In this thesis we study two different problems in the field of ultracold atoms: "Shock waves in Bose-Einstein condensates" and "Inelastic scattering of atoms in a double well".In the first problem, we study the supersonic flow of a Bose-Einstein condensate (BEC) past a slender impenetrable macroscopic obstacle in the framework of the twodimensional (2D) defocusing nonlinear Schrödinger equation (NLS).Assuming the oncoming flow speed sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched x-coordinate and the piston motion curve is defined by the spatial body profile.Two steady oblique spatial dispersive shock waves (DSWs) spreading from the pointed ends of the body are generated in both half-planes.These are described analytically by constructing appropriate exact solutions of the Whitham modulation equations for the front DSW and by using a generalized Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear DSW.We propose an extension of the traditional modulation description of DSWs to include the linear ship-wave pattern forming outside the nonlinear modulation region of the front DSW.Our analytic results are supported by direct 2D unsteady numerical simulations and are relevant to recent experiments on Bose-Einstein condensates freely expanding past obstacles.In the second problem, we study a mixture of two light spin-1/2 fermionic atoms and two heavy atoms in a double well potential.Inelastic scattering processes between both atomic species excite the heavy atoms and renormalize the tunneling rate and the interaction of the light atoms (polaron effect).The effective interaction of the light atoms changes its sign and becomes attractive for strong inelastic scattering.This is accompanied by a crossing of the energy levels from singly occupied sites at weak inelastic scattering to a doubly occupied and an empty site for stronger inelastic scattering.We are able to identify the polaron effect and the level crossing in the quantum dynamics.Em primeiro lugar, agradeço ao Prof. Arnaldo Gammal por TUDO! Raramente um orientador tem tanta paciência, boa vontade e principalmente, interesse e dedicação com seus orientandos.Agradeço-lhe pela oportunidade de poder trabalhar no seu grupo em temas tão interessantes.Também lhe agradeço por me introduzir na arte do cálculo numérico.Obrigado pela paciência de saber lidar com o profundo bom humor e eterno positivismo de Annibale.Ich danke Herrn Prof. Klaus Ziegler für die Gelegenheit, in Ihrer Gruppe in Augsburg -Deutschland arbeiten zu können.Dank Ihnen konnte ich an zwei interessante Projekte arbeiten, viel Physik lernen und zwei Jahre in der schönen Stadt Augsburg leben.
In this thesis we study two different problems in the field of ultracold atoms: "Shock waves in Bose-Einstein condensates" and "Inelastic scattering of atoms in a double well".In the first problem, we study the supersonic flow of a Bose-Einstein condensate (BEC) past a slender impenetrable macroscopic obstacle in the framework of the twodimensional (2D) defocusing nonlinear Schrödinger equation (NLS).Assuming the oncoming flow speed sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched x-coordinate and the piston motion curve is defined by the spatial body profile.Two steady oblique spatial dispersive shock waves (DSWs) spreading from the pointed ends of the body are generated in both half-planes.These are described analytically by constructing appropriate exact solutions of the Whitham modulation equations for the front DSW and by using a generalized Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear DSW.We propose an extension of the traditional modulation description of DSWs to include the linear ship-wave pattern forming outside the nonlinear modulation region of the front DSW.Our analytic results are supported by direct 2D unsteady numerical simulations and are relevant to recent experiments on Bose-Einstein condensates freely expanding past obstacles.In the second problem, we study a mixture of two light spin-1/2 fermionic atoms and two heavy atoms in a double well potential.Inelastic scattering processes between both atomic species excite the heavy atoms and renormalize the tunneling rate and the interaction of the light atoms (polaron effect).The effective interaction of the light atoms changes its sign and becomes attractive for strong inelastic scattering.This is accompanied by a crossing of the energy levels from singly occupied sites at weak inelastic scattering to a doubly occupied and an empty site for stronger inelastic scattering.We are able to identify the polaron effect and the level crossing in the quantum dynamics.Em primeiro lugar, agradeço ao Prof. Arnaldo Gammal por TUDO! Raramente um orientador tem tanta paciência, boa vontade e principalmente, interesse e dedicação com seus orientandos.Agradeço-lhe pela oportunidade de poder trabalhar no seu grupo em temas tão interessantes.Também lhe agradeço por me introduzir na arte do cálculo numérico.Obrigado pela paciência de saber lidar com o profundo bom humor e eterno positivismo de Annibale.Ich danke Herrn Prof. Klaus Ziegler für die Gelegenheit, in Ihrer Gruppe in Augsburg -Deutschland arbeiten zu können.Dank Ihnen konnte ich an zwei interessante Projekte arbeiten, viel Physik lernen und zwei Jahre in der schönen Stadt Augsburg leben.
Nesta tese estudamos dois problemas diferentes na área de átomos ultra frios: Ondas de choque em condensados de Bose-Einstein e Espalhamento inelástico de átomos em um potencial de dois poços. No primeiro problema, estudamos o fluxo supersônico de um condensado de Bose-Einstein (BEC) através de um obstáculo macroscópico impenetrável delgado no sistema da equação de Schrödinger não-linear (NLS) bidimensional. Assumindo-se que a velocidade do fluxo é suficientemente alta, reduzimos assintoticamente o problema bidimensional original de valor de contorno para o fluxo estacionário através de um obstáculo alongado ao problema do pistão dispersivo unidimensional descrito pela NLS 1D dependente do tempo, no qual a coordenada original x reescalonada faz o papel de tempo e o movimento do pistão está vinculado à geometria do obstáculo. Duas ondas de choque dispersivas (DSWs) são geradas no fluxo, cada uma sendo formada em uma extremidade (frontal e traseira) do obstáculo. A DSW frontal é descrita analiticamente construindo-se soluções de modulação exatas para as equações de Whitham e a para a DSW traseira, empregamos a regra de quantização de Bohr-Sommerfeld generalizada para descrever a distribuição dos sólitons escuros. Propomos uma extensão da solução de modulação tradicional, a fim de incluir o padrão de ship-wave linear formado fora da região da DSW frontal. Realizamos simulações numéricas 2D completas e verificamos a validade das previsões analíticas. Os resultados deste problema podem ser relevantes para experimentos recentes sobre o fluxo de BECs através de obstáculos. No segundo problema, estudamos uma mistura atômica de dois átomos fermiônicos leves de spin 1/2 e dois átomos pesados em um potencial de dois poços. Processos de espalhamento inelástico entre ambas as espécies atômicas excitam os átomos pesados e renormalizam a taxa de tunelamento e a interação entre os átomos leves (efeito polarônico). A interação efetiva dos átomos leves muda de sinal e se torna atrativa quando o espalhamento inelástico é forte. Observamos também o cruzamento de níveis de energia, de um estado onde cada poço contém apenas um férmion (espalhamento inelástico fraco) para um estado onde um poço contém um par de férmions e ou outra está vazio (espalhamento inelástico forte). Identificamos o efeito polarônico e o cruzamento dos níveis de energia estudando-se a dinâmica quântica do sistema.
We study a mixture of two light spin-1/2 fermionic atoms and two heavy atoms in a double-well potential. Inelastic scattering processes between both atomic species excite the heavy atoms and renormalize the tunneling rate and the interaction of the light atoms (polaron effect). The effective interaction of the light atoms changes its sign and becomes attractive for strong inelastic scattering. This is accompanied by a crossing of the energy levels from singly occupied sites at weak inelastic scattering to a doubly occupied and an empty site for stronger inelastic scattering. We are able to identify the polaron effect and the level crossing in the quantum dynamics.
The Bose-Einstein condensate (BEC) formed by ultracold atomic gases provides an ideal platform for studying various quantum phenomena. In this platform, researchers have explored in depth the important equilibrium and steady phenomena including superfluidity, vortices, and solitons, and recently started to study nonequilibrium problems. In a classical system, nonequilibrium problems, such as explosion, usually occur together with shock waves, which is presented when the explosion speed is larger than the local sound speed. For BEC systems which possess quantum properties, how to produce and understand the shock waves becomes a hot research topic. In this work, we systematically discuss the possibility of quantum shock wave and its essential mechanism in a one-dimensional BEC initially containing dark solitons through quenching interactions. When the system is quenched to the limit of non-interaction, we analytically obtain the post-quench dynamics of initially immobile dark solitons, and find the existence of shock wave, which can be explained through the quantum interference effect. When the system is quenched to finite interaction, we find similar phenomena through numerically solving the Gross-Pitaevskii equation, and analyze different situations. When the system is quenched to a finite weaker interaction, the situation is similar to a non-interaction case; when the system is quenched to a stronger interaction, the shock wave is accompanied by the splitting of the initial soliton, and the two objects can synchronously change; specifically when the quenched ratio of strength is an integer squared, the shock wave disappears, and the soliton is split perfectly. We further explore the properties of the shock wave including its amplitude and speed, and obtain the full scenario as the quenched interaction varies. This work provides theoretical guidance for realizing and measuring shock wave in experiment.
A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock-wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, e.g., traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation, hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical one-dimensional (1D) dissipative and dispersive shock problem shows significant differences in shock structure and shock-front speed. Numerical results associated with the three-dimensional experiment show that three- and two-dimensional approximations are in excellent agreement and 1D approximations are in good qualitative agreement. Using 1D DSW theory, it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.
No presente trabalho foi estudada a dinâmica de um sistema de muitas partículas no regime de temperaturas ultra-baixas. Realizamos um estudo dinâmico de sistemas condensados bidimensionais em uma rede óptica não-linear em uma direção e também na presença de uma armadilha harmônica assimétrica. Investigamos alguns aspectos sobre a estabilização e propagação de sólitons em condensados de Bose-Einstein. O colapso da função de onda é evitado pela não-linearidade periódica dissipativa, no caso de um meio com campo de fundo positivo (com sistemas atômicos atrativos). A variação adiabática do comprimento de espalhamento de fundo leva a existência de sólitons de onda de matéria metaestáveis. Um sóliton dissipativo pode existir no meio atrativo bidimensional (2D) com uma não-linearidade periódica unidimensional (1D), quando um mecanismo de alimentação atômica é utilizado. Um sóliton estável pode existir no caso de condensados repulsivos, em um campo de fundo negativo, com uma armadilha harmônica em uma direção e uma rede óptica não-linear na outra direção. Os resultados inteiramente numéricos, para a equação de Gross-Pitaevskii 2D, confirmam as simulações da abordagem variacional.
We investigate nonlinear excitations of Bose-Einstein condensates focusing on quasi one and two dimensional condensates. Different geometries of the external potential and effects of a spatial modulation of the interatomic interaction are studied. Our investigations incorporate the ground state and collective excitations of the condensate on a mean-field level by employing the Gross-Pitaevskii equation augmented by effective low-dimensional approaches.
Firstly, we demonstrate the effects of a spatially inhomogeneous scattering length on quasi-one-dimensional condensates trapped in a double-well potential. The collisional inhomogeneity produces a saddle-node bifurcation scenario between two additional solution branches. Moreover, the threshold of transition from Josephson matter-wave oscillations to nonlinearly induced self-trapping is shifted due to the collisional inhomogeneity. We develop an adjusted two-mode model which describes these effects accurately. In a second step, the effects of a collisional inhomogeneity on dark and bright matter-wave solitons are explored. In both cases appropriate versions of the soliton perturbation theory are shown to accurately describe the soliton dynamics. Moreover, we illustrate that modulating the interaction can be used to stabilize vortices against excitations due to finite temperature effects.
A second focal point of this thesis is the investigation of coherent excited states of Bose-Einstein condensates, i.e., solitonic and vortex states. The existence and stability of single and multiple dark soliton states in quasi-one-dimensional condensates trapped in a double-well potential and single as well as multiple vortex states in quasi-two-dimensional condensates are investigated systematically. We illustrate the connection of dark soliton stripes and multiple vortex states. Using a quasi-particle approach enables us to predict the motion of multiple solitons and vortices on an inhomogeneous background by simple ordinary differential equations.
As an extension to the investigations of dark and bright solitons, we study dark bright solitons in two component Bose Einstein condensates. A strong dependence of the oscillation frequency and of the stability of the dark-bright soliton on the atom number of its components is found theoretically and has been verified experimentally as well.
Moreover, methods for trapping of ultracold atoms are developed. We depict a way of trapping magnetically insensitive atoms by inter atomic collisions with atoms of the same species which are trapped magnetically. The controlled outcoupling of the trapped atoms by shaping the condensate's wave function is explored leading to a pulsed emission of atoms from the trapping region reminiscent of an atom laser. Moreover, we present a way of trapping atoms by applying near resonant laser light in a Raman configuration on a Ioffe-Pritchard like magnetic field configuration leading to versatile atom traps. The emerging potential surfaces are widely tunable and can be used for trapping two internal states.
Gegenstand dieser Arbeit ist die Untersuchung nichtlinearer Anregungen von Bose-Einstein Kondensaten mit Schwerpunkt auf ein- und zweidimensionalen Systemen. Unterschiedliche Fallenpotentiale und Effekte von einer ortsabhangigen Streulange werden studiert. Mittels der Gross-Pitaevskii Gleichung werden sowohl Grundzustand als auch kollektive Anregungen untersucht.
Als Erstes prasentieren wir Auswirkungen einer ortsabhangigen Streulange fur ein quasi eindimensionales Kondensat, welches in einem Doppeltopfpotential gefangen ist. Das inhomogene Streuverhalten fuhrt zu einer Sattelpunktbifurkation von zwei Zweigen von Losungen, die fur konstante Streulange nicht existieren. Des Weiteren ist der Ubergang von Josephson Schwingungen zum nichtlinear induziertem Selbstfangen verschoben. Wir entwickeln ein angepasstes Zweimodenmodell, welches diese Effekte exakt beschreibt. In einem zweiten Schritt werden die Auswirkungen einer ortsabhangigen Streulange auf dunkle und helle Materiewellen Solitonen untersucht. In beiden Fallen lasst sich die Solitonendynamik durch entsprechende Storungstheorie fur Solitonen exakt beschreiben. Daruber hinaus zeigen wir, dass die ortliche Modulation der Wechselwirkung dazu verwendet werden kann, um sogenannte Vortex Zustande bei endlicher Temperatur zu stabilisieren.
Ein zweiter Schwerpunkt dieser Arbeit ist die Untersuchung koharenter angeregter Zustande von Bose-Einstein Kondensaten, sog. Soliton und Vortex Zustande. Die Existenz und Stabilitat von einzelnen oder mehreren dunklen Solitonen in quasi-eindimensionalen Kondensaten, gefangen in einem Doppeltopf, sowie von einzelnen und mehreren Vortex Zustanden in quasi-zweidimensionalen Kondensaten werden systematisch untersucht. Wir veranschaulichen den Zusammenhang von dunklen Soliton-Streifen und mehreren Vortex Zustanden. Ein Quasi-Teilchen-Ansatz ermoglicht es uns, Potentiale fur die Wechselwirkung von mehreren Solitonen und Vortices auf einem inhomogenen Hintergrund herzuleiten und die entsprechende Dynamik durch einfache gewohnliche Differentialgleichungen zu beschreiben. Als Erweiterung zu den Untersuchungen von dunklen und hellen Solitonen studieren wir sogenannte dunkel-hell Solitonen in Zweikomponenten-Bose-Einstein Kondensaten. Eine starke Abhangigkeit der Schwingungsfrequenz und der Stabilitat der dunkel-hell Solitonen von der Atomzahl in ihren Komponenten wird gefunden.
Daruber hinaus werden Methoden zum Fangen von ultrakalten Atomen entwickelt. Wir prasentieren einen Weg zum Fangen von magnetisch unempfindlichen Atomen durch atomare Stose mit Atomen der gleichen Art, die magnetisch eingeschlossen werden. Die kontrollierte Auskopplung der gefangenen Atome durch entsprechende Gestaltung der Kondensatswellenfunktion wird erforscht. Dies fuhrt zu einer gepulsten Emission von Atomen aus der gefangenen Region, die an einen Atom-Laser erinnert. Daruber hinaus prasentieren wir eine Moglichkeit zum Fangen von Atomen durch die Anwendung von nahresonantem Laserlicht in einer Raman-Konfiguration in Kombination mit einer Ioffe-Pritchard Magnetfeld-Konfiguration. Die entstehende Potenzialflachen sind flexibel einstellbar und konnen zum gleichzeitigen Fangen zweier interner Atomzustande verwendet werden.