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