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arXiv:1401.0164 · Physical Review Letters

Periodic Thermodynamics of Isolated Quantum Systems

Achilleas Lazarides, Arnab Das, Roderich Moessner

Published Apr 16, 2014• 234 citations
arXivHTMLPDF

Abstract

The nature of the behavior of an isolated many-body quantum system periodically driven in time has been an open question since the beginning of quantum mechanics. After an initial transient period, such a system is known to synchronize with the driving; in contrast to the nondriven case, no fundamental principle has been proposed for constructing the resulting nonequilibrium state. Here, we analytically show that, for a class of integrable systems, the relevant ensemble is constructed by maximizing an appropriately defined entropy subject to constraints, which we explicitly identify. This result constitutes a generalization of the concepts of equilibrium statistical mechanics to a class of far-from-equilibrium systems, up to now mainly accessible using ad hoc methods.

DOI
10.1103/physrevlett.112.150401

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The nature of the behavior of an isolated many-body quantum system periodically driven in time has been an open question since the beginning of quantum mechanics. After an initial transient period, such a system is known to synchronize with the driving; in contrast to the nondriven case, no fundamental principle has been proposed for constructing the resulting nonequilibrium state. Here, we analytically show that, for a class of integrable systems, the relevant ensemble is constructed by maximizing an appropriately defined entropy subject to constraints, which we explicitly identify. This result constitutes a generalization of the concepts of equilibrium statistical mechanics to a class of far-from-equilibrium systems, up to now mainly accessible using ad hoc methods.
When a closed quantum system is driven periodically with period T, it approaches a periodic state synchronized with the drive in which any local observable measured stroboscopically approaches a steady value. For integrable systems, the resulting behavior is captured by a periodic version of a generalized Gibbs ensemble. By contrast, here we show that for generic nonintegrable interacting systems, local observables become independent of the initial state entirely. Essentially, this happens because Floquet eigenstates of the driven system at quasienergy ω(α) consist of a mixture of the exponentially many eigenstates of the undriven Hamiltonian, which are thus drawn from the entire extensive undriven spectrum. This is a form of equilibration which depends only on the Hilbert space of the undriven system and not on any details of its Hamiltonian.
Investigation on foundational aspects of quantum statistical mechanics recently entered a renaissance period due to novel intuitions from quantum information theory and to increasing attention on the dynamical aspects of single quantum systems. In the present contribution a simple but effective theoretical framework is introduced to clarify the connections between a purely mechanical description and the thermodynamic characterization of the equilibrium state of an isolated quantum system. A salient feature of our approach is the very transparent distinction between the statistical aspects and the dynamical aspects in the description of isolated quantum systems. Like in the classical statistical mechanics, the equilibrium distribution of any property is identified on the basis of the time evolution of the considered system. As a consequence equilibrium properties of quantum system appear to depend on the details of the initial state due to the abundance of constants of the motion in the Schrodinger dynamics. On the other hand the study of the probability distributions of some functions, such as the entropy or the equilibrium state of a subsystem, in statistical ensembles of pure states reveals the crucial role of typicality as the bridge between macroscopic thermodynamics and microscopic quantum dynamics. We shall consider two particular ensembles: the random pure state ensemble and the fixed expectation energy ensemble. The relation between the introduced ensembles, the properties of a given isolated system, and the standard quantum statistical description are discussed throughout the presentation. Finally we point out the conditions which should be satisfied by an ensemble in order to get meaningful thermodynamical characterization of an isolated quantum system.
Understanding how macroscopic thermodynamic behavior emerges from microscopic quantum dynamics remains an open problem. While equilibration of quantum observables is well established, thermodynamics also relies on variables not directly associated with linear operators, but which are defined instead as functions of expectation values. Whether and how such derived quantities inherit equilibration properties is an open question. Here, we establish that any continuously differentiable function of equilibrating expectation values also equilibrates. We apply this result to a bipartite isolated system, showing that the entropy and conjugate variables of each subsystem -- defined through Jaynes' maximum entropy principle -- equilibrate. Moreover, with the assumption that their equilibrium properties depend solely on local conserved quantities, we show the dynamical maximization of the total entropy, enforcing equality of conjugate variables across subsystems. These results provide a direct dynamical justification for entropy maximization and the emergence of thermodynamic equilibrium conditions, showing that fundamental principles of thermodynamics follow from the unitary evolution of quantum systems.
The description of the non-equilibrium dynamics of isolated quantum many-body systems within the framework of statistical mechanics is a fundamental open question. Conventional thermodynamical ensembles fail to describe the large class of systems that exhibit nontrivial conserved quantities, and generalized ensembles have been predicted to maximize entropy in these systems. We show experimentally that a degenerate one-dimensional Bose gas relaxes to a state that can be described by such a generalized ensemble. This is verified through a detailed study of correlation functions up to 10th order. The applicability of the generalized ensemble description for isolated quantum many-body systems points to a natural emergence of classical statistical properties from the microscopic unitary quantum evolution.
A central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.
The non-equilibrium dynamics of isolated quantum systems represent a theoretical and experimental challenge raising many fundamental questions with applications to different fields of modern physics. In these proceedings, we briefly review some of the recent findings on the subject, with particular emphasis to the existence of stationary expectation values of local observables and to their statistical mechanics description. It turns out that the appropriate statistical ensemble describing these asymptotic values depends on whether the Hamiltonian governing the time evolution is integrable or not.