## 2 Theory
### Hubbard Model for Excitons and Exciton-Polaritons
The photonic cavity mode \(\hbar\omega_{o}a^{\dagger}a\) with the resonance frequency \(\omega_{0}\) is coupled to the electron system as \(g\hat{\sum}_{i,\sigma}c_{i,\sigma}^{\dagger}c_{i,\sigma}(a^{\dagger}+a)\) with a coupling strength \(g\) in units of the standard hopping \(t\). The single band effective Hubbard model has been proven to be perfectly suited for the description of the movement of hole-bound electrons, excitons, as a quasiparticle through the lattice of metal ions [39; 40; 41], which is a lattice of Zn\({}^{2+}\) ions here. For the solution of the Hamiltonian for driven bulk matter (**i**) as well as for the bulk-cavity coupled system (**ii**) including the fifth and the sixth term of Equation (1), the explicit time dependence of the external field has to be accounted for as well as the dynamics of the system. It yields [PERSON]'s functions which depend on two separate time arguments. The double Fourier transform from time to frequency coordinates leads to two separate frequencies which are chosen as relative and center-of-mass frequency [30; 31; 34; 35] and we can thus expand the underlying physical procedures into Floquet modes, the graphical explanation is found in Figure 2a,  
\[G_{nm}^{\alpha\beta}(\omega) = \int\!\mathrm{d}\tau_{1}^{\alpha}\mathrm{d}\tau_{2}^{\beta}\ e^{ -i\Omega_{L}(m\tau_{1}^{\alpha}-m\tau_{2}^{\beta})}e^{i\omega(\tau_{1}^{\alpha }-\tau_{2}^{\beta})}G(\tau_{1}^{\alpha},\tau_{2}^{\beta})\] \[\equiv G^{\alpha\beta}(\omega-m\Omega_{L},\omega-n\Omega_{L}).\]  
In Equation (2) \((m,n)\) label the Floquet modes and \((\alpha,\beta)\) label the branch of the Keldysh contour \((\pm)\) where the respective time argument resides. Floquet modes in time-space an analogue to Bloch modes in real space. The physical meaning of the expansion into Floquet modes is however noteworthy, since it is the quantized absorption and emission of energy \(\hbar\Omega_{L}\) out of and into the classical external driving field. Figure 2: Theoretical setup (**a**) Floquet expansion of the [PERSON]’s function \(G\) into higher harmonics of \(\hbar\Omega_{L}\). (**b**) Hubbard model of the electronic system. The gap between bands is defined by the Coulomb interaction \(U\) symmetrically to the Fermi Energy \(E_{F}\). (**c**) Tight binding system of the crystal lattice. The hopping parameter \(t\) is renormalized by external time dependent excitations \(T(\tau)\). \(\varepsilon_{0}+E(\tau)\) are renormalizations of the local potential alone; (**d**) non-equilibrium DMFT. Pump photons \(\hbar\Omega_{L}\) enhance electronic hopping which is mapped on the single site on the background of the surrounding identical lattice sites, the bath. (**e**) integration over all sites yields an effective self-consistent theory including non-equilibrium effects. The electronic system is coupled to Mie resonances; (**f**) diagrammatic contributions of the second order IPT. In case of uncorrelated electrons, \(U\,=\,0\), an analytical solution for the Green's function \(G_{mn}(k,\omega)\) is found by solving the Hamiltonian. The retarded component of \(G_{mn}(k,\omega)\) is  
\[G_{mn}^{R}(k,\omega)=\sum_{\rho}\frac{\int_{\rho-m}\left(A_{0}\tilde{e}_{k} \right)\,J_{\rho-n}\left(A_{0}\tilde{e}_{k}\right)}{\omega-\rho\Omega_{L}- \epsilon_{k}+i0^{+}}. \tag{3}\]  
\(\tilde{e}_{k}\) represents the externally induced dispersion which is to be distinguished from the lattice dispersion \(\epsilon_{k}\). \(J_{n}\) are the cylindrical Bessel functions of integer order, \(A_{0}=\vec{d}\cdot\vec{E}_{0}\) and \(\Omega_{L}\) characterize the external driving laser. The physical Green's function for the optically excited band electron (LB) is derived  
\[G_{\text{LB}}^{R}(k,\omega)=\sum_{m,n}G_{mn}^{R}(k,\omega). \tag{4}\]