Orientador: Rafael de Freitas Leão / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T03:14:46Z (GMT). No. of bitstreams: 1 Mendes_Douglas_M.pdf: 1234399 bytes, checksum: 9934061cdc7cbbc1da3d2586302aac2e (MD5) Previous issue date: 2012 / Resumo: Os grupos Spin aparecem de várias formas em Matemática e em Física-Matemática, tendo grande importância na teoria de brados e de operadores diferenciais sobre os mesmos. O conceito de estrutura spin é deles derivado, sendo ele a base de toda uma teoria, conhecida como geometria spin. Esta dissertação introduz os primeiros conceitos necessários ao estudo de tais grupos, assim como alguns aspectos importantes relacionados a eles. Dada a natureza dos grupos Spin e dos problemas aos quais estão relacionados, vários tópicos na interface entre álgebra e geometria tiveram de ser abordados. Estudamos em um primeiro momento as álgebras de Clifford, sua representação adjunta torcida e os grupos Spin como subgrupos do grupo das unidades de tais álgebras. À estes estudos, seguiu-se uma análise detalhada da teoria de espaços de recobrimento e da classificação dos mesmos. Pudemos com isso entender o grupo Spin, via representação adjunta torcida, como o recobrimento universal do grupo especial ortogonal de um espaço quadrático não-degenerado. Nos concentramos daí na teoria de brados principais e a relação destes com as propriedades geométricas das variedades sobre as quais eles estão construídos. Para sintetizar o que foi estudado, construímos algebricamente a fibração de Hopf ao final desta dissertação, explicitando sua relação com a estrutura spin da esfera S² / Abstract: Spin groups come in many forms in Mathematics and Mathematical Physics, having great importance in the theory of fiber bundles and differential operators defined on them. The concept of spin structure is derived from them, being the basis of all a theory, known as spin geometry. This thesis introduces the first concepts necessary for the study of such groups, as well as important aspects related to them. Given the nature of the Spin groups and problems which they're related to, several topics at the interface between algebra and geometry had to be addressed. At first, we studied Clifford algebras, their twisted adjoint representation and Spin groups as subgroups of the group of units of such algebras. Followed these studies a detailed analysis of the theory of covering spaces and the classification of them. Done that, we were able to understand the group Spin, via the twisted adjoint representation, as the universal covering space of the special orthogonal group of a non-degenerate quadratic space. From there, we focused on the theory of principal bundles and their relationship with the geometric properties of manifolds on which they are built. To summarize what was studied, we algebraically construct the Hopf fibration at the end of this thesis, explaining its relationship with the spin structure of the sphere S² / Mestrado / Matematica / Mestre em Matemática

The primary study of this thesis is spin-nematic squeezing in a spin-1 condensate. The measurement of spin-nematic squeezing builds on the success of previous experiments of spin-mixing together with advances in low noise atom counting. The major contributions of this thesis are linking theoretical models to experimental results and the development of the intuition and tools to address the squeezed subspaces. Understanding how spin-nematic squeezing is generated and how to measure it has required a review of several theoretical models of spin-mixing as well as extending these existing models. This extension reveals that the squeezing is between quadratures of a spin moment and a nematic (quadrapole) moment in abstract subspaces of the SU(3) symmetry group of the spin-1 system. The identification of the subspaces within the SU(3) symmetry allowed the development of techniques using RF and microwave oscillating magnetic fields to manipulate the phase space in order to measure the spin-nematic squeezing. Spin-mixing from a classically meta-stable state, the phase space manipulation, and low noise atom counting form the core of the experiment to measure spin-nematic squeezing. Spin-nematic squeezing is also compared to its quantum optics analogue, two-mode squeezing generated by four-wave mixing. The other experimental study in this thesis is performing spin-dependent photo-association spectroscopy. Spin-mixing is known to depend on the difference of the strengths of the scattering channels of the atoms. Optical Feshbach resonances have been shown to be able to alter these scattering lengths but with prohibitive losses of atoms near the resonance. The possibility of using multiple nearby resonances from different scattering channels has been proposed to overcome this limitation. However there was no spectroscopy in the literature which analyzes for the different scattering channels of atoms for the same initial states. Through analysis of the initial atomic states, this thesis studies how the spin state of the atoms affects what photo-association resonances are available to the colliding atoms based on their scattering channel and how this affects the optical Feshbach resonances. From this analysis a prediction is made for the extent of alteration of spin-mixing achievable as well as the impact on the atom loss rate.

Orientador: Waldyr Alves Rodrigues Junior / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T23:09:32Z (GMT). No. of bitstreams: 1 Resende_AdrianaSouza_M.pdf: 17553204 bytes, checksum: a66cefe30e9957cc4351e03d3aec35b2 (MD5) Previous issue date: 2010 / Resumo: O presente trabalho tem a intenção de apresentar por intermédio de uma linguagem unificada alguns conceitos de cálculo vetorial, álgebra linear (matrizes e transformações lineares) e também algumas idéias elementares sobre os grupos de rotações em duas e três dimensões e seus grupos de recobrimento, que geralmente são tratados como "fragmentos" em várias modalidades de cursos no ensino superior. Acreditamos portanto que nosso texto possas ser útil para alunos dos cursos de graduação dos cursos de Engenharia, Física, Matemática e interessados em Matemática em geral. A linguagem unificada à que nos referimos acima é obtida com a introdução do conceitos das álgebras geométricas (ou de Clifford) onde, como veremos, é possível fornecer uma formulação algébrica elegante aos conceitos de vetores, planos e volumes orientados e definir para tais objetos o produto escalar, os produtos contraídos à esquerda e à direita, o produto exterior (associado, como veremos, em casos particulares ao produto vetorial) e finalmente o produto geométrico (Clifford), o que permite o uso desses conceitos para a solução de inúmeros problemas de geometria analítica no R ² e no R ³. Procuramos ilustrar todos estes conceitos com vários exemplos e exercícios com graus variáveis de dificuldades. Nossa apresentação é bem próxima àquela do livro de Lounesto, e de fato muitas seções são traduções (eventualmente seguidas de comentários) de seções daquele livro. Contudo, em muitos lugares, acreditamos que nossa apresentação esclarece e completa as correspondentes do livro de Lounesto / Abstract: This paper aims to present using an unified language a few concepts of vector calculus, linear algebra (matrices and linear transformations) and also some basic ideas about the groups of rotations in two and three dimensions and their covering group, which generally are treated as "fragments" in various types of courses in higher education. We believe therefore that our text should be useful to students of undergraduate courses like Engineering, Physics, Mathematics and people interested in Mathematics in general. The unified language that we refer to above is obtained by introducing the concept of geometric (or Clifford) algebra where, as we shall see, it is possible to give an elegant algebraic formulation to the concepts of vectors, oriented planes and oriented volumes, and to define to those objects the scalar product, the right and left contracted products, the exterior product (associated, as we shall see, in particular cases to the vector product) and finally the geometric (Clifford) product, and moreover, to use those concepts to solve may problems of analytic geometry in R ² and R ³. We illustrated all those concepts with several examples and exercises with variable degrees of difficulties. Our presentation is nearly the one in Lounesto's book, and in fact some sections are no more than translations (eventually with commentaries) from sections of that book. However, in many places, we believe that our presentation clarify nd completement the corresponding ones in Lounesto's book / Mestrado / Ágebra / Mestre em Matemática