Analyzing the dynamics of open quantum systems has a long history in mathematics and physics. Depending on the system at hand, basic physical phenomena that one would like to explain are, for example, convergence to equilibrium, the dynamics of quantum coherences (decoherence) and quantum correlations (entanglement), or the emergence of heat and particle fluxes in non-equilibrium situations. From the mathematical physics perspective, one of the main challenges is to derive the irreversible dynamics of the open system, starting from a unitary dynamics of the system and its environment. The repeated interactions systems considered in these notes are models of non-equilibrium quantum statistical mechanics. They are relevant in quantum optics, and more generally, serve as a relatively well treatable approximation of a more difficult quantum dynamics. In particular, the repeated interaction models allow to determine the large time (stationary) asymptotics of quantum systems out of equilibrium.
Analyzing the dynamics of open quantum systems has a long history in mathematics and physics. Depending on the system at hand, basic physical phenomena that one would like to explain are, for example, convergence to equilibrium, the dynamics of quantum coherences (decoherence) and quantum correlations (entanglement), or the emergence of heat and particle fluxes in non-equilibrium situations. From the mathematical physics perspective, one of the main challenges is to derive the irreversible dynamics of the open system, starting from a unitary dynamics of the system and its environment. The repeated interactions systems considered in these notes are models of non-equilibrium quantum statistical mechanics. They are relevant in quantum optics, and more generally, serve as a relatively well treatable approximation of a more difficult quantum dynamics. In particular, the repeated interaction models allow to determine the large time (stationary) asymptotics of quantum systems out of equilibrium.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.
Open quantum systems are often encountered in many different physical situations. From quantum optics to statistical mechanics, they are fundamental in the understanding of a great variety of different phenomena. Some of the most common examples are the relaxation to equilibrium, the existence of nonequilibrium stationary states, and the dynamics of atoms in interaction with electromagnetic fields. A crucial step in the analysis is to consider the quantum open system and its environment as the two mutually interacting components of a larger isolated system. Thereafter, the so‐called Markov approximation is often considered, which consists on assuming that the time scales associated to the dynamics of the quantum open system are larger than those of the environment. It is the interplay of the different time scales associated with the system and the environment what determines the validity of the different approximations made. In this paper we will discuss the dynamics of a open quantum system in contact with a reservoir when the Markov approximation is not valid, and we have to include some non‐Markovian or memory effects.
All real-world quantum systems interact with their surrounding environment to some extent. No matter how weak the coupling that prevents the system from being isolated, the evolution of an open quantum system is eventually plagued by non-unitary features, such as decoherence and dissipation. Decoherence is a quantum effect where the system loses its ability to exhibit coherent behavior. Nowadays, decoherence stands as a serious obstacle in quantum information processing. As the manipulation of quantum coherence in experiments has progressed, the need to better understand, control, and utilize dissipative non-equilibrium dynamics of quantum systems has grown in importance. This has ranged from fundamental questions associated with the quantum-to-classical transition and quantum thermodynamics innovation in quantum optics and cavity/circuit quantum electrodynamics. Current studies of many-body dynamics, ultracold atoms in optical lattices, trapped ions, superconducting systems, nanoelectro-mechanical systems, and quantum fields under the influence of external conditions offer new challenges to address questions regarding quantum systems far from equilibrium. On the theoretical side, quantum trajectory-based methods could be the key to both quantum nanodevice design and exploration of new regimes of quantum mechanics and quantum measurement, as well as applications to various quantum technologies.
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].
This book presents a number of basic concepts and mathematical techniques for analyzing quantum and classical dynamical systems on an equal basis, mainly but not solely aimed at quantifying randomizing dynamical behaviour. One of the central topics is the introduction of a quantum dynamical entropy based on successive observations of the system. This leads to a symbolic model in terms of a state on a quantum spin chain and its associated entropy that quantifies the complexity of the time evolution. Several model systems are analyzed, but more realistic dynamics and scaling limits are considered as well. Gradually the key concepts and mathematical results are presented and special attention is paid to their relevance for quantum dynamics. The book starts out with a reminder of standard Hilbert space quantum mechanics and then gently introduces the need for the more general and abstract C*-dynamical system setup. Basic examples like quantum spin chains, CAR- and CCR-formalism, and the irrational rotation algebra are described. Next the essential notions from ergodic theory and quantum irreversibility are provided along with the properties and use of von Neumann entropy as quantifiers for randomness. The last part of the book introduces a quantum dynamical entropy based on operational partitions: for classical dynamics the well-established KS-invariant is recovered, while for quantum dynamics a variety of systems are analyzed. The book ends with some open problems and speculations on the footprints of microscopic quantum dynamics on transport phenomena at the macroscopic scale.
The main goal of these notes is to give an introduction to the mathematics of quantum noise and some of its applications in non-equilibrium statistical mechanics. We start with some reminders from the theory of classical stochastic processes. We then provide a brief overview of quantum mechanics and quantum field theory, from the viewpoint of quantum probability and adopting the language of Hudson and Parthasarathy. We introduce quantum stochastic processes on a boson Fock space and their calculus. Whenever possible, we make connections with the relevant concepts in classical probability theory. As an application of the theory, we introduce the theory of open quantum systems, with emphasis on the physics and modeling aspects of these systems.