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 N atomic bosons trapped in an isotropic harmonic potential characterized by a frequency omega(0) and an oscillator length d(0) = root HBAR/m omega(0). The self-consistent static Bose condensate has macroscopic occupation number N-0 >> 1, with nonuniform spherical condensate density n(0)(r); by assumption, the depletion of the condensate is small (N' = N - N-0 << N-0). The linearized density fluctuation operator <(rho)over cap>' and velocity potential operator <(Phi)over cap>' 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 vl(r) that determine the appropriate linear combinations of particle operators, or in terms of hydrodynamic amplitudes rho l'(r) and Phi l'(r). The hydrodynamic picture suggests a simple variational approximation for l > 0 that provides an upper bound for the lowest eigenvalue w(l) and an estimate for the corresponding zero-temperature occupation number N-l'; both expressions closely resemble those for a uniform bulk Bose condensate.*