Experimental observation of oscillating and interacting matter wave dark solitons

We report on the generation, subsequent oscillation and interaction of a pair of matter wave dark solitons. These are created by releasing a Bose-Einstein condensate from a double well potential into a harmonic trap in the crossover regime between one dimension (1D) and three dimensions (3D). The oscillation of the solitons is observed and the frequency is in quantitative agreement with simulations using the Gross-Pitaevskii equation. An effective particle picture is developed and reveals that the deviation of the observed frequencies from the asymptotic prediction $\nu_{z}/\sqrt{2}$, where $\nu_{z}$ is the longitudinal trapping frequency, results from the dimensionality of the system and the interaction between the solitons.Comment: 5 pages, 3 figure

Multiple atomic dark solitons in cigar-shaped Bose-Einstein condensates

We consider the stability and dynamics of multiple dark solitons in cigar-shaped Bose-Einstein condensates (BECs). Our study is motivated by the fact that multiple matter-wave dark solitons may naturally form in such settings as per our recent work [Phys. Rev. Lett. 101, 130401 (2008)]. First, we study the dark soliton interactions and show that the dynamics of well-separated solitons (i.e., ones that undergo a collision with relatively low velocities) can be analyzed by means of particle-like equations of motion. The latter take into regard the repulsion between solitons (via an effective repulsive potential) and the confinement and dimensionality of the system (via an effective parabolic trap for each soliton). Next, based on the fact that stationary, well-separated dark multi-soliton states emerge as a nonlinear continuation of the appropriate excited eigensates of the quantum harmonic oscillator, we use a Bogoliubov-de Gennes analysis to systematically study the stability of such structures. We find that for a sufficiently large number of atoms, multiple soliton states may be dynamically stable, while for a small number of atoms, we predict a dynamical instability emerging from resonance effects between the eigenfrequencies of the soliton modes and the intrinsic excitation frequencies of the condensate. Finally we present experimental realizations of multi-soliton states including a three-soliton state consisting of two solitons oscillating around a stationary one.Comment: 17 pages, 11 figure

Rigorous mean-field dynamics of lattice bosons: Quenches from the Mott insulator

We provide a rigorous derivation of Gutzwiller mean-field dynamics for lattice bosons, showing that it is exact on fully connected lattices. We apply this formalism to quenches in the interaction parameter from the Mott insulator to the superfluid state. Although within mean-field the Mott insulator is a steady state, we show that a dynamical critical interaction $U_d$ exists, such that for final interaction parameter $U_f>U_d$ the Mott insulator is exponentially unstable towards emerging long-range superfluid order, whereas for $U_f<U_d$ the Mott insulating state is stable. We discuss the implications of this prediction for finite-dimensional systems.Comment: 6 pages, 3 figures, published versio

Dark solitons in atomic Bose-Einstein condensates: from theory to experiments

This review paper presents an overview of the theoretical and experimental progress on the study of matter-wave dark solitons in atomic Bose-Einstein condensates. Upon introducing the general framework, we discuss the statics and dynamics of single and multiple matter-wave dark solitons in the quasi one-dimensional setting, in higher-dimensional settings, as well as in the dimensionality crossover regime. Special attention is paid to the connection between theoretical results, obtained by various analytical approaches, and relevant experimental observations.Comment: 82 pages, 13 figures. To appear in J. Phys. A: Math. Theor

Nonlinear Waves in Bose-Einstein Condensates: Physical Relevance and Mathematical Techniques

The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.Comment: 69 pages, 10 figures, to appear in Nonlinearity, 2008. V2: new references added, fixed typo

Quantum transport in ultracold atoms

Ultracold atoms confined by engineered magnetic or optical potentials are ideal systems for studying phenomena otherwise difficult to realize or probe in the solid state because their atomic interaction strength, number of species, density, and geometry can be independently controlled. This review focuses on quantum transport phenomena in atomic gases that mirror and oftentimes either better elucidate or show fundamental differences with those observed in mesoscopic and nanoscopic systems. We discuss significant progress in performing transport experiments in atomic gases, contrast similarities and differences between transport in cold atoms and in condensed matter systems, and survey inspiring theoretical predictions that are difficult to verify in conventional setups. These results further demonstrate the versatility offered by atomic systems in the study of nonequilibrium phenomena and their promise for novel applications.Comment: 24 pages, 7 figures. A revie

Shape Parameterisation Based on Freeform Deformation in Aerodynamic Design Optimization

Shape parameterisation has been identified as an important task for aerodynamic design optimization. A method based on freeform deformation has been established, which enables in combination with grid generation software parameter-controlled variations of given shapes. The applicability and flexibility of the method for aerodynamic wing shape optimization has been demonstrated. For verification of the method airfoil recovery tests have been performed

Shape Parametrization Using Freeform Deformation

Shape parametrization has been identified as an import issue in aerodynamic design optimization based on high-fidelity CFD-methods. For given shapes, which are available as CAD-models, post-parametrization method, based on freeform deformation, has been established to simplify and to automate the generation of geometrical variants to be used for CFD analyses. To create the necessary deformation lattices, structured grid generation techniques of a grid generation system, developed at DLR, are utilized. As this grid generation system has the salient feature to store and to replay a sequence of processes with different parameter settings, modifications of shapes, given by polygonal curves and surfaces can be performed instantly. The present freeform deformation method has reached a state, where it can be integrated into design loops to handle a variety of shape optimization tasks. In two examples the applicability of the method for aerodynamic wing design and detailed design of a wing tip is demonstrated

Post-Parameterization of Complex CAD-Based Aircraft-Shapes Using Freeform Deformation

A post-parameterization method, based on free-form deformation techniques, has been established to simplify and to automate the generation of geometrical variants from existing CAD-models, to be used for CFD analyses. To create the necessary deformation lattices, structured grid generation techniques of a grid generation system, developed at DLR, are utilized. As this grid generation system has the salient feature to store and to replay a sequence of processes with different parameter settings, modifications of shapes, given by polygonal curves and surfaces, can be controlled quantitatively. Presently strategies and recipes are investigated for routinely use of this method in applied aerodynamics