Scalable Protocols for Probing $g^{(1)}$ and $g^{(2)}$ of Phonons in Quantum Simulations with Trapped Ions*

Simulation of bosonic systems with dense coupling graphs can be hard for classical computers and include interesting problems such as boson sampling, spin-boson Hamiltonians with long range interactions as well as light-matter interactions cite{H,O}. For general bosonic systems, the response functions can be used to extract information about the dynamics as well as identifying the critical points. For example, in quantum optics, measuring $g^{(1)}$ and $g^{(2)}$ are the standard toolbox used to quantify the classical and quantum correlations between photons. However, the utility of quantum simulations of bosonic systems can be limited by the exponential resources needed to characterize the dynamics of the infinite dimensional Fock space corresponding to the interacting collective phonons in the chain of ions. Here, we present efficient protocols to probe $g^{(1)}$ and $g^{(2)}$ of phonons which require polynomial number of local operations for a given number of collective phonons involved in the quantum simulation. 

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