Realization of three and four-body interaction in a cavity-QED system

Spin Hamiltonians in quantum simulation and quantum sensing have traditionally relied on pairwise (two-body) interactions among system constituents. Here, we present the experimental realization of an effective three-body Hamiltonian using laser-cooled atoms confined in a high-finesse optical cavity. The pseudo-spin-1/2 degrees of freedom are encoded in two atomic momentum states, and the interaction is engineered via two dressing tones that mediate photon exchange through the cavity. This configuration enables a virtual six-photon process while suppressing lower-order interactions through destructive interference. The resulting three-body interaction provides a powerful tool for rapid entanglement generation in quantum-enhanced sensing and for exploring exotic quantum phases. Moreover, the flexibility of this platform allows for extension to multi-level systems and higher-order interactions, such as four-body couplings mediated by virtual eight-photon processes. In addition, we report ongoing progress toward simulating a Higgs mode across the Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein condensate (BEC) crossover using cavity-mediated interactions.