## 2 Theory  
### Hubbard Model for Excitons and Exciton-Polaritons  
The interacting Hamiltonian for the complex driven semiconductor resonator can be written as follows  
\[H = \sum_{i,\sigma}\!\varepsilon_{i}c_{i,\sigma}^{\dagger}c_{i,\sigma} +\frac{U}{2}\sum_{i,\sigma}c_{i,\sigma}^{\dagger}c_{i,\sigma}c_{i,-\sigma}^{ \dagger}c_{i,-\sigma}-t\!\sum_{\langle ij\rangle,\sigma}\!c_{i,\sigma}^{ \dagger}c_{j,\sigma}\] \[+ i\vec{d}\cdot\vec{E}_{0}\cos(\Omega_{L}\tau)\sum_{<ij>,\sigma} \left(c_{i,\sigma}^{\dagger}c_{j,\sigma}-c_{j,\sigma}^{\dagger}c_{i,\sigma} \right)+\hbar\omega_{o}a^{\dagger}a\ \ +g\!\!\sum_{i,\sigma}c_{i,\sigma}^{\dagger}c_{i, \sigma}(a^{\dagger}\!+a).\]  
The electrons are described as a tight binding model, see Figure 2, where the splitting into valence and conduction band symmetrically to the Fermi level is included by means of the Coulomb interaction \(U\), see Figure 2b. The ZnO gap assumes the value of 3.38 eV [16; 17]. The first term in the Hamiltonian \(\sum_{i,\sigma}\!c_{i,\sigma}^{\dagger}c_{i,\sigma}\) denotes the local onsite potential. The term \(\frac{U}{2}\sum_{i,\sigma}c_{i,\sigma}^{\dagger}c_{i,\sigma}c_{i,\sigma}c_{i,- \sigma}^{\dagger}c_{i,-\sigma}\) is devoted to the onsite  
Figure 1: ZnO structure (ab-plane). (**a**) non-centrosymmetric, hexagonal, wurtzite configuration; (**b**) centrosymmetric, cubic, rocksalt configuration (Rochelle salt) [22; 25; 26]. The rocksalt configuration is distinguished by a tunable gap from 1.8 eV up to 6.1 eV, a gap value of 3.38 eV is typical for the monocrystal rocksalt configuration without oxygen vacancies [23; 24]. As such the rocksalt configuration could be suited for higher harmonics generation under non-equilibrium topological excitation [14; 15; 36]; (**c**) Typical setup of a solid state random laser: Densely packed disordered ZnO nano-resonators are embedded in a waveguide. The sample is externally pumped in (111) direction. Mie resonances [37] occur at the inner cylinder jacket and they couple to the density of bulk states in the sense of a Fano resonance. We consider in this article the exciton and exciton polariton dynamics of the single scatterer as the building block of the solid state random laser or the polariton random laser. Laser gain is attributed to the accumulation of excited states at a certain level along with a significantly increased life-time of the excited state, i.e., a stable laser in stationary state. In the random laser each scatterer of a well defined area or volume as part of an ensemble of several scatterers is going to lase. The investigation of the complex Mie-resonance in the non-equilibrium as the origin of pronounced laser levels for polariton lasing is on target. The formation of the polariton as a quasiparticle is characterized by the coupling strength \(\gamma\ =\ g/t\) of the bulk electronic procedures and the cavity resonance which is on the order of the band width or one order below and thus it is in the polariton range [38]. The waveguide principally plays only a subordinate role; it does not serve as the dominating conventional laser resonator cavity. Coulomb interaction \(U\) between electrons with opposite spins. The third term \(-t\sum_{\langle ij\rangle,\sigma}c_{i,\sigma}^{\dagger}c_{j,\sigma}\) is due to the hopping processes with the amplitude \(t\) between nearest neighbor sites. The classical external pumping is described in terms of the time dependent field \(\vec{E}_{0}\) with the laser frequency \(\Omega_{L}\), and \(\tau\) respectively. In the term \(i\vec{d}\cdot\vec{E}_{0}\cos(\Omega_{L}\tau)\sum_{<ij>,\sigma}\left(c_{i, \sigma}^{\dagger}c_{j,\sigma}-c_{j,\sigma}^{\dagger}c_{i,\sigma}\right)\) is noted the renormalization of the hopping processes due to interaction with the pump field. The electronic dipole operator \(\hat{d}\) is given with strength \(|\vec{d}|\).