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The Journal of Physical Chemistry Letters

Nonequilibrium Work Relations and Response Theories in Ensemble Quantum Systems

Amro Dodin, Adam P. Willard

Published Nov 10, 2021• 2 citations
Publisher

Abstract

High Resolution Image Download MS PowerPoint Slide We develop a nonequilibrium response theory for macroscopic quantum systems that separates the contributions of ensemble heterogeneity and intrinsic quantum uncertainty. To accomplish this, we describe systems with a quantum P -ensemble, which goes beyond the standard density matrix description by explicitly specifying the classical heterogeneity between individual quantum systems in an ensemble. We use the P -ensemble formalism to present quantum generalizations of linear response theory and the Jarzynski nonequilibrium work relation. We derive these generalizations from a Bochkov-Kuzovlev generating functional for quantum P -ensembles, which can be further utilized to derive all orders of response theory that apply to ensemble quantum systems. We contrast these developments with their ρ-ensemble analogs, and we discuss how these P -ensemble theories provide a guide for an effective application of single molecule experiments.

DOI
10.1021/acs.jpclett.1c01315

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High Resolution Image Download MS PowerPoint Slide We develop a nonequilibrium response theory for macroscopic quantum systems that separates the contributions of ensemble heterogeneity and intrinsic quantum uncertainty. To accomplish this, we describe systems with a quantum P -ensemble, which goes beyond the standard density matrix description by explicitly specifying the classical heterogeneity between individual quantum systems in an ensemble. We use the P -ensemble formalism to present quantum generalizations of linear response theory and the Jarzynski nonequilibrium work relation. We derive these generalizations from a Bochkov-Kuzovlev generating functional for quantum P -ensembles, which can be further utilized to derive all orders of response theory that apply to ensemble quantum systems. We contrast these developments with their ρ-ensemble analogs, and we discuss how these P -ensemble theories provide a guide for an effective application of single molecule experiments.
We briefly introduce the quantum Jarzynski and Bochkov-Kuzovlev equalities in isolated quantum Hamiltonian systems, which includes the origin of the equalities, their derivations using a quantum Feynman-Kac formula, the quantum Crooks equality, the evolution equations governing the characteristic functions of the probability density functions for the quantum work, the recent experimental verifications. Some results are given here first time. We particularly emphasize the formally structural consistence between these quantum equalities and their classical counterparts, which shall be useful in understanding the existing equalities and pursuing new fluctuation relations in other complex quantum systems.
The exact response theory based on the Dissipation Function applies to general dynamical systems and has yielded excellent results in various applications. In this article, we propose a method to apply it to quantum mechanics. In many quantum systems, it has not yet been possible to overcome the perturbative approach, and the most developed theory is the linear one. Extensions of the exact response theory developed in the field of nonequilibrium molecular dynamics could prove useful in quantum mechanics, as perturbations of small systems or far-from-equilibrium states cannot always be taken as small perturbations. Here, we introduce a quantum analogue of the classical Dissipation Function. We then derive a quantum expression for the exact calculation of time-dependent expectation values of observables, in a form analogous to that of the classical theory. We restrict our analysis to finite-dimensional Hilbert spaces, for the sake of simplicity, and we apply our method to specific examples, like qubit systems, for which exact results can be obtained by standard techniques. This way, we prove the consistency of our approach with the existing methods, where they apply. Although not required for open systems, we propose a self-adjoint version of our Dissipation Operator, obtaining a second equivalent expression of response, where the contribution of an anti-self-adjoint operator appears. We conclude by using new formalism to solve the Lindblad equations, obtaining exact results for a specific case of qubit decoherence, and suggesting possible future developments of this work.
Quantum response theory is a cornerstone of statistical physics.However, the standard Kubo formalism is restricted to isolated equilibrium systems.We here generalize Kubo's results to open quantum systems in nonequilibrium steady states.We derive three different, but equivalent, forms of the quantum response function.We discuss for each of them the role of the noncommutativity of quantum operators and introduce a steady-state extension of the Kubo transformation.We show in particular that the equilibrium response vanishes for perturbations that commute with the unperturbed Hamiltonian, while the steady-state response does not, highlighting the profound difference between the two linear response theories.
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 introduce a response theory for open quantum systems within nonequilibrium steady states subject to a Hamiltonian perturbation.Working in the weak system-bath coupling regime, our results are derived within the Lindblad-Gorini-Kossakowski-Sudarshan formalism.We find that the response of the system to a small perturbation is not simply related to a correlation function within the system, unlike traditional linear response theory in closed systems or expectations from the fluctuation-dissipation theorem.In limiting cases, when the perturbation is small relative to the coupling to the surroundings or when it does not lead to a change of the eigenstructure of the system, a perturbative expansion exists where the response function is related to a sum of a system correlation functions and additional forces induced by the surroundings.Away from these limiting regimes, however, the secular approximation results in a singular response that cannot be captured within the traditional approach but can be described by reverting to a microscopic Hamiltonian description.These findings are illustrated by explicit calculations in coupled qubits and anharmonic oscillators in contact with bosonic baths at different temperatures.