Quadrupole oscillation in a dipolar Fermi gas: Hydrodynamic versus collisionless regime

The surface quadrupole mode of an harmonically trapped dipolar Fermi gas is studied in both the hydrodynamic and collisionless regimes. The anisotropy and long-range effects of the dipolar force as well as the role of the trapping geometry are explicitly investigated. In the hydrodynamic regime the frequency is always slightly smaller than the $\sqrt{2}{\omega }_{\perp }$ value holding for gases interacting with contact interactions. In the collisionless regime the frequency can be either pretty smaller or larger than the noninteracting value $2{\omega }_{\perp }$, depending on the cloud aspect ratio. Our results suggest that the frequency of the surface quadrupole oscillation can provide a useful test for studying, at very low temperatures, the transition between the normal and the superfluid phase and, in the normal phase at higher temperatures, the crossover between the collisional and collisionless regimes. The consequences of the anisotropy of the dipolar force on the virial theorem are also discussed.

Figure 1

Density-dependent dipolar parameter ${\varepsilon }_{dd}$ as a function of the externally controllable dipolar paramenter ${\varepsilon }_{dd}^0$.

Figure 2

Gas aspect ratio normalized by the trap aspect ratio $\kappa /\lambda$, as a function of ${\varepsilon }_{dd}$. (Left panel) For a cigar-shaped trapping potential; (right panel) for a pancake-shaped trapping potential. Dashed line marks instability as predicted by including exchange interactions (see, for instance, [15]).

Figure 3

(Top panels) Quadrupole frequency as a function of the dipolar strength ${\varepsilon }_{dd}$, both for the CL and the HD regimes and for different trap aspect ratios. (Bottom panel) Quadrupole frequency as a function of $\lambda$ for ${\varepsilon }_{dd}^0{\lambda }^{-1/6}=0.5$. Dashed line marks instability as predicted by including exchange interactions (see, for instance, [15]).

Figure 4

Anisotropy function $f\left(\kappa \right)$ and its derivatives, $h\left(\kappa \right)$ and $g\left(\kappa \right)$, that appear in the expressions of the quadrupole frequency, as a function of $\kappa$.