In bipartite quantum systems commutation relations between the Hamiltonian of each subsystem and the interaction impose fundamental constraints on the dynamics of each partition. Here we investigate work, heat and entropy production in bipartite systems characterized by particular commutators between their local Hamiltonians and the interaction operator. We consider the formalism of (Weimer et al 2008 Europhys. Lett. 83 30008 ), in which heat (work) is identified with energy changes that (do not) alter the local von Neumann entropy, as observed in an effective local measurement basis. We demonstrate the consequences of the commutation relations on the work and heat fluxes into each partition, and extend the formalism to open quantum systems where one, or both, partitions are subject to a Markovian thermal bath. We also discuss the relation between heat and entropy in bipartite quantum systems out of thermal equilibrium, and reconcile the aforementioned approach with the second law of thermodynamics.
In bipartite quantum systems commutation relations between the Hamiltonian of each subsystem and the interaction impose fundamental constraints on the dynamics of each partition. Here we investigate work, heat and entropy production in bipartite systems characterized by particular commutators between their local Hamiltonians and the interaction operator. We consider the formalism of (Weimer et al 2008 Europhys. Lett. 83 30008 ), in which heat (work) is identified with energy changes that (do not) alter the local von Neumann entropy, as observed in an effective local measurement basis. We demonstrate the consequences of the commutation relations on the work and heat fluxes into each partition, and extend the formalism to open quantum systems where one, or both, partitions are subject to a Markovian thermal bath. We also discuss the relation between heat and entropy in bipartite quantum systems out of thermal equilibrium, and reconcile the aforementioned approach with the second law of thermodynamics.
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.
The question of with what we associate work and heat in a quantum thermodynamic process has been extensively discussed, mostly for systems with time-dependent Hamiltonians. In this paper, we aim to investigate the energy exchanged between two quantum systems through interaction where the Hamiltonian of the system is time-independent. An entropy-based re-definitions of heat and work are presented for these quantum thermodynamic systems therefore an entropy-based formalism of both the first and the second laws of thermodynamics are introduced. We will use the genuine reasoning based on which Clausius originally defined work and heat. The change in the energy which is accompanied by a change in the entropy is identified as heat, while any change in the energy which does not lead to a change in the entropy is known as work. It will be seen that quantum coherence does not allow all the energy exchanged between two quantum systems to be only of the heat form. Several examples will also be discussed. Finally, we will examine irreversibility from our entropy-based formalism of quantum thermodynamics.
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.
According to the first and second laws of thermodynamics and the definitions of work and heat, microscopic expressions for the non-equilibrium entropy production have been achieved. Recently, a redefinition of heat has been presented in [Nature Communications volume 8, Article number: 2180 (2017)]. Since thermal operations play an important role in the resource theory of thermodynamics, it would be very interesting to find out the effect of the above-mentioned definition on the expression of the entropy production for these kind of operations. This is one of the aims of the present paper. Using the new definition of heat, it is shown that the entropy production is the same as the mutual information between a system and a bath both for thermal operations and, if the system-bath initial state is factorized, for entropy-preserving operations. It is also discussed that how one can recognize the type of the correlation between a system and a bath through knowledge of the initial state of the system only. It is shown that if the initial state of a system is diagonal in the energy basis, the thermal operations cannot create a quantum correlation between the system and the bath, however, if the system initial state is coherent Gibbs state, there cannot be classical correlation due to the thermal operations.
Thermodynamics is a phenomenological theory describing the energy conversion of work and heat. At its origins thermodynamics was developed in order to understand and improve heat engines. In conventional thermodynamics, however, only such processes are completely describable which are slow enough to keep the system of interest in an equilibrium state with its thermal environment at all times. On the contrary, all real physical processes are accompanied by non-equilibrium phenomena. These are mathematically described with the help of the irreversible entropy production, i.e. the dissipated work divided by the temperature of the bath. The second law of thermodynamics merely states that the irreversible entropy production is on average always positive. The modern trend of miniaturization, however, leads to smaller and smaller devices. On short length scales thermal noise as well as quantum fluctuations become important. Thus, usual thermodynamic quantities as work and heat acquire stochastic nature. Moreover, in the quantum regime a completely new theory had to be invented, since classical notions of work and heat are no longer valid. The present dissertation contributes to this prevailing field by the derivation of analytical expressions for the entropy production in open and closed quantum system far from thermal equilibrium. The theoretical treatment is motivated by an experimental point of view. The obtained results were derived from the reduced dynamics of the quantum mechanical system, which may be coupled to a thermal bath. To this end, it was dealt with methods and approaches of statistical physics, conventional thermodynamics, quantum information theory and the theory of open quantum systems. In this dissertation the most important physical systems are treated. First, an isolated system is considered before later on a heat bath is included. The limit of weak as well as of strong coupling are analyzed in detail. For isolated systems it turns out that a geometric approach is able to capture the thermodynamic properties, completely. To this end, the irreversible entropy production is sharply estimated from below by the angle between the real non-equilibrium state and the corresponding equilibrium one. Further, it is shown that the Heisenberg uncertainty relation for energy and time can be formulated more precisely in terms of that angle. As experimental system cold ion traps are introduced, in which the theoretical predictions can be verified. For weakly coupled systems, then, the irreversible entropy production is identified and a fluctuation theorem is derived by merely thermodynamic arguments. As experimental system we, again, propose the cold ion traps, where the heat bath can be simulated with the help of LASER light. In the limit of strong coupling a semiclassical description of the reduced quantum system is discussed. Again, an analytically exact expression for the entropy production and the corresponding fluctuation theorem is derived. A physical system, in which strong coupling to the environment can be analyzed, is given by Josephson junctions. This dissertation discusses the entropy production for isolated, weakly coupled, and overdamped quantum systems. For these physical regimes analytical relation are derived and experimental systems proposed, with which the predictions can be verified.
We provide a characterization of energy in the form of exchanged heat and work between two interacting constituents of a closed, bipartite, correlated quantum system. By defining a binding energy we derive a consistent quantum formulation of the first law of thermodynamics, in which the role of correlations becomes evident, and this formulation reduces to the standard classical picture in relevant systems. We next discuss the emergence of the second law of thermodynamics under certain-but fairly general-conditions such as the Markovian assumption. We illustrate the role of correlations and interactions in thermodynamics through two examples.