Topological superfluid in one-dimensional spin-orbit-coupled atomic Fermi gases

We investigate theoretically the prospect of realizing a topological superfluid in one-dimensional spin-orbit-coupled atomic Fermi gases under a Zeeman field in harmonic traps. In the absence of spin-orbit coupling, it is well known that the system is either a Bardeen-Cooper-Schrieffer superfluid or an inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid. Here we show that with spin-orbit coupling it could be driven into a topological superfluid, which supports zero-energy Majorana modes. However, in the weakly interacting regime the topological superfluid prefers to stay at the trap edge, in contrast to a FFLO superfluid, which occurs near the trap center. As a result, it is unlikely to experimentally observe an inhomogeneous FFLO superfluid with topo-logical order without specifically tailoring the geometry or other parameters of the Fermi cloud.

Figure 1

Phase diagram at a given spin-orbit coupling $\lambda k_F/E_F=1$, determined from the behavior of the lowest eigenenergy of Bogoliubov quasiparticle spectrum $\min \left\{\right|E_{\eta }\left|\right\}$. As the Zeeman field increases, the system evolves from a conventional BCS superfluid to a topological superfluid and finally to a normal state. The insets on the top and bottom show the quasiparticle spectrum at $h/E_F=0.3$ and $0.5$, respectively.

Figure 2

Spatial dependence of the critical Zeeman field $h-h_c\left(x\right)$ (solid lines) and the superfluid order parameter $\Delta \left(x\right)$ (dot-dashed lines) at $\lambda k_F/E_F=1$ and at three Zeeman fields, $h/E_F=0.3$, $0.5$, and $0.8$. The hatching highlights the areas in which the atoms are in the topological superfluid state.

Figure 3

Wave functions of the paired Majorana modes at the inner wing of the trap, $x\simeq \pm 0.5x_F$: (a) energy $E_{\mathrm{ZES}}\simeq -3.3\times {10}^{-5}{E}_F$ and (b) $E_{\mathrm{ZES}}\simeq +3.3\times {10}^{-5}{E}_F$. Both modes satisfy the symmetry requirement for Majorana wave functions. The wave functions are in units of $a_{ho}^{-1/2}$. Here $h=0.5E_F$ and $\lambda k_F/E_F=1$. The system is in the topological superfluid phase.

Figure 4

Wave functions of the paired Majorana modes at the outer wing of the trap, $x\simeq \pm 1.1x_F$: (a) energy $E_{\mathrm{ZES}}\simeq -7.2\times {10}^{-10}{E}_F$ and (b) $E_{\mathrm{ZES}}\simeq +7.2\times {10}^{-10}{E}_F$. Other parameters are the same as in Fig. 3.

Figure 5

The spin-up and spin-down density distributions, $n_{\uparrow }\left(x\right)$ (dashed lines) and $n_{\uparrow }\left(x\right)$ (solid lines), and their difference $\Delta n\left(x\right)=n_{\uparrow }\left(x\right)-n_{\downarrow }\left(x\right)$ (dot-dashed lines) are shown at (a) the conventional superfluid phase, (b) topological superfluid state, and (c) normal state. The density distributions are in units of the Thomas-Fermi density $n_0=2N^{1/2}/\left(\pi a_{ho}\right)$. The spin-orbit coupling is $\lambda k_F/E_F=1$.

Figure 6

Linear contour plot (in arbitrary units) for the local density of states (a) of spin-up atoms ${\rho }_{\uparrow }(x,E)$ and (b) of spin-down atoms ${\rho }_{\downarrow }(x,E)$. The contributions from Majorana fermions are highlighted by circles. Here the Fermi cloud is in the topological superfluid state with parameters $h=0.5E_F$ and $\lambda k_F/E_F=1$. In the calculation, the $\delta$ function in ${\rho }_{\sigma }(x,E)$ has been simulated by a Lorentz distribution with a small energy broadening $\Gamma =0.01E_F$.

Figure 7

Phase diagram at a given Zeeman field $h=0.4E_F$, determined from the behavior of the lowest eigenenergy of Bogoliubov quasiparticle spectrum $\min \left\{\right|E_{\eta }\left|\right\}$. As the spin-orbit coupling increases, the system evolves from a FFLO superfluid to a topological superfluid. The inset shows the critical Zeeman field $h-h_c\left(x\right)$ and the order parameter $\Delta \left(x\right)$ at $\lambda k_F/E_F=0.3$, where the Fermi gas is in the FFLO superfluid state.

Figure 8

Density distributions and order parameter at $h=0.4E_F$ and at three different spin-orbit couplings: (a) $\lambda k_F/E_F=0.2$, (b) $0.6$, and (c) $1.0$. The density distribution $n_{\sigma }\left(x\right)$ is in units of the Thomas-Fermi density $n_0=2N^{1/2}/\left(\pi a_{ho}\right)$. The order parameter $\Delta \left(x\right)$ is in units of the Fermi energy $E_F$.