Fulde-Ferrell-Larkin-Ovchinnikov state to topological superfluidity transition in bilayer spin-orbit-coupled degenerate Fermi gases

Recently a scheme has been proposed for generating the two-dimensional Rashba-type spin-orbit coupling (SOC) for ultracold atomic bosons in a bilayer geometry [S.-W. Su et al., Phys. Rev. A 93, 053630 (2016)]. Here we investigate the superfluidity properties of a degenerate Fermi gas affected by the SOC in such a bilayer system. We demonstrate that a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state appears in the regime of small to moderate atom-light coupling. In contrast to the ordinary SOC, the FFLO state emerges in the bilayer system without adding any external fields or spin polarization. As the atom-light coupling increases, the system can transit from the FFLO state to a topological superfluid state. These findings are also confirmed by the Bogoliubov–-de Gennes simulations with a weak harmonic trap added.

Figure 1

Schematic representation of a Fermi gas in a bilayer structure. An asymmetric double-well potential along the $z$ axis traps the atoms in two layers separated by a distance $d$. The combined spin-layer atomic states $\left|j=1,2,\gamma =\uparrow ,\downarrow \right.〉$ are cyclically coupled by the intralayer Raman transitions and interlayer laser-assisted tunneling characterized by the matrix elements $\mathrm{\Omega }{e}^{i\kappa (x-y)\pm i\phi }$ and $J{e}^{i\kappa (x+y)}$, respectively.

Figure 2

Single-particle dispersion of the lowest two branches for various coupling strengths and $\phi =\pi /2$. (a) In a weak coupling regime $\mathrm{\Omega }=J\ll E_{\mathrm{rec}}$, the dispersion is a superimposition of four distinct paraboloids nearly centered at $(\pm \kappa ,\pm \kappa )$. (b) In the strong coupling regime $\mathrm{\Omega }=J\gg E_{\mathrm{rec}}$, the Rashba-ring minimum (red dashed line ) with a radius $\kappa /2$ emerges.

Figure 3

(a) Real-space density profile (left panel) and phase configurations (middle panel) of the order parameters ${\mathrm{\Delta }}_{1,2}$ for $\mathrm{\Omega }=J=0.05E_{\mathrm{rec}}$ and $\phi =\pi /2$. Here we have taken $N_{\mathrm{atom}}=100$ and $E_b=2E_{\mathrm{rec}}$. The right panel shows the corresponding momentum-space distribution. (b) An illustration of the FFLO-type of the Cooper pairing mechanism. The red and blue solid arrows represent the Cooper pairing momenta of atoms in the first and second layers, with ${\mathbf{Q}}_{1,2}$ denoting the total paring momentum.

Figure 4

(a) Profiles of the order parameter $\mathrm{\Delta }\left(\mathbf{r}\right)$ (up) and the corresponding momentum distributions (down) in the first and second layers for the $\mathrm{\Omega }=J=0.2,0.6,1.3E_{\mathrm{rec}}$. Other parameters are $E_b=2E_{\mathrm{rec}}$, $\phi =\pi /2$, and $N_{\mathrm{atom}}=100$. (b) Evolution of the order parameter ${\mathrm{\Delta }}_{1,2}$ (maximum of the order parameter in the whole region) (blue solid line) and the magnitude of the FFLO vector $|{\mathbf{Q}}_{1,2}|$ (red dashed-dotted line) are plotted as a function of the interlayer tunneling strength $J=\mathrm{\Omega }$.

Figure 5

Phases of the bilayer system for $\mathrm{\Omega }=J$ and $\phi =0.6\pi$. (a) Phase diagram in the $(E_b,J)$ plane for $N_{\mathrm{atom}}=100$. Three phases are formed: a FFLO superfluid represented by a dark red region, a normal superfluid (NS, white region) and a topological superfluid (TS, yellow region). The latter phase is characterized by a nonzero Chern number $C$ and a zero pairing momentum $Q$. (b) Chemical potential $\mu$ as a function of the interlayer tunneling $J$ at the binding energy $E_b=1.0E_{\mathrm{rec}}$. (c)–(e) Two lowest branches of the single-particle dispersion for different values of $J$ corresponding to different phases. Here the chemical potential is represented by a gray plane. The dispersion branches become lower with an increase of J, leading to a decrease of the chemical potential, as one can see in (b)–(e). The NS phase transforms to the TS phase when the chemical potential enters the energy gap, as one can see comparing (d) and (e). In (c)–(e) the momentum is measured in the units of the recoil momentum $\kappa$.

Figure 6

Evolution of parameters of the system across the topological phase boundary. (a) Behavior of the minimum excitation gap $E_{\mathrm{g}}$ (blue dashed line) and the order parameter ${\mathrm{\Delta }}_{1,2}=\mathrm{\Delta }$ (red solid line) with increasing chemical potential for $E_b=3E_{\mathrm{rec}}$, $J\equiv \mathrm{\Omega }=2E_{\mathrm{rec}}$, and $\phi =3/4\pi$. (b) Minimum excitation gap and pairing order parameter against the tunneling phase $\phi$ for $E_b=3E_{\mathrm{rec}}$, $J\equiv \mathrm{\Omega }=2E_{\mathrm{rec}}$, and $\mu =-0.5E_{\mathrm{rec}}$. The yellow filled circles indicate the integer Chern number $C$ which is a topological invariant.

Figure 7

Density profiles, phase configurations, and corresponding momentum-space distributions of the order parameter of the first layer at several temperatures: $T=0.15T_F$ (upper panel), $T=0.45T_F$ (middle panel), and $T=0.7T_F$ (bottom panel). Other parameters are $\mathrm{\Omega }=J=0.4E_{\mathrm{rec}}$, $E_b=1E_{\mathrm{rec}}$, $\phi =0.6\pi$, and $N_{\mathrm{atom}}=100$. The characteristic temperature scale $T_F=E_F/k_B$ is given by the Fermi energy. The momentum is measured in the units of the recoil momentum $\kappa$.