18 research outputs found
Vortices and dynamics in trapped Bose-Einstein condensates
I review the basic physics of ultracold dilute trapped atomic gases, with
emphasis on Bose-Einstein condensation and quantized vortices. The hydrodynamic
form of the Gross-Pitaevskii equation (a nonlinear Schr{\"o}dinger equation)
illuminates the role of the density and the quantum-mechanical phase. One
unique feature of these experimental systems is the opportunity to study the
dynamics of vortices in real time, in contrast to typical experiments on
superfluid He. I discuss three specific examples (precession of single
vortices, motion of vortex dipoles, and Tkachenko oscillations of a vortex
array). Other unusual features include the study of quantum turbulence and the
behavior for rapid rotation, when the vortices form dense regular arrays.
Ultimately, the system is predicted to make a quantum phase transition to
various highly correlated many-body states (analogous to bosonic quantum Hall
states) that are not superfluid and do not have condensate wave functions. At
present, this transition remains elusive. Conceivably, laser-induced synthetic
vector potentials can serve to reach this intriguing phase transition.Comment: Accepted for publication in Journal of Low Temperature Physics,
conference proceedings: Symposia on Superfluids under Rotation (Lammi,
Finland, April 2010
The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate
We study the Gross-Pitaevskii (GP) energy functional for a fast rotating
Bose-Einstein condensate on the unit disc in two dimensions. Writing the
coupling parameter as 1 / \eps^2 we consider the asymptotic regime \eps
\to 0 with the angular velocity proportional to
(\eps^2|\log\eps|)^{-1} . We prove that if \Omega = \Omega_0
(\eps^2|\log\eps|)^{-1} and then a minimizer of
the GP energy functional has no zeros in an annulus at the boundary of the disc
that contains the bulk of the mass. The vorticity resides in a complementary
`hole' around the center where the density is vanishingly small. Moreover, we
prove a lower bound to the ground state energy that matches, up to small
errors, the upper bound obtained from an optimal giant vortex trial function,
and also that the winding number of a GP minimizer around the disc is in accord
with the phase of this trial function.Comment: 52 pages, PDFLaTex. Minor corrections, sign convention modified. To
be published in Commun. Math. Phy
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
The excitation spectrum for weakly interacting bosons in a trap
We investigate the low-energy excitation spectrum of a Bose gas confined in a
trap, with weak long-range repulsive interactions. In particular, we prove that
the spectrum can be described in terms of the eigenvalues of an effective
one-particle operator, as predicted by the Bogoliubov approximation.Comment: LaTeX, 32 page
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
Vortex lattice of a Bose-Einstein Condensate in a rotating anisotropic trap
We study the vortex lattices in a Bose-Einstein Condensate in a rotating
anisotropic harmonic trap. We first investigate the single particle
wavefunctions obtained by the exact solution of the problem and give simple
expressions for these wavefunctions in the small anisotropy limit. Depending on
the strength of the interactions, a few or a large number of vortices can be
formed. In the limit of many vortices, we calculate the density profile of the
cloud and show that the vortex lattice stays triangular. We also find that the
vortex lattice planes align themselves with the weak axis of the external
potential. For a small number of vortices, we numerically solve the
Gross-Pitaevskii equation and find vortex configurations that are very
different from the vortex configurations in an axisymmetric rotating trap.Comment: 15 pages,4 figure
Rapidly Rotating Atomic Gases
This article reviews developments in the theory of rapidly rotating
degenerate atomic gases. The main focus is on the equilibrium properties of a
single component atomic Bose gas, which (at least at rest) forms a
Bose-Einstein condensate. Rotation leads to the formation of quantized vortices
which order into a vortex array, in close analogy with the behaviour of
superfluid helium. Under conditions of rapid rotation, when the vortex density
becomes large, atomic Bose gases offer the possibility to explore the physics
of quantized vortices in novel parameter regimes. First, there is an
interesting regime in which the vortices become sufficiently dense that their
cores -- as set by the healing length -- start to overlap. In this regime, the
theoretical description simplifies, allowing a reduction to single particle
states in the lowest Landau level. Second, one can envisage entering a regime
of very high vortex density, when the number of vortices becomes comparable to
the number of particles in the gas. In this regime, theory predicts the
appearance of a series of strongly correlated phases, which can be viewed as
{\it bosonic} versions of fractional quantum Hall states. This article
describes the equilibrium properties of rapidly rotating atomic Bose gases in
both the mean-field and the strongly correlated regimes, and related
theoretical developments for Bose gases in lattices, for multi-component Bose
gases, and for atomic Fermi gases. The current experimental situation and
outlook for the future are discussed in the light of these theoretical
developments.Comment: Published version + minor correction
Dimers, Effective Interactions, and Pauli Blocking Effects in a Bilayer of Cold Fermionic Polar Molecules
We consider a bilayer setup with two parallel planes of cold fermionic polar
molecules when the dipole moments are oriented perpendicular to the planes. The
binding energy of two-body states with one polar molecule in each layer is
determined and compared to various analytic approximation schemes in both
coordinate- and momentum-space. The effective interaction of two bound dimers
is obtained by integrating out the internal dimer bound state wave function and
its robustness under analytical approximations is studied. Furthermore, we
consider the effect of the background of other fermions on the dimer state
through Pauli blocking, and discuss implications for the zero-temperature
many-body phase diagram of this experimentally realizable system.Comment: 18 pages, 10 figures, accepted versio
Finite vortex numbers and symmetric vortex structures in a rotating trapped Fermi gas in the BCS-BEC crossover
The ground state of a three-dimensional (3D) rotating trapped
superfluid Fermi gas in the BCS-BEC crossover is mapped to finite
Nv-body vortex states by a simple ansatz. The total vortex energy
is measured from the ground-state energy of the system in the
absence of the vortices. The vortex state is stable since the vortex
potential and rotation energies are attractive while the vortex
kinetic energy and interaction between vortices are repulsive. By
combining the analytical and numerical works for the minimal vortex
energy, the 2D configurations of Nv vortices are studied by
taking into account of the finite size effects both on xy-plane and
on z-direction. The calculated vortex numbers as a function of the
interaction strength are appropriate to the renew experimental
results by Zwierlein in [High-temperature superfluidity in a ultracold Fermi
gas, Ph.D. thesis, Massachusetts Institute of Technology, 2006].
The numerical results show that there exist two types of vortex
structures: the trap center is occupied and unoccupied by a vortex,
even in the case of Nv < 10 with regular polygon and in the case of
Nv ≥ 10 with finite triangle lattice. The rotation frequency
dependent vortex numbers with different interaction strengths are
also discussed