The observation of giant Rydberg excitons in cuprous oxide (Cu2O) up to a principal quantum number of n = 25 by T. Kazimierczuk et al. [Nature 514, 343, (2014)] inevitably raises the question whether these quasi-particles must be described within a multi-polariton framework since excitons and photons are always coupled in the solid. In this paper we present the theory of excitonpolaritons in Cu2O. To this end we extend the Hamiltonian which includes the complete valence band structure, the exchange interaction, and the central-cell corrections effects, and which has been recently deduced by F. Schweiner et al. [Phys. Rev. B 95, 195201, (2017)], for finite values of the exciton momentum K. We derive formulas to calculate not only dipole but also quadrupole oscillator strengths when using the complete basis of F. Schweiner et al., which has recently been proven as a powerful tool to calculate exciton spectra. Very complex polariton spectra for the three orientations of K along the axes [001], [110], and [111] of high symmetry are obtained and a strong mixing of exciton states is reported. The main focus is on the 1S ortho exciton-polariton, for which pronounced polariton effects have been measured in experiments. We set up a 5 × 5 matrix model, which accounts for both the polariton effect and the K-dependent splitting, and which allows treating the anisotropic polariton dispersion for any direction of K. We especially discuss the dispersions for K being oriented in the planes perpendicular to [110] and [111], for which experimental transmission spectra have been measured. Furthermore, we compare our results with experimental values of the K-dependent splitting, the group velocity, and the oscillator strengths of this exciton-polariton. The results are in good agreement. This proves the validity of the 5 × 5 matrix model as a useful theoretical model for further investigations on the 1S ortho exciton-polariton.

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Experimental and theoretical investigations of excitons in cuprous oxide have revealed a significant fine-structure splitting of the excitonic Rydberg states caused by a strong impact of the valence band structure. We provide a semiclassical interpretation of that splitting by investigating the classical dynamics of the excitonic electron-hole pair beyond the hydrogen-like model. Considering the slow motion of Rydberg excitons in coordinate space compared to the fast dynamics of quasispin and hole spin we use an adiabatic approach and energy surfaces in momentum space for the computation of the exciton dynamics. We observe quasi-periodic motion on near-integrable tori. Semiclassical torus quantization yields the energy regions of the fine-structure splitting of n-manifolds in agreement with quantum mechanical computations.