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We use a self-assembled two-dimensional Coulomb crystal of $\sim 70$ ions inthe presence of an external transverse field to engineer a quantum simulator ofthe Dicke Hamiltonian. This Hamiltonian has spin and bosonic degrees of freedomwhich are encoded by two hyperfine states in each ion and the center of massmotional mode of the crystal, respectively. The Dicke model features a quantumcritical point separating two distinct phases: the superradiant (ferromagnetic)and normal (paramagnetic) phases. We experimentally explore protocols that aimto adiabatically prepare the superradiant ground state, a spin-boson cat statewith macroscopic phonon occupation, which is well-suited for enhanced metrologyand quantum information processing. We start in the normal phase, with allspins aligned along a large transverse field and ramp down the field across thecritical point following various protocols. We measure the spin observables,both experimentally and in our simulations to characterize the state of thesystem at the end of the ramp. We find that under current operating conditionsan optimally designed ramp is not sufficient to achieve significant fidelitywith the superradiant ground state. However, our theoretical investigationshows that slight modifications of experimental parameters, together withmodest reductions in decoherence rates and thermal noise can increase thecat-state fidelity to $\sim 75\%$ for $N \sim 20$ spins. Our results open apath for the use of large ensembles of trapped ions as powerful quantum sensorsand quantum computers.

Exploring adiabatic quantum dynamics of the Dicke model in a trapped ion quantum simulator