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) ≡ tr

_{BρT}(t); i.e., the partial trace of the total system and bath composite ρ

_{T}(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.

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