Dissertação (mestrado)—Universidade de Brasília, Instituto de Física, 2012. / Submitted by Albânia Cézar de Melo (albania@bce.unb.br) on 2012-11-09T14:18:52Z No. of bitstreams: 1 2012_ThiagoVelozoTrufini.pdf: 1101674 bytes, checksum: 77e002dd5667ce151a1e5fd3a3c3739c (MD5) / Approved for entry into archive by Guimaraes Jacqueline(jacqueline.guimaraes@bce.unb.br) on 2012-11-12T10:33:49Z (GMT) No. of bitstreams: 1 2012_ThiagoVelozoTrufini.pdf: 1101674 bytes, checksum: 77e002dd5667ce151a1e5fd3a3c3739c (MD5) / Made available in DSpace on 2012-11-12T10:33:49Z (GMT). No. of bitstreams: 1 2012_ThiagoVelozoTrufini.pdf: 1101674 bytes, checksum: 77e002dd5667ce151a1e5fd3a3c3739c (MD5) / Modelos bidimensionais de gravitação surgem naturalmente na descrição de diversos fenômenos físicos. Grande parte desses modelos visam entender o processo de criação do Universo no cenário pré Big-Bang, provendo também testes não-triviais no contexto da teoria quântica de campos. A gravitação bidimensional acoplada com o campo dilaton (G2dD) vem sendo bastante estudada. Ela se mostra como um sistema integrável na teoria clássica, assim, a estruturação da integrabilidade nos possibilita entender as soluções clássicas não-perturbativas e assim proceder com as quantização. A G2dD é bastante similar com o modelo sigma não linear (SNL), que também é integrável. No entanto, diferente da G2dD, o SNL não é quantizável usando a teoria de sistemas integráveis - isso ocorre graças ao comportamento ambíguo entre os parênteses de Poisson de suas matrizes de monodromia, que codificam todas as informações do modelo integrável. Recentemente, um interessante modelo da gravitação quântica foi proposta por Ho\v{r}ava. Esse modelo, chamado de Horava-Lifshitz (HL), vem sendo extensivamente estudado e explorado no meio científico. Ele propõe alterações na gravitação de Einstein de forma que a mesma seja renormalizável e, consequentemente, quantizável. Para ser uma teoria coerente, a HL deve tender a relatividade padrão de Einstein nos limites de baixas energias. Outro fato importante é que para altas energias a HL se transforma numa teoria bidimensional efetiva. Utilizando o HL como uma motivação, estudamos a gravitação bidimensional através do formalismo da Integrabilidade. Calculamos os parênteses de Poisson entre as matrizes de monodromia do modelo principal do campo quiral (PCM), que é um tipo de SNL bidimensional, e obtivemos a ambiguidade dessa estrutura. O mesmo foi feito para a G2dD, cujo resultado foi único e viável. Comparamos então as características e peculiaridades entre o PCM e a G2dD e observamos o papel fundamental que o dilaton possui: viabilizar a estrutura de Poisson, tornando o modelo quantizável pelo formalismo da integrabilidade. Visando tornar o trabalho acessível a todo público interessado nessa área da Física, fizemos uma revisão sobre grande parte dos pontos cruciais da teoria que são importantes para o entendimento do nosso resultado. Achamos necessário também preencher nosso trabalho com muita álgebra, levando em consideração o número elevado de pontos sutis. ______________________________________________________________________________ ABSTRACT / Two-dimensional models of gravitation arise naturally in the description of various physical phenomena. Most of these models aim to understand the process of creating the universe in the pre-Big Bang scenario, as well as provide various non-trivial tests in the context of the quantum eld theory. The two-dimensional gravity coupled with dilaton eld (G2dD) has been a subject of intense research in the recent years. It has integrable structure in the classical theory, and, therefore, allows one to understand both the classical non-perturbative solutions, as well as proceed with the quantization of the model. The G2dD has a similar to the nonlinear sigma model (SNL) structure, which is also integrable on the classical level. However, unlike G2dD, SNL is not quantizable easily using the theory of integrable systems - this is due to the ambiguous behavior of the algebra of the monodromy matrices, which encode the complete information of integrable structure. Recently, an interesting model of quantum gravity has been proposed by Ho rava. This model, called Ho rava-Lifshitz (HL), has been extensively studied and explored in the scienti c community in the last few years. He proposed changes in Einstein's gravitation theory so that it is renormalizable and therefore makes sense in the context of the quantum eld theory. To be a coherent theory, HL must tend to the standard Einstein's relativity in the low-energy limits, and one of the consequences of the HL theory is the important feature that for high energies HL becomes e ectively a two-dimensional theory. Using the HL as a motivation, we study two-dimensional gravitation utilizing the formalism of integrability. We calculate the Poisson brackets between monodromy matrices of the principal chiral model(PCM), which is a type of two-dimensional SNL, and show how the ambiguity of this structure appears. The same is done for G2dD, where we show that the ambiguity is removed, and the result is unique and well-de ned. We then compare the characteristics and peculiarities between the PCM and G2dD and note the key role played by the dilaton eld, due to which the Poisson structure becomes regularized, making the model quantizable in the framework of integrable models. Aiming to make the work accessible to everyone interested in this area of physics, we review in details most of the crucial points, that are important for understanding of our results, and include various non-trivial calculations explicitly, which should be especially useful for interested students.

This thesis is about some extensions of the ideas and techniques used in integrable field theories to deal with non-integrable theories. It is presented in two parts. The first part deals with gauge theories in 3 and 4 dimensional space-time; we propose what we call the integral formulation of them, which at the end give us a natural way of defining the conserved charges that are gauge invariant and do not depend on the parametrisation of space-time. The definition of gauge invariant conserved charges in non-Abelian gauge theories is an open issue in physics and we think our solution might be a first step into its full understanding. The integral formulation shows a deeper connection between different gauge theories: they share the same basic structure when written in the loop space. Moreover, in our construction the arguments leading to the conservation of the charges are dynamical and independent of the particular solution. In the second part we discuss the recently introduced concept called quasi-integrability: one observes soliton-like configurations evolving through non-integrable equations having properties similar to those expected for integrable theories. We study the case of a model which is a deformation of the non-linear Schr¨odinger equation consisting of a more general potential, connected in a way with the integrable one. The idea is to develop a mathematical approach to treat more realistic theories, which is in particular very important from the point of view of applications; the NLS model appears in many branches of physics, specially in optical fibres and Bose-Einstein condensation. The problem was treated analytically and numerically, and the results are interesting. Indeed, due to the fact that the model is not integrable one does not find an infinite number of conserved charges but, instead, a set of infinitely many charges that are asymptotically conserved, i.e., when two solitons undergo a scattering process the charges they carry before the collision change, but after the collision their values are recovered. / Essa tese discute algumas extensões de ideias e técnicas usadas em teorias de campos integráveis para tratar teorias que não são integráveis. Sua apresentação é feita em duas partes. A primeira tem como tema teorias de calibre em 3 e 4 dimensões; propomos o que chamamos de equação integral para uma tal teoria, o que nos permite de maneira natural a construção de suas cargas invariantes de calibre, e independentes da parametrização do espaço-tempo. A definição de cargas conservadas in variantes de calibre em teorias não-Abelianas ainda é um assunto em aberto e acreditamos que a nossa solução pode ser um primeiro passo em seu entendimento. A formulação integral mostra uma conexão profunda entre diferentes teorias de calibre: elas compartilham da mesma estrutura básica quando formuladas no espaço dos laços. Mais ainda, em nossa construção os argumentos que levam `a conservação das cargas são dinâmicos e independentes de qualquer solução particular. Na segunda parte discutimos o recentemente introduzido conceito de quasi-integrabilidade: em (1 + 1) dimensões existem modelos não integráveis que admitem soluções solitonicas com propriedades similares `aquelas de teorias integráveis. Estudamos o caso de um modelo que consiste de uma deformação (não-integrável) da equação de Schrödinger não-linear (NLS), proveniente de um potencial mais geral, obtido a partir do caso integrável. O que se busca é desenvolver uma abordagem matemática sistemática para tratar teorias mais realistas (e portanto não integráveis), algo bastante relevante do ponto de vista de aplicações; o modelo NLS aparece em diversas áreas da física, especialmente no contexto de fibra ótica e condensação de Bose-Einstein. O problema foi tratado de maneira analítica e numérica, e os resultados se mostram interessantes. De fato, sendo a teoria não integrável não é encontrado um conjunto com infinitas cargas conservadas, mas, pode-se encontrar um conjunto com infinitas cargas assintoticamente conservadas, i.e., quando dois solitons colidem as cargas que eles tinham antes tem os seus valores alterados, mas após a colisão, os valores inicias, de antes do espalhamento, são recobrados.

This thesis is divided in two parts. The first one is devoted to the dynamics of polariton condensates, with particular attention to their pattern-forming capabilities. In many configurations of physical interest, the dynamics of polariton condensates can be modelled by means of a non-linear PDE which is strictly related to the Gross-Pitaevskii and the complex Ginzburg-Landau equations. Numerical simulations of this equation are used to investigate the robustness of the rotating vortex lattice which is predicted to spontaneously form in a non-equilibrium trapped condensate. An idea for a polariton-based gyroscope is then presented. The device relies on peculiar properties of non-equilibrium condensates - the possibility of controlling the vortex emission mechanism and the use of pumping strength as a control parameter - and improves on existing proposals for superfluid-based gyroscopes. Finally, the important rôle played by quantum pressure in the recently observed transition from a phase-locked but freely flowing condensate to a spatially trapped one is discussed. The second part of this thesis presents work done in the context of the geometrical models of matter framework, which aims to describe particles in terms of 4-dimensional manifolds. Conserved quantum numbers of particles are encoded in the topology of the manifold, while dynamical quantities are to be described in terms of its geometry. Two infinite families of manifolds, namely ALF gravitational instantons of types A_k and D_k, are investigated as possible models for multi-particle systems. On the basis of their topological and geometrical properties it is concluded that A_k can model a system of k+1 electrons, and D_k a system of a proton and k-1 electrons. Energy functionals which successfully reproduce the Coulomb interaction energy, and in one case also the rest masses, of these particle systems are then constructed in terms of the area and Gaussian curvature of preferred representatives of middle dimension homology. Finally, an idea for constructing multi-particle models by gluing single-particle ones is discussed.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Meteorology, 1980. / Microfiche copy available in Archives and Science. / Vita. / Includes bibliographical references. / by William Allan Perrie. / Ph.D.

No description available.

Quadratic spatial soliton interactions were investigated in this Dissertation. The first part deals with characterizing the principal features of multi-soliton generation and soliton self-reflection. The second deals with two beam processes leading to soliton interactions and collisions. These subjects were investigated both theoretically and experimentally. The experiments were performed by using potassium niobate (KNBO3) and periodically poled potassium titanyl phosphate (KTP) crystals. These particular crystals were desirable for these experiments because of their large nonlinear coefficients and, more importantly, because the experiments could be performed under non-critical-phase-matching (NCPM) conditions. The single soliton generation measurements, performed on KNBO3 by launching the fundamental component only, showed a broad angular acceptance bandwidth which was important for the soliton collisions performed later. Furthermore, at high input intensities multi-soliton generation was observed for the first time. The influence on the multi-soliton patterns generated of the input intensity and beam symmetry was investigated. The combined experimental and theoretical efforts indicated that spatial and temporal noise on the input laser beam induced multi-soliton patterns. Another research direction pursued was intensity dependent soliton routing by using of a specially engineered quadratically nonlinear interface within a periodically poled KTP sample. This was the first time demonstration of the self-reflection phenomenon in a system with a quadratic nonlinearity. The feature investigated is believed to have a great potential for soliton routing and manipulation by engineered structures. A detailed investigation was conducted on two soliton interaction and collision processes. Birth of an additional soliton resulting from a two soliton collision was observed and characterized for the special case of a non-planar geometry. A small amount of spiraling, up to 30 degrees rotation, was measured in the experiments performed. The parameters relevant for characterizing soliton collision processes were also studied in detail. Measurements were performed for various collision angles (from 0.2 to 4 degrees), phase mismatch, relative phase between the solitons and the distance to the collision point within the sample (which affects soliton formation). Both the individual and combined effects of these collision variables were investigated. Based on the research conducted, several all-optical switching scenarios were proposed.

Discrete optical systems are a subgroup of periodic structures in which the evolution of a continuous electromagnetic field can be described by a discrete model. In this model, the total field is the sum of localized, discrete modes. Weakly coupled arrays of single mode channel waveguides have been known to fall into this class of systems since the late 1960's. Nonlinear discrete optics has received a considerable amount of interest in the last few years, triggered by the experimental realization of discrete solitons in a Kerr nonlinear AlGaAs waveguide array by H. Eisenberg and coworkers in 1998. In this work a detailed experimental investigation of discrete nonlinear wave propagation and the interactions between beams, including discrete solitons, in discrete systems is reported for the case of a strong Kerr nonlinearity. The possibility to completely overcome "discrete" diffraction and create highly localized solitons, in a scalar or vector geometry, as well as the limiting factors in the formation of such nonlinear waves is discussed. The reversal of the sign of diffraction over a range of propagation angles leads to the stability of plane waves in a material with positive nonlinearity. This behavior can not be found in continuous self-focusing materials where plane waves are unstable against perturbations. The stability of plane waves in the anomalous diffraction region, even at highest powers, has been experimentally verified. The interaction of high power beams and discrete solitons in arrays has been studied in detail. Of particular interest is the experimental verification of a theoretically predicted unique, all optical switching scheme, based on the interaction of a so called "blocker" soliton with a second beam. This switching method has been experimentally realized for both the coherent and incoherent case. Limitations of such schemes due to nonlinear losses at the required high powers are shown.

By nature discrete solitons represent self-trapped wavepackets in nonlinear periodic structures and result from the interplay between lattice diffraction (or dispersion) and material nonlinearity. In optics, this class of self-localized states has been successfully observed in both one-and two-dimensional nonlinear waveguide arrays. In recent years such lattice structures have been implemented or induced in a variety of material systems including those with cubic (Kerr), quadratic, photorefractive, and liquid-crystal nonlinearities. In all cases the underlying periodicity or discreteness leads to new families of optical solitons that have no counterpart whatsoever in continuous systems. In the first part of this dissertation, a theoretical investigation of linear and nonlinear optical wave propagation in semi-infinite waveguide arrays is presented. In particular, the properties and the stability of surface solitons at the edge of Kerr (AlGaAs) and quadratic (LiNbO3) lattices are examined. Hetero-structures of two dissimilar semi-infinite arrays are also considered. The existence of hybrid solitons in these latter types of structures is demonstrated. Rabi-type optical transitions in z-modulated waveguide arrays are theoretically demonstrated. The corresponding coupled mode equations, that govern the energy oscillations between two different transmission bands, are derived. The results are compared with direct beam propagation simulations and are found to be in excellent agreement with coupled mode theory formulations. In the second part of this thesis, the concept of parity-time-symmetry is introduced in the context of optics. More specifically, periodic potentials associated with PT-symmetric Hamiltonians are numerically explored. These new optical structures are found to exhibit surprising characteristics. These include the possibility of abrupt phase transitions, band merging, non-orthogonality, non-reciprocity, double refraction, secondary emissions, as well as power oscillations. Even though gain/loss is present in this class of periodic potentials, the propagation eigenvalues are entirely real. This is a direct outcome of the PT-symmetry. Finally, discrete solitons in PT-symmetric optical lattices are examined in detail.

This work is divided into two main parts. In the first part (chapters 2-7) we consider the nonlinear response of nano-particle colloidal systems. Starting from the Nernst-Planck and Smoluchowski equations, we demonstrate that in these arrangements the underlying nonlinearities as well as the nonlinear Rayleigh losses depend exponentially on optical intensity. Two different nonlinear regimes are identified depending on the refractive index contrast of the nanoparticles involved and the interesting prospect of self-induced transparency is demonstrated. Soliton stability is systematically analyzed for both 1D and 2D configurations and their propagation dynamics in the presence of Rayleigh losses is examined. We also investigate the modulation instability of plane waves and the transverse instabilities of soliton stripe beams propagating in nonlinear nano-suspensions. We show that in these systems, the process of modulational instability depends on the boundary conditions. On the other hand, the transverse instability of soliton stripes can exhibit new features as a result of 1D collapse caused by the exponential nonlinearity. Many-body effects on the systems' nonlinear response are also examined. Mayer cluster expansions are used in order to investigate particle-particle interactions. We show that the optical nonlinearity of these nano-suspensions can range anywhere from exponential to polynomial depending on the initial concentration and the chemistry of the electrolyte solution. The consequence of these inter-particle interactions on the soliton dynamics and their stability properties are also studied. The second part deals with linear and nonlinear properties of optical nano-wires and the coupled mode formalism of parity-time (PT) symmetric waveguides. Dispersion properties of AlGaAs nano-wires are studied and it is shown that the group velocity dispersion in such waveguides can be negative, thus enabling temporal solitons. We have also studied power flow in nano-waveguides and we have shown that under certain conditions, optical pulses propagating in such structures will exhibit power circulations. Finally PT symmetric waveguides were investigated and a suitable coupled mode theory to describe these systems was developed.

Over the years, non-spreading or non-diffracting wave configurations have been systematically investigated in optics. Perhaps the best known example of a diffraction-free optical wave is the so-called Bessel beam, first suggested and observed by Durnin et al. This work sparked considerable theoretical and experimental activity and paved the way toward the discovery of other interesting non-diffracting solutions. In 1979 Berry and Balazs made an important observation within the context of quantum mechanics: they theoretically demonstrated that the Schrodinger equation describing a free particle can exhibit a non-spreading Airy wavepacket solution. This work remained largely unnoticed in the literature-partly because such wavepackets cannot be readily synthesized in quantum mechanics. In this dissertation we investigate both theoretically and experimentally the acceleration dynamics of non-spreading optical Airy beams in both one- and two-dimensional configurations. We show that this class of finite energy waves can retain their intensity features over several diffraction lengths. The possibility of other physical realizations involving spatio-temporal Airy wavepackets is also considered. As demonstrated in our experiments, these Airy beams can exhibit unusual features such as the ability to remain quasi-diffraction-free over long distances while their intensity features tend to freely accelerate during propagation. We have demonstrated experimentally that optical Airy beams propagating in free space can perform ballistic dynamics akin to those of projectiles moving under the action of gravity. The parabolic trajectories of these beams as well as the motion of their center of gravity were observed in good agreement with theory. Another remarkable property of optical Airy beams is their resilience in amplitude and phase perturbations. We show that this class of waves tends to reform during propagation in spite of the severity of the imposed perturbations. In all occasions the reconstruction of these beams is interpreted through their internal transverse power flow. The robustness of these optical beams in scattering and turbulent environments was also studied. The experimental observation of self-trapped Airy beams in unbiased nonlinear photorefractive media is also reported. This new class of non-local self-localized beams owes its existence to carrier diffusion effects as opposed to self-focusing. These finite energy Airy states exhibit a highly asymmetric intensity profile that is determined by the inherent properties of the nonlinear crystal. In addition, these wavepackets self-bend during propagation at an acceleration rate that is independent of the thermal energy associated with two-wave mixing diffusion photorefractive nonlinearity.