Viscous relaxation and collective oscillations in a trapped Fermi gas near the unitarity limit

The viscous relaxation time of a trapped two-component gas of fermions in its normal phase is calculated as a function of temperature and scattering length, with the collision probability being determined by an energy-dependent $s$-wave cross section. The result is used for calculating the temperature dependence of the frequency and damping of collective modes studied in recent experiments, starting from the kinetic equation for the fermion distribution function with mean-field effects included in the streaming terms.

Figure 1

(Color online) The viscosity $\eta$ as a function of temperature for $k_{\mathrm{F}}\mid a\mid =4.5$, in units of ${\eta }_{\mathrm{cl}}\left(T_{\mathrm{F}}\right)=5{(mk{T}_{\mathrm{F}}∕\pi )}^{1∕2}∕32a^2$, the classical value of the viscosity for an energy-independent scattering cross section, evaluated at the Fermi temperature $T=T_{\mathrm{F}}$. The inset illustrates the low-temperature $T^{-2}$ dependence of the viscosity.

Figure 2

(Color online) The average viscous relaxation rate $1∕\tau$ divided by the transverse trap frequency ${\omega }_{\perp }$ as a function of temperature, for $k_{\mathrm{F}}\mid a\mid =0.01$. The asymptotic temperature dependences are indicated by the dashed lines. Note that the system is highly collisionless, since the maximum value of $1∕{\omega }_{\perp }\tau$ is about 0.000 25.

Figure 3

(Color online) The average viscous relaxation rate $1∕\tau$ divided by the transverse trap frequency ${\omega }_{\perp }$ as a function of temperature, for $k_{\mathrm{F}}\mid a\mid =5.5$, corresponding to the experiment of Ref. 6 at a magnetic field of $870\mathrm{G}$. The asymptotic temperature dependences are indicated by the dashed lines. The dotted line is the result obtained in the unitarity limit $\mid a\mid \to \infty$.

Figure 4

(Color online) The average viscous relaxation rate $1∕\tau$ divided by the transverse trap frequency ${\omega }_{\perp }$ as a function of $1∕k_{\mathrm{F}}\mid a\mid$ for four different temperatures. The parameters $T∕T_{\mathrm{F}}=0.03$ and $T∕T_{\mathrm{F}}=0.1$ correspond to the experimental conditions of Refs. 5, 7, respectively.

Figure 5

(Color online) The temperature dependence of the parameter $\xi$ given by Eq. (41) for different values of $k_{\mathrm{F}}a$. At high temperatures $\mid \xi \mid$ decreases as $T^{-5∕2}$.

Figure 6

(Color online) The calculated frequency of the transverse (+) mode as a function of temperature, with and without the mean-field correction for values of $\mid \xi \mid$ less than or equal to 0.5. The experimental values from Ref. 6 are indicated for comparison with the mean-field-corrected curve, with the estimated error bars included.

Figure 7

(Color online) The inverse damping rate of the transverse (+) mode as a function of temperature, with and without the mean-field correction. The mean-field-corrected curve is plotted for values of $\mid \xi \mid$ less than or equal to 0.5. The experimental values from Ref. 6 are indicated for comparison with the mean-field-corrected curve.

Figure 8

(Color online) The frequency of the axial (−) mode as a function of temperature.

Figure 9

(Color online) The inverse damping rate of the axial (−) mode as a function of temperature.