Low-lying excitation modes of trapped dipolar Fermi gases: From the collisionless to the hydrodynamic regime

By means of the Boltzmann-Vlasov kinetic equation we investigate dynamical properties of a trapped one-component Fermi gas at zero temperature, featuring the anisotropic and long-range dipole-dipole interaction. To this end, we determine an approximate solution by rescaling both space and momentum variables of the equilibrium distribution, thereby obtaining coupled ordinary differential equations for the corresponding scaling parameters. Based on previous results on how the Fermi sphere is deformed in the hydrodynamic regime of a dipolar Fermi gas, we are able to implement the relaxation-time approximation for the collision integral. Then, we proceed by linearizing the equations of motion around the equilibrium in order to study both the frequencies and the damping of the low-lying excitation modes all the way from the collisionless to the hydrodynamic regime. Our theoretical results are expected to be relevant for understanding current experiments with trapped dipolar Fermi gases.

Figure 1

Low-lying collective excitation frequencies within a cylinder-symmetric trap for the monopole (${\mathrm{\Omega }}_{+}$), the three-dimensional quadrupole (${\mathrm{\Omega }}_{-}$), and the radial quadrupole modes (${\mathrm{\Omega }}_{\mathrm{rq}}$). They are presented in both the collisionless regime (upper/red curves) and the hydrodynamic regime (lower/blue curves) as a function of the dimensionless dipolar interaction strength ${\epsilon }_{\mathrm{dd}}$ for cigar-shaped ($\lambda <1$) and pancake-shaped ($\lambda >1$) traps.

Figure 2

Low-lying excitation mode frequencies within a cylinder-symmetric trap with respect to the relaxation time $\tau$ for the trap aspect ratio $\lambda =5$ and the dimensionless interaction strength ${\epsilon }_{\mathrm{dd}}=1.33$: (a) monopole mode (red curve) and (b) radial quadrupole (lower/green curve) and three-dimensional quadrupole mode (upper/blue curve).

Figure 3

Damping of monopole mode (upper/red curve), radial-quadrupole mode (middle/green curve), and three-dimensional quadrupole mode (lower/blue curve) plotted with the same values for the trap aspect ratio and the dimensionless interaction strength as in Fig. 2.

Figure 4

Ratio of dissipative peak height and difference of collisionless and hydrodynamic frequency for the radial quadrupole mode (green/upper curve) according to Eq. (62), the three-dimensional quadrupole mode (blue/middle curve), and the monopole mode (red/lower curve) as a function of the trap aspect ratio $\lambda$ at the dimensionless interaction strength ${\epsilon }_{\mathrm{dd}}=1.33$.