Realization of cavity-mediated three and four-body interactions in a Bragg interferometer*

Spin Hamiltonians in quantum simulation and quantum sensing have traditionally relied on pairwise (two-body) interactions between system constituents. Here, we present an experimental realization of an effective three-body (n = 3) Hamiltonian in a system of laser-cooled atoms in a high-finesse optical cavity. The pseudo-spin-1/2 states are encoded in two atomic momentum states, and the interaction is achieved through two dressing tones that drive photon exchange via the cavity, enabling a virtual six-photon process while suppressing lower-order interactions through destructive interference. The resulting interactions provides a new tool for rapid entanglement generation for quantum enhanced sensing as well as for exploring exotic quantum phases. Moreover, the flexibility of our platform allows for extending to multi-level systems, higher-order interactions, such as a four-body (n = 4) interaction mediated by a virtual eight-photon process.