Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.29817

Observation of universal hierarchical relaxation in a quantum simulator

Jiaozi Wang, Manoj K. Joshi, Luca Capizzi, Rainer Blatt, Christian F. Roos, Leonardo Mazza, Dario Poletti

cond-mat.stat-mechcond-mat.quant-gasquant-phSubmitted Sep 24, 2026• Announced Sep 25, 2026• 0 citations
arXivHTMLPDF

Abstract

Autocorrelation functions play a key role in the theoretical characterization of the dynamical properties of interacting many-body quantum systems. Recently, bringing together the eigenstate thermalization hypothesis and hydrodynamics, it was theoretically predicted that the relaxation of autocorrelators can be described by the \textit{relaxation-overlap inequality}, which, when saturated, predicts a hierarchy of relaxation exponents for a set of operators that are easily identified and constructed. Here, we employ a trapped-ion quantum simulator to experimentally demonstrate it in the diffusive regime; to do so, we have extended the theory of the overlap-relaxation inequality to systems with multiple conservation laws. Our study thus opens the path to a thorough characterization of the relaxation to equilibrium and appearance of hydrodynamic behavior in quantum matter.

Comments
10 pages, 4 figures

Related papers

Ergodicity and hydrodynamics: From quantum to classical spin systems

Jiaozi Wang, Luca Capizzi, Dario Poletti +1 more

We show that in classical spin systems, the precise nature of the late-time hydrodynamic tails of the autocorrelation functions of a generic observable is determined by (i) the dynamical critical exponent and (ii) the equilibrium thermodynamic properties of the corresponding observable. We provide numerical results for one- and two-dimensional systems and present theoretical considerations that only rely on the notion of ergodicity. Our result extends to the classical framework the relaxation-overlap inequality, first introduced in [Phys. Rev. X 15, 011059 (2025)2160-330810.1103/PhysRevX.15.011059] for quantum many-body systems satisfying the eigenstate thermalization hypothesis.

arXiv:2509.04098•Physical review. E•May 15, 2026•0 citations

Hydrodynamics and the Eigenstate Thermalization Hypothesis

Luca Capizzi, Jiaozi Wang, Xiansong Xu +2 more

The eigenstate thermalization hypothesis (ETH) describes the properties of diagonal and off-diagonal matrix elements of local operators in the eigenenergy basis. In this work, we propose a relation between (i) the singular behavior of the off-diagonal part of ETH at small energy differences and (ii) the smooth profile of the diagonal part of ETH as a function of the energy density. We establish this connection from the decay of the autocorrelation functions of local operators, which is constrained by the presence of local conserved quantities whose evolution is described by hydrodynamics. We corroborate our predictions through numerical simulations of two distinct nonintegrable spin-1 Ising models, exhibiting diffusive and superdiffusive transport behaviors. The simulations are performed using dynamical quantum typicality up to 18 spins, for both infinite- and finite-temperature regimes.

arXiv:2405.16975•Physical Review X•Mar 14, 2025•20 citations

Probing eigenstate thermalization in quantum simulators via fluctuation-dissipation relations

Alexander Schuckert, Michael Knap

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 two-dimensional (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 nonthermal 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.

arXiv:2007.10347•Physical Review Research•Dec 4, 2020•22 citations

Observing emergent hydrodynamics in a long-range quantum magnet

Manoj K. Joshi, Florian Kranzl, Alexander Schuckert +5 more

Identifying universal properties of nonequilibrium quantum states is a major challenge in modern physics. A fascinating prediction is that classical hydrodynamics emerges universally in the evolution of any interacting quantum system. We experimentally probed the quantum dynamics of 51 individually controlled ions, realizing a long-range interacting spin chain. By measuring space-time-resolved correlation functions in an infinite temperature state, we observed a whole family of hydrodynamic universality classes, ranging from normal diffusion to anomalous superdiffusion, that are described by Lévy flights. We extracted the transport coefficients of the hydrodynamic theory, reflecting the microscopic properties of the system. Our observations demonstrate the potential for engineered quantum systems to provide key insights into universal properties of nonequilibrium states of quantum matter.

arXiv:2107.00033•Science•May 12, 2022•120 citations

Diffusive Relaxation of Participation Entropy in U(1)-symmetric Dynamics

Hanchen Liu, Tianci Zhou, Xiao Chen

Participation entropy (PE) quantifies the spread of a many-body wavefunction across configuration space. While PE relaxes rapidly in generic chaotic systems, we show that $\mathrm{U}(1)$ conservation laws slow it down by imprinting with the slow hydrodynamic modes. Using a cluster expansion around equilibrium, we show that, after local density inhomogeneities decay, the leading PE deficit is dominated by squared connected density correlations. The long time relaxation is therefore controlled by diffusive correlation spreading, giving $ΔS(t)\sim t^{-1/2}$ in the hydrodynamic regime and crossing over to $\sim \exp[-O(t/L^2)]$ when $t\geq L^2$. We confirm this entropy correlation relation using exact computation and infinite system tensor network simulations in various quantum $\mathrm{U}(1)$ conserving circuits. Our results establish PE as a sensitive probe of hydrodynamic memory and suggest that slow relaxation is a generic consequence of conservation laws.

quant-pharXiv:2606.11561•Jun 11, 2026•0 citations

Hierarchical hydrodynamics in long-range multipole-conserving systems

Jacopo Gliozzi, Julian May-Mann, Taylor L. Hughes +1 more

This work investigates the out-of-equilibrium dynamics of dipole and higher-moment-conserving systems with long-range interactions, drawing inspiration from trapped ion experiments in strongly tilted potentials. We introduce a hierarchical sequence of multipole-conserving models characterized by power-law decaying couplings. Although the moments are always globally conserved, adjusting the power-law exponents of the couplings induces various regimes in which only a subset of multipole moments are effectively locally conserved. We examine the late-time hydrodynamics analytically and numerically using an effective classical framework, uncovering a rich dynamical phase diagram that includes subdiffusion, conventional diffusion, and L\'evy flights. Our results are unified in an analytic reciprocal rule that captures the nested hierarchy of hydrodynamics in multipole conserving systems where only a subset of the moments are locally conserved. Moreover, we extend our findings to higher dimensions and explore the emergence of long timescales, reminiscent of prethermal regimes, in systems with low charge density. Lastly, we corroborate our results through state-of-the-art numerical simulations of a fully quantum long-range dipole-conserving system and discuss their relevance to trapped-ion experimental setups.

arXiv:2304.12342•Physical review. B./Physical review. B•Nov 6, 2023•16 citations
Autocorrelation functions play a key role in the theoretical characterization of the dynamical properties of interacting many-body quantum systems. Recently, bringing together the eigenstate thermalization hypothesis and hydrodynamics, it was theoretically predicted that the relaxation of autocorrelators can be described by the \textit{relaxation-overlap inequality}, which, when saturated, predicts a hierarchy of relaxation exponents for a set of operators that are easily identified and constructed. Here, we employ a trapped-ion quantum simulator to experimentally demonstrate it in the diffusive regime; to do so, we have extended the theory of the overlap-relaxation inequality to systems with multiple conservation laws. Our study thus opens the path to a thorough characterization of the relaxation to equilibrium and appearance of hydrodynamic behavior in quantum matter.
We show that in classical spin systems, the precise nature of the late-time hydrodynamic tails of the autocorrelation functions of a generic observable is determined by (i) the dynamical critical exponent and (ii) the equilibrium thermodynamic properties of the corresponding observable. We provide numerical results for one- and two-dimensional systems and present theoretical considerations that only rely on the notion of ergodicity. Our result extends to the classical framework the relaxation-overlap inequality, first introduced in [Phys. Rev. X 15, 011059 (2025)2160-330810.1103/PhysRevX.15.011059] for quantum many-body systems satisfying the eigenstate thermalization hypothesis.
The eigenstate thermalization hypothesis (ETH) describes the properties of diagonal and off-diagonal matrix elements of local operators in the eigenenergy basis. In this work, we propose a relation between (i) the singular behavior of the off-diagonal part of ETH at small energy differences and (ii) the smooth profile of the diagonal part of ETH as a function of the energy density. We establish this connection from the decay of the autocorrelation functions of local operators, which is constrained by the presence of local conserved quantities whose evolution is described by hydrodynamics. We corroborate our predictions through numerical simulations of two distinct nonintegrable spin-1 Ising models, exhibiting diffusive and superdiffusive transport behaviors. The simulations are performed using dynamical quantum typicality up to 18 spins, for both infinite- and finite-temperature regimes.
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 two-dimensional (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 nonthermal 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.
Identifying universal properties of nonequilibrium quantum states is a major challenge in modern physics. A fascinating prediction is that classical hydrodynamics emerges universally in the evolution of any interacting quantum system. We experimentally probed the quantum dynamics of 51 individually controlled ions, realizing a long-range interacting spin chain. By measuring space-time-resolved correlation functions in an infinite temperature state, we observed a whole family of hydrodynamic universality classes, ranging from normal diffusion to anomalous superdiffusion, that are described by Lévy flights. We extracted the transport coefficients of the hydrodynamic theory, reflecting the microscopic properties of the system. Our observations demonstrate the potential for engineered quantum systems to provide key insights into universal properties of nonequilibrium states of quantum matter.
Participation entropy (PE) quantifies the spread of a many-body wavefunction across configuration space. While PE relaxes rapidly in generic chaotic systems, we show that U(1)\mathrm{U}(1)U(1) conservation laws slow it down by imprinting with the slow hydrodynamic modes. Using a cluster expansion around equilibrium, we show that, after local density inhomogeneities decay, the leading PE deficit is dominated by squared connected density correlations. The long time relaxation is therefore controlled by diffusive correlation spreading, giving ΔS(t)∼t−1/2ΔS(t)\sim t^{-1/2}ΔS(t)∼t−1/2 in the hydrodynamic regime and crossing over to ∼exp⁡[−O(t/L2)]\sim \exp[-O(t/L^2)]∼exp[−O(t/L2)] when t≥L2t\geq L^2t≥L2. We confirm this entropy correlation relation using exact computation and infinite system tensor network simulations in various quantum U(1)\mathrm{U}(1)U(1) conserving circuits. Our results establish PE as a sensitive probe of hydrodynamic memory and suggest that slow relaxation is a generic consequence of conservation laws.
This work investigates the out-of-equilibrium dynamics of dipole and higher-moment-conserving systems with long-range interactions, drawing inspiration from trapped ion experiments in strongly tilted potentials. We introduce a hierarchical sequence of multipole-conserving models characterized by power-law decaying couplings. Although the moments are always globally conserved, adjusting the power-law exponents of the couplings induces various regimes in which only a subset of multipole moments are effectively locally conserved. We examine the late-time hydrodynamics analytically and numerically using an effective classical framework, uncovering a rich dynamical phase diagram that includes subdiffusion, conventional diffusion, and L\'evy flights. Our results are unified in an analytic reciprocal rule that captures the nested hierarchy of hydrodynamics in multipole conserving systems where only a subset of the moments are locally conserved. Moreover, we extend our findings to higher dimensions and explore the emergence of long timescales, reminiscent of prethermal regimes, in systems with low charge density. Lastly, we corroborate our results through state-of-the-art numerical simulations of a fully quantum long-range dipole-conserving system and discuss their relevance to trapped-ion experimental setups.