*5.2. Results*

Groups modeling the near resonance interaction of one cavity mode with a two level electronic system have expressed the importance of using a large enough, or the correct type, of a photon basis in the strongly interacting regime [87,88]. In many level systems where wavefunction and geometric effects are accounted for our experience is that convergence in numerical diagonalization is more sensitive to proper truncation of the electronic sector of the Fock many-body space. This reflects the polarizability of the electric charge by a cavity field in the construction of the photon-dressed electronic states. At the same time the inclusion of the diamagnetic interaction curbs the need for states with a very high photon number [42,50,89].

The polarizability of the first photon replica of the two-electron ground state is displayed in Figure 8 as a function of *g***EM**, the photon energy ¯*hω* and its polarization [50].

**Figure 8.** The expectation value (*r*/*aw*)<sup>2</sup> for the first photon replica of the two-electron ground state in the closed system at *t* = 0 for *x*- and *y*-polarization of the photon field. *h*¯ *ω* = 2.0 meV, *B* = 0.1 T. Two parallel quantum dots are embedded in the central system.

The polarizability is nonlinear, anisotropic, and largest for the cavity photon close to a resonance with the confinement energy in the *y*-direction.

A Rabi oscillation of two electrons in the double quantum dot system embedded in the short quantum wire leads to oscillating charge with time in the system. The oscillating probability of charge presence in the contact areas of the short wire thus lead to oscillations in the current leaving the system through the left and right leads [33], see Figure 9.

**Figure 9.** The left (black) and right (gold) currents and the mean electron number (blue) for initially fully entangled Rabi-split singlet two-electron states as the interacting system discharges in the transient regime. ¯*hω* = 2.0 meV, *B* = 0.1 T. Two parallel quantum dots are embedded in the central system.

Alternatively, one may view this as the consequence of the Rabi resonance entangling two states with different tunneling probability to the leads.

In the transient or the late transient regime we have used the non-Markovian GME to investigate several results: Thorsten Arnold et al. used a time-convolution-less (TCL) version of the GME to study the effects of magnetic field and photons [46] on the transport of interacting electrons through a quantum ring with spin-orbit interactions in a photon cavity with circular [86] and linear polarization [90]. Aharonov-Bohm oscillations were established in the time-dependent transport through a ring structure with additional vortexes in the contact region of the quantum wire. *x*-polarized photons with energy 0.3 meV attenuate the Aharonov-Bohm oscillations over a broad range of magnetic field, but *y*-polarized photons influence the transport in a more complex fashion. The oscillations

are generally attenuated, but one oscillation peak is split and the charge current is enhanced at a magnetic field corresponding to a half-integer flux quantum [46]. With the spin-orbit interactions the spin polarization and the spin photo currents of the quantum ring are largest for circularly polarized photon field and a destructive Aharonov–Casher (AC) phase interference. The dip in the charge current caused by the destructive AC phase becomes threefold under the circularly polarized photon field as the interaction of the angular momentum of the electron and the spin angular momentum of the light create a many-body level splitting [86]. The detailed balance between the para- and the diamagnetic electron-photon interactions has been studied for an electron in the quantum ring structure when excited by a short classical dipole pulse [47].

Nzar Rauf Abdullah et al. have used the GME formalism to investigate photon assisted transport [91], photon mediated switching in nanostructures [48,49,92], the balancing of magnetic and forces caused by cavity photons [93], cavity-photon affected thermal transport [94,95], and the influence of cavity photons on thermal spin currents in a system with spin orbit interactions [96,97].
