## (a) Spectrum of Bogoliubov excitations (red dots) for a homogeneous system with sharp boundaries, calculated for *J*_{z} = δ and *J* = 2 Δ^{(2)}

_{z}

**Figure 4.** (a) Spectrum of Bogoliubov excitations (red dots) for a homogeneous system with sharp boundaries, calculated for *J _{z}* = δ and

*J*= 2 Δ

^{(2)}. It exhibits a bulk gap Δ

_{bulk}= 0.18

*E*and a pair of zero-energy Majorana states with a residual splitting Δ

_{r}_{s}~ 10

^{−12}

*E*. (b) Evolution of the bulk gap amplitude Δ

_{r}_{bulk}as a function of the ratio

*J*/δ (red line), for

_{z}*J*= Δ

^{(2)}, with Δ

^{(2)}given by equation (20). The black line represents the prediction of second-order perturbation theory Δ

_{bulk}= 2 Δ

^{(2)}. (c) Density distribution along

*x*of a zero-energy Majorana state, in planes

*A*(red line) and

*B*(blue line, offset for clarity). In the strong coupling regime

*J*~ δ, the population in

*B*is not negligible. (d) Total density distribution along

*x*calculated at zero temperature. Majorana states are not visible in this almost uniform density profile.

**Abstract**

We propose an experimental implementation of a topological superfluid with ultracold fermionic atoms. An optical superlattice is used to juxtapose a 1D gas of fermionic atoms and a 2D conventional superfluid of condensed Feshbach molecules. The latter acts as a Cooper pair reservoir and effectively induces a superfluid gap in the 1D system. Combined with a spin-dependent optical lattice along the 1D tube and laser-induced atom tunnelling, we obtain a topological superfluid phase. In the regime of weak couplings to the molecular field and for a uniform gas, the atomic system is equivalent to Kitaev's model of a p-wave superfluid. Using a numerical calculation, we show that the topological superfluidity is robust beyond the perturbative limit and in the presence of a harmonic trap. Finally, we describe how to investigate some physical properties of the Majorana fermions located at the topological superfluid boundaries. In particular, we discuss how to prepare and detect a given Majorana edge state.