Probing Majorana fermions in spin-orbit-coupled atomic Fermi gases

We examine theoretically the visualization of Majorana fermions in a two-dimensional trapped ultracold atomic Fermi gas with spin-orbit coupling. By increasing an external Zeeman field, the trapped gas transits from a nontopological to a topological superfluid, via a mixed phase in which both types of superfluids coexist. We show that the zero-energy Majorana fermion, supported by the topological superfluid and localized at the vortex core, may be visible through (i) the core density and (ii) the local density of states, which are readily measurable in experiment. We present a realistic estimate on experimental parameters for ultracold ${}^{40}$K atoms.

Figure 1

(a) Phase diagram, along with the lowest eigenenergy of Bogoliubov spectrum. (b) Configuration of the mixed phase at $h=0.5E_F$. The local critical field is calculated in the vortex-free state. (c) The gap profile with and without a vortex at $h=0.5E_F$. (d), (e), and (f) Energy spectrum at $h/E_F=0.4$, $0.5$, and $0.6$ (where $E_F={\hslash }^2{k}_F^2/\left(2M\right)=\sqrt{N}\hslash {\omega }_{\perp }$ is the Fermi energy) in the presence of a single vortex. The color of the symbols indicates the mean radius $\sqrt{\langle r^2\rangle }/r_F$ [where $r_F={\left(4N\right)}^{1/4}\sqrt{\hslash /\left(M{\omega }_{\perp }\right)}$ is the Fermi radius] of the eigenstate, which is defined by $\langle r^2\rangle =\int r^2\left[\right|u_{\uparrow }{|}^2+|u_{\downarrow }{|}^2+|v_{\uparrow }{|}^2+|v_{\downarrow }{|}^2]d\mathbf{r}$. The color of the symbols changes from blue (dark gray) when the excited state is localized at the trap center to red (gray) when its mean radius approaches the Thomas-Fermi radius. The CdGM states are indicated by blue (dark gray) squares in (d) and (e), and by a solid line in (f). The dashed lines show the inner edge and outer edge branches.

Figure 2

Wave functions of the two lowest-energy modes in the full topological state at $h=0.6E_F$ [(a) and (b)] and in the partial topological state at $h=0.5E_F$ [(c) and (d)]. The wave functions are in units of $\sqrt{M{\omega }_{\perp }/\hslash }$.

Figure 3

Zeeman field dependence of spin-up (a) and spin-down (b) densities at the vortex core. The density is normalized by the Thomas-Fermi density $n_F=(\sqrt{N}/\pi )\sqrt{M{\omega }_{\perp }/\hslash }$. The insets show the core density distributions at $h=0.6E_F$. The red dotted-dashed lines show the result by excluding artificially the Majorana vortex-core state, whose contribution is shown by the shaded area.

Figure 4

Log-scale contour plot of the LDOS for spin-up (a) and spin-down (b) atoms at $h=0.6E_F$. Here we use $\Gamma =0.01E_F$. The LDOS is units of $n_F/E_F$.