Realization of three- and four-body interactions between momentum states in a cavity

Spin Hamiltonians in condensed matter and quantum sensing typically utilize pairwise or two-body interactions between constituents in the material or ensemble. However, there is growing interest in exploring more general <i>n</i>-body interactions for <i>n</i> > 2. In this study, we realized an effective <i>n</i> = 3-body Hamiltonian interaction using an ensemble of laser-cooled atoms in a high-finesse optical cavity with the pseudospin [Formula: see text] encoded by two atomic momentum states. We applied two dressing tones that induce the atoms to exchange photons via the cavity to realize a virtual six-photon process; lower-order interactions destructively interfered. We also observed signatures of a <i>n</i> = 4-body interaction mediated by a virtual eight-photon process. Our approach may be extensible to three-body interactions in multilevel systems or to even higher-order interactions.