Hydrodynamic excitations of Bose condensates in anisotropic traps

Abstract

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...

Collective excitations of degenerate Fermi gases in anisotropic parabolic traps

“…Apart from a trivial factor 2/3 the operatorB differs from its bosonic counterpart found in [8] merely by a term proportional to x · ∇. The eigenvalue-problem ofB can be easily solved simultaneously with that ofL z by the ansatz, in cylindrical coordinates ρ, φ, z,…”

“…The expressions as given apply for the case ω ⊥ < ω z . The case ω⊥ ≥ ω can be treated similarly with results which can also be obtained by the analytical continuation of the results given here, see [8]. The wave-equation can be transformed to these coordinates.…”

“…It has turned out that within the dynamic Thomas-Fermi approximation these modes could be calculated analytically by solving the Gross-Pitaevskii equation, linearized around the static Thomas-Fermi solution. This was done first [6,7] for isotropic parabolic traps, where the symmetries are sufficient for integrability, then for anisotropic axially symmetric parabolic traps [6,8,9], where an additional hidden symmetry could be identified [8], and finally also for completely anisotropic parabolic traps [10], where the resulting wave-equation could be related to various integrable dynamical systems and the Lamé equation.…”

Hidden symmetry of collisionless sound of Bose condensates in anisotropic traps

“…The equations governing these collisionless sound modes in a weakly interacting system in mean field approximation can be written in the form of hydrodynamic equations [9,10]. For isotropic [9], axially symmetric [14] and even fully anisotropic harmonic traps [16] the solutions of these equations can be classified in terms of three quantum numbers. We consider these mode functions in Fourier space where we scale all momenta by k i =k i mω 2 i /2µ for i = x, y, z with chemical potential µ and trap frequency ω i in direction i.…”

“…Due to the anisotropy of the potential new features show up: the azimuthal quantum number m l is no longer a good quantum number, at energies of the order of µ the system ceases to be strictly integrable, and a large chaotic component can be seen in the classical phase space [13]. However in the limit of low energies (hω ≪ µ) the systems reduces to an integrable one [16] and an additional constant of motion can be identified [14] …”