Este trabalho consiste em um estudo teórico da coexistência de condensados de Bose-Einstein atômicos acoplados. Esta mistura de condensados pode consistir em átomos de um mesmo elemento em vários estados hiperfinos (estado interno). Outra situação em que esta mistura ocorre é quando um átomo em um único estado interno pode ocupar mais de um estados externo. As semelhanças e diferenças entre condensados de Bose-Einstein com vários estados internos e externos serão analisadas cuidadosamente. Somente na aproximação de Bose-Hubbard os estados externos serão tratados como internos. Condensados envolvendo duas ou mais espécies serão estudados com a aproximação de Thomas-Fermi com estados coerentes. Esta aproximação implica em desprezar o termo de energia cinética. Muitos resultados analíticos serão exibidos para a condição em que todos os comprimentos de espalhamento são iguais entre si. Esta condição é bastante próxima da observada em um condensado de vários estados internos. Além disso serão apresentados alguns resultados anaíticos onde o comprimento de espalhamento para espécies diferentes é zero. Esta condição é verificada para a aproximação de Bose-Hubbard para misturas de condensados com estados externos. O objetivo principal destes cálculos é estudar o papel da fase relativa entre as funções de onda nas soluções estacionárias e na evolução temporal destes sistemas. / This work consists of a theoretical study of the coexistance of coupled atomic Bose-Einstein condensates. These mixed condensates can consist of atoms of the same element in different hyperfine internal states. A different situation is that in which an atom in one internal state can ocupate different single particle external states. The similarities and differences between mixed Bose-Einstein condensates involving internal and external states will be analysed carefully. Only in the Bose-Hubbard approximation the external states can be treated like internal states. Condensates involving two or more species will be studied with in a Thomas- Fermi\'s approximation with coherent states. This approximation involves in discarding of the Hamiltonian the kinetic energy term. A number of analytical results are given for the case in which the different channel scatering lenghts are equal to each other. This condition is which well approximated in condensates involving many internal states. Beside some results will also be given for situations where the scatering lenghts for different species vanishes. This condiction is verified for the Bose-Hubbard approximation to externally mixed condensates. The principal goal of this calculations is to study the role of relative phase between the wave functions in the stationary solutions and in the time evolution of the systems.

Uma análise dos processos colisionais homonucleares e heteronucleares, responsáveis por perdas em uma armadilha magneto-óptica, foi realizada neste trabalho. Resultados experimentais inéditos para as taxas de perdas, no sistema Rb-Cs, foram obtidos. Os dados experimentais em conjunto com os resultados do modelo sugerem que o processo de escape radiativo é dominante. O modelo utilizado é muito sensível à profundidade da armadilha, a qual depende de outros parâmetros (intensidade, dessintonia, gradiente de campo magnético, etc.). A colaboração com pesquisadores italianos possibilitou uma análise mais detalhada das taxas de perdas heteronucleares em outras regiões de intensidade. Estudamos a dependência das taxas heteronucleares com a razão das massas do par atômico, e uma razoável concordância foi observada. O estudo de processos colisionais em armadilhas magneto-ópticas é importante na obtenção do condensado de BoseEinstein com duas espécies atômicas distintas, e também em experiências de espectroscopia fotoassociativa. / In this work, we had investigated the heteronuclear trap loss rate in a two-species Rb-Cs magneto-optical trap. The experimental results suggest that radiative escape is the main collisional process responsible for heteronuclear losses. An addapted Gallagher - Pritchard model is compared with the data. The model is very sensitive to the trap depth, which depends on the trap parameters (intensity, detuning, magnetic - field gradient, etc.) This observation is also supported by experimental results from a Pisa\' group. We have compared experimental of heteronuclear rates as a function of the masses ratio of the atomic pair with the model, and a reasonable agreement is observed. These studies are relevant for high resolution atomic / molecular spectroscopy, and for the production of mixed - species Bose-Einstein condensates.

Made available in DSpace on 2014-07-29T15:15:47Z (GMT). No. of bitstreams: 1 Pages from Wesley Bueno1.pdf: 4320775 bytes, checksum: 75a8ed4cb0407f9831577bca544b704b (MD5) Previous issue date: 2010-07-09 / This work combines some of the results obtained on the study of solitons in relativistic fields and Bose-Einstein condensates. By using a first order formalism to solve the equations of motion of relativistic fields, introduced previously by our group, we construct several classes of lump solutions described by a single real scalar field. We show how these solutions can be controlled depending on a single parameter in the field potential. In condensed matter the solutions of the lump type correspond to bright solitons, very studied in the context of nonlinear crystals, fiber optics, Bose-Einstein condensates, etc. In all these cases, such solutions are obtained via a nonlinear Schr¨odinger equation, responsible for describing the propagation of pulses in optical fibers or crystals, or the atomic density in condensates. In this sense, our main goal is to study the soliton and breather modulations via nonlinear Schrodinger equation. We concentrate on the Bose-Einstein condensate in which the modulation of atomic density can be accomplished through the Feshbach resonance. We study cases where the nonlinearity is described by terms cubic, cubic and quintic, and purely quintic in the nonlinear Schr¨odinger equation. Also, situations where two interacting condensates in which the nonlinear Schr¨odinger equations are coupled, breather modulations, and the study of the soliton behavior under influence of chaotic, random and non-periodic perturbations in the nonlinearity of the system. In many cases we consider the condensate trapped in the cigarshaped configuration, i.e., with freedom in only one spatial dimension. Numerical simulations are performed to verify the stability of the solutions. / Este trabalho reúne alguns dos resultados obtidos sobre o estudo de sólitons em modelos de campos escalares relativísticos e em condensados de Bose-Einstein. Com o uso de um formalismo de primeira ordem para a equação de movimento de campos relativísticos, introduzido anteriormente por nosso grupo, construímos várias classes de soluções do tipo lump para um único campo escalar real. Mostramos que as formas dessas soluções podem ser controladas dependendo de um único parâmetro no potencial de campo utilizado. Em matéria condensada as soluções do tipo lump correspondem a sólitons brilhantes, muito estudados no contexto de cristais não-lineares, fibras ópticas, condensados de Bose-Einstein, etc. Em todos esses casos tais soluções são obtidas através da equação não-linear de Schrodinger, responsável por descrever a propagação de pulsos, em cristais ou fibras ópticas, ou a densidade atômica, no caso de condensados. Nesse sentido, nosso objetivo principal consiste no estudo da modulação de sólitons e breathers através da equação não-linear de Schrodinger. Nossa concentração é nos condensados de Bose-Einstein nos quais a modulação da densidade atômica pode ser realizada através da ressonância de Feshbach. Estudamos casos onde a não-linearidade é descrita por termos cúbico, cúbio e quíntico e puramente quíntico na equação não-linear de Schrodinger. Investigamos também situações onde dois condensados podem interagir e as ENLS estão acopladas, modulação de breathers e o estudo da influência de ruídos caóticos, aleatórios e não-periódicos na não-linearidade do sistema para o comportamento do sóliton. Na maior parte deste trabalho consideramos o condensado aprisionado na forma de charuto, isto é, com liberdade em apenas uma dimensão espacial. Simulações numéricas foram realizadas para verificar a estabilidade das soluções.

A extensão dos fenômenos quânticos em escala macroscópica é responsável por toda uma classe de efeitos como a supercondutividade, superfluidez, e condensação de Bose-Einstein, as quais desempenham um papel central na física ao longo do século passado. A produção dos primeiros condensados de Bose-Einstein tornou possível a realização de experimentos envolvendo fenômenos quânticos macroscópicos com um nível sem precedentes de controle dos parâmetros externos. As correntes persistentes em condensados estão intimamente relacionados com a nucleação de vórtices quantificados, que são defeitos topológicos como resposta à transferência de quanta de momento angular. Um método convencional para geração de tais defeitos consiste em confinar a nuvem atômica condensada em uma armadilha com rotação. Acontece que, para velocidades angulares acima de um valor crítico, estados de vórtice se tornam energeticamente favoráveis, induzindo assim a criação de vórtices quânticos. Realizações experimentais de condensados de átomos de metais alcalinos confinados por potenciais dependentes do tempo permitiram a observação não só de redes de vórtices, mas também de turbulência quântica. Uma vez que a turbulência quântica é caracterizada pela presença de um emaranhado de vórtices quânticos interagindo entre si, uma correta compreensão da dinâmica, formação e estabilidade de vórtices tem se mostrado de grande importância sendo objeto de muitos trabalhos teóricos. Em particular, o papel das excitações acústicas geradas pelo decaimento de vórtices de multipla carga no desenvolvimento de turbulência ainda é uma questão em aberto. Este trabalho tem como objetivo fornecer um conjunto de ferramentas que ajude a identificar a presença, como também a carga de vórtices em nuvens (não turbulentas) observadas utilizando imagens de tempo-de-voo. Temos feito um estudo detalhado de condensados contendo vórtices carga múltipla colocados no seu centro, onde a dinâmica do tempo-de-voo é apenas de nossos pontos de interesse. Devido ao controle que este sistema fornece experimentalmente, os modos coletivos tornam-se uma descrição importante, uma vez que podem ser excitadas usando métodos experimentais bem estabelecido tal como a modulação do comprimento de espalhamento de ondas-s, e que também pode ser responsável pelo decaimento do vórtice. Para tais fins, temos utilizado o método variacional (semi-analítico), e o cálculo totalmente numérico da equação de Gross-Pitaevskii. Assim, descrevemos os modos coletivos que acoplam a dinâmica do vórtice com as oscilações das componentes externas do condensado, bem como os efeitos em tempo-de-voo. O momento angular atua aumentando a energia cinética em torno do núcleo de vórtice, que implica em um aumento mais rápido da direção perpendicular a este. Esta situação desloca as freqüências de oscilações coletivas de um estado livre de vórtice, e gera modos coletivos mais ricos devido ao acoplamento. Agora, existem quatro modos possíveis, sendo dois tipos de modo monopolar e dois tipos de modos de quadrupolo. A diferença dentre tais modos é a fase de oscilação do vórtice. Quando se considera flutuações sem simetria polar, seus modos coletivos resultam no decaimento do vórtice. A fim de controlar e prevenir estes processos propusemos três mecanismos dinâmicos, tais como a modulação de comprimento de espalhamento, a modulação das frequências da armadilha harmônica e modulação da amplitude do potencial de Laguerre-Gauss. O último tem provado ser mais eficaz. / The extension of quantum phenomena into macroscopic scales is responsible for a whole class of effects such as superconductivity, superfluidity, and Bose-Einstein condensation, which played central roles in physics throughout the last century. The production of the first Bose-Einstein condensates made possible the realization of experiments involving macroscopic quantum phenomena with an unprecedented level of control of the external parameters. The persistent currents in condensates are intimately related to the nucleation of quantized vortices, which are topological defects as response to transference of quanta of angular momentum. A conventional method for generation of such defects consists in confining the condensed atomic cloud into a rotating trap. It turns out that, for angular velocities higher than a critical value, vortex states become energetically favorable, thus inducing the creation of quantized vortices. Experimental realizations of condensed alkali-metal atoms confined by more general time-dependent potentials allowed the observation not only of vortex lattices but also of quantum turbulence. Since quantum turbulence is characterized by the presence of a self-interacting tangle of quantized vortices, the correct understanding of dynamics, formation, and stability of vortices has shown to be of paramount importance being the subject of many theoretical works. In particular, the role of acoustic excitations generated by decaying multi-charged vortices in the development of turbulence is still an open question. This work aims to provide a set of tools that helps to identify the presence as well as the charge of vortices in non-turbulent clouds observed using time-of-flight pictures. We have done a detailed study of condensates containing multi-charged vortices placed at its center where time-of-flight dynamics is only one point of our interest. Due to the control that this system provides experimentally, the collective modes become an important description since they can be excited using well stablished experimental methods as such as modulation of the s-wave scattering length, and they can also be responsible to vortex decaying. For such purposes we have used the semi-analytical variational method, and the fully numerical calculation of Gross-Pitaevskii equation. Thus we have describes the collective modes that couples dynamics of vortex with the oscillation of external components of condensed atomic cloud as well as the effects in time-of-flight. The angular momentum acts increasing the kinetic energy around the vortex core, which results in a faster expansion of perpendicular direction to it. This situation shifts the frequencies of collective oscillations of a vortex-free state, and generates richer collective modes due the coupling. Now there are four possible modes, being two types of monopole mode and two types of quadrupole modes. The difference among these types is the phase of vortex oscillation. When one considers fluctuations without polar symmetry, their collective modes result in the vortex decaying. In order to control and prevent these processes we have proposed three dynamical mechanisms such as modulation of s-wave scattering length, modulation of frequencies of harmonic trap, and modulation of the amplitude of Laguerre-Gauss potential. The last one has proven to be more effective.

Dans cette thèse, nous faisons état de la construction d’un dispositif expérimentalcapable de piéger et refroidir un gaz d’ytterbium, dans l’optique de simuler des champsmagnétiques artificiels. Ce dispositif permettra, à terme, de produire et de caractériserdes états quantiques fortement corrélés, semblables aux états rencontrés dans la physiquede l’effet Hall quantique, entier ou fractionnaire.Dans un premier temps, nous décrivons la construction des parties de notre dispositifconsacrées au refroidissement optique de l’ytterbium (174Yb). En particulier, nousprésentons la conception d’un ralentisseur Zeeman, permettant le chargement direct d’unpiège magnéto-optique effectué sur la transition d’intercombinaison 1S0 ! 3P1 de l’ytterbium.Après transport dans un piège optique, une étape de refroidissement évaporatifnous permet d’obtenir des condensats de Bose-Einstein contenant environ 5×104 atomesd’ytterbium. Les condensats produits sont alors chargés dans un réseau optique verticalà la longueur d’onde « magique ».Nous présentons ensuite la construction d’un système laser étroit à 578nm capabled’exciter la transition « d’horloge » 1S0 ! 3P0 de l’ytterbium. Le laser est asservi surune cavité Fabry-Perot de grande finesse servant de référence de fréquence, dont nousavons caractérisé les différentes propriétés. Nous présentons en particulier une méthodepermettant de calibrer rapidement la fréquence absolue de la cavité par comparaison avecune transition de la molécule de diiode.Finalement, nous présentons les résultats d’expériences de spectroscopie effectuées surdes condensats d’ytterbium à l’aide du laser étroit, ainsi que la manipulation cohérentedu condensat sur la transition d’horloge au cours d’expériences d’oscillations de Rabi. Cesexpériences préliminaires ouvrent notamment la voie à la mesure des propriétés colisionnellesde l’ytterbium 174. / In this thesis, we report on the construction of an experiment aimed at trapping andcooling an ytterbium gaz, in order to realize artificial gauge fields. In the long term, thissetup will allow the study of strongly correlated quantum states which are atomic analogsof integer or fractional quantum Hall systems.We will first present the building of our experimental apparatus, and the optical coolingof ytterbium (174Yb). In particular, we will report on the design of a Zeeman slower,allowing for the direct loading of a magneto-optical trap operated on ytterbium’s intercombinationtransition 1S0 ! 3P1. The atomic cloud is then transported in an opticaldipole trap. A subsequent evaporative cooling stage results in the production of Bose-Einstein condensates of about 5 × 104 atoms.We then describe the construction of an ultra-narrow laser system at 578nm, able todrive ytterbium’s « clock » transition 1S0 ! 3P0. The laser frequency is stabilized using ahigh-finesse Fabry-Perot cavity, whose properties are precisely characterized in this work.Specifically, we present a method to calibrate the absolute frequency of the cavity bycomparison with an optical transition of molecular iodine.Finally, we show the results of spectroscopic measurements done on ytterbium condensatesusing the ultra-narrow laser. We also report on the coherent manipulation of thecondensate on the clock transition, consisting in the observation of Rabi oscillations.These preliminary experiments should allow for a measurement of ytterbium’s scatteringproperties.Keywords : cold atoms, optical lattices, Bose-Einstein condensates, ultra-stable lasers,clock transition, quantum simulation.

Dans ce manuscrit, nous présentons une étude expérimentale d'un gaz de Bose de spin 1 avec des interactions antiferromagnétiques avec des atomes de sodium ultra-froids dans l'état hyperfin F=1. Les trois composantes Zeeman sont piégées simultanément dans des pièges dipolaires optiques. Nous obtenons un condensat de Bose-Einstein spineur par refroidissement évaporatif et nous étudions ses propriétés magnétiques. Il y a deux types d’interactions dans le système: des interactions de contact qui ne changent pas les populations des composantes Zeeman et des interactions d'échange de spin qui les modifient. Une compétition entre l'énergie Zeeman et l'énergie d'échange impose l'ordre magnétique dans le système.Nous étudions dans un premier temps les phases magnétiques de condensats de Bose-Einstein spineurs a température quasi nulle. L'état fondamental comporte deux phases qui sont observées en variant le champ magnétique (donc l'énergie Zeeman quadratique) et la magnétisation de l'échantillon. Dans la phase antiferromagnétique, le spin de l'échantillon est simplement selon l'axe du champ magnétique. Dans la phase polaire, une composante transverse apparait pour minimiser l'énergie Zeeman. Pour une magnétisation nulle, le condensat spineur forme un nématique de spin. Cet état, nommé par analogie avec la phase nématique dans les cristaux liquides, est caractérisée par des fluctuations de spin orthogonales à un axe particulier, mais sans préférer une des deux direction sur cet axe. Dans chacune des deux phases, l'ordre nématique se manifeste par un minimisation de la longueur du spin transverse en imposant une valeur particulière ($pi$) de la phase relative des composantes Zeeman ${theta = phi_{+1} + phi_{-1} - 2 phi_{0}}$. Nous mesurons la longueur du spin transverse en analysant le bruit de spin après une rotation.Dans un second temps, nous étudions la thermodynamique d'un gaz de Bose de spin 1 près de la température critique pour la condensation de Bose-Einstein. Nous mesurons plusieurs scénarios de condensation séquentiels en fonction de la magnétisation et du champ magnétique. La température critique mesurée révèle que les interactions ont un effet important quand la condensation d'une composante se fait en présence d'un condensat dans une autre composante. Nous utilisons une théorie d'Hartree-Fock simplifiée, en négligeant les interactions d’échange de spin. Nous constatons que les résultats expérimentaux sont en bon accord. Cependant, pour de bas champs magnétiques, le diagramme de phase thermodynamique est largement modifié par les interactions d'échange de spin, ce qui pose de nouvelles questions sur leur rôle a température finie. / In this manuscript, we present an experimental study of a Spin 1 Bose gas with antiferromagnetic interactions with ultracold sodium atoms in the F=1 manifold. The three Zeeman components are trapped simultaneously in optical dipole traps. By performing evaporative cooling, we obtain quasi-pure spinor Bose-Einstein condensates of which we study the magnetic properties. There are two types of interactions between the constituents of the system: Contact interactions that do not change the Zeeman populations and spin-exchange contact interactions that do. A competition between Zeeman energy and the spin-exchange energy sets the magnetic ordering in the system.We first study the magnetic phases of spinor Bose-Einstein condensates near zero temperature. The ground state present two phases that are observed by varying the magnetic field (hence the quadratic Zeeman energy) and the magnetization of the sample. In the antiferromagnetic phase, the spin of the sample is purely along the direction of the magnetic field. In the broken-axisymmetry phase, a transverse component appears in order to minimize the Zeeman energy. For zero magnetization, the spinor condensate forms a spin nematic. This state, named in analogy with the liquid crystal nematic phase, is characterized by spin fluctuations orthogonal to a particular axis, with no preferred direction along that axis. In both phases, spin nematic order manifests as a minimization of the transverse spin length that is realized by enforcing a particular value ($pi$) of the relative phase of the Zeeman components $theta = phi_{+1} + phi_{-1} - 2 phi_0$. We measure the transverse spin length by analyzing spin noise after a spin rotation.Second, we study the thermodynamics of an antiferromagnetic spin 1 Bose gas next to the critical temperature for Bose-Einstein condensation. We measure several sequential condensation scenarii depending on the magnetization and the magnetic field. The measured critical temperatures reveal a large effect of interactions when one of the Zeeman component condenses in presence of a condensate in another component. We use a simplified Hartree-Fock theory, neglecting the spin exchange interactions and note a good agreement with our data. However, for low magnetic fields, the thermodynamic phase diagram is strongly modified which raises new open questions about the role of spin exchange interactions at finite temperatures.

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.

In 1981, Unruh suggested the possibility of simulating the dynamics of quantum fields in curved spacetimes using sound-waves propagating in moving fluids: a supersonic flow would indeed influence the dynamics of sound similarly to what happens to light when it’s dragged by the spacetime geometry in strong gravity environments. This simple yet groundbreaking observation has lead to the beginning of a whole new field of research, nowadays known as Analog Gravity. Due to their superfluid character, intrinsic quantum nature and impressive experimental tunability, Bose-Einstein condensates represent one of the most promising platforms to realize analog spacetimes, including black-hole geometries with horizons and ergoregions, as well as of time-dependent configurations relevant to cosmology. In this Thesis we go beyond the standard single-component BEC and focus on two-component mixtures of atomic condensates, possibly in the presence of a coherent coupling between the two-components: the availability of various branches of elementary excitations with different sound speed and effective mass may in fact lead to advantages in the implementation of interesting geometries and, eventually, to the exploration of a broader spectrum of physical processes. We first consider black-hole related phenomena (Hawking radiation and rotational superradiance) that have already been analysed with single-component systems, generalising the results to mixtures; we then proceed to tackle a problem (the decay from the false vacuum) which instead requires the additional degrees of freedom that only a mixture displays.

Investigamos o sistema formado por dois condensados aprisionados em estados hiperfinos diferentes do Rubídio, num potencial em forma de charuto, ou seja, num sistema físico real e quase-unidimensional. É investigada a dependência das soluções das equações de Gross-Pitaevski com a separação entre as armadilhas, bem como com o parâmetro de acoplamento de Josephson, para três valores diferentes do número total de átomos aprisionados. Para alguns conjuntos de parâmetros constatamos a existência de estados metaestáveis. O observável que escolhemos para caracterizar tal sistema físico foi a separação média entre os pacotes, pois os dois ramos de soluções encontramos correspondem a soluções mais juntas ou mais separadas espacialmente. / We study the system formed by two coupled condensates of different Rubidium hyperfine states trapped in a cigar shaped potential, that is, a real quasi one-dimensional system. The dependency of the solution of the Gross-Pitaevski equations is investigated as a function of trap displacement and Josephson coupling parameter for three different values of the total trapped atoms number. For some sets of parameters we report the existence of metastable states. The observable we chose to characterize this system was the mean separation between the packages, because we found two branches which correspond to closer or more separated solutions.

Ultra-cold dilute gases of neutral atoms are attractive candidates for creating controlled mesoscopic quantum systems. In particular, quantum degenerate gases of bosonic and fermionic atoms can be used to model the correlated many-body behaviour of Bose and Fermi condensed matter systems, and to study matter wave interference and coherence. This thesis describes the experimental realization and manipulation of Bose-Einstein condensates (BECs) of 87Rb and degenerate Fermi gases (DFGs) of 40K using static and dynamic magnetic atom chip traps. Atom chips are versatile modern tools used to manipulate atomic gases. The chips consist of micrometre-scale conductors supported by a planar insulating substrate, and can be used to create confining potentials for neutral atoms tens or hundreds of micrometres from the chip surface. We demonstrate for the first time that a DFG can be produced via sympathetic cooling with a BEC using a simple single-vacuum-chamber apparatus. The large 40K-87Rb collision rate afforded by the strongly confining atom chip potential permits rapid cooling of 40K to quantum degeneracy via sympathetic cooling with 87Rb. By studying 40K-87Rb cross-thermalization as a function of temperature, we observe the Ramsauer-Townsend reduction in the 40K-87Rb elastic scattering cross-section. We achieve DFG temperatures as low as T = 0.1TF , and observe Fermi pressure in the time-of-flight expansion of the gas. This thesis also describes the radio-frequency (RF) manipulation of trapped atoms to create dressed state double-well potentials for BEC and DFG.We demonstrate for the first time that RF-dressed potentials are species-selective, permitting the formation of simultaneous 87Rb double-well and 40K single-well potentials using a 40K-87Rb mixture. We also develop tools to measure fluctuations of the relative atom number and relative phase of a dynamically split 87Rb BEC. In particular, we observe atom number fluctuations at the shot-noise level using time-of-flight absorption imaging. These measurement tools lay the foundation for future investigations of number squeezing and matter wave coherence in BEC and DFG systems.