Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions.
The transition temperature, the depletion of the condensate atoms and the collective excitations of a Bose–Einstein condensation (BEC) with two- and three-body interactions in an anharmonic trap at finite temperature are studied in detail. By using the Popov version of the Hartree–Fock–Bogoliubov (HFB) approximation, an extended self-consistent model describing BEC with both two- and three-body interactions in a distorted harmonic potential at finite temperature is obtained and solved numerically. The results show that the transition temperature, the condensed atom number and the collective excitations are modified dramatically by the atomic three-body interactions and the distortion of the harmonic trap.
We report results for the Thomas-Fermi ground state of a spin-polarized dipolar interacting Bose-Einstein condensate for the case when the external magnetic field
B is not orientated parallel to a principal axis but is aligned parallel to a symmetry plane of a harmonic anisotropic trap. For a dipole interaction strength parameter ${\ensuremath{\varepsilon}}_{D}\ensuremath{\ne}0$ the release energy of the condensate depends on the trap orientation angle ${\ensuremath{\vartheta}}_{T}$ between the principal axis
ez,T of the trap and the field
B. From the quasiclassical Josephson equation of macroscopic quantum physics we determine the low-lying eigenfrequencies of small-amplitude collective modes of the condensate density for various trap frequencies ${\ensuremath{\omega}}_{a}$ and trap orientation angles ${\ensuremath{\vartheta}}_{T}$. For the special case of a spherical harmonic trap with trap frequency $\ensuremath{\omega}$ it is rigorously shown for $\ensuremath{-}\frac{1}{2}<{\ensuremath{\varepsilon}}_{D}<1$ that a pure
s-wave symmetry breather excitation of the condensate density exists, that oscillates at a constant frequency ${\ensuremath{\Omega}}_{s}=\sqrt{5}\ensuremath{\omega}$ around the ground-state cloud, despite the well-known fact that the shape of the ground-state cloud of a spin-polarized dipolar condensate is for ${\ensuremath{\varepsilon}}_{D}\ensuremath{\ne}0$ not isotropic. For ${\ensuremath{\vartheta}}_{T}\ensuremath{\ne}0$ the small-amplitude modes of the particle density with isotropic and quadrupolar symmetry consist of two groups. There exist four modes that are combinations of basis functions, with
s-wave, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$- and
dz2-wave, and
dxz-wave symmetry, and two modes that are combinations of basis functions with
dyz- and
dxy-wave symmetry. A characteristic difference in the dependence of the frequencies of these six collective modes on the dipole interaction strength parameter ${\ensuremath{\varepsilon}}_{D}$ for prolate and oblate harmonic triaxial traps, respectively, is suggested to be used as an experimental method to measure the
s-wave scattering length
as of the atoms.
We discuss our numerical studies of the low energy excitations of trapped Bose condensates using a Bogoliubov-Hartree treatment.In the zero temperature limit, the lowest few excitation frequencies calculated within the Bogoliubov approximation agree well with the experimental data.Finite temperature results obtained using the Popov approximation display qualitative differences from the experimental data close to the critical temperature region.Details of our numerical approach are presented and comparison with other results is discussed.
The ground and excited states of a weakly interacting and dilute Bose-Einstein condensed gas, confined in a completely anisotropic harmonic oscillator potential, are determined at zero temperature within the Bogoliubov approximation. The numerical calculations employ a computationally efficient procedure based on a discrete variable representation (DVR) of the Hamiltonian. The DVR is efficient for problems where the interaction potential may be expressed as a local function of interparticle coordinates. In order to address condensates that are both very large ($\ensuremath{\sim}{10}^{6}$ atoms) and fully anisotropic, the ground state is found using a self-consistent field approach. Experience has demonstrated, however, that standard iterative techniques applied to the solution of the nonlinear partial differential equation for the condensate are nonconvergent. This limitation is overcome using the method of direct inversion in the iterated subspace (DIIS). In addition, the sparse structure of the DVR enables the efficient application of iterative techniques such as the Davidson and/or Lanczos methods, to extract the eigenvalues of physical interest. The results are compared with recent experimental data obtained for Bose-Einstein condensed alkali-metal vapors confined in magnetic traps.
Based on the energy functional and variational method, we present a new method to investigate the ground state properties for a weakly interacting Bose-condensed gas in an anisotropic harmonic trap at zero temperature. With this method we are able to find the analytic expression of the ground-state wavefunction and to explore the relevant quantities, such as energy, chemical potential, and the aspect ratio of the velocity distribution. These results agree well with previous ground state numerical solutions of the Gross–Pitaevskii equation given by Dalfovo et al. [Phys. Rev. A 53 (1996) 2477] This new method is simple compared to other methods used to solve numerically the Gross–Pitaevskii equation, and one can obtain analytic and reliable results.