Signatures of macroscopic quantum coherence in ultracold dilute Fermi gases

We propose a double-well configuration for optical trapping of ultracold two-species Fermi-Bose atomic mixtures. Two signatures of macroscopic quantum coherence attributable to a superfluid phase transition for the Fermi gas are analyzed. The first signature is based upon tunneling of Fermi pairs when the power of the deconfining laser beam is significantly reduced. The second relies on the observation of interference fringes in a regime where the fermions are trapped in two sharply separated minima of the potential. Both signatures rely on small decoherence times for the Fermi samples, which should be possible by reaching low temperatures using a Bose gas as a refrigerator, and a bichromatic optical dipole trap for confinement, with optimal heat-capacity matching between the two species.

Figure 1

Bistable potential for optically trapped Fermi-Bose mixtures. The plots represent the equipotential surfaces above the minima of the potential $U_{\alpha }^{\min }$ by an amount $\Delta U_{\alpha }=0.5$, 2.5, 5, 10, $20\mathrm{nK}$ (from inner to outer shells, respectively) for Fermionic ${}^6\mathrm{Li}$ (top) and Bosonic ${}^{23}\mathrm{Na}$ (bottom). We assume a laser power of $P_1=10\mathrm{mW}$ at ${\lambda }_1=1064\mathrm{nm}$, $P_2∕P_1=2.5\times {10}^{-3}$ at ${\lambda }_2=532\mathrm{nm}$, and waists $w_1=w_2=10\mu \mathrm{m}$. The atomic transition wavelengths are ${\lambda }_{\mathrm{f}}=671\mathrm{nm}$ and ${\lambda }_{\mathrm{b}}=589\mathrm{nm}$.

Figure 2

Equipotential profiles in the $x\text{‐}y$ plane of $U_{\alpha }(x,y,0)$ (left), for $\Delta U_{\alpha }=0.5$, 2.5, 5, 10, 20, 30, 40, 50, 60, $70\mathrm{nK}$ (from inner to outer shells, respectively), and potential energy along the $x$ axis for the two species (right). The minima of the bistable potentials have been shifted to zero for the sake of comparison, their values being $U_{\mathrm{f}}^{\min }=-3.87\mu \mathrm{K}$ and $U_{\mathrm{b}}^{\min }=-3.31\mu \mathrm{K}$. All trap parameters as in Fig. 1.

Figure 3

Macroscopic coherence of the Fermi gas through tunneling experiments. Dependence of tunneling current amplitude per unit of available atoms $A_{\alpha }∕N_{\alpha }$ (solid line) and tunneling angular frequency $\Delta E_{\alpha }∕\hslash$ (dashed line) for ${}^{23}\mathrm{Na}$ and ${}^6\mathrm{Li}\text{‐}{}^6\mathrm{Li}$ Cooper pairs versus bias strength $b$. For the sake of comparison, amplitude and frequency of bosons have been multiplied by ${10}^5$. The trap parameters are as in Fig. 1.

Figure 4

Macroscopic coherence of the Fermi gas through interference experiments. Dependence of fringe spacing ${\ell }_{\alpha }$ for ${}^{23}\mathrm{Na}$ bosons and ${}^6\mathrm{Li}\text{‐}{}^6\mathrm{Li}$ Cooper pairs vs $P_2∕P_1$. We assume a time of flight $t=10\mathrm{ms}$.

Figure 5

Temperature of the cloud (dashed line), and critical temperatures $T_{\mathrm{BCS}}$, $T_{\mathrm{BEC}}$ (continuous lines) as a function of time $t$ in the case of a freely expanding ${}^6\mathrm{Li}$ cloud with $N_{\mathrm{F}}=3\times {10}^6$ atoms. The initial density is $n_0=3.5\times {10}^{13}{\mathrm{cm}}^{-3}$ and we use an elastic scattering length for fermions $a=-230\mathrm{nm}$ [1]. The dashed lines close to the critical temperature continuous lines are the asymptotic values for $T_{\mathrm{BCS}}$, $T_{\mathrm{BEC}}$ obtained by substituting $n\left(t\right)$ with $n_{\infty }\left(t\right)$ into Eqs. (10, 11).