Made available in DSpace on 2014-08-27T14:36:44Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-02-26Bitstream added on 2014-08-27T15:57:04Z : No. of bitstreams: 1 000781512.pdf: 7049961 bytes, checksum: 5670d7d0c2e14f1533ef292642ea6f6d (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Os gases atômicos ultrafrios têm proporcionado um importante ambiente no estudo de sistemas quânticos de muitas partículas nas duas últimas décadas. Em 2005, a realização experimental dum condensado de Bose-Einstein de 'INTPOT. 52 Cr' com interação magnética dipolo dipolo inter-atômica abriu a porta para um novo nível na pesquisa de gases quânticos degenerados. Ao contrário da interacção de contacto, esta nova interação é de longo alcance e anisotrópica sendo em parte repulsiva ou atrativa. Na aproximação de campomeio, inicialmente, são introduzidas as principais questões sobre condesados com especial interesse no regime atrativo (a< 0) onde é possível a formação de solitons brilhantes e a existência da instabilidade por colapso além de um certo valor crítico. O estudo é realizado,principalmente,usandoummétodonuméricoeumvariacional. Posteriormente, o condensado de Bose-Einstein dipolar é descrito através da equação não-local de GrossPitaevskii. A partir do cenário não-dipolar, por meio da extensão no método numérico e no método variacional é determinada a formação de solitons brilhantes na equação de Gross-Pitaevskii nos modelos tridimensional e quasi-unidimensional para três diferentes condensados dipolares de relevância experimental, isto é 'INTPOT. 52 Cr', 'INTPOT. 168 Er' e 'INTPOT. 164 Dy'. Grá?cos do potencial químico e a raiz quadrática média (rms) dos solitons são obtidos. Finalmente, estuda-se a dinâmica da colisão de dois solitons brilhantes no modelo dipolar quasi-1D de cada condensado acima / The ultracold atomic gases have provided an important environment for studying quantum many-particle systems in the last two decades. In 2005 the experimental realization of a 'INTPOT. 52 Cr' Bose-Einstein condensate with inter-atomic magnetic dipole dipole interaction opened the door to a new level in the research of degenerate quantum gases. As opposed to the usual contact interaction, this new interaction is long-range and anisotropic being partially repulsive or attractive. In the mean-?eld approximation initially are introduced the main issues about non-dipolar condensates with particular interest in the attractive regime (a< 0) where is possible the formation of bright solitons and the existence of instability by collapse beyond a certain critical value. The study is carried out mainly using a numerical method and a variational one. Later, the dipolar Bose-Einstein condensate is depicted by means of the non-local Gross-Pitaevskii equation. From the non-dipolar scenario by means of the extension in the numerical and the variational method is determined the formation of bright solitons in the GPE in the three-dimensional model and thequasi-one-dimensionaltothreedi?erentdipolarcondensatesofexperimentalrelevance, namely 'INTPOT. 52 Cr', 'INTPOT. 168 Er' and 'INTPOT. 164 Dy'. Plots of chemical potential and rms sizes of solitons are obtained. Finally, it is studied collision dynamics of two bright solitons in the quasi-1D dipolar model of every condensate above

Orientador: Sadhan Kumar Adhikari / Banca: Arnaldo Gammal / Banca: Lauro Tomio / Resumo: Os gases atômicos ultrafrios têm proporcionado um importante ambiente no estudo de sistemas quânticos de muitas partículas nas duas últimas décadas. Em 2005, a realização experimental dum condensado de Bose-Einstein de 'INTPOT. 52 Cr' com interação magnética dipolo dipolo inter-atômica abriu a porta para um novo nível na pesquisa de gases quânticos degenerados. Ao contrário da interacção de contacto, esta nova interação é de longo alcance e anisotrópica sendo em parte repulsiva ou atrativa. Na aproximação de campomeio, inicialmente, são introduzidas as principais questões sobre condesados com especial interesse no regime atrativo (a< 0) onde é possível a formação de solitons brilhantes e a existência da instabilidade por colapso além de um certo valor crítico. O estudo é realizado,principalmente,usandoummétodonuméricoeumvariacional. Posteriormente, o condensado de Bose-Einstein dipolar é descrito através da equação não-local de GrossPitaevskii. A partir do cenário não-dipolar, por meio da extensão no método numérico e no método variacional é determinada a formação de solitons brilhantes na equação de Gross-Pitaevskii nos modelos tridimensional e quasi-unidimensional para três diferentes condensados dipolares de relevância experimental, isto é 'INTPOT. 52 Cr', 'INTPOT. 168 Er' e 'INTPOT. 164 Dy'. Gráficos do potencial químico e a raiz quadrática média (rms) dos solitons são obtidos. Finalmente, estuda-se a dinâmica da colisão de dois solitons brilhantes no modelo dipolar quasi-1D de cada condensado acima / Abstract: The ultracold atomic gases have provided an important environment for studying quantum many-particle systems in the last two decades. In 2005 the experimental realization of a 'INTPOT. 52 Cr' Bose-Einstein condensate with inter-atomic magnetic dipole dipole interaction opened the door to a new level in the research of degenerate quantum gases. As opposed to the usual contact interaction, this new interaction is long-range and anisotropic being partially repulsive or attractive. In the mean-field approximation initially are introduced the main issues about non-dipolar condensates with particular interest in the attractive regime (a< 0) where is possible the formation of bright solitons and the existence of instability by collapse beyond a certain critical value. The study is carried out mainly using a numerical method and a variational one. Later, the dipolar Bose-Einstein condensate is depicted by means of the non-local Gross-Pitaevskii equation. From the non-dipolar scenario by means of the extension in the numerical and the variational method is determined the formation of bright solitons in the GPE in the three-dimensional model and thequasi-one-dimensionaltothreedifferentdipolarcondensatesofexperimentalrelevance, namely 'INTPOT. 52 Cr', 'INTPOT. 168 Er' and 'INTPOT. 164 Dy'. Plots of chemical potential and rms sizes of solitons are obtained. Finally, it is studied collision dynamics of two bright solitons in the quasi-1D dipolar model of every condensate above / Mestre

Turbulence is a young field of research which is characterized by chaotic spinning flow regimes which appears in many important processes in nature. Vorticity, in superfluid systems, may present the simplest form of turbulence, and be a gateway to the study of this phenomenon in quantum gases. A 87Rb Bose condensate was used to observe and investigate the emergence of quantum turbulence, a few years back in our group. The vortices are created on the condensed-thermal interface and propagate across the cloud, setting up the experimental conditions favorable to the emergence of turbulence. Once the turbulent regime is set, the condensate is released and expands under free fall. Then, the atomic density profile is acquired, after some time-of-flight, and used to determine the in situ momentum distribution of the BEC. In this work, we have observed that, the perturbed density profiles are characteristic and different from the standard, non-perturbed ones. We have seen evidences of power law in the studied momentum and energy distributions and also coupling of quadrupolar mode to the momentum distribution of the excited condensate which is the main part of our findings. Additional features of the system, such as the condensates excited collective modes which plays a very important role on the roadmap to the turbulence regime, are discussed. We are currently setting up an experiment to be able to further investigate such features, and also to unfold the effects of interactions on the energy and momentum spectra associated to the density profiles. In doing so, we will further develop the tools and techniques needed to acquire more accurate and reliable results. / Um dos tópicos recentes das pesquisas em superfluidos atômicos é o estudo da turbulência quântica. Em fluídos, a turbulência é caracterizada pelo regime caótico no escoamento dos fluidos e aparece em muitos importantes processos na natureza. Em sistemas superfluidos, a forma mais simples da turbulência é apresentada pelo enovelamento de vórtices. Assim, o estudo de vórtices nesses sistemas torna-se um ponto de partida para estudar o fenômeno da turbulência em gases quânticos. Há alguns anos atrás, em nosso grupo de pesquisa, um condensado de Bose-Einstein de 87Rb foi usado para observar e investigar a emergência de turbulência quântica. Em continuidade a esses estudos, aplicamos uma excitação oscilatória na nuvem atômica aprisionada e os vórtices são criados na interface entre o condensado e a nuvem térmica, que se propagam para o interior da nuvem, atingindo as condições ideais para o aparecimento de um regime turbulento. Uma vez que esse regime é atingido, o condensado é diagnosticado através de uma imagem de absorção obtida após a sua expansão balística em tempo de voo. O perfil de densidade obtido é usado para determinar a distribuição de momento do condensado aprisionado. Neste trabalho, observamos que os perfis de densidade dos condensados excitados possuem uma forma característica e diferente dos condensados não-excitados. Nos estudos da distribuição de momento e energia dessas nuvens excitadas, vimos uma evidência de uma lei de potência (parecida com a lei de Kolmogorov para turbulência) e, além disso, um acoplamento entre o modo quadrupolar de oscilação da nuvem e a distribuição de momentos dessa nuvem. Também discutimos algumas propriedades adicionais do sistema, por exemplo, os modos coletivos de excitação do condensado, o que tem um papel muito importante na rota para o regime de turbulência quântica. Para continuarmos com os estudos neste tópico de pesquisa, estamos melhorando nosso sistema experimental a fim de investigarmos melhor estas propriedades dinâmicas do superfluido, através dos efeitos dos modos coletivos no espectro de momentos da nuvem atômica. Para isso, pretendemos desenvolver novas técnicas e ferramentas necessárias para realizar medidas mais precisas e reprodutivas.

CoordenaÃÃo de AperfeiÃoamento de NÃvel Superior / O principal objetivo deste trabalho à apresentar uma generalizaÃÃo das mÃtricas quasi-Einstein generalizadas para campos de vetores suaves quaisquer. AlÃm disso, serÃo apresentadas algumas fÃrmulas integrais para mÃtricas quasi-Einstein gradiente generalizadas definidas em uma variedade compacta. / The main objective of this paper is to present a generalization of generalized quasi-Einstein metrics to any smooth vector fields. Moreover, we will present some integral formulae for quasi-Einstein metrics defined in a compact manifolds.

In this thesis a number of aspects on possible manipulation of Bose-Einstein condensate in trapped atomic gases is investigated. First, a model for atom optical experiments involving Bose condensates is proposed and numerical simulations are presented to illustrate its characteristics. We demonstrate ways of focusing and splitting the condensate by modifying experimentally adjustable parameters. We show that there are at least two ways of implementing atom optical elements: one may modulate the interatomic scattering length in space, or alternatively, use a sinusoidal, externally applied potential. The temporal evolution of quasiparticle excitations is studied via the Gross-Pitaevskii Equation. Nonlinear mode mixing of quasiparticles is introduced, and is observed using a quasiparticle projection method. This is used as a basis of time-dependent finite temperature simulations, which we argue to be valid under regimes of high occupation number. An illustration via a closely related evaporative cooling simulation is provided. A phenomenological damping formalism for superfluidity near the λ point is adopted to describe the damping of excitations in a Bose-Einstein condensate. An estimate for the damping parameter is found. The damping formalism as a numerical tool to calculate the ground eigenstate of the condensate is explored. A novel, experimentally realisable interferometry for Bose-Einstein condensates using near-field diffraction is proposed. The scheme is based on the phenomenon of intermode traces or quantum carpets; we demonstrate the structured spatio-temporal pattern for the dilute, atomic Bose-Einstein condensate. The pattern is found to change with temperature, which allows us to perform interferometric temperature measurements. Finally, an output coupler for Bose-Einstein Condensates based on stimulated Raman transition is investigated. The spectrum and coherence are calculated for an atomic beam slowly coupled out of a trap containing a partially condensed Bose gas at finite temperatures. A number conserving Hartree-Fock-Bogliubov formalism has been used to incorporate finite temperature effects. Various different processes are found to become dominant for a suitable choice of the coupling parameters.

Bose-Einstein Condensation is a macroscopic occupation of bosons in the lowest energy state. For atoms, extremely low temperatures are required to observe this phenomenon. For photons, condensation has been demonstrated at room temperature, requiring a large number of particles (N ∼ 77000) and very complicated setup. Here we study the possibility of observing BEC of light at room temperature with a much lower number of particles by leveraging disorder in a dielectric material. There is no constraint in the number of photons in the system like in the previous research. We investigate what happens to photons once they are put inside a cavity with a disorder. The analysis is carried out by using time-dependent quantum Langevin equations, complemented by a thermodynamic analysis on quantum photons. Both approaches give the same expression for the critical temperature of condensation. We demonstrate that photons in a disordered cavity with arbitrary initial statistical distribution reach thermal equilibrium and undergo a Bose-Einstein Condensation if the temperature is sufficiently reduced. In our model we demonstrate that the temperature is related to the losses of the system. At this state, photons follow Boltzmann distribution. It is demonstrated that by only varying the strength of disorder, it is possible to change the critical temperature of the phase transition, thus making condensation possible at room temperature. This work opens up the possibility to create new types of light condensate by using disorder.

Die Bose-Einstein-Kondensation (BEK) und die damit verbundenen Effekte wie Superfluidität und Supraleitung sind faszinierende Resultate der Quantennatur von Bosonen. Nachdem die Bose-Einstein-Kondensation für Atom-Systeme nur bei Temperaturen nahe dem absoluten Nullpunkt realisierbar ist, was einen enormen technologischen Aufwand benötigt, wurden Bosonen mit wesentlich kleineren Massen zur Untersuchung der BEK gesucht. Hierfür bieten sich Quasiteilchen in Festkörpern wie Magnonen oder Exzitonen an, da deren effektive Massen sehr klein sind und die Kondensationstemperatur dementsprechend höher ist als für ein atomares System. Ein weiteres Quasiteilchen ist das Exziton-Polariton als Resultat der starken Licht-Materie-Wechselwirkung in Halbleitermikrokavitäten, welches sowohl Materie- als auch Photoneigenschaften hat und dessen Masse theoretisch eine BEK bis Raumtemperatur erlaubt. Ein weiterer Vorteil dieses System ist die einfache Erzeugung des Bose-Einstein-Kondensats in diesen Systemen durch elektrisches oder optisches Injizieren von Exzitonen in die Halbleiter-Quantenfilme der Struktur. Außerdem kann die Impulsraumverteilung dieser Quasiteilchen leicht durch einfache experimentelle Methoden mittels eines Fourierraumspektroskopie-Aufbaus bestimmt werden. Durch die winkelabhängige Messung der Emission kann direkt auf die Impulsverteilung der Exziton-Polaritonen in der Quantenfilmebene zurückgerechnet werden, die zur Identifikation der BEK hilfreich ist. Deshalb wird das Exziton-Polariton als ein Modellsystem für die Untersuchung von Bose-Einstein-Kondensation in Festkörpern und den damit in Relation stehenden Effekten angesehen. In dieser Arbeit wird die Grundzustandskondensation von Exziton-Polaritonen in Halbleitermikrokavitäten verschiedener Dimensionen realisiert und deren Emissionseigenschaften untersucht. Dabei wird vor allem die Wechselwirkung des Polariton-Kondensats mit der der unkondensierten Polaritonen bzw. der Quantenfilm-Exzitonen im externen Magnetfeld verglichen und ein Nachweis zum Erhalt der starken Kopplung über die Polariton-Kondensationsschwelle hinaus entwickelt. Außerdem werden die Kohärenzeigenschaften von null- und eindimensionalen Polariton-Kondensaten durch Bestimmung der Korrelationsfunktion erster beziehungsweise zweiter Ordnung analysiert. Als Materialsystem werden hierbei die III/V-Halbleiter gewählt und die Quantenfilme bestehen bei allen Messungen aus GaAs, die von einer AlAs Kavität umgeben sind. Eindimensionale Polariton-Kondensation - räumliche Kohärenz der Polariton-Drähte Im ersten experimentellen Teil dieser Arbeit (Kapitel 1) wird die Kondensation der Polaritonen in eindimensionalen Drähten unter nicht-resonanter optischer Anregung untersucht. Dabei werden verschiedene Drahtlängen und -breiten verwendet, um den Einfluss des zusätzlichen Einschlusses auf die Polariton-Dispersion bestimmen zu können. Ziel dieser Arbeit ist es, ein eindimensionales Bose-Einstein-Kondensat mit einer konstanten räumlichen Kohärenz nach dem zentralen Abfall der g^(1)(r)-Funktion für große Abstände r in diesen Drähten zu realisieren (sogenannte langreichweitige Ordnung im System, ODLRO (Abkürzung aus dem Englischen off-diagonal long-range order). Durch Analyse der Fernfeldemissionseigenschaften können mehrere Polariton-Äste, der eindimensionale Charakter und die Polariton-Kondensation in 1D-Systemen nachgewiesen werden. Daraufhin wird die räumliche Kohärenzfunktion g^(1)(r) mithilfe eines hochpräzisen Michelson-Interferometer, das im Rahmen dieser Arbeit aufgebaut wurde, bestimmt. Die g^(1)(r)-Funktion nimmt hierbei über große Abstände im Vergleich zur thermischen De-Broglie-Wellenlänge einen konstanten Plateauwert an, der abhängig von der Anregungsleistung ist. Unterhalb der Polariton-Kondensationsschwelle (Schwellleistung P_S) ist kein Plateau sichtbar und die räumliche Kohärenz ist nur im zentralen Bereich von unter |r| < 1 µm vorhanden. Mit ansteigender Anregungsleistung nimmt das zentrale Maximum in der Weite zu und es bildet sich das Plateau der g^(1)(r)-Funktion aus, das nur außerhalb des Drahtes auf Null abfällt. Bei P=1,6P_S ist das Plateau maximal und beträgt circa 0,15. Außerdem kann nachgewiesen werden, dass mit steigender Temperatur die Plateauhöhe abnimmt und schließlich bei T=25K nicht mehr gemessen werden kann. Hierbei ist dann nur noch das zentrale Maximum der Kohärenzfunktion g^(1)(r) sichtbar. Weiterhin werden die Ergebnisse mit einer modernen mikroskopischen Theorie, die auf einem stochastischen Mastergleichungssystem basiert, verglichen, wodurch die experimentellen Daten reproduziert werden können. Im letzten Teil des Kapitels wird noch die Kohärenzfunktion g^(1)(r) im 1D-Fall mit der eines planaren Polariton-Kondensats verglichen (2D). Nulldimensionale Polariton-Kondensation - Kondensation und Magnetfeldwechselwirkung in einer Hybridkavität Im zweiten Teil der Arbeit wird die Polariton-Kondensation in einer neuartigen Hybridkavität untersucht. Der Aufbau des unteren Spiegels und der Kavität inklusive der 12 verwendeten Quantenfilme ist analog zu den gewöhnlichen Mikrokavitäten auf Halbleiterbasis. Der obere Spiegel jedoch besteht aus einer Kombination von einem DBR (Abkürzung aus dem Englischen distributed Bragg reflector) und einem Brechungsindexkontrast-Gitter mit einem Luft-Halbleiterübergang (größt möglichster Brechungsindexkontrast). Durch die quadratische Strukturgröße des Gitters (Seitenlänge 5µm) sind die Polaritonen zusätzlich zur Wachstumsrichtung noch in der Quantenfilmebene eingesperrt, so dass sie als nulldimensional angesehen werden können (Einschluss auf der ungefähren Größe der thermischen De-Broglie-Wellenlänge). Um den Erhalt der starken Kopplung über die Kondensationsschwelle hinaus nachweisen zu können, wird ein Magnetfeld in Wachstumsrichtung angelegt und die diamagnetische Verschiebung des Quantenfilms mit der des 0D-Polariton-Kondensats verglichen. Hierdurch kann das Polariton-Kondensat von dem konventionellen Photonlasing in solchen Strukturen unterschieden werden. Weiterhin wird als letztes Unterscheidungsmerkmal zwischen Photonlasing und Polariton-Kondensation eine Messung der Autokorrelationsfunktion zweiter Ordnung g^(2)(t) durchgeführt. Dabei kann ein Wiederanstieg des g^(2)(t = 0)-Werts mit ansteigender Anregungsleistung nachgewiesen werden, nachdem an der Kondensationsschwelle der g^(2)(t = 0)-Wert auf 1 abgefallen ist, was auf eine zeitliche Kohärenzzunahme im System hinweist. Oberhalb der Polariton-Kondensationsschwelle P_S steigt der g^(2)(t = 0)-Wert wieder aufgrund zunehmender Dekohärenzprozesse, verursacht durch die im System ansteigende Polariton-Polariton-Wechselwirkung, auf Werte größer als 1 an. Für einen gewöhnlichen Photon-Laser (VCSEL, Abkürzung aus dem Englischen vertical-cavity surface-emitting laser) im monomodigen Betrieb kann mit steigender Anregungsleistung kein Wiederanstieg des g^(2)(t = 0)-Werts gemessen werden. Somit stellt dies ein weiteres Unterscheidungsmerkmal zwischen Polariton-Kondensation und Photonlasing dar. Zweidimensionale Polariton-Kondensation - Wechselwirkung mit externem Magnetfeld Im letzten experimentellen Kapitel dieser Arbeit wird die Magnetfeldwechselwirkung der drei möglichen Regime der Mikrokavitätsemission einer planaren Struktur (zweidimensional) untersucht. Dazu werden zuerst durch eine Leistungsserie bei einer Verstimmung des Photons und des Quantenfilm-Exzitons von d =-6,5meV das lineare, polaritonische Regime, das Polariton-Kondensat und bei weiterer Erhöhung der Anregungsleistung das Photonlasing identifiziert. Diese drei unterschiedlichen Regime werden daraufhin im Magnetfeld von B=0T-5T auf ihre Zeeman-Aufspaltung und ihre diamagnetische Verschiebung untersucht und die Ergebnisse der Magnetfeldwechselwirkung werden anschließend miteinander verglichen. Im linearen Regime kann die Abhängigkeit der Zeeman-Aufspaltung und der diamagnetischen Verschiebung vom exzitonischen Anteils des Polaritons bestätigt werden. Oberhalb der Polariton-Kondensationsschwelle wird eine größere diamagnetische Verschiebung gemessen als für die gleiche Verstimmung im linearen Regime. Dieses Verhalten wird durch Abschirmungseffekte der Coulomb-Anziehung von Elektronen und Löchern erklärt, was in einer Erhöhung des Bohrradius der Exzitonen resultiert. Auch die Zeeman-Aufspaltung oberhalb der Polariton-Kondensationsschwelle zeigt ein vom unkondensierten Polariton abweichendes Verhalten, es kommt sogar zu einer Vorzeichenumkehr der Aufspaltung im Magnetfeld. Aufgrund der langen Spin-Relaxationszeiten von 300ps wird eine Theorie basierend auf der im thermischen Gleichgewichtsfall entwickelt, die nur ein partielles anstatt eines vollständigen thermischen Gleichgewicht annimmt. So befinden sich die einzelnen Spin-Komponenten im Gleichgewicht, während zwischen den beiden Spin-Komponenten kein Gleichgewicht vorhanden ist. Dadurch kann die Vorzeichenumkehr als ein Zusammenspiel einer dichteabhängigen Blauverschiebung jeder einzelner Spin-Komponente und der Orientierung der Spins im Magnetfeld angesehen werden. Für das Photonlasing kann keine Magnetfeldwechselwirkung festgestellt werden, wodurch verdeutlicht wird, dass die Messung der Zeeman-Aufspaltung beziehungsweise der diamagnetischen Verschiebung im Magnetfeld als ein eindeutiges Werkzeug zur Unterscheidung zwischen Polariton-Kondensation und Photonlasing verwendet werden kann. / Bose-Einstein-Condensation (BEC) and the associated effects, for instance superfluidity and superconductivity, are fascinating results of the bosonic quantum nature. Since Bose-Einstein-Condensation for atomic systems can only be realized with enormous technical and experimental effort at sufficient low temperatures near absolute zero, people are looking for bosons with smaller masses. For this quasi-particles in solid state systems such as magnons and excitons are perfect candidates, due to their smaller effective masses and as a consequence thereof the higher Bose-Einstein-Condensation temperatures in comparison to atomic systems. Another quasi-particle is the exciton-polariton, originating from strong light-matter coupling in semiconductor microcavities. This particle has properties of the light as well as of the matter part and the mass is so small that in theory room temperature Bose-Einstein-Condensation is possible. A further benefit of this system is the relatively easy realization of the BEC by injecting excitons via optical or electrical excitation in the quantum wells of the structure. Additionally, the momentum space distribution of these quasi-particles can be measured via a straightforward Fourier-spectroscopy setup. By determination of the angular distribution of the emitted light of the microcavity, the momentum of the particle in the quantum-well plane can be defined. This information helps to identify the BEC-phase. On these grounds the exciton-polaritons are model systems for studying Bose-Einstein-Condensation and attributed phenomena in matter. In this work the ground-state condensation of exciton-polaritons is realized in microcavities of various dimensions and the emission properties of these are investigated. Thereby, especially the magnetic field interaction of the uncondensated and condensated polaritons will probed and resulting from this a proof concept for the persistence of the strong-coupling across the condensation threshold is developed. Additionally, the coherence properties of the polariton-condensates in different dimensions will be studied by determination of the first and second order correlation functions. For all experiments a III/V-semiconductor system consisting of GaAs quantum-wells and a surrounding AlAs cavity is used. One-dimensional polariton-condensation - spatial coherence of polariton-wires In the first experimental part (chapter 1), the polariton-condensation of one-dimensional wires under non-resonant, optical excitation is studied. For this, different length and width of wires are used to determine the influence on the polariton-dispersion of additional confinement. The aim of this investigation is to realize a one-dimensional Bose-Einstein-Condensate with a nearly constant plateau of the spatial coherence for high distances r after the central drop-down of the first order correlation g(1)(r)-function, which is a characteristic of the off-diagonal long-range order (ODLRO) of the system. By analysis of the farfield emission characteristics several polariton branches, the one-dimensionality and the polariton-condensation can be proved in this wires. After this, the spatial coherence function g^(1)(r) is measured with a high precision Michelson-interferometer, which is established during this work. The g^(1)(r)-function has, for distances essential greater than the thermal De-Broglie-wavelength of the polaritons, a plateau value which depends on the pumping power of the system. Below the polariton-condensation threshold (power P_S) no plateau is visible and coherence is only established within the central part for |r|<1µm of the interferograms. With increasing power, the width of the central coherence peak growth and the plateau of the g^(1)(r)-function appears, which is only absent away from the wire. At a power of P = 1.6P_S the plateau reaches its maximal value and it amounts to 0.15. Furthermore, it can be demonstrated, that as the temperature of the crystal lattice increases the plateau decreases and it vanishes at T = 25K indicating the loss of the off-diagonal long range coherence. This behavior can be explained by a modern microscopic theory based on the master-equation approach which fits the experimental data well. The last part of this chapter compares the characteristics of the first order correlation function g^(1)(r) in the 1D case with the spatial coherence function of the planar polariton-condensate of a similar structure (2D). Zero-dimensional polariton-condensation - condensation and magnetic field interaction in a hybrid cavity In the second part of this thesis, polariton-condensation in novel hybrid-cavities is investigated. The layout of the lower mirror and of the cavity with 12 embedded quantum wells is analog to normal semiconductor microcavities. However, the upper mirror consists of a combination of a DBR (distributed Bragg reflector) and a high refractive index contrast grating with a semiconductor air transition (highest possible refractive index contrast). Due to the fact that the grating has a quadratic structure (side length 5µm), the polaritons are additionally confined in the plane of the quantum wells perpendicular to the growth direction and can be treated as zero-dimensional particles. To prove the persistence of the strong coupling across the condensation threshold, a magnetic field is applied in the growth direction, in order that the diamagnetic shift of the polaritons below and above the condensation thresholds can be measured comparatively. With this the polariton-condensate can be distinguished from the conventional photonic lasing of the microcavity. As another way to distinguish between polariton- and photon-lasing the second order autocorrelation function g^(2)(t) of the system is determined. Here a re-increase of the g^(2)(t=0)-value can be shown with increasing excitation power, after the value dropped down to 1 at the condensation threshold of the system indicating the increase of temporal coherence. Far above the condensation-threshold P_S the g^(2)(t=0)-value increases to values higher than 1 due to the appearance of decoherence processes in the system caused by rising polariton-polariton-interaction. For a conventional single-mode photon-laser, as for instance VCSEL (vertical-cavity surface-emitting laser), this behavior is not expected and it is an additional criterion to distinguish between polariton-condensation and photon-lasing. Two-dimensional polariton-condensation - interaction with external magnetic fields In the last experimental chapter of this thesis, the magnetic field interaction of the three possible working regimes of the planar microcavity (2D) emission is analyzed. First power series at a exciton-photon detuning of d= -6.5meV are performed to identify the linear polaritonic regime, the polariton-condensate phase and the photonic lasing at sufficient high excitation powers. After that, the Zeeman splitting and the diamagnetic shift of these three regimes are investigated in an external magnetic field applied again in the growth direction and ranging from B= 0T to B=5T. The results are compared to each other. For the linear regime, the theoretic expected dependence of the Zeeman splitting and diamagnetic shift is confirmed. However, above the threshold in the polariton-condensate phase a higher diamagnetic shift compared to the linear regime is measured. This behavior can be explained by taking into account bleaching effects of the Coulomb interaction due to the high carrier density resulting in an increase of the Bohr radius of the excitons. Also for the Zeeman splitting a different behavior to the equilibrium theory is found. The sign of the magnetic field splitting is reversed in comparison to the linear regime. Due to long spin relaxation time in red detuned systems of about 300ps, a theory is developed based on a partial thermal equilibrium of the spin components of the condensate. Here the spin components are in equilibrium with themselves, but not with the other component. As a consequence, the sign reversal can be interpreted as an interplay of the density dependent blueshift of the single spin components and the orientation of the spins in the magnetic field. For the photonic lasing no magnetic field interaction is found, indicating that the measurement of the Zeeman splitting and the diamagnetic shift in an external magnetic field is a unique tool to distinguish between polariton-condensation and photon-lasing.

<p> The aim of this thesis is to construct some smooth Einstein manifolds with nonzero Einstein constant, and then to investigate their topological and geometric properties.</p> <p> In the negative case, we are able to construct conformally compact Einstein metrics on 1. products of an arbitrary closed Einstein manifold and a certain even-dimensional ball bundle over products of Hodge Kähler-Einstein manifolds, 2. certain solid torus bundles over a single Fano Kähler-Einstein manifold. We compute the associated conformal invariants, i.e., the renormalized volume in even dimensions and the conformal anomaly in odd dimensions. As by-products, we obtain many Riemannian manifolds with vanishing Q-curvature.</p> <p> In the positive case, we are able to construct complete Einstein metrics on certain 3-sphere bundles over a Fano Kähler-Einstein manifold. We classify the homeomorphism and diffeomorphism types of the total spaces when the base manifold is the complex projective plane.</p> / Thesis / Doctor of Philosophy (PhD)

This thesis is dedicated to the study of the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for the existence of Einstein metrics with totally geodesic fibers in terms of Casimir operators. Some particular cases are studied, for instance, for normal base or fiber, symmetric fiber, Einstein base or fiber, for which the Einstein equations are manageable. We investigate the existence of such Einstein metrics for invariant bisymmetric fibrations of maximal rank, i.e., when both the base and the fiber are symmetric spaces and the base is an isotropy irreducible space of maximal rank. We find this way new Einstein metrics. For such spaces we describe explicitly the isotropy representation in terms subsets of roots and compute the eigenvalues of the Casimir operators of the fiber along the horizontal direction. Results for compact simply connected 4-symmetric spaces of maximal rank follow from this. Also, new invariant Einstein metrics are found on Kowalski n-symmetric spaces.

In this thesis we solve the Gross-Pitaevskii equation numerically in order to model the response of trapped Bose-Einstein condensed gases to perturbations by electromagnetic fields. First, we simulate output coupling of pulses from the condensate and compare our results to experiments. The excitation and separation of eigen-modes on flow through a constriction is also studied. We then move on to the main theme of this thesis: the important subject of quantised vortices in Bose condensates, and the relation between Bose-Einstein condensation and superfluidity. We propose methods of producing vortex pairs and rings by controlled motion of objects. Full three-dimensional simulations under realistic experimental conditions are performed in order to test the validity of these ideas. We link vortex formation to drag forces on the object, which in turn is connected with energy transfer to the condensate. We therefore argue that vortex formation by moving objects is intimately related to the onset of dissipation in superfluids. We discuss this idea in the context of a recent experiment, using simulations to provide evidence of vortex formation in the experimental scenario. Superfluidity is also manifest in the property of persistent currents, which is linked to vortex stability and dynamics. We simulate vortex line and ring motion, and find in both cases precessional motion and thermodynamic instability to dissipation. Strictly speaking, the Gross-Pitaevskii equation is valid only for temperatures far below the BEG transition. We end the thesis by describing a simple finite- temperature model to describe mean-field coupling between condensed and non- condensed components of the gas. We show that our hybrid Monte-Carlo/FFT technique can describe damping of the lowest energy excitations of the system. Extensions to this model and future research directions are discussed in the conclusion.