Experimental observation of a generalized Gibbs ensemble
Tim Langen, Sebastian Erne, Remi Geiger, Bernhard Rauer, Thomas Schweigler, Maximilian Kuhnert, Wolfgang Rohringer, I. E. Mazets, Thomas Gasenzer, Jörg Schmiedmayer
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.
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.
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.
In this Letter we pose the question of whether a many-body quantum system with a full set of conserved quantities can relax to an equilibrium state, and, if it can, what the properties of such a state are. We confirm the relaxation hypothesis through an ab initio numerical investigation of the dynamics of hard-core bosons on a one-dimensional lattice. Further, a natural extension of the Gibbs ensemble to integrable systems results in a theory that is able to predict the mean values of physical observables after relaxation. Finally, we show that our generalized equilibrium carries more memory of the initial conditions than the usual thermodynamic one. This effect may have many experimental consequences, some of which have already been observed in the recent experiment on the nonequilibrium dynamics of one-dimensional hard-core bosons in a harmonic potential [T. Kinoshita et al., Nature (London) 440, 900 (2006)10.1038/nature04693].
Statistical mechanics is one of the most comprehensive theories in physics. From a boiling pot of water to the complex dynamics of quantum many-body systems it provides a successful connection between the microscopic dynamics of atoms and molecules and the macroscopic properties of matter. However, statistical mechanics only describes the thermal equilibrium situation of a system, and there is no general framework to describe how equilibrium is reached or under which circumstances it can be reached at all. This problem is particularly challenging in quantum mechanics, where unitarity appears to render the very concept of thermalization counterintuitive. With the rapid experimental progress in the control and probing of ultracold quantum gases this question has become within reach of detailed experimental investigations. In these notes we present a series of experiments with ultracold one-dimensional Bose gases, which provide novel insights into this fundamental question.
One of the distinguishing achievements of physics has been the ability to describe the Universe based upon underlying microscopic principles. When there are many participating objects being governed by these microscopic principles, the resultant behavior often harbors emergent collective phenomena that may not be obviously related to the microscopic description. Phase transitions, quantum or classical, are a canonical example of such many-body phenomena that are understood in equilibrium through a framework of thermodynamic constraints, statistical ensembles, and a growth of system-wide correlations at the transition. However, a new class of quantum phase transitions has recently emerged for non-equilibrium systems with behavior that is not captured by this traditional framework. In the case of isolated quantum systems, these transitions occur among excited eigenstates and requires knowledge of the structure of these individual eigenstates in the quantum system. Understanding what drives such a phase transition requires a deeper microscopic knowledge of the role of correlations at the transition. \nThe use of ultracold atoms in optical lattices has paved the way for faithfully realizing quantum phase transitions. In particular, the advent of the quantum gas microscope realized an experimental platform that is highly controllable and provides microscopic access to the many-bodied wave functions. All experiments presented in this thesis use such a quantum gas microscope architecture where a quantum degenerate gas of Rb is used to probe the behavior of non-equilibrium quantum systems in one spatial dimension.\nThis thesis briefly discusses some studies of entanglement in equilibrium quantum phase transitions near the ground state of the system. In the first experiment, we directly measure the second-order R\\'enyi entanglement entropy that develops as a Bose-Hubbard system transitions from a Mott insulator to a superfluid. This is followed by a study of entanglement growth at the transition in the transverse Ising model. \nThese studies progress to the investigation of two particular non-equilibrium quantum phases and their transition into one another: quantum thermalization and many-body localization. We investigate the development of entanglement entropy that mimics thermodynamic entropy for an isolated quantum system which locally thermalizes under its own unitary dynamics. We then investigate the only known robust exception to this thermalizing behavior by studying the onset of many-body localization due to disorder. We observe the breakdown of thermalization in this phase and the logarithmically slow growth of entanglement throughout the system, which identifies the localized system to be an interacting one. Lastly, we investigate the role of correlations in the critical dynamics that emerge at high-orders where the system transitions from thermalizing behavior to many-body localization.
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.
Predicting the stationary behavior of observables in isolated many-body quantum systems is a central challenge in quantum statistical mechanics. While one can often use the Gibbs ensemble, which is simple to compute, there are many scenarios where this is not possible and one must instead use another ensemble, such as the diagonal, microcanonical or generalized Gibbs ensembles. However, these all require detailed information about the energy or other conserved quantities to be constructed. Here we propose a general and computationally easy approach to determine the stationary probability distribution of observables with few outcomes. Interpreting coarse measurements at equilibrium as noisy communication channels, we provide general analytical arguments in favor of the applicability of a maximum entropy principle for this class of observables. We show that the resulting theory accurately predicts stationary probability distributions without detailed microscopic information like the energy eigenstates. Extensive numerical experiments on 7 non-weakly interacting spin-1/2 Hamiltonians demonstrate the broad applicability and robustness of this framework in both quantum integrable and chaotic models.