No description available.

In this Thesis, we study some of the classical properties of an extension of the Skyrme model defined by adding a sixth order derivative term to the Lagrangian. In chapter 1, we review the physical as well as the mathematical motivation behind the study of the Skyrme model and in chapter 2, we give a brief summary of various extended Skyrme models that have been proposed over the last few years. We then define a new sixth order Skyrme model by introducing a dimensionless parameter λ that denotes the mixing between the two higher order terms, the Skyrme term and the sixth order term. In chapter 3 we compute numerically the multi-skyrmion solutions of this extended model and show that they have the same symmetries with the usual skyrmion solutions. In addition, we analyse the dependence of the energy and radius of these classical solutions with respect to the coupling constant λ. We compare our results with experimental data and determine whether this modified model can provide us with better theoretical predictions than the original one. In chapter 4, we use the rational map ansatz, introduced by Houghton, Manton and Sutcliffe, to approximate minimum energy multi-skyrmion solutions with B ≤ 9 of the SU(2) model and with B ≤ 6 of the SU(3) model. We compare our results with the ones obtained numerically and show that the rational map ansatz works just as well for the generalised model as for the pure Skyrme model, at least for B ≤ 5. In chapter 5, we use a generalisation of the rational map ansatz, introduced by loannidou, Piette and Zakrzewski, to construct analytically some topologically non-trivial solutions of the extended model in SU(3). These solutions are spherically symmetric and some of them can be interpreted as bound states of skyrmions. Finally, we use the same ansatz to construct low energy configurations of the SU(N) sixth order Skyrme model.

In Introduction, we review integrable and topological solitons. In Numerical Methods, we describe how to minimize functionals, time-integrate configurations and solve eigenvalue problems. We also present the Simulated Annealing scheme for minimisation in solitonic systems. In Classical Aspects, we analyse the effect of the potential term on the structure of minimal- energy solutions for any topological charge n. The simplest holomorphic baby Skyrme model has no known stable minimal-energy solution for n > 1. The one-vacuum baby Skyrme model possesses non-radially symmetric multi-skyrmions that look like 'skyrmion lattices' formed by skyrmions with n = 2. The two-vacua baby Skyrme model has radially symmetric multi- skyrmions. We implement Simulated Annealing and it works well for higher order terms. We find that the spatial part of the six-derivative term is zero. In Quantum Aspects, we find the first order quantum mass correction for the Ф(^4) kink using the semi-classical expansion. We derive a trace formula which gives the mass correction by using the eigenmodes and values of the soliton and vacuum perturbations. We show that the zero mode is the most important contribution. We compute the mass correction of Ф(^4) kink and Sine-Gordon numerically by solving the eigenvalue equations and substituting into the trace formula.

In this thesis we study the physics of quantum many-body systems confined to one-dimensional geometries. The work was motivated by the recent success of experimentalists in developing atom traps, capable of restricting the motion of the individual atoms to a single spatial dimension. Specifically, we look at aspects of the one-dimensional Bose gas including; excitation spectrum, correlation functions, and dynamical behaviour. In Chapter \ref{ch:excitation1D} we consider the Lieb-Liniger model of interacting bosons in one-dimension. We numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss novel features of the solutions, including deviations from the well-known string solutions due to finite size effects. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Our results are compared to those obtained via exact diagonalization of the Hamiltonian in a truncated basis. In Chapter \ref{ch:g2} we analytically calculate the spatial nonlocal pair correlation function for an interacting uniform one dimensional Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects. In Chapter \ref{ch:casimirdrag} we study the drag force below the critical velocity for obstacles moving in a superfluid. The absence of drag is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a non-zero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytic solutions to the Bogoliubov-de Gennes equations for a gray soliton. In Chapter \ref{ch:solitons} we study the effect of quantum noise on the stability of a soliton. We find the soliton solutions exactly define the reflectionless potentials of the Bogoliubov-de Gennes equations. This results in complete stability of the solitons in a purely one dimensional system. We look at the modifications to the density profile of a black soliton due to quantum fluctuations.

In this thesis we study the physics of quantum many-body systems confined to one-dimensional geometries. The work was motivated by the recent success of experimentalists in developing atom traps, capable of restricting the motion of the individual atoms to a single spatial dimension. Specifically, we look at aspects of the one-dimensional Bose gas including; excitation spectrum, correlation functions, and dynamical behaviour. In Chapter \ref{ch:excitation1D} we consider the Lieb-Liniger model of interacting bosons in one-dimension. We numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss novel features of the solutions, including deviations from the well-known string solutions due to finite size effects. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Our results are compared to those obtained via exact diagonalization of the Hamiltonian in a truncated basis. In Chapter \ref{ch:g2} we analytically calculate the spatial nonlocal pair correlation function for an interacting uniform one dimensional Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects. In Chapter \ref{ch:casimirdrag} we study the drag force below the critical velocity for obstacles moving in a superfluid. The absence of drag is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a non-zero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytic solutions to the Bogoliubov-de Gennes equations for a gray soliton. In Chapter \ref{ch:solitons} we study the effect of quantum noise on the stability of a soliton. We find the soliton solutions exactly define the reflectionless potentials of the Bogoliubov-de Gennes equations. This results in complete stability of the solitons in a purely one dimensional system. We look at the modifications to the density profile of a black soliton due to quantum fluctuations.

In this work, we study the nonlinear propagation of light in liquid crystals (LCs) and the optical gain provided by LCs when they are polymer- or dye-doped.We will focus on nematic LCs, which are characterized by a mean orientation (also called director) of the elongated molecules and by a subsequent birefringence. After a general introduction on LCs, we focus on the nonlinear propagation of light in nematic LCs, and in particular the soliton-like propagation (nematicon). Indeed, if the light injected in the cell is intense enough, it can create a waveguide that counteracts the diffraction of the light. The light then propagates with an almost constant (or periodic) transverse profile.Our contribution to the subject starts with the numerical modeling of the thermal noise that characterizes the nematic LCs and the study of spatial instabilities of the soliton propagation caused by that noise. In Ch.3 we show that, by explicitly implementing the spatial correlation of the director in the LC thermal noise, it is possible to reproduce some of the features that characterize the LC response, such as the speckle generation or the fluctuating trajectory of the spatial optical soliton in LCs. Indeed, when the nematicon diameter is of the same order ofmagnitude as or smaller than the refractive index perturbations caused by the thermal noise, the nematicon starts to fluctuate in space. These fluctuations are not present when the noise is not correlated, indicating that the long-range interactions in LCs are crucial to explain the fluctuations. The model also allows us to introduce the propagation losses experienced by the nematicon without the use of an ad-hoc term. The simulations are in agreement with the experimental results. This method could also help the modeling of complex nonlinear phenomena in LCs that rely on noise, such as modulation instabilities or filamentation.Then, the optical gain is included in the LCs by dissolving photoluminescent polymers or dyes in it. In particular, we show that a particular polymer, the polyfuorene, when dissolved in nematic LCs, creates an intricate supramolecular pattern composed by homogeneous LC-rich regions surrounded by polymer-rich boundaries. The study of these structures through an ultra-fast spectroscopic technique (the pump-probe technique) and confocal microscopy reveals that the boundaries are composed by ordered and isolated chains of polymers. This particular morphology allows the observation of the optical gain from an oxidized unit of the polymeric chain (keto defects). This signal is usually covered by the absorption caused by the chain aggregation in solid state samples, while in LCs it is clearly visible. The optical gain from the keto defects appears also to be polarized orthogonal to the LC director, which is also the orientation of most of the boundaries. When a dye, one of the pyrromethenes, is dissolved in the LCs, the sample appears to be homogeneous. The optical gain from the dye ispolarized along the LC director and it shows an important spectral blue-shift (10 nm) passing from a polarization parallel to orthogonal to the LC director. The amplified spontaneous emission (ASE) shows the same shift when changing the direction of the sample excitation.When the ASE and the nematicon are generated in the same sample, it is possible to study the interaction between the two. In particular, the waveguide induced by the soliton can be used to guide another signal at another wavelength. We show that the nematicon can collect the ASE generated in the same device and guide it to the same fiber used to inject the nematicon in the LC cell. The extraction of the ASE from the device increases almost one order of magnitude when the soliton is present. However, due to the nematicon spatial fluctuations in LCs, an optimal nematicon power has to be found. Indeed, by increasing the soliton power, the light guiding is improved since the refractive index contrast of the nematicon-induced waveguide is increased. However, very high soliton powers have to be avoided, since the power-dependent soliton fluctuations prevent an optimal collection of the light. The nematicon is also found to increase the spectral purity and the polarization degree of the guided signal.Another LC system is studied, the chiral nematic LCs. In this system, the molecules are disposed following an helicoidal distribution. Due to their optical anisotropy and the periodic distribution, the system presents an optical band-gap. If the LC is also dye-doped, the combination of optical band-gap and gain generates laser emission. We are interested in a fast (sub-ms) reorientation of the helix, with the aim of studying the effect of this reorientation on the laser emission. The first step is the alignment of the LC helix (without the dye) with its axis parallel to the glass plates that constitute the cell, which is difficult to obtain with a high optical quality. For this reason, an innovative method is developed to align LCs through directional solvent evaporation. The solvent-induced method allows us to obtain particularly homogeneous textures, with a contrast ratio between the bright and the dark states that is a factor of 4 greater than that obtained with traditional methods. The LC samples based on solvent-induced alignment are then stabilized via two-photon photo-polymerization. This technique allows us to polymerize small regions of the device while the rest of the sample can be washed out in a solvent bath. When an achiral material is used to refill the device, it assumes a chiral alignment in the polymerized regions and an achiral nematic distribution in the rest. The first characterization of the laser emission is then presented in the last Chapter, with the aim of achieving sub-ms electrical tuning in future works.In this work a wide range of aspects have been investigated, leading to the realization of novel techniques for the fabrication of liquid crystal devices, the demonstration of novel phenomena for light amplification in liquid crystals and the experimental verification of new numerical modeling tools for light propagation in liquid crystals. The three key aspects of the work are nonlinear propagation, optical amplification and electrical response of different LC-based mixtures. Although the first few chapters deal with some of the aspects separately, in the last chapter these aspects are combined, revealing interesting new phenomena and pointing out a number of new aspects that could be part of future work. The results in this work have potential applications in fast tunable lasers, optical communication systems and lab-on-chip components. / Doctorat en Sciences de l'ingénieur et technologie / info:eu-repo/semantics/nonPublished

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-07-17T13:07:11Z No. of bitstreams: 2 Tese - Luciano Calaça Alves - 2018.pdf: 4699371 bytes, checksum: 706846314ebb74648be890b03e18d4cc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-07-17T13:55:46Z (GMT) No. of bitstreams: 2 Tese - Luciano Calaça Alves - 2018.pdf: 4699371 bytes, checksum: 706846314ebb74648be890b03e18d4cc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-07-17T13:55:46Z (GMT). No. of bitstreams: 2 Tese - Luciano Calaça Alves - 2018.pdf: 4699371 bytes, checksum: 706846314ebb74648be890b03e18d4cc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-06-07 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / This work presents the study of solitary wave solutions, known as solitons, in non-linear and non- homogeneous media using non-linear Schrödinger equations. Three cases are studied: first considering a logarithmic nonlinear term; second with saturation effect and finally including effects of high orders (Raman scattering). Solutions are modulated by three different types of potential. First, linear in the spatial and oscillatory coordinate in the temporal coordinate. The second, quadratic in the spatial and oscillatory in the temporal coordinates. Finally, it is also modulated using a mixed potential, which is the junction of the two potentials presented above. After including inomogeneities in linear and nonlinear coefficients, the similarity transformation technique is used to convert the non-linear, non-autonomous equation into an autonomous one that will be solved analytically. This field of study has potential applications in crystals, optical fibers and in Bose- Einstein condensates, also serving to understand the fundamentals related to this state of matter. The stability of the solutions are checked by numerical simulations. / Este trabalho apresenta o estudo de soluções de ondas solitárias, conhecidas como sólitons, em meios não lineares e não homogêneos por meio de equações não lineares de Schrödinger. São estudados três casos: primeiro considerando um termo não linear do tipo logarítmico; segundo com efeito de saturação e por último incluindo efeitos de altas ordens (espalhamento Raman). As soluções são moduladas por três tipos diferentes de potencial. O primeiro, linear na coordenada espacial e oscilatório na coordenada temporal. O segundo, quadrático na coordenada espacial e oscilatório na temporal. Por fim, modula-se também utilizando um potencial misto, que é a junção dos dois potenciais apresentados anteriormente. Depois de incluir heterogeneidades nos coeficientes lineares e não lineares, é utilizada a técnica da transformação de similaridade para converter a equação não linear, não autônoma em uma autônoma que será resolvida analiticamente. Esse campo de estudo tem potenciais aplicações em cristais, fibras ópticas e em condensados de Bose-Einstein, servindo também para o entendimento dos fundamentos relacionados a esse estado da matéria. A estabilidade das soluções são checadas por meio de simulações numéricas.

Neste trabalho, propomos um novo mÃtodo de modulaÃÃo Ãptica usando um filtro acÃstoÃptico sintonizÃvel (AOTF) convencional e um filtro acÃsto-Ãptico com polarizaÃÃo sintonizÃvel (AOTPF) baseado em fibra de cristal fotÃnico (PCF), onde portas lÃgicas Ãpticas E e OU sÃo obtidas pela operaÃÃo simultÃnea da modulaÃÃo em dupla banda lateral Ãptica (ODSB) e da modulaÃÃo por posiÃÃo de pulso no domÃnio da frequÃncia (PPFDM). Estes dispositivos sÃo operados com pulsos sÃlitons ultracurtos de 100 ps e 55,5 fs para o AOTF convencional e para o AOTPF baseado em PCF, respectivamente. Desta forma, um pulso leva dois bits de informaÃÃo apÃs ser criptografado pela modulaÃÃo proposta aqui. Em seguida, analisamos a modulaÃÃo ODSB-PPFDM para os pulsos de entrada, polarizados nos dois modos de entrada, permitindo uma variaÃÃo no parÃmetro de codificaÃÃo εcod para cada pulso de entrada. Para o AOTF convencional, consideramos uma diferenÃa de fase de dfi = pi rad entre ambos os pulsos de entrada. Como resultado, obtivemos vÃrios valores do parÃmetro de codificaÃÃo |εcod| onde as operaÃÃes lÃgicas E e OU foram possÃveis. JÃ para o AOTPF baseado em PCF, consideramos uma diferenÃa de fase de dfi = 1,28pi rad entre ambos os pulsos de entrada para obtermos valores do parÃmetro de codificaÃÃo |εcod| onde as operaÃÃes lÃgicas E e OU foram possÃveis. / We propose a new method of optical modulation using a conventional Acousto-Optic Tunable Filter (AOTF) and an Acoustic Optic Tunable Polarization Filter (AOTPF) based on Photonic Crystal Fiber (PCF). In both devices the all-optical logic gates, namely AND and OR, are obtained by simultaneously operation of Optical Double Sideband (ODSB) modulation and a Pulse Position Frequency Domain Modulation (PPFDM). These devices shall operate with ultrashort soliton light pulses 100 ps and 55.5 fs for conventional AOTF and all-fiber AOTPF based on PCF, respectively. In this way, a pulse will carry two bits of information after been encoded by the modulation proposed here. We then analyze the modulation ODSB - PPFDM for input pulses, polarized in the two input modes, allowing a variation in the modulation parameter εcod for each input pulse. For conventional AOTF, a phase difference dfi = pi rad was considered between both input pulses, obtaining various values of the coding parameter offset |εcod| where the AND and OR logic operations were possible. For the all-fiber AOTPF based on PCF, a phase difference of dfi = 1,28pi rad was necessary between both input pulses to generate values of coding parameter offset |εcod|, for which AND and OR logic operations were possible.

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2016-06-15T19:51:42Z No. of bitstreams: 2 Dissertação - Elismar Dias Batista - 2016.pdf: 1518873 bytes, checksum: 8375db389a2056c5849ee168f5efa5ce (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-06-28T12:21:16Z (GMT) No. of bitstreams: 2 Dissertação - Elismar Dias Batista - 2016.pdf: 1518873 bytes, checksum: 8375db389a2056c5849ee168f5efa5ce (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Made available in DSpace on 2016-06-28T12:21:16Z (GMT). No. of bitstreams: 2 Dissertação - Elismar Dias Batista - 2016.pdf: 1518873 bytes, checksum: 8375db389a2056c5849ee168f5efa5ce (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) Previous issue date: 2016-03-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is based on the articles [26] and [27], where we studied Einstein manifolds and gradient Ricci soliton with twisted product structure. As a result, we prove the following: if M is an Einstein warped product space with nonpositive scalar curvature and compact base, then M is a Riemannian product space. Besides, we show that the Riemannian product Rp×F is a gradient Ricci soliton if and only if F is Ricci soliton gradient. Then, we show that the warped product R×f B is gradient Ricci solitons with f ′′ 6= 0, therefore F is Einstein. By using these results, we build nontrivial examples of gradient Ricci soliton where the fiber is either an Einstein manifold or a nontrivial gradient Ricci soliton. / Este trabalho está baseado nos artigos [26] e [27], onde estudamos Variedades de Einstein e gradiente Ricci solitons com estrutura de produto torcido. Provamos que: se M é um produto torcido Einstein com curvatura escalar não positiva e base compacta, então a função torção é constante, ou seja, o produto torcido é Riemanniano. Mostramos ainda que o produto Riemanniano Rp ×F é um gradiente Ricci soliton se e somente se F for gradiente Ricci soliton. Em seguida, mostramos que se o produto torcido R×f F for gradiente Ricci soliton com f ′′(t) 6= 0, então F é Einstein. Usando estes resultados construímos exemplos de gradiente Ricci soliton não trivial com a fibra sendo Einstein ou gradiente Ricci soliton não trivial. Finalmente consideramos o produto torcido Lorentziano sendo gradiente Ricci soliton e obtivemos critérios análogos ao Riemanniano para que F seja Einstein ou gradiente Ricci soliton.

No description available.