2,725 research outputs found

    Ground state solution of Bose-Einstein condensate by directly minimizing the energy functional

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    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

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    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

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    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

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    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

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    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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