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Boettcher V, Hartmann R, Beyer K, Strunz WT. Dynamics of a strongly coupled quantum heat engine-Computing bath observables from the hierarchy of pure states. J Chem Phys 2024; 160:094108. [PMID: 38436445 DOI: 10.1063/5.0192075] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2023] [Accepted: 02/07/2024] [Indexed: 03/05/2024] Open
Abstract
We present a fully quantum dynamical treatment of a quantum heat engine and its baths based on the Hierarchy of Pure States (HOPS), an exact and general method for open quantum system dynamics. We show how the change of the bath energy and the interaction energy can be determined within HOPS for arbitrary coupling strength and smooth time dependence of the modulation protocol. The dynamics of all energetic contributions during the operation can be carefully examined both in its initial transient phase and, also later, in its periodic steady state. A quantum Otto engine with a qubit as an inherently nonlinear work medium is studied in a regime where the energy associated with the interaction Hamiltonian plays an important role for the global energy balance and, thus, must not be neglected when calculating its power and efficiency. We confirm that the work required to drive the coupling with the baths sensitively depends on the speed of the modulation protocol. Remarkably, departing from the conventional scheme of well-separated phases by allowing for temporal overlap, we discover that one can even gain energy from the modulation of bath interactions. We visualize these various work contributions using the analog of state change diagrams of thermodynamic cycles. We offer a concise, full presentation of HOPS with its extension to bath observables, as it serves as a universal tool for the numerically exact description of general quantum dynamical (thermodynamic) scenarios far from the weak-coupling limit.
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Affiliation(s)
- Valentin Boettcher
- Institute of Theoretical Physics, TUD Dresden University of Technology, 01062 Dresden, Germany
- Department of Physics, McGill University, Montréal, Québec H3A 2T8, Canada
| | - Richard Hartmann
- Institute of Theoretical Physics, TUD Dresden University of Technology, 01062 Dresden, Germany
| | - Konstantin Beyer
- Institute of Theoretical Physics, TUD Dresden University of Technology, 01062 Dresden, Germany
- Department of Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA
| | - Walter T Strunz
- Institute of Theoretical Physics, TUD Dresden University of Technology, 01062 Dresden, Germany
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Chen T, Cheng YC. Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems. J Chem Phys 2022; 157:064106. [PMID: 35963728 DOI: 10.1063/5.0099761] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We describe a numerical algorithm for approximating the equilibrium-reduced density matrix and the effective (mean force) Hamiltonian for a set of system spins coupled strongly to a set of bath spins when the total system (system + bath) is held in canonical thermal equilibrium by weak coupling with a "super-bath". Our approach is a generalization of now standard typicality algorithms for computing the quantum expectation value of observables of bare quantum systems via trace estimators and Krylov subspace methods. In particular, our algorithm makes use of the fact that the reduced system density, when the bath is measured in a given random state, tends to concentrate about the corresponding thermodynamic averaged reduced system density. Theoretical error analysis and numerical experiments are given to validate the accuracy of our algorithm. Further numerical experiments demonstrate the potential of our approach for applications including the study of quantum phase transitions and entanglement entropy for long range interaction systems.
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Affiliation(s)
- Tyler Chen
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
| | - Yu-Chen Cheng
- Department of Data Science, Dana-Farber Cancer Institute, Boston, Massachusetts 02215, USA
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Dong H, Reiche D, Hsiang JT, Hu BL. Quantum Thermodynamic Uncertainties in Nonequilibrium Systems from Robertson-Schrödinger Relations. ENTROPY 2022; 24:e24070870. [PMID: 35885093 PMCID: PMC9324490 DOI: 10.3390/e24070870] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/30/2022] [Revised: 06/15/2022] [Accepted: 06/19/2022] [Indexed: 02/05/2023]
Abstract
Thermodynamic uncertainty principles make up one of the few rare anchors in the largely uncharted waters of nonequilibrium systems, the fluctuation theorems being the more familiar. In this work we aim to trace the uncertainties of thermodynamic quantities in nonequilibrium systems to their quantum origins, namely, to the quantum uncertainty principles. Our results enable us to make this categorical statement: For Gaussian systems, thermodynamic functions are functionals of the Robertson-Schrödinger uncertainty function, which is always non-negative for quantum systems, but not necessarily so for classical systems. Here, quantum refers to noncommutativity of the canonical operator pairs. From the nonequilibrium free energy, we succeeded in deriving several inequalities between certain thermodynamic quantities. They assume the same forms as those in conventional thermodynamics, but these are nonequilibrium in nature and they hold for all times and at strong coupling. In addition we show that a fluctuation-dissipation inequality exists at all times in the nonequilibrium dynamics of the system. For nonequilibrium systems which relax to an equilibrium state at late times, this fluctuation-dissipation inequality leads to the Robertson-Schrödinger uncertainty principle with the help of the Cauchy-Schwarz inequality. This work provides the microscopic quantum basis to certain important thermodynamic properties of macroscopic nonequilibrium systems.
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Affiliation(s)
- Hang Dong
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China;
| | - Daniel Reiche
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany;
| | - Jen-Tsung Hsiang
- Center for High Energy and High Field Physics, National Central University, Taoyuan 320317, Taiwan
- Correspondence:
| | - Bei-Lok Hu
- Maryland Center for Fundamental Physics and Joint Quantum Institute, University of Maryland, College Park, MD 20742, USA;
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Hsiang JT, Hu BL. Intrinsic Entropy of Squeezed Quantum Fields and Nonequilibrium Quantum Dynamics of Cosmological Perturbations. ENTROPY 2021; 23:e23111544. [PMID: 34828242 PMCID: PMC8621705 DOI: 10.3390/e23111544] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/07/2021] [Revised: 11/02/2021] [Accepted: 11/11/2021] [Indexed: 11/30/2022]
Abstract
Density contrasts in the universe are governed by scalar cosmological perturbations which, when expressed in terms of gauge-invariant variables, contain a classical component from scalar metric perturbations and a quantum component from inflaton field fluctuations. It has long been known that the effect of cosmological expansion on a quantum field amounts to squeezing. Thus, the entropy of cosmological perturbations can be studied by treating them in the framework of squeezed quantum systems. Entropy of a free quantum field is a seemingly simple yet subtle issue. In this paper, different from previous treatments, we tackle this issue with a fully developed nonequilibrium quantum field theory formalism for such systems. We compute the covariance matrix elements of the parametric quantum field and solve for the evolution of the density matrix elements and the Wigner functions, and, from them, derive the von Neumann entropy. We then show explicitly why the entropy for the squeezed yet closed system is zero, but is proportional to the particle number produced upon coarse-graining out the correlation between the particle pairs. We also construct the bridge between our quantum field-theoretic results and those using the probability distribution of classical stochastic fields by earlier authors, preserving some important quantum properties, such as entanglement and coherence, of the quantum field.
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Affiliation(s)
- Jen-Tsung Hsiang
- Center for High Energy and High Field Physics, National Central University, Taoyuan 32001, Taiwan
- Correspondence:
| | - Bei-Lok Hu
- Maryland Center for Fundamental Physics and Joint Quantum Institute, University of Maryland, College Park, MD 20742, USA;
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Bernardo BDL. Relating heat and entanglement in strong-coupling thermodynamics. Phys Rev E 2021; 104:044111. [PMID: 34781427 DOI: 10.1103/physreve.104.044111] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2021] [Accepted: 09/27/2021] [Indexed: 11/07/2022]
Abstract
Explaining the influence of strong coupling in the dynamics of open quantum systems is one of the most challenging issues in the rapidly growing field of quantum thermodynamics. By using a particular definition of heat, we develop an approach to study the thermodynamics in the strong-coupling regime, which takes into account quantum resources such as coherence and entanglement. We apply the method to calculate the time-dependent thermodynamic properties of a system and an environment interacting via the generalized amplitude-damping channel. The results indicate that the transient imbalance between heat dissipated and heat absorbed that occurs in the process is responsible for the generation of system-environment entanglement.
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Affiliation(s)
- Bertúlio de Lima Bernardo
- Departamento de Física, Universidade Federal da Paraíba, 58051-900 João Pessoa, PB, Brazil and Departamento de Física, Universidade Federal de Campina Grande, Caixa Postal 10071, 58109-970 Campina Grande, PB, Brazil
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Hsiang JT, Hu BL. Nonequilibrium nonlinear open quantum systems: Functional perturbative analysis of a weakly anharmonic oscillator. Int J Clin Exp Med 2020. [DOI: 10.1103/physrevd.101.125002] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Strasberg P, Esposito M. Measurability of nonequilibrium thermodynamics in terms of the Hamiltonian of mean force. Phys Rev E 2020; 101:050101. [PMID: 32575212 DOI: 10.1103/physreve.101.050101] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2020] [Accepted: 04/22/2020] [Indexed: 11/07/2022]
Abstract
The nonequilibrium thermodynamics of an open (classical or quantum) system in strong contact with a single heat bath can be conveniently described in terms of the Hamiltonian of mean force. However, the conventional formulation is limited by the necessity to measure differences in equilibrium properties of the system-bath composite. We make use of the freedom involved in defining thermodynamic quantities, which leaves the thermodynamics unchanged, to show that the Hamiltonian of mean force can be inferred from measurements on the system alone, up to that irrelevant freedom. In doing so, we refute a key criticism expressed in the works by P. Talkner and P. Hänggi [Phys. Rev. E 94, 022143 (2016)10.1103/PhysRevE.94.022143 and arXiv:1911.11660]. We also discuss the remaining part of the criticism.
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Affiliation(s)
- Philipp Strasberg
- Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - Massimiliano Esposito
- Complex Systems and Statistical Mechanics, Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
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Rivas Á. Strong Coupling Thermodynamics of Open Quantum Systems. PHYSICAL REVIEW LETTERS 2020; 124:160601. [PMID: 32383934 DOI: 10.1103/physrevlett.124.160601] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2019] [Accepted: 03/17/2020] [Indexed: 06/11/2023]
Abstract
A general thermodynamic framework is presented for open quantum systems in fixed contact with a thermal reservoir. The first and second law are obtained for arbitrary system-reservoir coupling strengths, and including both factorized and correlated initial conditions. The thermodynamic properties are adapted to the generally strong coupling regime, approaching the ones defined for equilibrium, and their standard weak-coupling counterparts as appropriate limits. Moreover, they can be inferred from measurements involving only system observables. Finally, a thermodynamic signature of non-Markovianity is formulated in the form of a negative entropy production rate.
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Affiliation(s)
- Ángel Rivas
- Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain and CCS-Center for Computational Simulation, Campus de Montegancedo UPM, 28660 Boadilla del Monte, Madrid, Spain
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Quantum Thermodynamics in the Refined Weak Coupling Limit. ENTROPY 2019; 21:e21080725. [PMID: 33267439 PMCID: PMC7515254 DOI: 10.3390/e21080725] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/31/2019] [Revised: 07/17/2019] [Accepted: 07/24/2019] [Indexed: 12/22/2022]
Abstract
We present a thermodynamic framework for the refined weak coupling limit. In this limit, the interaction between system and environment is weak, but not negligible. As a result, the system dynamics becomes non-Markovian breaking divisibility conditions. Nevertheless, we propose a derivation of the first and second law just in terms of the reduced system dynamics. To this end, we extend the refined weak coupling limit for allowing slowly-varying external drivings and reconsider the definition of internal energy due to the non-negligible interaction.
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Strasberg P, Esposito M. Non-Markovianity and negative entropy production rates. Phys Rev E 2019; 99:012120. [PMID: 30780330 DOI: 10.1103/physreve.99.012120] [Citation(s) in RCA: 48] [Impact Index Per Article: 9.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/12/2018] [Indexed: 11/07/2022]
Abstract
Entropy production plays a fundamental role in nonequilibrium thermodynamics to quantify the irreversibility of open systems. Its positivity can be ensured for a wide class of setups, but the entropy production rate can become negative sometimes. This is often taken as an indicator of non-Markovianity. We make this link precise by showing under which conditions a negative entropy production rate implies non-Markovianity and when it does not. For a system coupled to a single heat bath, this can be established within a unified language for two setups: (i) the dynamics resulting from a coarse-grained description of a Markovian master equation and (ii) the classical Hamiltonian dynamics of a system coupled to a bath. The quantum version of the latter result is shown not to hold despite the fact that the integrated thermodynamic description is formally equivalent to the classical case. The instantaneous fixed point of a non-Markovian dynamics plays an important role in our study. Our key contribution is to provide a consistent theoretical framework to study the finite-time thermodynamics of a large class of dynamics with a precise link to its non-Markovianity.
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Affiliation(s)
- Philipp Strasberg
- Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Massimiliano Esposito
- Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
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Emergent Quantum Mechanics: David Bohm Centennial Perspectives. ENTROPY 2019; 21:e21020113. [PMID: 33266829 PMCID: PMC7514595 DOI: 10.3390/e21020113] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 01/19/2019] [Accepted: 01/24/2019] [Indexed: 12/16/2022]
Abstract
Emergent quantum mechanics (EmQM) explores the possibility of an ontology for quantum mechanics. The resurgence of interest in realist approaches to quantum mechanics challenges the standard textbook view, which represents an operationalist approach. The possibility of an ontological, i.e., realist, quantum mechanics was first introduced with the original de Broglie–Bohm theory, which has also been developed in another context as Bohmian mechanics. This Editorial introduces a Special Issue featuring contributions which were invited as part of the David Bohm Centennial symposium of the EmQM conference series (www.emqm17.org). Questions directing the EmQM research agenda are: Is reality intrinsically random or fundamentally interconnected? Is the universe local or nonlocal? Might a radically new conception of reality include a form of quantum causality or quantum ontology? What is the role of the experimenter agent in ontological quantum mechanics? The Special Issue also includes research examining ontological propositions that are not based on the Bohm-type nonlocality. These include, for example, local, yet time-symmetric, ontologies, such as quantum models based upon retrocausality. This Editorial provides topical overviews of thirty-one contributions which are organized into seven categories to provide orientation.
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