Scale Invariance and Viscosity of a Two-Dimensional Fermi Gas

We investigate collective excitations of a harmonically trapped two-dimensional Fermi gas from the collisionless (zero sound) to the hydrodynamic (first sound) regime. The breathing mode, which is sensitive to the equation of state, is observed with an undamped amplitude at a frequency 2 times the dipole mode frequency for a large range of interaction strengths and different temperatures. This provides evidence for a dynamical SO(2,1) scaling symmetry of the two-dimensional Fermi gas. Moreover, we investigate the quadrupole mode to measure the shear viscosity of the two-dimensional gas and study its temperature dependence.

Figure 1

(a) Frequency of the quadrupole mode at $T/T_F=0.47$ with $E_F=h\times (8.2\pm 0.7)\mathrm{kHz}$. The dotted line shows the theoretical prediction of 19 in the collisionless limit and the dashed line at ${\omega }_Q/{\omega }_{\perp }=\sqrt{2}$ is the hydrodynamic case. (b) Damping of the quadrupole mode. The solid line shows the fit for ${\Gamma }_0$ (zero sound) and the dashed line for ${\Gamma }_1$ (first sound), see text. The dash-dotted line is the damping rate of the dipole mode. (c) Frequency of the breathing mode. For the strong excitation we have $T/T_F=0.37$ and $E_F=h\times (5.4\pm 0.8)\mathrm{kHz}$ and for the weak excitation $T/T_F=0.42$ and $E_F=h\times (8.1\pm 1.1)\mathrm{kHz}$. (d) Damping of the breathing mode. The dash-dotted line is the damping rate of the dipole mode.

Figure 2

Difference between the breathing mode frequencies of the two different axes at the collisionless-hydrodynamic crossover in a slightly anisotropic trap. The insets show the correlation plot of the cloud widths along the two axes.

Figure 3

Temperature dependent damping of the quadrupole mode. (a) Damping rate as a function of temperature for various interaction strengths. $E_F/h$ for the data sets are (from lowest to highest temperature): 6.4, 8.2, 9.0, and 9.1 kHz. Inset: Derived values of $\alpha (T/T_F)$ and power-law fits. (b) Viscosity amplitude ${\alpha }_0$. (c) Power-law exponent $\beta$.