Probing eigenstate thermalization with the emergence of fluctuation-dissipation relations in quantum simulators

We propose a theory-independent route to probe the full eigenstate thermalization hypothesis in quantum simulators by observing the emergence of fluctuation-dissipation relations (FDRs). We present protocols to independently measure fluctuations and dissipations as well as higher-order time ordered correlation functions. We first show how the emergence of FDRs from a nonequilibrium initial state can be observed for the 2D Bose-Hubbard model in superconducting qubits or quantum gas microscopes. In the long range transverse field Ising model implementable by trapped ions, we show that FDRs can be used to observe prethermalization to a Hamiltonian with an approximately conserved quantity at large transverse fields. Furthermore, our protocols make it possible to directly identify non-thermal excitations in experiment. The FDRs enable an experimental diagonalization of the Hamiltonian in integrable models. Our work paves the way to quantum simulate condensed matter pump-probe experiments.