Transportless equilibration in isolated many-body quantum systems

A general analytical theory of temporal relaxation processes in isolated quantum systems with many degrees of freedom is elaborated, which unifies and substantially amends several previous approximations. Specifically, the Fourier transform of the initial energy distribution is found to play a key role, which is furthermore equivalent to the so-called survival probability in case of a pure initial state. The main prerequisite is the absence of any notable transport currents, caused for instance by some initially unbalanced local densities of particles, energy, and so on. In particular, such a transportless relaxation scenario naturally arises when both the system Hamiltonian and the initial non-equilibrium state do not exhibit any spatial inhomogeneities on macroscopic scales. A further requirement is that the relaxation must not be notably influenced by any approximate (but not exact) constant of motion or metastable state. The theoretical predictions are compared with various experimental and numerical results from the literature.A general analytical theory of temporal relaxation processes in isolated quantum systems with many degrees of freedom is elaborated, which unifies and substantially amends several previous approximations. Specifically, the Fourier transform of the initial energy distribution is found to play a key role, which is furthermore equivalent to the so-called survival probability in case of a pure initial state. The main prerequisite is the absence of any notable transport currents, caused for instance by some initially unbalanced local densities of particles, energy, and so on. In particular, such a transportless relaxation scenario naturally arises when both the system Hamiltonian and the initial non-equilibrium state do not exhibit any spatial inhomogeneities on macroscopic scales. A further requirement is that the relaxation must not be notably influenced by any approximate (but not exact) constant of motion or metastable state. The theoretical predictions are compared with various experimental and numerical results from the literature.

arXiv:1905.03114New Journal of PhysicsApr 17, 201917 citations

Typical Relaxation of Isolated Many-Body Systems Which Do Not Thermalize

Ben N. Balz, Peter Reimann

We consider isolated many-body quantum systems which do not thermalize; i.e., expectation values approach an (approximately) steady longtime limit which disagrees with the microcanonical prediction of equilibrium statistical mechanics. A general analytical theory is worked out for the typical temporal relaxation behavior in such cases. The main prerequisites are initial conditions which appreciably populate many energy levels and do not give rise to significant spatial inhomogeneities on macroscopic scales. The theory explains very well the experimental and numerical findings in a trapped-ion quantum simulator exhibiting many-body localization, in ultracold atomic gases, and in integrable hard-core boson and XXZ models.We consider isolated many-body quantum systems which do not thermalize; i.e., expectation values approach an (approximately) steady longtime limit which disagrees with the microcanonical prediction of equilibrium statistical mechanics. A general analytical theory is worked out for the typical temporal relaxation behavior in such cases. The main prerequisites are initial conditions which appreciably populate many energy levels and do not give rise to significant spatial inhomogeneities on macroscopic scales. The theory explains very well the experimental and numerical findings in a trapped-ion quantum simulator exhibiting many-body localization, in ultracold atomic gases, and in integrable hard-core boson and XXZ models.

arXiv:1705.07014Physical Review LettersMay 12, 201723 citations

Temporal relaxation of gapped many-body quantum systems

Peter Reimann, Ben N. Balz, Jonas Richter +1 more

Typicality of the orthogonal dynamics (TOD) is established as a generic feature of the temporal relaxation processes in isolated many-body quantum systems. The basic idea in the simplest case is that the transient nonequilibrium behavior is mainly governed by the component of the time-evolved system state parallel to the initial state, while the orthogonal component appears as equilibrated right from the beginning. The main emphasis is laid on the largely unexplored and particularly challenging case that one energy level exhibits a much larger population than all the others. Important examples are gapped many-body systems at low energies, for instance, due to a quantum quench. A general analytical prediction is derived and is found to compare very well with various numerically exact results.Typicality of the orthogonal dynamics (TOD) is established as a generic feature of the temporal relaxation processes in isolated many-body quantum systems. The basic idea in the simplest case is that the transient nonequilibrium behavior is mainly governed by the component of the time-evolved system state parallel to the initial state, while the orthogonal component appears as equilibrated right from the beginning. The main emphasis is laid on the largely unexplored and particularly challenging case that one energy level exhibits a much larger population than all the others. Important examples are gapped many-body systems at low energies, for instance, due to a quantum quench. A general analytical prediction is derived and is found to compare very well with various numerically exact results.

arXiv:2002.08137Physical review. B./Physical review. BMar 4, 20208 citations

Mechanism of macroscopic equilibration of isolated quantum systems

Anatoly Y. Dymarsky

We discuss the approach toward equilibrium of an isolated quantum system. For a wide class of systems, in particular, for chaotic systems, we argue that the expectation value of a local operator averaged over a time interval of length $T$ is bounded by the so-called deviation function, which characterizes maximal deviation from the equilibrium of all states with a given value of energy fluctuations. This result applies to any initial state with a well-defined effective temperature. We provide numerical evidence that the bound is approximately saturated by the initial configurations with spatial inhomogeneities at a macroscopic level. In this way the deviation function establishes an explicit connection between the macroscopically observed timescales associated with the transport and the properties of microscopic matrix elements. The form of the deviation function indicates that among the ``slowest'' states which saturate the bound there are also those with arbitrarily long equilibration times.We discuss the approach toward equilibrium of an isolated quantum system. For a wide class of systems, in particular, for chaotic systems, we argue that the expectation value of a local operator averaged over a time interval of length $T$ is bounded by the so-called deviation function, which characterizes maximal deviation from the equilibrium of all states with a given value of energy fluctuations. This result applies to any initial state with a well-defined effective temperature. We provide numerical evidence that the bound is approximately saturated by the initial configurations with spatial inhomogeneities at a macroscopic level. In this way the deviation function establishes an explicit connection between the macroscopically observed timescales associated with the transport and the properties of microscopic matrix elements. The form of the deviation function indicates that among the ``slowest'' states which saturate the bound there are also those with arbitrarily long equilibration times.

arXiv:1806.04187Physical review. B./Physical review. BJun 6, 201920 citations

Typical fast thermalization processes in closed many-body systems

Peter Reimann

The lack of knowledge about the detailed many-particle motion on the microscopic scale is a key issue in any theoretical description of a macroscopic experiment. For systems at or close to thermal equilibrium, statistical mechanics provides a very successful general framework to cope with this problem. However, far from equilibrium, only very few quantitative and comparably universal results are known. Here a quantum mechanical prediction of this type is derived and verified against various experimental and numerical data from the literature. It quantitatively describes the entire temporal relaxation towards thermal equilibrium for a large class (in a mathematically precisely defined sense) of closed many-body systems, whose initial state may be arbitrarily far from equilibrium.The lack of knowledge about the detailed many-particle motion on the microscopic scale is a key issue in any theoretical description of a macroscopic experiment. For systems at or close to thermal equilibrium, statistical mechanics provides a very successful general framework to cope with this problem. However, far from equilibrium, only very few quantitative and comparably universal results are known. Here a quantum mechanical prediction of this type is derived and verified against various experimental and numerical data from the literature. It quantitatively describes the entire temporal relaxation towards thermal equilibrium for a large class (in a mathematically precisely defined sense) of closed many-body systems, whose initial state may be arbitrarily far from equilibrium.

arXiv:1603.00669Nature CommunicationsMar 1, 2016135 citations

General features of the relaxation dynamics of interacting quantum systems

Eduardo Jonathan Torres-Herrera, Manan Vyas, Léa Ferreira dos Santos

We study numerically and analytically isolated interacting quantum systems that are taken out of equilibrium instantaneously (quenched). The probability of finding the initial state in time, the so-called fidelity, decays fastest for systems described by full random matrices, where simultaneous many-body interactions are implied. In the realm of realistic systems with two-body interactions, the dynamics is slower and depends on the interplay between the initial state and the Hamiltonian characterizing the system. The fastest fidelity decay in this case is Gaussian and can persist until saturation. A simple general picture, in which the fidelity plays a central role, is also achieved for the short-time dynamics of few-body observables. It holds for initial states that are eigenstates of the observables. We also discuss the need to reassess analytical expressions that were previously proposed to describe the evolution of the Shannon entropy. Our analyses are mainly developed for initial states that can be prepared in experiments with cold atoms in optical lattices.We study numerically and analytically isolated interacting quantum systems that are taken out of equilibrium instantaneously (quenched). The probability of finding the initial state in time, the so-called fidelity, decays fastest for systems described by full random matrices, where simultaneous many-body interactions are implied. In the realm of realistic systems with two-body interactions, the dynamics is slower and depends on the interplay between the initial state and the Hamiltonian characterizing the system. The fastest fidelity decay in this case is Gaussian and can persist until saturation. A simple general picture, in which the fidelity plays a central role, is also achieved for the short-time dynamics of few-body observables. It holds for initial states that are eigenstates of the observables. We also discuss the need to reassess analytical expressions that were previously proposed to describe the evolution of the Shannon entropy. Our analyses are mainly developed for initial states that can be prepared in experiments with cold atoms in optical lattices.

arXiv:1402.3299New Journal of PhysicsJun 5, 201478 citations

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is bounded by the so-called deviation function, which characterizes maximal deviation from the equilibrium of all states with a given value of energy fluctuations. This result applies to any initial state with a well-defined effective temperature. We provide numerical evidence that the bound is approximately saturated by the initial configurations with spatial inhomogeneities at a macroscopic level. In this way the deviation function establishes an explicit connection between the macroscopically observed timescales associated with the transport and the properties of microscopic matrix elements. The form of the deviation function indicates that among the ``slowest'' states which saturate the bound there are also those with arbitrarily long equilibration times.