Spin-dipole mode in a trapped Fermi gas near unitarity

We theoretically investigate the spin-dipole oscillation of a strongly interacting Fermi gas confined by a harmonic trapping potential. By using a diagrammatic strong-coupling theory combined with a local density approximation and a sum rule approach, we study the temperature dependence of the spin-dipole frequency near unitarity. The connection of the spin-dipole frequency with the spin susceptibility and the pairing correlations is exploited. While the spin-dipole frequency exactly coincides with the trap frequency in a noninteracting Fermi gas, it is shown to be strongly enhanced in the superfluid state, because of the suppression of the spin degree of freedom due to the spin-singlet Cooper-pair formation. In strongly interacting Fermi gases, such enhancement occurs even above the superfluid phase transition temperature, due to the strong pairing correlations.

Figure 1

Feynman diagram for the self-energy $\widehat{\mathrm{\Sigma }}$, as well as the many-body $T$ matrix $\widehat{\mathrm{\Gamma }}$ (double solid lines). The box represents a contact-type attractive interaction $-U$. The solid and dashed lines show the dressed and bare Green's functions $\widehat{G}$ and ${\widehat{G}}^0$, respectively.

Figure 2

Calculated spin-dipole frequency in a trapped unitary Fermi gas at finite temperature. The solid, dotted, and dashed lines represent the results of ETMA, BCS, and an ideal Fermi gas. The LDA critical temperatures of ETMA $\left(T_{\mathrm{c}}=0.29T_{\mathrm{F}}\right)$ and BCS $\left(T_{\mathrm{c}}=0.37T_{\mathrm{F}}\right)$ are shown, where $T_{\mathrm{F}}={\left(3N\right)}^{1/3}{\omega }_{\mathrm{tr}}$ and ${\omega }_{\mathrm{tr}}$ are the Fermi temperature in a two-component ideal gas and the trap frequency, respectively. The black circle shows the recent experimental result on a ${}^6\mathrm{Li}$ unitary Fermi gas [26].

Figure 3

Local spin susceptibility $\chi (r,T)$ at (a1) $T=0.1T_{\mathrm{F}}$ (superfluid phase), (b1) $T=0.4T_{\mathrm{F}}$, and (c1) $T=0.8T_{\mathrm{F}}$ in a trapped unitary Fermi gas. (a2), (b2), and (c2) show the corresponding local densities $n_{\uparrow }\left(r\right)+n_{\downarrow }\left(r\right)$. The inset in (a2) shows the LDA superfluid order parameter $\mathrm{\Delta }\left(r\right)$. The solid and dashed lines represent the results of the ETMA and of the mean-field BCS theory, respectively. ${\chi }_0(r,T)$ is the Pauli susceptibility in a homogeneous gas with the number density $n_{\sigma }(r,T)$ (see text). The vertical dotted lines indicate the boundary between the normal state and the superfluid state in each calculation.

Figure 4

Comparison of the spin-dipole frequencies at ${\left(k_{\mathrm{F}}{a}_s\right)}^{-1}=-0.6$ (dashed line), 0 (solid line), and 0.6 (dot-dashed line). $k_{\mathrm{F}}=\sqrt{2m{\omega }_{\mathrm{tr}}{\left(3N\right)}^{1/3}}$ is the LDA Fermi momentum.

Figure 5

Phase diagram of an attractively interacting trapped Fermi gas. The solid line shows the LDA superfluid critical temperature $T_{\mathrm{c}}$, below which the system undergoes the superfluid phase (“SF”). While the spin-dipole frequency can be explained by the Kohn (dipole) mode in the high-temperature region (“KM”), it is strongly enhanced at low temperatures (fast oscillation region, “FO”). Although there are no clear boundaries between “KM” and “FO,” the crossover between these two regimes can be characterized by the temperatures where ${\omega }_{\mathrm{SD}}/{\omega }_{\mathrm{tr}}=1.05$ (dotted line), 1.10 (dashed line), and 1.20 (long-dashed line). For comparison, we also plot the peak temperature $T_{\mathrm{p}}$ of the trap-averaged spin susceptibility shown in [46].

Figure 6

Calculated spin-dipole frequency ${\omega }_{\mathrm{SD}}^{\mathrm{cl}}$ in the classical atom-molecule mixture at $T=T_{\mathrm{p}}$. The inset shows the peak temperature $T_{\mathrm{p}}$ of the trap-averaged spin susceptibility [46], as a function of ${\left(k_{\mathrm{F}}{a}_s\right)}^{-1}$, in the strong-coupling limit.

Figure 7

(a) Calculated frequency shift $\delta {\omega }_{\mathrm{SD}}=({\omega }_{\mathrm{SD}}^{\mathrm{ETMA}}-{\omega }_{\mathrm{SD}}^{\mathrm{BCS}})/{\omega }_{\mathrm{SD}}^{\mathrm{ETMA}}$ due to the pairing correlations beyond the mean-field approximation near $T_{\mathrm{c}}$. (b) LDA superfluid order parameter $\mathrm{\Delta }(r=0)$ in the trap center.