Nesta tese estudamos dois problemas diferentes na área de átomos ultra frios: Ondas de choque em condensados de Bose-Einstein e Espalhamento inelástico de átomos em um potencial de dois poços. No primeiro problema, estudamos o fluxo supersônico de um condensado de Bose-Einstein (BEC) através de um obstáculo macroscópico impenetrável delgado no sistema da equação de Schrödinger não-linear (NLS) bidimensional. Assumindo-se que a velocidade do fluxo é suficientemente alta, reduzimos assintoticamente o problema bidimensional original de valor de contorno para o fluxo estacionário através de um obstáculo alongado ao problema do pistão dispersivo unidimensional descrito pela NLS 1D dependente do tempo, no qual a coordenada original x reescalonada faz o papel de tempo e o movimento do pistão está vinculado à geometria do obstáculo. Duas ondas de choque dispersivas (DSWs) são geradas no fluxo, cada uma sendo formada em uma extremidade (frontal e traseira) do obstáculo. A DSW frontal é descrita analiticamente construindo-se soluções de modulação exatas para as equações de Whitham e a para a DSW traseira, empregamos a regra de quantização de Bohr-Sommerfeld generalizada para descrever a distribuição dos sólitons escuros. Propomos uma extensão da solução de modulação tradicional, a fim de incluir o padrão de ship-wave linear formado fora da região da DSW frontal. Realizamos simulações numéricas 2D completas e verificamos a validade das previsões analíticas. Os resultados deste problema podem ser relevantes para experimentos recentes sobre o fluxo de BECs através de obstáculos. No segundo problema, estudamos uma mistura atômica de dois átomos fermiônicos leves de spin 1/2 e dois átomos pesados em um potencial de dois poços. Processos de espalhamento inelástico entre ambas as espécies atômicas excitam os átomos pesados e renormalizam a taxa de tunelamento e a interação entre os átomos leves (efeito polarônico). A interação efetiva dos átomos leves muda de sinal e se torna atrativa quando o espalhamento inelástico é forte. Observamos também o cruzamento de níveis de energia, de um estado onde cada poço contém apenas um férmion (espalhamento inelástico fraco) para um estado onde um poço contém um par de férmions e ou outra está vazio (espalhamento inelástico forte). Identificamos o efeito polarônico e o cruzamento dos níveis de energia estudando-se a dinâmica quântica do sistema. / In this thesis we study two different problems in the field of ultracold atoms: Shock waves in Bose-Einstein condensates and Inelastic scattering of atoms in a double well. In the first problem, we study the supersonic flow of a Bose-Einstein condensate (BEC) past a slender impenetrable macroscopic obstacle in the framework of the twodimensional (2D) defocusing nonlinear Schr¨odinger equation (NLS). Assuming the oncoming flow speed sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched x-coordinate and the piston motion curve is defined by the spatial body profile. Two steady oblique spatial dispersive shock waves (DSWs) spreading from the pointed ends of the body are generated in both half-planes. These are described analytically by constructing appropriate exact solutions of the Whitham modulation equations for the front DSW and by using a generalized Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear DSW. We propose an extension of the traditional modulation description of DSWs to include the linear ship-wave pattern forming outside the nonlinear modulation region of the front DSW. Our analytic results are supported by direct 2D unsteady numerical simulations and are relevant to recent experiments on Bose-Einstein condensates freely expanding past obstacles. In the second problem, we study a mixture of two light spin-1/2 fermionic atoms and two heavy atoms in a double well potential. Inelastic scattering processes between both atomic species excite the heavy atoms and renormalize the tunneling rate and the interaction of the light atoms (polaron effect). The effective interaction of the light atoms changes its sign and becomes attractive for strong inelastic scattering. This is accompanied by a crossing of the energy levels from singly occupied sites at weak inelastic scattering to a doubly occupied and an empty site for stronger inelastic scattering. We are able to identify the polaron effect and the level crossing in the quantum dynamics.

Made available in DSpace on 2016-03-15T19:43:01Z (GMT). No. of bitstreams: 1 Luis Henrique Beust - EAHC2006.pdf: 1468553 bytes, checksum: 0b1bab36dcb4bfab15d9a6293929a4a6 (MD5) Previous issue date: 2006-08-14 / Fundo Mackenzie de Pesquisa / The present work makes a case for the relevance of Albert Einstein and Sigmund Freud s exchange on war and peace for current efforts towards establishing a Culture of Peace as designated by the UN/UNESCO International Decade for a Culture of Peace (2001-2010). Two open letters exchanged between Einstein and Freud in 1932 served as the principle source for this investigation. In these letters exchanged under the auspices of the International Committee on Intellectual Cooperation of the League of Nations Einstein and Freud describe their rationale regarding the causes of war and the possible paths which might lead to the establishment of a lasting peace between the Nations. This paper analyzes the perspectives of Einstein and Freud in relation to war and peace in light of the theoretical framework given by Norberto Bobbio s characterization of the contemporary modes of pacifism ( paths to peace ). In spite of the fact that seventy years have passed since Einstein and Freud discussed their views on war and peace and notwithstanding the fact that some of Freud´s thinking has been challenged since, the author concludes by suggesting that the exchange between Einstein and Freud is directly in line with, and contributes towards, current efforts to promote a Culture of Peace particularly regarding their shared view that war is not intrinsic to human nature and therefore can be eliminated. In retrospect, the contribution that Einstein and Freud can make towards the goal of establishing a Culture of Peace stems as much from their exchange as it does from the fact that these two thinkers have become icon figures of the 20th century. / O presente trabalho busca situar a relevância das idéias de Albert Einstein e de Sigmund Freud sobre a guerra e a paz para os esforços contemporâneos de construção de uma Cultura de Paz, conforme postulados pela ONU/UNESCO para a Década Internacional da Cultura de Paz (2001-2010). Como corpus para esta investigação, tem-se as duas cartas-abertas que Einstein e Freud trocaram entre si em 1932 - sob os auspícios do Instituto Internacional de Cooperação Intelectual da Liga das Nações - nas quais apresentavam suas visões sobre as causas da guerra e os possíveis caminhos para o estabelecimento de uma paz definitiva entre as nações. Os postulados de Einstein e de Freud sobre a guerra e a paz são analisados à luz da categorização contemporânea das formas de pacifismo ( as vias da paz ) elaborada por Norberto Bobbio. Conclui-se que as idéias apresentadas por Einstein e Freud, apesar de passados setenta anos de sua redação, ainda se mantêm relevantes na contemporaneidade, apesar de alguns dos pressupostos, especialmente de Freud, terem sofrido desgaste com o passar do tempo. Argumenta-se que suas contribuições são especialmente relevantes para fortalecer a idéia fundamental por trás dos esforços de educação por uma Cultura de Paz, qual seja, de que a guerra não é intrínseca à natureza humana, e que, portanto, pode ser eliminada. Em retrospectiva, percebe-se que a contribuição que Einstein e Freud podem prestar à Cultura de Paz advém tanto de suas idéias quanto do fato de terem se tornado figuras icônicas da cultura do século XX.

Ultra cold quantum gas is a convenient system to study fundamental questions of modern physics, such as phase transitions and critical phenomena. This master thesis is devoted to experimental investigation of the thermodynamics susceptibilities, such as the isothermal compressibility and the thermal expansion coefficient of a trapped Bose-Einstein condensate (BEC) of 87Rb atoms. The critical phenomena and the critical exponents across the transition can explain the behavior of the isothermal compressibility and the thermal expansion coefficient near the critical temperature TC. By employing the developed formalism of global thermodynamics variables, we carry out a statistical treatment of Bose gas in a 3D harmonic potential. After that, comparison of obtained results reveals the most appropriate state variables describing the system, namely volume and pressure parameter V and Π respectively. The both are related with the confining frequencies and BEC density distribution. We apply this approach to define the set of new thermodynamic variables of BEC, and also to construct the isobaric phase diagram V T. Its allows us to extract the compressibility κT and the thermal expansion coefficient βΠ. The behavior of the isothermal compressibility corresponds to the second-order phase transition, while the thermal expansion coefficient around the critical point behaves as β ∼ tr-α, where tr is reduced temperature of the system and α is the critical exponent on the basic of these. Results we have obtained the critical exponent α = 0.15±0.09, which allows us to determine the system dimensionality by means of the scaling theory, relating the critical exponents with the dimensionality. As a result, we found out that the dimensionality of the system to be d ∼ 3, one is in agreement with the real dimension of the system. / Amostras atômicas ultrafrias de um gás de Bose são convenientes para estudar questões fundamentais da física moderna, como as transições de fase e fenômenos críticos em condensados de Bose-Einstein (BEC). A minha dissertação dedica se à investigação das susceptibilidades termodinâmicas como a compressibilidade isotérmica e o coeficiente de expansão térmica de a traves da transição de um BEC de 87Rb. Os fenômenos críticos e os exponentes críticos a traves da transição podem explicar o comportamento da compressibilidade isotérmica e do coeficiente de expansão térmica perto da temperatura crítica TC. Ao empregar o desenvolvido formalismo das variáveis termodinâmicas globais, levamos a cabo o tratamento estatístico de um gás de Bose num potencial harmônico 3D. Depois da comparação dos resultados obtidos, revelam as mais apropriadas variáveis de estado descrevendo o sistema, chamadas parâmetro de volume e pressão, V e Π respectivamente. As duas estão relacionadas com as frequências de confinamento e a distribuição de densidade do BEC. Nós aplicamos esta abordagem para definir um conjunto de novas variáveis termodinâmicas do BEC, e também para construir o diagrama de fase isobárico V T. O anterior nós permite extrair a compressibilidade κT e o coeficiente de expansão termina βΠ. O comportamento da compressibilidade isotérmica corresponde a uma transição de fase de segunda ordem enquanto que o coeficiente de expansão térmica ao redor do ponto crítico comporta se como β ∼ tr-α, onde tr é a temperatura reduzida do sistema, e α o exponente crítico. Deste resultado nós obtemos um exponente critico, α = 0.15 ± 0.09, que permite determinar a dimensionalidade do sistema a traves da teoria de escala, relacionando os exponentes críticos com a dimensionalidade. Como resultado, encontramos que a dimensionalidade do sistema é d ∼ 3 que está de acordo como a dimensão real do sistema.

Nesta dissertação estudaremos sistemas de bósons ultrafrios armadilhados em uma rede ótica quadrada bidimensional sem levar em consideração o confinamento harmônico. A dinâmica desses sistemas é bem descrita pelo modelo de Bose-Hubbard, que prevê uma transição de fase quântica de um superfluido para um isolante de Mott a temperaturas baixas, e pode ser induzida variando a profundidade do potencial da rede ótica. Apresentaremos o diagrama de fases dessa transição construído a partir de uma aproximação de campo médio e também com um cálculo numérico usando um algoritmo de Monte Carlo Quântico, denominado algoritmo Worm. Encontramos o ponto crítico para o primeiro lobo de Mott em ambos os casos, concordando com trabalhos anteriores. / This work study the two-dimensional ultracold bosonic atoms loaded in a square optical lattice, without harmonic confinement. The dynamics of this system is described by the Bose-Hubbard model, which predicts a quantum phase transition from a superfluid to a Mott-insulator at low temperatures that can be induced by varying the depth of the optical potential. We present here the phase diagram of this transition built from a mean field approach and from a numerical calculation using a Quantum Monte Carlo algorithm, namely the Worm algorithm. We found the critical transition point for the first Mott lobe in both cases, in agreement with the standard literature.

En 1916 Albert Einstein énonce la théorie de la relativité : véritable révolution scientifique du XXe siècle, elle aura des répercussions touchant à tous les domaines de la connaissance. L'objectif de notre étude est de tracer un cadre des relations existantes entre la littérature européenne et la théorie de la relativité, sur l'interaction de leurs deux langues et sur l'échange de leurs images. Si la science et la littérature dialoguent entre elles, elles le font au sein d'une atmosphère, l'esprit du temps. Les deux disciplines prennent leurs images et leurs thèmes d'un terroir commun : elles deviennent ainsi l'expression d'une époque et en même temps la source de laquelle l'époque se nourrit. Le dialogue entre science et littérature se traduit dans notre méthodologie : le paradoxe des jumeaux, qu'Einstein utilise pour expliquer la déformation spatio-temporelle, se traduit dans le thème du double à partir duquel on a sélectionné un corpus européen. Ensuite, on a procède à une analyse sur deux niveaux, l'un narratologique et l'autre symbolique. Pour le premier niveau on a analysé l'espace-temps en relation au double : on a vu comment les deux sont entrelacés et comment l'espace-temps ne se limite pas à assumer une fonction de décor. Pour le deuxième niveau d'analyse on a emprunté certaines images utilisées par Einstein dans ses textes de vulgarisation, comme le train et la lumière, pour construire une grille critique que nous avons appelé « constellation einsteinienne ». Grace à cette grille de symboles, qui se superpose à celle narratologique, nous pouvons voir comme les éléments évoqués par Einstein se transforment en littérature en gardant toute leur valeur scientifique. De notre analyse résulte un cadre où des éléments scientifiques et des textes de fiction s'entrelacent dans l'expression de l'esprit du temps. La relativité scientifique trouve son contrepartie dans la relativité littéraire. / In 1916 Albert Einstein formulated the theory of relativity, a great XX century scientific revolution, which will have repercussions affecting all areas of human knowledge. The aim of our study is to draw a framework of the existing relations between European literature and the theory of relativity, about the interaction of their two languages and the exchange of their images. If science and literature interact to each other, they do so within an atmosphere, the spirit of the times. The two disciplines take their images and their themes from a common ground: they thus become the expression of an era and at the same time the source from which the era nourishes itself. The dialogue between science and literature is reflected in our methodology: the paradox of twins, which Einstein uses to explain spatio-temporal deformation, is transformed in the double theme from which a European corpus was selected. Then, we proceeded to an analysis on two levels, the one narratological and the other symbolic. For the first level, we have focused on space-time double relation, discovering how the two are interlaced and how space-time is not limited to assuming a decor function. For the second level of analysis we have borrowed certain images used by Einstein in his popularization texts, such as train and light, to build a critical grid that we have called the “Einsteinian constellation”. Thanks to this grid of symbols, which is superimposed on that of narratology, we can see that the elements evoked by Einstein transform into literature while retaining all their scientific value. From our analysis results a framework where scientific elements and fiction texts intertwine in the expression of the spirit of the time. Scientific relativity finds its counterpart in literary relativity.

Atominterferometer erlauben es, Beschleunigungen mit bisher nicht erreichter Präzision zu messen. Anwendungen in der Grundlagenforschung beinhalten Gravitationswellendetektoren, die Bestimmung von Naturkonstanten oder Tests des schwachen Äquivalenzprinzips. Die Sensitivität eines Sensors für Tests des schwachen Äquivalenzprinzips skaliert quadratisch mit der Zeit der freien Entwicklung der Atome während der Interferometersequenz. Durch die Verwendung von Bose-Einstein-Kondensaten mit stark reduzierter Ausdehnungsgeschwindigkeit sowie dem Betrieb in Schwerelosigkeit kann die Sensitivität um Größenordnungen verbessert werden. Das QUANTUS-2 Experiment stellt die zweite Generation eines mobilen Atominterferometers dar, welches am Fallturm in Bremen zum Einsatz kommt und dient als Wegbereiter für zukünftige Experimente mit kalten Atomen auf Satelliten. Durch differentielle Messung der Beschleunigung von Rubidium und Kalium mit Hilfe der Atominterferometrie soll das schwache Äquivalenzprinzip getestet werden. Im Rahmen dieser Arbeit wurde das auf mikro-integrierten Diodenlasern sowie einer kompakten Elektronik basierende Rubidiumlasersystem aufgebaut und qualifiziert. Nach erfolgter Integration in die QUANTUS-2 Kapsel, wurden über 200 Abwürfe und Katapultflüge am Fallturm absolviert. Diese demonstrieren die Robustheit des Experimentes unter Beschleunigungen von bis zu 43 g während eines Katapultfluges. Die Dynamik des Kondensates wurde in Schwerelosigkeit untersucht und die Ausbreitungsgeschwindigkeit in allen drei Raumrichtungen mit Hilfe einer magnetischen Linse verringert. Die dabei erreichten Ausbreitungsgeschwindigkeiten entsprechen effektiven Temperaturen von unter 120 pK eines thermischen Ensembles. Dieser stellt den niedrigsten in allen drei Raumrichtungen erreichten Wert dar. Die gezeigten Ergebnisse demonstrieren somit die Verfügbarkeit wichtiger Schlüsselkonzepte zukünftiger hochpräziser Quantensensoren auf Satelliten. / Atom interferometers offer the possibility to measure accelerations with unprecedented precision. Applications in fundamental research include gravitational wave detectors, the determination of physical constants, or tests of the weak equivalence principle. The sensitivity of an atom interferometer testing the weak equivalence principle scales quadratically with the time of free evolution of the atoms during the interferometer sequence. By using Bose-Einstein condensates with ultra-low expansion rates as test masses and operating the experiment in microgravity, one can enhance the sensitivity by orders of magnitude. QUANTUS-2 is the second generation mobile atom interferometer to be operated at the drop tower in Bremen and serves as a pathfinder for future cold atom experiments in space. It is envisaged to test the weak equivalence principle by a differential measurement of the acceleration of rubidium and potassium by means of atom interferometry. Within this thesis, the rubidium laser system was set up and qualified. It is based on micro-integrated laser modules and compact electronics. After integration into the QUANTUS-2 capsule, 200 drops and catapult flights were conducted at the drop tower. These are demonstrating the robustness of the complete experiment when being subjected to accelerations of up to 43 g during a catapult flight. The dynamics of the condensate were analyzed and the mean kinetic energy was reduced in all three dimensions by means of a magnetic lens. Expansion rates equivalent to a thermal ensemble having a temperature below 120 pK have been reached and represent the lowest value ever achieved in all three dimensions. The results prove the availability of relevant key concepts for future high-precision quantum sensors on a satellite platform.

Depuis la réalisation expérimentale en 1995 des premiers condensats gazeux de<br />Bose-Einstein (B-E) d'atomes alcalins : $\left(<br />^(87)Rb\,\ , \ ^(23)Na\,\ , \ ^(7)Li\right) $, ultra-froids ($T=0\,^(\mathrm(o))\!\mathrm(K)$) et confinés dans des pièges magnétiques 3D, la physique des condensats de Bose-Einstein et<br />de Fermi a connu un développement remarquable aussi bien expérimental que<br />théorique. L'objectif de ce mémoire de thèse a été fixé dans le cadre général du progrès récemment accompli dans l'étude de l'évolution dynamique des condensats<br />de B-E répulsifs, et de la réduction de leur dimensionnalité. Le manuscrit de<br />cette thèse comprend deux parties. La première a été consacrée, d'une part, à la<br />présentation du phénomène de la condensation de B-E depuis sa prédiction en 1925 par Einstein, dans un gaz idéal de Bose, jusqu'à sa réalisation en 1995, et<br />d'autre part, à la description de la dynamique des condensats dilués de B-E, à<br />la température $T=0\,^(\mathrm(o))\!\mathrm(K)$, par l'équation nonlinaire de Schr\"(o)dinger (ENLS), connue aussi sous le nom : équation de Gross-Pitaevskii (EGP). La seconde partie<br />comprend les résultats numériques de notre étude portant sur la dynamique d'un<br />condensat de B-E répulsif, quasi-1D et confiné dans un piège non-harmonique<br />(piège allongé avec des bords paraboliques), et sur son comportement dissipatif<br />et superfluide. Notre étude a montré que: i) les bords paraboliques du piège<br />considéré, ainsi qu'un obstacle en forme d'une bosse gaussienne, placé dans la partie plate<br />de ce piège, ont un effet d'anti-amortissement sur la propagation uniforme d'un<br />soliton gris dans le condensat, et cet effet se manifeste par une émission spontanée des<br />phonons; ii) le mouvement uniforme et rectiligne (en va-et-vient) d'un obstacle gaussien dans le condensat considéré conduit, lorsque la vitesse constante de l'obstacle<br />dépasse une certaine valeur critique ( vitesse critique ), à la création des solitons gris et des phonons dans ce<br />condensat qui devient un milieu dissipatif.<br /> Nous avons montré que le comportement dissipatif du condensat croît avec l'augmentation de la vitesse de<br />l'obstacle, atteint son maximum et finit par disparaître quasi-totalement pour<br />de grandes valeurs de la vitesse constante de l'obstacle, pour lesquelles le condensat se comporte comme un quasi-superfluide.

Die Dynamik anfänglich aus dem Gleichgewicht gebrachter wechselwirkender Quantenvielteilchensysteme wirft aktuell noch spannende Fragen auf. In Bezug auf die Thermalisierung ist z.B. nach wie vor ungeklärt, in welcher Form sie überhaupt stattfindet und in welchen Observablen bzw. auf welcher Zeitskala sie zu beobachten ist. Eine ideale Grundlage zur Erforschung von Relaxationsdynamiken in wechselwirkenden Vielteilchensystemen bieten ultrakalte Quantengase aufgrund ihrer guten Kontrollier- und Variierbarkeit. Ein allgemeiner theoretischer Rahmen, auf dessen Basis solche Prozesse zu untersuchen sind, steht jedoch infolge der großen Anzahl der beteiligten Freiheitsgrade bisher nicht zur Verfügung. Für ultrakalte bosonische Gase stellt die Gross-Pitaevskii-Gleichung eines der wichtigsten theoretischen Werkzeuge dar, eine klassische Feldgleichung für die Kondensatwellenfunktion in Molekularfeldnäherung. Die ihr zugrunde liegende Näherung erlaubt jedoch keine nicht-trivialen Aussagen über den vollen N-Teilchenzustand, dessen Kenntnis für die Untersuchung einer möglichen Relaxationsdynamik unabdingbar ist. Um der theoretischen Beschreibung des vollen bosonischen Feldes einen Schritt näher zu kommen, untersucht die vorliegende Arbeit die Anwendung semiklassischer Methoden auf ultrakalte Bosegase. Diese sind in der Regel dann sehr genau, wenn die beteiligten Wirkungen groß gegenüber dem Planckschen Wirkungsquantum sind. Für bosonische Felder wird dieser Grenzfall durch die Bedingung einer großen Teilchenzahl ersetzt. Die immense Anzahl an Teilchen in den hier behandelten Vielteilchensystemen macht die Anwendung semiklassischer Methoden auf diesem Gebiet also vielversprechend. Als zentrales Modellsystem wird ein anfänglich aus dem Gleichgewicht gebrachtes ultrakaltes bosonisches Doppelmuldensystem betrachtet, das eine hochinteressante Dynamik aufweist, die auf das Wechselspiel der Tunneldynamik einerseits und der Wechselwirkung der Teilchen untereinander andererseits zurückzuführen ist. Als Referenz lassen sich aufgrund der speziellen Fallengeometrie im Rahmen der Zwei-Moden-Näherung die Ergebnisse einer numerisch exakten Untersuchung heranziehen. Durch den Einsatz der namhaften WKB-Quantisierung und des besonders aus der Molekülphysik bekannten Reflexionsprinzips wird hier ein geschlossener analytischer Ausdruck für die sogenannte Populationsdifferenz im Doppelminimum hergeleitet, der ausschließlich von den wenigen relevanten Systemparametern abhängt. Diese mächtige Formel erlaubt es nun zum ersten Mal, in quantitativer Weise die charakteristische Sequenz aus Oszillationen, Kollapsen und Revivals in Abhängigkeit der vorausgesetzten Parameter zu untersuchen. Nach dieser ersten erfolgreichen Anwendung semiklassischer Methoden im Modellsystem wird über die reduzierte Dynamik der Populationsdifferenz hinausgegangen. Mithilfe des semiklassischen Herman-Kluk-Propagators lässt sich selbst der volle N-Teilchenzustand untersuchen. Da es letztlich um die Beschreibung ultrakalter Bosonen in beliebigen Potentialen gehen soll, wird zunächst der Herman-Kluk-Propagator für eine Feldtheorie vorgestellt. Im Doppelmuldensystem zeigt sich dann in der Anwendung die semiklassische Propagation in der Lage, für alle untersuchten Parameterregime gute Übereinstimmung mit den numerisch exakten Ergebnissen zu liefern. Zusätzlich findet ein Abgleich der Resultate mit der Truncated Wigner Approximation statt, auf die im Forschungsgebiet ultrakalter Bosonen häufig zurück gegriffen wird. Diese beschreibt die Zeitentwicklung einer Wignerverteilung unter Aussparung der Quanteninterferenzen. In der vorliegenden Arbeit wird gezeigt, dass die Herman-Kluk-Propagation unter Berücksichtigung der Phasen weit über die Truncated Wigner Approximation hinausgeht: Sie gibt alle wichtigen Charakteristika der Dynamik im Doppelmuldensystem wieder. Um die Semiklassik auf ihre Aussagefähigkeit in Bezug auf eine noch komplexere Dynamik zu untersuchen, wird zum Abschluss das Drei-Topf-System betrachtet, das zusätzlich chaotische Regionen im Phasenraum aufweist. Auch hier zeigt sich, dass die semiklassische Berücksichtigung der Phasen die Truncated Wigner Approximation in den Schatten stellt. Allerdings ergeben sich durch die Instabilität der Trajektorien für stark chaotische Regime numerische Probleme, die es in der Zukunft zu lösen gilt. / The dynamics of initially non equilibrium interacting quantum many body systems is an ongoing and interesting field of research. It is still an open question in which form relaxation occurs in such systems, and in which observables and on which timescales a possible thermalization might appear. A perfect playground for the investigations of relaxation dynamics in interacting many body schemes is provided by ultracold quantum gases, which are easily to be controlled and varied in experiments. However, a general theoretical framework for the investigation of such processes is still missing, due to the huge amount of involved degrees of freedom. One of the main theoretical tools in the field of ultracold bosonic gases represents the famous Gross-Pitaevskii equation, a field equation for the Bose-Einstein condensate wave function in terms of a mean-field approximation. However, the underlying approximation prevents the possibility to draw non-trivial conclusions about the full N-particle state, the information of which is necessary for the analysis of relaxation processes. To gain the theoretical description of the full bosonic field, the present thesis deals with the application of semiclassical methods to ultracold boson gases. Those techniques become in general exact, as long as the involved actions are large compared to Planck's constant. For many body systems it turns out that semiclassics are expected to give good results also for the condition of high particle numbers, which is precisely fulfilled in these schemes, making the semiclassical approaches promising. As an essential model system an initially out of equilibrium ultracold bosonic double-well system is investigated. This configuration provides highly interesting dynamics due to the interplay of the tunneling dynamics on the one hand and the interaction amongst the particles on the other. The special trap geometry makes exact numerical calculations in the framework of the two-mode approximation available, which serve in the following as reference data. By applying the common semiclassical WKB approximation and the reflection principle known from molecule physics, a closed analytical expression for the so-called population imbalance of the bosons in the double-well is derived, depending only on the few relevant system parameters. This mighty formula allows for the first time the quantitative investigation of the characteristic sequence consisting of oscillations, collapse and revivals in dependence on the parameters of the system. Since the semiclassical approaches succeeded for the double-well model so far the so-called Herman-Kluk propagator is adopted, to go beyond the reduced dynamics of the population imbalance. The propagator provides the possibility to treat the full N-particle state theoretically and is introduced for the most general case of a bosonic quantum field. Its application to the double-well system yields for all investigated parameter regimes very good agreement with the numerical exact results. Furthermore the outcomes are compared to the Truncated Wigner approximation, which is frequently used in the research field of ultracold bosons. This approach pictures the time evolution of a Wigner distribution, without taking into account the quantum interferences. In the present thesis it is shown that the Herman-Kluk propagation goes clearly beyond the truncated Wigner approach by considering in addition the quantum phases: The propagator is able to reproduce all of the distinctive features of the double-well dynamics. In order to test the performance of semiclassical methods in matters of even more complex systems, the ultracold bosonic triple-well model is finally considered, which exhibits unlike the double-well scheme chaotic regions in phase space. It turns out that the semiclassical propagation outplays again the truncated Wigner approximation. On the other hand the instability of the highly chaotic trajectories causes numerical problems, which have to be solved in the future.

The vacuum of quantum field theory is not empty space but filled with quantum vacuum fluctuations, which give rise to many intriguing effects. The first part of this Thesis addresses cosmic inflation, where the quantum fluctuations of the inflaton field freeze and get amplified in the expanding universe. Afterwards, we turn our attention towards Bose-Einstein condensates, a laboratory system. Since most of our calculations are performed using a mean-field expansion, we will study the accuracy of a finite-range interaction potential onto such an expansion. Exploiting the universality of quantum fluctuations, several aspects of cosmic inflation will be identified in ballistically expanding Bose-Einstein condensates. The effective action technique for calculating the quantum backreaction will be scrutinized. Finally, we consider dynamic quantum phase transitions in the last part of this Thesis. To this end two specific scenarios will be investigated: firstly, the structure formation during the superfluid to Mott-insulator transition in the Bose-Hubbard model; and secondly, the formation of spin domains as a two-dimensional spin-one Bose gas is quenched from the (polar) paramagnetic to the (planar) ferromagnetic phase. During this quench, the symmetry of the ground state is spontaneously broken and vortices (topological defects) form.

Σκοπός της εργασίας μας είναι η μελέτη κάποιων αναγωγικών χώρων που παρουσιάζουν ενδιαφέρουσα γεωμετρία. Συγκεκριμένα, μελετάμε τη γεωμετρία της σφαίρας S^n όταν αυτή είναι αμφιδιαφορική με έναν χώρο πηλίκο G/K και την γεωμετρία των πολλαπλοτήτων Stiefel SO(n)/SO(n-k) (το σύνολο όλων των k-πλαισίων του R^n). Ένας ομογενής χώρος αποτελεί επέκταση των ομάδων Lie, καθώς είναι μια λεία πολλαπλότητα M στην οποία δρα μεταβατικά μια ομάδα Lie G. Κάθε τέτοιος χώρος δίνεται ως M = G/K, όπου K = {g\in G : gp = p} (p \in M). Η βασική γεωμετρική ιδιότητα των ομογενών χώρων είναι ότι αν γνωρίζουμε την τιμή κάποιου γεωμετρικού μεγέθους σε ένα σημείο του χώρου, τότε μπορούμε να υπολογίσουμε την τιμή του μεγέθους αυτού σε οποιοδήποτε άλλο σημείο. Το ιδιαίτερο χαρακτηριστικό των αναγωγικών χώρων G/K είναι ότι υπάρχει ένας Ad(K)-αναλλοίωτος υπόχωρος της άλγεβρας Lie(G). Η περιγραφή όλων των μεταβατικών δράσεων μιας ομάδας Lie σε μια πολλαπλότητα M αποτελεί ένα δύσκολο πρόβλημα. Για την περίπτωση των σφαιρών αυτές έχουν περιγραφτεί το 1953 από τους Montgomery-Samelson-Borel. Στην εργασία μας μελετάμε τη γεωμετρία (καμπυλότητες, μετρικές Einstein) των σφαιρών S^3, S^5 όταν αυτές είναι αμφιδιαφορικές με τα πηλίκα S^3 = SO(4)/SO(3) = SU(2) και S^5 = SO(6)/SO(5) = SU(3)/SU(2). Αντίστοιχα προβλήματα εξετάζονται για τις πολλαπλότητες Stiefel SO(n)/SO(n-k), όπου η περιγραφή όλων των SO(n)-αναλλοίωτων μετρικών παρουσιάζει δυσκολία, λόγω του ότι η ισοτροπική αναπαράστασή τους περιέχει ισοδύναμα υποπρότυπα. Μελετάμε για ποιές από τις συγκεκριμένες πολλαπλότητες η μετρική που επάγεται από τη μορφή Killing είναι μετρική Einstein και περιγράφουμε αναλυτικά τις διαγώνιες SO(n)-αναλλοίωτες μετρικές Einstein στις πολλαπλότητες SO(n)/SO(n-2). Επιπλέον παρουσιάζουμε και ένα καινούργιο αποτέλεσμα, ότι στην πολλαπλότητα SO(5)/SO(2) οι μοναδικές SO(5)-αναλλοίωτες μετρικές Einstein είναι οι μετρικές που είχαν βρεθεί από τον Jensen το 1973. / The purpose of our work is to study homogeneous spaces that present interesting geometry. These include the geometry of the sphere S^n diffeomorphic to a quotient space G/K and the geometry of Stiefel manifolds SO(n)/SO(n-k) (the set of all k-planes in R^n). A homogeneous space is a smooth manifold M in which a Lie group acts transitively. Any such space is given as M = G/K where K = {g\in G : gp = p} (p\in M). The basic geometric property of homogeneous space is that if we know the value of a geometrical object at a point of the space, then we can estimate the value of thiw quantity at any other point. The special feature of reductive homogeneous space G/K is that there exists an Ad(K)-invariant subspace of the Lie algebra Lie(G). The description of all transitive actions of a Lie group into a manifold M is a difficult problem. In the case of spheres such actions have been described in 1953 by the Montgomery, Samelson and Borel. In our work we study the geometry (curvature, Einstein metrics) of the sphere S^3 = SO(4)/SO(3) = SU(2), S^5 = SO(6)/SO(5) = SU(3)/SU(2). Similar problems are examined for the Stiefel manifolds SO(n)/SO(n-k). The description of all SO(n)-invariant metrics presents serious difficulties because the isotropy representation contains equivalent submodules. We study for which of the manifolds SO(n)/SO(n-k) the metric induced by the Killing form is an Einstein metric and we describe in detail the diagonal SO(n)-invariant Einstein metrics on the Stiefel manifolds SO(n)/SO(n-2). In addition, we give the new result that for the Stiefel manifold SO(5)/SO(2) the unique SO(5)-invariant Einstein metrics are the metrics found by Jensen in 1973.