Quantum hydrodynamic models for semiconductors with and without collisions

In dieser Arbeit werden Quantum-Hydrodynamische (QHD) Modelle betrachtet, die ihren Einsatz besonders in der Modellierung von Halbleiterbauteilen finden. Das QHD Modell besteht aus den Erhaltungsgleichungen für die Teilchendichte, das Momentum und die Energiedichte, inklusive der Quanten-Korrekturen durch das Bohmsche Potential. Zu Beginn wird eine Übersicht über die bekannten Ergebnisse der QHD Modelle unter Vernachlässigung von Kollisionseffekten gegeben, die aus ein­em Schrödinger-System für den gemischten-Zustand oder aus der Wigner-Glei­chung hergeleitet werden können. Nach der Reformulierung der eindimensionalen QHD Gleichungen mit linearem Potential als stationäre Schrö­din­ger-Gleichung werden die semianalytischen Fassungen der QHD Gleichungen für die Gleichspannungs-Kurve betrachtet. Weiterhin werden die viskosen Stabilisierungen des QHD Modells be­rück­sich­tigt, sowie die von Gardner vorgeschlagene numerische Viskosität für das {sf upwind} Finite-Differenzen Schema berechnet. Im Weiteren wird das viskose QHD Modell aus der Wigner-Glei­chung mit Fokker-Planck Kollisions-Ope­ra­tor hergeleitet. Dieses Modell enthält die physikalische Viskosität, die durch den Kollision-Operator eingeführt wird. Die Existenz der Lösungen (mit strikt positiver Teilchendichte) für das isotherme, stationäre, eindimensionale, viskose Modell für allgemeine Daten und nichthomogene Randbedingungen wird gezeigt. Die dafür notwendigen Abschätzungen hängen von der Viskosität ab und erlauben daher den Grenzübergang zum nicht-viskosen Fall nicht. Numerische Simulationen der Resonanz-Tunneldiode modelliert mit dem nichtisothermen, stationären, eindimensionalen, viskosen QHD Modell zeigen den Einfluss der Viskosität auf die Lösung. Unter Verwendung des von Degond und Ringhofer entwickelten Quanten-Entropie-Minimierungs-Verfahren werden die allgemeinen QHD-Gleichungen aus der Wigner-Boltzmann-Gleichung mit dem BGK-Kollisions-Operator hergeleitet. Die Herleitung basiert auf der vorsichtige Entwicklung des Quanten-Max­well­ians in Potenzen der skalierten Plankschen Konstante. Das so erhaltene Modell enthält auch vertex-Terme und dispersive Terme für die Ge­schwin­dig­keit. Dadurch bleibt die Gleichspannungs-Kurve für die Re­so­nanz-Tunnel­diode unter Verwendung des allgemeinen QHD Modells in einer Dimension numerisch erhalten. Die Ergebnisse zeigen, dass der dispersive Ge­schwin­dig­keits-Term die Lösung des Systems stabilisiert.

Quantum Correlations in Field Theory and Integrable Systems

In this thesis we will investigate some properties of one-dimensional quantum systems. From a theoretical point of view quantum models in one dimension are particularly interesting because they are strongly interacting, since particles cannot avoid each other in their motion, and you we can never ignore collisions. Yet, integrable models often generate new and non-trivial solutions, which could not be found perturbatively. In this dissertation we shall focus on two important aspects of integrable one- dimensional models: Their entanglement properties at equilibrium and their dynamical correlators after a quantum quench. The first part of the thesis will be therefore devoted to the study of the entanglement entropy in one- dimensional integrable systems, with a special focus on the XYZ spin-1/2 chain, which, in addition to being integrable, is also an interacting model. We will derive its Renyi entropies in the thermodynamic limit and its behaviour in different phases and for different values of the mass-gap will be analysed. In the second part of the thesis we will instead study the dynamics of correlators after a quantum quench , which represent a powerful tool to measure how perturbations and signals propagate through a quantum chain. The emphasis will be on the Transverse Field Ising Chain and the O(3) non-linear sigma model, which will be both studied by means of a semi-classical approach. Moreover in the last chapter we will demonstrate a general result about the dynamics of correlation functions of local observables after a quantum quench in integrable systems. In particular we will show that if there are not long-range interactions in the final Hamiltonian, then the dynamics of the model (non equal- time correlations) is described by the same statistical ensemble that describes its statical properties (equal-time correlations).

Quantum Integrability in Non-Linear Sigma Models related to Gauge/String Correspondences

The Thermodynamic Bethe Ansatz analysis is carried out for the extended-CP^N class of integrable 2-dimensional Non-Linear Sigma Models related to the low energy limit of the AdS_4xCP^3 type IIA superstring theory. The principal aim of this program is to obtain further non-perturbative consistency check to the S-matrix proposed to describe the scattering processes between the fundamental excitations of the theory by analyzing the structure of the Renormalization Group flow. As a noteworthy byproduct we eventually obtain a novel class of TBA models which fits in the known classification but with several important differences. The TBA framework allows the evaluation of some exact quantities related to the conformal UV limit of the model: effective central charge, conformal dimension of the perturbing operator and field content of the underlying CFT. The knowledge of this physical quantities has led to the possibility of conjecturing a perturbed CFT realization of the integrable models in terms of coset Kac-Moody CFT. The set of numerical tools and programs developed ad hoc to solve the problem at hand is also discussed in some detail with references to the code.

Supersymmetric quantum mechanics on the lattice: II. Exact results

Simulations of supersymmetric field theories with spontaneously broken supersymmetry require in addition to the ultraviolet regularisation also an infrared one, due to the emergence of the massless Goldstino. The intricate interplay between ultraviolet and infrared effects towards the continuum and infinite volume limit demands careful investigations to avoid potential problems. In this paper – the second in a series of three – we present such an investigation for N=2 supersymmetric quantum mechanics formulated on the lattice in terms of bosonic and fermionic bonds. In one dimension, the bond formulation allows to solve the system exactly, even at finite lattice spacing, through the construction and analysis of transfer matrices. In the present paper we elaborate on this approach and discuss a range of exact results for observables such as the Witten index, the mass spectra and Ward identities.

Solution of the relativistic Schrödinger equation for the δ ′ -Function potential in one dimension using cutoff regularization

We study the relativistic version of the Schrödinger equation for a point particle in one dimension with the potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states. The initial calculations show that the reciprocal of the bare coupling constant is ultraviolet divergent, and the resultant expression cannot be renormalized in the usual sense, where the divergent terms can just be omitted. Therefore, a general procedure has been developed to derive different physical properties of the system. The procedure is used first in the nonrelativistic case for the purpose of clarification and comparisons. For the relativistic case, the results show that this system behaves exactly like the delta function potential, which means that this system also shares features with quantum filed theories, like being asymptotically free. In addition, in the massless limit, it undergoes dimensional transmutation, and it possesses an infrared conformal fixed point. The comparison of the solution with the relativistic delta function potential solution shows evidence of universality.

Virtual-bound, filamentary and layered states in a box-shaped quantum dot of square potential form the exact numerical solution of the effective mass Schrodinger equation

The effective mass Schrodinger equation of a QD of parallelepipedic shape with a square potential well is solved by diagonalizing the exact Hamiltonian matrix developed in a basis of separation-of-variables wavefunctions. The expected below bandgap bound states are found not to differ very much from the former approximate calculations. In addition, the presence of bound states within the conduction band is confirmed. Furthermore, filamentary states bounded in two dimensions and extended in one dimension and layered states with only one dimension bounded, all within the conduction band which are similar to those originated in quantum wires and quantum wells coexist with the ordinary continuum spectrum of plane waves. All these subtleties are absent in spherically shaped quantum dots, often used for modeling.

Testing integrability with a single bit of quantum information

We show that deterministic quantum computing with a single bit can determine whether the classical limit of a quantum system is chaotic or integrable using O(N) physical resources, where N is the dimension of the Hilbert space of the system under study. This is a square-root improvement over all known classical procedures. Our study relies strictly on the random matrix conjecture. We also present numerical results for the nonlinear kicked top.

Cluster-state quantum computation

This article is a short introduction to and review of the cluster-state model of quantum computation, in which coherent quantum information processing is accomplished via a sequence of single-qubit measurements applied to a fixed quantum state known as a cluster state. We also discuss a few novel properties of the model, including a proof that the cluster state cannot occur as the exact ground state of any naturally occurring physical system, and a proof that measurements on any quantum state which is linearly prepared in one dimension can be efficiently simulated on a classical computer, and thus are not candidates for use as a substrate for quantum computation.

Quantum nondemolition measurements for quantum information

We discuss the characterization and properties of quantum nondemolition (QND) measurements on qubit systems. We introduce figures of merit which can be applied to systems of any Hilbert space dimension, thus providing universal criteria for characterizing QND measurements. The controlled-NOT gate and an optical implementation are examined as examples of QND devices for qubits. We also consider the QND measurement of weak values.

Studies on Wilson loops, correlators and localization in supersymmetric quantum field theories

In this thesis we study at perturbative level correlation functions of Wilson loops (and local operators) and their relations to localization, integrability and other quantities of interest as the cusp anomalous dimension and the Bremsstrahlung function. First of all we consider a general class of 1/8 BPS Wilson loops and chiral primaries in N=4 Super Yang-Mills theory. We perform explicit two-loop computations, for some particular but still rather general configuration, that confirm the elegant results expected from localization procedure. We find notably full consistency with the multi-matrix model averages, obtained from 2D Yang-Mills theory on the sphere, when interacting diagrams do not cancel and contribute non-trivially to the final answer. We also discuss the near BPS expansion of the generalized cusp anomalous dimension with L units of R-charge. Integrability provides an exact solution, obtained by solving a general TBA equation in the appropriate limit: we propose here an alternative method based on supersymmetric localization. The basic idea is to relate the computation to the vacuum expectation value of certain 1/8 BPS Wilson loops with local operator insertions along the contour. Also these observables localize on a two-dimensional gauge theory on S^2, opening the possibility of exact calculations. As a test of our proposal, we reproduce the leading Luscher correction at weak coupling to the generalized cusp anomalous dimension. This result is also checked against a genuine Feynman diagram approach in N=4 super Yang-Mills theory. Finally we study the cusp anomalous dimension in N=6 ABJ(M) theory, identifying a scaling limit in which the ladder diagrams dominate. The resummation is encoded into a Bethe-Salpeter equation that is mapped to a Schroedinger problem, exactly solvable due to the surprising supersymmetry of the effective Hamiltonian. In the ABJ case the solution implies the diagonalization of the U(N) and U(M) building blocks, suggesting the existence of two independent cusp anomalous dimensions and an unexpected exponentation structure for the related Wilson loops.

Exotic meson decay widths using lattice quantum chromodynamics

Most experiments in particle physics are scattering experiments, the analysis of which leads to masses, scattering phases, decay widths and other properties of one or multi-particle systems. Until the advent of Lattice Quantum Chromodynamics (LQCD) it was difficult to compare experimental results on low energy hadron-hadron scattering processes to the predictions of QCD, the current theory of strong interactions. The reason being, at low energies the QCD coupling constant becomes large and the perturbation expansion for scattering; amplitudes does not converge. To overcome this, one puts the theory onto a lattice, imposes a momentum cutoff, and computes the integral numerically. For particle masses, predictions of LQCD agree with experiment, but the area of decay widths is largely unexplored. ^ LQCD provides ab initio access to unusual hadrons like exotic mesons that are predicted to contain real gluonic structure. To study decays of these type resonances the energy spectra of a two-particle decay state in a finite volume of dimension L can be related to the associated scattering phase shift δ(k) at momentum k through exact formulae derived by Lüscher. Because the spectra can be computed using numerical Monte Carlo techniques, the scattering phases can thus be determined using Lüscher's formulae, and the corresponding decay widths can be found by fitting Breit-Wigner functions. ^ Results of such a decay width calculation for an exotic hybrid( h) meson (JPC = 1-+) are presented for the decay channel h → πa 1. This calculation employed Lüscher's formulae and an approximation of LQCD called the quenched approximation. Energy spectra for the h and πa1 systems were extracted using eigenvalues of a correlation matrix, and the corresponding scattering phase shifts were determined for a discrete set of πa1 momenta. Although the number of phase shift data points was sparse, fits to a Breit-Wigner model were made, resulting in a decay width of about 60 MeV. ^

MOVPE growth and characterization of Al(Ga)N and InAlN/AlGaN quantum wells for UV LED applications

The study of III-nitride materials (InN, GaN and AlN) gained huge research momentum after breakthroughs in the production light emitting diodes (LEDs) and laser diodes (LDs) over the past two decades. Last year, the Nobel Prize in Physics was awarded jointly to Isamu Akasaki, Hiroshi Amano and Shuji Nakamura for inventing a new energy efficient and environmental friendly light source: blue light-emitting diode (LED) from III-nitride semiconductors in the early 1990s. Nowadays, III-nitride materials not only play an increasingly important role in the lighting technology, but also become prospective candidates in other areas, for example, the high frequency (RF) high electron mobility transistor (HEMT) and photovoltaics. These devices require the growth of high quality III-nitride films, which can be prepared using metal organic vapour phase epitaxy (MOVPE). The main aim of my thesis is to study and develop the growth of III-nitride films, including AlN, u-AlGaN, Si-doped AlGaN, and InAlN, serving as sample wafers for fabrication of ultraviolet (UV) LEDs, in order to replace the conventional bulky, expensive and environmentally harmful mercury lamp as new UV light sources. For application to UV LEDs, reducing the threading dislocation density (TDD) in AlN epilayers on sapphire substrates is a key parameter for achieving high-efficiency AlGaNbased UV emitters. In Chapter 4, after careful and systematic optimisation， a working set of conditions, the screw and edge type dislocation density in the AlN were reduced to around 2.2×108 cm-2 and 1.3×109 cm-2 , respectively, using an optimized three-step process, as estimated by TEM. An atomically smooth surface with an RMS roughness of around 0.3 nm achieved over 5×5 µm 2 AFM scale. Furthermore, the motion of the steps in a one dimension model has been proposed to describe surface morphology evolution, especially the step bunching feature found under non-optimal conditions. In Chapter 5, control of alloy composition and the maintenance of compositional uniformity across a growing epilayer surface were demonstrated for the development of u-AlGaN epilayers. Optimized conditions (i.e. a high growth temperature of 1245 °C) produced uniform and smooth film with a low RMS roughness of around 2 nm achieved in 20×20 µm 2 AFM scan. The dopant that is most commonly used to obtain n-type conductivity in AlxGa1-xN is Si. However, the incorporation of Si has been found to increase the strain relaxation and promote unintentional incorporation of other impurities (O and C) during Si-doped AlGaN growth. In Chapter 6, reducing edge-type TDs is observed to be an effective appoach to improve the electric and optical properties of Si-doped AlGaN epilayers. In addition, the maximum electron concentration of 1.3×1019 cm-3 and 6.4×1018 cm-3 were achieved in Si-doped Al0.48Ga0.52N and Al0.6Ga0.4N epilayers as measured using Hall effect. Finally, in Chapter 7, studies on the growth of InAlN/AlGaN multiple quantum well (MQW) structures were performed, and exposing InAlN QW to a higher temperature during the ramp to the growth temperature of AlGaN barrier (around 1100 °C) will suffer a significant indium (In) desorption. To overcome this issue, quasi-two-tempeature (Q2T) technique was applied to protect InAlN QW. After optimization, an intense UV emission from MQWs has been observed in the UV spectral range from 320 to 350 nm measured by room temperature photoluminescence.

A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)

We propose a family of local CSS stabilizer codes as possible candidates for self-correcting quantum memories in 3D. The construction is inspired by the classical Ising model on a Sierpinski carpet fractal, which acts as a classical self-correcting memory. Our models are naturally defined on fractal subsets of a 4D hypercubic lattice with Hausdorff dimension less than 3. Though this does not imply that these models can be realized with local interactions in R3, we also discuss this possibility. The X and Z sectors of the code are dual to one another, and we show that there exists a finite temperature phase transition associated with each of these sectors, providing evidence that the system may robustly store quantum information at finite temperature.

Disorder driven transitions in non-equilibrium quantum systems

This thesis presents studies of the role of disorder in non-equilibrium quantum systems. The quantum states relevant to dynamics in these systems are very different from the ground state of the Hamiltonian. Two distinct systems are studied, (i) periodically driven Hamiltonians in two dimensions, and (ii) electrons in a one-dimensional lattice with power-law decaying hopping amplitudes. In the first system, the novel phases that are induced from the interplay of periodic driving, topology and disorder are studied. In the second system, the Anderson transition in all the eigenstates of the Hamiltonian are studied, as a function of the power-law exponent of the hopping amplitude.

In periodically driven systems the study focuses on the effect of disorder in the nature of the topology of the steady states. First, we investigate the robustness to disorder of Floquet topological insulators (FTIs) occurring in semiconductor quantum wells. Such FTIs are generated by resonantly driving a transition between the valence and conduction band. We show that when disorder is added, the topological nature of such FTIs persists as long as there is a gap at the resonant quasienergy. For strong enough disorder, this gap closes and all the states become localized as the system undergoes a transition to a trivial insulator.

Interestingly, the effects of disorder are not necessarily adverse, disorder can also induce a transition from a trivial to a topological system, thereby establishing a Floquet Topological Anderson Insulator (FTAI). Such a state would be a dynamical realization of the topological Anderson insulator. We identify the conditions on the driving field necessary for observing such a transition. We realize such a disorder induced topological Floquet spectrum in the driven honeycomb lattice and quantum well models.

Finally, we show that two-dimensional periodically driven quantum systems with spatial disorder admit a unique topological phase, which we call the anomalous Floquet-Anderson insulator (AFAI). The AFAI is characterized by a quasienergy spectrum featuring chiral edge modes coexisting with a fully localized bulk. Such a spectrum is impossible for a time-independent, local Hamiltonian. These unique characteristics of the AFAI give rise to a new topologically protected nonequilibrium transport phenomenon: quantized, yet nonadiabatic, charge pumping. We identify the topological invariants that distinguish the AFAI from a trivial, fully localized phase, and show that the two phases are separated by a phase transition.

The thesis also present the study of disordered systems using Wegner's Flow equations. The Flow Equation Method was proposed as a technique for studying excited states in an interacting system in one dimension. We apply this method to a one-dimensional tight binding problem with power-law decaying hoppings. This model presents a transition as a function of the exponent of the decay. It is shown that the the entire phase diagram, i.e. the delocalized, critical and localized phases in these systems can be studied using this technique. Based on this technique, we develop a strong-bond renormalization group that procedure where we solve the Flow Equations iteratively. This renormalization group approach provides a new framework to study the transition in this system.

On Ordinal VC-Dimension and Some Notions of Complexity

We generalize the classical notion of Vapnik–Chernovenkis (VC) dimension to ordinal VC-dimension, in the context of logical learning paradigms. Logical learning paradigms encompass the numerical learning paradigms commonly studied in Inductive Inference. A logical learning paradigm is defined as a set W of structures over some vocabulary, and a set D of first-order formulas that represent data. The sets of models of ϕ in W, where ϕ varies over D, generate a natural topology W over W. We show that if D is closed under boolean operators, then the notion of ordinal VC-dimension offers a perfect characterization for the problem of predicting the truth of the members of D in a member of W, with an ordinal bound on the number of mistakes. This shows that the notion of VC-dimension has a natural interpretation in Inductive Inference, when cast into a logical setting. We also study the relationships between predictive complexity, selective complexity—a variation on predictive complexity—and mind change complexity. The assumptions that D is closed under boolean operators and that W is compact often play a crucial role to establish connections between these concepts. We then consider a computable setting with effective versions of the complexity measures, and show that the equivalence between ordinal VC-dimension and predictive complexity fails. More precisely, we prove that the effective ordinal VC-dimension of a paradigm can be defined when all other effective notions of complexity are undefined. On a better note, when W is compact, all effective notions of complexity are defined, though they are not related as in the noncomputable version of the framework.