In the two preceding chapters of this book we have analyzed many interesting physical phenomena in magnetic and superconducting systems which could adequately be described by phenomenological dynamical equations in terms of collective classical variables. One unavoidable consequence of this approach is that, as we are always dealing with variables that describe only part of the whole system, the interaction with the remaining degrees of freedom shows up through the presence of non-conservative terms which describe the relaxation of those variables to equilibrium. Those phenomenological equations are able to describe a very rich diversity of physical phenomena, in particular, those which can be studied in the context of quantum mechanics. Since these are genuine dynamical equations, there is no reason why they should be restricted to classical physics. However, as we do not yet know how to treat dissipative effects in quantum mechanics, we have deliberately neglected those terms when trying to describe quantum mechanical effects of our collective variables. In this chapter we will describe the general approach to dealing with dissipation in quantum mechanics. However, before we embark on this enterprise we should spend some time learning a little bit about the classical behavior of dissipative systems. In this way we can develop some intuition on how systems evolve during a dissipative process and, hopefully, this will be useful later on when we deal with quantum mechanical systems. The immediate problem we have to face concerns the choice of dissipative system to be studied.
In the two preceding chapters of this book we have analyzed many interesting physical phenomena in magnetic and superconducting systems which could adequately be described by phenomenological dynamical equations in terms of collective classical variables. One unavoidable consequence of this approach is that, as we are always dealing with variables that describe only part of the whole system, the interaction with the remaining degrees of freedom shows up through the presence of non-conservative terms which describe the relaxation of those variables to equilibrium. Those phenomenological equations are able to describe a very rich diversity of physical phenomena, in particular, those which can be studied in the context of quantum mechanics. Since these are genuine dynamical equations, there is no reason why they should be restricted to classical physics. However, as we do not yet know how to treat dissipative effects in quantum mechanics, we have deliberately neglected those terms when trying to describe quantum mechanical effects of our collective variables. In this chapter we will describe the general approach to dealing with dissipation in quantum mechanics. However, before we embark on this enterprise we should spend some time learning a little bit about the classical behavior of dissipative systems. In this way we can develop some intuition on how systems evolve during a dissipative process and, hopefully, this will be useful later on when we deal with quantum mechanical systems. The immediate problem we have to face concerns the choice of dissipative system to be studied.
Having achieved this point we hope to have accomplished, at least partly, our main aim which was to convince the reader that once appropriate systems have been found (or built) they can present a very peculiar combination of microscopic parameters in such a way that quantum mechanics should be applied to general macroscopic variables to describe the collective effects therein. Moreover, the very nature of these macroscopic variables does not allow them to be treated in an isolated fashion. They must rather be considered coupled to uncontrollable microscopic degrees of freedom which is the ultimate origin of dissipative phenomena. The latter, at least in the great majority of cases, play a very deleterious role in the dynamics of the macroscopic variables and we hope to have introduced minimal phenomenological techniques in order to quantify this. We have concentrated our discussions on questions originating from a few examples of superconducting or magnetic systems where quantum mechanics and dissipative effects coexist. In particular, superconducting devices which present the possibility of displaying several different quantum effects (quantum interference, decay by quantum tunneling, or coherent tunneling) are of special importance, as we will see below. Prior to development of the modern cryogenic techniques and/or the ability to build nanometric devices, it was unthinkable to imagine the existence of subtle quantum mechanical effects such as the entanglement of macroscopically distinct quantum states.
Reviewing macroscopic quantum phenomena and quantum dissipation, from the phenomenology of magnetism and superconductivity to the presentation of alternative models for quantum dissipation, this book develops the basic material necessary to understand the quantum dynamics of macroscopic variables. Macroscopic quantum phenomena are presented through several examples in magnetism and superconductivity, developed from general phenomenological approaches to each area. Dissipation naturally plays an important role in these phenomena, and therefore semi-empirical models for quantum dissipation are introduced and applied to the study of a few important quantum mechanical effects. The book also discusses the relevance of macroscopic quantum phenomena to the control of meso- or nanoscopic devices, particularly those with potential applications in quantum computation or quantum information. It is ideal for graduate students and researchers.
In the previous chapter we have seen that departures from equilibrium generate a plethora of new phenomena in statistical physics. While our discussion so far has been limited to classical phenomena, in this chapter we want to ask how quantum mechanics interferes with the conditions of nonequilibrium. Experience shows that, in some cases, nonequilibrium quantum systems can be understood in terms of reasonably straightforward quantization of their classical limits. In others, an interplay of quantum coherence and nonequilibrium driving leads to entirely new phenomena – such as lasing, a paradigm for a steady state nonequilibrium quantum system. At any rate, we have every reason to anticipate that quantum nonequilibrium physics will be as diverse and colourful as its classical counterpart. Even minor departures from equilibrium call for a theoretical description entirely different from the ‘Matsubara formalism’ developed in earlier chapters of this book. The applicability of the latter is rigidly tied to the existence of a quantum grand canonical density operator, the hallmark of a many particle equilibrium system. But how then, can an functional theory of nonequilibrium quantum systems be developed? An obvious idea would be to once more go through the different elements of classical nonequilibrium theory developed in the previous chapter, and subject every one of them to an individual quantization scheme. Luckily, there are more efficient ways to achieve our goal: some decades ago, a many particle formalism suitable to describe nonequilibrium systems under the most general conditions was introduced by Keldysh.
The outline of this work will be as follows: in chapter 1, after a brief overview of the systems studied and a preview of the results obtained, we will introduce the general models and methods we are going to use to tackle the systems considered, hopefully providing enough theory to understand our later extensive treatment, without entering in too much detail in the widely studied general theories of our models. In chapter 2 we will show our first application: the description of a spin-sensitive dissipation channel compatible with the experimental findings of an atomic force microscopy experiment. We will describe in some detail the experiment and the system under study and why some direct approaches are unable to account for the observed effect; we will then specialize the previously described path integral technique to obtain a numerical description of the system, highlighting our proposed dissipation mechanism. In chapter 3 we will consider electron current pumping in a threesite system and how it can be affected by the presence of an environment. We will first obtain and solve a simple equation for the isolated system, then couple a bath to this and specialize the master equation theory to obtain an analytical result for the system in presence of an environment, observing some interesting changes it its behavior. The experimental feasibility of the proposed setup will be explored. Finally, in chapter 4 we will briefly present another model for the description of dissipative systems which has been investigated but, for now, is not complete enough to produce interesting results: inspired from another atomic force microscopy experiment where frictional effects of hydrogen atoms on a surface are observed, we will try to investigate the possibility of inherently quantum effects in a similar system, where a light particle is coupled to other heavier atoms, treated classically. We will propose a model and a technique for its simulation, though this will prove too computationally demanding to be of practical use.
This part of the book is fully devoted to the physics of (1 + 1)-dimensional quantum or twodimensional classical models. As usual, I shall distinguish between classical and quantum systems only if it is necessary. Those models where the difference between quantum and classical is not relevant will be called ‘two-dimensional’; those where for some reason or other I want to emphasize the quantum aspect will be called ‘one-dimensional’. The reason I spend so much time discussing two-dimensional physics is that here one finds a kind of paradise for strong interactions and nonperturbative effects. Those effects which are tricky or even impossible to achieve in higher dimensions appear in almost every two-dimensional model. To our greatest satisfaction there are theoretical tools at our disposal which allow us to solve many of the corresponding problems and describe these phenomena. The availability of these strong mathematical tools is a unique feature of two-dimensional physics. Even if it turns out to be impossible to generalize these tools for higher dimensions, I hope that the reader will be rewarded for the time spent by the pleasure obtained from contemplating their beauty. The world we are about to enter has certain distinct features which are worth mentioning in the introduction. The first one concerns the second-order phase transitions. In (1 + 1)-dimensional systems such transitions may occur only at zero temperature.
Among the few lessons that Physics teaches us on a daily basis, one in particular is hard to miss: thermalization processes are ubiquitous. With the same degree of certainty, one can predict an apple to fall onto the ground if its stalk is cut, or a cup of tea to be found cold if left on the table for too long. In this respect, generic classical and quantum mechanical systems seem to behave in the same way. This is also one of the main difficulties in the experimental observation of the bizarre effects predicted from the quantum theory, as several of these are known to disappear at finite temperatures. It has been known for a long time that thermalization is associated with “chaotic” behavior at the microscopic level, but recently systematic efforts, both theoretical and experimental, have allowed us to understand its mechanisms at an unprecedented level of accuracy. Indeed, it has been realized that thermalization generally occurs also in isolated systems, where the absence of interactions with the environment allows for first principle theoretical investigations1. In this case thermalization takes place locally, as the whole system acts as a thermal bath for its own subsystems. Theoretical research has been motivated and paralleled by exciting experimental progress in cold-atom physics, which has provided robust, nearly ideal realizations of several theoretical models2. As an established piece of knowledge, recent research has now confirmed that two outstanding exceptions actually exist in the quantum realm, where thermalization is prevented to occur. These are many-body localized systems3, where disorder plays a crucial role, and integrable ones, which are protected by the existence of higher conservation laws. By their own nature, these systems display exceptional non-equilibrium features, which are not washed out by the onset of thermalization as relaxation occurs. At the core of the present thesis lie the remarkable properties of integrable systems out of equilibrium, with a special attention to physical effects which could be observed in cold-atom realizations. The work collected here is part of a large theoretical effort4 which in the past decade has focused on the study of relatively simple protocols to bring a quantum system out of equilibrium, such as the so-called quantum quench. In these “theoretical laboratories” it has been possible to provide quantitative predictions which helped us to develop an intuition on general questions regarding non-equilibrium and thermalization processes. Furthermore, in some cases these studies have highlighted interesting physical effects which could be directly probed experimentally within cold-atom settings. The contribution of the present thesis is two-fold. On the one hand, we have developed new technical tools for the study of integrable systems out of equilibrium, and for the computation of measurable physical quantities such as correlation functions. The techniques employed are mainly analytical and rooted within the mathematical structures of integrability. On the other hand, we have singled out physically relevant nonequilibrium situations where exotic, non-thermal states of matter emerge after relaxation occurs, and provided quantitative analytical predictions in these cases. A comprehensive discussion of the motivations and results of our work will be presented in Chapter 1, where we provide a complete overview of this thesis and discuss the organization of its content.