Spin- and atom-interactions in multimode cavity QED

Optical cavity QED provides a versatile platform with which to explore quantum many-body physics in driven-dissipative systems. Multimode cavities are particularly useful for exploring beyond mean-field physics. We highlight experimental progress towards simulation of driven-dissipative spin glasses. This involves demonstrating a strong, tunable-range, photon-mediated atom-atom interaction, a sign-changing interaction due to Gouy phases, and a spin-spin interaction in a spinful Bose-Einstein condensate. The joint spin-spatial degrees of freedom exhibit spinor self-organization as well as dynamical spin-orbit coupling. We highlight near-term experimental studies of few-body interacting systems exploring exotic, strongly correlated phenomena such as driven-dissipative spin glasses and quantum neural networks.