5,344 research outputs found
Semiclassical Corrections to a Static Bose-Einstein Condensate at Zero Temperature
In the mean-field approximation, a trapped Bose-Einstein condensate at zero
temperature is described by the Gross-Pitaevskii equation for the condensate,
or equivalently, by the hydrodynamic equations for the number density and the
current density. These equations receive corrections from quantum field
fluctuations around the mean field. We calculate the semiclassical corrections
to these equations for a general time-independent state of the condensate,
extending previous work to include vortex states as well as the ground state.
In the Thomas-Fermi limit, the semiclassical corrections can be taken into
account by adding a local correction term to the Gross-Pitaevskii equation. At
second order in the Thomas-Fermi expansion, the semiclassical corrections can
be taken into account by adding local correction terms to the hydrodynamic
equations
Dark soliton states of Bose-Einstein condensates in anisotropic traps
Dark soliton states of Bose-Einstein condensates in harmonic traps are
studied both analytically and computationally by the direct solution of the
Gross-Pitaevskii equation in three dimensions. The ground and self-consistent
excited states are found numerically by relaxation in imaginary time. The
energy of a stationary soliton in a harmonic trap is shown to be independent of
density and geometry for large numbers of atoms. Large amplitude field
modulation at a frequency resonant with the energy of a dark soliton is found
to give rise to a state with multiple vortices. The Bogoliubov excitation
spectrum of the soliton state contains complex frequencies, which disappear for
sufficiently small numbers of atoms or large transverse confinement. The
relationship between these complex modes and the snake instability is
investigated numerically by propagation in real time.Comment: 11 pages, 8 embedded figures (two in color
A particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas
The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is
generalized to apply to a gas with an exact large number of particles.
This generalization yields a description of the Schr\"odinger picture field
operators as the product of an annihilation operator for the total number
of particles and the sum of a ``condensate wavefunction'' and a phonon
field operator in the form when the field operator acts on the N particle subspace. It
is then possible to expand the Hamiltonian in decreasing powers of ,
an thus obtain solutions for eigenvalues and eigenstates as an asymptotic
expansion of the same kind. It is also possible to compute all matrix elements
of field operators between states of different N.Comment: RevTeX, 11 page
Solitary-wave description of condensate micro-motion in a time-averaged orbiting potential trap
We present a detailed theoretical analysis of micro-motion in a time-averaged
orbiting potential trap. Our treatment is based on the Gross-Pitaevskii
equation, with the full time dependent behaviour of the trap systematically
approximated to reduce the trapping potential to its dominant terms. We show
that within some well specified approximations, the dynamic trap has
solitary-wave solutions, and we identify a moving frame of reference which
provides the most natural description of the system. In that frame eigenstates
of the time-averaged orbiting potential trap can be found, all of which must be
solitary-wave solutions with identical, circular centre of mass motion in the
lab frame. The validity regime for our treatment is carefully defined, and is
shown to be satisfied by existing experimental systems.Comment: 12 pages, 2 figure
Bifurcations, order, and chaos in the Bose-Einstein condensation of dipolar gases
We apply a variational technique to solve the time-dependent Gross-Pitaevskii
equation for Bose-Einstein condensates in which an additional dipole-dipole
interaction between the atoms is present with the goal of modelling the
dynamics of such condensates. We show that universal stability thresholds for
the collapse of the condensates correspond to bifurcation points where always
two stationary solutions of the Gross-Pitaevskii equation disappear in a
tangent bifurcation, one dynamically stable and the other unstable. We point
out that the thresholds also correspond to "exceptional points," i.e. branching
singularities of the Hamiltonian. We analyse the dynamics of excited condensate
wave functions via Poincare surfaces of section for the condensate parameters
and find both regular and chaotic motion, corresponding to (quasi-)
periodically oscillating and irregularly fluctuating condensates, respectively.
Stable islands are found to persist up to energies well above the saddle point
of the mean-field energy, alongside with collapsing modes. The results are
applicable when the shape of the condensate is axisymmetric.Comment: 10 pages, 4 figures, minor changes in the text and additional
reference adde
The ground state of a Gross–Pitaevskii energy with general potential in the Thomas–Fermi limit
We study the ground state which minimizes a Gross–Pitaevskii
energy with general non-radial trapping potential, under the unit mass constraint, in the Thomas–Fermi limit where a small parameter tends to 0. This ground state plays an important role in the mathematical treatment of recent
experiments on the phenomenon of Bose–Einstein condensation, and in the study of various types of solutions of nonhomogeneous defocusing nonlinear Schrodinger equations. Many of these applications require delicate estimates
for the behavior of the ground state near the boundary of the condensate, as the singular parameter tends to zero, in the vicinity of which the ground state has irregular behavior in the form of a steep corner layer. In particular, the role of this layer is important in order to detect the presence of vortices in the small density region of the
condensate, understand the superfluid flow around an obstacle, and also has a leading order contribution in the energy. In contrast to previous approaches, we utilize a perturbation argument to go beyond the classical Thomas–Fermi
approximation and accurately approximate the layer by the Hastings–McLeod solution of the Painleve–II equation. This settles an open problem, answered very recently only for the special case of the model harmonic potential. In fact, we even improve upon previous results that relied heavily on the radial symmetry of the potential trap. Moreover, we show that the ground state has the maximal regularity available,
namely it remains uniformly bounded in the
1/2-Holder norm, which is the exact Holder regularity of the singular limit profile, as the singular parameter tends to zero. Our study is highly motivated by an interesting open problem posed recently by Aftalion, Jerrard, and Royo-Letelier, and an open question of Gallo and Pelinovsky,
concerning the removal of the radial symmetry assumption from the potential trap
Bosons in anisotropic traps: ground state and vortices
We solve the Gross-Pitaevskii equations for a dilute atomic gas in a magnetic
trap, modeled by an anisotropic harmonic potential. We evaluate the wave
function and the energy of the Bose Einstein condensate as a function of the
particle number, both for positive and negative scattering length. The results
for the transverse and vertical size of the cloud of atoms, as well as for the
kinetic and potential energy per particle, are compared with the predictions of
approximated models. We also compare the aspect ratio of the velocity
distribution with first experimental estimates available for Rb. Vortex
states are considered and the critical angular velocity for production of
vortices is calculated. We show that the presence of vortices significantly
increases the stability of the condensate in the case of attractive
interactions.Comment: 22 pages, REVTEX, 8 figures available upon request or at
http://anubis.science.unitn.it/~dalfovo/papers/papers.htm
Ground state solution of Bose-Einstein condensate by directly minimizing the energy functional
In this paper, we propose a new numerical method to compute the ground state
solution of trapped interacting Bose-Einstein condensation (BEC) at zero or
very low temperature by directly minimizing the energy functional via finite
element approximation. As preparatory steps we begin with the 3d
Gross-Pitaevskii equation (GPE), scale it to get a three-parameter model and
show how to reduce it to 2d and 1d GPEs. The ground state solution is
formulated by minimizing the energy functional under a constraint, which is
discretized by the finite element method. The finite element approximation for
1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical
symmetry are presented in detail and approximate ground state solutions, which
are used as initial guess in our practical numerical computation of the
minimization problem, of the GPE in two extreme regimes: very weak interactions
and strong repulsive interactions are provided. Numerical results in 1d, 2d
with radial symmetry and 3d with spherical symmetry and cylindrical symmetry
for atoms ranging up to millions in the condensation are reported to
demonstrate the novel numerical method. Furthermore, comparisons between the
ground state solutions and their Thomas-Fermi approximations are also reported.
Extension of the numerical method to compute the excited states of GPE is also
presented.Comment: 33 pages, 22 figure
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