The doctorand will be integrated in the experimental research activity of the Quantum Optics Group of the Wigner Research Centre for Physics. The group aims at the implementation of atomic quantum technologies for quantum sensing and for quantum communication. The thesis work consists in the development of an operating cold atom cavity QED setup. The objective is the realization of various non-linear optical measurements in an exotic medium of quantum matter, which is an intra-cavity optical lattice of atoms. The experiments also aim at the observation of collective effects in the radiation of the atomic cloud coupled to a high-finesse optical resonator. The proposed research requires substantial theoretical work in the field of quantum optics, in particular, the description of many-body open quantum systems.