Concomitant modulated superfluidity in polarized Fermi gases

Recent ground-breaking experiments studying the effects of spin polarization on pairing in unitary Fermi gases encountered mutual qualitative and quantitative discrepancies which seem to be a function of the confining geometry. Using numerical algorithms we study the solution space for a three-dimensional fully self-consistent formulation of realistic systems with up to ${10}^5$ atoms. A study of the three types of solutions obtained demonstrates a tendency toward metastability as the confining geometry is elongated. One of these solutions, which is consistent with Rice experiments at high trap aspect ratio, supports a state strikingly similar to the long sought Fulde-Ferrel-Larkin-Ovchinnikov state. Our study helps to resolve the long-standing controversy concerning the discrepancies between the findings from two different experimental groups and highlights the versatility of actual-size numerical calculations for investigating inhomogeneous fermionic superfluids.

Figure 1

(a) Axial profiles of the gap (in units of $E_F$) showing the P-N (blue dashed line), P-SF (red solid line) andSF (green dotted line) states. The LDA solution (not shown) almost completely overlaps with the SF result. The free energies perparticle are: $0.67(0)E_F$, $0.65(8)E_F$, $0.65(5)E_F$ and $0.64(4)E_F$ for the P-N, P-SF, SF and LDA states, respectively. (b) Local polarization $p(\mathrm{r⃗})$ within the partially polarized region of the P-SF(red solid line) and P-N(blue dashed line) solutions. (c) An $r-z$ plot of the normalized density difference $\delta \rho =({\rho }_{\uparrow }-{\rho }_{\downarrow })/{\rho }_F$ of the partially polarized region of the P-SF state $({\rho }_{\text{F}}=\sqrt{(2E_F){}^3}/6{\pi }^2)$. All the results shown in this paper are obtained at a small temperature $T=0.02E_F/k_B$, and with $N=50\hspace{0.5em}000$, $\alpha =50$,and $P=0.3$.

Figure 2

Plots showing the doubly integrated axial spin density $\delta {\rho }_{1d}(z)$ for, from top down, the SF, P-N, and P-SF states shown in Fig. 1a.

Figure 3

Column densities (rescaled to have aspect ratio 5 for clarity) $\int {\rho }_{\uparrow }\mathit{dx}$, $\int {\rho }_{\downarrow }\mathit{dx}$, and $\int ({\rho }_{\uparrow }-{\rho }_{\downarrow })\mathit{dx}$ and the axial spin density $\delta {\rho }_{1d}$, respectively. The states represented are (a) the SF, (b) the P-SF, and (c) the P-N states illustrated in Fig. 1a. In (d) we plot the Rice experimental results for $N\approx 260\hspace{0.5em}000$, $P\approx 0.35$, $\alpha =45.23,$ and $T<0.05E_F/k_B$.