  Optical cavity QED provides a platform with which to explore quantum
many-body physics in driven-dissipative systems. Single-mode cavities provide
strong, infinite-range photon-mediated interactions among intracavity atoms.
However, these global all-to-all couplings are limiting from the perspective of
exploring quantum many-body physics beyond the mean-field approximation. The
present work demonstrates that local couplings can be created using multimode
cavity QED. This is established through measurements of the threshold of a
superradiant, self-organization phase transition versus atomic position.
Specifically, we experimentally show that the interference of near-degenerate
cavity modes leads to both a strong and tunable-range interaction between
Bose-Einstein condensates (BECs) trapped within the cavity. We exploit the
symmetry of a confocal cavity to measure the interaction between real BECs and
their virtual images without unwanted contributions arising from the merger of
real BECs. Atom-atom coupling may be tuned from short range to long range. This
capability paves the way toward future explorations of exotic, strongly
correlated systems such as quantum liquid crystals and driven-dissipative spin
glasses.
