Collective Excitations of a Dipolar Bose-Einstein Condensate

We have measured the effect of dipole-dipole interactions on the frequency of a collective mode of a Bose-Einstein condensate. At relatively large numbers of atoms, the experimental measurements are in good agreement with zero temperature theoretical predictions based on the Thomas-Fermi approach. Experimental results obtained for the dipolar shift of a collective mode show a larger dependency to both the trap geometry and the atom number than the ones obtained when measuring the modification of the condensate aspect ratio due to dipolar forces. These findings are in good agreement with simulations based on a Gaussian ansatz.

Figure 1

Influence of DDIs on the intermediate collective mode frequency. Free oscillations of the mode after an excitation in a trap set by $\varphi =27°$ ($\lambda ={\omega }_x/{\omega }_z=0.79$) are monitored by measuring the aspect ratio of the BEC, given by the two experimental TF radii along the $y$ and $z$ axes. The magnetic field is either vertical (black diamonds) or parallel to the horizontal imaging beam (red dots). We plot as well the best fits from damped sinusoidal forms. The residuals to the fit are shown below. Inset: The ellipse qualitatively represents the atomic cloud, and the thick arrows show that the TF radius along $y$ oscillates out of phase compared to the other two directions.

Figure 2

Variation with the magnetic field orientation of the vertical dipole mode frequency, as a function of the trap geometry. The experimental relative shifts ${\delta }_D$ are plotted versus the angle $\varphi$ (bottom axis) (see text) and versus the corresponding trap anisotropy $\lambda ={\omega }_x/{\omega }_z$ (top axis). A fit assuming a (constant) tensorial shift and a (trap geometry dependent) magnetic gradient origin for the shifts is also shown.

Figure 3

Effect of DDIs on the experimental relative shifts induced by a 90° $B$ field rotation, as a function of the trap geometry: ${\delta }_Q$ (red squares) is the shift of the intermediate collective mode frequency, while ${\delta }_{\sigma }$ (black dots) is the shift of the expanded BEC aspect ratio. Results of numerical simulations based on the TF approximation and with no adjustable parameter are also shown.

Figure 4

Variation with the BEC atom number of the relative shifts induced by a 90° $B$ field rotation, for two trap geometries [blue circles, $\varphi =27°$ ($\lambda =0.79$); red squares, $\varphi =26°$ ($\lambda =0.73$)]. (a) Variation of ${\delta }_Q$ and comparison with simulation (for $\varphi =26°$). (b) Variation of ${\delta }_{\sigma }$, with the corresponding results of our simulations. For both (a) and (b), TF predictions are recovered in the large atom number limit of our numerical simulations. The experimental intermediate collective mode frequency is also plotted (black diamonds). The black line is a guide for the eye.