The spatial evolution of a pair of resonant Bragg modes through a medium characterized by a complex one-dimensional PT -symmetric periodic relative electric permittivity is thoroughly investigated. By using the two wave model, analytic solutions of Maxwell’s equations are derived in the nonparaxial regime in order to investigate the periodic energy exchange between the Bragg modes for the Hermitian optical lattices as well as for complex lattices and also to investigate the spatial evolution of the real part of the electric field that propagates through this medium. Three regimes defined by the symmetry breaking point are discussed: below it, above it and at it. These regimes are determined by the existence of four complex eigenvalues below the symmetry breaking point, which collide and coalesce into a pair of complex doubly degenerate eigenvalues at the breaking point. Above the critical value each member of the pair bifurcates into a pair of complex values and now they have a nonzero real part. In the Hermitian case, it is demonstrated that a complete reciprocal energy transfer between the pair of Bragg modes, in a manner similiar to the Pendellösung effect known from X-ray diffraction by a crystal, takes place. When the optical lattice is complex, the dynamics of the energy transfer is completely different from the Hermitian case: below the symmetry breaking point there exist a very nonreciprocal beating-like oscillatory behavior of the energy transfer between the Bragg modes; above the symmetry breaking point the spatial evolution of the energy transfer grows unlimited but an oscillatory evolution still takes place; and at the symmetry breaking point the phenomenon of mode trapping does not occur in this nonparaxial regime (previously seen in PT -symmetric optical lattices in the paraxial regime). For the complex lattice, all these regimes share the common features: existence of a preferable mode for which the energy is transferred and a spatial evolution of this transfer in a nonreciprocal fashion, some of the characteristics very well known of PT -symmetric optical systems. / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / A evolução espacial de um par de modos de Bragg ressonantes através de um meio descrito por uma permissividade elétrica relativa PT -simétrica é completamente investigada. Usando o modelo de duas ondas, soluções analíticas para as equações de Maxwell são derivadas no regime não-paraxial a fim de investigar a transferência de energia entre esses modos de Bragg para uma rede ótica Hermitiana bem como para uma rede ótica complexa e também estudar a evolução espacial da parte real do campo elétrico que se propaga através desse meio. Três regimes definidos pela quebra de simetria são discutidos: abaixo, acima e no ponto de quebra de simetria. Estes regimes são determinados pela existência de quatro autovalores complexos distintos abaixo da quebra de simetria, que colidem e coalescem num par de autovalores valores complexos duplamente degenerados no ponto de quebra de simetria. Acima do ponto de quebra, quatro autovalores complexos distintos voltam a existir e agora cada um possui uma parte real não-nula. No caso Hermitiano, é demonstrado que uma transferência de energia completamente recíproca entre o par de modos de Bragg, numa maneira similiar ao efeito Pendellösung conhecido da difração de raios X por cristais, ocorre. Quando a rede ótica é complexa, a dinâmica da transferência de energia é completamente diferente do caso Hermitiano: abaixo do ponto de quebra de simetria existe um comportamento oscilatório do tipo batimento muito não-recíproco para a transferência de energia entre os modos de Bragg; acima do ponto de quebra de simetria a evolução espacial da transferência de energia cresce ilimitadamente mas um evolução oscilatória ainda ocorre; no ponto de quebra o fenômeno do aprisionamento de modo não ocorre nesse regime não-paraxial (anteriormente visto em redes óticas PT -simétricas no regime paraxial). Para a rede ótica complexa, todos estes regimes compartilham características em comum: existência de um modo preferencial para o qual a energia é transferida e uma evolução espacial dessa transferência de maneira não-recíproca, algumas das características muito bem conhecidas de sistemas óticos PT simétricos.

Indiana University-Purdue University Indianapolis (IUPUI) / More than a decade ago, it was shown that non-Hermitian Hamiltonians with combined parity (P) and time-reversal (T ) symmetry exhibit real eigenvalues over a range of parameters. Since then, the field of PT symmetry has seen rapid progress on both the theoretical and experimental fronts. These effective Hamiltonians are excellent candidates for describing open quantum systems with balanced gain and loss. Nature seems to be replete with examples of PT -symmetric systems; in fact, recent experimental investigations have observed the effects of PT symmetry breaking in systems as diverse as coupled mechanical pendula, coupled optical waveguides, and coupled electrical circuits. Recently, PT -symmetric Hamiltonians for tight-binding lattice models have been extensively investigated. Lattice models, in general, have been widely used in physics due to their analytical and numerical tractability. Perhaps one of the best systems for experimentally observing the effects of PT symmetry breaking in a one-dimensional lattice with tunable hopping is an array of evanescently-coupled optical waveguides. The tunneling between adjacent waveguides is tuned by adjusting the width of the barrier between them, and the imaginary part of the local refractive index provides the loss or gain in the respective waveguide. Calculating the time evolution of a wave packet on a lattice is relatively straightforward in the tight-binding model, allowing us to make predictions about the behavior of light propagating down an array of PT -symmetric waveguides. In this thesis, I investigate the the strength of the PT -symmetric phase (the region over which the eigenvalues are purely real) in lattices with a variety of PT - symmetric potentials. In Chapter 1, I begin with a brief review of the postulates of quantum mechanics, followed by an outline of the fundamental principles of PT - symmetric systems. Chapter 2 focuses on one-dimensional uniform lattices with a pair of PT -symmetric impurities in the case of open boundary conditions. I find that the PT phase is algebraically fragile except in the case of closest impurities, where the PT phase remains nonzero. In Chapter 3, I examine the case of periodic boundary conditions in uniform lattices, finding that the PT phase is not only nonzero, but also independent of the impurity spacing on the lattice. In addition, I explore the time evolution of a single-particle wave packet initially localized at a site. I find that in the case of periodic boundary conditions, the wave packet undergoes a preferential clockwise or counterclockwise motion around the ring. This behavior is quantified by a discrete momentum operator which assumes a maximum value at the PT -symmetry- breaking threshold. In Chapter 4, I investigate nonuniform lattices where the parity-symmetric hop- ping between neighboring sites can be tuned. I find that the PT phase remains strong in the case of closest impurities and fragile elsewhere. Chapter 5 explores the effects of the competition between localized and extended PT potentials on a lattice. I show that when the short-range impurities are maximally separated on the lattice, the PT phase is strengthened by adding short-range loss in the broad-loss region. Consequently, I predict that a broken PT symmetry can be restored by increasing the strength of the short-range impurities. Lastly, Chapter 6 summarizes my salient results and discusses areas which can be further developed in future research.

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Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2016-05-02T10:49:21Z No. of bitstreams: 1 arquivototal.pdf: 1791577 bytes, checksum: ecd7a6eb8a18230d0145c24932d9e248 (MD5) / Made available in DSpace on 2016-05-02T10:49:22Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1791577 bytes, checksum: ecd7a6eb8a18230d0145c24932d9e248 (MD5) Previous issue date: 2015-08-03 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / The Hanbury Brown-Twiss experiment is very well established in quantum optics literature, so we devoted this dissertation in order to embed the parity and temporal reversal symmetry in the former experiment. Therefore, we developed the scattering matrix formalism which allow us use some techniques such as the scattering matrices' concatenation of di erent sections in terms of one matrix and the scattering problem of a parity and temporal reverse symmetric system. In this manner, we could derive the scattering matrix of a parity and temporal reverse symmetric Hanbury Brown- Twiss experiment(HBT-PT). With the possession of this matrix, we proposed a theoretical model which provides how to measure the symmetry of this system, which we called correlation functions formalism. In order to stablish the former formalism, we studied B uttiker formalism, which we veri ed how the correlation between 2 incident particles in a system relates to the noise due transport of this particles and what kind of noise we are treating in a given regime which the system is operating. Then, we found the input states in terms of the output states of two particles inciding in a multiterminal system, which we used it's scattering matrix to stablish the relation between the states. Thereat, we derived all the possibles correlations(therefore, the noise) of two incident particles in the former system. Thereby, we analysed the extreme cases of a barrier coupled to the HBT-PT experiment for the purpose of stablish which regime the experiment is symmetric adjusted, so, demonstrating the Hanbury Brown-Twiss E ect with parity and temporal reverse symmetries. / Sendo o experimento Hanbury Brown-Twiss bem estabelecido na literatura da otica quântica, dedicamos esta dissertação a embutir no mesmo experimento a questão da simetria por paridade e reversão temporal. Para tanto, abrimos mão do formalismo da matriz de espalhamento que permitiu nos utilizar tecnicas como a concatenação de matrizes de espalhamento de sec ções diferentes em termos de apenas uma matriz e o problema do espalhamento em um sistema simétrico por paridade e reversão temporal. Dessa forma, pudemos derivar a matriz de espalhamento para o experimento Hanbury Brown-Twiss com simetria de paridade e revers~ao temporal(HBT-PT). De posse dessa matriz, propusemos um modelo teórico que propicia a medição experimental a simetria desse sistema, o qual chamamos de formalismo das fun c~oes de correla c~ao. Para estabelecermos o formalismo supracitado, estudamos o formalismo de Buttiker, onde veri case como a correlaçãao entre 2 partículas incidentes em um sistema relaciona-se com o ru do devido ao transporte dessas part culas e que tipo de ru do estamos tratando, dado o regime em que o sistema está operando. Em seguida, encontramos os estados de entrada em termos dos estados de saída de 2 partículas incidindo em um sistema multiterminal, onde utilizamos a sua matriz de espalhamento para estabelecer a rela c~ao entre os estados. Com isso, derivamos todas as possíveis correlações (e, por conseguinte, o ruído) de 2 partículas incidentes nesse sistema. Assim, analisamos os casos extremos de uma barreira acoplada ao experimento HBT-PT, a m de estabelecer o regime em que o experimento está simetricamente ajustado e demonstramos o Efeito Hanbury Brown-Twiss por paridade e reversão temporal.