In this thesis, I present the work performed towards a proposal to couple a piezoelectric, nanomechanical beam to a radio frequency single electron transistor (RF-SET). Lumped element RF circuit theory is applied to 50 kOhm single electron transistors acting as the resistor in an RLC circuit. It is shown that for the expected inductances and stray capacitances, at an operating frequency of 1.25 GHz, the RF-SET is expected to have a usable half-bandwidth of 175-200 MHz and a charge sensitivity on the order of 10^(−5) e/√Hz. A fabricated RF-SET device is cryogenically cooled and used to find experimental values of the stray capacitance. A heterostructure made of gallium arsenide and aluminum gallium arsenide from which piezoelectric beams can be made is designed to contain a 2-dimensional electron gas (2DEG). Quantum Hall effect samples are fabricated from the wafer, and magnetoresistance measurements for each sample are presented. It is shown that the 2DEG has a high electron concentration of about 8 × 10^11 cm−2 but a low mobility of about 3.5 × 10^4 cm^2/(V·s) for this type of heterostructure. / Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2010-03-01 22:55:56.427

Thesis (M.Sc.). / Written for the Dept. of Physics. Title from title page of PDF (viewed 2008/12/04). Includes bibliographical references.

Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (leaves [108]-114).

Thesis (Ph. D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (May 2, 2007) Vita. Includes bibliographical references.

Le futur Système International d'unités, fondé sur des constantes fondamentales, va permettre de profiter pleinement des étalons quantiques de résistance, de courant et de tension qui sont reliés à la constante de planck et à la charge élémentaire. Dans cette thèse, nous avons développé et étudié un étalon de résistance fondé sur l'effet Hall quantique (EHQ) dans du graphène obtenu par dépôt chimique en phase vapeur (propane/hydrogène) sur substrat de carbure de silicium. Nous avons réussi à montrer, pour la première fois, qu'un étalon de résistance en graphène pouvait fonctionner à des conditions expérimentales plus pratiques que son homologue en GaAs/AlGaAs, c'est-à-dire à des températures plus élevées (T⋍10 K), des champs magnétiques plus faibles (B ⋍ 3,5 T) et des courants de mesures plus importants (I⋍500 μA). Dans une optique de compréhension et d'amélioration, nous avons analysé la reproductibilité du processus de fabrication de barres de Hall, testé une méthode de modification de la densité électronique et étudié les mécanismes de dissipation en régime d'EHQ.Dans une seconde partie, nous avons démontré qu'il était possible de réaliser une source de courant quantique programmable et versatile, directement reliée à la charge élémentaire, en combinant les deux étalons quantiques de tension et de résistance dans un circuit quantique intégrant un comparateur cryogénique de courant. Des courants ont ainsi pu être générés dans une gamme allant de 1 μA jusqu'à 5 mA avec une incertitude relative jamais atteinte de 10⁻⁸. Nous avons également prouvé que cet étalon de courant, réalisant la nouvelle définition de l'ampère, pouvait être utilisé pour étalonner un ampèremètre. / The future International System of Units, based on fundamental constants, will allow to take full advantage of the quantum standards of resistance, current and voltage that are linked to the planck constant and the elementary charge only.In this thesis, we have developed and studied a resistance standard based on the quantum Hall effect in graphene obtained by chemical vapor deposition (propane/hydrogen) on silicon carbide substrate. For the first time we were able to show that a graphene resistance standard could operate at more practical experimental conditions than its counterpart in GaAs/AlGaAs, ie at higher temperatures (T⋍10 K), weaker magnetics fields (B ⋍ 3,5 T) and larger measurement currents (I⋍500 μA). From an understanding and improvement perspective, we have analyzed the fabrication process of the Hall bar and its reproducibility, tested a method to modify the electronic density, and investigated the quantum Hall effect dissipation mechanisms.In a second part, we have demonstrated that it was possible torealize a programmable and versatile quantum current source from the elementary charge, by combining the two quantum standards of voltage and resistance in a quantum circuit integrating a cryogenic current comparator. Currents were generated in the range from 1 μA to 5 mA, with a relative uncertainty never achieved before of 10⁻⁸. We have also showed that this current standard, realizing the new definition of the ampere, could be used to calibrate an ammeter.

This work represents a first-principles calculation of the electron tunneling current into quantum dots in the fractional quantum Hall effect regime. The system under consideration consists of an idealized Scanning Tunneling Microscope (STM) tip and a quantum dot with disk geometry and interacting electrons in a transverse magnetic field. Within the context of this model the tunneling current between the STM tip and the dot is examined for spin-polarized electrons at and around a filling factor of 1/3. The current expression is based on a second-quantized Hamiltonian in which electrons in the dot are interacting, confined, and restricted to the lowest Landau level, necessary to capture the physics of the fractional quantum Hall effect. The Hamiltonian includes simple approximations for the STM tip and the tip-dot tunneling. An exact analytic expression for the first-order tunneling current is derived using a Green's function approach. To calculate the tunneling current numerically the infinite Hilbert space of the dot is truncated to have a finite dimension within the lowest Landau level. This simplification is appropriate for a low temperature system in the fractional quantum Hall regime because of the finite size of the quantum dot and the large energy gap between Landau levels. The tunneling current is then solved in two steps. First, many-electron energy eigenstates are calculated from the truncated Hamiltonian by numerical diagonalization. This is carried out for varying numbers of electrons N. The energy eigenstates form a set of complete basis states of the system and are used in the expression for the tunneling current. In the second step, the chemical potential in the dot is chosen to select a desired number of electrons and the tunneling current evaluated. We have carried out this program for filling factors near 1=3 while modulating the system parameters of interest to determine functional dependencies.

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The discovery of the fractional quantum Hall effect for two-dimensional electron gases immersed in a strong orthogonal magnetic field represents a cornerstone of modern physics. The states responsible for the appearance of the fractional quantum Hall effect have been found to be part of a whole new class of phases of matter, characterized by an internal order with unprecedented properties and known as topological order. This fact opened up a completely new territory for physical studies, paving the way towards many of the current hot topics in physics, such as topological phases of matter, topological order and topological quantum computing. As it happens for most topologically-ordered phases, fractional quantum Hall states are breeding ground for the observation of many exotic physical phenomena. Important examples include the appearance of degenerate ground states when the system in placed on a space with non-trivial topology, the existence of chiral gapless edge excitations which unidirectionally propagate without suffering of back-scattering processes, and the possibility of hosting elementary excitations, known as quasiparticles and quasiholes, carrying fractional charge and anyonic statistics. Even though for years since their discovery fractional quantum Hall states have been studied only in electronic systems, the recent advances made in the domains of quantum simulators and artificial gauge fields opened the possibility to realize bosonic analogs of these states in platforms based on ultracold atoms and photons. Reaching the appropriate conditions for the simulation of the fractional quantum Hall effect with neutral particles (such as atoms and photons) has required decades of both theoretical and experimental efforts and passed through the implementation of many topological models at the single-particle level. However, we strongly believe that the stage is set finally and that bosonic fractional quantum Hall states will be realized soon in different set-ups. Motivated by this fact, we dedicate this Thesis to the study of the edge and quasihole excitations of bosonic fractional quantum Hall states with the goal of guiding near future experiments towards exciting discoveries such as the observation of anyons. In the first part of the Thesis we focus our attention on the behavior of the edge excitations of the bosonic $ u=1/2$ Laughlin state (a paradigmatic wave function for the fractional quantum Hall effect) in the presence of cylindrically symmetric hard-wall confining potentials. With respect to electronic devices, atomic and photonic platforms offers indeed a more precise control on the external potential confining the systems, as confirmed by the recent realization of flat-bottomed traps for ultracold atoms and by the flexibility in designing optical cavities. At the same time, most of the theoretical works in this direction have considered harmonic confinements, for which the edge states have been found to display the standard chiral Luttinger liquid behavior, leaving the field open for our analysis of new physics beyond the Luttinger paradigm. In the second part we propose a novel method to probe the statistical properties of the quasihole excitations on top of a fractional quantum Hall state. As compared to the previous proposals, it does not rely on any form of interference and it has the undeniable advantage of requiring only the measurements of density-related observables. As we have already mentioned, although the existence of anyons have been theoretically predicted long time ago, it still lacks a clear-cut experimental evidence and this motivated people working with ultracold atoms and photons to push their systems into the fractional quantum Hall regime. However, while there exist plenty of proposals for the detection of anyons in solid-state systems (mostly based on interferometric schemes in which currents are injected into the system and anyons travel along its edges), in the present literature the number of detection schemes applicable in ultracold atomic and/or photonic set-ups is much smaller and they are typically as demanding as those proposed in the electronic context. Finally, in the last part of the Thesis we move to the lattice counterparts of the fractional quantum Hall states, the so-called fractional Chern insulators. Still with the purpose of paving the way for future experimental studies with quantum simulators, we focus our attention of the simplest bosonic version of these states and, in particular, on the properties of its quasihole excitations. Although this topic has already been the subject of intense studies, most of the previous works were limited either to system sizes which are too small to host anyonic excitations, or to unphysical conditions, such as periodic geometries and non-local Hamiltonians. Our study investigates for the first time the properties of genuine quasihole excitations in experimentally relevant situations.

This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.