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

Paradigmatic spin Hamiltonians in condensed matter and quantum sensing typically utilize pair-wise or 2-body interactions between constituents in the material or ensemble. However, there is growing interest in exploring more general $n$-body interactions for $n >2$, with examples including more efficient quantum gates or the realization of exotic many-body fracton states. Here we realize an effective $n=3$-body Hamiltonian interaction using an ensemble of laser-cooled atoms in a high finesse optical cavity with the pseudo-spin 1/2 encoded by two atomic momentum states. To realize this interaction, we apply two dressing tones that coax the atoms to exchange photons via the cavity to realize a virtual 6-photon process, while the lower-order interactions destructively interfere. The resulting photon mediated interactions are not only $n>2$-body but also all-to-all(-to-all) and therefore of great interest for fast entanglement generation and quantum simulation of exotic phases such as the long sought but not yet observed charge-Qe superconductors, with $Q=2n$ . The versatility of our experimental system can also allow for extending to 3-body interactions in multi-level systems or to higher-order interactions, such as the signature of a $n=4$-body interaction mediated by a virtual eight photon process that we also observe.