Quantum simulations with atoms and photons

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. In this talk, I will describe these setups, and some of the intriguing consequences of the interplay of strong atom-atom and strong light-matter coupling. I will then present the use of the cavity to induce long-range interactions between atoms, and show our recent results on density-wave ordering and magnetism induced by this interaction.  Last I will outline the perspectives open by the convergence of cavity QED with complex quantum matter, in particular the possibility of programming cavity-mediated interactions at will and the application of these ideas for condensed matter and high-energy physics. 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. In this talk, I will describe these setups, and some of the intriguing consequences of the interplay of strong atom-atom and strong light-matter coupling. I will then present the use of the cavity to induce long-range interactions between atoms, and show our recent results on density-wave ordering and magnetism induced by this interaction.  Last I will outline the perspectives open by the convergence of cavity QED with complex quantum matter, in particular the possibility of programming cavity-mediated interactions at will and the application of these ideas for condensed matter and high-energy physics.