In this thesis we develop an approach to nonequilibrium quantum-statistical mechanics which is based on the consideration of the thermodynamic entropy as being a quantum observable with an associated hermitian operator and with its own equation of motion, from which the rate of entropy production can be studied. The relationship of this quantum observable---the expectation value of which is proved to obey the laws of thermodynamics and is thus called the thermodynamic entropy---with heat dissipation in quantum many-body systems is investigated in detail. After showing how the classical theory of nonequilibrium thermodynamics is obtained from our entropy-production formalism in the limit of very weakly coupled subsystems of a larger isolated quantum system, the solution of the equation of motion for the thermodynamic entropy operator is formally obtained and applied to the case of electrons interacting with phonons and being driven by an external electric field, arriving at an explicit expression for the Joule heat without any a priori consideration of the rate of change of the energy of the system. Finally, the formalism is applied to solve a puzzle introduced with the most basic model for atom-light interactions: the Jaynes-Cummings model. Without using the correct thermodynamic arguments implied by our entropy-production theory, this model leads to the conclusion that a stream of two-level atoms sent to a cavity filled with monochromatic photons---in the so called one-atom maser configuration---thermalize, under off-resonant conditions, to a temperature different from that of the photons, casting doubts on the validity of the principle of conservation of energy.
In this thesis we develop an approach to nonequilibrium quantum-statistical mechanics which is based on the consideration of the thermodynamic entropy as being a quantum observable with an associated hermitian operator and with its own equation of motion, from which the rate of entropy production can be studied. The relationship of this quantum observable---the expectation value of which is proved to obey the laws of thermodynamics and is thus called the thermodynamic entropy---with heat dissipation in quantum many-body systems is investigated in detail. After showing how the classical theory of nonequilibrium thermodynamics is obtained from our entropy-production formalism in the limit of very weakly coupled subsystems of a larger isolated quantum system, the solution of the equation of motion for the thermodynamic entropy operator is formally obtained and applied to the case of electrons interacting with phonons and being driven by an external electric field, arriving at an explicit expression for the Joule heat without any a priori consideration of the rate of change of the energy of the system. Finally, the formalism is applied to solve a puzzle introduced with the most basic model for atom-light interactions: the Jaynes-Cummings model. Without using the correct thermodynamic arguments implied by our entropy-production theory, this model leads to the conclusion that a stream of two-level atoms sent to a cavity filled with monochromatic photons---in the so called one-atom maser configuration---thermalize, under off-resonant conditions, to a temperature different from that of the photons, casting doubts on the validity of the principle of conservation of energy.
We define the entropy operator as the negative of the logarithm of the density matrix, give a prescription for extracting its thermodynamically measurable part, and discuss its dynamics. For an isolated system we derive the first, second, and third laws of thermodynamics. For weakly coupled subsystems of an isolated system, an expression for the long-time limit of the expectation value of the rate of change of the thermodynamically measurable part of the entropy operator is derived and interpreted in terms of entropy production and entropy transport terms. The interpretation is justified by comparison to the known expression for the entropy production in an aged classical Markovian system with Gaussian fluctuations and by a calculation of the current-induced entropy production in a conductor with electron-phonon scattering.
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.
A central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.
To understand in detail the relation between unitary quantum theory that describes our world at the microscopic scale and thermodynamics, which was long believed only to apply to macroscopic objects, is one of the most interesting and long-standing problems in physics. Recently, this problem has received renewed attention, in particular from the community of quantum information theory, but also from the field of statistical mechanics, inspired from stochastic thermodynamics. These results suggest that thermodynamics is also relevant for individual quantum systems, provided that they can be brought into contact with thermal baths. In this thesis, I use recently developed tools to provide new results both on fundamental questions, but also on questions which are of practical relevance for potential miniaturized thermal machines. In terms of fundamental questions, the results in this thesis contribute to understanding how the basic laws of thermodynamics and statistical mechanics can be understood directly from unitary quantum mechanics. In particular, I discuss and answer the following questions: i) How can we quantify the third law of thermodynamics using information theoretic methods? ii) How can we quantify the "thermodynamic value" of a state-transition in quantum systems? iii) How can we axiomatically characterize the non-equilibrium free energy and relative entropy? iv) How can we justify statistical ensembles from an operational perspective, without having to introduce either some probability measures or an information theoretic entropy measure? v) How can we understand the equilibration of closed quantum systems, how long does it take and how difficult is it to avoid? In the second part of the thesis I discuss in detail how experimental restrictions, which become important at the quantum scale, influence the ultimate thermodynamic bounds for thermal machines. In particular, the results in this thesis provide thermodynamic bounds on work-extraction and efficiencies of thermal machines in situations where 1.) an experimenter only has limited field strengths available, 2.) an experimenter cannot control the interactions between particles, but external fields arbitrarily well, and 3.) situations in which a small quantum system can only be strongly coupled to heat baths. These bounds are tight and I provide explicit examples illustrating the different behaviours. Finally, I come back to a classic problem in statistical physics: The emergence of spontaneous symmetry breaking. Here, I provide general and rigorous new results that show how symmetry-breaking stationary states emerge from fluctuations in order parameters in dissipative lattice models.
We consider the temporal evolution of the entropy S of an isolated system from an initial non-equilibrium state 2· of entropy S = S i to the equilibrium state of maximum entropy S = S max > S i. The application of usual density matrix theory of quantum mechanics leads us to the surprising result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, because the entropy of the isolated system could not spontaneously increase from S i to S max by entropy production P > 0. This contradicts the second law of thermodynamics. We avoid this discrepancy by changing from quantum mechanics to quantum thermodynamics. Now the entropy production P = dS/dt and the time t are the relevant observables. In order to treat the temporal development of S and P, we at first prepare an initial state i of entropy production P i at time t < 0. Subsequently at ϑ t ≥ 0 the initial state i interacts with a continuum of final states f, which transfers the entropy production from i to f. The temporal evolution of the entropy production P(t) of our system is treated by means of the time-dependent perturbation theory. For that reason we split the operator of entropy production P = P 0 + W into two parts. The operator P 0 describes the unperturbed basic vectors of entropy production | i > and | f >, while the perturbation operator W determines the interaction between | i > and | f >. We expand the time-dependent state function < t | ψ > of the isolated system into the complete set of stationary eigenvectors | i > and | f > and calculate the time-dependent expansion coefficients c i(t) and c(f, t) by means of the dynamic equation of quantum thermodynamics − i k (d/dt) < t | ψ > = < t | P | ψ >. Here k means the Boltzmann constant, which is characteristic of quantum thermodynamics and represents the atomic entropy unit. The expansion coefficients are given by a system of coupled differential equations, from which we deduce an integral-differential equation for the coefficient c i(t) of the initial state i. Unfortunately it has been impossible to obtain an analytic solution of the complicated equation. However approximations are available. Accordingly, in the vicinity of the equilibrium state, the probability of still finding the initial state i of entropy production P i at time t is given by ω i(t) = | c i(t) |2 = exp (−Lt). The rate constant L ≥ σ depends on the density ρ(P f) of final states f of entropy production P f near by P i, and on the squared modulus of the perturbation matrix elements W if = < i | W | f >, which determines the interaction between | i > and | f >. The entropy production P(t) = P i ω i(t) = P i exp (−ϑLt) of the isolated system decreases exponentially and irreversibly with proceeding time t. Its rate-determining step is the transfer of entropy production from i to f. The integral over the entropy production of the final states f vanishes. Therefore the continuum states f do not contribute to the total entropy production P(t). The temporal evolution of the entropy S(t) can easily be calculated by integration of P(t). In conclusion, the entropy production of a non-equilibrium state is finite P(t) > 0 and only vanishes in the case of the equilibrium state P(∞) = 0. Thus, quantum thermodynamics agrees with the second law of 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.