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Recent work proposed a variant on Ramsey interferometry for coupledspin-$1/2$ systems that directly measures the retarded spin-spin Green'sfunction. We expand on that work by investigating nonequilibrium retardedspin-spin Green's functions within the transverse-field Ising model. We derivethe lowest four spectral moments to understand the short-time behavior and weemploy a Lehmann-like representation to determine the spectral behavior. Wesimulate a Ramsey protocol for a nonequilibrium quantum spin system thatconsists of a coherent superposition of the ground state and diabaticallyexcited higher-energy states via a temporally ramped transverse magnetic field.We then apply the Ramsey spectroscopy protocol to the final Hamiltonian, whichhas a constant transverse field. The short-time behavior directly relates toLieb-Robinson bounds for the transport of many-body correlations, while thelong-time behavior relates to the excitation spectra of the Hamiltonian.Compressive sensing is employed in the data analysis to efficiently extractthat spectra.

Measuring nonequilibrium retarded spin-spin Green's functions in an ion-trap based quantum simulator