206 research outputs found
Three-dimensional vortex configurations in a rotating Bose Einstein condensate
We consider a rotating Bose-Einstein condensate in a harmonic trap and
investigate numerically the behavior of the wave function which solves the
Gross Pitaevskii equation. Following recent experiments [Rosenbuch et al, Phys.
Rev. Lett., 89, 200403 (2002)], we study in detail the line of a single
quantized vortex, which has a U or S shape. We find that a single vortex can
lie only in the x-z or y-z plane. S type vortices exist for all values of the
angular velocity Omega while U vortices exist for Omega sufficiently large. We
compute the energy of the various configurations with several vortices and
study the three-dimensional structure of vortices
Dissipative flow and vortex shedding in the Painlev\'e boundary layer of a Bose Einstein condensate
Raman et al. have found experimental evidence for a critical velocity under
which there is no dissipation when a detuned laser beam is moved in a
Bose-Einstein condensate. We analyze the origin of this critical velocity in
the low density region close to the boundary layer of the cloud. In the frame
of the laser beam, we do a blow up on this low density region which can be
described by a Painlev\'e equation and write the approximate equation satisfied
by the wave function in this region. We find that there is always a drag around
the laser beam. Though the beam passes through the surface of the cloud and the
sound velocity is small in the Painlev\'e boundary layer, the shedding of
vortices starts only when a threshold velocity is reached. This critical
velocity is lower than the critical velocity computed for the corresponding 2D
problem at the center of the cloud. At low velocity, there is a stationary
solution without vortex and the drag is small. At the onset of vortex shedding,
that is above the critical velocity, there is a drastic increase in drag.Comment: 4 pages, 4 figures (with 9 ps files
On the shape of vortices for a rotating Bose Einstein condensate
For a Bose-Einstein condensate placed in a rotating trap, we study the
simplified energy of a vortex line derived in Aftalion-Riviere Phys. Rev. A 64,
043611 (2001) in order to determine the shape of the vortex line according to
the rotational velocity and the elongation of the condensate. The energy
reflects the competition between the length of the vortex which needs to be
minimized taking into account the anisotropy of the trap and the rotation term
which pushes the vortex along the z axis. We prove that if the condensate has
the shape of a pancake, the vortex stays straight along the z axis while in the
case of a cigar, the vortex is bent
On mathematical models for Bose-Einstein condensates in optical lattices (expanded version)
Our aim is to analyze the various energy functionals appearing in the physics
literature and describing the behavior of a Bose-Einstein condensate in an
optical lattice. We want to justify the use of some reduced models. For that
purpose, we will use the semi-classical analysis developed for linear problems
related to the Schr\"odinger operator with periodic potential or multiple wells
potentials. We justify, in some asymptotic regimes, the reduction to low
dimensional problems and analyze the reduced problems
Rapidly Rotating Bose-Einstein Condensates in Strongly Anharmonic Traps
We study a rotating Bose-Einstein Condensate in a strongly anharmonic trap
(flat trap with a finite radius) in the framework of 2D Gross-Pitaevskii
theory. We write the coupling constant for the interactions between the gas
atoms as and we are interested in the limit (TF
limit) with the angular velocity depending on . We derive
rigorously the leading asymptotics of the ground state energy and the density
profile when tends to infinity as a power of . If
a ``hole'' (i.e., a region where the
density becomes exponentially small as ) develops for
above a certain critical value. If
the hole essentially exhausts the container and a ``giant vortex'' develops
with the density concentrated in a thin layer at the boundary. While we do not
analyse the detailed vortex structure we prove that rotational symmetry is
broken in the ground state for .Comment: LaTex2e, 28 pages, revised version to be published in Journal of
Mathematical Physic
Energy and Vorticity in Fast Rotating Bose-Einstein Condensates
We study a rapidly rotating Bose-Einstein condensate confined to a finite
trap in the framework of two-dimensional Gross-Pitaevskii theory in the strong
coupling (Thomas-Fermi) limit. Denoting the coupling parameter by 1/\eps^2
and the rotational velocity by , we evaluate exactly the next to
leading order contribution to the ground state energy in the parameter regime
|\log\eps|\ll \Omega\ll 1/(\eps^2|\log\eps|) with \eps\to 0. While the TF
energy includes only the contribution of the centrifugal forces the next order
corresponds to a lattice of vortices whose density is proportional to the
rotational velocity.Comment: 19 pages, LaTeX; typos corrected, clarifying remarks added, some
rearrangements in the tex
Asymptotic analysis of a secondary bifurcation of the one-dimensional Ginzburg-Landau equations of superconductivity
The bifurcation of asymmetric superconducting solutions from the normal solution is considered for the one-dimensional Ginzburg--Landau equations by the methods of formal asymptotics. The behavior of the bifurcating branch depends on the parameters d, the size of the superconducting slab, and , the Ginzburg--Landau parameter. The secondary bifurcation in which the asymmetric solution branches reconnect with the symmetric solution branch is studied for values of for which it is close to the primary bifurcation from the normal state. These values of form a curve in the -plane, which is determined. At one point on this curve, called the quintuple point, the primary bifurcations switch from being subcritical to supercritical, requiring a separate analysis. The results answer some of the conjectures of [A. Aftalion and W. C. Troy, Phys. D, 132 (1999), pp. 214--232]
A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates
Numerical computations of stationary states of fast-rotating Bose-Einstein
condensates require high spatial resolution due to the presence of a large
number of quantized vortices. In this paper we propose a low-order finite
element method with mesh adaptivity by metric control, as an alternative
approach to the commonly used high order (finite difference or spectral)
approximation methods. The mesh adaptivity is used with two different numerical
algorithms to compute stationary vortex states: an imaginary time propagation
method and a Sobolev gradient descent method. We first address the basic issue
of the choice of the variable used to compute new metrics for the mesh
adaptivity and show that simultaneously refinement using the real and imaginary
part of the solution is successful. Mesh refinement using only the modulus of
the solution as adaptivity variable fails for complicated test cases. Then we
suggest an optimized algorithm for adapting the mesh during the evolution of
the solution towards the equilibrium state. Considerable computational time
saving is obtained compared to uniform mesh computations. The new method is
applied to compute difficult cases relevant for physical experiments (large
nonlinear interaction constant and high rotation rates).Comment: to appear in J. Computational Physic
Vortex density models for superconductivity and superfluidity
We study some functionals that describe the density of vortex lines in
superconductors subject to an applied magnetic field, and in Bose-Einstein
condensates subject to rotational forcing, in quite general domains in 3
dimensions. These functionals are derived from more basic models via
Gamma-convergence, here and in a companion paper. In our main results, we use
these functionals to obtain descriptions of the critical applied magnetic field
(for superconductors) and forcing (for Bose-Einstein), above which ground
states exhibit nontrivial vorticity, as well as a characterization of the
vortex density in terms of a non local vector-valued generalization of the
classical obstacle problem.Comment: 34 page
Rotating superfluids in anharmonic traps: From vortex lattices to giant vortices
We study a superfluid in a rotating anharmonic trap and explicate a rigorous
proof of a transition from a vortex lattice to a giant vortex state as the
rotation is increased beyond a limiting speed determined by the interaction
strength. The transition is characterized by the disappearance of the vortices
from the annulus where the bulk of the superfluid is concentrated due to
centrifugal forces while a macroscopic phase circulation remains. The analysis
is carried out within two-dimensional Gross-Pitaevskii theory at large coupling
constant and reveals significant differences between 'soft' anharmonic traps
(like a quartic plus quadratic trapping potential) and traps with a fixed
boundary: In the latter case the transition takes place in a parameter regime
where the size of vortices is very small relative to the width of the annulus
whereas in 'soft' traps the vortex lattice persists until the width of the
annulus becomes comparable to the vortex cores. Moreover, the density profile
in the annulus where the bulk is concentrated is, in the 'soft' case,
approximately gaussian with long tails and not of the Thomas-Fermi type like in
a trap with a fixed boundary.Comment: Published version. Typos corrected, references adde
- …