Relaxation at finite temperaturesThe weak coupling theory of chapter 17 is the workhorse of quantum optics and serves very well in practice, also at nonzero temperatures.From the viewpoint of the theory one might wonder about the structure at fixed small, but nonzero, coupling strength, which needs to go beyond the analysis of the weak coupling theory.Much effort has been invested to achieve this goal, basically by trying to identify corrections within time-dependent perturbation theory.Unfortunately, since the long-time behavior must be extracted, the details quickly become unwieldy and one has to rely on ad hoc approximations.Over recent years a novel approach has been pursued which investigates the pole structure of the analytic continuation of the resolvent of HA. across the real axis through complex dilations; compare with section 17.3.The techniques are demanding but simplify substantially at finite temperatures when the Hamiltonian is replaced by the Liouvillean and, since its spectrum is the full real line, complex dilations are replaced by complex translations which can be handled more easily.From the pole structure a fairly complete picture of the long-time dynamics can be extracted with the potential of computing systematically higher corrections to the weak coupling theory.The finite temperature relaxation is a digression into the realm of timedependent statistical mechanics with small deviations from thermal equilibrium.While this is of independent interest and has important applications in quantum optics and condensed matter, our goal is merely to illustrate the power of complex translations and make the connection to the weak coupling theory.For completeness we recall once again the set-up.In the dipole approximation and N -level approximation the Hamiltonian is (18.1)see ( 17 .I 0).Hat acts on eN.Diverging from our previous convention, the energies c: j are labeled as cJ ::S c:2 ... ::S c: N allowing for
Relaxation at finite temperaturesThe weak coupling theory of chapter 17 is the workhorse of quantum optics and serves very well in practice, also at nonzero temperatures.From the viewpoint of the theory one might wonder about the structure at fixed small, but nonzero, coupling strength, which needs to go beyond the analysis of the weak coupling theory.Much effort has been invested to achieve this goal, basically by trying to identify corrections within time-dependent perturbation theory.Unfortunately, since the long-time behavior must be extracted, the details quickly become unwieldy and one has to rely on ad hoc approximations.Over recent years a novel approach has been pursued which investigates the pole structure of the analytic continuation of the resolvent of HA. across the real axis through complex dilations; compare with section 17.3.The techniques are demanding but simplify substantially at finite temperatures when the Hamiltonian is replaced by the Liouvillean and, since its spectrum is the full real line, complex dilations are replaced by complex translations which can be handled more easily.From the pole structure a fairly complete picture of the long-time dynamics can be extracted with the potential of computing systematically higher corrections to the weak coupling theory.The finite temperature relaxation is a digression into the realm of timedependent statistical mechanics with small deviations from thermal equilibrium.While this is of independent interest and has important applications in quantum optics and condensed matter, our goal is merely to illustrate the power of complex translations and make the connection to the weak coupling theory.For completeness we recall once again the set-up.In the dipole approximation and N -level approximation the Hamiltonian is (18.1)see ( 17 .I 0).Hat acts on eN.Diverging from our previous convention, the energies c: j are labeled as cJ ::S c:2 ... ::S c: N allowing for
In Chapter 3, we introduced a simple but reasonable approximation for the nonequilibrium dynamics of a Bose-condensed gas based on a generalized GP equation coupled to a kinetic equation. In Chapters 4–7, we turn to the question of how to derive such coupled equations for the condensate and noncondensate components in a way that gives a deeper understanding of the ZNG theory. Chapters 4–7 involve an introduction to Green's functions and field theoretic techniques for nonequilibrium problems. These provide the natural language and formalism to deal with the many subtle aspects of a Bose-condensed gas at finite temperatures. These four chapters are fairly technical. This chapter is mainly based on Kadanoff and Baym (1962) and Imamović-Tomasović (2001). Readers who are not interested in these questions can go straight to Chapter 8, which begins the discussion of applications of the ZNG coupled equations given in Chapter 3. Overview of Green's function approach To derive a microscopic theory of the nonequilibrium behaviour of a dilute weakly interacting Bose-condensed gas at finite temperatures, there are several different approaches available in the literature. We will use the wellknown Kadanoff–Baym (KB) nonequilibrium Green's function method. The generalization of this formalism to a Bose-condensed system was first considered by Kane and Kadanoff (1965), whose goal was to derive the Landau two-fluid hydrodynamic equations for a system with a Bose broken symmetry. The general problem consists of how to calculate the nonequilibrium response of a system induced by an external (space- and time-dependent) perturbation. In response to such an external perturbation, many interesting physical phenomena appear, including the excitation of collective modes and various transport processes.
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.
The theoretical description of trapped weakly-interacting Bose-Einstein condensates is characterized by a large number of seemingly very different approaches which have been developed over the course of time by researchers with very distinct backgrounds. Newcomers to this field, experimentalists and young researchers all face a considerable challenge in navigating through the ‘maze’ of abundant theoretical models, and simple correspondences between existing approaches are not always very transparent. This Tutorial provides a generic introduction to such theories, in an attempt to single out common features and deficiencies of certain ‘classes of approaches ’ identified by their physical content, rather than their particular mathematical implementation. This Tutorial is structured in a manner accessible to a non-specialist with a good working knowledge of quantum mechanics. Although some familiarity with concepts of quantum field theory would be an advantage, key notions such as the occupation number representation of second quantization are nonetheless briefly reviewed. Following a general introduction, the complexity of models is gradually built up, starting from the basic zero-temperature formalism of the Gross-Pitaevskii equation. This structure enables readers to probe different levels of theoretical developments (mean-field, number-conserving and stochastic) according to their particular needs. In addition to its ‘training element’, we hope that this Tutorial will prove useful to active researchers in this field, both in terms of the correspondences made between different theoretical models, and as a source of reference for existing and developing finite-temperature theoretical models.
The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.
This work concerns itself with the exact study of the dynamical properties of two model systems. After a brief summary of theory and concepts previous work is discussed, and this provides the motivation for the formulation of the first model. This quantum mechanical lattice model and some of its equilibrium properties are described in Chapter II. The dynamical problem to be studied is formulated in Chapter III; this is essentially the study of the time evolution (generated by the Hamiltonian) of a finite system at temperature T1 coupled to an infinite copy of itself at a temperature T2 and acting as a heat bath for the system. The problem is solved for the special case, when the coupling as scaled by the parameter γ, takes the value γ = 1. The general case for arbitrary γ values is treated in Chapter IV. It is shown that the system approaches the equilibrium state of the heat bath in a non-exponential manner, provided the spectrum of the Hamiltonian is continuous and does not have a discrete part. This result is in complete accord with the findings of other work summarised in Chapter I. The mixing properties of the model and behaviour of the relaxation rate in the weak coupling limit are studied in Chapter V. The model is shown to fail to behave as a calorimeter and in view of this result the relevance of the concept of mixing to irreversible behaviour is discussed. The main conclusions and results for the model are summarised at the end of Chapter V. The second model discussed, was first introduced by R.J. Glauber to study the dynamics of the Ising chain. The main feature here is that the time evolution is defined through a Master equation, and the associated stochastic operator. It is shown in Chapter VI that exploiting fully the free fermion character of the stochastic operator for the Glauber model, it is possible to provide a simple method to the study of the dynamics of the Ising Chain.
In this thesis we present an investigation of the influence of quantum correlations on a quantum system’s dynamical behaviour. Our focus is specifically on the time dependence of quantum spins in an environment of itinerant electrons. This is an archetype for strong correlation physics, whose dynamical onset is the central correlation effect investigated in this work. We derive an analytic result for the time evolution of a single localised quantum spin in weak contact with conduction electrons. This result is obtained from a detailed analysis of the pole structure of the Nakajima-Zwanzig equation for the reduced density matrix in Laplace space. We provide a description of the full time range, from very short times in which a novel result for non-Markovian behaviour is obtained, to long times in which we recover the well-known exponential decay expressions. For the short times we show how the non-Markovian memory effects of the spin’s dynamics arise from the backaction of coherent electronic particle-hole fluctuations. As an application of the fast dynamics we propose a cooling protocol going beyond the paradigm of thermodynamic cycles. The protocol relies on a rapid pump scheme with a repeated reinitialisation of the fast quantum coherent dynamics, with each repetition carrying away a small amount of heat from the electronic environment. This protocol is temperature independent and designed to circumvent a natural bottleneck in standard demagnetisation cooling due to long relaxation times at low temperatures. Finally we extend the dynamics to a pair of localised spins coupled through the same electronic environment, using a self-consistent projection operator framework. In contrast to the conventional RKKY coupling we derive a set of coupled equations including the temporal and spatial correlations. This set becomes finite through a meticulous identification of the electronic fluctuations responsible for the coupled dynamics, allowing for a numerical solution.