Title: Exploring quantum matter with atoms and photons

Abstract: 
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 Fermi gases. After describing this system, I will present the use of the cavity to engineer long-range interactions between Fermions or between spins encoded atoms. In a first experiment, we realize a doubly-tunable Fermi gas, with strong contact interactions and strong long-range, photon mediated interactions. This system exhibits superfluidity without long-range interactions and a density-wave order above a critical long-range interaction strength, opening a new window into the complex interplay of phases in quantum matter. In a second experiment, we realize an all-to-all connected quantum magnet and study the competition between ferromagnetism and a controlled disorder, by controlling locally the spatial structure of light-matter interaction. This opens the perspective to engineer of arbitrary strength and range of interactions in quantum gases.