Thermodynamics and statistic physics developed as one of the most fruitful physical concepts. Seen from the scientific point of view their range of application extends over far areas, not only in physics. The progressive nano-technology makes it necessary, the applicability for that already admitted concepts from classical thermodynamics on the microscopic level to examine. Questions about the extensivity of the energy and entropy, as well as about the existence of a temperature are just as important as the investigation of thermodynamic cyclic processes for quantum systems. With the help of the theory of open quantum systems stationary nonequilibrium states are simulated. If the quantum system under consideration is not driven heat transport can be examined. The question about local characteristics of individual subsystems is inherent. Because only if the local change of energy of a subsystem is known, a local heat or energy current can be determined. The interaction between the subsystems is for the characterisation of the energy transfer of crucial importance. A quantum-mechanical analysis shows, that by the choice of the measuring basis in an experiment specifies variables such as work and heat. For driven quantum systems in nonequilibrium it shows up that from classical thermodynamics admitted cyclic processes are realizable. The simplest quantum system, a spin can undergo, e.g., an Otto-cycle if suitable embedded and operates thus as heat pump or heat engine. The work mode depends on the local energy splitting as well as on the temperature gradient.
Thermodynamics and statistic physics developed as one of the most fruitful physical concepts. Seen from the scientific point of view their range of application extends over far areas, not only in physics. The progressive nano-technology makes it necessary, the applicability for that already admitted concepts from classical thermodynamics on the microscopic level to examine. Questions about the extensivity of the energy and entropy, as well as about the existence of a temperature are just as important as the investigation of thermodynamic cyclic processes for quantum systems. With the help of the theory of open quantum systems stationary nonequilibrium states are simulated. If the quantum system under consideration is not driven heat transport can be examined. The question about local characteristics of individual subsystems is inherent. Because only if the local change of energy of a subsystem is known, a local heat or energy current can be determined. The interaction between the subsystems is for the characterisation of the energy transfer of crucial importance. A quantum-mechanical analysis shows, that by the choice of the measuring basis in an experiment specifies variables such as work and heat. For driven quantum systems in nonequilibrium it shows up that from classical thermodynamics admitted cyclic processes are realizable. The simplest quantum system, a spin can undergo, e.g., an Otto-cycle if suitable embedded and operates thus as heat pump or heat engine. The work mode depends on the local energy splitting as well as on the temperature gradient.
Thermodynamics is a theory of impressive success and a wide range of applicability. Nevertheless, it took about two hundred years after the basic formulation of phenomenological thermodynamics until Boltzmann and Maxwell gave a foundation in terms statistical mechanics based on classical mechanics. The invention of quantum mechanics has triggered various attempts to establish a theory of quantum thermodynamics, i.e., to explain and derive thermodynamics by quantum mechanics only. Recently, a new approach to this question has been developed by Gemmer et al. focusing on the role of the partitioning of the universe into "system" and "environment" and on how the entanglement between those parts and the properties of typical environments lead to a thermal state in the system for almost any instant in time. Within this approach, the emergence of thermodynamic behavior in subsystems of a wide class of autonomous quantum networks could be established although the state of the total system is a pure. This turns our understanding of the emergence of thermodynamics upside down: No longer the presence of ideal (infinitely large and always in equilibrium) makes the system thermodynamic, but thermodynamic behavior of the system itself induced by an appropriate embedding - which by itself does not need to be thermodynamic in any way - becomes the crucial ingredient. Moreover, it was found that not only macroscopic embeddings but also quantum networks as small as 10 spins may serve already as excellent thermal embeddings. In this context, this thesis approaches a number of questions: First, complementary to the emergence of relaxation (heat), how does mechanical control over a system (work) emerge in autonomous quantum systems? What types of embeddings allow for this control, and is there a lower limit of their size? The answer is discussed by making the connection between work, parametric control of a Hamiltonian, and classical driving with the help of the factorization approximation. Using this link, we propose the local effective measurement basis method to determine heat and work currents in arbitrary bipartite quantum systems, and discuss measures of how to assess the thermodynamic functionality of an arbitrary embedding. We then apply these concepts to a minimal model consisting of only a spin and a quantum harmonic oscillator (spin-oscillator model) to demonstrate that even a single oscillator may act as an ideal work reservoir. A slight variation of the model then illustrates the pros and cons of the proposed measures of thermodynamic functionality. Second, since thermodynamics is first and foremost a theory of processes, which need both, thermal and mechanical control, we ask if autonomous quantum networks can implement thermodynamic cycles. We do so by first reviewing some specialties of quantum thermodynamic processes in general and then presenting the autonomous dynamic three spin machine, an autonomous quantum network implementing a thermodynamic cycle, which is driven by a emergent quantum work reservoir as discussed in the first part. Finally, we discuss quantum thermodynamic pseudomachines, a class of models that exhibit machine-like functionality without use of a thermodynamic cycle. This gives an answer to the question how new forms of control that are only found in quantum systems lead to new thermodynamic functionalities. The presented models can all be understood as thermodynamic laser models. In particular, we discuss the extended dissipative Jaynes-Cummings model and its thermodynamic properties. We resolve a conflict between different interpretations of the model by a careful analysis of the model itself and the thermodynamic concepts used by different authors. It is found that the model is an intricate heat transport model and no machine and that its thermodynamic functionality relies on the transition-selective coupling of heat baths to a few-level quantum system.
Questions about the route from a nonequilibrium initial state to the final global equilibrium have played an important role since the early days of phenomenological thermodynamics and statistical mechanics. Nowadays, their implications reach from central technical devices of the contemporary human society, like heat engines, refrigerators and computers to recent physics at almost all length scales, from Bose-Einstein-condensation and superconductors to black holes. This work addresses the foundation of macroscopic laws concerning the decay to equilibrium, e.g. the celebrated Fourier's Law, on microscopic Schrödingerian quantum dynamics. Here, a proper treatment requires the usage of modern methods in theoretical physics such as the Theory of Open Quantum Systems, the Kubo Formula in Liouville Space and the novel Hilbert Space Average Method. It turns out that both the relaxation to equilibrium as well as the transport of heat is mainly determined by quantum effects comparable to the role of entanglement in considerations of the global equilibrium within Quantum Thermodynamics. Finally, the foundation of phenomenological thermodynamics on a microscopic theory will hopefully improve our understanding of those most impressive and far-reaching theories and their background and will possibly open the way to overcoming their nanoscopic limits.
In this thesis focus is laid on fluctuation theorems in autonomous modular quantum systems. Moreover, non-equilibrium cyclic processes using a quantum system as working agent are investigated. The first part is concerned with possible pitfalls in adopting idealized model systems from quasistatic thermodynamics for processes driving the system far away from equilibrium. Investigations of a one-dimensional quantum analogue of a gas confined in a cylinder show that the idealization of the cylinder as being absolutely impenetrable leads to questionable results when the gas is in a highly non-equilibrium state during an enforced process. Only when the quasistatic case is recovered, this idealization, well proven in equilibrium thermodynamics, regains its validity. For the investigations concerning fluctuation theorems this thesis focuses on the Jarzynski relation which relates the work performed during a non-equilibrium process with the free energy difference of two equilibrium states. In order to formulate the Jarzynski relation for a specific local part of a closed modular quantum system, the energy transfer between different subparts has to be divided into work and heat, respectively. This separation is performed using LEMBAS, a method to distinguish between heat and work for quantum systems in arbitrary states. In the second part functional environments for a system of interest are identified by using LEMBAS. These functional environments are classified as work or heat sources, respectively. Using these work and heat sources in the third part, different estimation methods for the local effective free energy for the system of interest are studied. This way it is shown that it is possible to generalize the Jarzynski relation to local effective dynamics, giving good estimates for the free energy difference if the correct estimation method is used. In this context, LEMBAS provides a feasible separation of heat and work. The last part of this thesis is concerned with a quantum generalization of the classical Otto cycle. The working agent is to be taken as a quantum system whereas the heat baths are assumed to be classical. For this setting the efficiency of cyclic processes is shown to be limited by the corresponding classical efficiency.
Accurately describing work extraction from a quantum system is a central objective for the extension of thermodynamics to individual quantum systems. The concepts of work and heat are surprisingly subtle when generalizations are made to arbitrary quantum states. We formulate an operational thermodynamics suitable for application to an open quantum systems undergoing a general quantum evolution. We derive the first law of ther- modynamics for a process described by a completely-positive and trace-preserving map and show consistency with the Hatano-Sasa statement of the second law. We show that heat, from the first law, is positive when the input state of the map majorises the output state. Moreover, the change in entropy is also positive for the same majorisation condition. This makes a strong connection between the two operational laws of thermodynamics. Introduction.— The laws of thermodynamics were forged in the furnaces of the industrial revolution, as engineers and scientists refined their picture of energy, studying heat and its interconversion to mechanical work with a view to powering the mines and factories of this new era of human endeavor. Followed by the development of statistical mechanics at the change of the centuries (1), far from its pragmatic inception, thermodynamics is now a theory with a remarkable range of applicability, successfully describing the properties of macro- scopic systems ranging from refrigerators to black holes (2). Moving on to the 21st century with both the industrial and electronic revolutions behind us, we are currently pushing technology towards and beyond the microscopic scale. With a view to devices operating at a scale where quantum mech- anical laws become important we may ask whether the solid and some combination of the two (8). Finally, central to the work presented here is a work extraction formalism for non-passivity of quantum states (9). Despite the range of ap- proaches a more general picture for the thermodynamics of general quantum evolutions is far from clear. In this Letter, we take an operational approach to character- izing the energy change of an open quantum process described by a completely-positive trace-preserving (CPTP) map. In analogy to the first law of thermodynamics we discuss work done, extractable work, and heat. The concepts of ergotropy and adiabatic work allow us to state our main result: An op- erational first law for general quantum processes. We show that our operational first law is in agreement with widely used Hatano-Sasa version of the second law for CPTP maps (10, 11) by explicitly stating the Clausius inequality for unital and thermal maps. We then show that both operational heat and the change in von Neumann entropy are positive when the input state of the map majorises the output state. Thermodynamics of quantum systems.— The first law of thermodynamics states that the internal energy change in a thermodynamic process can be split up into two contributions - work and heat: dE = Q + W. For a generic quantum system, the internal energy at time t is E(t) = tr( (t)H(t)), implying that the change in the internal energy dE depends only on the end points. Heat and work on the other hand are path-dependent—thus the different notation for the 'differen- tials'. As an illustration we may consider the heat expended when pushing a piston into a cylinder filled with gas: It not only depends on the initial and final positions of the piston but also on how fast it is pushed. Using the derivative of the internal energy with respect to time the following two expres- sions are motivated (10):
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.
Quantum thermodynamics (QT) is an emerging field of research that aims to investigate how the laws of thermodynamics and quantum mechanics merge together in small quantum systems.With advances at the necessary technology to control and measure those small physical systems this field has acquired even more importance, not only in the sense that it can be tested, which is a good thing for basic research, but that new applications could be implemented at these scales, so a better comprehension of the limitations imposed by quantum thermodynamics turns out to be of crucial importance for these goals, which forces theoreticians to produce experimentally relevant versions of these new concepts.Another important aspect present at those systems is that part of them work in a regime where its constituents are described by non-thermal states, and in particular non-thermal steady states, which brings to light a different thermodynamic description, usually called steady-state thermodynamics, therefore one of the goals that we are willing to achieve with this thesis is to give an introduction to QT of systems out-of-equilibrium.One of the related subtopics that physicists deal with in QT and the one that we will be focusing on this work are the use of non-thermal stationary states to build heat engines in the quantum domain, and the analyses of the features that this new regime could possibly allow, like the use of quantum resources as a way to overcome classical limitations imposed on its performance, like to attain efficiencies higher than Carnot's or operate in certain regimes unattainable using only classical resources.Therefore, in order to clarify the underlying physics of those systems in a non-thermal regime, any experimentally well suited content is more than welcome.So keeping that in mind we devised an experimentally relevant thermodynamic cycle for a transmon qubit WS interacting with a non-thermal environment composed by two subsystems, an externally excited cavity and a classical heat bath with temperature T. The WS undergoes a non-conventional cycle (different from Otto, Carnot, etc.) through a succession of non-thermal stationary states obtained by slowly varying its bare frequency and the amplitude of the field applied on the cavity.The efficiency of this engine obtains a maximum value up to 47% in the regime of operation used.We also wanted to look for the role played by the different types of coherences, present at the WS, on the behavior of the engine and its efficiency.By different types of coherence we mean the so called modes of coherence, whose definition is based on how they respond to symmetry transformations.We did that for the trivial case of the qubit, that only contains the modes 1 and -1, and that has shown to be extremely important for the efficiency of the machine.The same procedure was repeated for a 3-level system WS.The modes 1 and -1 was again very important, not only to the absolute value of the engine's efficiency but to the regime of operation of the machine.The additional modes, 2 and -2, had a negative impact on the efficiency, reducing its absolute value.This result appears to show some evidences that quantumness won't necessarily bring improvements to the operation of those machines.