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 \ensuremath{\omega} and oscillator length ${\mathit{d}}_{0}$=\ensuremath{\surd}\ensuremath{\Elzxh}/m\ensuremath{\omega}. By assumption, the self-consistent condensate has a macroscopic occupation number ${\mathit{N}}_{0}$\ensuremath{\gg}1, with N-${\mathit{N}}_{0}$\ensuremath{\ll}${\mathit{N}}_{0}$. 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 \ensuremath{\rho}^\ensuremath{'} and velocity potential operator \ensuremath{\Phi}^\ensuremath{'}, along with the condensate density ${\mathit{n}}_{0}$ and velocity ${\mathbf{v}}_{0}$. For positive scattering length a and large stationary condensate (${\mathit{N}}_{0}$\ensuremath{\gg}${\mathit{d}}_{0}$/a and ${\mathbf{v}}_{0}$=0), the spherical condensate has a well-defined radius ${\mathit{R}}_{0}$\ensuremath{\gg}${\mathit{d}}_{0}$, 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 ${\mathit{E}}_{0\mathit{l}}$\ensuremath{\approxeq}${\mathit{E}}_{00}$+1/2${\mathrm{\ensuremath{\Elzxh}}}^{2}$l(l+1)/${\mathrm{mR}}_{0}^{2}$, with radial zero-point energy ${\mathit{E}}_{00}$\ensuremath{\propto}\ensuremath{\Elzxh}\ensuremath{\omega}(${\mathit{R}}_{0}$/${\mathit{d}}_{0}$${)}^{2/3}$=(${\mathrm{\ensuremath{\Elzxh}}}^{2}$ m${\mathrm{\ensuremath{\omega}}}^{4}$${\mathit{R}}_{0}^{2}$${)}^{1/3}$. \textcopyright{} 1996 The American Physical Society.