Traditionally, (charge) transport phenomena describe the average current flow in response to a constant applied driving force. Additional information is provided by current-current correlations or noise, telling about the granularity, the statistical properties, and the interactions between the particles. The complete characterization of transport is given by the full counting statistics, quantifying the probability that n particles pass the device during a time window t. The desire for a complete characterization of transport in a mesoscopic device goes together with the development of new single-particle sources and the availability of quantum bits as measurement devices. In this thesis, we will describe how to combine these three elements, full counting statistics, single-particle sources, and qubit measurement devices and how these concepts relate to the description of charge transport of noninteracting electrons in a mesoscopic device. In the work on full counting statistics, we develop a new wave-packet formalism that provides access to an entire class of interesting results. The wavepacket formalism is derived from an insight regarding the equivalence between the fidelity of a quantum system and the generating function of full counting statistics; furthermore, we show how qubits can be used to measure both quantities, fidelity and full counting statistics, exploiting the induced ‘decoherence’ as a signal; thereby, we transfer the measurement of the fidelity and of the full counting statistics from the realm of a ‘Gedanken’ experiment to a practical proposal realizable with today’s qubit technology. Subsequently, we study the full counting statistics for the transmission of two identical particles with positive or negative symmetry under exchange, for the situation where the scattering depends on energy. We make use of our firstquantized wave-packet formalism and find interesting results. While it is expected that the noise as well as higher-order correlators will be sensitive to the exchange symmetry, we find the astounding result that exchange can hugely enhance (or suppress) the average charge already. The effect is a consequence of the symmetry-induced reshuffling of weight in the momentum distribution of the
Traditionally, (charge) transport phenomena describe the average current flow in response to a constant applied driving force. Additional information is provided by current-current correlations or noise, telling about the granularity, the statistical properties, and the interactions between the particles. The complete characterization of transport is given by the full counting statistics, quantifying the probability that n particles pass the device during a time window t. The desire for a complete characterization of transport in a mesoscopic device goes together with the development of new single-particle sources and the availability of quantum bits as measurement devices. In this thesis, we will describe how to combine these three elements, full counting statistics, single-particle sources, and qubit measurement devices and how these concepts relate to the description of charge transport of noninteracting electrons in a mesoscopic device. In the work on full counting statistics, we develop a new wave-packet formalism that provides access to an entire class of interesting results. The wavepacket formalism is derived from an insight regarding the equivalence between the fidelity of a quantum system and the generating function of full counting statistics; furthermore, we show how qubits can be used to measure both quantities, fidelity and full counting statistics, exploiting the induced ‘decoherence’ as a signal; thereby, we transfer the measurement of the fidelity and of the full counting statistics from the realm of a ‘Gedanken’ experiment to a practical proposal realizable with today’s qubit technology. Subsequently, we study the full counting statistics for the transmission of two identical particles with positive or negative symmetry under exchange, for the situation where the scattering depends on energy. We make use of our firstquantized wave-packet formalism and find interesting results. While it is expected that the noise as well as higher-order correlators will be sensitive to the exchange symmetry, we find the astounding result that exchange can hugely enhance (or suppress) the average charge already. The effect is a consequence of the symmetry-induced reshuffling of weight in the momentum distribution of the
The study of electrical currents in a regime where a description as a continuous flow fails and the particle nature becomes important, but where notions such as particle density or currents continue to be meaningful, will be at the heart of the present dissertation. Given such a mesoscopic system through which electrons flow, either actively through quantum pumping or passively due to a potential difference, we investigate quantities such as transferred charge, noise, current fluctuations, and relations between them. More generally we shall examine the statistics of particle transport. All discussions will concentrate on compact devices transferring charge between two one-dimensional reservoirs (the leads) of independent electrons. Firstly, we consider the generating function of the moments of transferred charge and we prove that it is given by the determinant of a single-particle operator. It is formally equivalent to a formula of Lesovik and Levitov, but is shown to be well-defined in the situations under consideration, whereas the former is not, in general. The mechanism leading to regularization is typical for mesoscopic systems: we consider a limiting case that reveals the fundamental structure of the problem. In fact, as suited for an open system, the leads contain an infinite number of charges. It follows mathematically that the second quantization ought to be carried out upon the Fermi sea and not upon the vacuum, a procedure known as the GNS construction. We shall rely heavily on operator algebraic techniques. The set of observables, the C*-algebra of canonical anticommutation relations, admits different representations for systems in the thermodynamic limit, which are described by their reduced one-particle density matrix. An important question will be the existence of operators such as the charge or the time evolution in a given representation, which induce so-called Bogoliubov transformations. These prerequisites at hand, we shall pose the problem of counting statistics clearly, prove that the regularized determinant emerges naturally, and show that it describes the transport of integer charges with non-negative probabilities. Secondly, we introduce a model of instantaneous scattering in which various examples can be computed explicitely. In particular, we show that the above theorem applies for scattering through a smoothly operating pump over a finite time interval. We also exhibit situations where the propagator does exist but the generating function is ambiguous. In a more computational approach, we then turn to the relation of individual moments of transferred charge to current correlators. It turns out that the former can indeed be expressed as integrals of the latter over time, provided the current operators are ordered according to Matthews’ T∗ ordering. It is given by first arranging the charge operators according to their time indices, and only then taking the time derivatives, which induce contact terms whose singular support lies along time coincidences. We shall give two derivations of this result, because the first one is transparent, and the second one is related to contradictory computations in the literature where the system is considered to be measured by an external spin. In the model of constant instantaneous scattering, we then argue from the Lesovik-Levitov determinant that the transport can be interpreted in terms of a binomial process in the long time limit. In a parallel computation, we concentrate on the third cumulant as defined by integration of current correlators and show how the contact terms of the T∗ ordering play an essential role in providing the expected binomial result, too. To close this thesis, we suggest an algebraic framework that lays the foundations for a rigorous discussion of the Casimir effect. The problem of infinites arising in the comparison of two physical situations for a bosonic field is interpreted as the non-equivalence of states on the algebra of canonical commutation relations. We exhibit sufficient conditions on the one-particle Hamiltonians which ensure that the Casimir energy is finite and give an expression for it. We discuss its formal equivalence with the result obtained from the standard mode summing method.
We study the full counting statistics for the transmission of two identical particles with positive or negative symmetry under exchange for the situation where the scattering depends on energy. We find that, in addition to the expected sensitivity of the noise and higher cumulants, the exchange symmetry has a huge effect on the average transmitted charge; for equal-spin exchange-correlated electrons, the average transmitted charge can be orders of magnitude larger than the corresponding value for independent electrons.
The complete characterization of the charge transport in a mesoscopic device is provided by the full counting statistics (FCS) Pt(m), describing the amount of charge Q=me transmitted during the time t. Although numerous systems have been theoretically characterized by their FCS, the experimental measurement of the distribution function Pt(m) or its moments $\ensuremath{\langle}{Q}^{n}\ensuremath{\rangle}$ is rare and often plagued by strong back-action. Here, we present a strategy for the measurement of the FCS, more specifically its characteristic function $\ensuremath{\chi}(\ensuremath{\lambda})$ and moments $\ensuremath{\langle}{Q}^{n}\ensuremath{\rangle}$, by a qubit with a set of different couplings ${\ensuremath{\lambda}}_{j}$, $j=1,\ensuremath{\cdots},k,\ensuremath{\cdots}k+p$, $k=\ensuremath{\lceil}n/2\ensuremath{\rceil}$, $p\ensuremath{\ge}0$, to the mesoscopic conductor. The scheme involves multiple readings of Ramsey sequences at the different coupling strengths ${\ensuremath{\lambda}}_{j}$, and we find the optimal distribution for these couplings ${\ensuremath{\lambda}}_{j}$ as well as the optimal distribution Nj of $N=\ensuremath{\sum}{N}_{j}$ measurements among the different couplings ${\ensuremath{\lambda}}_{j}$. We determine the precision scaling for the moments $\ensuremath{\langle}{Q}^{n}\ensuremath{\rangle}$ with the number N of invested resources and show that the standard quantum limit can be approached when many additional couplings $p\ensuremath{\gg}1$ are included in the measurement scheme.
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.
Quantum transport is commonly studied with the use of quasi-particle infinite- extended states. This leads to a powerful formalism, the scattering-states theory, able to capture in compact formulas quantities of interest, such as average current, noise, etc.. However, when investigating the spatial size-dependence of quasi-particle wave packets in quantum noise with exchange and tunneling, unexpected new terms appear in the quantum noise expression. For this purpose, the two particle transmission and reflection probabilities for two initial one-particle wave packets (with opposite central momentums) spatially localized at each side of a potential barrier are studied. After the interaction, each wave packet splits into a transmitted and a reflected component. It can be shown that the probability of detecting two (identically injected) electrons at the same side of the barrier is different from zero in very common (single or double barrier) scenarios. This originates an increase of quantum noise which cannot be obtained through the scattering states formalism.