Anisotropic Excitation Spectrum of a Dipolar Quantum Bose Gas

We measure the excitation spectrum of a dipolar chromium Bose-Einstein condensate with Raman-Bragg spectroscopy. The energy spectrum depends on the orientation of the dipoles with respect to the excitation momentum, demonstrating an anisotropy that originates from the dipole-dipole interactions between the atoms. We compare our results with the Bogoliubov theory based on the local density approximation and, at large excitation wavelengths, with the numerical simulations of the time-dependent Gross-Pitaevskii equation. Our results show an anisotropy of the speed of sound.

Figure 1

The principle of the experiment and absorption pictures after a Bragg pulse. The two Bragg beams are coupled to the BEC with an angle $\varphi$, and transfer a momentum ${\hslash }_{\mathbf{q}}$ and an energy $hf$ to the excited fraction. Because of DDIs, the excitation spectrum depends on the angle $\theta$ between $\mathbf{q}$ and $\mathbf{B}$. The two absorption pictures, in false color, show the density after Bragg pulse and time of flight, and the solid lines indicate the direction of momentum transfer. In (a), the small value of $\varphi$ (14°), and hence of $q$, barely allows spatial separation of the excited fraction, contrary to the case of large $\varphi$ (82°) in (b).

Figure 2

Excitation spectra for $q{\xi }_0=0.8$ ($\varphi =14\mathrm{deg}$). The excited fraction $P_{\mathrm{exc}}$ is plotted versus the detuning frequency $f$ between the two Bragg beams for $\theta =0$ (filled circles) and $\theta =\pi /2$ (open circles). The two solid lines are asymmetric Gaussian fits to the data (see text). The error bars represent the 1-sigma statistical uncertainty associated to three measurements. The two spectra are shown separately below (a) $\theta =\pi /2$, (b) $\theta =0$ along with the results of our linear response theory (dashed lines), which has no adjustable parameters, and of our numerical simulations (diamonds).

Figure 3

(a) Energy spectra: the energy $ϵ(\mathbf{q})$ divided by $h$ is plotted as a function of Bragg-transferred momentum $q$. The circles correspond to the experimental data: for a given value of $q$, the top (bottom) circle corresponds to the parallel (perpendicular) case. The point at $q{\xi }_0=1.6$ corresponds to a four-photon process (see text). The two solid lines are the results of LDA calculations for parallel (top line) and orthogonal (bottom line) cases. The dashed line gives the results for a free particle. Inset: zoom for low $q$ values. (b) Relative shift $\Delta (q)$ of the excitation energies (see text) as a function of $q$. Black points: results of fits of the experimental data, with error bars. Solid line: results of LDA calculations. The diamonds show the results of numerical simulations. For (a) and (b), the vertical error bars represent $1\sigma$ statistical uncertainty, while the horizontal ones correspond to uncertainties on the values of $\varphi$.