Emission Dynamics of Rydberg Excitons in $\mathbf{\mathrm{Cu_2O}}$: Distinguishing Second Harmonic Generation from Secondary Emission
Rydberg excitons in $\mathrm{Cu_2O}$ simultaneously give rise to two very different optical responses under resonant two-photon excitation: a coherent second-harmonic signal mediated by the excitonic second order susceptibility tensor $\chi^{(2)}$, and a secondary emission originating from the radiative decay of real exciton populations. Distinguishing these two channels is essential for interpreting nonlinear and quantum-optical experiments based on high-$n$ states, yet their temporal, spectral, and power-dependent signatures often overlap. Here we use time-resolved resonant two-photon excitation to cleanly separate SHG and SE and to map how each depends on $n$, temperature, excitation power, and crystal quality. This approach reveals the markedly different sensitivities of the two processes to phonons, defects, and many-body effects, and establishes practical criteria for identifying SE and SHG in a wide range of experimental conditions. Our results provide a unified framework for interpreting emission from Rydberg excitons and offer guidelines for future studies aiming to exploit their nonlinear response and long-range interactions.