Collective excitations of a trapped Bose-condensed gas

TL;DR

This paper derives an explicit solution for collective excitations in a trapped Bose-condensed gas at zero temperature, revealing how interactions modify the excitation spectrum compared to the noninteracting case.

Contribution

The authors provide an analytical solution for the excitation spectrum of a trapped Bose gas including interactions, extending previous models by incorporating the effects of repulsive forces and trap deformation.

Findings

Derived the dispersion law for elementary excitations in a trapped Bose gas.

Compared the interacting case with the noninteracting harmonic oscillator model.

Estimated the effects of kinetic energy pressure using a sum rule approach.

Abstract

By taking the hydrodynamic limit we derive, at $T=0$, an explicit solution of the linearized time dependent Gross-Pitaevskii equation for the order parameter of a Bose gas confined in a harmonic trap and interacting with repulsive forces. The dispersion law $\omega=\omega_0(2n^2+2n\ell+3n+\ell)^{1/2}$ for the elementary excitations is obtained, to be compared with the prediction $\omega=\omega_0(2n+\ell)$ of the noninteracting harmonic oscillator model. Here $n$ is the number of radial nodes and $\ell$ is the orbital angular momentum. The effects of the kinetic energy pressure, neglected in the hydrodynamic approximation, are estimated using a sum rule approach. Results are also presented for deformed traps and attractive forces.