The collective excitations of Bose condensates in anisotropic axially symmetric harmonic traps are investigated in the hydrodynamic and Thomas-Fermi limit. We identify an additional conserved quantity, besides the axial angular momentum and the total energy, and separate the wave equation in elliptic coordinates. The solution is thereby reduced to the algebraic problem of diagonalizing finite dimensional matrices. The classical quasi-particle dynamics in the local density approximation for energies of the order of the chemical potential is shown to be chaotic. 03.75.Fi,05.30.Jp,32.80.Pj,67.90.+z The new Bose condensates of alkali atoms in magnetic traps [1][2][3] offer a unique way to investigate the low-lying collective excitations in Bose condensates [4][5][6][7][8][9][10][11][12][13][14][15][16]. Experimentally collective modes with a given symmetry have been excited by time-dependent modulations of the trapping potential, and their evolution has been followed in real time by measurements of the resulting shape oscillations of the condensates. The measurements performed so far have involved turning off the trap after a given time [4][5][6], but in the future they could even be performed non-destructively by elastic off-resonant light scattering [7]. Theoretically the collective modes have been analyzed by using the Bogoliubov-equations or by linearizing the time-dependent Gross-Pitaevskii equation around the time-independent condensate and solving these equations numerically [8,9] or analytically in various approximations [10][11][12][13][14][15][16]. Very good agreement between the numerical and the experimental results has been found.In a seminal paper Stringari [16] has shown how the coupled wave equations for the collective excitations are simplified in the hydrodynamic limit to become a single second-order wave equation for density waves, and he obtained analytical solutions for all its modes in spherically symmetric harmonic traps, and, remarkably, also for some of its modes in axially symmetric harmonic traps. The latter are particularly important, because all experiments have been performed with traps of this symmetry [4][5][6].In the present paper it is our goal to study in more detail by analytical means the hydrodynamic wave equation in the axially symmetric case. We wish to find an explanation why at least some analytical solutions have been possible in this case and intend to use this insight to construct more solutions in a systematic way.In principle the collective mode problem looks very different for isotropic and for axially symmetric traps: In the isotropic case the rotational symmetry ensures that angular momentum conservation gives two good quantum numbers, and therefore the wave equation is separable in spherical coordinates. For axial symmetry, however, only the axial component of angular momentum remains a good quantum number, besides the energy, and one may expect that the system, having three degrees of freedom, is not integrable. In fact, this expectation is born out for collect...
The dielectric formalism is used to set up an approximate description of a spatially homogeneous weakly interacting Bose gas in the collision-less regime, which is both conserving and gap-less, and has coinciding poles of the single-particle Green's function and the density autocorrelation function in the Bose-condensed regime. The approximation takes into account the direct and the exchange interaction in a consistent way. The fulfillment of the generalized Ward identities related to the conservation of particle-number and the breaking of the gauge-symmetry is demonstrated. The dynamics at long wavelengths is considered in detail below and above the phase-transition, numerically and in certain limits also in analytical approximations. The explicit form of the density autocorrelation function and the Green's function is exhibited and discussed.Comment: 26 pages revtex including 11 figure
The dynamics of quasi-particles in repulsive Bose condensates in a harmonic trap is studied in the classical limit. In isotropic traps the classical motion is integrable and separable in spherical coordinates. In anisotropic traps the classical dynamics is found, in general, to be nonintegrable. For quasi-particle energies E much smaller than the chemical potential µ, besides the conserved quasiparticle energy, we identify two additional nearly conserved phase-space functions. These render the dynamics inside the condensate (collective dynamics) integrable asymptotically for E/µ → 0. However, there coexists at the same energy a dynamics confined to the surface of the condensate, which is governed by a classical Hartree-Fock Hamiltonian. We find that also this dynamics becomes integrable for E/µ → 0 because of the appearance of an adiabatic invariant. For E/µ of order 1 a large portion of the phase-space supports chaotic motion, both, for the Bogoliubov Hamiltonian and its Hartree-Fock approximant. To exemplify this we exhibit Poincaré surface of sections for harmonic traps with the cylindrical symmetry and anisotropy found in TOP traps. For E/µ ≫ 1 the dynamics is again governed by the Hartree-Fock Hamiltonian. In the case with cylindrical symmetry it becomes quasi-integrable because the remaining small chaotic components in phase space are tightly confined by tori. 03.75Fi,67.40Db,03.65Sq
We present a quantum theory of low-lying excitations in a trapped Bose condensate with finite particle numbers. We find that even at zero temperature condensate number fluctuations and/or fluctuations of the excitation frequency due to quantum uncertainties of the mode occupation lead to a collapse of the collective modes due to dephasing. Coherent revivals of the collective excitations are predicted on a much longer timescale. Depletion of collective modes due to second-harmonic generation is discussed.
In the short wavelength limit the Bogoliubov quasiparticles of trapped Bose-Einstein condensates can be described as classical particles and antiparticles with dynamics in a mixed phase-space. For anisotropic parabolic traps we determine the location of the resonances and study the influence of the sharpness of the condensate surface on the appearance of chaos as the energy of the quasiparticles is lowered from values much larger than to values comparable with the chemical potential.Comment: 20 pages, 4 figure
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