We derive a generalization of the Ehrenfest theorem valid for open quantum systems. From this result, we identify three contributions to the evolution of expected values: (i) the explicit time dependence of the observable, (ii) the incompatibility between the observable and an operator which plays the role of an effective Hamiltonian, and (iii) entropy changes. Considering the local Hamiltonian as the observable, and adopting a specific interpretation of the nature of thermal interactions, we obtain an alternative version of the first law of thermodynamics. Within this framework, we show that in some cases the power performed by the system can be considered as a quantum observable. As an application, the pure-dephasing process is reinterpreted from this perspective.
We derive a generalization of the Ehrenfest theorem valid for open quantum systems. From this result, we identify three contributions to the evolution of expected values: (i) the explicit time dependence of the observable, (ii) the incompatibility between the observable and an operator which plays the role of an effective Hamiltonian, and (iii) entropy changes. Considering the local Hamiltonian as the observable, and adopting a specific interpretation of the nature of thermal interactions, we obtain an alternative version of the first law of thermodynamics. Within this framework, we show that in some cases the power performed by the system can be considered as a quantum observable. As an application, the pure-dephasing process is reinterpreted from this perspective.
By computing the local energy expectation values with respect to some local measurement basis we show that for any quantum system there are two fundamentally different contributions: changes in energy that do not alter the local von Neumann entropy and changes that do. We identify the former as work and the latter as heat. Since our derivation makes no assumptions on the system Hamiltonian or its state, the result is valid even for states arbitrarily far from equilibrium. Examples are discussed ranging from the classical limit to purely quantum-mechanical scenarios, i.e. where the Hamiltonian and the density operator do not commute.
We study the semiclassical Ehrenfest trajectories in open quantum systems. We first give a self-contained derivation of the Fokker--Planck equation that governs the time evolution of the mixing measure for a Gaussian mixture. Then, we embed the generalized Ehrenfest theorem recently obtained for open quantum systems into this phase-space picture to study the time evolution of the expectations of observables with respect to the Gaussian mixture. We show how the coherent and irreversible contributions are separated in the frictionless harmonic regime relevant to the semiclassical approximation. Our work provides a transparent phase-space interpretation of the emergence of classical trajectories in open quantum dynamics.
The second law of thermodynamics states that the entropy of an isolated system can only increase over time. This appears to conflict with the reversible evolution of isolated quantum systems under the Schrödinger equation, which preserves the von Neumann entropy. Nonetheless, one finds that with respect to many observables, expectation values approach a fixed value—their equilibrium value. This ultimately raises the question: ? For classical systems, one introduces the assumption of a low-entropy initial state along with the concept of ignorance about the microscopic details of the physical system, leading to a statistical interpretation of the second law. By considering the observables through which we examine quantum systems, both these assumptions can be incorporated, building upon recent studies of the of observables. While the statistical behavior of observable expectation values is well established, a quantitative connection to entropy increase has been lacking so far. In deriving novel bounds for the equilibration of observables, and considering the entropy of the system relative to observables, we recover a variant of the second law: the entropy with respect to a given observable tends toward its equilibrium value in the course of the system’s unitary evolution. These results also support recent findings that question the necessity of nonintegrability for equilibration in quantum systems. We further illustrate our bounds using numerical results from the paradigmatic example of a quantum Ising model on a chain of spins. There, we observe entropy increasing up to equilibrium values, as well as fluctuations, which expose the underlying reversible evolution in accordance with the derived bounds.
Given the evolution of an arbitrary open quantum system, we formulate a general and unambiguous method to separate the internal energy change of the system into an entropy-related contribution and a part causing no entropy change, identified as heat and work, respectively. We also demonstrate that heat and work admit geometric and dynamical descriptions by developing a universal dynamical equation for the given trajectory of the system. The dissipative and coherent parts of this equation contribute exclusively to heat and work, where the specific role of a work contribution from a counterdiabatic drive is underlined. Next we define an expression for the irreversible entropy production of the system which does not have explicit dependence on the properties of the ambient environment; rather, it depends on a set of the system's observables excluding its Hamiltonian and is independent of internal energy change. We illustrate our results with three examples.
A consistent theory of non-equilibrium thermodynamics for Markovian open quantum systems has been developed in the late seventies in analogy with Classical Irreversible Thermodynamics. The time-evolution of these open systems is usually described by means of effective master equations in Lindblad form, that turn out to be reliable when there is a separation of time-scales between system and environment, such that memory effects are negligible and the so-called Markovian approximation is justified. In this framework, the variations of energy and entropy in the system are consistently described, distinguishing between heat and work contributions and providing a statement of the second law of thermodynamics as positivity of the entropy production.\nHowever, there is empirical evidence that many physical systems like photosynthetic complexes,\nopto-mechanical resonators and superconducting qubits, just to mention a few, experience more general non-Markovian dynamics. The formulation of the laws of thermodynamics in a non-Markovian setting is matter of current research and represents the main topic of the present work. As a first step, we show that the entropy production defined as in the Markovian case can be negative for a class of non-Markovian dynamics. We argue that this outcome should not be interpreted as a violation of the second law of thermodynamics because the environment must be explicitly taken into account in the balance of entropy in the non-Markovian setting. In order to justify this claim we adopt a more general point of view, studying a closed bipartite quantum system, such that each of the two subsystems plays the role of a finite environment out of equilibrium for the other one. We concentrate on the balance of energy first and construct an effective Hamiltonian for each subsystem using physically reasonable requirements; then we define heat and work as in the standard Markovian treatment, with the effective Hamiltonian replacing the free Hamiltonian. It turns out that, in our framework, the work power is perfectly balanced between subsystems, while the correlations can store a part of energy locally inaccessible and exchange it with both subsystems in the form of heat.\nConcerning the balance of entropy, a quite general formulation of the second law of thermodynamics can be given as follows: under the assumption of a factorized initial state for the compound system, the sum of the total variations of the entropies in the two subsystems is always nonnegative. We show with an explicit example that this general formulation does not correspond to the statement presented in the framework of Markovian master equations, which should not be considered a priori the second law of thermodynamics.\nIn the last part of the thesis we concentrate of the so-called fluctuation relations, that are results extending the thermodynamic formalism beyond the behavior of average quantities. After reviewing the main theoretical outcomes, such as the Jarzynski equality and the Crooks fluctuation theorem, we describe a proposal to access experimentally the work performed on an ensemble of diatomic molecules by a time-dependent electric field coupled with their vibrational degree of freedom. This procedure could then be used to test the quantum Jarzynski equality. With respect to the results so far appeared in the literature, in which the left-hand side of the equality is inferred from an experiment and the right-hand side is computed according to a model, in our proposed setting we should be able to estimate from the experiment both the left-hand side and the right-hand side of the equality, independently.
Deriving the laws of thermodynamics from an underlying microscopic picture is a central quest of statistical mechanics. The present article focuses on the derivation of the first and second law of thermodynamics for closed and open quantum systems, where such foundational questions in addition become practically relevant for emergent nanotechnologies. Recent progress is presented in a self-contained way, with an emphasis on general ideas and useful tools instead of particular applications. We identify limits and conceptual shortcomings of previous approaches. Then, we propose a novel, unifying perspective, which starts from a microscopic definition of nonequilibrium thermodynamic entropy. The change of this entropy is identified with the entropy production, which satisfies a fluctuation theorem for a large class of initial states. Moreover, this entropy production can be naturally separated into a quantum and a classical component. Within our framework, we introduce the notions of recoverable work and remaining heat. In the case of a weakly perturbed ideal thermal bath this approach is in quantitative agreement with previous ones, thus ensuring their thermodynamic consistency.