## 3 Results  
For driven ZnO bulk we display in Figure 3a the local density of excitonic states (LDOS) for 10.0 MW/cm\({}^{2}\) pump power and the varying external laser frequency \(\Omega_{L}\). The systems parameters are found in the caption of Figure 3. A multitude of Floquet bands arises due to the AC Stark effect and sub-gaps are formed whereas the original gap closes. These results are computed for a cubic lattice structure and their physical interpretation besides the formation of exciton-quasiparticles is the generation of bands of even higher harmonics due to the symmetry of the system. For the fixed value of \(\hbar\Omega_{L}\,=\,1.75\,\)eV, Figure 3b, we find an increasing LDOS for excitons in the ZnO gap region. The solid black line indicates the undriven bandstructure with a half band width of 1.7 eV. In the valence and the conduction band sub gaps towards the near-gap band edges at \(-0.67\) eV and \(4.05\) eV are formed, while at the inner band edges a step-like decrease is found. A smooth mid-gap feature is building up with increasing pump strength. In the results for the inverse lifetime, \(1/\tau\), of excitons \(\mathrm{Im}\Sigma\), Figure 3c, which characterizes the dynamics of the excitons we find a sharp peak at the Fermi edge, i.e., small lifetimes of excitons as we expected it. \(\mathrm{Im}\Sigma\) is increasing in the gap near the equilibrium band edges. We find an increasing LDOS for excitons with the increase of the external excitation strength. The lifetimes of the near-edge states are therefore progressively decreased. A plateauing of exciton lifetimes is observed in the band gap otherwise, whereas deep in the band at e.g., \(4.8\) eV we find for increasing external pump intensity a crossover of the corresponding excitonic lifetimes. Here \(1/\tau\) exhibits a local maximum around a pump intensity of \(|E_{0}|^{2}\,=\,3.8\) MW/cm\({}^{2}\) which then transforms with further increasing pump strength into a local minimum (orange dash-dotted line) and eventually reaches lifetimes almost as high as in the un-driven equilibrium system (black solid line). For the cavity coupled system, Figure 3d, we find a qualitatively different behavior. It should be noted first that the presence of the cavity also for the weakly coupled case in the non-equilibrium always has an influence on the full spectrum of the LDOS. This is necessarily the case for energy and particle conservation. The semiconductor pillars with a full diameter of \(d\,=\,260.0\) nm are chosen with a geometrical Mie resonance as a single whispering gallery mode on the inner jacket of the resonator. The coat thickness may vary with the fineness of the ZnO material. As such it may determine the Q-factor of the single Mie resonator in principle [48]; however, in the non-equilibrium regime and in the corresponding physical time range microscopic interactions in the sense of impurity scattering in the selvedge have no fundamental influence. The Mie resonance of \(\hbar\omega_{0}\)\(=2.45\) eV corresponds to the wavelength \(\lambda\,=\,505.34\) nm [37]; effectively this whispering gallery is populated in the resonance by photons. Thus, the classical picture of the Mie resonance [49] of a sole geometric resonance of otherwise passive matter is replaced here by the Fano resonance [50] of the single mode with the continuum of excitons which is embedded in a Floquet matrix [28; 29; 30; 31]. The Floquet matrix, see Figure 2a, represents mathematically the physics of high amplitude excitations and higher harmonics [36] under the presumption of energy conservation. We display the exciton-polariton lifetimes for \((g/t)\)=\(0.2,0.4,0.9\) at zero temperature and at half filling. The coupling strength as such is characteristic for the formation of polaritons [38]. The results are derived for an external optical pump intensity of \(2.4\) MW/cm\({}^{2}\) which on the other side is a typical pump strength for solid state random lasers [51; 52; 53; 54]. We find a sharp dip in the lifetimes of polaritonic states at the Fermi energy.