Electromagnetically induced transparency with Cu$_2$O excitons in the presence of phonon coupling

Highly excited Rydberg states of excitons in Cu$_2$O semiconductors provide a promising approach to explore and control strong particle interactions in a solid-state environment. A major obstacle has been the substantial absorption background that stems from exciton-phonon coupling and lies under the Rydberg excitation spectrum, weakening the effects of exciton interactions. Here, we demonstrate that two-photon excitation of Rydberg excitons under conditions of electromagnetically induced transparency (EIT) can be used to control this background. Based on a microscopic theory that describes the known single-photon absorption spectrum, we analyze the conditions under which two-photon EIT permits separating the optical Rydberg excitation from the phonon-induced absorption background, and even suppressing it entirely. Our findings thereby pave the way for the exploitation of Rydberg blockade with Cu$_2$O excitons in nonlinear optics and other applications.