Observation of three and four-body interactions between momentum states in a high-finesse cavity

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 rubidium atoms in a high-finesse optical cavity1 . 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. This enables a virtual six-photon process while suppressing lower-order interactions through destructive interference. By tuning the desired process into resonance and detuning the unwanted processes, pure n-body interactions can be generated with the lower-order processes canceled via symmetry. The n-body interaction can be understood as n different atoms flipping their momentum states in concert from p0 − ℏk to p0 + ℏk. The n-body interactions are experimentally observed both via a spectroscopic signal and through mean-field dynamics measured over the Bloch sphere. The resulting interactions exhibit an all-to-all connectivity, making them suitable for rapid entanglement generation and the exploration of exotic quantum phases. The flexibility of our platform also allows for extension to multi-level systems and higher-order interactions, such as a four-body (n = 4) interaction mediated by a virtual eight-photon process.