Dans ce manuscrit, nous présentons notre avancement pour réaliser une méthode spectroscopique pour étudier la transition d’Anderson avec des atomes froids. Cela repose sur la réalisation d'un potentiel désordonné sélectif en état de spin, le désordre n'étant significatif que pour l'un des deux états de spin impliqués. En combinant cela avec la technique de transfert par radiofréquence d’un état insensible au désordre à un état exclusivement sensible au désordre, il devient possible de charger une onde de matière dans le désordre dans des états d’énergie bien définies. Pour prouver le concept, nous avons effectué des mesures des fonctions spectrales d’atomes ultra-froids dans des potentiels désordonnés, qui sont directement proportionnels au taux de transfert des atomes. Nous présentons les résultats en montrant un excellent accord avec les calculs numériques. Cela a ouvert des perspectives pour d’autres études sur la transition d’Anderson. En particulier, nous cherchons à observer la transition entre les états diffusifs et les états localisés séparés par une énergie critique, appelée le seuil de mobilité. Une telle étude nécessite la réalisation d’un désordre sélectif en état de spin qui permet un long temps de propagation dans le désordre afin de distinguer les deux phases. À cette fin, nous présentons un nouveau schéma du désordre sélectif en état de spin avec deux lasers du speckle (speckle bichromatique). Cela ouvre la voie à une approche spectroscopique de la transition d’Anderson avec des atomes froids avec une résolution en énergie bien supérieure à celles des expériences précédentes. / In this manuscript, we present our progress towards realizing a spectroscopic method to study of Anderson transition with ultracold atoms. This relies on the realization of state-dependent disordered potential whereby the disorder is significant only for one of two involved spin-states. Combined with technique of radio-frequency transfer from the disorder-free state to the state with controlled disorder, it becomes possible to load a matter wave in the disorder in a well-defined energy states. As a proof of principle, we have performed measurements of the spectral functions of ultracold atoms in disordered potentials, which are directly proportional to the transfer rate of the atoms. We present the results showing excellent agreement with numerical calculations. This has opened up prospects for further studies of the Anderson transition. In particular we seek to observe transition between the diffusive and the localized states separated by a critical energy, the so-called mobility edge. Such study requires realization of state-dependent disorder which allows long propagation time in the disorder in order to distinguish the two phases. For this purpose, we present a new scheme of the state-dependent disorder with two laser speckles (bichromatic laser speckle). This paves the way towards spectroscopic approach of Anderson transition with ultracold atoms with energy resolution much higher than those in the previous experiments.

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.

The ground state of a quantum system is the minimizer of the total energy of that system. The aim of this thesis is to present and numerically solve the Gross-Pitaevskii eigenvalue problem (GPE) as a physical model for the formation of ground states of dilute Bose gases at ultra-low temperatures in a disordered potential. The first part of the report introduces the quantum mechanical phenomenon that arises at ground states of the Bose gases; the Anderson localization, and presents the nonlinear eigenvalue problem and the finite element method (FEM) used to discretize the GPE. The numerical method used to solve the eigenvalue problem for the smallest eigenvalue is called the inverse power iteration method, which is presented and explained. In the second part of the report, the smallest eigenvalue of a linear Schrödinger equation is compared with the numerically computed smallest eigenvalue (ground state) in order to evaluate the accuracy of a linear numerical scheme constructed as first step for numerically solving the non-linear problem. In the next part of the report, the numerical methods are implemented to solve for the eigenvalue and eigenfunction of the (non-linear) GPE at ground state (smallest eigenvalue). The mathematical expression for the quantum energy and smallest eigenvalue of the non-linear system are presented in the report. The methods used to solve the GPE are the FEM and inverse power iteration method and different instances of the Anderson localization are produced. The study shows that the error of the smallest eigenvalue approximation for the linear case without disorder is satisfying when using FEM and Power iteration method. The accuracy of the approximation obtained for the linear case without disorder is satisfying, even for a low numbers of iterations. The methods require many more iterations for solving the GPE with a strong disorder. On the other hand, pronounced instances of Anderson localizations are produced in a certain scaling regime. The study shows that the GPE indeed works well as a physical model for the Anderson localization. / Syftet med denna avhandling är att undersöka hur väl Gross-Pitaevskii egenvärdesekvation (GPE) passar som en fysisk modell för bildandet av stationära elektronstater i utspädda Bose-gaser vid extremt låga temperaturer. Fenomenet som skall undersökas heter Anderson lokalisering och uppstår när potentialfältets styrka och störning i systemet är tillräckligt hög. Undersökningen görs i denna avhandling genom att numeriskt lösa GPE samt illustrera olika utfall av Anderson lokaliseringen vid olika numeriska värden. Den första delen av rapporten introducerar det icke-linjära matematiska uttrycket för GPE samt de numeriska metoderna som används för att lösa problemet numerisk: finita elementmetoden (FEM) samt egenvärdesalgoritmen som heter inversiiteration. Finita elementmetoden används för att diskretisera variationsproblemet av GPE och ta fram en enkel algebraisk ekvation. Egenvärdesalgoritmen tillämpas på den algebraiska ekvation för att iterativt beräkna egenfunktionen som motsvarar det minsta egenvärdet. Det minsta egenvärdet av en fullt definierad (linjär) Schrödinger ekvation löses i rapportens andra del. Den linjära ekvationen löses för att ta fram en förenklad numerisk algoritm att utgå ifrån innan den icke-linjära algoritmen tas fram. För att försäkra sig att den linjära algoritmen stämmer bra jämförs det exakta egenvärdet för problemet med ett numeriskt framtaget värde. Undersökningen av den linjära algoritmen visar att vi får en bra uppskattning av egenvärdet - även vid få iterationer. Vidare konstrueras den ickelinjära algoritmen baserat på den linjära. Ekvationen löses och undersökes. Egenfunktionen som motsvarar minsta egenvärdet framtas och beskriver kvantsystemet i lägsta energitillståndet, så kallade grundtillståndet. Undersökningen av GPE visar att de numeriska metoderna kräver många fler iterationer innan en tillräckligt bra uppskattning av egenvärdet fås. Å andra sidan fås markanta Anderson lokaliseringar för ett skalningsområde som beskrivs av styrkan av potentialfältet i relation till dess störning. Slutsatsen är att Gross-Pitaevskii egenvärdesekvation passar bra som en fysisk modell för detta kvantsystem.

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.

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.

The Kibble-Zurek mechanism (KZM) primarily characterizes scaling in the formation of topological defects when a system crosses a continuous phase transition. The KZM was first used to study the evolution of the early universe, describing the topology of cosmic domains and strings as the symmetry-breaking phase transitions acted on the vacuum fields during the initial cooling. A ferromagnetic spin-1 $^{87}$Rb Bose-Einstein condensate (BEC) exhibits a second-order gapless quantum phase transition due to a competition between the magnetic and collisional spin interaction energies. Unlike extended systems where the KZM is illustrated by topological defects, we focus our study on the temporal evolution of the spin populations and observe how the scaling of the spin dynamics depend on how fast the system is driven through the critical point. In our case, the excitations are manifest in the temporal evolution of the spin populations illustrating a Kibble-Zurek type scaling, where the dynamics of slow quenches through the critical point are predicted to exhibit universal scaling as a function of quench speed. The KZM has been studied theoretically and experimentally in a large variety of systems. There has also been a tremendous interest in the KZM in the cold atoms community in recent years. It has been observed not only in ion chains and in atomic gases in optical lattices, but also in Bose gases through the formation of vortices or solitons. The KZM in the context of crossing the quantum phase transition in a ferromagnetic BEC has been theoretically studied, but this thesis is the first experimental investigation of this phenomenon.

Thesis (PhD)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: Please refer to full text. / AFRIKAANSE OPSOMMING: Sien asb volteks vir opsomming

Ultracold atoms have been successfully used to study numerous systems, previously unaccessible, but a precise control over the atom number of the sample still remains a challenge. This dissertation describes our progress towards achieving Fock states of atoms. The first three chapters cover the basic physics necessary to understand the techniques we use in our lab to manipulate atoms. We then summarize our experimental results from an earlier setup where we did two experiments. In the first experiment we compare the transport of cold atoms and a Bose Einstein Condensate (BEC) in a periodic potential. We find a critical potential height beyond which the condensate behavior deviates significantly from that of thermal atoms. In the second experiment we study the effect of periodic temporal kicks by a spatially periodic potential on a BEC in a quasi one dimensional trap. We observe a limit on the energy that the system can absorb from the kicks, which we conclude is due to the finite height of the trap rather than quantum effects. The majority of the dissertation discusses our experimental setup designed to produce Fock states. The setup is designed to use the method of laser culling to produce Fock states. We are able to create a BEC and transport it into a glass cell 25 cm away. We tried different innovative methods to reduce vibrations during transport before finally settling to a commercial air bearing translation stage. We create a high confinement one dimensional optical trap using the Hermite Gaussian TEM₀₁ mode of a laser beam. Such a trap gives trapping frequencies comparable to an optical lattice and allows us to create a single one dimensional trap. We creating the TEM₀₁ mode using an appropriate phase object (phase plate) in the path of a TEM₀₀ mode beam. The method for producing the phase plate was very well controlled to obtain a good quality mode. Once the atoms are loaded into this one dimensional trap we can proceed to do laser culling to observe Sub-Poissonian number statistics and eventually create Fock states of few atoms. Finally, we describe a novel method to create a real time tunable optical lattice which would provide us with the ability of spatially resolved single atom detection. The majority of the dissertation discusses our experimental setup designed to produce Fock states. The setup is designed to use the method of laser culling to produce Fock states. We are able to create a BEC and transport it into a glass cell 25 cm away. We tried different innovative methods to reduce vibrations during tr₀ansport before finally settling to a commercial air bearing translation stage. We create a high confinement one dimensional optical trap using the Hermite Gaussian TEM₀₁ mode of a laser beam. Such a trap gives trapping frequencies comparable to an optical lattice and allows us to create a single one dimensional trap. We creating the TEM₀₁ mode using an appropriate phase object (phase plate) in the path of a TEM₀₀ mode beam. The method for producing the phase plate was very well controlled to obtain a good quality mode. Once the atoms are loaded into this one dimensional trap we can proceed to do laser culling to observe Sub-Poissonian number statistics and eventually create Fock states of few atoms. Finally, we describe a novel method to create a real time tunable optical lattice which would provide us with the ability of spatially resolved single atom detection. The majority of the dissertation discusses our experimental setup designed to produce Fock states. The setup is designed to use the method of laser culling to produce Fock states. We are able to create a BEC and transport it into a glass cell 25 cm away. We tried different innovative methods to reduce vibrations during transport before finally settling to a commercial air bearing translation stage. We create a high confinement one dimensional optical trap using the Hermite Gaussian TEM₀₁ mode of a laser beam. Such a trap gives trapping frequencies comparable to an optical lattice and allows us to create a single one dimensional trap. We creating the TEM₀₁ mode using an appropriate phase object (phase plate) in the path of a TEM₀₀ mode beam. The method for producing the phase plate was very well controlled to obtain a good quality mode. Once the atoms are loaded into this one dimensional trap we can proceed to do laser culling to observe Sub-Poissonian number statistics and eventually create Fock states of few atoms. Finally, we describe a novel method to create a real time tunable optical lattice which would provide us with the ability of spatially resolved single atom detection. The majority of the dissertation discusses our experimental setup designed to produce Fock states. The setup is designed to use the method of laser culling to produce Fock states. We are able to create a BEC and transport it into a glass cell 25 cm away. We tried different innovative methods to reduce vibrations during transport before finally settling to a commercial air bearing translation stage. We create a high confinement one dimensional optical trap using the Hermite Gaussian TEM₀₁ mode of a laser beam. Such a trap gives trapping frequencies comparable to an optical lattice and allows us to create a single one dimensional trap. We creating the TEM₀₁ mode using an appropriate phase object (phase plate) in the path of a TEM₀₀ mode beam. The method for producing the phase plate was very well controlled to obtain a good quality mode. Once the atoms are loaded into this one dimensional trap we can proceed to do laser culling to observe Sub-Poissonian number statistics and eventually create Fock states of few atoms. Finally, we describe a novel method to create a real time tunable optical lattice which would provide us with the ability of spatially resolved single atom detection. / text

Mon projet de thèse a eu pour objectif l'étude des propriétés magnétiques de condensats de Bose-Einstein d'atomes de Sodium confinés dans un piège optique. Dans la première partie, nous présentons le dispositif expérimental et le protocole suivi pour la production tout-optique de condensats. La première étape consiste dans le chargement d'un piège dipolaire croisé désaccordé vers le rouge à partir d'atomes pré-refroidis dans un piège magnéto-optique. La deuxième étape est le refroidissement évaporatif dans un piège dipolaire composite, combinaison du piège dipolaire croisé avec un faisceau fortement focalisé. Nous sommes ainsi capables de réaliser des condensats de Bose-Einstein quasi-purs contenant environ 3000 atomes. Dans la deuxième partie, nous nous intéressons aux propriétés magnétiques qui découlent de la présence de trois espèces de spin simultanément piégées. Nous présentons des méthodes de contrôle de la magnétisation des nuages ultra-froids, ainsi que des procédures de diagnostic de la composition de spin. Nous utilisons ces échantillons pour explorer le diagramme de phase à basse température, en fonction de la magnétisation et du champ magnétique. Nous montrons l'accord satisfaisant de ces résultats expérimentaux avec une théorie de champ champ moyen dans l'approximation de mode commun. Enfin, nous observons des fluctuations anormales des populations à bas champ et basse magnétisation. On les relie à des fluctuations collectives tendant à restaurer la symmétrie de spin, qui disparaissent à la limite thermodynamique mais sont présentes dans nos échantillons de taille finie.

This dissertation presents several experimental methods that were devised to generate or manipulate quantized vortices in highly oblate dilute-gas Bose-Einstein condensates (BECs). Studies that involve single vortex dynamics, vortex-vortex interactions, and vortex-impurity interactions are essential in developing a deeper understanding of the nature of superfluidity and in particular, superfluid turbulence. In highly oblate systems, vortex dynamics have a two-dimensional (2D) nature and the resulting superfluid characteristics may be substantially different from those in three-dimensional (3D) superfluids. However, there have been remarkably few experimental studies of 2D vortex dynamics in superfluids. Therefore, to study 2D vortex dynamics and interactions, it is necessary to first develop experimental methods that can generate vortices and vortex distributions in nominally 2D systems, such as highly oblate BECs. Four main experiments are discussed in this dissertation. Two of these experiments generate multiple singly quantized vortices in a relatively stochastic manner leading to disordered vortex distributions. From these two vortex methods, the physics of high vorticity and highly disordered systems may be observed and studied in a highly oblate system. These methods may prove useful in studies of 2D quantum turbulence. The other two experiments involve newly developed techniques for controlled generation and manipulation of vortices. One of these methods creates multiply quantized pinned vortices with a control in the generated vorticity. The other method reliably creates a pair of singly quantized vortices of opposite circulation, whose positions can be easily manipulated after creation, such that they can be placed in any location within the BEC. The two techniques may be scalable to higher number of vortices and may prove useful in superfluid dynamics and vortex interactions that require repeatable vortex distributions. Taken together, these tools and methods may be applicable to many further studies of vortex physics in highly oblate BECs.