Exploring and Controlling Fermi Gases with Light in a High-Finesse Cavity

Cavity quantum electrodynamics (QED) is one of the most powerful framework to observe and leverage quantum phenomena. While it has been thoroughly studied for simple quantum systems such as two-level systems or harmonic oscillators, it has only recently become available for complex, correlated quantum many-body systems. In the last five years, we have developed systems combining cavity QED with ultra-cold Fermi gases [1]. In this talk, I will describe several intriguing consequences of the interplay of strong atom-atom and strong light-matter coupling, such as the onset of coherent excitations mixing Fermion-pairs and photons, pair-polaritons [2] or the coupling of density fluctuations with light [3]. I will then present the use of the cavity to induce  long-range interactions in a strongly-interacting Fermi gas, leading to density-wave order, a system of direct relevance to condensed matter physics. Last I will outline the perspectives open by the convergence of cavity QED with Fermionic quantum matter, in particular the possibility of programming cavity-mediated interactions at will between atoms.