Dynamics of Cu$_2$O Rydberg excitons -- real density matrix approach

Basing on recent reliable measurements of the life- and coherence time of Cu$_2$O Rydberg excitons, we present an analytical method which enables to describe the experimental results. The real-density matrix approach (RDMA) is applied to describe two-photon absorption in Cu$_2$O through the excitonic response to the exciting fields. This approach produces an analytical expression for the emission intensity time evolution, and can be used to explain trends in reported experimental data. Our approach takes into account the effect of the coherence between the electron-hole pair and the electromagnetic fields, the quantum beats and the life- and coherence time dependence on the applied laser power. Adding a separation of slow- and rapid dynamics components we find excellent agreement with the experimental data.