Programmable multi-mode entanglement via dissipative engineering in vibrating trapped ions

Generating multi-partite quantum entangled states amid dissipative environment is essential for advancing quantum simulation, metrology, and fundamental quantum research. Here, we use controlled dissipation as a resource and present an approach to generate programmable multimode entangled states in the vibrational modes of trapped ions, free of stringent requirements of initial state preparation. We experimentally demonstrate the generation of multimode squeezed states across two, three, and five modes out of initial thermal states by controlled couplings to dissipative internal spins. We characterize the output states by fidelity estimates and two-body correlations, confirming genuine multipartite entanglement using the van Loock-Furusawa inseparability criteria. This work outlines a generic path to create entangled nonclassical states with quantum harmonic oscillators, applicable for quantum information processing in continuous-variable quantum systems.