Probing the Collective Modes of Spherical Shell-Shaped Condensates with Quench Numerics

We explore the collective modes of Bose-Einstein condensates by numerical solution of the Gross-Pitaevskii equation with an external ``bubble trap'' potential ($V_{trap}=\sqrt{(r^2-\Delta)^2/4-\Omega^2}$) that can be continuously tuned between a thin spherical shell-shaped condensate (at large $\Delta$) and an ordinary spherical condensate in a harmonic trap (at $\Delta=\Omega=0$). We excite the condensate's collective modes by making a small sudden change to the trapping potential and analyzing the subsequent time evolution of the condensate wavefunction. We observe the evolution of the frequency of the low-lying collective modes between the limits of a thin-shell condensate and a filled-spherical condensate.