Collective modes of ground and excited states of Bose-Einstein condensates in axially symmetric anisotropic traps

Collective modes of the ground and excited states of Bose-Einstein condensates are calculated by diagonalization of the Bogoliubov-de Gennes matrix in cylindrical coordinates. For an isotropic magnetic trap, the first breathing mode frequencies of the ground and dark soliton excited states are found to be in excellent agreement with a direct solution of the Gross-Pitaevskii equation with a time-dependent driving potential. In addition, the matrix diagonalization yields densities and frequencies for second breathing modes and higher order multipole modes. For an axially symmetric anisotropic trap, a considerable number of collective modes of the ground and dark soliton excited states are calculated. Prolate traps are found to have many more low-frequency multipolar modes than found in oblate traps for both ground and dark soliton excited states.