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Physical Review A

Many-body interferometry of one-dimensional integrable systems

Min Liu, Maksims Arzamasovs, Yue Zhang, Rui Tian, Zehou Li, Shuai Li, Bo Liu

Published May 16, 2025• 1 citation
Publisher

Abstract

We propose using many-body Ramsey interferometry to measure correlation functions of one-dimensional (1D) integrable systems. The 1D transverse-field Ising model, which is conjectured to equilibrate into nonthermal generalized Gibbs ensemble (GGE) steady states, is studied. It is shown that retarded Green's functions, as opposed to ordinary spin-spin correlators considered previously, convincingly distinguish between the GGE and thermal postquench steady states, justifying the assumption of convergence towards the GGE as the system equilibrates. We also propose the experimental protocol for measuring response functions with Ramsey interferometry, which can be used to distinguish between different postquench phases of the model. Our proposal can be realized with current ultracold atom techniques and opens up the possibility to study dynamics in nonthermal ensembles.

DOI
10.1103/physreva.111.053310

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We propose using many-body Ramsey interferometry to measure correlation functions of one-dimensional (1D) integrable systems. The 1D transverse-field Ising model, which is conjectured to equilibrate into nonthermal generalized Gibbs ensemble (GGE) steady states, is studied. It is shown that retarded Green's functions, as opposed to ordinary spin-spin correlators considered previously, convincingly distinguish between the GGE and thermal postquench steady states, justifying the assumption of convergence towards the GGE as the system equilibrates. We also propose the experimental protocol for measuring response functions with Ramsey interferometry, which can be used to distinguish between different postquench phases of the model. Our proposal can be realized with current ultracold atom techniques and opens up the possibility to study dynamics in nonthermal ensembles.
We propose using many-body Ramsey interferometry to measure non-equilibrium correlation functions of one-dimensional (1D) integrable systems. The 1D transverse-field Ising model, which is conjectured to equilibrate into non-thermal Gibbs ensemble (GGE) steady states, is studied. It is shown that retarded Green's functions, as opposed to ordinary spin-spin correlators considered previously, can convincingly distinguish between the GGE and thermal post-quench steady states, justifying the assumption of convergence towards the GGE as the system equilibrates. We also propose the experimental protocol for measuring the response functions with Ramsey interferometry, which can be used to distinguish between different post-quench phases of the model. Our proposal can be realized with current ultracold atom techniques, and opens up the possibility to study dynamics in non-thermal ensembles.
Abstract The generalized Gibbs ensemble (GGE) was introduced ten years ago to describe observables in isolated integrable quantum systems after equilibration. Since then, the GGE has been demonstrated to be a powerful tool to predict the outcome of the relaxation dynamics of few-body observables in a variety of integrable models, a process we call generalized thermalization. This review discusses several fundamental aspects of the GGE and generalized thermalization in integrable systems. In particular, we focus on questions such as: which observables equilibrate to the GGE predictions and who should play the role of the bath; what conserved quantities can be used to construct the GGE; what are the differences between generalized thermalization in noninteracting systems and in interacting systems mappable to noninteracting ones; why is it that the GGE works when traditional ensembles of statistical mechanics fail. Despite a lot of interest in these questions in recent years, no definite answers have been given. We review results for the XX model and for the transverse field Ising model. For the latter model, we also report original results and show that the GGE describes spin–spin correlations over the entire system. This makes apparent that there is no need to trace out a part of the system in real space for equilibration to occur and for the GGE to apply. In the past, a spectral decomposition of the weights of various statistical ensembles revealed that generalized eigenstate thermalization occurs in the XX model (hard-core bosons). Namely, eigenstates of the Hamiltonian with similar distributions of conserved quantities have similar expectation values of few-spin observables. Here we show that generalized eigenstate thermalization also occurs in the transverse field Ising model.
We propose to use Ramsey interferometry and single-site addressability, available in synthetic matter such as cold atoms or trapped ions, to measure real-space and time-resolved spin correlation functions. These correlation functions directly probe the excitations of the system, which makes it possible to characterize the underlying many-body states. Moreover, they contain valuable information about phase transitions where they exhibit scale invariance. We also discuss experimental imperfections and show that a spin-echo protocol can be used to cancel slow fluctuations in the magnetic field. We explicitly consider examples of the two-dimensional, antiferromagnetic Heisenberg model and the one-dimensional, long-range transverse field Ising model to illustrate the technique.
Recently a variant on Ramsey interferometry for coupled spin-1/21/21/2 systems was proposed to directly measure the retarded spin-spin Green's function. In conventional experimental situations, the spin system is initially in a nonequilibrium state before the Ramsey interferometry is performed, so we examine the nonequilibrium retarded spin-spin Green's functions within the transverse-field Ising model. We derive the lowest four spectral moments to understand the short-time behavior and we employ a Lehmann-like representation to determine the spectral behavior. We simulate a Ramsey protocol for a nonequilibrium quantum spin system that consists of a coherent superposition of the ground state and diabatically excited higher-energy states via a temporally ramped transverse magnetic field. We then apply the Ramsey spectroscopy protocol to the final Hamiltonian, which has a constant transverse field. The short time allows us to extract the initial transport of many-body correlations, while the long-time behavior relates to the excitation spectra of the Hamiltonian. Compressive sensing is employed in the data analysis to efficiently extract that spectra.
The nonequilibrium dynamics of integrable systems are highly constrained by the conservation of certain charges. There is substantial evidence that after a quantum quench they do not thermalize but their asymptotic steady state can be described by a generalized Gibbs ensemble (GGE) built from the conserved charges. Most of the studies on the GGE so far have focused on models that can be mapped to quadratic systems, while analytic treatment in nonquadratic systems remained elusive. We obtain results on interaction quenches in a nonquadratic continuum system, the one-dimensional (1D) Bose gas described by the integrable Lieb-Liniger model. The direct implementation of the GGE prescription is prohibited by the divergence of the conserved charges, which we conjecture to be endemic to any continuum integrable systems with contact interactions undergoing a sudden quench. We compute local correlators for a noninteracting initial state and arbitrary final interactions as well as two-point functions for quenches to the Tonks-Girardeau regime. We show that in the long time limit integrability leads to significant deviations from the predictions of the grand canonical ensemble, allowing for an experimental verification in cold-atom systems.
The integrable system is constrained strictly by the conservation law during the time evolution, and the prethermal state from the nearly integrable system is also constrained by the conserved parameters (the constants of motion) with the corresponding generalized Gibbs ensemble (GGE), which is indubitably a powerful tool in the prediction of the relaxation dynamics. For stochastic evolution dynamics with considerable noise, the two-point correlation of local operators (like the density of kinks or transverse magnetization correlators), which do not exhibit the thermal features, display the behaviors of nonthermalization and an asymptotic GGE. In fact it is an asymptotic quasi-steady state with an infinite temperature, therefore the required distance to the nonthermal steady state is in an infinite time average. In this paper, we unambiguously investigate the relaxation of a nonequilibrium system in a canonical ensemble for integrable and nonintegrable systems. Temporal behavior of the many-body quantum system and the corresponding linear-coupling between the harmonic oscillators are discussed. The matrix-method in entropy ensemble is utilized to discuss the boundary and the diagonalization algebraically. The approximation results for nonintegrable system under the considerable perturbations are also presented.