A star graph consists of an arbitrary number of segments that are joined at a point which is called the vertex. In this work it is investigated from a pure theoretical point of view, in the framework of quantum field theory. As a concrete physical application, the electric conductance tensor is obtained. In particular it is shown that this conductance behaves differently according to whether the scattering matrix associated with the vertex of the graph has bound-state poles or not.

Thesis (Ph. D.)--Carleton University, 1992. / Also available in electronic format on the Internet.

No description available.

<div><div><div><p>Numerous fundamental quests and technological advances require trapping light waves. Generally, light is trapped by the absence of radiation channels or by forbid- ding access to them. Unconventional bound states of light, called bound states in the continuum (BICs), have recently gained tremendous interest due to their peculiar and extreme capabilities of trapping light in open structures with access to radiation. A BIC is a localized state of an open structure with access to radiation channels, yet it remains highly confined with, in theory, infinite lifetime and quality factor. There have been many realizations of such exceptional states in dielectric systems without loss. However, realizing BICs in lossy systems such as those in plasmonics remains a challenge. This thesis explores the realization of BICs in a hybrid plasmonic-photonic structure consisting of a plasmonic grating coupled to a dielectric optical waveguide with diverging radiative quality factors. The plasmonic-photonic system supports two distinct groups of BICs: symmetry protected BICs and Friedrich-Wintgen BICs. The photonic waveguide modes are strongly coupled to the gap plasmons in the grating leading to an avoided crossing behavior with a high value of Rabi splitting of 150 meV . Additionally, it is shown that the strong coupling significantly alters the band diagram of the hybrid system, revealing opportunities for supporting stopped light at an off-Γ wide angular span.</p><p>In another study, we demonstrate the design of a BIC-based all-dielectric metasurface and its application as a nanolaser. Metasurfaces have received an ever-growing interest due to their unprecedented ability to control light using subwavelength structures arranged in an ultrathin planar profile. However, the spectral response of meta- surfaces is generally broad, limiting their use in applications requiring high quality (Q) factors. In this study, we design, fabricate, and optically characterize metasur- faces with very high Q-factors operating near the BIC regime. The metasurfaces are coated with an organic lasing dye as an active medium, and their lasing action is experimentally characterized. The proposed BIC-based metasurfaces nanolaser have very favorable characteristics including low threshold, easily tunable resonances, polarization-independent response, and room temperature operation.</p><div><div><div><p>The second part of the thesis deals with the nonlinear phenomenon in nanopho- tonic structures. We developed an advanced full-wave framework to model nonlinear light-matter interactions. Rate equations, describing atomic relaxations and excita- tion dynamics, are coupled to the Maxwell equations using a Lorentzian oscillator that models the kinetics-dependent light-matter interaction in the form of averaged polarization. The coupled equations are discretized in space and time using a finite- difference time-domain method that provides a versatile multiphysics framework for designing complex structures and integrating diverse material models. The proposed framework is used to study gain dynamics in silver nanohole array, reverse saturable absorption dynamic in optical limiters, and saturable absorption in random lasers. This framework provides critical insights into the design of photonic devices and their complementary optical characterization, and serve as an invaluable utility for guiding the development of synthetic materials. It allows accurate physics-based numerical modeling and optimization of the devices with complex micro- and nano-structured materials and complex illumination sources such as non-paraxial structured beams.</p></div></div></div></div></div></div>

Indiana University-Purdue University Indianapolis (IUPUI) / In this thesis, we focus on two broad categories of problems, exciton condensation and bound states, and two complimentary approaches, real and momentum space, to solve these problems. In chapter 2 we begin by developing the self-consistent mean field equations, in momentum space, used to calculate exciton condensation in semiconductor heterostructures/double quantum wells and graphene. In the double quantum well case, where we have one layer containing electrons and the other layer with holes separated by a distance $d$, we extend the analytical solution to the two dimensional hydrogen atom in order to provide a semi-quantitative measure of when a system of excitons can be considered dilute. Next we focus on the problem of electron-electron screening, using the random phase approximation, in double layer graphene. The literature contains calculations showing that when screening is not taken into account the temperature at which excitons in double layer graphene condense is approximately room temperature. Also in the literature is a calculation showing that under certain assumptions the transition temperature is approximately \unit{mK}. The essential result is that the condensate is exponentially suppressed by the number of electron species in the system. Our mean field calculations show that the condensate, is in fact, not exponentially suppressed. Next, in chapter 3, we show the use of momentum space to solve the Schr\"{o}dinger equation for a class of potentials that are not usually a part of a quantum mechanics courses. Our approach avoids the typical pitfalls that exist when one tries to discretize the real space Schr\"{o}dinger equation. This technique widens the number of problems that can presented in an introductory quantum mechanics course while at the same time, because of the ease of its implementation, provides a simple introduction to numerical techniques and programming in general to students. We have furthered this idea by creating a modular program that allows students to choose the potential they wish to solve for while abstracting away the details of how the solution is found. In chapter 4 we revisit the single exciton and exciton condensation in double layer graphene problems through the use of real space lattice models. In the first section, we once again develop the equations needed to solve the problem of exciton condensation in a double layer graphene system. In addition to this we show that by using this technique, we find that for a non-interacting system with a finite non-zero tunneling between the layers that the on-site exciton density is proportional to the tunneling amplitude. The second section returns to the single exciton problem. In agreement with our momentum space calculations, we find that as the layer separation distance is increased the bound state wave function broadens. Finally, an interesting consequence of the lattice model is explored briefly. We show that for a system containing an electron in a periodic potential, there exists a bound state for both an attractive as well as repulsive potential. The bound state for the repulsive potential has as its energy $-E_0$ where $E_0$ is the ground state energy of the attractive potential with the same strength.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: The goal of this thesis is to investigate the effects of distorting a graphene lattice and connect- ing this distorted graphene sheet to a superconductor. At low energies the possible excitation states in graphene are restricted to two distinct regions in momentum space called valleys. Many electronic applications are possible if one can design a graphene system where excitations can be forced to occupy a single valley in a controllable way. Investigating the spectrum of the distorted graphene sheet reveals that, if the chemical potential is chosen to coincide with a bulk Landau level, the normal-superconductor interface always supports propagating modes in both directions. Excitations from opposite valleys travel in opposite directions along the interface. The spectrum of a distorted graphene sheet terminated by an armchair edge, in contrast, is dis- persionless. We verify this insulating nature of the armchair edge for finite samples by numerical means. Furthermore, we verify previous analytical results pertaining to a graphene sheet with NS interface and an applied perpendicular real magnetic field numerically. In the process, it is shown that considering graphene sheets of perfect width is not necessary, as long as the width a few magnetic lengths away from the interface is well-defined. By then considering a finite graphene sheet, terminated by armchair edges, that is distorted and connected to a superconductor, we find bound states near the NS interface that can be changed by distorting the graphene lattice further. / AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is om die uitwerking van die vervorming van ’n grafeenrooster te ondersoek wanneer die met ’n supergeleier verbind word. By lae energieë word die moontlike opwekkings in grafeen beperk tot twee aparte gebiede van momentumruimte — die sogenaamde valleie. Verskeie elektroniese toepassings is moontlik indien ’n grafeenstelsel ontwerp kan word waar opwekkings slegs ’n enkele vallei beset en die besetting beheer kan word. Deur die spektrum van die vervormde grafeenrooster te ondersoek word daar gevind dat, indien die chemiese potensiaal gekies word om saam te val met ’n Landauvlak, die NS-tussenvlak geleiding in beide rigtings ondersteun. Opwekkings van verskillende valleie beweeg in teenoorgestelde rigtings langs die tussenvlak. Daarteenoor is die spektrum van ’n vervormde grafeenroster met ’n leunstoelrand dispersieloos. Ons bevestig hierdie insulerende gedrag van ’n leunstoelrand vir eindige grafeen- roosters deur middel van ’n numeriese berekening. Verder word vorige analitiese resultate wat verband hou met ’n grafeenrooster met normaal-supergeleiertussenvlakstelsel en loodregte mag- neetveld op die vlak bevestig deur middel van numeriese berekeninge. In die proses word dit ook aangedui dat die grafeenrooster nie ’n perfekte wydte hoef te hˆe nie, solank die wydte goed gedefinieer is vir ’n paar magnetiese lengtes in die omgewing van die tussenvlak. Deur dan die eindige grafeenrooster met leunstoelrande te koppel aan ’n supergeleier word gebonde toe- stande naby aan die NS tussenvalk gevind. Hierdie toestande kan gemanipuleer word deur die grafeenrooster verder te vervorm.

Neste trabalho estudamos uma rede hexagonal com uma cadeia de impurezas nas bordas e com supercondutividade induzida, de forma a mostrar a existência de fases com férmions de Majorana. Para tal, começamos introduzindo invariantes topológicos, número de Chern e Z2 e mostramos dois modelos para rede hexagonal. O primeiro, modelo de Haldane, fazemos como motivação histórica. O segundo, modelo de Kane-Mele, é usado como base para todo o trabalho. Seguimos introduzindo supercondutividade e como ela ocorre quando aplicada junto do Modelo de Kane-Mele, o método auto-consistente e quais as condições necessárias para termos supercondutividade apenas nas bordas. Continuamos com efeitos de impurezas magnéticas nas bordas e introduzimos férmions de Majorana que são os alvos principais dos resultados. Mostramos então, que existe fases topológicas em cadeias de impureza magnética, com momentos em espiral, contudo o diagrama de fase depende de várias condições. Por fim, mostramos que a variação da fase topológica se deve a oscilações nos níveis de energia em que o invariante topológico também varia, contrariando resultados obtidos para a rede quadrada. Concluímos esse trabalho com implicações experimentais desse resultado e possíveis caminhos que podem ser seguidos. / In this work, we study a honeycomb lattice with induced superconductivity and edge impurity in order to show the existence of a phase that host Majorana bound state. To do so, we start introducing topological invariants, Chern number and Z2, and we show two models for honeycomb lattice. The first, Haldane\'s Model, due its historical importance. The second, Kane-Mele model, because it will be used during all this work. Then we review superconductivity, showing the self-consistent method, and we apply it to Kane-Mele model, in which we find some necessary conditions to induce superconductivity only at the edges. From this point, we study the effect of magnetic impurities at the edges, and we introduce Majorana bound state, that will be the main objective of our results. In our results, we show the existence of topological non-trivial phases for spiral magnetic chain in the zigzag edge. With this we make a phase diagram. We also find oscillation in the energy spectrum and the topological phase changes with the oscillation, this is different from square lattice in which we should not have a change in the topological phase. We conclude this work with experimental implications of our result and possible developments.

by Chan Kwan Leung. / Parallel title in Chinese characters. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 168-169). / by Chan Kwan Leung. / Acknowledgement --- p.i / Abstract --- p.ii / Chapter 1. --- Introduction / Chapter 1.1 --- Objective of our variational method --- p.2 / Chapter 1.2 --- Outline of the content --- p.5 / Chapter 2. --- Formulation of the new variational method / Chapter 2.1 --- Formulation --- p.14 / Chapter 2.2 --- Motivation --- p.15 / Chapter 3. --- The variational method applied to the anharmonic oscillator problem / Chapter 3.1 --- Formalism --- p.18 / Chapter 3.2 --- Relationship with usual variational method --- p.32 / Chapter 3.3 --- Relationship with W.K.B. approximation --- p.37 / Chapter 3.4 --- Perturbative corrections --- p.45 / Chapter 3.5 --- Diagonalization of non-orthogonal basis --- p.57 / Chapter 3.6 --- Perturbative corrections using the non-orthogonal basis --- p.72 / Chapter 3.7 --- Some previous works on the anharmonic oscillator problem --- p.85 / Chapter 4. --- The variational method applied to the helium-like atomic problem / Chapter 4.1 --- Previous work on the problem --- p.90 / Chapter 4.2 --- Formulation of the variational method on the problem --- p.95 / Chapter 4.3 --- Zeroth order results for atomic helium --- p.103 / Chapter 4.4 --- Diagonalization using the non-orthogonal basis --- p.109 / Chapter 4.5 --- Results for some helium-like ions --- p.136 / Chapter 4.6 --- Possibility of generalization to systems with more electrons --- p.140 / Chapter 5 --- Concluding remarks / Chapter 5.1 --- Range of applicability of our variational method --- p.164 / Chapter 5.2 --- Ground state problem --- p.165 / Chapter 5.3 --- Completeness of our 'basis' --- p.166 / References --- p.168

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Physics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Includes bibliographical references. / by Jens Hoppe. / Ph.D.

Orientador: Antonio Soares de Castro / Banca: Denis Dalmazi / Banca: Manuel Máximo B. Malheiro de Oliveira / Resumo: O estudo dos estados ligados das equações relativísticas têm muitas aplicações em física nuclear e em outras áreas da física, portanto as soluções que apresentam estados ligados usando as equações de Dirac e Klei-Gordon(KG) desempenham um papel importante no interesse dos pesquisadores. A redução a uma dimensão espacial das equações relativisticas permite melhorar o entendimento dos problemas equivalentes em três dimensões de uma maneira fisicamente mais transparente. Esta redução espacial traz como consequência alterações nas estruturas relativísticas dos potenciais que devem ser estudadas. A tarefa de achar estados ligados dessas emoções não é fácil para formas funcionais gerais para os potenciais externos. Neste trabalho as equações de Dirac e KG em um dimensão espacial são investigadas para diferentes tipos de acoplamento e formas funcionais para o potenciais externos que apresentam estados ligados.Um resultado relevantes que para certas misturas convenientes dos potenciais externos as equações de Dirac e KG apresentam as mesmas auto-energias mas diferentes autofunções. O problema em geral pode ser mapeado num problema de Sturm-Liouville encontrando-se soluções de estados ligados exatamente. Discutimos detalhadamente o comportamento das autofunções e auto-energias para partículas e antipartículas obtidas do problema de Sturm-Liouvillee as possíveis soluções isoladas no caso da equação de Dirac. Uma aparente violação do princípio da incerteza em algns casos é remediada com a introdução do conceito de comprimento de onda Compton efetivo, mostrando que a partícula pode ser localizada numa região do espaço arbitrariamente pequena sem a produção de pares partícula-antipartícula. / Abstract: The study of bound states of the relativistic equations as many applications in nuclear physics and other areas of the physics, therefor the solutions that present bound states for the Dirac and the Klein-Gordon (KG) equations play an important role in the interest of the researches. The reduction to one space dimension of the relativistic equations allows to improve our understanding of the equivalent problems in three dimensionin a physically more transparent way. This space reduction brings as a consequence alterations in the relativistic structures of the potentials that must be studied. The task of findingof these equations is not easy for general functional forms for the external potentials. In this work the Dirac and the KG equationsin one space dimensions for different tpes of coupling and functional forms for the external potentials that present bound states are investigated. A relevant result is that for certain convenient mixtures of the external potentials the Dirac and the KG equations present the same eigenvalues but different eigenfunctins. The problem in general can be mapped into an exactly solvable Sturm-Liouville problem. The behavior of the eigenfunctions and eigenvalues for particles and antiparticles of the Sturn-Liouville problem and the possible isolated solutions in the case of the Dirac equation is discussed in detail. An apparent breaking of the uncertainty priniple in some cases is remedied by the introduction of the concept of effective Compton wavelength, showing that the particle can be located into a region of space arbitrarily small without producing particle-antiparticle pairs. / Mestre