Indiana University-Purdue University Indianapolis (IUPUI) / This research investigates characteristics of PT (parity-time) symmetry breaking in a system of two optically-coupled, time-delayed semiconductor lasers. A theoretical rate equation model for the lasers' electric fields is presented and then reduced to a 2x2 Hamiltonian model, which, in the absence of time-delay, is PT-symmetric. The important parameters we control are the temporal separation of the lasers, the frequency detuning, and the coupling strength. The detuning is experimentally controlled by varying the lasers' temperatures, and intensity vs. detuning behavior are examined, specifically how the PT-transition and the period and amplitude of sideband intensity oscillations change with coupling and delay. Experiments are compared to analytic predictions and numerical results, and all are found to be in good agreement. Eigenvalues, eigenvectors, and exceptional points of the reduced Hamiltonian model are numerically and analytically investigated, specifically how nonzero delay affects existing exceptional points.

Made available in DSpace on 2014-06-11T19:32:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-02-04Bitstream added on 2014-06-13T21:03:27Z : No. of bitstreams: 1 santos_vgcs_dr_guara.pdf: 603356 bytes, checksum: 48d0890069648043a713c383f62ba614 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Nesta tese procuramos veri car e aprofundar os limites de validade dos chamados sistemas quânticos com simetria PT. Nestes tem-se, por exemplo, sistemas cuja hamiltoniana é não-hermitiana mas apresenta um espectro de energia real. Tal característica é usualmente justi cada pela presença da simetria PT (paridade e inversão temporal), muito embora não haja ainda uma demonstração bem aceita na literatutra desta propriedade de tais sistemas. Inicialmente estudamos sistemas quânticos não-relativísticos dependentes do tempo, sistemas em mais dimensões espaciais, a m de veri car possíveis limites da simetria PT na garantia da realidade do espectro. Logo depois estudamos sistemas quânticos relativísticos em 1+1D que possuem simetria PT com uma mistura adequada de potenciais: vetor, escalar e pseudo-escalar, sendo o potencial vetor complexo. Em seguida trabalhamos com densidades de lagrangiana com potenciais não-hermitianos em 1+1 dimensões espaço-temporais e em dimensões mais altas. A vantagem das baixas dimensões é que alguns sistemas possuem soluções não-perturbativas exatas. Finalmente, mostramos que não somente é possível ter um modelo consistente com dois campos escalares, mas também que a introdução de um número maior de campos permite que a densidade de energia também permaneça real. / In this thesis we verify and try to deepen the limits of validity of the so called quantum systems with PT-symmetry. These are systems whose Hamiltonians are non-Hermitian but present real energy spectra. Such characteristic usually is justi ed by the presence of PT symmetry (parity and time inversion), despite of the fact that there is no well accepted demonstration in literature of this property of such systems yet. Initially we study timedependent non-relativistic quantum systems in one spatial dimension in order to verify possible limits for which the PT symmetry grants the reality of the spectra. Soon later we study relativistic quantum systems in 1+1D that they possess symmetry PT with an convenient mixing of complex vector plus scalar plus pseudoscalar potentials is considered. After that, we work with a Lagrangian density with such features in 1+1 space-time dimensions and higher dimensions, in the context of eld theory. The advantage of working in low dimensions is that, in such dimensions, some systems possess exact nonperturbative solutions. Finally, we show that not only it is possible to have a consistent model with two scalar elds, but also that the introduction of a bigger number of elds allows that the energy density also remains real.

Symmetry plays a crucial role in exploring the laws of nature. By exploiting some of the underlying analogies between the mathematical formalism of quantum mechanics and that of electrodynamics, in this dissertation we show that optics can provide a fertile ground for studying, observing, and utilizing some of the peculiar symmetries that are currently out of reach in other areas of physics. In particular, in this work, we investigate two important classes of symmetries, parity-time symmetry (PT) and supersymmetry (SUSY), within the context of classical optics. The presence of PT symmetry can lead to entirely real spectra in non-Hermitian systems. In optics, PT-symmetric structures involving balanced regions of gain and loss exhibit intriguing properties which are otherwise unattainable in traditional Hermitian systems. We show that selective PT symmetry breaking offers a new method for achieving single mode operation in laser cavities. Other interesting phenomena also arise in connection with PT periodic structures. Along these lines, we introduce a new class of optical lattices, the so called mesh lattices. Such arrays provide an ideal platform for observing a range of PT-related phenomena. We show that defect sates and solitons exist in such periodic environments exhibiting unusual behavior. We also investigate the scattering properties of PT-symmetric particles and we show that such structures can deflect light in a controllable manner. In the second part of this dissertation, we introduce the concept of supersymmetric optics. In this regard, we show that any optical structure can be paired with a superpartner with similar guided wave and scattering properties. As a result, the guided mode spectra of these optical waveguide systems can be judiciously engineered so as to realize new families of mode filters and mode division multiplexers and demultiplexers. We also present the first experimental demonstration of light dynamics in SUSY ladders of photonic lattices. In addition a new type of transformation optics based on supersymmetry is also explored. Finally, using the SUSY formalism in non-Hermitian settings, we identify more general families of complex optical potentials with real spectra.

Indiana University-Purdue University Indianapolis (IUPUI) / Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semi-classical models with mode selective losses, and lossy quantum systems. The rapidly growing research on these systems has mainly focused on the wide range of novel functionalities they demonstrate. In this thesis, I intend to present some intriguing properties of a class of open systems which possess parity (P) and time-reversal (T) symmetry with a theoretical background, accompanied by the experimental platform these are realized on. These systems show distinct regions of broken and unbroken symmetries separated by a special phase boundary in the parameter space. This separating boundary is called the PT-breaking threshold or the PT transition threshold. We investigate non-Hermitian systems in two settings: tight binding lattice models, and electrical circuits, with the help of theoretical and numerical techniques. With lattice models, we explore the PT-symmetry breaking threshold in discrete realizations of systems with balanced gain and loss which is determined by the effective coupling between the gain and loss sites. In one-dimensional chains, this threshold is maximum when the two sites are closest to each other or the farthest. We investigate the fate of this threshold in the presence of parallel, strongly coupled, Hermitian (neutral) chains, and find that it is increased by a factor proportional to the number of neutral chains. These results provide a surprising way to engineer the PT threshold in experimentally accessible samples. In another example, we investigate the PT-threshold for a one-dimensional, finite Kitaev chain—a prototype for a p-wave superconductor— in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, onsite potential, and the distance between the gain and loss sites. In addition to a robust, non-local threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band-structure of the Hermitian Kitaev model. Finally, with electrical circuits, we propose a protocol to study the properties of a PT-symmetric system in a single LC oscillator circuit which is contrary to the notion that these systems require a pair of spatially separated balanced gain and loss elements. With a dynamically tunable LC oscillator with synthetically constructed circuit elements, we demonstrate static and Floquet PT breaking transitions by tracking the energy of the circuit. Distinct from traditional mechanisms to implement gain and loss, our protocol enables parity-time symmetry in a minimal classical system.

Orientador: Alvaro de Souza Dutra / Banca: Denis Dalmazi / Banca: Marcelo Batista Hotti / Banca: Alexandre Grezzi de Miranda Schmidt / Banca: Elso Drigo Filho / Resumo: Nesta tese procuramos veri car e aprofundar os limites de validade dos chamados sistemas quânticos com simetria PT. Nestes tem-se, por exemplo, sistemas cuja hamiltoniana é não-hermitiana mas apresenta um espectro de energia real. Tal característica é usualmente justi cada pela presença da simetria PT (paridade e inversão temporal), muito embora não haja ainda uma demonstração bem aceita na literatutra desta propriedade de tais sistemas. Inicialmente estudamos sistemas quânticos não-relativísticos dependentes do tempo, sistemas em mais dimensões espaciais, a m de veri car possíveis limites da simetria PT na garantia da realidade do espectro. Logo depois estudamos sistemas quânticos relativísticos em 1+1D que possuem simetria PT com uma mistura adequada de potenciais: vetor, escalar e pseudo-escalar, sendo o potencial vetor complexo. Em seguida trabalhamos com densidades de lagrangiana com potenciais não-hermitianos em 1+1 dimensões espaço-temporais e em dimensões mais altas. A vantagem das baixas dimensões é que alguns sistemas possuem soluções não-perturbativas exatas. Finalmente, mostramos que não somente é possível ter um modelo consistente com dois campos escalares, mas também que a introdução de um número maior de campos permite que a densidade de energia também permaneça real. / Abstract: In this thesis we verify and try to deepen the limits of validity of the so called quantum systems with PT-symmetry. These are systems whose Hamiltonians are non-Hermitian but present real energy spectra. Such characteristic usually is justi ed by the presence of PT symmetry (parity and time inversion), despite of the fact that there is no well accepted demonstration in literature of this property of such systems yet. Initially we study timedependent non-relativistic quantum systems in one spatial dimension in order to verify possible limits for which the PT symmetry grants the reality of the spectra. Soon later we study relativistic quantum systems in 1+1D that they possess symmetry PT with an convenient mixing of complex vector plus scalar plus pseudoscalar potentials is considered. After that, we work with a Lagrangian density with such features in 1+1 space-time dimensions and higher dimensions, in the context of eld theory. The advantage of working in low dimensions is that, in such dimensions, some systems possess exact nonperturbative solutions. Finally, we show that not only it is possible to have a consistent model with two scalar elds, but also that the introduction of a bigger number of elds allows that the energy density also remains real. / Doutor

This research investigates characteristics of PT (parity-time) symmetry breaking in a system of two optically-coupled, time-delayed semiconductor lasers. A theoretical rate equation model for the lasers’ electric fields is presented and then reduced to a 2x2 Hamiltonian model, which, in the absence of time-delay, is PT-symmetric. The important parameters we control are the temporal separation of the lasers (τ), the frequency detuning (∆ω), and the coupling strength (κ). The detuning is experimentally controlled by varying the lasers’ temperatures, and intensity vs. ∆ωbehavior are examined, specifically how the PT-transition and the period and amplitude of sideband intensity oscillations change withκandτ. Experiments are compared to analytic predictions and numerical results, and all are found to be in good agreement. Eigenvalues, eigenvectors, and exceptional points of the reduced Hamiltonian model are numerically and analytically investigated, specifically how nonzero delay affects existing exceptional points.

<p>Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semi-classical models with mode selective losses, and lossy quantum systems. The rapidly growing research on these systems has mainly focused on the wide range of novel functionalities they demonstrate. In this thesis, I intend to present some intriguing properties of a class of open systems which possess parity (P) and time-reversal (T) symmetry with a theoretical background, accompanied by the experimental platform these are realized on. These systems show distinct regions of broken and unbroken symmetries separated by a special phase boundary in the parameter space. This separating boundary is called the PT-breaking threshold or the PT transition threshold.</p> <p>We investigate non-Hermitian systems in two settings: tight binding lattice models, and electrical circuits, with the help of theoretical and numerical techniques. </p> <p><br></p> <p>With lattice models, we explore the PT-symmetry breaking threshold in discrete realizations of systems with balanced gain and loss which is determined by the effective coupling between the gain and loss sites. In one-dimensional chains, this threshold is maximum when the two sites are closest to each other or the farthest. We investigate the fate of this threshold in the presence of parallel, strongly coupled, Hermitian (neutral) chains, and find that it is increased by a factor proportional to the number of neutral chains. These results provide a surprising way to engineer the PT threshold in experimentally accessible samples.</p> <p>In another example, we investigate the PT-threshold for a one-dimensional, finite Kitaev chain—a prototype for a p-wave superconductor— in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, onsite potential, and the distance between the gain and loss sites. In addition to a robust, non-local</p> <p>threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band-structure of the Hermitian Kitaev model.</p> <p>Finally, with electrical circuits, we propose a protocol to study the properties of a PT-symmetric system in a single LC oscillator circuit which is contrary to the notion that these systems require a pair of spatially separated balanced gain and loss elements. With a dynamically tunable LC oscillator with synthetically constructed circuit elements, we demonstrate static and</p> <p>Floquet PT breaking transitions by tracking the energy of the circuit. Distinct from traditional mechanisms to implement gain and loss, our protocol enables parity-time symmetry in a minimal classical system.</p>

Topologische optische Systeme ziehen als Gegenstand aktueller Forschung große Aufmerksamkeit auf sich. Bemerkenswert sind dabei Phänomene wie die streu- und verlustfreie Lichtausbreitung mit Unempfindlichkeit gegenüber Defekten oder die einseitig gerichtete Lichtausbreitung. Das wissenschaftliche Verständnis topologischer Systeme ist jedoch noch nicht vollständig. Ziel dieser Doktorarbeit ist es, topologische Systeme in einer Dimension sowohl aus experimenteller wie auch aus theoretischer Sicht besser zu verstehen. Grundlage für alle Untersuchungen sind Mikrokavitäten mit einer optischen Dicke von der Hälfte der Designwellenlänge 1/2·λ_D = 1/2·620 nm. Diese werden umschlossen von Braggreflektoren und erreichen Qualitätsfaktoren in der Größenordnung von 10^3. Aufgrund der starken Lokalisierung des elektrischen Feldes in Kombination mit zahlreichen Möglichkeiten zur Durchführung optischer Messungen bieten Mikrokavitäten sowohl ein System zur Realisierung topologischer Zustände als auch Nachweismethoden für diese Zustände. Die Kavitäten sind mit der organischen Matrix tris-(8-hydroxy quinoline) aluminum (Alq_3) und darin eingebetteten kleinen organischen Farbstoffmolekülen gefüllt. In einem ansonsten symmetrischen Probenaufbau wechseln sich Kavitäten mit 4-(dicyanomethylene)-2-methyl-6-(p-dimethylaminostyryl)-4H-pyran (DCM) als optisch aktivem Medium (Gewinn) und zinc phthalocyanine (ZnPc) als Absorber (Verlust) ab. Bei sorgfältig austariertem Gewinn und Verlust ermöglicht eine Kombination aus Raumspiegelungs- und Zeitumkehrsymmetrie (PT-Symmetrie) eine spontane Symmetriebrechung, die für das Auftreten nicht-trivialer topologischer Eigenschaften erforderlich ist. Auf theoretischer Seite wird ein Tight-Binding-Modell für optische Mikrokavitäten hergeleitet. Das elektrische Feld ist stark in den Kavitäten lokalisiert und gebunden; die Transmission durch die Spiegel, welche die Kavitäten trennen, wird mittels eines Hüpfterms im Hamiltonoperator beschrieben. Mit dem entwickelten Modell wird ein Probenaufbau mit gekoppelten Kavitäten, die in einer PT -symmetrischen Su-Schrieffer-Heeger-Kette (SSH-Kette) angeordnet sind, betrachtet. Die Auswertung des Hamiltonoperators sagt die Ausbildung topologisch nicht-trivialer Randzustände ab sechs gekoppelten Kavitäten voraus. Die Analyse einer nicht-trivialen topologischen Kette mit zehn gekoppelten Kavitäten zeigt das Auftreten von Randzuständen und simuliert die daraus resultierenden Eigenschaften der Reflexionsmessungen. Im Experiment werden Proben mit zwei gekoppelten Kavitäten (eine Einheitszelle der SSH-Kette) hergestellt und das Transmissions- und Laserverhalten analysiert. Sowohl die symmetrischen als auch die antisymmetrischen Moden des gekoppelten Systems zeigen Lasing. Oberhalb der Laserschwelle zeigt das gekoppelte System mit austariertem Gewinn und Verlust nicht-reziprokes Verhalten. Die Messungen unterscheiden sich in Abhängigkeit von der Pump- und Detektionsrichtung in der Intensität, was auf eine gebrochene PT -Symmetrie hinweist.:1 Introduction 2 Principles of microcavity lasers 3 Physical models of light as particle and wave 4 Sample preparation and measurement setups 5 Theoretical modeling with the tight-binding approximation 6 Experimental results 7 Summary and Outlook Bibliography / Topological photonics has attracted tremendous research interest in recent years due to remarkable phenomena, like scatter-free and lossless light propagation with immunity to defects or directional light propagation. However, many questions regarding non-trivial topological systems are still open. This thesis aims to deepen the understanding of non-trivial topological systems in one dimension from both the experimental and theoretical points of view. The basis for all investigations are microcavities with an optical thickness of half of the design wavelength 1/2·λ_D = 1/2·620 nm. These are enclosed by Bragg reflectors and achieve quality factors in the order of 10^3. Due to the strong confinement of the electric field in combination with numerous possibilities to conduct optical measurements, microcavities offer a system for both realizing topological states as well as detection methods for these states. The cavities are filled with the organic matrix tris-(8-hydroxy quinoline) aluminum (Alq_3) and therein embedded small organic dye molecules. In an otherwise symmetric sample design, coupled cavities are doped alternating with 4-(dicyanomethylene)-2-methyl-6-(p-dimethyl\-amino\-styryl)-4H-pyran (DCM) as optically active medium (gain) and zinc phthalocyanine (ZnPc) as absorber (loss). With balanced gain and loss, parity-time (PT) symmetry provides the spontaneous breaking of symmetry necessary for the emergence of non-trivial topological signatures. From the theoretical side, a tight-binding model for optical microcavities is developed. The electric field is strongly confined in the cavities; transmission of the electric field through the mirrors separating the cavities is explained with the help of a hopping mechanism. This model is then applied to a sample design with coupled cavities arranged in a PT-symmetric Su-Schrieffer-Heeger (SSH) chain. The evaluation of the Hamiltonian predicts topological non-trivial edge states starting from a minimum of six coupled cavities. The analysis of a non-trivial topological chain with ten coupled cavities shows the emergence of edge states and predicts the implications on reflection measurements. In the experiment, samples with two coupled cavities (one unit cell in the SSH chain) are fabricated, and transmission and lasing behavior are analyzed. Both the symmetric and antisymmetric modes of the coupled system show lasing. Above the lasing threshold, the coupled system with balanced gain and loss shows non-reciprocal behavior. The measurements differ in intensity as a function of the pump and detection directions, pointing to the achieved broken PT-symmetric phase.:1 Introduction 2 Principles of microcavity lasers 3 Physical models of light as particle and wave 4 Sample preparation and measurement setups 5 Theoretical modeling with the tight-binding approximation 6 Experimental results 7 Summary and Outlook Bibliography

This thesis is concerned with analysis of spectral and orbital stability of solitary wave solutions to discrete and continuous PT-symmetric nonlinear Schroedinger equations. The main tools of this analysis are inspired by Hamiltonian systems, where conserved quantities can be used for proving orbital stability and Krein signature can be computed for prediction of instabilities in the spectrum of linearization. The main results are obtained for the chain of coupled pendula represented by a discrete NLS model, and for the trapped atomic gas represented by a continuous NLS model. Analytical results are illustrated with various numerical examples. / Thesis / Doctor of Philosophy (PhD)

Simetrias compõe parte integral da análise na Teoria das Matrizes Aleatórias (RMT). As simetrias de inversão temporal e rotacional são aspectos-chave do Ensemble Gaussiano Ortogonal (GOE), enquanto esta última é quebrada no Ensemble Gaussiano Simplético (GSE) e ambas são quebradas no Conjunto Unitário Gaussiano (GUE). Desde o final da década de 1990, o crescente interesse no campo dos sistemas quânticos PT-simétricos levou os pesquisadores a considerar o efeito, em matrizes aleatórias, dessa classe de simetrias, bem como simetrias pseudo-hermitianas. A principal questão a ser respondida pela pesquisa apresentada nesta tese é se a simetria PT ou, de forma mais geral, a pseudo-Hermiticidade implica alguma distribuição de probabilidade específica para os autovalores. Ou, em outras palavras, se há um aspecto comum transmitido por tal simetria que pode ser usada para modelar alguma classe particular de sistemas físicos. A abordagem inicial considerada consistiu na introdução de um conjunto pseudo-hermitiano, isospectral ao conjunto -Hermite, que apresentaria o tipo de quebra de realidade típico dos sistemas PT-simétricos. Nesse modelo, a primeira abordagem adotada foi a introdução de perturbações que quebraram a realidade dos espectros. Os resultados obtidos permitem concluir que a transformação em seu similar pseudo-hermitiano conduz a um sistema assintoticamente instável. Esse modelo foi extendido ao considerar um pseudo-hermitiano não positivo, que leva a uma quebra similar na realidade dos espectros. Este caso apresenta um comportamento mais próximo do típico dos sistemas PT-simétricos presentes na literatura. Um modelo denso geral baseado em projetores foi proposto, e duas realizações particulares deste modelo receberam atenção mais detalhada. O comportamento espectral também foi similar àquele típico da simetria PT para as duas realizações consideradas, e seus limites assintóticos foram conectados a conjuntos clássicos de teoria de matriz aleatória. Além disso, as propriedades de seus polinômios característicos médios foram obtidas e os limites assintóticos desses polinômios também foram considerados e relacionados a polinômios clássicos. O comportamento estatístico deste conjunto foi estudado e comparado com o destes polinômios. Impor a pseudo-Hermiticidade não parece implicar qualquer distribuição particular de autovalores, sendo a característica comum a quebra da realidade dos autovalores comumente encontrados na literatura de simetria PT. O resultado mais notável dos estudos apresentados nesta tese é o fato de que uma interação pseudo-hermitiana pode ser construída de tal forma que o comportamento espectral médio possa ser controlado calibrando-se o mecanismo de interação, bem como sua intensidade. / The role of symmetries is an integral part of the analysis in Random Matrix Theory (RMT). Time reversal and rotational symmetries are key aspects of the Gaussian Orthogonal Ensemble (GOE), whereas the latter is broken in the Gaussian Sympletic Ensemble (GSE) and both are broken in the Gaussian Unitary Ensemble (GUE). Since the late 1990s, growing interest in the field of PT symmetric quantum systems has led researchers to consider the effect, in random matrices, of this class of symmetries, as well as that of pseudo-Hermitian symmetries. The primary question to be answered by the research presented in this thesis is whether PT-symmetry or, more generally, pseudo-Hermiticity implies some specific probability distribution for the eigenvalues. Or, in other words, whether there is a common aspect imparted by such a symmetry which may be used to model some particular class of physical systems. The initial approach considered consisted of introducing an pseudo-Hermitian ensemble, isospectral to the -Hermite ensemble, which would present the type of reality-breaking typical of PT-symmetrical systems. In this model, the first approach taken was to introduce perturbation which broke the reality of the spectra. The results obtained allow the conclusion that the transformation into its pseudo-Hermitian similar leads into a system which is asymptotically unstable. An extension of this model was to consider a non-positive pseudo-Hermitian , which lead to similar breaking in the reality of the spectra. This case displays behavior closer to that typical of the PT-symmetric systems present in the literature. A general dense projector model was proposed, and two particular realizations of this model were given more detailed attention. The spectral behavior was also similar to that typical of PT-symmetry for the two realizations considered, and their asymptotic limits were shown to connect to classical ensembles of random matrix theory. Furthermore, the properties of their average characteristic polynomials were obtained and the asymptotic limits of these polynomials were also considered and were related to classical polynomials. The statistical behavior of this ensemble was studied and compared to that of these polynomials. Imposing the pseudo-Hermitian does seem not imply any particular eigenvalue distribution, the common feature being the breaking of the reality of the eigenvalues commonly found in PT-symmetry literature. The most notable result of the studies presented herein is the fact that a pseudo-Hermitian interaction may be constructed such that the average spectral behavior may be controlled by calibrating the mechanism of interaction as well as its intensity.