Equation of State and Collective Frequencies of a Trapped Fermi Gas Along the BEC-Unitarity Crossover

We show that the study of the collective oscillations in a harmonic trap provides a very sensitive test of the equation of state of a Fermi gas near a Feshbach resonance. Using a scaling approach, whose high accuracy is proven by comparison with exact hydrodynamic solutions, the frequencies of the lowest compressional modes are calculated at $T=0$ in terms of a dimensionless parameter characterizing the equation of state. The predictions for the collective frequencies, obtained from the equations of state of mean-field BCS theory and of recent Monte Carlo calculations, are discussed in detail.

Figure 1

Chemical potential $\mu (n)$ (shifted by half the molecular binding energy ${\hslash }^2/2m{a}^2$) in units of the Fermi energy $E_F$ as a function of the coupling parameter $1/k_{F}a$. The Monte Carlo data are extracted from Ref. 7.

Figure 2

Reduced square ${\nu }^2={\omega }^2/{\Omega }^2$ of the lowest monopole frequency in isotropic trap for the MC and the MF-BCS equation of states as a function of $1/k_F(0)a$, where $k_F(0)$ is the Fermi wave vector at the trap center. Solid lines correspond to the results of all our methods, which cannot be distinguished at this scale. Inset: (uncorrected) quasipolynomial (dot-dashed line), corrected polytropic (solid line), and scaling (dashed line), all compared to the corrected quasipolynomial model.

Figure 3

Reduced square ${\nu }^2={\omega }^2/{\omega }_z^2$ of the axial mode frequency in very elongated trap for the MC and the MF-BCS equation of states, as a function of $k_F^{0}a$, where $k_F^0=1.7N^{1/6}/a_{\mathrm{ho}}$ is the Fermi wave vector of the ideal Fermi gas in a harmonic trap and $a_{\mathrm{ho}}=(\hslash /m\overline{\omega }{)}^{1/2}$, where $\overline{\omega }=({\omega }_x{\omega }_y{\omega }_z{)}^{1/3}$. Experimental results are from Ref. 17. The inset gives the relation between $k_F^0$ and $k_F(0)$, used in Fig. 2.