• gases
• Enrico Fermi
• cavity
• ultracold

Main aim of the present thesis is to investigate the quantum phases arising from the interaction of an ultra-cold Fermi gas with the quantized electromagnetic field of a single-mode optical cavity. Both numerical and analytical methods are used. In particular, we suggest that the same microscopic mechanism, originated by exchange of photons inside the optical cavity, induces interactions between the spins of the fermionic atoms and, indirectly, a superfluid pairing. As a result, a phase diagram can be established, where different order parameters emerge, related to spin and density degrees of freedom, which evolve into each other after tuning the sign and strength of the interactions across the cavity resonance.