2,725 research outputs found
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
Exact dynamics and decoherence of two cold bosons in a 1D harmonic trap
We study dynamics of two interacting ultra cold Bose atoms in a harmonic
oscillator potential in one spatial dimension. Making use of the exact solution
of the eigenvalue problem of a particle in the delta-like potential we study
time evolution of initially separable state of two particles. The corresponding
time dependent single particle density matrix is obtained and diagonalized and
single particle orbitals are found. This allows to study decoherence as well as
creation of entanglement during the dynamics. The evolution of the orbital
corresponding to the largest eigenvalue is then compared to the evolution
according to the Gross-Pitaevskii equation. We show that if initially the
center of mass and relative degrees of freedom are entangled then the
Gross-Pitaevskii equation fails to reproduce the exact dynamics and
entanglement is produced dynamically. We stress that predictions of our study
can be verified experimentally in an optical lattice in the low-tunneling
limit.Comment: 9 figures, 5 movies available on-lin
Coupled eigenmodes in a two-component Bose-Einstein condensate
We have studied the elementary excitations in a two-component Bose-Einstein
condensate. We concentrate on the breathing modes and find the elementary
excitations to possess avoided crossings and regions of coalescing oscillations
where both components of the condensates oscillate with same frequency. For
large repulsive interactions between the condensates, their oscillational modes
tend to decouple due to decreased overlap. A thorough investigation of the
eigenmodes near the avoided crossings is presented.Comment: Replacement, 17 pages, 9 figure
Stability of matter-wave solitons in a density-dependent gauge theory
We consider the linear stability of chiral matter-wave solitons described by
a density-dependent gauge theory. By studying the associated Bogoliubov-de
Gennes equations both numerically and analytically, we find that the stability
problem effectively reduces to that of the standard Gross-Pitaevskii equation,
proving that the solitons are stable to linear perturbations. In addition, we
formulate the stability problem in the framework of the Vakhitov-Kolokolov
criterion and provide supplementary numerical simulations which illustrate the
absence of instabilities when the soliton is initially perturbed.Comment: 12 pages, 4 figures. Comments are welcom
Two-Level discretization techniques for ground state computations of Bose-Einstein condensates
This work presents a new methodology for computing ground states of
Bose-Einstein condensates based on finite element discretizations on two
different scales of numerical resolution. In a pre-processing step, a
low-dimensional (coarse) generalized finite element space is constructed. It is
based on a local orthogonal decomposition and exhibits high approximation
properties. The non-linear eigenvalue problem that characterizes the ground
state is solved by some suitable iterative solver exclusively in this
low-dimensional space, without loss of accuracy when compared with the solution
of the full fine scale problem. The pre-processing step is independent of the
types and numbers of bosons. A post-processing step further improves the
accuracy of the method. We present rigorous a priori error estimates that
predict convergence rates H^3 for the ground state eigenfunction and H^4 for
the corresponding eigenvalue without pre-asymptotic effects; H being the coarse
scale discretization parameter. Numerical experiments indicate that these high
rates may still be pessimistic.Comment: Accepted for publication in SIAM J. Numer. Anal., 201
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