We provide complete phase diagrams describing the ground state of a trapped spinor BEC under the combined effects of rotation and a Rashba spin-orbit coupling. The interplay between the different parameters (magnitude of rotation, strength of the spi
n-orbit coupling and interaction) leads to a rich ground state physics that we classify. We explain some features analytically in the Thomas-Fermi approximation, writing the problem in terms of the total density, total phase and spin. In particular, we analyze the giant skyrmion, and find that it is of degree 1 in the strong segregation case. In some regions of the phase diagrams, we relate the patterns to a ferromagnetic energy.
Several misprints and small mistakes were in the initial version. They have been corrected. Following the recent experimental realization of synthetic gauge magnetic forces, Jean Dalibard adressed the question whether the adiabatic ansatz could be ma
th- ematically justified for a model of an atom in 2 internal states, shone by a quasi resonant laser beam. In this paper, we derive rigorously the asymptotic model guessed by the physicists, and show that this asymptotic analysis contains the in- formation about the presence of vortices. Surprisingly the main difficulties do not come from the nonlinear part but from the linear Hamiltonian. More precisely, the analysis of the nonlinear minimization problem and its asymptotic reduction to simpler ones, relies on an accurate partition of low and high frequencies (or mo- menta). This requires to reconsider carefully previous mathematical works about the adiabatic limit. Although the estimates are not sharp, this asymptotic analysis provides a good insight about the validity of the asymptotic picture, with respect to the size of the many parameters initially put in the complete model.
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 place
d 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.
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 trapp
ing 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.
We investigate the effect of the anisotropy of a harmonic trap on the behaviour of a fast rotating Bose-Einstein condensate. Fast rotation is reached when the rotational velocity is close to the smallest trapping frequency, thereby deconfining the co
ndensate in the corresponding direction. A striking new feature is the non-existence of visible vortices for the ground state. The condensate can be described with the lowest Landau level set of states, but using distorted complex coordinates. We find that the coarse grained atomic density behaves like an inverted parabola with large radius in the deconfined direction, and like a fixed Gaussian in the other direction. It has no visible vortices, but invisible vortices which are needed to recover the mixed Thomas-Fermi Gaussian profile. There is a regime of small anisotropy and intermediate rotational velocity where the behaviour is similar to the isotropic case: a hexagonal Abrikosov lattice of vortices, with an inverted parabola profile.
We investigate the effect of the anisotropy of a harmonic trap on the behaviour of a fast rotating Bose-Einstein condensate. This is done in the framework of the 2D Gross-Pitaevskii equation and requires a symplectic reduction of the quadratic form d
efining the energy. This reduction allows us to simplify the energy on a Bargmann space and study the asymptotics of large rotational velocity. We characterize two regimes of velocity and anisotropy; in the first one where the behaviour is similar to the isotropic case, we construct an upper bound: a hexagonal Abrikosov lattice of vortices, with an inverted parabola profile. The second regime deals with very large velocities, a case in which we prove that the ground state does not display vortices in the bulk, with a 1D limiting problem. In that case, we show that the coarse grained atomic density behaves like an inverted parabola with large radius in the deconfined direction but keeps a fixed profile given by a Gaussian in the other direction. The features of this second regime appear as new phenomena.
