Journal Article•

# Irreversible dynamics in quantum many-body systems

TL;DR: In this paper, the authors study the dynamics of observable echoes in quantum many-body systems occurring when the time-evolution is effectively inverted, and they find that small imperfections in the timereversal procedure that alter the state of the system lead to exponentially decaying echoes.

Abstract: The concept of irreversibility is necessary to justify the statistical description of many-body systems. Here, the authors study the dynamics of observable echoes in quantum many-body systems occurring when the time-evolution is effectively inverted. For nonintegrable systems they find that small imperfections in the time-reversal procedure that alter the state of the system lead to exponentially decaying echoes. As the corresponding decay rate is largely independent of the strength of the perturbation, an intrinsic sensitivity to the perturbation is revealed. This implies irreversibility of the dynamics for all practical purposes, similar to the situation in chaotic classical systems.

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TL;DR: In this paper, the authors show that a homogeneous 1D Bose gas with point-like collisional interactions is integrable, and that it is possible to construct a system with many degrees of freedom that does not reach thermal equilibrium even after thousands of collisions.

Abstract: It is a fundamental assumption of statistical mechanics that a closed system with many degrees of freedom ergodically samples all equal energy points in phase space. To understand the limits of this assumption, it is important to find and study systems that are not ergodic, and thus do not reach thermal equilibrium. A few complex systems have been proposed that are expected not to thermalize because their dynamics are integrable. Some nearly integrable systems of many particles have been studied numerically, and shown not to ergodically sample phase space. However, there has been no experimental demonstration of such a system with many degrees of freedom that does not approach thermal equilibrium. Here we report the preparation of out-of-equilibrium arrays of trapped one-dimensional (1D) Bose gases, each containing from 40 to 250 87Rb atoms, which do not noticeably equilibrate even after thousands of collisions. Our results are probably explainable by the well-known fact that a homogeneous 1D Bose gas with point-like collisional interactions is integrable. Until now, however, the time evolution of out-of-equilibrium 1D Bose gases has been a theoretically unsettled issue, as practical factors such as harmonic trapping and imperfectly point-like interactions may compromise integrability. The absence of damping in 1D Bose gases may lead to potential applications in force sensing and atom interferometry.

941 citations

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TL;DR: In this paper, the authors numerically calculate the out-of-time-order correlation functions in the one-dimensional Bose-Hubbard model and derive the Lyapunov exponent in the BH model across the quantum critical regime at finite temperature.

Abstract: In this paper we numerically calculate the out-of-time-order correlation functions in the one-dimensional Bose-Hubbard model. Our study is motivated by the conjecture that a system with Lyapunov exponent saturating the upper bound $2\\pi/\\beta$ will have a holographic dual to a black hole at finite temperature. We further conjecture that for a many-body quantum system with a quantum phase transition, the Lyapunov exponent will have a peak in the quantum critical region where there exists an emergent conformal symmetry and is absent of well-defined quasi-particles. With the help of a relation between the Renyi entropy and the out-of-time-order correlation function, we argue that the out-of-time-order correlation function of the Bose-Hubbard model will also exhibit an exponential behavior at the scrambling time. By fitting the numerical results with an exponential function, we extract the Lyapunov exponents in the one-dimensional Bose-Hubbard model across the quantum critical regime at finite temperature. Our results on the Bose-Hubbard model support the conjecture. We also compute the butterfly velocity and propose how the echo type measurement of this correlator in the cold atom realizations of the Bose-Hubbard model without inverting the Hamiltonian.

59 citations

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TL;DR: In this paper, single-site imaging in a two-dimensional optical lattice filled with interacting rubidium atoms shows that disorder can prevent thermalization in many-body localized systems.

Abstract: Bosons refusing to thermalize in 2D Messy, interacting quantum-mechanical systems are difficult to analyze theoretically. In a single spatial dimension, the calculations are still tractable, and experiments have recently confirmed the prediction that sufficiently strong disorder can disrupt the transport of interacting particles. In two dimensions, however, the theoretical blueprint is missing. Choi et al. used single-site imaging of cold 87Rb atoms in an optical lattice to show that similar localization occurs in two-dimensional (2D) systems. The study highlights the power of quantum simulation to solve problems that are currently inaccessible to classical computing techniques. Science, this issue p. 1547 Single-site imaging in a two-dimensional optical lattice filled with interacting rubidium atoms shows that disorder can prevent thermalization. A fundamental assumption in statistical physics is that generic closed quantum many-body systems thermalize under their own dynamics. Recently, the emergence of many-body localized systems has questioned this concept and challenged our understanding of the connection between statistical physics and quantum mechanics. Here we report on the observation of a many-body localization transition between thermal and localized phases for bosons in a two-dimensional disordered optical lattice. With our single-site–resolved measurements, we track the relaxation dynamics of an initially prepared out-of-equilibrium density pattern and find strong evidence for a diverging length scale when approaching the localization transition. Our experiments represent a demonstration and in-depth characterization of many-body localization in a regime not accessible with state-of-the-art simulations on classical computers.

46 citations

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TL;DR: In this article, the authors show that the tripartite mutual information (TMI) can be negative or positive for both integrable and non-integrable systems, and that TMI is an independent property of the integrability of Hamiltonians.

Abstract: We systematically investigate scrambling (or delocalizing) processes of quantum information encoded in quantum many-body systems by using numerical exact diagonalization. As a measure of scrambling, we adopt the tripartite mutual information (TMI) that becomes negative when quantum information is delocalized. We clarify that scrambling is an independent property of the integrability of Hamiltonians; TMI can be negative or positive for both integrable and nonintegrable systems. This implies that scrambling is a separate concept from conventional quantum chaos characterized by nonintegrability. Specifically, we argue that there are a few exceptional initial states that do not exhibit scrambling, and show that such exceptional initial states have small effective dimensions. Furthermore, we calculate TMI in the Sachdev-Ye-Kitaev (SYK) model, a fermionic toy model of quantum gravity. We find that disorder does not make scrambling slower but makes it smoother in the SYK model, in contrast to many-body localization in spin chains.

25 citations

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TL;DR: In this article , the authors review the prominent role that the out of time ordered correlator (OTOC) plays in quantum many-body systems, with no obvious classical meaning.

Abstract: Quantum Chaos has originally emerged as the field which studies how the properties of classical chaotic systems arise in their quantum counterparts. The growing interest in quantum many-body systems, with no obvious classical meaning has led to consider time-dependent quantities that can help to characterize and redefine Quantum Chaos. This article reviews the prominent role that the out of time ordered correlator (OTOC) plays to achieve such goal.

12 citations

##### References

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TL;DR: This work observes a quantum phase transition in a Bose–Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential, and can induce reversible changes between the two ground states of the system.

Abstract: For a system at a temperature of absolute zero, all thermal fluctuations are frozen out, while quantum fluctuations prevail. These microscopic quantum fluctuations can induce a macroscopic phase transition in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value. Here we observe such a quantum phase transition in a Bose-Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential. As the potential depth of the lattice is increased, a transition is observed from a superfluid to a Mott insulator phase. In the superfluid phase, each atom is spread out over the entire lattice, with long-range phase coherence. But in the insulating phase, exact numbers of atoms are localized at individual lattice sites, with no phase coherence across the lattice; this phase is characterized by a gap in the excitation spectrum. We can induce reversible changes between the two ground states of the system.

4,467 citations

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TL;DR: It is shown that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey Berry's conjecture, and argued that these results constitute a sound foundation for quantum statistical mechanics.

Abstract: We show that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey Berry's conjecture. Berry's conjecture is expected to hold only if the corresponding classical system is chaotic, and essentially states that the energy eigenfunctions behave as if they were Gaussian random variables. We review the existing evidence, and show that previously neglected effects substantially strengthen the case for Berry's conjecture. We study a rarefied hard-sphere gas as an explicit example of a many-body system which is known to be classically chaotic, and show that an energy eigenstate which obeys Berry's conjecture predicts a Maxwell-Boltzmann, Bose-Einstein, or Fermi-Dirac distribution for the momentum of each constituent particle, depending on whether the wave functions are taken to be nonsymmetric, completely symmetric, or completely antisymmetric functions of the positions of the particles. We call this phenomenon eigenstate thermalization. We show that a generic initial state will approach thermal equilibrium at least as fast as O(\ensuremath{\Elzxh}/\ensuremath{\Delta})${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$, where \ensuremath{\Delta} is the uncertainty in the total energy of the gas. This result holds for an individual initial state; in contrast to the classical theory, no averaging over an ensemble of initial states is needed. We argue that these results constitute a sound foundation for quantum statistical mechanics.

2,649 citations

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TL;DR: It is demonstrated that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription, and it is shown that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch and Srednicki.

Abstract: It is demonstrated that an isolated generic quantum many-body system does relax to a state well described by the standard statistical mechanical prescription The thermalization happens at the level of individual eigenstates, allowing the computation of thermal averages from knowledge of any eigenstate in the microcanonical energy window An understanding of the temporal evolution of isolated many-body quantum systems has long been elusive Recently, meaningful experimental studies1,2 of the problem have become possible, stimulating theoretical interest3,4,5,6,7 In generic isolated systems, non-equilibrium dynamics is expected8,9 to result in thermalization: a relaxation to states in which the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable using statistical mechanics However, it is not obvious what feature of many-body quantum mechanics makes quantum thermalization possible in a sense analogous to that in which dynamical chaos makes classical thermalization possible10 For example, dynamical chaos itself cannot occur in an isolated quantum system, in which the time evolution is linear and the spectrum is discrete11 Some recent studies4,5 even suggest that statistical mechanics may give incorrect predictions for the outcomes of relaxation in such systems Here we demonstrate that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription Moreover, we show that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch12 and Srednicki13 A striking consequence of this eigenstate-thermalization scenario, confirmed for our system, is that knowledge of a single many-body eigenstate is sufficient to compute thermal averages—any eigenstate in the microcanonical energy window will do, because they all give the same result

2,598 citations

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TL;DR: A closed quantum-mechanical system with a large number of degrees of freedom does not necessarily give time averages in agreement with the microcanonical distribution, so by adding a finite but very small perturbation in the form of a random matrix, the results of quantum statistical mechanics are recovered.

Abstract: A closed quantum-mechanical system with a large number of degrees of freedom does not necessarily give time averages in agreement with the microcanonical distribution. For systems where the different degrees of freedom are uncoupled, situations are discussed that show a violation of the usual statistical-mechanical rules. By adding a finite but very small perturbation in the form of a random matrix, it is shown that the results of quantum statistical mechanics are recovered. Expectation values in energy eigenstates for this perturbed system are also discussed, and deviations from the microcanonical result are shown to become exponentially small in the number of degrees of freedom.

2,390 citations

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TL;DR: This work demonstrates a method for creating controlled many-body quantum matter that combines deterministically prepared, reconfigurable arrays of individually trapped cold atoms with strong, coherent interactions enabled by excitation to Rydberg states, and realizes a programmable Ising-type quantum spin model with tunable interactions and system sizes of up to 51 qubits.

Abstract: Controllable, coherent many-body systems can provide insights into the fundamental properties of quantum matter, enable the realization of new quantum phases and could ultimately lead to computational systems that outperform existing computers based on classical approaches. Here we demonstrate a method for creating controlled many-body quantum matter that combines deterministically prepared, reconfigurable arrays of individually trapped cold atoms with strong, coherent interactions enabled by excitation to Rydberg states. We realize a programmable Ising-type quantum spin model with tunable interactions and system sizes of up to 51 qubits. Within this model, we observe phase transitions into spatially ordered states that break various discrete symmetries, verify the high-fidelity preparation of these states and investigate the dynamics across the phase transition in large arrays of atoms. In particular, we observe robust many-body dynamics corresponding to persistent oscillations of the order after a rapid quantum quench that results from a sudden transition across the phase boundary. Our method provides a way of exploring many-body phenomena on a programmable quantum simulator and could enable realizations of new quantum algorithms.

2,026 citations