Ultracold atoms coupled to optical cavities offer a powerful platform for studying strongly correlated many-body physics. Here, we propose an experimental scheme for creating diatomic molecules via cavity-enhanced photoassociation from an atomic condensate. The resulting system features long-range three-body interactions mediated by tripartite cavity-atom-molecule coupling. Beyond a critical pump strength, a self-organized square lattice phase for molecular condensate emerges, resulting in hybrid atom-molecule superradiance with spontaneous $U(1)$ symmetry breaking. Distinct from previously reported ultracold bosonic (fermionic) atomic superradiance, our findings demonstrate bosonic enhancement with a cubic scaling of steady-state photon number with total atom number. Additionally, strong photon-matter entanglement serves as a sensitive probe of superradiant quantum phase transition and may enable the study of super-Heisenberg-limited metrology enhanced by cavity-mediated three-body interactions. Our findings advance the frontier of quantum superchemistry and nonequilibrium many-body dynamics in cavity-coupled quantum gases.

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• Cavity quantum electrodynamics
• Cold and ultracold molecules
• Light-matter interaction
• Long-range interactions
• Photoassociation
• Quantum correlations in quantum information
• Quantum information with atoms & light
• Superradiance & subradiance