The excitonic polariton concept was introduced already in 1958 by J. J. Hopfield. Although its description was based on a full quantum theory including light quantization, the investigations of the optical properties of excitons developed mainly independently of quantum optics. In this chapter we shall review exciton polariton quantum optical effects by means of some recent works and results that have appeared in the literature in both bulk semiconductors and in cavity embedded quantum wells. The first manifestation of excitonic quantum-optical coherent dynamics was observed experimentally 20 years later, in 1978, exploiting resonant hyper-parametric scattering. On the other hand, the possibility of generating entangled photon pairs by means of this resonant process was theoretically pointed out only lately in 1999, whereas the experimental evidence for the generation of ultraviolet polarization-entangled photon pairs by means of biexciton resonant parametric emission in a single crystal of semiconductor CuCl was reported only in 2004. The demonstrations of parametric amplification and parametric emission in semiconductor microcavities, together with the possibility of ultrafast optical manipulation and ease of integration of these micro-devices, have increased the interest in possible realization of nonclassical cavity-polariton states. In 2005 an experiment that probes polariton quantum correlations by exploiting quantum complementarity was proposed and realized. These results unequivocally proved that quantum optical effects at single photon level, arising from the interaction of light with electronic excitations of semiconductors and semiconductor nanostructures, were possible within these solid state systems, despite being far from isolated systems. The theoretical predictions that we review are based on a microscopic quantum theory of the nonlinear optical response of interacting electron systems relying on the dynamics controlled truncation scheme extended to include light quantization. In addition, the environment decoherence is taken into account microscopically by Markov noise sources within a quantum Langevin framework.
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