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Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: High-dimensional quantum entanglement offers an increase in information capacity per photon; a highly desirable property for quantum information processes such as quantum communication, computation and teleportation. As the orbital angular momentum (OAM) modes of light span an infinite-dimensional Hilbert space, they have become frontrunners in achieving entanglement in higher dimensions. In light of this, we investigate the potential of OAM entanglement of photons by controlling the parameters in both the generation and measurement systems. We show the experimental procedures and apparatus involved in generating and measuring entangled photons in two-dimensions. We verify important quantum tests such as the Einstein, Podolsky and Rosen (EPR) paradox using OAM and angle correlations, as well as a violation of a Bell-type inequality. By performing a full state tomography, we characterise our quantum state and show we have a pure, highly entangled quantum state. We demonstrate that this method can be extended to higher dimensions. The experimental techniques used to generate and measure OAM entanglement place an upper bound on the number of accessible OAM modes. As such, we investigate new methods in which to increase the spiral bandwidth of our generated quantum state. We alter the shape of the pump beam in spontaneous parametric down-conversion and demonstrate an effect on both OAM and angle correlations. We also made changes to the measurement scheme by projecting the photon pairs into the Bessel-Gaussian (BG) basis and demonstrate entanglement in this basis. We show that this method allows the measured spiral bandwidth to be optimised by simply varying the continuous radial parameter of the BG modes. We demonstrate that BG modes can be entangled in higher dimensions compared with the commonly used helical modes by calculating and comparing the linear entropy and fidelity for both modes. We also show that quantum entanglement can be accurately simulated using classical light using back-projection, which allows the study of projective measurements and predicts the strength of the coincidence correlations in an entanglement experiment. Finally, we make use of each of the techniques to demonstrate the effect of a perturbation on OAM entanglement measured in the BG basis. We investigate the self-healing property of BG beams and show that the classical property is translated to the quantum regime. By calculating the concurrence, we see that measured entanglement recovers after encountering an obstruction. / AFRIKAANSE OPSOMMING: Hoë-dimensionele kwantumverstrengeldheid bied ’n toename in inligtingskapasiteit per foton. Hierdie is ’n hoogs wenslike eienskap vir kwantum inligting prosesse soos kwantum kommunikasie, berekening en teleportasie. Omdat die orbitale hoekmomentum (OAM) modusse van lig ’n oneindig dimensionele Hilbertruimte beslaan, het dit voorlopers geword in die verkryging van verstrengeling in hoër dimensies. In die lig hiervan, ondersoek ons die potensiaal van OAM verstrengeling van fotone deur die parameters in beide die generering en meting stelsels te beheer. Ons toon die eksperimentele prosedures en apparaat wat betrokke is by die generering en die meet van verstrengelde fotone in twee dimensies. Ons verifieer kwantumtoetse, soos die Einstein, Podolsky en Rosen (EPR) paradoks vir OAM en die hoekkorrelasies, sowel as ’n skending van ’n Bell-tipe ongelykheid. Deur middel van ’n volledige toestand tomografie, karakteriseer ons die kwantum toestand en wys ons dat dit ’n suiwer, hoogs verstrengel kwantum toestand is. Ons toon ook dat hierdie metode uitgebrei kan word na hoër dimensies. Die eksperimentele tegnieke wat tydens die generasie en meet van OAM verstrengeling gebruik is, plaas ’n bogrens op die aantal toeganklik OAM modusse. Dus ondersoek ons nuwe metodes om die spiraal bandwydte van ons gegenereerde kwantum toestand te verhoog. Ons verander die vorm van die pomp bundel in spontane parametriese af-omskakeling en demonstreer die uitwerking daarvan op beide OAM en die hoekkorrelasies. Ons het ook veranderinge aan die meting skema gemaak deur die foton pare op die Bessel-Gauss (BG) basis te projekteer. Ons wys dat hierdie metode die gemeetde spiraal bandwydte kan optimeer deur eenvoudig die kontinue radiale parameter van die BG modes te verander. Ons demonstreer dat BG modusse verstrengel kan word in hoër dimensies as die heliese modusse, wat algemeen gebruik word, deur berekeninge te maak en te vergelyk met lineêre entropie en vir beide modusse. Ons wys ook dat kwantumverstrengling akkuraat nageboots kan word, met behulp van die klassieke lig terug-projeksie, wat die studie van projeksie metings toelaat en voorspel die krag van die saamval korrelasies in ’n verstrengeling eksperiment. Ten slotte, gebruik ons elk van die tegnieke om die effek van ’n storing op OAM verstrengling wat in die BG basis gemeet is, te demonstreer. Ons ondersoek die self-genesingseienskap van BG bundels en wys dat die klassieke eienskap vertaal na die kwantum-gebied. Deur die berekening van die konkurrensie (concurrence), sien ons dat die gemeetde verstrengeling herstel word nadat ’n obstruksie ondervind is.

The interplay between different self-organized electronically ordered states and their relation to unconventional electronic properties like superconductivity constitutes one of the most exciting challenges of modern condensed matter physics. In the present thesis this issue is thoroughly investigated for the prototypical layered material 1T-TaS2 both experimentally and theoretically. At first the static charge density wave order in 1T-TaS2 is investigated as a function of pressure and temperature by means of X-ray diffraction. These data indeed reveal that the superconductivity in this material coexists with an inhomogeneous charge density wave on a macroscopic scale in real space. This result is fundamentally different from a previously proposed separation of superconducting and insulating regions in real space. Furthermore, the X-ray diffraction data uncover the important role of interlayer correlations in 1T-TaS2. Based on the detailed insights into the charge density wave structure obtained by the X-ray diffraction experiments, density functional theory models are deduced in order to describe the electronic structure of 1T-TaS2 in the second part of this thesis. As opposed to most previous studies, these calculations take the three-dimensional character of the charge density wave into account. Indeed the electronic structure calculations uncover complex orbital textures, which are interwoven with the charge density wave order and cause dramatic differences in the electronic structure depending on the alignment of the orbitals between neighboring layers. Furthermore, it is demonstrated that these orbital-mediated effects provide a route to drive semiconductor-to-metal transitions with technologically pertinent gaps and on ultrafast timescales. These results are particularly relevant for the ongoing development of novel, miniaturized and ultrafast devices based on layered transition metal dichalcogenides. The discovery of orbital textures also helps to explain a number of long-standing puzzles concerning the electronic self-organization in 1T-TaS2 : the ultrafast response to optical excitations, the high sensitivity to pressure as well as a mysterious commensurate phase that is commonly thought to be a special phase a so-called “Mott phase” and that is not found in any other isostructural modification.

To address key challenges for both quantum communication and quantum computing applications in a simultaneous manner, we propose to employ the photon angular momentum approach by invoking the well-known fact that photons carry both the spin angular momentum (SAM) and the orbital angular momentum (OAM). SAM is associated with polarization, while OAM is associated with azimuthal phase dependence of the complex electric field. Given that OAM eigenstates are mutually orthogonal, in principle, an arbitrary number of bits per single photon can be transmitted. The ability to generate/analyze states with different photon angular momentum, by using either holographic or interferometric methods, allows the realization of quantum states in multidimensional Hilbert space. Because OAM states provide an infinite basis state, while SAM states are 2-D only, the OAM can also be used to increase the security for quantum key distribution (QKD) applications and improve computational power for quantum computing applications. The goal of this paper is to describe photon angular momentum based deterministic universal quantum qudit gates, namely, {generalized-X, generalized-Z, generalized-CNOT} qudit gates, and different quantum modules of importance for various applications, including (fault-tolerant) quantum computing, teleportation, QKD, and quantum error correction. For instance, the basic quantum modules for quantum teleportation applications include the generalized-Bell-state generation module and the QFT-module. The basic quantum module for quantum error correction and fault-tolerant computing is the nonbinary syndrome calculator module. The basic module for entanglement assisted QKD is either the generalized-Bell-state generation module or the Weyl-operator-module. The possibility of implementing all these modules in integrated optics is discussed as well. Finally, we provide security analysis of entanglement assisted multidimensional QKD protocols, employing the proposed qudit modules, by taking into account the imperfect generation of OAM modes.

This dissertation focuses on the design and study of charged aromatic molecules where weak π-π interactions synergize with electrostatic interactions to enhance the overall interaction between aromatic moieties. Each chapter investigates some aspect of this hypothetical synergy between electrostatics and π-face aromatic cohesion. The first chapter unveiled the importance of electrostatics in the intramolecular stacking of flexible aromatic molecular templates 1-2Br and 2a. While our previous studies found dicationic molecular template 1-2Br to have intramolecular π-stacking between electron poor pyridinium and electron rich xylylene moieties, no such stacking interaction was observed in the neutral analog 2a. Chapter two systematically explored the stacking pattern of electron poor aromatics in the form of oxygen- and / or nitrogen- substituted triangulenium cations, [1(NR)3]+ and [1(O)3(OH)3]+. As indicated in the chemical literature, triazatriangulenium cations [1(NR)3]+ with N- ethyl (and longer alkyl chains) chains were found to pack as face-to-face dimers. This study found the formation of columnar, face-to-face, n-meric association between aromatic cations in the structures with decreased steric interactions of the side chains in the stacking planes ([1(NMe)3]+ and [1(O)3(OH)3]+). Similar iso-structural triangulene based aromatic anions, (2)- and (3)2- didn’t indicate any facial interactions in the solid states. The possible synergy between unit charge electrostatics and π-face aromatic interactions was explored in aromatic ion pairs 1•2 of triangulene based aromatic cations and aromatic anions. This charge-assisted π-π stacking seems to be the novel way of getting strong π-system interactions where the strongest non-covalent force and the weakest non-covalent force: ionic bonding and π-stacking respectively synergize together. The π-π interaction between ionic aromatics in the solid state was investigated by means of single crystal x-ray diffraction and powder x-ray diffraction (PXRD). The interaction in the solution state was examined by UV-Vis spectroscopy, electrospray ionization mass spectroscopy (ESI-MS) and electrochemical studies. Studies found that optimal synergy was possible only in the ion pairs with no steric interactions of alkyl (or aryl) side chains in the stacking planes (1(O)3•2 & 1(NMe)3•2) and the interaction was found to be comparable with the strongest radical-assisted π-stacking described in the chemical literature.

Ca2RuO4 is a structurally-driven Mott insulator with a metal-insulator (MI) transition at TMI = 357K, followed by a well-separated antiferromagnetic order at TN = 110 K. Slightly substituting Ru with a 3d transition metal ion M effectively shifts TMI and induces exotic magnetic behavior below TN. Moreover, M doping for Ru produces negative thermal expansion in Ca2Ru1-xMxO4 (M = Cr, Mn, Fe or Cu); the lattice volume expands on cooling with a total volume expansion ratio reaching as high as 1%. The onset of the negative thermal expansion closely tracks TMI and TN, sharply contrasting classic negative thermal expansion that shows no relevance to electronic properties. In addition, the observed negative thermal expansion occurs near room temperature and extends over a wide temperature interval. These findings underscores new physics driven by a complex interplay between orbital, spin and lattice degrees of freedom. These materials constitute a new class of Negative Thermal Expansion (NTE) materials with novel electronic and magnetic functions.

Spectroscopic orbital elements and/or related parameters have been determined for eight binary systems, using radial-velocity measurements that have a typical precision of about 15 ms⁻¹. The orbital periods of these systems range from about 10 days to 26 years, with a median of about 6 years. Orbital solutions were determined for the seven systems with shorter periods. The measurement of the mass ratio of the longest-period system, HD217166, demonstrates that this important astrophysical quantity can be estimated in a model-free manner with less than 10% of the orbital cycle observed spectroscopically.\\ Single-lined orbital solutions have been derived for five of the binaries. Two of these systems are astrometric binaries: β Ret and ν Oct. The other SB1 systems were 94 Aqr A, θ Ant, and the 10-day system, HD159656. The preliminary spectroscopic solution for θ Ant (P~18 years), is the first one derived for this system. The improvement to the precision achieved for the elements of the other four systems was typically between 1--2 orders of magnitude. The very high precision with which the spectroscopic solution for HD159656 has been measured should allow an investigation into possible apsidal motion in the near future. In addition to the variable radial velocity owing to its orbital motion, the K-giant, ν Oct, has been found to have an additional long-term irregular periodicity, attributed, for the time being, to the rotation of a large surface feature.\\ Double-lined solutions were obtained for HD206804 (K7V+K7V), which previously had two competing astrometric solutions but no spectroscopic solution, and a newly discovered seventh-magnitude system, HD181958 (F6V+F7V). This latter system has the distinction of having components and orbital characteristics whose study should be possible with present ground-based interferometers. All eight of the binary systems have had their mass ratio and the masses of their components estimated.\\ The following comments summarize the motivation for getting these results, and the manner in which the research was carried out. \\ The majority of stars exist in binary systems rather than singly as does the Sun. These systems provide astronomers with the most reliable and proven means to determine many of the fundamental properties of stars. One of these properties is the stellar mass, which is regarded as being the most important of all, since most other stellar characteristics are very sensitive to the mass. Therefore, empirical masses, combined with measurements of other stellar properties, such as radii and luminosities, are an excellent test for competing models of stellar structure and evolution.\\ Binary stars also provide opportunities to observe and investigate many extraordinary astrophysical processes that do not occur in isolated stars. These processes often arise as a result of direct and indirect interactions between the components, when they are sufficiently close to each other. Some of the interactions are relatively passive, such as the circularization of the mutual orbits, whilst others result from much more active processes, such as mass exchange leading to intense radiation emissions. \\ A complete understanding of a binary system's orbital characteristics, as well as the measurement of the all-important stellar masses, is almost always only achieved after the binary system has been studied using two or more complementary observing techniques. Two of the suitable techniques are astrometry and spectroscopy. In favourable circumstances, astrometry can deduce the angular dimensions of the orbit, the total mass of the system, and sometimes, its distance from us. Spectroscopy, on the other hand, can determine the linear scale of the orbit and the ratio of the stellar masses, based on the changing radial velocities of both stars. When a resolved astrometric orbital solution is also available, the velocities of both stars can allow the binary system's parallax to be determined, and the velocities of one star can provide a measure of the system mass ratio.\\ Unfortunately, relatively few binary systems are suited to these complementary studies. Underlying this difficulty are the facts that, typically, astrometrically-determined orbits favour those with periods of years or decades, whereas spectroscopic orbital solutions are more often measured for systems with periods of days to months. With the development of high-resolution astrometric and spectroscopic techniques in recent years, it is hoped that many more binary systems will be amenable to these complementary strategies.\\ Several months after this thesis began, a high-resolution spectrograph, HERCULES, commenced operations at the Mt John University Observatory, to be used in conjuction with the 1-metre McLellan telescope. For late-type stars, the anticipated velocity precision was ≲10 ms⁻¹. The primary goals of this thesis were: 1.~to assess the performance of HERCULES and the related reduction software that subsequently followed, 2.~to carry out an observational programme of 20 or so binary systems, and 3.~to determine the orbital and stellar parameters which characterize some of these systems. The particular focus was on those binaries that have resolved or unresolved astrometric orbital solutions, which therefore may be suited to complementary investigations.\\ HERCULES was used to acquire spectra of the programme stars, usually every few weeks, over a timespan of about three years. High-resolution spectra were acquired for the purpose of measuring precise radial velocities of the stars. When possible, orbital solutions were derived from these velocities, using the method of differential corrections.

The combined observations of a planet’s transits and the radial velocity variations of its host star allow the determination of the planet’s orbital parameters, and most inter- estingly of its radius and mass, and hence its mean density. Observed densities provide important constraints to planet structure and evolution models. The uncertainties on the parameters of large exoplanets mainly arise from those on stellar masses and radii. For small exoplanets, the treatment of stellar variability limits the accuracy on the de- rived parameters. The goal of this PhD thesis was to reduce these sources of uncertainty by developing new techniques for stellar variability filtering and for the determination of stellar temperatures, and by robustly fitting the transits taking into account external constraints on the planet’s host star. To this end, I developed the Iterative Reconstruction Filter (IRF), a new post-detection stellar variability filter. By exploiting the prior knowledge of the planet’s orbital period, it simultaneously estimates the transit signal and the stellar variability signal, using a com- bination of moving average and median filters. The IRF was tested on simulated CoRoT light curves, where it significantly improved the estimate of the transit signal, particu- lary in the case of light curves with strong stellar variability. It was then applied to the light curves of the first seven planets discovered by CoRoT, a space mission designed to search for planetary transits, to obtain refined estimates of their parameters. As the IRF preserves all signal at the planet’s orbital period, t can also be used to search for secondary eclipses and orbital phase variations for the most promising cases. This en- abled the detection of the secondary eclipses of CoRoT-1b and CoRoT-2b in the white (300–1000 nm) CoRoT bandpass, as well as a marginal detection of CoRoT-1b’s orbital phase variations. The wide optical bandpass of CoRoT limits the distinction between thermal emission and reflected light contributions to the secondary eclipse. I developed a method to derive precise stellar relative temperatures using equiv- alent width ratios and applied it to the host stars of the first eight CoRoT planets. For stars with temperature within the calibrated range, the derived temperatures are con- sistent with the literature, but have smaller formal uncertainties. I then used a Markov Chain Monte Carlo technique to explore the correlations between planet parameters derived from transits, and the impact of external constraints (e.g. the spectroscopically derived stellar temperature, which is linked to the stellar density). Globally, this PhD thesis highlights, and in part addresses, the complexity of perform- ing detailed characterisation of transit light curves. Many low amplitude effects must be taken into account: residual stellar activity and systematics, stellar limb darkening, and the interplay of all available constraints on transit fitting. Several promising areas for further improvements and applications were identified. Current and future high precision photometry missions will discover increasing numbers of small planets around relatively active stars, and the IRF is expected to be useful in characterising them.

For certain nonlinear Schroedinger equations there exist solutions which are called solitary waves. Addition of a potential $V$ changes the dynamics, but for small enough $||V||_{L^\infty}$ we can still obtain stability (and approximately Newtonian motion of the solitary wave's center of mass) for soliton-like solutions up to a finite time that depends on the size and scale of the potential $V$. Our method is an adaptation of the well-known Lyapunov method. For the sake of completeness, we also prove long-time stability of traveling solitons in the case $V=0$.