Theoretical studies of optical non-linear effects in ultracold Rydberg gases

Light propagation in non-linear media generates effective photon-photon interactions, which are necessary to the implementation of two-qubit optical logic gates. In this talk I will present a study of quantum optical non-linear effects induced in an atomic ensemble in a cavity by the electromagnetically induced transparency phenomenon combined with the excitation blockade due to the strong dipole-dipole interactions between Rydberg atoms. I will first describe the dispersive regime, i.e. when atoms can be effectively considered two-level, in the Rydberg bubble approximation, and show that the transmitted light can be either “bunched” or “antibunched”, according to the cavity parameters. Then I present an alternative treatment of the system, based on the factorization of operator product averages, which allows us to study the resonant regime at the lowest non-vanishing perturbative order and reveals new features of the transmitted and reflected light correlation function. The Schwinger-Keldysh formalism allows us to push further the perturbative expansion with respect to the cavity feeding Hamiltonian: we obtain, in particular, the analytic expression of the elastic and inelastic components of the cavity transmission spectrum and identify a new polaritonic resonance structure. Finally, I will describe a new high fidelity photonic phase gate protocol based on the Rydberg blockade in an atomic ensemble placed in an optical cavity.