  We devise a phase-coherent three-pulse protocol to probe the polariton
dynamics in a trapped-ion quantum simulation. In contrast to conventional
nonlinear signals, the presented scheme does not change the number of
excitations in the system, allowing for the investigation of the dynamics
within an $N$-excitation manifold. In the particular case of a filling factor
one ($N$ excitations in an $N$-ion chain), the proposed interaction induces
coherent transitions between a delocalized phonon superfluid and a localized
atomic insulator phase. Numerical simulations of a two-ion chain demonstrate
that the resulting two-dimensional spectra allow for the unambiguous
identification of the distinct phases, and the two-dimensional lineshapes
efficiently characterize the relevant decoherence mechanism.
