Typical fast thermalization processes in closed many-body systems

Heat Bath in a Quantum Circuit

We discuss the concept and realization of a heat bath in solid state quantum systems. We demonstrate that, unlike a true resistor, a finite one-dimensional Josephson junction array or analogously a transmission line with non-vanishing frequency spacing, commonly considered as a reservoir of a quantum circuit, does not strictly qualify as a Caldeira-Leggett type dissipative environment. We then consider a set of quantum two-level systems as a bath, which can be realized as a collection of qubits. We show that only a dense and wide distribution of energies of the two-level systems can secure long Poincare recurrence times characteristic of a proper heat bath. An alternative for this bath is a collection of harmonic oscillators, for instance, in the form of superconducting resonators.

Exact Universal Bounds on Quantum Dynamics and Fast Scrambling

Quantum speed limits such as the Mandelstam-Tamm or Margolus-Levitin bounds offer a quantitative formulation of the energy-time uncertainty principle that constrains dynamics over short times. We show that the spectral form factor, a central quantity in quantum chaos, sets a universal state-independent bound on the quantum dynamics of a complete set of initial states over arbitrarily long times, which is tighter than the corresponding state-independent bounds set by known speed limits. This bound further generalizes naturally to the real-time dynamics of time-dependent or dissipative systems where no energy spectrum exists. We use this result to constrain the scrambling of information in interacting many-body systems. For Hamiltonian systems, we show that the fundamental question of the fastest possible scrambling time-without any restrictions on the structure of interactions-maps to a purely mathematical property of the density of states involving the non-negativity of Fourier transforms. We illustrate these bounds in the Sachdev-Ye-Kitaev model, where we show that despite its "maximally chaotic" nature, the sustained scrambling of sufficiently large fermion subsystems via entanglement generation requires an exponentially long time in the subsystem size.

Asymptotic Quantum Many-Body Scars

We consider a quantum lattice spin model featuring exact quasiparticle towers of eigenstates with low entanglement at finite size, known as quantum many-body scars (QMBS). We show that the states in the neighboring part of the energy spectrum can be superposed to construct entire families of low-entanglement states whose energy variance decreases asymptotically to zero as the lattice size is increased. As a consequence, they have a relaxation time that diverges in the thermodynamic limit, and therefore exhibit the typical behavior of exact QMBS, although they are not exact eigenstates of the Hamiltonian for any finite size. We refer to such states as asymptotic QMBS. These states are orthogonal to any exact QMBS at any finite size, and their existence shows that the presence of an exact QMBS leaves important signatures of nonthermalness in the rest of the spectrum; therefore, QMBS-like phenomena can hide in what is typically considered the thermal part of the spectrum. We support our study using numerical simulations in the spin-1 XY model, a paradigmatic model for QMBS, and we conclude by presenting a weak perturbation of the model that destroys the exact QMBS while keeping the asymptotic QMBS.

Eigenstate Thermalization Hypothesis and Free Probability

Quantum thermalization is well understood via the eigenstate thermalization hypothesis (ETH). The general form of ETH, describing all the relevant correlations of matrix elements, may be derived on the basis of a "typicality" argument of invariance with respect to local rotations involving nearby energy levels. In this Letter, we uncover the close relation between this perspective on ETH and free probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to reduce in a straightforward way higher-order correlation functions to a decomposition given by minimal blocks, identified as free cumulants, for which we give an explicit formula. This perspective naturally incorporates the consistency property that local functions of ETH operators also satisfy ETH. The present results uncover a direct connection between the eigenstate thermalization hypothesis and the structure of free probability, widening considerably the latter's scope and highlighting its relevance to quantum thermalization.

Eigenstate fluctuation theorem in the short- and long-time regimes

The canonical ensemble plays a crucial role in statistical mechanics in and out of equilibrium. For example, the standard derivation of the fluctuation theorem relies on the assumption that the initial state of the heat bath is the canonical ensemble. On the other hand, the recent progress in the foundation of statistical mechanics has revealed that a thermal equilibrium state is not necessarily described by the canonical ensemble but can be a quantum pure state or even a single energy eigenstate, as formulated by the eigenstate thermalization hypothesis (ETH). Then a question raised is how these two pictures, the canonical ensemble and a single energy eigenstate as a thermal equilibrium state, are compatible in the fluctuation theorem. In this paper, we theoretically and numerically show that the fluctuation theorem holds in both of the long- and short-time regimes, even when the initial state of the bath is a single energy eigenstate of a many-body system. Our proof of the fluctuation theorem in the long-time regime is based on the ETH, while it was previously shown in the short-time regime on the basis of the Lieb-Robinson bound and the ETH [Phys. Rev. Lett. 119, 100601 (2017)0031-900710.1103/PhysRevLett.119.100601]. The proofs for these time regimes are theoretically independent and complementary, implying the fluctuation theorem in the entire time domain. We also perform a systematic numerical simulation of hard-core bosons by exact diagonalization and verify the fluctuation theorem in both of the time regimes by focusing on the finite-size scaling. Our results contribute to the understanding of the mechanism that the fluctuation theorem emerges from unitary dynamics of quantum many-body systems and can be tested by experiments with, e.g., ultracold atoms.

Asymmetric temperature equilibration with heat flow from cold to hot in a quantum thermodynamic system

A model computational quantum thermodynamic network is constructed with two variable temperature baths coupled by a linker system, with an asymmetry in the coupling of the linker to the two baths. It is found in computational simulations that the baths come to "thermal equilibrium" at different bath energies and temperatures. In a sense, heat is observed to flow from cold to hot. A description is given in which a recently defined quantum entropy S_{univ}^{Q} for a pure state "universe" continues to increase after passing through the classical equilibrium point of equal temperatures, reaching a maximum at the asymmetric equilibrium. Thus, a second law account ΔS_{univ}^{Q}≥0 holds for the asymmetric quantum process. In contrast, a von Neumann entropy description fails to uphold the entropy law, with a maximum near when the two temperatures are equal, then a decrease ΔS^{vN}<0 on the way to the asymmetric equilibrium.

Stiffness of probability distributions of work and Jarzynski relation for initial microcanonical and energy eigenstates

We consider closed quantum systems which are driven such that only negligible heating occurs. If driving only affects small parts of the system, it may nonetheless be strong. Our analysis aims at clarifying under which conditions the Jarzynski relation (JR) holds in such setups, if the initial states are microcanonical or even energy eigenstates. We find that the validity of the JR for the microcanonical initial state hinges on an exponential density of states and on stiffness. The latter indicates an independence of the probability density functions (PDFs) of work of the energy of the respective microcanonical initial state. The validity of the JR for initial energy eigenstates is found to additionally require smoothness. The latter indicates an independence of the work PDFs of the specific energy eigenstates within a microcanonical energy shell. As the validity of the JR for pure initial energy eigenstates has no analog in classical systems, we consider it a genuine quantum phenomenon.

Test of the Eigenstate Thermalization Hypothesis Based on Local Random Matrix Theory

We verify that the eigenstate thermalization hypothesis (ETH) holds universally for locally interacting quantum many-body systems. Introducing random matrix ensembles with interactions, we numerically obtain a distribution of maximum fluctuations of eigenstate expectation values for different realizations of interactions. This distribution, which cannot be obtained from the conventional random matrix theory involving nonlocal correlations, demonstrates that an overwhelming majority of pairs of local Hamiltonians and observables satisfy the ETH with exponentially small fluctuations. The ergodicity of our random matrix ensembles breaks down because of locality.

Taking snapshots of a quantum thermalization process: Emergent classicality in quantum jump trajectories

We investigate theoretically the emergence of classical statistical physics in a finite quantum system that is either totally isolated or otherwise subjected to a quantum measurement process. We show via a random matrix theory approach to nonintegrable quantum systems that the set of outcomes of the measurement of a macroscopic observable evolve in time like stochastic variables, whose variance satisfies the celebrated Einstein relation for Brownian diffusion. Our results show how to extend the framework of eigenstate thermalization to the prediction of properties of quantum measurements on an otherwise closed quantum system. We show numerically the validity of the random matrix approach in quantum chain models.

Universal fluctuations around typicality for quantum ergodic systems

For a quantum system in a macroscopically large volume V, prepared in a pure state and subject to maximally noisy or ergodic unitary dynamics, the reduced density matrix of any sub-system v≪V is almost surely totally mixed. We show that the fluctuations around this limiting value, evaluated according to the invariant measure of these unitary flows, are captured by the Gaussian unitary ensemble (GUE) of random matrix theory. An extension of this statement, applicable when the unitary transformations conserve the energy but are maximally noisy or ergodic on any energy shell, allows to decipher the fluctuations around canonical typicality. According to typicality, if the large system is prepared in a generic pure state in a given energy shell, the reduced density matrix of the sub-system is almost surely the canonical Gibbs state of that sub-system. We show that the fluctuations around the Gibbs state are encoded in a deformation of the GUE whose covariance is specified by the Gibbs state. Contact with the eigenstate thermalization hypothesis is discussed.

Simulating quantum thermodynamics of a finite system and bath with variable temperature

We construct a finite bath with variable temperature for quantum thermodynamic simulations in which heat flows between a system S and the bath environment E in time evolution of an initial SE pure state. The bath consists of harmonic oscillators that are not necessarily identical. Baths of various numbers of oscillators are considered; a bath with five oscillators is used in the simulations. The bath has a temperaturelike level distribution. This leads to definition of a system-environment microcanonical temperature T_{SE}(t) which varies with time. The quantum state evolves toward an equilibrium state which is thermal-like, but there is significant deviation from the ordinary energy-temperature relation that holds for an infinite quantum bath, e.g., an infinite system of identical oscillators. There are also deviations from the Einstein quantum heat capacity. The temperature of the finite bath is systematically greater for a given energy than the infinite bath temperature, and asymptotically approaches the latter as the number of oscillators increases. It is suggested that realizations of these finite-size effects may be attained in computational and experimental dynamics of small molecules.

Entanglement-Ergodic Quantum Systems Equilibrate Exponentially Well

One of the outstanding problems in nonequilibrium physics is to precisely understand when and how physically relevant observables in many-body systems equilibrate under unitary time evolution. General equilibration results show that equilibration is generic provided that the initial state has overlap with sufficiently many energy levels. But results not referring to typicality which show that natural initial states actually fulfill this condition are lacking. In this work, we present stringent results for equilibration for systems in which Rényi entanglement entropies in energy eigenstates with finite energy density are extensive for at least some, not necessarily connected, subsystems. Our results reverse the logic of common arguments, in that we derive equilibration from a weak condition akin to the eigenstate thermalization hypothesis, which is usually attributed to thermalization in systems that are assumed to equilibrate in the first place. We put the findings into the context of studies of many-body localization and many-body scars.

Relaxation to Gaussian generalized Gibbs ensembles in quadratic bosonic systems in the thermodynamic limit

Integrable quantum many-body systems are considered to equilibrate to generalized Gibbs ensembles (GGEs) characterized by the expectation values of integrals of motion. We study the dynamics of exactly solvable quadratic bosonic systems in the thermodynamic limit, and show a general mechanism for the relaxation to GGEs, in terms of the diagonal singularity. We show analytically and explicitly that a free bosonic system relaxes from a general (not necessarily Gaussian) initial state under certain physical conditions to a Gaussian GGE. We also show the relaxation to a Gaussian GGE in an exactly solvable coupled system, a harmonic oscillator linearly interacting with bosonic reservoirs.

Dynamical signatures of quantum chaos and relaxation time scales in a spin-boson system

Quantum systems whose classical counterparts are chaotic typically have highly correlated eigenvalues and level statistics that coincide with those from ensembles of full random matrices. A dynamical manifestation of these correlations comes in the form of the so-called correlation hole, which is a dip below the saturation point of the survival probability's time evolution. In this work, we study the correlation hole in the spin-boson (Dicke) model, which presents a chaotic regime and can be realized in experiments with ultracold atoms and ion traps. We derive an analytical expression that describes the entire evolution of the survival probability and allows us to determine the time scales of its relaxation to equilibrium. This expression shows remarkable agreement with our numerical results. While the initial decay and the time to reach the minimum of the correlation hole depend on the initial state, the dynamics beyond the hole up to equilibration is universal. We find that the relaxation time of the survival probability for the Dicke model increases linearly with system size.

Quantum chaotic fluctuation-dissipation theorem: Effective Brownian motion in closed quantum systems

We analytically describe the decay to equilibrium of generic observables of a nonintegrable system after a perturbation in the form of a random matrix. We further obtain an analytic form for the time-averaged fluctuations of an observable in terms of the rate of decay to equilibrium. Our result shows the emergence of a fluctuation-dissipation theorem corresponding to a classical Brownian process, specifically, the Ornstein-Uhlenbeck process. Our predictions can be tested in quantum simulation experiments, thus helping to bridge the gap between theoretical and experimental research in quantum thermalization. We test our analytic results by exact numerical experiments in a spin chain. We argue that our fluctuation-dissipation relation can be used to measure the density of states involved in the nonequilibrium dynamics of an isolated quantum system.

Classical Lieb-Robinson Bound for Estimating Equilibration Timescales of Isolated Quantum Systems

We study equilibration of an isolated quantum system by mapping it onto a network of classical oscillators in Hilbert space. By choosing a suitable basis for this mapping, the degree of locality of the quantum system reflects in the sparseness of the network. We derive a Lieb-Robinson bound on the speed of propagation across the classical network, which allows us to estimate the timescale at which the quantum system equilibrates. The bound contains a parameter that quantifies the degree of locality of the Hamiltonian and the observable. Locality was disregarded in earlier studies of equilibration times, and it is believed to be a key ingredient for making contact with the majority of physically realistic models. The more local the Hamiltonian and observables, the longer the equilibration timescale predicted by the bound.

Impact of eigenstate thermalization on the route to equilibrium

The eigenstate thermalization hypothesis (ETH) and the theory of linear response (LRT) are celebrated cornerstones of our understanding of the physics of many-body quantum systems out of equilibrium. While the ETH provides a generic mechanism of thermalization for states arbitrarily far from equilibrium, LRT extends the successful concepts of statistical mechanics to situations close to equilibrium. In our work, we connect these cornerstones to shed light on the route to equilibrium for a class of properly prepared states. We unveil that, if the off-diagonal part of the ETH applies, then the relaxation process can become independent of whether or not a state is close to equilibrium. Moreover, in this case, the dynamics is generated by a single correlation function, i.e., the relaxation function in the context of LRT. Our analytical arguments are illustrated by numerical results for idealized models of random-matrix type and more realistic models of interacting spins on a lattice. Remarkably, our arguments also apply to integrable quantum systems where the diagonal part of the ETH may break down.

Random-matrix behavior of quantum nonintegrable many-body systems with Dyson's three symmetries

We propose a one-dimensional nonintegrable spin model with local interactions that covers Dyson's three symmetry classes (classes A, AI, and AII) depending on the values of parameters. We show that the nearest-neighbor spacing distribution in each of these classes agrees with that of random matrices having the same symmetry. By investigating the ratios between the standard deviations of diagonal and off-diagonal matrix elements, we numerically find that they are universal, depending only on symmetries of the Hamiltonian and an observable, as predicted by random matrix theory. These universal ratios are evaluated from long-time dynamics of small isolated quantum systems.

Eigenstate thermalization hypothesis and out of time order correlators

The eigenstate thermalization hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems. Another signal of chaos is a positive Lyapunov exponent, defined on the basis of Loschmidt echo or out of time order correlators. For this exponent to be positive, correlations between matrix elements unrelated by symmetry, usually neglected, have to exist. The same is true for the peak of the dynamic heterogeneity length χ_{4}, relevant for systems with slow dynamics. These correlations, as well as those between elements of different operators, are encompassed in a generalized form of ETH.

Typicality of Prethermalization

Prethermalization refers to the remarkable relaxation behavior which an integrable many-body system in the presence of a weak integrability-breaking perturbation may exhibit: After initial transients have died out, it stays for a long time close to some nonthermal steady state, but on even much larger time scales, it ultimately switches over to the proper thermal equilibrium behavior. By extending Deutsch's conceptual framework from Phys. Rev. A 43, 2046 (1991)PLRAAN1050-294710.1103/PhysRevA.43.2046, we analytically predict that prethermalization is a typical feature for a very general class of such weakly perturbed systems.

Quantum Microcanonical Entropy, Boltzmann's Equation, and the Second Law

A recent proposal for a quantum entropy S _univ ^Q for a pure state of a system-environment "universe" is developed to encompass a much more realistic temperature bath. Microcanonical entropy is formulated in the context of the idea of a quantum microcanonical shell. The fundamental relation that holds for the classical microcanonical ensemble - TΔ S _univ = Δ F _sys is tested for the quantum entropy Δ S _univ ^Q in numerical simulations. It is found that there is "excess entropy production" Δ S ^x due to quantum time-energy uncertainty and spreading of states in the zero-order basis. The excess entropy production is found numerically to become small as the magnitude of the system-environment coupling nears zero, as one would hope for in the limit of the classical microcanonical ensemble. The quantum microcanonical ensemble and the new "universe entropy" thereby appear as well-founded concepts poised to serve as a point of departure for time-dependent processes in which excess entropy production is physically significant.

Relation between far-from-equilibrium dynamics and equilibrium correlation functions for binary operators

Linear response theory (LRT) is one of the main approaches to the dynamics of quantum many-body systems. However, this approach has limitations and requires, e.g., that the initial state is (i) mixed and (ii) close to equilibrium. In this paper, we discuss these limitations and study the nonequilibrium dynamics for a certain class of properly prepared initial states. Specifically, we consider thermal states of the quantum system in the presence of an additional static force which, however, become nonequilibrium states when this static force is eventually removed. While for weak forces the relaxation dynamics is well captured by LRT, much less is known in the case of strong forces, i.e., initial states far away from equilibrium. Summarizing our main results, we unveil that, for high temperatures, the nonequilibrium dynamics of so-called binary operators is always generated by an equilibrium correlation function. In particular, this statement holds true for states in the far-from-equilibrium limit, i.e., outside the linear response regime. In addition, we confirm our analytical results by numerically studying the dynamics of local fermionic occupation numbers and local energy densities in the spin-1/2 Heisenberg chain. Remarkably, these simulations also provide evidence that our results qualitatively apply in a more general setting, e.g., in the anisotropic XXZ model where the local energy is a non-binary operator, as well as for a wider range of temperature. Furthermore, exploiting the concept of quantum typicality, all of our findings are not restricted to mixed states, but are valid for pure initial states as well.

Thermalization and Heating Dynamics in Open Generic Many-Body Systems

The last decade has witnessed remarkable progress in our understanding of thermalization in isolated quantum systems. Combining the eigenstate thermalization hypothesis with quantum measurement theory, we extend the framework of quantum thermalization to open many-body systems. A generic many-body system subject to continuous observation is shown to thermalize at a single trajectory level. We show that the nonunitary nature of quantum measurement causes several unique thermalization mechanisms that are unseen in isolated systems. We present numerical evidence for our findings by applying our theory to specific models that can be experimentally realized in atom-cavity systems and with quantum gas microscopy. Our theory provides a general method to determine an effective temperature of quantum many-body systems subject to the Lindblad master equation and thus should be applicable to noisy dynamics or dissipative systems coupled to nonthermal Markovian environments as well as continuously monitored systems. Our work provides yet another insight into why thermodynamics emerges so universally.

New Equilibrium Ensembles for Isolated Quantum Systems

The unitary dynamics of isolated quantum systems does not allow a pure state to thermalize. Because of that, if an isolated quantum system equilibrates, it will do so to the predictions of the so-called "diagonal ensemble" ρ DE . Building on the intuition provided by Jaynes' maximum entropy principle, in this paper we present a novel technique to generate progressively better approximations to ρ DE . As an example, we write down a hierarchical set of ensembles which can be used to describe the equilibrium physics of small isolated quantum systems, going beyond the "thermal ansatz" of Gibbs ensembles.

Stiffness of probability distributions of work and Jarzynski relation for non-Gibbsian initial states

We consider closed quantum systems (into which baths may be integrated) that are driven, i.e., subject to time-dependent Hamiltonians. Our point of departure is the assumption that if systems start in non-Gibbsian states at some initial energies, the resulting probability distributions of work may be largely independent of the specific initial energies. It is demonstrated that this assumption has some far-reaching consequences, e.g., it implies the validity of the Jarzynski relation for a large class of non-Gibbsian initial states. By performing numerical analysis on integrable and nonintegrable spin systems, we find the above assumption fulfilled for all examples considered. Through an analysis based on Fermi's golden rule, we partially relate these findings to the applicability of the eigenstate thermalization ansatz to the respective driving operators.

Eigenstate Thermalization for Degenerate Observables

Under unitary time evolution, expectation values of physically reasonable observables often evolve towards the predictions of equilibrium statistical mechanics. The eigenstate thermalization hypothesis (ETH) states that this is also true already for individual energy eigenstates. Here we aim at elucidating the emergence of the ETH for observables that can realistically be measured due to their high degeneracy, such as local, extensive, or macroscopic observables. We bisect this problem into two parts, a condition on the relative overlaps and one on the relative phases between the eigenbases of the observable and Hamiltonian. We show that the relative overlaps are unbiased for highly degenerate observables and demonstrate that unless relative phases conspire to cumulative effects, this makes such observables verify the ETH. Through this we elucidate potential pathways towards proofs of thermalization.

Relaxation, thermalization, and Markovian dynamics of two spins coupled to a spin bath

It is shown that by fitting a Markovian quantum master equation to the numerical solution of the time-dependent Schrödinger equation of a system of two spin-1/2 particles interacting with a bath of up to 34 spin-1/2 particles, the former can describe the dynamics of the two-spin system rather well. The fitting procedure that yields this Markovian quantum master equation accounts for all non-Markovian effects in as much the general structure of this equation allows and yields a description that is incompatible with the Lindblad equation.

Typical equilibrium state of an embedded quantum system

We consider an arbitrary quantum system coupled nonperturbatively to a large arbitrary and fully quantum environment. In the work by Ithier and Benaych-Georges [Phys. Rev. A 96, 012108 (2017)2469-992610.1103/PhysRevA.96.012108] the typicality of the dynamics of such an embedded quantum system was established for several classes of random interactions. In other words, the time evolution of its quantum state does not depend on the microscopic details of the interaction. Focusing on the long-time regime, we use this property to calculate analytically a partition function characterizing the stationary state and involving the overlaps between eigenvectors of a bare and a dressed Hamiltonian. This partition function provides a thermodynamical ensemble which includes the microcanonical and canonical ensembles as particular cases. We check our predictions with numerical simulations.

Internal temperature of quantum chaotic systems at the nanoscale

The extent to which a temperature can be appropriately assigned to a small quantum system, as an internal property but not as a property of any large environment, is still an open problem. In this paper, a method is proposed for solving this problem, by which a studied small system is coupled to a two-level system as a probe, the latter of which can be measured by measurement devices. A main difficulty in the determination of possible temperature of the studied system comes from the back-action of the probe-system coupling to the system. For small quantum chaotic systems, we show that a temperature can be determined, the value of which is sensitive to neither the form, location, and strength of the probe-system coupling, nor the Hamiltonian and initial state of the probe. The temperature thus obtained turns out to have the form of Boltzmann temperature.

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

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.

Thermalization and Return to Equilibrium on Finite Quantum Lattice Systems

Thermal states are the bedrock of statistical physics. Nevertheless, when and how they actually arise in closed quantum systems is not fully understood. We consider this question for systems with local Hamiltonians on finite quantum lattices. In a first step, we show that states with exponentially decaying correlations equilibrate after a quantum quench. Then, we show that the equilibrium state is locally equivalent to a thermal state, provided that the free energy of the equilibrium state is sufficiently small and the thermal state has exponentially decaying correlations. As an application, we look at a related important question: When are thermal states stable against noise? In other words, if we locally disturb a closed quantum system in a thermal state, will it return to thermal equilibrium? We rigorously show that this occurs when the correlations in the thermal state are exponentially decaying. All our results come with finite-size bounds, which are crucial for the growing field of quantum thermodynamics and other physical applications.

Equilibration of isolated many-body quantum systems with respect to general distinguishability measures

We demonstrate equilibration of isolated many-body systems in the sense that, after initial transients have died out, the system behaves practically indistinguishable from a time-independent steady state, i.e., non-negligible deviations are unimaginably rare in time. Measuring the distinguishability in terms of quantum mechanical expectation values, results of this type have been previously established under increasingly weak assumptions about the initial disequilibrium, the many-body Hamiltonian, and the considered observables. Here, we further extend these results with respect to generalized distinguishability measures which fully take into account the fact that the actually observed, primary data are not expectation values but rather the probabilistic occurrence of different possible measurement outcomes.

Eigenstate thermalization hypothesis and quantum Jarzynski relation for pure initial states

Since the first suggestion of the Jarzynski equality many derivations of this equality have been presented in both the classical and the quantum context. While the approaches and settings differ greatly from one another, they all appear to rely on the condition that the initial state is a thermal Gibbs state. Here, we present an investigation of work distributions in driven isolated quantum systems, starting from pure states that are close to energy eigenstates of the initial Hamiltonian. We find that, for the nonintegrable quantum ladder studied, the Jarzynski equality is fulfilled to a good accuracy.