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Nonlinear optical properties and Kerr nonlinearity of Rydberg excitons in Cu$_2$O quantum wells

The quantum confiment of Rydberg excitons (REs) in quantum structures opens the way towards considering nonlinear interactions in such systems. We present a theoretical calculation of optical functions in the case of a nonlinear coupling between REs in a quantum well with an electromagnetic wave. Using the Real Density Matrix Approach (RDMA), the analytical expressions for a linear and nonlinear absorption are derived and numerical calculations for Cu$_2$0 quantum wells are performed. The results indicate the conditions in which quantum well confinement states can be observed in linear and nonlinear optical spectra. The Kerr nonlinearity and self-phase modulation in such a system are studied. The effect of Rydberg blockade and the associated optical bleaching are also discussed and confronted with available experimental data.