Bose-Einstein condensate: critical velocities and energy diagrams in the Thomas-Fermi regime

For a Bose-Einstein condensate placed in a rotating trap and confined in the z axis, we set a framework of study for the Gross-Pitaevskii energy in the Thomas Fermi regime. We investigate an asymptotic development of the energy, the critical velocities of nucleation of vortices with respect to a small parameter \ep and the location of vortices. The limit \ep going to zero corresponds to the Thomas Fermi regime. The non-dimensionalized energy is similar to the Ginzburg-Landau energy for superconductors in the high-kappa high-field limit and our estimates rely on techniques developed for this latter problem. We also take the advantage of this similarity to develop a numerical algorithm for computing the Bose-Einstein vortices. Numerical results and energy diagrams are presented.Comment: 10pages 9 figure

Rotation of a Bose-Einstein Condensate held under a toroidal trap

The aim of this paper is to perform a numerical and analytical study of a rotating Bose Einstein condensate placed in a harmonic plus Gaussian trap, following the experiments of \cite{bssd}. The rotational frequency $\Omega$ has to stay below the trapping frequency of the harmonic potential and we find that the condensate has an annular shape containing a triangular vortex lattice. As $\Omega$ approaches $\omega$, the width of the condensate and the circulation inside the central hole get large. We are able to provide analytical estimates of the size of the condensate and the circulation both in the lowest Landau level limit and the Thomas-Fermi limit, providing an analysis that is consistent with experiment

Bifurcation problems for Ginzburg-Landau equations and applications to Bose Einstein condensates

3rd cycl

Pinning phenomena in the Ginzburg-Landau Model of Superconductivity

We study the Ginzburg-Landau energy of superconductors with a term a_\ep modelling the pinning of vortices by impurities in the limit of a large Ginzburg-Landau parameter \kappa=1/\ep. The function a_\ep is oscillating between 1/2 and 1 with a scale which may tend to 0 as $\kappa$ tends to infinity. Our aim is to understand that in the large $\kappa$ limit, stable configurations should correspond to vortices pinned at the minimum of a_\ep and to derive the limiting homogenized free-boundary problem which arises for the magnetic field in replacement of the London equation. The method and techniques that we use are inspired from those of Sandier-Serfaty (in which the case a_\ep \equiv 1 was treated) and based on energy estimates, convergence of measures and construction of approximate solutions. Because of the term a_\ep(x) in the equations, we also need homogenization theory to describe the fact that the impurities, hence the vortices, form a homogenized medium in the material.Comment: 40 page

A classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate

We classify the ground states and topological defects of a rotating two-component condensate when varying several parameters: the intracomponent coupling strengths, the intercomponent coupling strength and the particle numbers.No restriction is placed on the masses or trapping frequencies of the individual components. We present numerical phase diagrams which show the boundaries between the regions of coexistence, spatial separation and symmetry breaking. Defects such as triangular coreless vortex lattices, square coreless vortex lattices and giant skyrmions are classified. Various aspects of the phase diagrams are analytically justified thanks to a non-linear $\sigma$ model that describes the condensate in terms of the total density and a pseudo-spin representation