In this thesis, we consider a Gross–Pitaevskii (GP) energy functional as a model for rotating one component and two components Bose–Einstein condensates (BEC) in two dimensions. This model can be non-dimensionalized to highlight a strong confinement regime with strong interaction between the two components. We introduce a new discretization of this energy, featuring both finite difference and fast Fourier transform approaches, in a bounded domain of R 2 using Dirichlet boundary conditions. We develop an explicit gradient method algorithm with adaptive step and projection (EPG) over the constraints manifold for the minimization of the discrete energy. This method allows for the derivation of a stopping criterion. Moreover, we propose two post processing algorithms for the numerical minimizers. One is aimed for single vortices while the other is aimed for vortex sheets. Both algorithms detect these structures and compute their indices. In a recent article [6], the authors study the behaviour of a segregated two components BEC set into rotation. They are able to prove that for large rotation, the interface between the components gets long, conjecturing the possibility towards vortex sheets. They also study the vortex structures of BEC in a segregated regime. In this thesis, we produce numerical simulations using EPG, validating these recent theoretical results, supporting conjectures, and covering different physical cases (the cases of one component and two components in coexistence regime [45, 39]). We also illustrate the efficiency of EPG compared to the well-known GPELab method [11] which requires to solve a linearly implicit system at each step. Finally, we adapt a few theorems found in the literature to our discrete setting. We prove the existence of a global minimizer of the GP energy functional for the finite difference scheme and study some of its properties. We also work on symmetrical problems we encountered in some of the numerical simulations.
In this thesis, we consider a Gross–Pitaevskii (GP) energy functional as a model for rotating one component and two components Bose–Einstein condensates (BEC) in two dimensions. This model can be non-dimensionalized to highlight a strong confinement regime with strong interaction between the two components. We introduce a new discretization of this energy, featuring both finite difference and fast Fourier transform approaches, in a bounded domain of R 2 using Dirichlet boundary conditions. We develop an explicit gradient method algorithm with adaptive step and projection (EPG) over the constraints manifold for the minimization of the discrete energy. This method allows for the derivation of a stopping criterion. Moreover, we propose two post processing algorithms for the numerical minimizers. One is aimed for single vortices while the other is aimed for vortex sheets. Both algorithms detect these structures and compute their indices. In a recent article [6], the authors study the behaviour of a segregated two components BEC set into rotation. They are able to prove that for large rotation, the interface between the components gets long, conjecturing the possibility towards vortex sheets. They also study the vortex structures of BEC in a segregated regime. In this thesis, we produce numerical simulations using EPG, validating these recent theoretical results, supporting conjectures, and covering different physical cases (the cases of one component and two components in coexistence regime [45, 39]). We also illustrate the efficiency of EPG compared to the well-known GPELab method [11] which requires to solve a linearly implicit system at each step. Finally, we adapt a few theorems found in the literature to our discrete setting. We prove the existence of a global minimizer of the GP energy functional for the finite difference scheme and study some of its properties. We also work on symmetrical problems we encountered in some of the numerical simulations.
This article introduces a new numerical method for the minimization under constraints of a discrete energy modeling multicomponents rotating Bose-Einstein condensates in the regime of strong confinement and with rotation. Moreover, we consider both segregation and coexistence regimes between the components. The method includes a discretization of a continuous energy in space dimension 2 and a gradient algorithm with adaptive time step and projection for the minimization. It is well known that, depending on the regime, the minimizers may display different structures, sometimes with vorticity (from singly quantized vortices, to vortex sheets and giant holes). In order to study numerically the structures of the minimizers, we introduce in this paper a numerical algorithm for the computation of the indices of the vortices, as well as an algorithm for the computation of the indices of vortex sheets. Several computations are carried out, to illustrate the efficiency of the method, to cover different physical cases, to validate recent theoretical results as well as to support conjectures. Moreover, we compare this method with an alternative method from the literature.
Simulation numérique de condensats de Bose-Einstein Dans cette thèse, nous considérons une fonctionnelle d'énergie Gross--Pitaevskii (GP) comme modèle pour la rotation d'un espèce et de deux espèces de condensats de Bose--Einstein (BEC) en deux dimensions. Ce modèle peut être adimensionné pour mettre en évidence un régime de confinement fort avec une forte interaction entre les deux composants. Nous introduisons une nouvelle discrétisation de cette énergie, comportant à la fois des schémas de différence finie et de Fourier, dans un domaine borné dans R² en utilisant des conditions aux bords de Dirichlet. Nous développons un algorithme de méthode de gradient explicite avec pas adaptatif et projection (EPG) sur la variété de contraintes pour la minimisation de l'énergie. Cette méthode permet de dériver un critère d'arrêt. Nous proposons également deux algorithmes de post-traitement pour les minimiseurs numériques. L'un est destiné aux vortex simples tandis que l'autre est destiné aux nappes de vortex. Les deux algorithmes détectent ces structures et calculent leurs indices.Dans un article récent intitulé "Vortex patterns and sheets in segregated two component Bose-Einstein condensates", les auteurs étudient le comportement d'un BEC ségrégué à deux espèces mis en rotation. Ils ont pu prouver que pour une grande rotation, l'interface entre les deux composants s'allonge, conduisant éventuellement à des nappes de vortex. Ils ont également étudié les structures des vortex du BEC dans un régime ségrégué. Dans cette thèse, nous avons pu produire des simulations numériques à l'aide d'EPG, validant ces résultats théoriques récents, supportant des conjectures et couvrant différents cas physiques (les cas d'un espèce et de deux espèces en régime de coexistence qui existent déjà dans la littérature mathématique). Nous illustrons également l'efficacité d'EPG par rapport à la méthode bien connue de GPELab qui consiste à résoudre un système linéairement implicite à chaque pas de temps.Enfin, nous avons pu adapter quelques théorèmes trouvés dans la littérature à notre problème discret. Nous prouvons l'existence d'un minimiseur global de la fonctionnelle d'énergie Gross--Pitaevskii pour le schéma aux différences finies et étudions certaines de ses propriétés. Nous travaillons également sur des problèmes symétriques rencontrés dans certaines simulations numériques.
In this paper, we mainly review recent results on mathematical theory andnumerical methods for Bose-Einstein condensation (BEC),based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are thencarried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models forspin-1 BEC, Bogoliubov excitation and BEC at finite temperature.
We study ground, symmetric and central vortex states, as well as their energy and chemical potential diagrams, in rotating Bose-Einstein condensates (BEC) analytically and numerically.We start from the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term, scale it to obtain a four-parameter model, reduce it to a 2D GPE in the limiting regime of strong anisotropic confinement and present its semiclassical scaling and geometrical optics.We discuss the existence/nonexistence problem for ground states (depending on the angular velocity) and find that symmetric and central vortex states are independent of the angular rotational momentum.We perform numerical experiments computing these states using a continuous normalized gradient flow (CNGF) method with a backward Euler finite difference (BEFD) discretization.Ground, symmetric and central vortex states, as well as their energy configurations, are reported in 2D and 3D for a rotating BEC.Through our numerical study, we find various configurations with several vortices in both 2D and 3D structures, energy asymptotics in some limiting regimes and ratios between energies of different states in a strong replusive interaction regime.Finally we report the critical angular velocity at which the ground state loses symmetry, numerical verification of dimension reduction from 3D to 2D, errors for the Thomas-Fermi approximation, and spourous numerical ground states when the rotation speed is larger than the minimal trapping frequency in the xy plane.
静态相耦合的Gross-Pitaevskii方程组的解描述了双原子的玻色爱因斯坦凝聚体在极低温度下的基态现象。我们提出一种十分有效的数值方法——梯度法来求解此基态解。我们提出的梯度法在数值上既保持总模量守恒又能使总能量递减；我们严格地证明我们提出的梯度法是一种获得能量函数在给定限制性条件下的最小值(也即基态解)的十分有效的方法。我们通过大量的例子来显示该方法的优点并且应用该方法去研究处于旋转状态下的双原子–玻色爱因斯坦凝聚体在极低温度下所呈现的复杂涡旋现象。 The ground states of rotating two-component Bose-Einstein condensates (BEC) at extremely low temperature are solutions of time-independent coupled Gross-Pitaevskii equations. To compute the ground states, we propose an efficient numerical method—gradient flows with discrete nor-malization. In linear cases and under properly chosen initial data, we can prove that the gradient flows converge into the ground state which has the lowest energy. We show that the method is quite efficient and apply the method to study complicated vortex structure in the ground state solutions of rotating two-component BEC at extremely low temperature.
We investigate vortex states of immiscible two-component Bose-Einstein condensates under rotation through numerical simulations of the coupled Gross-Pitaevskii equations. For strong intercomponent repulsion, the two components undergo phase separation to form several density domains of the same component. In the presence of the rotation, the nucleated vortices are aligned between the domains to make up winding chains of singly quantized vortices, a vortex sheet, instead of periodic vortex lattices. The vortices of one component are located at the region of the density domains of the other component, which results in the serpentine domain structure. The sheet configuration is stable as long as the imbalance of the intracomponent parameter is small. We employ a planar sheet model to estimate the distance between neighboring sheets, determined by the competition between the surface tension of the domain wall and the kinetic energy of the superflow via quantized vortices. By comparing the several length scales in this system, the phase diagram of the vortex state is obtained.