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Journal of the Physical Society of Japan

Typical Pure States and Nonequilibrium Processes in Quantum Many-Body Systems

Takaaki Monnai, Ayumu Sugita

Published Aug 7, 2014• 30 citations
Publisher

Abstract

We show that it is possible to calculate equilibrium expectation values of many-body correlated quantities such as the characteristic functions and probability distributions with the use of only a single typical pure state. It also means that we can apply the pure state approach to Heisenberg operators and their spectral fluctuation, and hence to nonequilibrium processes starting from equilibrium. In particular, we can accurately analyze the full statistics of entropy production in nonequilibrium mesoscopic systems. In this way, we can access the full information on higher-order fluctuations in the large deviation regime far from equilibrium.

DOI
10.7566/jpsj.83.094001

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We show that it is possible to calculate equilibrium expectation values of many-body correlated quantities such as the characteristic functions and probability distributions with the use of only a single typical pure state. It also means that we can apply the pure state approach to Heisenberg operators and their spectral fluctuation, and hence to nonequilibrium processes starting from equilibrium. In particular, we can accurately analyze the full statistics of entropy production in nonequilibrium mesoscopic systems. In this way, we can access the full information on higher-order fluctuations in the large deviation regime far from equilibrium.
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We study static and dynamical expectation values for a class of pure states. This class mimics non-equilibrium density matrices via the concept of quantum typicality and allows for a controlled preparation of expectation values close to or far away from their equilibrium values at arbitrary temperatures. While these pure states do not require specific types of observables, we investigate so-called binary operators such as local fermionic occupation numbers or local energy densities in the isotropic Heisenberg spin-1/2 chain. For these two examples, we use numerical forward-propagation methods in real and imaginary time to demonstrate the accuracy of the pure-state approach within and outside the linear-response regime. At high temperatures, our numerical results show that the dynamics in all regimes is generated by a single correlation function. These numerical results are in excellent agreement with our analytical predictions for binary operators.
Controlling dynamical fluctuations in open quantum systems is essential both for our comprehension of quantum nonequilibrium behavior and for its possible application in near-term quantum technologies. However, understanding these fluctuations is extremely challenging due, to a large extent, to a lack of efficient important sampling methods for quantum systems. Here, we devise a unified framework-based on population-dynamics methods-for the evaluation of the full probability distribution of generic time-integrated observables in Markovian quantum jump processes. These include quantities carrying information about genuine quantum features, such as quantum superposition or entanglement, not accessible with existing numerical techniques. The algorithm we propose provides dynamical free-energy and entropy functionals which, akin to their equilibrium counterpart, permit one to unveil intriguing phase-transition behavior in quantum trajectories. We discuss some applications and further disclose coexistence and hysteresis, between a highly entangled phase and a low entangled one, in large fluctuations of a strongly interacting few-body system.
Equilibrium statistical mechanics provides powerful tools to understand physics at the macroscale. Yet, the question remains how this can be justified based on a microscopic quantum description. Here, we extend the ideas of pure state quantum statistical mechanics, which focus on single time statistics, to show the equilibration of isolated quantum processes. Namely, we show that most multitime observables for sufficiently large times cannot distinguish a nonequilibrium process from an equilibrium one, unless the system is probed for an extremely large number of times or the observable is particularly fine-grained. A corollary of our results is that the size of non-Markovianity and other multitime characteristics of a nonequilibrium process also equilibrate.
Schrödinger suggested that thermodynamical functions cannot be based on the gratuitous allegation that quantum-mechanical levels (typically the orthogonal eigenstates of the Hamiltonian operator) are the only allowed states for a quantum system [E. Schrödinger, Statistical Thermodynamics (Courier Dover, Mineola, 1967)]. Different authors have interpreted this statement by introducing density distributions on the space of quantum pure states with weights obtained as functions of the expectation value of the Hamiltonian of the system. In this work we focus on one of the best known of these distributions and prove that, when considered in composite quantum systems, it defines partition functions that do not factorize as products of partition functions of the noninteracting subsystems, even in the thermodynamical regime. This implies that it is not possible to define extensive thermodynamical magnitudes such as the free energy, the internal energy, or the thermodynamic entropy by using these models. Therefore, we conclude that this distribution inspired by Schrödinger's idea cannot be used to construct an appropriate quantum equilibrium thermodynamics.