Spin-orbit-coupled topological Fulde-Ferrell states of fermions in a harmonic trap

Motivated by recent experimental breakthroughs in generating spin-orbit coupling in ultracold Fermi gases using Raman laser beams, we present a systematic study of spin-orbit-coupled Fermi gases confined in a quasi-one-dimensional trap in the presence of an in-plane Zeeman field (which can be realized using a finite two-photon Raman detuning). We find that a topological Fulde-Ferrell state will emerge, featuring finite-momentum Cooper pairing and zero-energy Majorana excitations localized near the edge of the trap based on the self-consistent Bogoliubov–de Gennes (BdG) equations. We find analytically the wave functions of the Majorana modes. Finally, using the time-dependent BdG, we show how the finite-momentum pairing field manifests itself in the expansion dynamics of the atomic cloud.

Figure 1

(a) Density profiles $n_{\sigma }\left(x\right)$ in units of $n_0$ (left y axis) and order parameter profile $\left|\Delta \right(x\left)\right|$ in units of $E_F$ (right y axis). (b) The real and imaginary parts of the order parameter profile. The inset shows the pairing momentum $q$ as a function of $\nu$. (c) Low-lying spectrum of quasiparticle excitation. The inset shows the corresponding spectrum of a topologically trivial phase with $\nu =0$ and $h=0.5E_F$ with other parameters the same as those of the rest of the figure. (d) Wave function of the Majorana modes. The thin (thick) lines represent numerical (analytical) results. The parameters used for this figure are $\gamma =2.2$, $\lambda =1.5E_F/k_F$, $h=0.8E_F$, and $\nu =0.2E_F$ unless otherwise noted.

Figure 2

Density of states in real space (left panel) and in momentum space (right panel). The parameters are the same as in Fig. 1. The brighter color represents a higher density of states.

Figure 3

Expansion dynamics of the Fermi cloud released at time $t=0$. Top row: An interacting case with $\gamma =2.2$. Bottom row: A noninteracting case with $\gamma =0$. Other parameters are $\lambda =1.5E_F/k_F$, $h=0.8E_F$, and $\nu =0.2E_F$. From left to right are plots of density profiles at $t=0$, $0.88/{\omega }_0$, and $2.04/{\omega }_0$, respectively.

Figure 4

Center-of-mass position of the atomic cloud during free expansion. In (a), different curves represent different interaction strengths for $\nu =0.2E_F$. In (b), different curves represent different in-plane Zeeman fields for $\gamma =2.2$. Other parameters are the same as in Fig. 3.