Excited states of a static dilute spherical Bose condensate in a trap


The Bogoliubov approximation is used to study the excited states of a dilute gas of NN atomic bosons trapped in an isotropic harmonic potential characterized by a frequency Ο‰0\omega_0 and an oscillator length d0=ℏ/mΟ‰0d_0 = \sqrt{\hbar/m\omega_0}. The self-consistent static Bose condensate has macroscopic occupation number N0≫1N_0 \gg 1, with nonuniform spherical condensate density n0(r)n_0(r); by assumption, the depletion of the condensate is small (N′≑Nβˆ’N0β‰ͺN0N' \equiv N - N_0\ll N_0). The linearized density fluctuation operator ρ^β€²\hat \rho' and velocity potential operator Ξ¦^β€²\hat\Phi' satisfy coupled equations that embody particle conservation and Bernoulli's theorem. For each angular momentum ll, introduction of quasiparticle operators yields coupled eigenvalue equations for the excited states; they can be expressed either in terms of Bogoliubov coherence amplitudes ul(r)u_l(r) and vl(r)v_l(r) that determine the appropriate linear combinations of particle operators, or in terms of hydrodynamic amplitudes ρlβ€²(r)\rho_l'(r) and Ξ¦lβ€²(r)\Phi_l'(r). The hydrodynamic picture suggests a simple variational approximation for l>0l >0 that provides an upper bound for the lowest eigenvalue Ο‰l\omega_l and an estimate for the corresponding zero-temperature occupation number Nlβ€²N_l'; both expressions closely resemble those for a uniform bulk Bose condensate.Comment: 5 pages, RevTeX, contributed paper accepted for Low Temperature Conference, LT21, August, 199

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