The study of dissipative systems holds significant interest in physics. This thesis aims to explore these phenomena in both classical and quantum regimes. In the first part, we utilize the system-reservoir approach to investigate the dynamics of two Brownian particles immersed in the same bath. Two methods are employed to address this problem: one with bilinear coupling and the other with nonlinear coupling between the particles and the bath. The extension of the system to include two particles with bilinear coupling yields unphysical results, such as free-particle motion for the relative coordinate and a lack of interaction between closely spaced particles. To address this issue, authors have been introduced an exponential coupling. In this work, we propose a method to reconcile both linear and nonlinear couplings. We demonstrate how to derive the same nonlinear dissipation rates starting from the bilinear Lagrangian, achieved through a modified spectral function that explicitly depends on the distance between the particles. Additionally, we implement a modified spectral function to mitigate anomalous diffusion observed in the standard nonlinear model, along with a phenomenological model describing hydrodynamic interaction between a pair of Brownian particles in a viscous fluid. In the quantum regime, we adopt the same system-reservoir approach to investigate the dissipative adaptation hypothesis proposed by Jeremy England. This hypothesis proposes a general thermodynamic mechanism that explains the self-organization of systems through the dissipation of work absorbed from an external drive. In the second part of this thesis, we explore the quantum dynamics of systems subjected to an external drive, evaluating the thermodynamic quantities of a self-organization process. To do so, we utilize a time-dependent spin-boson Hamiltonian that characterizes a particle subject to a metastable double potential with time-dependent parameters controlling the asymmetry of the wells. Our objective is to demonstrate that the asymmetric potential can localize the particle in the unstable side of the well and verify that this transition results in the highest energy absorption. In conclusion, we propose further investigations into the driven spin-boson model to establish a comprehensive theory of the systems evolution and its thermodynamic implications.
The study of dissipative systems holds significant interest in physics. This thesis aims to explore these phenomena in both classical and quantum regimes. In the first part, we utilize the system-reservoir approach to investigate the dynamics of two Brownian particles immersed in the same bath. Two methods are employed to address this problem: one with bilinear coupling and the other with nonlinear coupling between the particles and the bath. The extension of the system to include two particles with bilinear coupling yields unphysical results, such as free-particle motion for the relative coordinate and a lack of interaction between closely spaced particles. To address this issue, authors have been introduced an exponential coupling. In this work, we propose a method to reconcile both linear and nonlinear couplings. We demonstrate how to derive the same nonlinear dissipation rates starting from the bilinear Lagrangian, achieved through a modified spectral function that explicitly depends on the distance between the particles. Additionally, we implement a modified spectral function to mitigate anomalous diffusion observed in the standard nonlinear model, along with a phenomenological model describing hydrodynamic interaction between a pair of Brownian particles in a viscous fluid. In the quantum regime, we adopt the same system-reservoir approach to investigate the dissipative adaptation hypothesis proposed by Jeremy England. This hypothesis proposes a general thermodynamic mechanism that explains the self-organization of systems through the dissipation of work absorbed from an external drive. In the second part of this thesis, we explore the quantum dynamics of systems subjected to an external drive, evaluating the thermodynamic quantities of a self-organization process. To do so, we utilize a time-dependent spin-boson Hamiltonian that characterizes a particle subject to a metastable double potential with time-dependent parameters controlling the asymmetry of the wells. Our objective is to demonstrate that the asymmetric potential can localize the particle in the unstable side of the well and verify that this transition results in the highest energy absorption. In conclusion, we propose further investigations into the driven spin-boson model to establish a comprehensive theory of the systems evolution and its thermodynamic implications.
The study of dissipative systems holds significant interest in physics.This thesis aims to explore these phenomena in both classical and quantum regimes.In the first part, we utilize the system-reservoir approach to investigate the dynamics of two Brownian particles immersed in the same bath.Two methods are employed to address this problem: one with bilinear coupling and the other with nonlinear coupling between the particles and the bath.The extension of the system to include two particles with bilinear coupling yields unphysical results, such as free-particle motion for the relative coordinate and a lack of interaction between closely spaced particles.To address this issue, authors have been introduced an exponential coupling.In this work, we propose a method to reconcile both linear and nonlinear couplings.We demonstrate how to derive the same nonlinear dissipation rates starting from the bilinear Lagrangian, achieved through a modified spectral function that explicitly depends on the distance between the particles.Additionally, we implement a modified spectral function to mitigate anomalous diffusion observed in the standard nonlinear model, along with a phenomenological model describing hydrodynamic interaction between a pair of Brownian particles in a viscous fluid.In the quantum regime, we adopt the same system-reservoir approach to investigate the dissipative adaptation hypothesis proposed by Jeremy England.This hypothesis proposes a general thermodynamic mechanism that explains the self-organization of systems through the dissipation of work absorbed from an external drive.In the second part of this thesis, we explore the quantum dynamics of systems subjected to an external drive, evaluating the thermodynamic quantities of a self-organization process.To do so, we utilize a time-dependent spin-boson Hamiltonian that characterizes a particle subject to a metastable double potential with time-dependent parameters controlling the asymmetry of the wells.Our objective is to demonstrate that the asymmetric potential can localize the particle in the unstable side of the well and verify that this transition results in the highest energy absorption.In conclusion, we propose further investigations into the driven spin-boson model to establish a comprehensive theory of the system's evolution and its thermodynamic implications.
We investigate the dissipative adaptation hypothesis in a quantum regime using a system-reservoir approach. This hypothesis proposes that self-organization arises from a system's ability to dissipate the work transiently absorbed from an external drive. We analyze the quantum dynamics of a driven open system described by a time-dependent spin-boson Hamiltonian modeling a particle in a metastable double-well potential with controllable asymmetry. We explore how the work provided by the dynamic potential is related to the transition probability between the two ground states of the double well. These studies motivate further investigations of the driven spin-boson model toward an understanding of the system's evolution and its thermodynamic implications.
The thesis comprises two major themes of quantum statistical dynamics. One is the development of quantum dissipation theory (QDT). It covers the establishment of some basic relations of quantum statistical dynamics, the construction of several nonequivalent complete second-order formulations, and the development of exact QDT. Another is related to the applications of quantum statistical dynamics to a variety of research fields. In particular, unconventional but novel theories of the electron transfer in Debye solvents, quantum transport, and quantum measurement are developed on the basis of QDT formulations. The thesis is organized as follows. In Chapter 1, we present some background knowledge in relation to the afore-mentioned two themes of this thesis. The key quantity in QDT is the reduced density operator ρ(t) ≡ trBρT(t); i.e., the partial trace of the total system and bath composite ρT(t) over the bath degrees of freedom. QDT governs the evolu-tion of reduced density operator, where the effects of bath are treated in a quantum statistical manner. In principle, the reduced density operator contains all dynam-ics information of interest. However, the conventional quantum transport theory is formulated in terms of nonequilibrium Green's function. The newly emerging field of quantum measurement in relation to quantum information and quantum computing does exploit a sort of QDT formalism. Besides the background of the relevant theoretical development, some representative experiments on molecular nanojunctions are also briefly discussed. In Chapter 2, we outline some basic (including new) relations that highlight several important issues on QDT. The content includes the background of nonequi-librium quantum statistical mechanics, the general description of the total compos-ite Hamiltonian with stochastic system-bath interaction, a novel parameterization scheme for bath correlation functions, a newly developed exact theory of driven Brownian oscillator (DBO) systems, and its closely related solvation mode trans-formation of system-bath coupling Hamiltonian in general. The exact QDT of DBO systems is also used to clarify the validity of conventional QDT formulations that involve Markovian approximation. In Chapter 3, we develop three nonequivalent but all complete second-order QDT (CS-QDT) formulations. Two of them are of the conventional prescriptions in terms of time-local dissipation and memory kernel, respectively. The third one is called the correlated driving-dissipation equations of motion (CODDE). This novel CS-QDT combines the merits of the former two for its advantages in both the application and numerical implementation aspects. Also highlighted is the importance of correlated driving-dissipation effects on the dynamics of the reduced system. In Chapter 4, we construct an exact QDT formalism via the calculus on path integrals. The new theory aims at the efficient evaluation of non-Markovian dis-sipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. By adopting exponential-like expansions for bath corre-lation function, hierarchical equations of motion formalism and continued fraction Liouville-space Green's function formalism are established. The latter will soon be used together with the Dyson equation technique for an efficient evaluation of non-perturbative reduced density matrix dynamics. The interplay between system-bath interaction strength, non-Markovian property, and the required level of hierarchy is also studied with the aid of simple spin-boson systems, together with the three proposed schemes to truncate the infinite hierarchy. In Chapter 5, we develop a nonperturbative theory of electron transfer (ET) in Debye solvents. The resulting exact and analytical rate expression is constructed on the basis of the aforementioned continued fraction Liouville-space Green's func-tion formalism, together with the Dyson equation technique. Not only does it re-cover the celebrated Marcus' inversion and Kramers' turnover behaviors, the new theory also shows some distinct quantum solvation effects that can alter the ET mechanism. Moreover, the present theory predicts further for the ET reaction ther-modynamics, such as equilibrium Gibbs free-energy and entropy, some interesting solvent-dependent features that are calling for experimental verification. In Chapter 6, we discuss the constructed QDTs, in terms of their unified math-ematical structure that supports a linear dynamics space, and thus facilitates their applications to various physical problems. The involving details are exemplified with the CODDE form of QDT. As the linear space is concerned, we identify the Schrödinger versus Heisenberg picture and the forward versus backward propaga-tion of the reduced, dissipative Liouville dynamics. For applications we discuss the reduced linear response theory and the optimal control problems, in which the correlated effects of non-Markovian dissipation and field driving are shown to be important. In Chapter 7, we turn to quantum transport, i.e. electric current through molecular or mesoscopic systems under finite applied voltage. By viewing the nonequilibrium transport setup as a quantum open system, we develop a reduced-density-matrix approach to quantum transport. The resulting current is explicitly expressed in terms of the molecular reduced density matrix by tracing out the degrees of freedom of the electrodes at finite bias and temperature. We propose a conditional quantum master equation theory, which is an extension of the conven-tional (or unconditional) QDT by tracing out the well-defined bath subsets indi-vidually, instead of the entire bath degrees of freedom. Both the current and the noise spectrum can be conveniently analyzed in terms of the conditional reduced density matrix dynamics. By far, the QDT (including the conditional one) has only been exploited in a second-order form. A self-consistent Born approximation for the system-electrode coupling is further proposed to recover all existing nonlinear current-voltage behaviors including the nonequilibrium Kondo effect. Transport theory based on the exact QDT formalism will be developed in future. In Chapter 8, we study the quantum measurement of a qubit with a quantum-point-contact detector. On the basis of a unified quantum master equation (a form of QDT), we study the measurement-induced relaxation and dephasing of the qubit . Our treatment pays particular attention on the detailed-balance relation, which is a consequence of properly accounting for the energy exchange between the qubit and detector during the measurement process. We also derive a conditional quantum master equation for quantum measurement in general, and study the readout characteristics of the qubit measurement. Our theory is applicable to the quantum measurement at arbitrary voltage and temperature. A number of remarkable new features are found and highlighted in concern with their possible relevance to future experiments. In Chapter 9, we discuss the further development of QDT, aiming at an effi-cient evaluation of many-electron systems. This will be carried out by reducing the many-particle (Fermion or Boson) QDT to a single-particle one by exploring, e.g. the Wick's contraction theorem. It also results in a time-dependent density func-tional theory (TDDFT) for transport through complex large-scale (e.g. molecules) systems. Primary results of the TDDFT-QDT are reported. In Chapter 10, we summary the thesis, and comment and remark on the future work on both the theoretical and application aspects of QDT.
Abstract Dissipative adaptation is a general thermodynamic mechanism that explains self-organization in a broad class of driven classical many-body systems. It establishes how the most likely (adapted) states of a system subjected to a given drive tend to be those following trajectories of highest work absorption, followed by dissipated heat to the reservoir. Here, we extend the dissipative adaptation phenomenon to the quantum realm. We employ a fully-quantized exactly solvable model, where the source of work on a three-level system is a single-photon pulse added to a zero-temperature infinite environment, a scenario that cannot be treated by the classical framework. We find a set of equalities relating adaptation likelihood, absorbed work, heat dissipation and variation of the informational entropy of the environment. Our proof of principle provides the starting point towards a quantum thermodynamics of driven self-organization.
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.
We use the system-plus-reservoir model to study the dynamics of a system of two particles that interact with a heat bath in thermal equilibrium, but do not interact with each other.We present an extension of the bath of oscillators capable of inducing an eective coupling between the brownian particles depending on the choice made to the spectral function of the oscillators components of the bath.The coupling is non-linear in the variables of interest and an exponential dependence is imposed in order to guarantee the translational invariance of the model.The resultant eective interaction is presented using the classical equations of motion and the quantum limit is studied through the time evolution of reduced density operator of the two particles.We nd that, in the limit of long times and short distances, our model reproduces the results of the linear model for the case of only one particle.Finally, we describe the time evolution of an initial state given by the product of two simple packets.v B Propriedades da força utuante C Aproximação local do superpropagador 1 Capítulo 1 Introdução Existem muitos fenômenos físicos nos quais alguma grandeza está constantemente sujeita a pequenas utuações aleatórias.O estudo destas utuações, tanto no regime clássico quanto no quântico, tem sido amplamente desenvolvido desde que Einstein apresentou a sua teoria quantitativa do movimento browniano em 1905 baseando-se nas idéias da teoria cinética dos gases.No regime quântico estas pesquisas têm encontrado variadas aplicações em diversas áreas da física da matéria condensada e na físico-química.Começando com o trabalho de Callen e Welton [2℄ que generaliza as relações de Einstein para incluir efeitos quânticos, são muitos os esforços dedicados a conciliar as equações dissipativas com o processo de quantização.Estas pesquisas podem ser classicadas em dois grandes grupos.No primeiro, a idéia principal é procurar novos esquemas de quantização.Embora os modelos empregados sejam de uma sosticação matemática admirável, eles apresentam problemas com relação, por exemplo, ao princípio de incerteza e ao princípio de superposição [3, 4, 6, 7, 5℄.As pesquisas do segundo grupo, baseiam-se no fato de que nenhum sistema na natureza se encontra totalmente isolado e, portanto, considera o sistema de interesse acoplado a um reservatório térmico responsável pelas perdas de energia e mantém o esquema de quantização usual.A abordagem usada no último grupo tem sido a mais bem sucedida [8, 9, 10, 12, 11℄ e será a adotada neste trabalho.Se a informação que possuimos do banho for suciente podemos, dentro das aproximações adequadas para o acoplamento particular que queremos estudar, encontrar a dinâmica efetiva do sistema de interesse partindo das equações microscópicas.Por outro lado, se não tivermos acesso a estas informações ou não estivermos interessados nas características microscópicas do banho nem nos detalhes do acoplamento, podemos escolher um modelo simples que facilite o tratamento matemático subsequente e impôr sobre ele algumas condições que nos permitam, no limite clássico, recuperar as equações fenomenológicas de movimento.O movimento browniano quântico tem sido estudado através do acoplamento da partícula de interesse com um reservatório escolhido fenomenologicamente como um conjunto de osciladores harmônicos desacoplados entre si.O acoplamento é bilinear nas coordenadas de ambos os sistemas e escolhe-se uma função espectral que carrega a informação sobre a distribuição dos modos do banho e cuja forma determina o tipo de dissipação que queremos estudar.Este modelo tem sido usado com(C.27)