the electrons cascade through on their way
through the system. Like expected the electron-photon interaction is well
described in the dipole approximation when it is augmented by the lowest order
diamagnetic part for a nanoscale system in a cavity in an external magnetic
field.
