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

Abstract

The Bogoliubov approximation is used to study the excited states of a dilute gas of $N$ atomic bosons trapped in an isotropic harmonic potential characterized by a frequency ${\omega }_0$ and an oscillator length $d_0=\sqrt{\hslash /m{\omega }_0}$. The self-consistent static Bose condensate has macroscopic occupation number $N_0\gg 1$, with nonuniform spherical condensate density $n_0(r)$; by assumption, the depletion of the condensate is small ($N^{\prime }\equiv N-N_0\ll N_0$). The linearized density fluctuation operator ${\hat{\rho }}^{\prime }$ and velocity potential operator ${\hat{\Phi }}^{\prime }$ satisfy coupled equations that embody particle conservation and Bernoulli's theorem. For each angular momentum $l$, introduction of quasiparticle operators yields coupled eigenvalue equations for the excited states; they can be expressed either in terms of Bogoliubov coherence amplitudes $u_l(r)$ and $v_l(r)$ that determine the appropriate linear combinations of particle operators, or in terms of hydrodynamic amplitudes ${\rho }_l^{\prime }(r)$ and ${\Phi }_l^{\prime }(r)$. The hydrodynamic picture suggests a simple variational approximation for $l>0$ that provides an upper bound for the lowest eigenvalue ${\omega }_l$ and an estimate for the corresponding zero-temperature occupation number $N_l^{\prime }$; both expressions closely resemble those for a uniform bulk Bose condensate.

arXiv comment5 pages, RevTeX, contributed paper accepted for Low Temperature Conference, LT21, August, 1996

References in corpus (1)

• Ground-State Properties of Magnetically Trapped Bose-Condensed Rubidium Gas