Collective modes of a two-dimensional spin-$1/2$ Fermi gas in a harmonic trap

We derive analytical expressions for the frequency and damping of the lowest collective modes of a two-dimensional Fermi gas using kinetic theory. For strong coupling, we furthermore show that pairing correlations overcompensate the effects of Pauli blocking on the collision rate for a large range of temperatures, resulting in a rate which is larger than that of a classical gas. Our results agree well with experimental data, and they recover the observed crossover from collisionless to hydrodynamic behavior with increasing coupling for the quadruple mode. Finally, we show that a trap anisotropy within the experimental bounds results in a damping of the breathing mode which is comparable to what is observed, even for a scale-invariant system.

Figure 1

(a) Plot of damping $\Gamma$ versus frequency ${\omega }_{\text{QM}}$ for the quadrupole mode. The theory curve (solid line) relies only on the moment method and is independent of a microscopic model for the collision time $\tau$. The dots are the experimental data. (b) The collision rate $1/\tau$ as a function of interaction strength at $T/T_F=0.47$. The inset shows the relaxation rate as a function of $T$ at fixed interaction strength $\ln \left(k_F{a}_{2\text{D}}\right)=0.5$. All theory curves were calculated with (solid black line) and without (dashed blue line) Pauli blocking and with Pauli blocking and medium corrections (red dotted line). The collision rate including medium corrections diverges when $T^{*}$ as given by the Thouless criterion (indicated by the vertical dashed line) is approached from above.

Figure 2

(a),(b) Quadrupole-mode frequency (a) and damping (b) in units of ${\Omega }_{\perp }={\left({\omega }_x{\omega }_y\right)}^{1/2}$ compared to the experimental data of Ref. [12]. In the gray shaded area, the normal state is unstable towards pairing (within BCS mean-field theory). We have added an offset of $0.1{\Omega }_{\perp }$ to all damping theory curves to account for other sources of damping present in the experiment of Ref. [12]. The insets show the eigenmode frequencies and damping rates for an anisotropic trap (${\omega }_y=1.05{\omega }_x$), which leads to a better agreement between theory and experimental data.

Figure 3

Frequency (a) and damping (b) of the lowest collective modes of a two-component Fermi gas in a slightly anisotropic quasi-2D harmonic trap (${\omega }_y=1.05{\omega }_x$) normalized by the geometric mean of the frequencies ${\Omega }_{\perp }={\left({\omega }_x{\omega }_y\right)}^{1/2}$ as a function of collision time $\tau$. The thin gray lines in (a) denote the isotropic limits of the breathing- and quadrupole-mode frequencies ${\omega }_{B,Q}$ in the collisionless ${\omega }_{B,Q}/{\Omega }_{\perp }\equiv 2$ and the hydrodynamic ${\omega }_B/{\Omega }_{\perp }\equiv 2,{\omega }_Q/{\Omega }_{\perp }=\sqrt{2}$, regimes, respectively. In an anisotropic trap, the breathing (dotted red line) and quadrupole (solid black line) modes are coupled. The blue dashed curves correspond to the scissors mode (see the Appendix). Insets: (a) Decoupled eigenmodes in the collisionless regime [$\ln \left(k_F{a}_{2\mathrm{D}}\right)\gg 10$] start to lock towards the strongly interacting regime [$\ln \left(k_F{a}_{2\mathrm{D}}\right)\lesssim 5$], resulting in a single collective mode with strong breathing-mode character. (b) Same plots for the parameters of the experiment [12] as a function of interaction strength at $T=0.47T_F$ (here shown for the collision time obtained from the theory with Pauli blocking). The dots are the experimental data for the breathing-mode damping rate for two different excitation strengths [40].