A Variational Sum-Rule Approach to Collective Excitations of a Trapped Bose-Einstein Condensate

It is found that combining an excitation-energy sum rule with Fetter’s trial wave function gives almost exact low-lying collective-mode frequencies of a trapped Bose-Einstein condensate at zero temperature. Realizations of Bose-Einstein condensation (BEC) in trapped atomic gases1–4 have enabled us to study many-body physics of weakly interacting bosons with unprecedented precision. Experiments on elementary excitations of BEC5–8 present a new challenge because inhomogeneity and rotational symmetries of the system must be explicitly taken into account. A number of theoretical studies have been reported in literature.9–17 The results of numerical analyses based on the Bogoliubov approximation have so far found the best agreement with those of experiments. Stringari obtained exact analytic expressions of collective-mode frequencies in the Thomas-Fermi (TF) limit.20–24 Pérez-Garćıa et al. assumed a Gaussian trial wave function which yields correct collective-mode frequencies at the weak and strong coupling limits but overestimates them at an intermediate regime. In this Letter, it is found that combining an excitation-energy sum rule with Fetter’s trial wave function gives the almost exact frequencies of a trapped Bose system at zero temperature. Our method will be shown to yield collective-mode frequencies in excellent agreement with experimental results as well as numerical ones based on the Bogoliubov approximation.