173 research outputs found
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
Giant vortices in combined harmonic and quartic traps
We consider a rotating Bose-Einstein condensate confined in combined harmonic
and quartic traps, following recent experiments [V. Bretin, S. Stock, Y. Seurin
and J. Dalibard, cond-mat/0307464]. We investigate numerically the behavior of
the wave function which solves the three-dimensional Gross Pitaevskii equation.
When the harmonic part of the potential is dominant, as the angular velocities
increases, the vortex lattice evolves into a giant vortex. We also
investigate a case not covered by the experiments or the previous numerical
works: for strong quartic potentials, the giant vortex is obtained for lower
, before the lattice is formed. We analyze in detail the three
dimensional structure of vortices
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
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]
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
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
Vortex Rings in Fast Rotating Bose-Einstein Condensates
When Bose-Eintein condensates are rotated sufficiently fast, a giant vortex
phase appears, that is the condensate becomes annular with no vortices in the
bulk but a macroscopic phase circulation around the central hole. In a former
paper [M. Correggi, N. Rougerie, J. Yngvason, {\it arXiv:1005.0686}] we have
studied this phenomenon by minimizing the two dimensional Gross-Pitaevskii
energy on the unit disc. In particular we computed an upper bound to the
critical speed for the transition to the giant vortex phase. In this paper we
confirm that this upper bound is optimal by proving that if the rotation speed
is taken slightly below the threshold there are vortices in the condensate. We
prove that they gather along a particular circle on which they are evenly
distributed. This is done by providing new upper and lower bounds to the GP
energy.Comment: to appear in Archive of Rational Mechanics and Analysi
Inhomogeneous Vortex Patterns in Rotating Bose-Einstein Condensates
We consider a 2D rotating Bose gas described by the Gross-Pitaevskii (GP)
theory and investigate the properties of the ground state of the theory for
rotational speeds close to the critical speed for vortex nucleation. While one
could expect that the vortex distribution should be homogeneous within the
condensate we prove by means of an asymptotic analysis in the strongly
interacting (Thomas-Fermi) regime that it is not. More precisely we rigorously
derive a formula due to Sheehy and Radzihovsky [Phys. Rev. A 70, 063620(R)
(2004)] for the vortex distribution, a consequence of which is that the vortex
distribution is strongly inhomogeneous close to the critical speed and
gradually homogenizes when the rotation speed is increased. From the
mathematical point of view, a novelty of our approach is that we do not use any
compactness argument in the proof, but instead provide explicit estimates on
the difference between the vorticity measure of the GP ground state and the
minimizer of a certain renormalized energy functional.Comment: 41 pages, journal ref: Communications in Mathematical Physics: Volume
321, Issue 3 (2013), Page 817-860, DOI : 10.1007/s00220-013-1697-
Shape oscillation of a rotating Bose-Einstein condensate
We present a theoretical and experimental analysis of the transverse monopole
mode of a fast rotating Bose-Einstein condensate. The condensate's rotation
frequency is similar to the trapping frequency and the effective confinement is
only ensured by a weak quartic potential. We show that the non-harmonic
character of the potential has a clear influence on the mode frequency, thus
making the monopole mode a precise tool for the investigation of the fast
rotation regime
Rapidly Rotating Fermions in an Anisotropic Trap
We consider a cold gas of non-interacting fermions in a two dimensional
harmonic trap with two different trapping frequencies ,
and discuss the effect of rotation on the density profile. Depending on the
rotation frequency and the trap anisotropy , the
density profile assumes two qualitatively different shapes. For small
anisotropy (), the
density consists of elliptical plateaus of constant density, corresponding to
Landau levels and is well described by a two dimensional local density
approximation. For large anisotropy (), the density profile is Gaussian in the strong confining
direction and semicircular with prominent Friedel oscillations in the weak
direction. In this regime, a one dimensional local density approximation is
well suited to describe the system. The crossover between the two regimes is
smooth where the step structure between the Landau level edges turn into
Friedel oscillations. Increasing the temperature causes the step structure or
the Friedel oscillations to wash out leaving a Boltzmann gas density profile.Comment: 14 pages, 7 figure
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