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Generalised Quantum Fluctuation Relations

Jordi Mur-Petit, Armando Relaño, Rafael A. Molina, Dieter Jaksch

Published Nov 2, 2017• 0 citations
OpenAlex

Abstract

The non-equilibrium dynamics of quantum many-body systems is one of the most fascinating problems in physics. Open fundamental and practical questions range from how they relax to equilibrium, to how to extract useful work from them. Here, we derive a set of exact results that relate out-of-equilibrium fluctuations in the energy and other observables of a quantum system to its equilibrium properties for a very general family of initial conditions. These quantum fluctuation relations generalise the Jarzynski and Crooks relations to quantum systems with conserved quantities, and can be applied to protocols driving the system between integrable and chaotic regimes, or coupling it to different reservoirs. We illustrate our results with simulations of an integrable model subject to quenches realisable with current technology. Our findings will help guiding research on the interplay of quantum and thermal fluctuations in quantum simulation, and their exploitation in the design of new quantum devices.

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Fluctuation relations allow for the computation of equilibrium properties, like free energy, from an ensemble of non-equilibrium dynamics simulations. Computing them for quantum systems, however, can be difficult, as performing dynamic simulations of such systems is exponentially hard on classical computers. Quantum computers can alleviate this hurdle, as they can efficiently simulate quantum systems. Here, we present an algorithm utilizing a fluctuation relation known as the Jarzynski equality to approximate free energy differences of quantum systems on a quantum computer. We discuss under which conditions our approximation becomes exact, and under which conditions it serves as a strict upper bound. Furthermore, we successfully demonstrate a proof-of-concept of our algorithm using the transverse field Ising model on a real quantum processor. The free energy is a central thermodynamic property that allows one to compute virtually any equilibrium property of a physical system. Thus, as quantum hardware continues to improve, our algorithm may serve as a valuable tool in a wide range of applications including the construction of phase diagrams, prediction of transport properties and reaction constants, and computer-aided drug design in the future.

arXiv:2103.09846•Mar 17, 2021•3 citations
The non-equilibrium dynamics of quantum many-body systems is one of the most fascinating problems in physics. Open fundamental and practical questions range from how they relax to equilibrium, to how to extract useful work from them. Here, we derive a set of exact results that relate out-of-equilibrium fluctuations in the energy and other observables of a quantum system to its equilibrium properties for a very general family of initial conditions. These quantum fluctuation relations generalise the Jarzynski and Crooks relations to quantum systems with conserved quantities, and can be applied to protocols driving the system between integrable and chaotic regimes, or coupling it to different reservoirs. We illustrate our results with simulations of an integrable model subject to quenches realisable with current technology. Our findings will help guiding research on the interplay of quantum and thermal fluctuations in quantum simulation, and their exploitation in the design of new quantum devices.
The non-equilibrium dynamics of quantum many-body systems is one of the most fascinating problems in physics. Open questions range from how they relax to equilibrium to how to extract useful work from them. A critical point lies in assessing whether a system has conserved quantities (or 'charges'), as these can drastically influence its dynamics. Here we propose a general protocol to reveal the existence of charges based on a set of exact relations between out-of-equilibrium fluctuations and equilibrium properties of a quantum system. We apply these generalised quantum fluctuation relations to a driven quantum simulator, demonstrating their relevance to obtain unbiased temperature estimates from non-equilibrium measurements. Our findings will help guide research on the interplay of quantum and thermal fluctuations in quantum simulation, in studying the transition from integrability to chaos and in the design of new quantum devices.
Abstract The non-equilibrium dynamics of quantum many-body systems is one of the most fascinating problems in physics. Open questions range from how they relax to equilibrium to how to extract useful work from them. A critical point lies in assessing whether a system has conserved quantities (or ‘charges’), as these can drastically influence its dynamics. Here we propose a general protocol to reveal the existence of charges based on a set of exact relations between out-of-equilibrium fluctuations and equilibrium properties of a quantum system. We apply these generalised quantum fluctuation relations to a driven quantum simulator, demonstrating their relevance to obtain unbiased temperature estimates from non-equilibrium measurements. Our findings will help guide research on the interplay of quantum and thermal fluctuations in quantum simulation, in studying the transition from integrability to chaos and in the design of new quantum devices.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.
We derive fluctuation relations for a many-body quantum system prepared in a generalized Gibbs ensemble subject to a general nonequilibrium protocol. By considering isolated integrable systems, we find generalizations to the Tasaki-Crooks and Jarzynski relations. Our approach is illustrated by studying the one-dimensional quantum Ising model subject to a sudden change in the transverse field, where we find that the statistics of the work done and irreversible entropy show signatures of quantum criticality. We discuss these fluctuation relations in the context of thermalization.
This Ph.D. thesis concerns a very active area of current research in theoretical physics, which is non-equilibrium statistical mechanics. More specifically, the focus will be on the study of the non-equilibrium dynamics and fluctuations in isolated many body systems driven out of equilibrium. We shall consider primarily quantum systems, albeit many of the theoretical techniques that will be used can be applied to both classical and quantum statistical systems. The novel contribution of this work is two-fold. First, we have developed new analytical techniques to study exactly the large-deviation statistics of physical quantities relevant for the description of systems driven out of equilibrium, like the work done in a quantum quench or the time-integrated current characterizing transport of matter, energy or charge etc, discussed in Part I of the thesis. Second, we have singled out a class of experimentally relevant nondisordered quantum systems showing a dramatically slow approach to thermal equilibrium and we have provided quantitative predictions for this behavior, which is discussed in Part II.
Fluctuation relations allow for the computation of equilibrium properties, like free energy, from an ensemble of non-equilibrium dynamics simulations. Computing them for quantum systems, however, can be difficult, as performing dynamic simulations of such systems is exponentially hard on classical computers. Quantum computers can alleviate this hurdle, as they can efficiently simulate quantum systems. Here, we present an algorithm utilizing a fluctuation relation known as the Jarzynski equality to approximate free energy differences of quantum systems on a quantum computer. We discuss under which conditions our approximation becomes exact, and under which conditions it serves as a strict upper bound. Furthermore, we successfully demonstrate a proof-of-concept of our algorithm using the transverse field Ising model on a real quantum processor. The free energy is a central thermodynamic property that allows one to compute virtually any equilibrium property of a physical system. Thus, as quantum hardware continues to improve, our algorithm may serve as a valuable tool in a wide range of applications including the construction of phase diagrams, prediction of transport properties and reaction constants, and computer-aided drug design in the future.