In this thesis, certain aspects of time-dependent quantum transport properties in nanoscale structures are investigated using the non-equilibrium Green’s function(NEGF). For the transient behaviors of the electric current and heat current after a quantum quench, the exact analytical solution beyond the wide band limit (WBL) is obtained for the first time which paves a way to simulate realistic physical systems. The time-dependent thermoelectric effects are evaluated based on this general solution and it is shown that the time-dependent thermopower is significantly enhanced in the transient regime. By taking initial correlation into account, the combined Caroli and Cini scheme in quantum transport is proposed and its exact solution beyond WBL is derived for the first time as well. Furthermore, besides the average electric current, a current conserving and gauge invariant formalism for the total electric current operator is established at equilibrium state with the help of the electric displacement current operator formulated by taking the Coulomb interaction into account at the operator level. By means of the NEGF and the scattering matrix theory (SMT), a current conserving theory for equilibrium noise power at finite frequencies is obtained which solves one of the fundamental problems in quantum transport. A generalized equilibrium fluctuation-dissipation theorem is presented. Moreover, the entanglement entropy \nfluctuation and its distribution are investigated by introducing the entanglement entropy operator. The essential feature of the entanglement entropy generated by quantum transport is that it is a stochastic quantity whose fluctuation and distribution can be obtained from the generating function (GF). Combining the NEGF with the Grassmann algebras, it is found that the GF of entanglement \nentropy can be expressed in terms of the lesser Green’s function of the subsystem. The general relation connecting the full counting statistics (FCS) of charge transfer with the GF of entanglement entropy is found which indicates that, from the experimental point of view, the entanglement entropy fluctuation can be measured indirectly. As an illustration of the general theoretical framework of entanglement entropy fluctuation, the quantum point contact (QPC) system as well as the quantum dot (QD) system are studied. It is shown that for a QPC under dc bias, the entanglement entropy is maximized and fluctuationless when the transmission coefficient T is one-half. For the quantum dot system, at short times, a universal scaling relation among transient peak values of cumulants of entanglement entropy is shown.
In this thesis, certain aspects of time-dependent quantum transport properties in nanoscale structures are investigated using the non-equilibrium Green’s function(NEGF). For the transient behaviors of the electric current and heat current after a quantum quench, the exact analytical solution beyond the wide band limit (WBL) is obtained for the first time which paves a way to simulate realistic physical systems. The time-dependent thermoelectric effects are evaluated based on this general solution and it is shown that the time-dependent thermopower is significantly enhanced in the transient regime. By taking initial correlation into account, the combined Caroli and Cini scheme in quantum transport is proposed and its exact solution beyond WBL is derived for the first time as well. Furthermore, besides the average electric current, a current conserving and gauge invariant formalism for the total electric current operator is established at equilibrium state with the help of the electric displacement current operator formulated by taking the Coulomb interaction into account at the operator level. By means of the NEGF and the scattering matrix theory (SMT), a current conserving theory for equilibrium noise power at finite frequencies is obtained which solves one of the fundamental problems in quantum transport. A generalized equilibrium fluctuation-dissipation theorem is presented. Moreover, the entanglement entropy \nfluctuation and its distribution are investigated by introducing the entanglement entropy operator. The essential feature of the entanglement entropy generated by quantum transport is that it is a stochastic quantity whose fluctuation and distribution can be obtained from the generating function (GF). Combining the NEGF with the Grassmann algebras, it is found that the GF of entanglement \nentropy can be expressed in terms of the lesser Green’s function of the subsystem. The general relation connecting the full counting statistics (FCS) of charge transfer with the GF of entanglement entropy is found which indicates that, from the experimental point of view, the entanglement entropy fluctuation can be measured indirectly. As an illustration of the general theoretical framework of entanglement entropy fluctuation, the quantum point contact (QPC) system as well as the quantum dot (QD) system are studied. It is shown that for a QPC under dc bias, the entanglement entropy is maximized and fluctuationless when the transmission coefficient T is one-half. For the quantum dot system, at short times, a universal scaling relation among transient peak values of cumulants of entanglement entropy is shown.
Abstract We review one of the most versatile theoretical approaches to the study of time-dependent correlated quantum transport in nano-systems: the non-equilibrium Green’s function (NEGF) formalism. Within this formalism, one can treat, on the same footing, inter-particle interactions, external drives and/or perturbations, and coupling to baths with a (piece-wise) continuum set of degrees of freedom. After a historical overview on the theory of transport in quantum systems, we present a modern introduction of the NEGF approach to quantum transport. We discuss the inclusion of inter-particle interactions using diagrammatic techniques, and the use of the so-called embedding and inbedding techniques which take the bath couplings into account non-perturbatively. In various limits, such as the non-interacting limit and the steady-state limit, we then show how the NEGF formalism elegantly reduces to well-known formulae in quantum transport as special cases. We then discuss non-equilibrium transport in general, for both particle and energy currents. Under the presence of a time-dependent drive—encompassing pump–probe scenarios as well as driven quantum systems—we discuss the transient as well as asymptotic behavior, and also how to use NEGF to infer information on the out-of-equilibrium system. As illustrative examples, we consider model systems general enough to pave the way to realistic systems. These examples encompass one- and two-dimensional electronic systems, systems with electron–phonon couplings, topological superconductors, and optically responsive molecular junctions where electron–photon couplings are relevant.
The entanglement entropy generated by quantum transport, similar to any physical observable quantity, is a stochastic variable that has its distribution and can fluctuate. The fundamental question is how to define the entanglement entropy operator which allows one to discuss entanglement entropy fluctuation. By introducing the entanglement entropy operator, we develop a theoretical framework to calculate the entanglement entropy fluctuation as well as its higher order cumulants generated by electronic transport in open systems. The distribution of entanglement entropy generated by opening or closing a quantum point contact (QPC) is solved exactly. When the transmission coefficient of QPC is one-half, the entanglement entropy is maximized and fluctuationless. We also establish a general relation between the generated entanglement entropy fluctuation and charge fluctuation. We apply our theory to electronic transport through a quantum dot and study the generated entanglement entropy in the transient regime. Universal behavior is found for the cumulants of entanglement entropy at short times.
As experimental probes have matured to observe ultrafast transient and high frequency responses of materials and devices, so to have the theoretical methods to numerically and analytically simulate time- and frequency-resolved transport. In this review article, we discuss recent progress in the development of the time-dependent and frequency-dependent non-equilibrium Green function (NEGF) technique. We begin with an overview of the theoretical underpinnings of the underlying Kadanoff-Baym equations and derive the fundamental NEGF equations in the time and frequency domains. We discuss how these methods have been applied to a variety of condensed matter systems such as molecular electronics, nanoscale transistors, and superconductors. In addition, we survey the application of NEGF in fields beyond condensed matter, where it has been used to study thermalization in ultra-cold atoms and to understand leptogenesis in the early universe. Throughout, we pay special attention to the challenges of incorporating contacts and interactions, as the NEGF method is uniquely capable of accounting for such features.
We propose a time-dependent many-body approach to study the short-time dynamics of correlated electrons in quantum transport through nanoscale systems contacted to metallic leads. This approach is based on the time propagation of the Kadanoff-Baym equations for the nonequilibrium many-body Green's function of open and interacting systems out of equilibrium. An important feature of the method is that it takes full account of electronic correlations and embedding effects in the presence of time-dependent external fields, while at the same time satisfying the charge conservation law. The method further extends the Meir-Wingreen formula to the time domain for initially correlated states. We study the electron dynamics of a correlated quantum wire attached to two-dimensional leads exposed to a sudden switch on of a bias voltage using conserving many-body approximations at Hartree-Fock, second Born and GW level. We obtain detailed results for the transient currents, dipole moments, spectral functions, charging times, and the many-body screening of the quantum wire as well as for the time-dependent density pattern in the leads, and we show how the time dependence of these observables provides a wealth of information on the energy level structure of the quantum wire out of equilibrium. For moderate interaction strengths the second Born and GW results are in excellent agreement at all times. We find that many-body effects beyond the Hartree-Fock approximation have a large effect on the qualitative behavior of the system and lead to a bias-dependent gap closing and quasiparticle broadening, shortening of the transient times and washing out of the step features in the current-voltage curves.
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.
This thesis investigates the full-counting statistics (FCS) of charge and spin transport using nonequilibrium Green's function (NEGF) in non-interacting systems. \nGenerating function (GF) encodes all the distribution information and is the key of FCS. For the charge statistics, GF is expressed with respect to the modified evolution operator on the complex contour using the two time measurement scheme, the modified evolution operator is expressed in terms of the modified Hamiltonian. Using Keldysh NEGF technique, GF has the form of Fredholm determinant in terms of Greens functions and self-energies in the time domain. In the long time limit, the celebrated Levitov-Lesovik's formula and the fluctuation relation are derived from this formalism. GF of charge statistics for the quantum point contact system in the transient regime is obtained as well. The transient dynamics in lead-dot-lead transport system, including the cumulants of transferred charges and waiting time distribution defined under the transient regime for different temperatures is investigated numerically. A detailed description on how to calculate the GF in the time domain is presented. Generalizations of the formalism to the ferromagnet-normal-ferromagnet system to investigate the FCS of charge current, spin current and spin transfer torque (STT) are presented. GFs in time domain are Fredholm determinants expressed by NEGF as well. As an application of FCS, a formalism using FCS of STT in the long time limit to calculate the switching probability of a nanomagnet system is proposed. The formalism enables us to calculate the switching probability for nonequilibrium systems which are non-Gaussian. From the stochastic Landau-Lifshitz-Gilbert (LLG) equation, the contributions to the change of the anisotropic energy, one is from the power gain due to Gilbert damping and the other is from power dissipation due to the spin transfer torque, were derived. Optimal path approximation which requires the nanomagnet a modestly large volume is presented and the approximation greatly reduces the numerical complexities. This formalism could be used to predict the switching probability numerically for real systems in combination with the first principle calculation.