  Quantum simulation and sensing hold great promise for providing new insights
into nature, from understanding complex interacting systems to searching for
undiscovered physics. Large ensembles of laser-cooled atoms interacting via
infinite-range photon mediated interactions are a powerful platform for both
endeavours. Here, we realize for the first time momentum-exchange interactions
in which atoms exchange their momentum states via collective emission and
absorption of photons from a common cavity mode. The momentum-exchange
interaction leads to an observed all-to-all Ising-like interaction in a
matter-wave interferometer, which is useful for entanglement generation. A
many-body energy gap also emerges, effectively binding interferometer
matter-wave packets together to suppress Doppler dephasing, akin to M\"ossbauer
spectroscopy. The tunable momentum-exchange interaction provides a new
capability for quantum interaction-enhanced matter-wave interferometry and for
realizing exotic behaviors including simulations of superconductors and
dynamical gauge fields.
