Thermal Equilibrium of a Macroscopic Quantum System in a Pure State

Entropy Reorders Polariton States

The main result is that the long-range phase coherence of the polariton states formed by strong coupling between a photon mode in a cavity and an ensemble of molecules leads to exceptionally low entropy of the upper and lower polariton states, starkly contrasting with the dark states. That result means that spectroscopy does not correctly order the free energy of the excited states because there is a significant entropic contribution to the free energy, which turns out to comparable to the electronic energy gap between the lower polariton state and the dark-state manifold. The reordered states, according to their free energy, is important to predict the potential of polariton states for reactivity, to predict spontaneous photophysical dynamics, or to understand their decoherence. The entropic contribution adds to the polariton electronic gap, rendering states surprisingly more reactive than anticipated from the input excitation energy. This apparently "additional" reactivity, evident from the thermodynamics, suggests how the low entropy of highly coherent states can be exploited as a resource.

The fourth law of thermodynamics: steepest entropy ascent

When thermodynamics is understood as the science (or art) of constructing effective models of natural phenomena by choosing a minimal level of description capable of capturing the essential features of the physical reality of interest, the scientific community has identified a set of general rules that the model must incorporate if it aspires to be consistent with the body of known experimental evidence. Some of these rules are believed to be so general that we think of them as laws of Nature, such as the great conservation principles, whose 'greatness' derives from their generality, as masterfully explained by Feynman in one of his legendary lectures. The second law of thermodynamics is universally contemplated among the great laws of Nature. In this paper, we show that in the past four decades, an enormous body of scientific research devoted to modelling the essential features of non-equilibrium natural phenomena has converged from many different directions and frameworks towards the general recognition (albeit still expressed in different but equivalent forms and language) that another rule is also indispensable and reveals another great law of Nature that we propose to call the 'fourth law of thermodynamics'. We state it as follows: every non-equilibrium state of a system or local subsystem for which entropy is well defined must be equipped with a metric in state space with respect to which the irreversible component of its time evolution is in the direction of steepest entropy ascent compatible with the conservation constraints. To illustrate the power of the fourth law, we derive (nonlinear) extensions of Onsager reciprocity and fluctuation-dissipation relations to the far-non-equilibrium realm within the framework of the rate-controlled constrained-equilibrium approximation (also known as the quasi-equilibrium approximation). This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

Maxwell's Demon in Quantum Mechanics

Maxwell’s Demon is a thought experiment devised by J. C. Maxwell in 1867 in order to show that the Second Law of thermodynamics is not universal, since it has a counter-example. Since the Second Law is taken by many to provide an arrow of time, the threat to its universality threatens the account of temporal directionality as well. Various attempts to “exorcise” the Demon, by proving that it is impossible for one reason or another, have been made throughout the years, but none of them were successful. We have shown (in a number of publications) by a general state-space argument that Maxwell’s Demon is compatible with classical mechanics, and that the most recent solutions, based on Landauer’s thesis, are not general. In this paper we demonstrate that Maxwell’s Demon is also compatible with quantum mechanics. We do so by analyzing a particular (but highly idealized) experimental setup and proving that it violates the Second Law. Our discussion is in the framework of standard quantum mechanics; we give two separate arguments in the framework of quantum mechanics with and without the projection postulate. We address in our analysis the connection between measurement and erasure interactions and we show how these notions are applicable in the microscopic quantum mechanical structure. We discuss what might be the quantum mechanical counterpart of the classical notion of “macrostates”, thus explaining why our Quantum Demon setup works not only at the micro level but also at the macro level, properly understood. One implication of our analysis is that the Second Law cannot provide a universal lawlike basis for an account of the arrow of time; this account has to be sought elsewhere.

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.

New characteristic of quantum many-body chaotic systems

An isolated quantum system in a pure state may be perceived as thermal if only a substantially small fraction of all degrees of freedom is probed. We propose that in a quantum chaotic many-body system all states with sufficiently small energy fluctuations are approximately thermal. We refer to this hypothesis as canonical universality (CU). The CU hypothesis complements the eigenstate thermalization hypothesis which proposes that for chaotic systems individual energy eigenstates are thermal. Integrable and many-body localization systems do not satisfy CU. We provide theoretical and numerical evidence supporting the CU hypothesis.

Saturation of entropy production in quantum many-body systems

Bridging the second law of thermodynamics and microscopic reversible dynamics has been a longstanding problem in statistical physics. Here, we address this problem on the basis of quantum many-body physics, and discuss how the entropy production saturates in isolated quantum systems under unitary dynamics. First, we rigorously prove that the entropy production does indeed saturate in the long time regime, even when the total system is in a pure state. Second, we discuss the non-negativity of the entropy production at saturation, implying the second law of thermodynamics. This is based on the eigenstate thermalization hypothesis, which states that even a single energy eigenstate is thermal. We also numerically demonstrate that the entropy production saturates at a non-negative value even when the initial state of a heat bath is a single energy eigenstate. Our results reveal fundamental properties of the entropy production in isolated quantum systems at late times.

Thermalization without eigenstate thermalization hypothesis after a quantum quench

Nonequilibrium dynamics of a nonintegrable system without the eigenstate thermalization hypothesis is studied. It is shown that, in the thermodynamic limit, this model thermalizes after an arbitrary quantum quench at finite temperature, although it does not satisfy the eigenstate thermalization hypothesis. In contrast, when the system size is finite and the temperature is low enough, the system may not thermalize. In this case, the steady state is well described by the generalized Gibbs ensemble constructed by using highly nonlocal conserved quantities. We also show that this model exhibits prethermalization, in which the prethermalized state is characterized by nonthermal energy eigenstates.

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.

Typical pure nonequilibrium steady states and irreversibility for quantum transport

It is known that each single typical pure state in an energy shell of a large isolated quantum system well represents a thermal equilibrium state of the system. We show that such typicality holds also for nonequilibrium steady states (NESS's). We consider a small quantum system coupled to multiple infinite reservoirs. In the long run, the total system reaches a unique NESS. We identify a large Hilbert space from which pure states of the system are to be sampled randomly and show that the typical pure states well describe the NESS. We also point out that the irreversible relaxation to the unique NESS is important to the typicality of the pure NESS's.

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.

Typical fast thermalization processes in closed many-body systems

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 relaxation of closed macroscopic systems towards thermal equilibrium is an ubiquitous experimental fact, but very difficult to characterize theoretically. Here, the author establishes a quantitative description of such relaxation under arbitrary typical conditions, capturing well experimental data.

Generalized Gibbs ensemble in a nonintegrable system with an extensive number of local symmetries

We numerically study the unitary time evolution of a nonintegrable model of hard-core bosons with an extensive number of local Z(2) symmetries. We find that the expectation values of local observables in the stationary state are described better by the generalized Gibbs ensemble (GGE) than by the canonical ensemble. We also find that the eigenstate thermalization hypothesis fails for the entire spectrum but holds true within each symmetry sector, which justifies the GGE. In contrast, if the model has only one global Z(2) symmetry or a size-independent number of local Z(2) symmetries, we find that the stationary state is described by the canonical ensemble. Thus, the GGE is necessary to describe the stationary state even in a nonintegrable system if it has an extensive number of local symmetries.