Effect of the Casimir-Polder force on the collective oscillations of a trapped Bose-Einstein condensate

We calculate the effect of the interaction between an optically active material and a Bose-Einstein condensate on the collective oscillations of the condensate. We provide explicit expressions for the frequency shift of the center-of-mass oscillation in terms of the potential generated by the substrate and of the density profile of the gas. The form of the potential is discussed in detail and various regimes (van der Waals–London, Casimir-Polder, and thermal regimes) are identified as a function of the distance of atoms from the surface. Numerical results for the frequency shifts are given for the case of a sapphire dielectric substrate interacting with a harmonically trapped condensate of ${}^{87}\mathrm{R}\mathrm{b}$ atoms. We find that at distances of $4–8\mu \mathrm{m}$, where thermal effects become visible, the relative frequency shifts produced by the substrate are of the order ${10}^{-4}$ and hence accessible experimentally. The effects of nonlinearities due to the finite amplitude of the oscillation are explicitly discussed. Predictions are also given for the radial breathing mode.