Abstract
The eigenstate thermalization hypothesis (ETH) offers a universal mechanism for the approach to equilibrium of closed quantum many-body systems. So far, however, experimental studies have focused on the relaxation dynamics of observables as described by the diagonal part of ETH, whose verification requires substantial numerical input. This leaves many of the general assumptions of ETH untested. Here, we propose a theory-independent route to probe the full ETH in quantum simulators by observing the emergence of fluctuation-dissipation relations, which directly probe the off-diagonal part of ETH. We discuss and propose protocols to independently measure fluctuations and dissipations as well as higher-order time ordered correlation functions. We first show how the emergence of fluctuation dissipation relations from a nonequilibrium initial state can be observed for the 2D Bose-Hubbard model in superconducting qubits or quantum gas microscopes. Then we focus on the long-range transverse field Ising model (LTFI), which can be realized with trapped ions. The LTFI exhibits rich thermalization phenomena: For strong transverse fields, we observe prethermalization to an effective magnetization-conserving Hamiltonian in the fluctuation dissipation relations. For weak transverse fields, confined excitations lead to non-thermal features resulting in a violation of the fluctuation-dissipation relations up to long times. Moreover, in an integrable region of the LTFI, thermalization to a generalized Gibbs ensemble occurs and the fluctuation-dissipation relations enable an experimental diagonalization of the Hamiltonian. Our work presents a theory-independent way to characterize thermalization in quantum simulators and paves the way to quantum simulate condensed matter pump-probe experiments.
Highlights
The long coherence timescales accessible in quantum simulators made it possible to experimentally observe thermalization in isolated quantum systems [1,2,3,4], the absence thereof in the presence of disorder [5,6,7,8,9], and integrability in reduced dimensions [10,11]
(a) Central-time averaged equal-site density-density spectral function ρ as a function of central time JT and frequency. (b) Late time spectral and statistical functions (JT = 40, dark) compared to early times (JT = 2, bright). (c) Fluctuation dissipation relation function defined in Eq (23) at time JT = 40 compared to the equilibrium expectation, with the inverse temperature β set by energy of the initial state according to Eq (24)
We have shown how to probe the off-diagonal part of eigenstate thermalization with two-time functions in quantum simulators, which is an open experimental challenge
Summary
The long coherence timescales accessible in quantum simulators made it possible to experimentally observe thermalization in isolated quantum systems [1,2,3,4], the absence thereof in the presence of disorder [5,6,7,8,9], and integrability in reduced dimensions [10,11] These observations were based on probing equal-time correlation functions [12,13,14], concluding the observation of equilibration by comparison to the expected microcanonical expectation values at the same energy density as the initial state. At small transverse fields, confined excitations can be directly observed in the spectral function and lead to genuine nonthermal features including a violation of the FDR observable up to long times
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