  Strong quantum correlations in matter are responsible for some of the most
extraordinary properties of material, from magnetism to high-temperature
superconductivity, but their integration in quantum devices requires a strong,
coherent coupling with photons, which still represents a formidable technical
challenge in solid state systems. In cavity quantum electrodynamics, quantum
gases such as Bose-Einstein condensates or lattice gases have been strongly
coupled with light. However, neither Fermionic quantum matter, comparable to
electrons in solids, nor atomic systems with controlled interactions, have thus
far been strongly coupled with photons. Here we report on the strong coupling
of a quantum-degenerate unitary Fermi gas with light in a high finesse cavity.
We map out the spectrum of the coupled system and observe well resolved dressed
states, resulting from the strong coupling of cavity photons with each spin
component of the gas. We investigate spin-balanced and spin-polarized gases and
find quantitative agreement with ab-initio calculation describing light-matter
interaction. Our system offers complete and simultaneous control of atom-atom
and atom-photon interactions in the quantum degenerate regime, opening a wide
range of perspectives for quantum simulation.
