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Journal of Physics B Atomic Molecular and Optical Physics

Collective modes of ground and dark-soliton states of Bose Einstein condensates in anisotropic traps

M. S. Pindzola, Frederic Texier

Published Apr 14, 2003• 4 citations
Publisher

Abstract

Collective modes of the ground and excited states of Bose–Einstein condensates are studied by numerical solution of the time-dependent Gross–Pitaevskii equation in three spatial dimensions. Scissors modes are studied by quadrupole excitation of the ground and excited states of 23 Na condensates in a fully anisotropic magnetic trap. The frequencies of the collective modes for the ground state condensates are found to be in good agreement with previous linear response theory. New results are presented for the frequencies of the collective modes for excited state condensates which exhibit solitary density depressions. A key to the proper characterization of the wide variety of dark-soliton states in atomic condensates is the experimental observation of collective modes.

DOI
10.1088/0953-4075/36/9/310

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Collective modes of the ground and excited states of Bose–Einstein condensates are studied by numerical solution of the time-dependent Gross–Pitaevskii equation in three spatial dimensions. Scissors modes are studied by quadrupole excitation of the ground and excited states of 23 Na condensates in a fully anisotropic magnetic trap. The frequencies of the collective modes for the ground state condensates are found to be in good agreement with previous linear response theory. New results are presented for the frequencies of the collective modes for excited state condensates which exhibit solitary density depressions. A key to the proper characterization of the wide variety of dark-soliton states in atomic condensates is the experimental observation of collective modes.
Collective modes of the ground and excited states of Bose?Einstein condensates are calculated by diagonalization of the Bogoliubov?de Gennes matrix in cylindrical coordinates. For an isotropic magnetic trap, the first breathing mode frequencies of the ground and dark soliton excited states are found to be in excellent agreement with a direct solution of the Gross?Pitaevskii equation with a time-dependent driving potential. In addition, the matrix diagonalization yields densities and frequencies for second breathing modes and higher order multipole modes. For an axially symmetric anisotropic trap, a considerable number of collective modes of the ground and dark soliton excited states are calculated. Prolate traps are found to have many more low-frequency multipolar modes than found in oblate traps for both ground and dark soliton excited states.
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.
Abstract We investigate the collective mode of a self-gravitating Bose–Einstein condensate (BEC) described by the Gross–Pitaevskii–Poisson (GPP) equations. The self-gravitating BEC has garnered considerable attention in cosmology and astrophysics, being proposed as a plausible candidate for dark matter. Our inquiry delves into the breathing and anisotropic collective modes by numerically solving the GPP equations and using the variational method. The breathing mode demonstrates a reduction in period with increasing total mass due to the density dependence of the self-gravitating BEC, attributed to the density-dependent nature of self-gravitating BECs, aligning quantitatively with our analytical findings. Additionally, we investigate an anisotropic collective mode in which the quadrupole mode intertwines with the breathing mode. The period of the quadrupole mode exhibits similar total mass dependence to that of the breathing mode. The characteristics of these periods differ from those of a conventional BEC confined by an external potential. Despite the differences in density dependence, the ratio of their periods equals that of the BEC confined by an isotropic harmonic potential. Furthermore, an extension of the variational method to a spheroidal configuration enables the isolation of solely the quadrupole mode from the anisotropic collective mode.
We develop an analytical approach for calculating the Landau damping of collective modes in the Bose-Einstein condensate trapped by an anisotropic harmonic potential. Based on a variational ground-state wave function obtained by solving the time-independent Gross-Pitaevskii equation beyond Thomas-Fermi approximation, we solve the Bogoliubov--de Gennes equations describing thermal excited quasiparticles and provide divergence-free analytical solutions for the Bogoliubov amplitudes and coupling matrix elements that characterize the interaction between the collective modes and the quasiparticles. With these analytical results we evaluate the Landau damping rates of several collective modes for various anisotropic parameters of the trapping potential in terms of the formulas derived from a time-dependent mean-field theory. In addition, we discuss the dependence of the damping rates on temperature, particle number, trapping frequency, and anisotropic parameter of the system, and compare our theoretical results with experimental and numerical data reported in literature.
We theoretically investigate quadrupolar collective excitations in two-component Bose-Einstein condensates and their nonlinear dynamics associated with harmonic generation and mode coupling. Under the Thomas-Fermi approximation and the quadratic polynomial ansatz for density fluctuations, the linear analysis of the superfluid hydrodynamic equations predicts excitation frequencies of three normal modes constituted from monopole and quadrupole oscillations, and those of three scissors modes. These six modes are bifurcated into in-phase and out-of-phase modes by the intercomponent interaction, yielding the nonlinear dynamics that are absent in a single-component condensate. We obtain analytically the resonance conditions for the second-harmonic generation in terms of the trap aspect ratio and the strength of intercomponent interaction. The numerical simulation of the coupled Gross-Pitaevskii equations vindicates the validity of the analytical results and reveals the dynamics of the second-harmonic generation and nonlinear mode coupling that lead to nonlinear oscillations of the condensate with damping and recurrence reminiscent of the Fermi-Pasta-Ulam problem.
We experimentally investigate the collective excitation of 87 Rb Bose–Einstein condensates confined in a cigar-shaped magnetic trap (QUIC trap). Using a method of magnetic perturbation, the center-of-mass oscillation of the condensate is excited, so that the radial trapping frequency of the QUIC trap can be precisely determined. A high-order excitation, characterized by a fast shape oscillation, also occurs simultaneously, with a noticeable damping in the oscillation amplitude compared with the oscillation of the center of mass. The measured oscillation frequencies, associated with these two low-energy excitation modes, agree well with theoretical predictions based on the Gross–Pitaevskii equation.