Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to weak perturbations is now particularly well advanced, and has allowed for the development of a theory of weakly connected neural networks. However, the strong-attraction assumption may well not be the natural one for many neural oscillator models. For example, the popular conductance based Morris-Lecar model is known to respond to periodic pulsatile stimulation in a chaotic fashion that cannot be adequately described with a phase reduction. In this paper, we generalise the phase description that allows one to track the evolution of distance from the cycle as well as phase on cycle. We use a classical technique from the theory of ordinary differential equations that makes use of a moving coordinate system to analyse periodic orbits. The subsequent phase-amplitude description is shown to be very well suited to understanding the response of the oscillator to external stimuli (which are not necessarily weak). We consider a number of examples of neural oscillator models, ranging from planar through to high dimensional models, to illustrate the effectiveness of this approach in providing an improvement over the standard phase-reduction technique. As an explicit application of this phase-amplitude framework, we consider in some detail the response of a generic planar model where the strong-attraction assumption does not hold, and examine the response of the system to periodic pulsatile forcing. In addition, we explore how the presence of dynamical shear can lead to a chaotic response.

The purpose of this study was to investigate some theoretical approaches to the scattering of positive k-mesons by nucleons in an attempt to explain the experimental data. In this work the problem has been investigated by the technique of the weak coupling approximation.

Exchange rate determination allows precise modeling of chemical systems, and allows one to infer properties relevant to tumor biology such as enzyme activity and pH. Current exchange rate determination methods found via Contrast Enhanced Saturation Transfer agents are not effective for fast exchanging protons and use non-linear models. A comparison of their effectiveness has not been performed. In this thesis, I compare the effectiveness of current exchange rate measurement methods. I also develop exchange rate measurement methods that are effective for fast exchanging CEST agents and use linear models instead of non-linear models. In chapter 1 I review current exchange rate measurement methods. In chapter 2 I compare several of the current methods of exchange rate measurement, along with several techniques we develop. In chapter 3 I linearize the Quantifying Exchange through Saturation Transfer (QUEST) measurement method analogously to the Omega Plot method, and compare its effectiveness to the QUEST method. In chapter 4, I compare the effectiveness of current exchange rate theories (Transition State Theory and Landau-Zener theory) in the moderate coupling regime, and propose our own combined Eyring-Landau-Zener theory for this intermediate regime. In chapter 5 I discuss future directions for method development and experiments involving exchange rate determination.

<p> The influence of a logarithmically dependent (van Hove singularity) electronic density of states is studied in the weak-coupling limit. Through analytic and numerical analysis it is found that the model can give rise to temperatures in the 100 K range, and that universal BCS ratios such as 2Δ/kBTC and ΔC/γTC do not change essentially from their constant BCS values. The consequences of this model on the calculation of the isotope effect and specific heat are discussed in detail and compared to recent experimental results.</p> / Thesis / Master of Science (MSc)

Superradiance is an enhanced decay of an excited system of emitters resulting from their mutual coupling. This thesis is focused on superradiance of the emitters coupled via their interaction with a plasmonic nanoparticle. So-called plasmon-mediated superradiance results in even stronger enhancement of the decay rate as the nanoparticle serves as an additional decay chanel. We have developed a quantum model of the system of emitters coupled to a plasmonic nanoparticle, which allows us to differentiate between a pure dephasing and decay processes. We show that the pure dephasing can destroy the cooperative effect leading to superradiance. Furthermore, we have studied how the direct mutual coupling between emitters affects time evolution of the system in dependence on its configuration, and we show conditions when a decay of the system is dramatically decreased by direct coupling.

In this thesis, we study the unconventional superconductivity in Sr2RuO4 (SRO) and related systems. The superconducting state in SRO remains a puzzle after more than 28 years of study. Early experiments had pointed toward a topological non-trivial time-reversal symmetry breaking (TRSB) chiral p-wave order. This pairing candidate has attracted a large amount of attention, partly in relation to the possibility of topological quantum computation, and has stimulated studies on higher chirality superconducting systems. In the first part of this thesis, we study the spontaneous edge current in chiral d- and f-wave superconductors. We show that these currents, which vanish in the continuum limit at zero temperature, are generally non-vanishing but tiny, compared to the simplest chiral p-wave case. In the presence of strong surface roughness, the direction of the edge current in the chiral d-wave case can be reversed, compared with that of a specular ideal surface with specular scattering. However, it is shown that this current reversal is non-universal beyond the continuum limit. The chiral p-wave scenario in SRO is overturned by recent Knight shift measurements, highlighting the importance of exploring different pairing symmetries for SRO. Recently, $d_{x^2-y^2} \pm ig_{(x^2-y^2)xy}$, $s' \pm id_{xy}$ and mixed helical p-wave pairings have been proposed as order parameter candidates. However, the stability of these states, especially of the $d_{x^2-y^2} \pm ig$ pairing, remains unclear. In the second part of the thesis, we study the leading superconducting instabilities in SRO in the presence of sizable atomic spin-orbit coupling (SOC), non-local SOC, and non-local interactions. We find that it is difficult to stabilize chiral p-wave pairing in SRO models; this is because, among the triplet p-wave states, the atomic SOC favors helical states over the chiral state. The presence of both d- and g-wave pairings, including a $d_{x^2-y^2} \pm ig$ state, is found when the second nearest neighbor (in-plane) repulsions, together with orbital-anisotropy of the non-local interactions and/or the B2g channel non-local SOC are included. We further analyze the properties, such as nodal structures, in-plane field spin-susceptibility, and spontaneous edge current, of the realized $d_{x^2-y^2} \pm ig$ pairing and find that this state is more compatible with existing experimental measurements than the $s' \pm id_{xy}$ and the mixed helical p-wave proposals. / Dissertation / Doctor of Philosophy (PhD)

Les systèmes électroniques bidimensionnels sont d'une grande actualité tout particulièrement depuis la découverte de la supraconductivité à haute température. Ici, on se restreint à l'étude d'un modèle de Hubbard étendu, à la limite d'un couplage faible. En général, le gaz électronique subit une instabilité supraconductrice même sans phonons. Cependant, dans le cas spécial d'une bande demi-remplie, la surface de Fermi est emboîtée et se trouve à une singularité de Van Hove. Cette situation conduit à une compétition entre six instabilités différentes. Outre la supraconductivité en onde $s$ et $d$, on trouve des ondes de densités de spin et de charge ainsi que deux phases qui sont caractérisées par des courants circulaires de charge et de spin respectivement. Le formalisme du groupe de renormalisation est présenté en reliant l'idée de la "< sommation parquet "> au concept plus moderne de l'action effective de Wilson. Comme résultat on obtient un diagramme de phases riche en fonction de l'interaction du modèle. Ce diagramme de phase est exact dans la limite d'une interaction infiniment faible, puisque dans ce cas les lignes de transitions sont fixées par des symétries du modèle. Les comportements à basse température de la susceptibilité de spin ainsi que de la compressibilité de charge complètent l'image physique de ces instabilités. Il s'avère que la surface de Fermi à une tendence générale de se déformer spontanément, mais l'emboîtement n'est pas détruit. En résumé, le modèle de Hubbard à couplage faible reproduit deux propriétés essentielles des cuprates: une phase antiferromagnetique à demi remplissage et la supraconductivité en onde $d$ dans le cas dopé. Mais elle n'éxplique pas les propriétés inhabituelles de l'état métallique dans le régime sous-dopé. Une extension systématique de l'approche perturbative pourrait aider à mieux comprendre ces propriétés, mais reste difficile puisque les techniques nécessaires ne sont pas encore complètement développées.

The idea that we live in a higher-dimensional space was first introduced almost 100 years ago. In the past two decades many extra-dimensional models have been proposed in order to solve fundamental problems of nature such as the hierarchy problem. Most of them need exploration via non-perturbative approaches and Lattice Gauge Theory provides a tool for doing this. In this thesis, we make attempts to find a non-perturbative way to localize gauge fields that arise from five-dimensional SU(2) gauge theories on 3-branes. In 1984, it was proposed that the phase diagram of anisotropic extra-dimensional lattice gauge theories inherits a new phase, called the "layered" phase, where the gauge fields behave as four-dimensional ones. This was shown for the abelian case, but the existence of this new phase for the simplest non-abelian group, SU(2), was still in doubt. We investigated this system in large volumes using Monte Carlo simulations and we could not find a second order phase transition from a five-dimensional to a continuous four-dimensional theory when all directions were kept large. This made the model unattractive for further exploration as nothing suggests that a non-trivial fixed point could exist. The above investigation was done in a flat background metric. We extended the previous work by putting our theory into a slice of AdS5 space, usually called the warped background. The motivation for this is that our SU(2) theory looks like the gauge-sector of the Randall-Sundrum model, which does not have a concrete solution to the problem of localization of the gauge fields on a 3-brane. We carried out our investigation using the Mean-Field Approach and we present novel results for the phase diagram and measurements of important observables. In our implementation we have a finite extent of the extra dimension and one layer (or 3-brane) on each extra-dimensional coordinate. At weak coupling, we observed that each layer decouples one at a time in the transition to the fully layered phase of the system, forming a mixed phase, whereas there is a strong and sharp transition between the fully layered and the strong-coupling phase. Within the mixed phase, close to the transition into the layered phase, we found evidence that the system is four-dimensional acquiring a Yukawa mass and resembling a Higgs-like phase. The mixed phase grows as the curvature increases suggesting that for an infinite extra dimension the entire weak-coupling phase is mixed.

L'objet de cette thèse est le couplage entre des nanocristaux luminescents et des nanostructures métalliques. Ces structures présentent nombre d'intérêts dans un large panel d'applications de par l'apparition de modes électromagnétiques de surface (dénommés plasmons) que l'on contrôle via la géométrie de ces structures. Dans cette thèse, j'étudie deux types de nanostructures métalliques différentes : les plasmons « long-range» et les nanoantennes plasmoniques.Dans un premier temps je me suis intéressé à une géométrie qui couple deux plasmons propagatifs en deux modes hybrides au travers d'une fine couche de métal pour former des plasmons à forte longueur de propagations. En couplant des nanocristaux luminescents à ces modes, la répartition en énergie de l'émission dans les différents canaux de désexcitation disponibles a été étudiée. J'ai aussi montré que le métal pouvait augmenter leur taux d'émission spontanée d'un facteur 1,7. La contribution non négligeable des modes de guide conventionnels à l'émission dans ces structure a également été mise en évidence.Dans un second temps, j'ai étudié la potentielle utilisation de nanoparticules métalliques comme nanoantennes pour exalter et rediriger l'émission spontanée. La structure sera composée d'un dimère métallique créant un «point chaud » placé à proximité d'un plot métallique permettant la redirection. Des calculs FDTD montrent qu'une géométrie en pilier permet à la fois des pertes faibles (<10%), une forte augmentation de la cadence d'émission(>x80), une redirection de la lumière et ouvre la possibilité de multiplexage directif en longueur d'onde de l'information. Ces structures présentent l'avantage d'être compatibles avec les techniques modernes d'élaboration en couche mince. Des réalisations préliminaires ont alors été présentées / The object of this thesis is the coupling between luminescent nanocristals and metallic nanostructures. These structures show numerous interest in a large variety of applications thanks to the apparition of electromagnetic surface wave known as plasmons whose properties are tailored with the geometry of these structures. In this thesis, two types of geometry will be adressed : the long-range plasmons, and plasmonic nanoantennas. In a first time, the study focuses on a geometry in which two propagative surface plasmons are coupled through a thin metal film; creating a new type of plasmons with extended propagation lenghts. By coupling the emission of nanocristals in such a geometry, the energy repartition in the different desexcitation channels available has been adressed. The viccinity of the metal has also proved to increase the spontaneous decay rate up to 1.7. The non trivial contribution of conventional waveguide modes has also been demonstrated. In a second time, the potential of using metallic nanoparticles in a pillar geometry as nanoantennas to enhance and redirect the spontaneous emission has been investigated. The structure is composed of a metallic dimer creating a hotspot on top of which another metallic nanoparticles has been placed. FDTD simulations has shown that this kind of geometry can lead to few loss (<10%), a strong enhancement of the emission rate (>x80), a redirection of the emission and paves the way to wavelenght multiplexing possibilities. Besides, these structures present the advantage to be compatible with modern thin film elaboration techniques. Preliminary realisations have then been introduced

Cette thèse expérimentale s’est attachée à l’étude des effets induits par l’extension spatiale desplasmons de surface sur l’émission de matériaux organiques et inorganiques. Le système estformé d’un ensemble d’émetteurs localisés émettant principalement des plasmons de surfacedélocalisés. Dans un premier temps, nous nous sommes intéressés à l’imagerie par microscopieplasmon, technique de plus en plus utilisée dans divers domaines, notamment la biologie. Nousavons montré que l’émission détectée en un point provient essentiellement de l’environnementet non du point observé, définissant ainsi un cercle d’influence lié à la longueur de propagationdu plasmon de surface. Quand le plasmon interagit plus fortement avec des émetteurs, ilpeut entrer en régime de couplage fort. Ce couplage fort se traduit par un changement dansles énergies du système et par l’apparition de nouveaux états hybrides excitons-plasmons, lespolaritons. Les différents émetteurs localisés (des chaines de colorants agrégés) ne sont alorsplus indépendants entre eux. Des mesures de diffusion montrent un effet collectif induit par lecouplage fort. Ces expériences ont été confirmées par des mesures de cohérence spatiale, réaliséesen ajoutant une expérience de fentes d’Young au dispositif de microscopie plasmon. Ilapparait qu’un état cohérent étendu sur plusieurs microns se forme, conformément aux prévisionsthéoriques. L’ensemble d’émetteurs se comporte alors comme une macromolécule, dontl’interaction est induite par le plasmon de surface. / This experimental thesis studies effects induced by the spatial extension of surface plasmonpolaritons on the emission properties of organic and inorganic materials. First, we focused onleakage radiation microscopy images, a technic which is now widely used in a lot of differentscientific fields, as biology for exemple. We showed that the detected emission at a given point ofthe fluorescence image of an assembly of emitters mostly comes from the environment and notfrom the observed point, defining an influence circle related to the surface plasmon propagationlength. When the surface plasmon strongly interact with emitters, the strong coupling leadsto energy modifications in the system and new hybride states excitons-plasmons appear calledpolaritons. All the different localized emitters (aggregated dye chains) are not independantanymore. Diffusion measurments showed a collective effect induced by the strong-coupling.Two Young’s slits experiment added on the optical system confirm that an extended coherentstate of several micrometers is created as predicted by theory. All emitters behave as only onemacromolecule where the interaction is mediated by the surface plasmon.