Dissertations / Theses on the topic 'Boset'

The term boset was coined by Patrick Jordan, both as an abbreviation of biordered set, and as a generalisation of poset, itself an abbreviation of partially ordered set. A boset is a set equipped with a partial multiplication and two intertwining reflexive and transitive arrow relations which satisfy certain axioms. When the arrow relations coincide the boset becomes a poset. Bosets were invented by Nambooripad (in the 1970s) who developed his own version of the theory of fundamental regular semigroups, including the classical theory of fundamental inverse semigroups using semilattices, due to Munn (in the 1960s). A semigroup is fundamental if it cannot be shrunk homomorphically without collapsing its skeleton of idempotents, which is a boset. Nambooripad constructed the maximum fundamental regular semigroup with a given boset of idempotents. Fundamental semigroups and bosets are natural candidates for basic building blocks in semigroup theory because every semigroup is a coextension of a fundamental semigroup in which the boset of idempotents is undisturbed. Recently Jordan reproved Nambooripad's results using a new construction based on arbitrary bosets. In this thesis we prove that this construction is always fundamental, which was previously known only for regular bosets, and also that it possesses a certain maximality property with respect to semigroups which are generated by regular elements. For nonregular bosets this constuction may be regular or nonregular. We introduce a class of bosets, called sawtooth bosets, which contain many regular and nonregular examples, and correct a criterion of Jordan's for the regularity of this construction for sawtooth bosets with two teeth. We also introduce a subclass, called cyclic sawtooth bosets, also containing many regular and nonregular examples, for which the construction is always regular.

Doctor of Philosphy (PhD)
The term boset was coined by Patrick Jordan, both as an abbreviation of biordered set, and as a generalisation of poset, itself an abbreviation of partially ordered set. A boset is a set equipped with a partial multiplication and two intertwining reflexive and transitive arrow relations which satisfy certain axioms. When the arrow relations coincide the boset becomes a poset. Bosets were invented by Nambooripad (in the 1970s) who developed his own version of the theory of fundamental regular semigroups, including the classical theory of fundamental inverse semigroups using semilattices, due to Munn (in the 1960s). A semigroup is fundamental if it cannot be shrunk homomorphically without collapsing its skeleton of idempotents, which is a boset. Nambooripad constructed the maximum fundamental regular semigroup with a given boset of idempotents. Fundamental semigroups and bosets are natural candidates for basic building blocks in semigroup theory because every semigroup is a coextension of a fundamental semigroup in which the boset of idempotents is undisturbed. Recently Jordan reproved Nambooripad's results using a new construction based on arbitrary bosets. In this thesis we prove that this construction is always fundamental, which was previously known only for regular bosets, and also that it possesses a certain maximality property with respect to semigroups which are generated by regular elements. For nonregular bosets this constuction may be regular or nonregular. We introduce a class of bosets, called sawtooth bosets, which contain many regular and nonregular examples, and correct a criterion of Jordan's for the regularity of this construction for sawtooth bosets with two teeth. We also introduce a subclass, called cyclic sawtooth bosets, also containing many regular and nonregular examples, for which the construction is always regular.

The primary goal of this thesis is to find the distribution function of number of particles imbalance for ultra cold bosons trapped in a double well potential elongated in one direction for different strength of interaction between particles. This distribution function has been found to be Gaussian distribution function in two different limits. The first limit is weak interaction limit, where we only consider one energy level per well that is called "two mode approximation" regime. The second limit is when the interaction energy is in the same order as gap between energy levels where in this case, we consider finite number of levels per well. The standard deviation of number of particles distribution also has been found in both limits to be proportional to the square root of the total number of particles. In addition, some numerical work is done to find these distribution functions and the results are in agreement with analytical results.

El poblado HIFRENSA es un conjunto residencial formado por la agrupación de viviendas para alojar a los trabajadores de la central nuclear de Vandellòs-I. Cuenta además con una serie de equipamientos a escala de barrio. El conjunto está conectado mediante pasos para peatones con plazas públicas con esquema viario en cul-de-sac. Antonio Bonet recibe el encargo de la construcción del poblado en 1967, así como la realización de las dependencias administrativas y otros edificios de carácter técnico en el emplazamiento de la central nuclear. La tesis revela a Bonet como arquitecto investigador. En el poblado HIFRENSA no se puede establecer una línea clara que separe la arquitectura del urbanismo. Bonet busca una solución universal con la definición de un tipo o patrón, como elemento susceptible de ser perfeccionado, sistematizado y prefabricado. El patrón puede mutar, ya sea para adaptarse a un sistema constructivo o por necesidades topográficas, climáticas o programáticas. Particularmente en la obra de Bonet, también para satisfacer unos requerimientos de orden superior, su soporte urbanístico. La mayor dificultad radica en la definición de un criterio ordenador que permita dar forma superior a las unidades: ya sean viviendas o equipamientos sin que el conjunto urbanístico acuse monotonía y falta de identidad.
The HIFRENSA settlement is a residential complex built to accommodate workers of Vandellòs-I nuclear power plant. It also has a number of facilities at neighbourhood scale. The set is connected by pedestrian crossings with public squares and road scheme in cul-de-sac. In 1967 Antonio Bonet was commissioned to build the settlement, the administrative offices and other technical buildings on nuclear power plant site . The thesis exposes Bonet as architect researcher. At HIFRENSA settlement, a clear line between architecture and urbanism cannot be drawn. Bonet searches an universal solution. with a type or pattern definition as an element liable to be improved, refined, systematized and prefabricated. The pattern can mutate either to accommodate a construction system or for another needs, topographic, climatic or programmatic. Particularly in Bonet’s work, also to meet a higher requirement: its urban planning support. The greatest difficulty lies on the definition of an order judgment that can give a higher shape to the units, dwellings or facilities , with the intention of avoiding a monotonous urban complex and lack of identity.

Sabe-se que o arquiteto catalão Antonio Bonet Castellana é reconhecido por suas obras na costa uruguaia, sendo a urbanização de Punta Ballena, e o conjunto de casas, de grande importância para a Arquitetura Moderna no sul da América Latina. Contudo, este trabalho entende que a atuação de Bonet no continente, sobretudo em relação a sua contribuição urbanística, não pode ser compreendida apenas através de tal obra, e visa complementar essa perspectiva. Assim, em vez da uruguaia Punta Ballena, é a cidade não construída em Buenos Aires, na Argentina, que se pretende investigar. Para tanto, esta dissertação analisa os projetos não construídos de conjunto habitacional para Casa Amarilla (1943), o bairro proposto em Bajo Belgrano (1948-49) e a remodelação de Barrio Sur (1956), pensados para tal cidade, reconhecendo-os como uma série. O intuito deste estudo é o de melhor conhecer a atuação de Bonet, sua contribuição ao projeto da cidade moderna e às soluções de habitação coletiva que dela derivam. Também é intenção do trabalho compreender as relações entre a cidade pensada por Bonet e a cidade que existe, colaborando para um estudo mais detalhado sobre a cidade da Arquitetura Moderna.
It is known that the Catalan architect Antonio Bonet Castellana is recognized for his masterpiece work in the Uruguayan coast, which include the Punta Ballena settlement and a set of houses – of great importance to the Modern Architecture in South America. Nonetheless, this text assumes that Bonet’s contribution on modern urbanism cannot be fully understood only through such work and aims to complement that statement. Thus, instead of the built Punta Ballena it is the unbuilt city in Buenos Aires, Argentina, that this text intends to investigate.Therefore, this dissertation analyzes the unbuilt projects of Casa Amarilla housing (1943), Bajo Belgrano neighborhood (1948-49) and Barrio Sur quarter (1956), conceived for the city of Buenos Aires, recognizing them as a series. It is intended to investigate Bonet’s design and his contribution to the modern city project, including its solutions of collective housing. It is also an intention to understand the relationship between the city planned by Bonet and the existing one, collaborating on a more detailed study about the city of the Modern Architecture.
Es sabido que el arquitecto catalán Antonio Bonet Castellana es reconocido por sus obras en la costa uruguaya, siendo la urbanización de Punta Ballena, y el conjunto de casas, de gran importancia para la Arquitectura Moderna en el sur de Latinoamérica. Sin embargo, esta disertación entiende que la actuación de Bonet en el continente, sobre todo con relación a su contribución urbanística, no puede ser comprendida solamente a través de tal obra, y procura complementar dicha perspectiva. Así, en vez de la uruguaya Punta Ballena, es la ciudad no construida en Buenos Aires, Argentina, que se pretende investigar. De esta forma, el texto analiza los proyectos no construidos del conjunto residencial en Casa Amarilla (1943), el barrio propuesto en Bajo Belgrano (1948-49) y la remodelación de Barrio Sur (1956), pensados para tal ciudad, reconociéndolos como una serie. La finalidad de este estudio es conocer mejor la obra de Bonet, su contribución al proyecto de la ciudad moderna y las soluciones de vivienda colectiva que de ella derivan. También es objetivo del trabajo comprender las relaciones entre la ciudad pensada por Bonet y la ciudad existente, colaborando para un estudio más detallado sobre la ciudad de la Arquitectura Moderna.

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

My dissertation treats the physics of ultracold gases, in particular of Bose-Einstein condensates with long-ranged interactions induced by admixing a small fraction of a Rydberg state to the atomic ground state. The resulting interaction leads to the emergence of supersolid states and to the self-trapping of a Bose-Einstein condensate.

A Bose-Einstein condensate can be made of photons. The photons are held at thermodynamic equilibrium in a dye-filled microcavity and pumped with a laser. Thermalisation can be demonstrated and above the threshold a Bose-Einstein condensate will form. A Mach-Zehnder interferometer is built and used to measure the spatial and temporal first-order coherence under various conditions. We build a momentum-resolved spectrometer and use it to obtain views into the phase-space distribution of the photon condensate. We put an upper bound on the value of the interaction strength parameter and find that the microcavity system is ergodic even when not at thermal equilibrium. We build a setup to stabilise the pump laser power with the aim to observe the λ-point of the condensate.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Physics, 2019
Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 147-151).
Quantum simulation is a very active and growing field. Various quantum systems can be used to emulate existing materials in an accurate and controllable way, as well as to generate new states of matter that have not been found in the real world but the existence of which does not contradict the fundamental laws of physics. Ultracold atoms form a perfect system to realize idealized models and study physical mechanisms that stand out clearly in them. Recent efforts have been made to simulate artificial gauge fields with ultracold atoms, including spin-dependent gauge fields, such as spin-orbit coupling. Motivated by this goal our lab explored several approaches to generate a one-dimensional spinorbit coupling interaction, which has a rich phase diagram and plays an important role for topological insulators, the quantum spin Hall effect and spintronics.
The first method we developed allowed us to detect a stripe phase by dressing Bose-Einstein condensates with an optical superlattice and Raman beams. The observed density modulation in the ground state meets the definition of the long-awaited supersolid state of matter. The second approach we took was to generate spin-orbit coupling without use of lasers. The method is based on the idea of periodic driving of the quantum system and dressing its evolution with fast micromotion, often refered to as Floquet engineering. Our experiment provided an insightful pedagogical example of what Floquet engineering is capable of. In the experiment we endowed a low energy radio-frequency photon with tunable momentum. When dressed with recoil momentum, the interaction of a radio-frequency photon with an atom occurred in a Doppler-sensitive way. We also demonstrated how to tune the momentum and flip its direction. In this thesis, I first describe the experiments done in the optical superlattice.
Then I discuss the behavior of periodically driven classical and quantum systems and provide detailed analysis of how a radio-frequency photon can be magnetically dressed with tunable momentum. The experiments we carried out demonstrated novel methods of generation for spin-dependent gauge fields and showed pedagogical examples and interpretations of evolution of periodically driven systems. The scheme of periodically driven atoms inspired a theoretical study of heating in Floquet systems.
by Boris Shteynas.
Ph. D.
Ph.D. Massachusetts Institute of Technology, Department of Physics

Ultra-kalte atomare Gase werden in zahlreichen Laboren weltweit untersucht und finden unter anderem Anwendung in Atomuhren und in Atominterferometer. Die Einsatzgebiete erstrecken sich von der Geodäsie über die Metrologie bis hin zu wichtigen Fragestellungen der Fundamentalphysik, wie z.B. Tests des Äquivalenzprinzips. Doch die beispiellose Messgenauigkeit ist durch die irdische Gravitation eingeschränkt. Zum einen verzerrt die Schwerkraft das Fallenpotential und macht damit die Reduktion der atomaren Energie unter einem bestimmten Limit unmöglich. Zum anderen werden die aus einer Falle frei gelassenen Teilchen durch die Erdanziehung beschleunigt und so ist deren Beobachtungszeit begrenzt. Im Rahmen dieser Arbeit werden die Ergebnisse des Projektes QUANTUS (Quantengase Unter Schwerelosigkeit) dargestellt. Auf dem Weg zur Implementierung eines Quantengasexperimentes im Weltraum wurde innerhalb einer deutschlandweiten Zusammenarbeit eine kompakte, portable und mechanisch stabile Apparatur zur Erzeugung und Untersuchung eines Bose-Einstein-Kondensats (BEC) unter Schwerelosigkeit im Fallturm Bremen entwickelt. Sowohl die Abbremsbeschleunigung von bis zu 50 g als auch das begrenzte Volumen der Fallkapsel stellen hohe Ansprüche an die mechanische Stabilität und die Miniaturisierung von optischen und elektronischen Komponenten. Der Aufbau besteht aus einer im ultra-hoch Vakuum geschlossenen magnetischen Mikrofalle (Atomchip) und einem kompakten auf DFB-Dioden basierenden Lasersystem. Mit diesem Aufbau ließ sich das erste BEC unter Schwerelosigkeit realisieren und nach 1 Sekunde freier Expansion zu beobachten. Weder die schwache Krümmung des Fallenpotentials noch die lange Beobachtungszeit würden in einem erdgebundenen Experiment realisierbar. Die erfolgreiche Umsetzung des Projektes eröffnet ein innovatives Forschungsgebiet - degenerierte Quantengase bei ultratiefen Temperaturen im pK-Bereich, mit großen freien Evolutions- und Beobachtungszeiten von mehreren Sekunden.
Recently, cooling, trapping and manipulation of neutral atoms and ions has become an especially active field of quantum physics. The main motivation for the cooling is to reduce motional effects in high precision measurements including spectroscopy, atomic clocks and matter interferometry. The spectrum of applications of these quantum devices cover a broad area from geodesy, through metrology up to addressing the fundamental questions in physics, as for instance testing the Einstein’s equivalence principle. However, the unprecedented precision of the quantum sensors is limited in terrestial laboratories. Freezing atomic motion can be nowadays put to the limit at which gravity becomes a major perturbation in a system. Gravity can significantly affect and disturb the trapping potential. This limits the use of ultra-shallow traps for low energetic particles. Moreover, free particles are accelerated by gravitational force, which substantially limits the observation time. Targeting the long-term goal of studying cold quantum gases on a space platform, we currently focus on the implementation of a Bose-Einstein condensate (BEC) experiment under microgravity conditions at the drop tower in Bremen. Special challenges in the construction of the experimental setup are posed by a low volume of the drop capsule as well as critical decelerations up to 50g during recapture at the bottom of the tower. All mechanical and electronic components were thus been designed with stringent demands on miniaturization and mechanical stability. This work reports on the observation of a BEC released from an ultra-shallow magnetic potential and freely expanding for one second. Both, the low trapping frequency and long expansion time are not achievable in any earthbound laboratory. This unprecedented time of free evolution leads to new possibilities for the study of BEC-coherence. It can also be applied to enhance the sensitivity of inertial quantum sensors based on ultra-cold matter waves.

Questo lavoro ripercorre gli aspetti principali della teoria del condensato di Bose-Einstein, utilizzando nozioni di fisica statistica, meccanica quantistica non relativistica e idrodinamica. Nella prima parte si ricava dalla statistica di Bose il fenomeno della condensazione, ovvero una transizione nel sistema che porta a un’occupazione macroscopica del livello energetico. Si discutono anche alcuni fenomeni fisici caratteristici che emergono da una prima osservazione del condensato in una trappola con potenziale armonico anisotropo. Dopo aver introdotto gli urti tra i bosoni, si calcola, mediante la teoria di campo medio, l’equazione di Gross-Pitaevskii, soddisfatta dalla funzione d’onda del condensato. Sfruttando il calcolo variazionale, è possibile inoltre valutare l’importanza del termine energetico di interazione e dell’energia cinetica, e introdurre opportune approssimazioni alla funzione d’onda a seconda delle condizioni fisiche in cui si realizza il condensato. Nell’ultimo capitolo si considera la dinamica del condensato generalizzando l’equazione di Gross-Pitaevskii al caso dipendente dal tempo. In particolare, sfruttando alcune nozioni della teoria di Bogoliubov, si mettono in luce le connessioni tra la teoria dello stato condensato e il fenomeno della superfluidità.

La presente trattazione prende in esame i condensati di Bose-Einstein (BEC) e la formazione di vortici in tali strutture. Per analizzare questi aspetti si fa ricorso sia alla descrizione fisica, ripercorrendo il lavoro di Boltzmann, di Bose e di Einstein, sia al formalismo matematico tipico della topologia algebrica. Inoltre viene analizzato il fenomeno della superfluidità, segnatamente nel caso di He liquido, e vengono messe in evidenza analogie e diversità con la condensazione di Bose-Einstein. La transizione BKT rappresenta un punto imprescindibile per la comprensione del meccanismo alla base della formazione dei vortici. Infine ci si sofferma sugli aspetti sperimentali sottesi alla realizzazione dei BEC e alla produzione dei vortici in questi ultimi, vale a dire trappole magnetiche e raffreddamento laser per quanto riguarda i primi ed effetto Stark per quanto concerne i secondi.

This work reports on the construction of a new-generation system capable to create single-mode spinor Bose-Einstein condensates of 87Rb, and non-destructively probe them using optical Faraday rotation. This system brings together many of the stateof-the-art technologies in ultra-cold physics in a minimalist design which was possible due to the prolific advances in the field respect to the pioneering experiments (Cornell's, Ketterle's, and Chapman's groups). There is rich phenomena that can be potentially studied in this system from the study of predicted novel quantum phases and topologies to entanglement and spin squeezing which are useful for quantum information and interferometry. The potential of this system make it suitable to answer fundamental questions on the phase transition to a condensed and ferromagnetic state. In particular, this work describes theoretically and experimentally, the atomic spin coherence, which is relevant for applications like coherent sensing of magnetic fields. In this direction, our findings demonstrate the characteristics of our system make it a sensor with the best predicted energy resolution per unit bandwidth (~10^-2 h) among all the different technologies applied to magnetometry. The thesis is structured as follows: Part I is dedicated to the mathematical description of the relevant interactions. First, the interaction of optical polarization and atomic spin polarization is reviewed, with special attention to ac-Stark shifts, which are used to generate a conservative trapping potential and Faraday rotation effects that are used for non-destructive spin detection. Second, the interaction of the atoms with a magnetic field is presented. And finally, the mean-filed theory of spinor Bose-Einstein condensation is summarized. The dynamics of a spin-1system in this picture is described by a three-component Gross-Pitaevskii equation. Part II contains three chapters describing the implemented technologies and techniques used in the experiment to create and characterize a spinor condensate. The first chapter describes the ultrahigh vacuum, magnetic fields, lasers, spectroscopy and imaging needed to create a magneto optical trap (MOT) and transfer those atoms into an optical dipole trap (ODT). We implemented a non-standard loading technique based on the semicompensation of the strong differential lightshift induced by the ODT which profits from the effective dark-MOT created at the trap position. In the second chapter we detail, theoretically and experimentally, the all-optical evaporation process employed to achieve condensation in less than five second after the loading. In the final chapter the spin manipulation and read-out techniques are presented. Because there is no observable associated to the spin angle, we exploit the Faraday rotation effect and Stern-Gerlach imaging in order to retrieve information about the spin dynamics. Finally in Part III, we consider the potential of a spinor BEC as a magnetic sensor. The measurement of fundamental properties defining the sensitivity of the sensor are detailed. Those properties are the volume, the temporal coherence and the readout noise. We present a model of the magnetic field environment and its repercussion on the noise of the magnetometer. In the last chapter we present our perspectives to the possible applications of our system.
Este trabajo compila los detalles experimentales de un aparato de "nueva generación" capaz de crear condensados Espinoriales de 87Rb en un único dominio magnético, y de obtener información del estado de espín en una forma no destructiva explotando el efecto Faraday. Este aparato conjunta algunas de las tecnologías de punta aplicadas a física de gases ultrafrios en un diseño minimalista. Estas tecnologías se han podido desarrollar debido a los prolíficos avances en el campo, respecto a los experimentos pioneros en los grupos de Cornell, Ketterle y Chapman. Una rica cantidad de fenómenos pueden ser estudiados en este sistema, desde el estudio de novedosas fases y topologías cuánticas hasta la aplicación de entrelazamiento y estados comprimidos relevantes en información cuántica e interferometría. Su potencial lo hace un buen candidato para responder preguntas acerca de la naturaleza de las transiciones ferromagnética y de condensación. En particular, este trabajo describe teorética y experimentalmente la coherencia del estado de espín, el cual, es relevante en aplicaciones como la medición coherente de campos magnéticos. En este sentido, nuestros resultados demuestran que las características de nuestro condensado espinorial lo hacen el sensor con la mejor resolución en energía por unidad de ancho de banda (~10^-2 h ), de entre todas las tecnologías aplicadas a magnetometría. Esta tesis se estructura de la siguiente manera: Part I está dedicada a la descripción matemática de las interacciones relevantes. Primero la interacción entre la luz y el espín atómico es revisada, con especial énfasis en el desplazamiento ac-Stark, que es explotado para generar un potencial conservador, así como en las medidas no destructivas del espín via efecto Faraday. En segundo lugar, estudiamos la dinámica de espín bajo la interacción Zeeman entre los átomos y un campo magnético que varía en el tiempo. Finalmente es brevemente tratada la teoría de campo medio (mean-field theory) que describe los condensados espinoriales en la forma de una ecuación de Gross-Pitaevskii multicomponente. Part II contiene tres capítulos que detallan la tecnologías y técnicas usadas en el experimento para crear y caracterizar el condensado. El primer capítulo describe el ultra-alto vacío, los campos magnéticos, láseres, espectroscopía e imaging usados para crear una trampa magneto-óptica (MOT), y para transferir esos átomos en una trampa dipolar óptica (ODT). Nosotros implementamos una técnica poco estandard para cargar la ODT, la cual se basa en compensar medianamente el excesivo lightshift diferencial inducido por nuestra ODT. Esta técnica nos ayuda a crear una dark-MOT efectiva con la que podemos conseguir altas densidades de átoms en la ODT. En el segundo capítulo detallamos la evaporación que es "all-optical", con la que podemos conseguir un condensado en menos de 5 s de evaporación. En el capítulo final describimos las técnicas para crear arbitrarios estados de espín y cómo detectarlos. Para esto último explotamos el efecto Faraday y capturamos imágenes Stern-Gerlach. Finalmente en Part III, estudiamos las propiedades de coherencia, tiempo de vida y extensión espacial del condensado. Detallamos el sistema especialmente en el contexto de sensores magnéticos. Además, presentamos un modelo del campo magnético ambiental y sus repercusiones en el ruido del magnetómetro. En el último capítulo hablamos de algunas de las alternativas aplicaciones de nuestro sistema.

Ce mémoire de thèse présente une série d'expériences visant à caractériser la dynamique d'un condensat de Bose-Einstein dilué, en nous concentrant plus particulièrement sur sa mise en rotation. En utilisant un laser très désaccordé, nous avons pu réaliser un potentiel dipolaire simulant l'expérience du seau tournant effectuée précédemment sur l'hélium II : la mise en rotation s'accompagne alors de la formation de tourbillons quantifiés caractérisés par un défaut de phase en leur coeur. Ce défaut topologique a pu être mis en évidence par une expérience d'interférences atomiques entre deux paquets d'ondes séparés spatialement. De plus, nous avons mesuré le moment cinétique du nuage en étudiant le spectre de ses modes de surface, nous permettant ainsi de préciser le mécanisme de nucléation des tourbillons. Ceux-ci se forment suite à une instabilité dynamique du condensat, dépendant fortement de la géométrie du potentiel tournant.

Ce mémoire de thèse présente les résultats d'un série
d'expériences sur des condensats de Bose-Einstein en rotation. Du
fait de la cohérence d'un condensat, la rotation s'introduit
généralement sous la forme de singularités de phase appelées
vortex. L'isolation et l'observation d'une unique ligne de vortex
ont permis d'étudier sa dynamique dans un condensat allongé. La
forme d'équilibre de la ligne de vortex ainsi que son évolution
sur des échelles de temps longues ont été caractérisées. Par leur
effet sur l'amortissement des modes de surface, les modes de
vibration de la ligne ont été mis en évidence. La limite des
rotations rapides a également été étudiée. Pour des vitesses de
rotation intermédiaires, elle est caractérisée par un grand nombre
de vortex arrangés en réseaux d'Abrikosov. Pour des vitesses
encore plus grandes, différentes phases sont attendues selon la
forme du potentiel. Cette limite a été étudiée dans le cas d'un
potentiel hybride contenant un petit terme quartique. Les nuages
en rotation, que nous avons caractérisés par plusieurs méthodes,
se révèlent avoir un comportement original. Leur observation
constitue un premier pas vers celle de phases corrélées proches du
régime Hall quantique fractionnaire.

In this thesis, the properties of the Bose-Einstein condensation (BEC) in low dimensions are reviewed. Three dimensional weakly interacting Bose systems are examined by the variational method. The effects of both the attractive and the repulsive interatomic forces are studied. Thomas-Fermi approximation is applied to find the ground state energy and the chemical potential. The occurrence of the BEC in low dimensional systems, is studied for ideal gases confined by both harmonic and power-law potentials. The properties of BEC in highly anisotropic trap are investigated and the conditions for reduced dimensionality are derived.

The differences between classical and quantum mechanics were highlighted early in the development of quantum mechanics when Schrodinger proposed the thought experiment of a cat in a superposition of alive and dead. In this thesis I try to understand these differences by considering superpositions of large objects at a single particle level. Research in the field of superconductors has provided evidence for macroscopic quantum superpositions (or cat states) of currents in superconducting loops. Bose-Einstein condensates of ultracold atoms provide another promising system for experimentally producing similar results. I begin by describing two straightforward schemes that make macroscopic superpositions of superfluid flow states of Bose-Einstein condensates trapped in optical lattice rings. The first scheme achieves a superposition of three flow states by nonadiabatically evolving the barrier heights between the sites. The second scheme produces a superposition of two flow states by applying a 7f phase around the ring. This could be experimentally achieved by physically rotating the sites or imparting angular momentum from two co-propagating lasers. The next part of the thesis investigates why it is difficult to produce macroscopic superpositions. By treating the interaction strength between the atoms as a perturbation I show three reasons, other than decoherence, why macroscopic superpositions are hard to make. Firstly, the energy of the two distinct flow states must be sufficiently close. Secondly, coupling between the two states must be sufficiently strong, and thirdly, other states must be well separated from those two flow states. To make larger superpositions I look at a Josephson junction coupled to a superfluid loop. This shows that making superpositions depends on the number of atoms in the junction rather than the whole system. Finally I propose ways of developing the work. This concentrates on how the systems could experimentally create macroscopic superpositions and how we could measure signatures of these states. I then suggest ways of using the systems, such as quantum information and precision measurement schemes.

Entanglement in many-body systems of identical particles often resides between modes of the field occupied by particles rather than between the particles themselves, or their internal degrees of freedom. Spatial field modes are an important choice of modes and investigating entanglement between them for a non-interacting Bose gas is the concern of this thesis. A particularly interesting phase of a Bose gas is a Bose-Einstein condensate (BEC). Long-range correlations between all points in space exist in a BEC, which are detected by the single-particle reduced density matrix.

Cavity polaritons are particles which are formed from photons confined to a cavity coupled to electronic excitations such as semiconductor excitons. Since the observation of cavity polaritons in 1992, there has been considerable interest in the quantum statistical behaviour of cavity polaritons. This thesis is a theoretical study of one of the most spectacular quantum statistical behaviours, Bose condensation, for cavity polaritons. In this thesis, we investigate Bose condensation of cavity polaritons in a generic model of photons interacting with electronic excitations. The model we consider is a generalisation of the Dicke model, familiar from quantum optics. It consists of a single bosonic oscillator, describing the electromagnetic field in a cavity, interacting with a large number of two-state oscillators with a distribution of energies. These oscillators could represent, for example, excitons bound to traps in a disordered semiconductor. Bose condensation is a phenomenon associated with conserved particles in thermal equilibrium. Thus to investigate Bose condensed polaritons we study the thermodynamics of the model at a fixed number of polaritons. We do this using two techniques: a variational approach, and a more powerful path-integral technique. The latter allows us to give an essentially exact description of both the thermodynamics and the excitations of the model.

This thesis is a comprehensive study of the charged Bose-gas (CBG). Special attention has been paid to the role of Bose-Einstein condensation, and it's effect on the superconducting properties of the system. The main motivation for this work has been the relevance of the CBG as a model for the high temperature superconductors, within the context of the bipolaron theory[1], the experimental and theoretical evidence for which is discussed in the introductory chapters. By using standard many body theory relevant properties of the CBG have been derived. Much attention has been placed on how to deal with the condensate in an interacting system. The Bogoliubov-de Gennes (BdG) equations are formulated for the Charged Bose Gas (CBG), and their extension to the Ginzburg-Landau-Abrikosov-Gor'kov (GLAG) type theory is discussed. The temperature dependence of the condensate density for the Coulomb Bose-gas (CBG) was studied using the Bogoliubov approximation. This enabled the calculation of the London penetration depth as a function of temperature. Other thermodynamic variables, such as the free energy and specific heat, were also calculated. The nature of the Bose-Einstein condensation of the CBG in a magnetic field was investigated for ultra-low temperatures and ultra-high magnetic fields. The temperature dependence of the upper critical field was found to have positive curvature, one of the remarkable features of the high temperature superconductors. The difference between the resistive transition and the peak in the specific heat is also discussed.

The quantum phase of a Bose-Einstein condensate has long been a subject fraught with misunderstanding and confusion. In this thesis we provide a consis- tent description of this phenomenon and, in particular, discuss how phase may be defined, created, manipulated, and controlled. We begin by describing how it is possible to set up a reference condensate against which the phase of other condensates can be compared. This allows us to think of relative phases as if they were absolute and gives a clear and precise definition to 'the phase of a condensate'. A relative phase may also be established by coupling condensates and we show how this can be controlled. We then extend this model to explain how the phase along a chain of coupled condensates can lock naturally without the need for any measurements. The second part of the thesis deals primarily with the link between entangle- ment and phase. We show that, in general, the more entangled a state is, the better its phase resolution. This leads us to consider schemes by which maximally entangled states may be able to be created since these should give the best prac- tical advantages over their classical counterparts. We consider two such states: a number correlated pair of condensates and a Schrodinger cat state. Both schemes are shown to be remarkably robust to loss. A comparison of the merits of these two states, as the inputs to an interferom- eter, reveals very different behaviours. In particular, the number correlated state performs significantly better than the cat state in the presence of loss, which means that it might be useful in interferometry and frequency standard schemes where phase resolution is of the utmost importance. Finally, we propose a scheme for concentrating the entanglement between con- densates, which is an important step in quantum communication protocols. This, along with the ability to manipulate phase and entanglement, suggests that the future for condensates holds not only academic interest but great potential for practical applications.

The generation of dark solitons in Bose-Einstein condensates has been an area of interest since the first experimental condensates were produced. The ubiquity of solitons in the natural world makes them an important phenomenon to understand. Despite excellent theoretical work in two dimensional dark solitons, few experiments have had the opportunity to investigate this regime. The work presented investigates the generation of dark solitons in a Rb-87 Bose-Einstein condensate. The evolution and decay of these topological excitations are investigated. The decay of the dark solitons is found to vary with the phase-step used to generate them. Dark solitons created with a phase-step width of 0.60 ±0.15 μm are found to decay into vortices after 10 ms. Dark solitons generated with larger phase-steps are found not to exhibit this vortex decay, instead dissipating over 10-15 ms back into the condensate. The first experimental generation of two dimensional Jones-Roberts solitons is reported in this work. These dark solitons differ from the standard planar dark soliton in that they are finite in extent and are found to be more dynamically stable. The Jones-Roberts solitons are observed for 40 ms with no observed change in energy.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2006.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
"June 2006."
Includes bibliographical references (p. 205-209).
As the coldest form of matter known to exist, atomic Bose-Einstein condensates are unique forms of matter where the constituent atoms lose their individual identities, becoming absorbed into the cloud as a whole. Effectively, these gases become a single macroscopic object that inherits its properties directly from the quantum world. In this work, I describe the quantum properties of a zero temperature condensate where the atoms have a propensity to pair, thereby leading to a molecular character that coexists with the atoms. Remarkably, the addition of this molecular component is found to induce a quantum instability that manifests itself as a collective decay of the assembly as a whole. As a signature of this phenomenon, there arises a complex chemical potential in which the imaginary part quantifies a coherent decay into collective phonon excitations of a collapsing ground state. The unique decay rate dependencies on both the scattering length and the density can be experimentally tested by tuning near a Feshbach resonance. Being a purely quantum mechanical effect, there exists no mechanical picture corresponding to this coherent many-body process. The results presented can serve as a model for other systems with similar underlying physics.
by George E. Cragg.
Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2007.
Includes bibliographical references (p. 133-147).
Recent developments in atom optics have brought Bose-Einstein condensates within 1 pm of solid surfaces where the atom-surface interactions can no longer be ignored. At long- range, the atom-surface interaction is described by the weakly attractive Casimir-Polder potential which is classically predicted to accelerate an incident atom toward the surface where it will interact strongly with the internal modes of the surface, lose energy, and land in a bound state of the surface. When the incident atom is very cold, on the order of a few nanokelvin, however, the acceleration of the atomic wavefunction is so abrupt that the atom may partially reflect from the attractive tail in a process known as quantum reflection. This work presents experimental evidence for quantum reflection from a solid surface at normal incidence. Using atoms from a 23Na BEC, cooled to a few nanokelvin in a recently demonstrated single-coil trap, controlled collisions were induced between atoms and solid silicon surface. A maximum reflection probability of - 12% was observed for an incident velocity of 1 mm/s. Atoms confined against the surface at low density exhibited an enhanced lifetime due to quantum reflection. A surprising aspect of quantum reflection is that nano-structured surfaces are predicted to exhibit enhanced quantum reflection due to the reduction of the atom-surface interaction from reduced density surfaces. Using a pillared surface with an density reduced to 1% of bulk density, we observe an enhancement of the reflection probability to ' 60%. At velocities from 2-25 mm/s, predicted threshold dependence of the reflection probability was observed. At velocities below 2 mm/s, the reflection probability was observed to saturate. We develop a simple model which predicts the saturation as a result of mean-field interactions between atoms in the incident Bose-Einstein condensate.
by Thomas A. Pasquini.
Ph.D.