Cold Fermions with long-range, all-to-all interactions*

Cavity quantum electrodynamics offers the possibilty to control and enhance light-matter interactions, such that atoms trapped inside a cavity can exchange photons with each other over the entire volume of the cavity mode. As a result, for clouds of atoms inside macroscopic cavities, this photon exchange interaction is naturally of all-to-all character. In this talk, I will present our realization of ultracold Fermions with all-to-all interactions induced by cavity photons. I will then present our progresses in controlling the spatial structure of this interaction, first in a proof-of-principle experiment where these interactions are made random, then in a new cavity-microscope device, where light-matter interactions are controlled and programmed in space and in time down to the smallest microscopic length and time scales. Last I will outline the results of simulations demonstrating how this device can be leveraged to implement the SYK interaction in a scalable way on a mesoscopic gas of Fermionic atoms.