Ground state and excited states of a confined condensed Bose gas

Abstract

The Bogoliubov approximation is used to study the ground state and low-lying excited states of a dilute gas of N atomic bosons held in an isotropic harmonic potential characterized by frequency ω and oscillator length ${\mathit{d}}_0$=√ħ/mω. By assumption, the self-consistent condensate has a macroscopic occupation number ${\mathit{N}}_0$≫1, with N-${\mathit{N}}_0$≪${\mathit{N}}_0$. A linearized hydrodynamic description yields operator forms of the particle-conservation law and Bernoulli’s theorem, expressed in terms of the small density fluctuation operator ρ^′ and velocity potential operator Φ^′, along with the condensate density ${\mathit{n}}_0$ and velocity ${\mathbf{v}}_0$. For positive scattering length a and large stationary condensate (${\mathit{N}}_0$≫${\mathit{d}}_0$/a and ${\mathbf{v}}_0$=0), the spherical condensate has a well-defined radius ${\mathit{R}}_0$≫${\mathit{d}}_0$, and the low-lying excited states are irrotational compressional waves localized near the surface. Approximate variational energies ${\mathit{E}}_{0\mathit{l}}$ of the lowest radial modes (n=0) for successive values of orbital angular momentum l form a rotational band given by ${\mathit{E}}_{0\mathit{l}}$≊${\mathit{E}}_{00}$+1/2${\mathrm{ħ}}^2$l(l+1)/${\mathrm{mR}}_0^2$, with radial zero-point energy ${\mathit{E}}_{00}$∝ħω(${\mathit{R}}_0$/${\mathit{d}}_0$ ${)}^{2/3}$=(${\mathrm{ħ}}^2$ m${\mathrm{\omega }}^4$ ${\mathit{R}}_0^2$ ${)}^{1/3}$. © 1996 The American Physical Society.