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arXiv:2205.04457 · Physical Review A

Constraint on local definitions of quantum internal energy

Luis Rodrigo Torres Neves, Frederico Brito

Published Oct 13, 2023• 4 citations
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Abstract

Recent advances in quantum thermodynamics have been focusing on ever more elementary systems of interest, approaching the limit of a single qubit, with correlations, strong coupling, and far-from-equilibrium environments coming into play. Under such scenarios, it is clear that fundamental physical quantities must be revisited. This article questions whether a universal definition of internal energy for open quantum systems can be devised, setting limits on its possible properties. We argue that, for such a definition to be regarded as local, it should be determined by using only local resources, i.e., the open system's reduced density operator $\ensuremath{\varrho}$ and its time derivatives. The simplest construction, then, would be a functional $U(\ensuremath{\varrho},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varrho}})$. We adopt the minimalist implementation of a bipartite quantum universe, namely, two qubits in a pure joint state, and show that the functional relationship cannot be that simple if it is to generally recover the well-established internal energy of the universe. No further hypothesis or approximation scheme was assumed. An illustrative counterexample is explored, and possible implications of the general constraint are discussed.

DOI
10.1103/physreva.108.042209

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Recent advances in quantum thermodynamics have been focusing on ever more elementary systems of interest, approaching the limit of a single qubit, with correlations, strong coupling, and far-from-equilibrium environments coming into play. Under such scenarios, it is clear that fundamental physical quantities must be revisited. This article questions whether a universal definition of internal energy for open quantum systems can be devised, setting limits on its possible properties. We argue that, for such a definition to be regarded as local, it should be determined by using only local resources, i.e., the open system's reduced density operator \ensuremathϱ\ensuremath{\varrho}\ensuremathϱ and its time derivatives. The simplest construction, then, would be a functional U(\ensuremathϱ,\ensuremathϱ\ifmmode˙\else˙\fi)U(\ensuremath{\varrho},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varrho}})U(\ensuremathϱ,\ensuremathϱ\ifmmode˙\else˙\fi​). We adopt the minimalist implementation of a bipartite quantum universe, namely, two qubits in a pure joint state, and show that the functional relationship cannot be that simple if it is to generally recover the well-established internal energy of the universe. No further hypothesis or approximation scheme was assumed. An illustrative counterexample is explored, and possible implications of the general constraint are discussed.
The quest to develop a general framework for thermodynamics, suitable for the regime of strong coupling and correlations between subsystems of an autonomous quantum "universe," has entailed diverging definitions for basic quantities, including internal energy. While most approaches focus solely on the system of interest, we propose that a universal notion of internal energy should also account for the environment in order to keep consistency with the closed-system energy of the universe. We introduce an abstract framework to describe all effective Hamiltonian-based approaches and address a rigorous definition of energy additivity in this context, in both a weak and a strong form, discussing the underlying subtleties. As an illustration, we study a particular two-qubit universe model, obtaining the exact master equations for both parties and calculating their effective Hamiltonians and internal energies as given by the recently devised minimal dissipation approach. In this case, we show that internal energies are neither additive nor conservative, which leads to unphysical features.
The development of a self-consistent thermodynamic theory of quantum systems is of fundamental importance for modern physics. Still, despite its essential role in quantum science and technology, there is no unifying formalism for characterizing the thermodynamics within general autonomous quantum systems, and many fundamental open questions remain unanswered. Along these lines, most current efforts and approaches restrict the analysis to particular scenarios of approximative descriptions and semi-classical regimes. Here, we propose a novel approach to describe the thermodynamics of arbitrary bipartite autonomous quantum systems based on the well-known Schmidt decomposition. This formalism provides a simple, exact, and symmetrical framework for expressing the energetics between interacting systems, including scenarios beyond the standard description regimes, such as strong coupling. We show that this procedure allows straightforward identification of local effective operators suitable for characterizing the physical local internal energies. We also demonstrate that these quantities naturally satisfy the usual thermodynamic notion of energy additivity.
Abstract Understanding how to assign internal energy, heat, and work in quantum systems beyond weak coupling remains a central problem in quantum thermodynamics, particularly as the difference between competing definitions becomes increasingly relevant. We identify two common sets of definitions for first-law quantities that are used to describe the thermodynamics of quantum systems coupled to thermal environments. Both are conceptually non-symmetric, treating one part of the bipartition (the ‘system’) differently from the other (the ‘bath’). We analyze these in a setting where such roles are not easily assigned—two large (but finite) sets of thermal harmonic oscillators interacting with each other. We further compare them with a third set of definitions based on a local, conceptually symmetric open-system approach (‘minimal dissipation’) and discuss their quantitative and structural differences. In particular, we observe that all three sets of definitions differ substantially even when the two subsystems are weakly coupled and far detuned, and that the minimal dissipation approach features distinct work peaks that increase with the coupling strength.
Abstract We compare definitions of the internal energy of an open quantum system and strategies to split the internal energy into work and heat contributions as given by four different approaches from the autonomous system framework. Our discussion focuses on methods that allow for arbitrary environments (not just heat baths) and driving by a quantum mechanical system. As a simple application we consider an atom as the system of interest and an oscillator field mode as the environment. Three different types of coupling are analyzed. We discuss ambiguities in the definitions and highlight differences that appear if one aims at constructing environments that act as pure heat or work reservoirs. Further, we identify different sources of work (e.g. coherence, correlations, or frequency offset), depending on the underlying framework. Finally, we give arguments to favour the approach based on minimal dissipation.
In a series of papers, we intend to take the perspective of open quantum systems and examine from their nonequilibrium dynamics the conditions when the physical quantities, their relations, and the laws of thermodynamics become well defined and viable for quantum many-body systems. We first describe how an open-system nonequilibrium dynamics (ONEq) approach is different from the closed combined system + environment in a global thermal state (CGTs) setup. Only after the open system equilibrates will it be amenable to conventional thermodynamics descriptions, thus quantum thermodynamics (QTD) comes at the end rather than assumed in the beginning. The linkage between the two comes from the reduced density matrix of ONEq in that stage having the same form as that of the system in the CGTs. We see the open-system approach having the advantage of dealing with nonequilibrium processes as many experiments in the near future will call for. Because it spells out the conditions of QTD's existence, it can also aid us in addressing the basic issues in quantum thermodynamics from first principles in a systematic way. We then study one broad class of open quantum systems where the full nonequilibrium dynamics can be solved exactly, that of the quantum Brownian motion of N strongly coupled harmonic oscillators, interacting strongly with a scalar-field environment. In this paper, we focus on the internal energy, heat capacity, and the third law. We show for this class of physical models, amongst other findings, the extensive property of the internal energy, the positivity of the heat capacity, and the validity of the third law from the perspective of the behavior of the heat capacity toward zero temperature. These conclusions obtained from exact solutions and quantitative analysis clearly disprove claims of negative specific heat in such systems and dispel allegations that in such systems the validity of the third law of thermodynamics relies on quantum entanglement. They are conceptually and factually unrelated issues. Entropy and entanglement will be the main theme of our second paper on this subject matter.
We study the nonequilibrium thermodynamics of pure decoherence processes in open quantum systems coupled to a thermal reservoir. We review various definitions of central quantities, such as internal energy, work, heat, and entropy production, developed within local and global approaches to quantum thermodynamics. Within local approaches, thermodynamic quantities only refer to the open system's degrees of freedom, while in the global approaches, certain quantities are defined by referring explicitly to the reservoir degrees of freedom. Employing a microscopic, analytically solvable model, we perform a comparison of these two perspectives, revealing substantial differences in the thermodynamic quantities and in the formulations of the first and second laws. The main reason for these discrepancies is the fact that the global approaches involve the system-reservoir interaction which exchanges a large amount of energy with the environment, while the average open system energy is constant in time because the dynamics represents pure decoherence and does not affect the open system populations.