Rapid progress in the fabrication and manipulation of micro and nanoscale devices has made it necessary to extend the concepts of thermodynamics to small systems and finally to quantum systems. In such systems the extensive thermodynamic quantities, such as entropy, heat, and work, are not described by their average values alone but by distributions. \n \nRemarkably, it has been shown that the stochastic thermodynamic variables often obey fluctuation relations which usually appear in the form of relations between exponential averages. While the two-measurement protocol of thermodynamic variables, especially work, is now well studied in closed quantum systems, there have been conceptual and experimental problems in open quantum systems. \n \nIn this thesis, we theoretically study work statistics and fluctuation relations in open quantum systems. To calculate the work statistics, we use the quantum jump method and direct master equation calculations. We show that work definitions equivalent in closed systems can lead to mutually different results in the case of open quantum systems due to a different order of approximations. Moreover, we show that the fluctuation relations can be extended to nearly adiabatically driven systems by using the adiabatic renormalization procedure. \n \nLast, we focus on how a finite size of the environment affects the dynamics and work statistics of open quantum systems. This is important from the experimental point of view because a finite-size environment allows the detection of heat exchange between the system and the environment. We derive a master equation that takes into account these effects and develop a stochastic model that unravels the master equation. We explicitly show that the finite size of the environment influences the dynamics and the work statistics. We also show that the common fluctuation relations are still valid.
Rapid progress in the fabrication and manipulation of micro and nanoscale devices has made it necessary to extend the concepts of thermodynamics to small systems and finally to quantum systems. In such systems the extensive thermodynamic quantities, such as entropy, heat, and work, are not described by their average values alone but by distributions. \n \nRemarkably, it has been shown that the stochastic thermodynamic variables often obey fluctuation relations which usually appear in the form of relations between exponential averages. While the two-measurement protocol of thermodynamic variables, especially work, is now well studied in closed quantum systems, there have been conceptual and experimental problems in open quantum systems. \n \nIn this thesis, we theoretically study work statistics and fluctuation relations in open quantum systems. To calculate the work statistics, we use the quantum jump method and direct master equation calculations. We show that work definitions equivalent in closed systems can lead to mutually different results in the case of open quantum systems due to a different order of approximations. Moreover, we show that the fluctuation relations can be extended to nearly adiabatically driven systems by using the adiabatic renormalization procedure. \n \nLast, we focus on how a finite size of the environment affects the dynamics and work statistics of open quantum systems. This is important from the experimental point of view because a finite-size environment allows the detection of heat exchange between the system and the environment. We derive a master equation that takes into account these effects and develop a stochastic model that unravels the master equation. We explicitly show that the finite size of the environment influences the dynamics and the work statistics. We also show that the common fluctuation relations are still valid.
Technological advances over the past few decades have made it possible to address quantum systems down to individual quanta of energy. Artificial atoms are now routinely fabricated and manipulated using solid-state technology. Exciting new applications are now within grasp, such as quantum heat engines, which have been theorised to achieve energy efficiencies not possible for their classical counterparts. Energy fluctuations are an inherent part of these devices and a detailed understanding of them is required. However, the proper theoretical framework under which such fluctuations are to be understood is still debated. In this thesis, we study energy fluctuations in open quantum systems at the level of individual runs of an experimental protocol. This work has been carried out in two parts. In the first part, we focus on calorimetric measurements – a promising new experimental protocol to continuously measure energy fluctuations of a qubit interacting with a thermal environment. Applying the protocol to more complex systems is not straightforward. Using a simple two-level approximation, we show that for some experimental parameters this protocol can be extended to a driven harmonic oscillator. Another aspect of the calorimetric protocol is that it can induce non-Markovian effects which affect how work and heat are defined. We quantify the degree of non-Markovianity induced by this measurement protocol, when applied to a qubit, based on a previously developed theoretical model. Importantly, we show that non-Markovianity does not necessarily decrease monotonically with the environment size. The attempt to extend the calorimetric protocol to general systems has led to the second part of this thesis. The calorimetric detection is a particular case of a continuously monitored system. However, other schemes exist that lead to different definitions of work and heat. Reconciling these approaches is not straightforward, as evidenced by the many work definitions present in the literature. The difficulty is centred at the fact that, in the orthodox interpretation of quantum mechanics, energy is not always a well-defined quantity. Here, based on the Bohmian interpretation of quantum mechanics, we put forward a general framework to remove this problem in a clear conceptual manner. In particular, we show that the classical definition of work for closed systems can be straightforwardly generalised to the quantum case. For an open system, we used the Bohmian conditional wave function which provides a stochastic state evolution for the open system. We show that this approach can lead to new insights regarding the role of entanglement in energy fluctuations and has the potential to provide a general and observer-independent framework of energy fluctuations in open quantum systems. The results presented in this thesis give a small contribution to the developing field of quantum thermodynamics.
Measuring the thermodynamic properties of open quantum systems poses a major challenge. A calorimetric detection has been proposed as a feasible experimental scheme to measure work and fluctuation relations in open quantum systems. However, the detection requires a finite size for the environment, which influences the system dynamics. This process cannot be modeled with the standard stochastic approaches. We develop a quantum jump model suitable for systems coupled to a finite-size environment. We use the method to study the common fluctuation relations and prove that they are satisfied.
Thermodynamics typically concerns the physical behaviour of macroscopic systems comprised of many particles. However, recent theoretical progress has extended the theory to both the classical-stochastic and quantum regimes, where systems are comprised of just a small number of particles. In this thesis I investigate a range of situations in which new thermodynamic phenomena emerge due to the reduced size of the systems involved. One central assumption in macroscopic thermodynamics is the weak coupling approximation, which posits that the equilibrium properties of a system are not influenced by the interactions with its surrounding environment. However, for microscopic systems this assumption can break down, and I derive new fluctuation relations that provide a refined form of the second law of thermodynamics in this strong-coupling regime, taking into account corrections stemming from these interactions. In this work I provide a characterisation of stochastic heat and entropy production for small scale classical systems that are defined regardless of the strength of interaction. I then show that these definitions lead to a consistent thermodynamic framework valid beyond the usual weak-coupling regime. The thesis also concerns the effect of interactions on the equilibrium properties of strongly-coupled quantum systems, and I investigate how these interactions can influence the resulting temperature fluctuations in this regime. Using tools from quantum estimation theory, I derive an uncertainty relation between energy and temperature valid at all coupling strengths and system sizes. The relation reveals how quantum energy coherences contribute to statistical fluctuations in the estimated temperature of small- scale systems. Finally, I investigate how quantum fluctuations influence the statistics of work extracted from slowly-driven quantum systems. I prove that, unlike in classical systems, the work dissipated by a general quantum system is no longer proportional to its statistical fluctuations. This result reveals new subtleties involved in designing optimal quantum thermodynamic processes.
Markovian quantum master equations (MQMEs) were established nearly half a century ago. They have often been used in the study of irreversible thermodynamics. However, the previous results were mainly concerned about ensemble averages; the stochastic thermodynamics of these systems went unnoticed for a very long time. This situation remained unchanged until a variety of fluctuation theorems in classical and quantum regimes were found in the past two decades. In this paper, we systematically summarize the current understanding on the stochastic heat and work in MQMEs using two distinct strategies. One strategy is to treat the system and its surrounding heat bath as a closed quantum system, to suppose that the evolution of the composite system is unitary under a time-dependent total Hamiltonian and to define the heat and work as the changes in energy by applying two energy measurements scheme to the composite system. The other strategy is to unravel these MQMEs into random quantum jump trajectories (QJTs) and to define the stochastic heat and work along the individual trajectories. Many important physical concepts, mathematical techniques, and fluctuation theorems at different descriptive levels are given in as detailed a manner as possible. We also use concrete models to illustrate these results.
We extend the quantum jump method to nearly adiabatically driven open quantum systems in a way that allows for an accurate account of the external driving in the system-environment interaction. Using this framework, we construct the corresponding trajectory-dependent work performed on the system and derive the integral fluctuation theorem and the Jarzynski equality for nearly adiabatic driving. We show that such identities hold as long as the stochastic dynamics and work variable are consistently defined. We numerically study the emerging work statistics for a two-level quantum system and find that the conventional diabatic approximation is unable to capture some prominent features arising from driving, such as the continuity of the probability density of work. Our results reveal the necessity of using accurate expressions for the drive-dressed heat exchange in future experiments probing jump time distributions.
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.