BCS-BEC Crossover in a Gas of Fermi Atoms with a $p$-Wave Feshbach Resonance

We investigate unconventional superfluidity in a gas of Fermi atoms with an anisotropic $p$-wave Feshbach resonance. Including the $p$-wave Feshbach resonance as well as the associated three kinds of quasimolecules with finite orbital angular momenta $L_z=\pm 1,0$, we calculate the transition temperature of the superfluid phase. As one passes through the $p$-wave Feshbach resonance, we find the usual BCS-BEC crossover phenomenon. The $p$-wave BCS state continuously changes into the BEC of bound molecules with $L=1$. Our calculation includes the effect of fluctuations associated with Cooper pairs and molecules which are not Bose condensed.

Figure 1

(a) Particle-particle scattering matrix in the $t$-matrix approximation in terms of the nonresonant interaction $U$ (first line) and the $p$-wave Feshbach resonance $g_{\mathrm{r}}$ (second line). $G_0$ and $D_0$ are the single-particle Fermi and Bose Green functions, respectively. (b) Corrections to the thermodynamic potential originating from fluctuations in the $p$-wave Cooper channels (left diagram) and the Feshbach resonance (right diagram).

Figure 2

(a) $T_{\mathrm{c}}$ in the $p$-wave BCS-BEC crossover. $\uparrow \uparrow$: single-component Fermi gas, with $\overline{U}\equiv N{p}_{\mathrm{F}}^{2}U=0.4{\epsilon }_{\mathrm{F}}$. $\uparrow \downarrow$: two-component Fermi gas, with $\overline{U}=0.8{\epsilon }_{\mathrm{F}}$. Results for a narrow Feshbach resonance (${\overline{g}}_{\mathrm{r}}\equiv \sqrt{N{p}_{\mathrm{F}}^2}{g}_{\mathrm{r}}=0.6{\epsilon }_{\mathrm{F}}$) and a broad Feshbach resonance $({\overline{g}}_{\mathrm{r}}=5{\epsilon }_{\mathrm{F}})$ are shown. BCS shows a weak-coupling result in the two-component case with $\mu$ being fixed at the value at $\nu =2.5{\epsilon }_{\mathrm{F}}$. (b) Chemical potential $\mu (T_{\mathrm{c}})$ in a single-component Fermi gas. The solid and dashed lines show the results for a narrow and a broad Feshbach resonance, respectively. The inset shows $\mu (T_{\mathrm{c}})$ as a function of the threshold energy $2\nu$ for a narrow Feshbach resonance.

Figure 3

Numbers for various kinds of particles at $T_{\mathrm{c}}$ in a single-component Fermi gas. (a) Narrow Feshbach resonance and (b) broad Feshbach resonance. $N_{\mathrm{M}}\equiv N_{\mathrm{B}}+N_{\mathrm{C}}$, where $N_{\mathrm{B}}$ describes Feshbach molecules and $N_{\mathrm{C}}$ gives the contribution from Cooper pairs (stable and unstable).