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Physical Review A

Coexistence of Majorana zero modes and the Fulde-Ferrell-Larkin-Ovchinnikov state in a dipolar fermionic gas with laser-assisted interwire tunneling

Jinxin Li, Xing-Dong Zhao, Xue-Jing Feng, Ying-Ying Zhang, Jingxue Liu, Lu Qin, Zunlue Zhu, Wu-Ming Liu, Liang‐Liang Wang

Published May 16, 2025• 0 citations
Publisher

Abstract

Spin-orbit coupling (SOC) changes the Fermi surface and creates topological states with non-Abelian Majorana zero modes. However, it also drastically suppresses the Cooper pairing with finite momentum and thus the coexistence of Majorana zero modes and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is still challenging to be observed. In this work we introduce more complex interactions to the platform and investigate the unconventional superfluidity of a dipolar fermionic gas in a biwire configuration with the SOC induced by laser-assisted interwire tunneling. Employing a nonlocal self-consistent Bogoliubov--de Gennes formalism, we reveal a complex phase diagram featuring various topological-superfluid phases. In particular, due to the peculiar competition between the long-range dipolar interaction and the SOC, a phase separation structure emerges, exhibiting a stripe FFLO phase core encompassed by the topological-superfluid states. Moreover, the long-range dipolar interaction induces a remarkable oscillatory structure in the resultant Majorana zero modes at the FFLO phase boundary and enables direct manipulation of these zero modes. Finally, we provide an experimental proposal to realize our system and give the method to detect the novel quantum phase. The discovery of the Majorana zero modes at the interface between the FFLO state and the topological phase, along with their unique properties, deepens our understanding of Majorana fermions and enriches the zoo of Majorana quasiparticles.

DOI
10.1103/physreva.111.053311

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Spin-orbit coupling (SOC) changes the Fermi surface and creates topological states with non-Abelian Majorana zero modes. However, it also drastically suppresses the Cooper pairing with finite momentum and thus the coexistence of Majorana zero modes and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is still challenging to be observed. In this work we introduce more complex interactions to the platform and investigate the unconventional superfluidity of a dipolar fermionic gas in a biwire configuration with the SOC induced by laser-assisted interwire tunneling. Employing a nonlocal self-consistent Bogoliubov--de Gennes formalism, we reveal a complex phase diagram featuring various topological-superfluid phases. In particular, due to the peculiar competition between the long-range dipolar interaction and the SOC, a phase separation structure emerges, exhibiting a stripe FFLO phase core encompassed by the topological-superfluid states. Moreover, the long-range dipolar interaction induces a remarkable oscillatory structure in the resultant Majorana zero modes at the FFLO phase boundary and enables direct manipulation of these zero modes. Finally, we provide an experimental proposal to realize our system and give the method to detect the novel quantum phase. The discovery of the Majorana zero modes at the interface between the FFLO state and the topological phase, along with their unique properties, deepens our understanding of Majorana fermions and enriches the zoo of Majorana quasiparticles.
We systematically investigate unconventional superfluid phases of fermionic dipolar particles lying in a double-wire setup with laser-assisted interwire tunneling. Our numerical simulations, based on the nonlocal Kohn-Sham Bogoliubov-de Gennes equation, reveal the existence of a large Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) region with a stripe phase under an imbalance of particle densities between two wires. When the laser-assisted interwire tunneling is present, it induces a transition from the FFLO phase to the topological superfluid phase and the associated Majorana zero modes exhibit an oscillation structure, which is significantly enhanced by the long-range nature of the interwire dipolar interaction. This distinguishes itself from the results obtained with usual contact interaction and offers new opportunities for manipulating and reshaping Majorana zero modes by adjusting the degree of the nonlocality and the interwire separation.
Pairings are the heart for exploring novel superfluid physics of degenerate fermi gases. In the presence of spin-orbit coupling (SOC), some exotic pairings, including the triplet ppp-wave pairing, have been predicted, which paves a new way to search topological superfluids and associated Majorana fermions. In this work, we reveal unconventional pairings induced by SOC in a one-dimensional optical lattice, using the exact density-matrix renormalization group. In the weak SOC, we find the interband Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) and intraband Bardeen-Cooper-Schrieffer (BCS) pairings may coexist, when both bands are partially occupied. This interesting coexistence can lead to a new quantum state, called the FFLO-BCS phase, which exhibits a unique three-peak structure in pairing momentum distribution. However, in the intermediate and strong SOCs, the intraband BCS pairing dominates in the whole parameter regime. These results shed great light on the realization of topological superfluids in the spin-orbit coupled lattice.
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.
The recent experimental realization of spin-orbit (SO) coupling for ultracold atoms opens a completely new avenue for exploring new quantum matters. In experiments, the SO coupling is implemented simultaneously with a Zeeman field. Such SO-coupled Fermi gases are predicted to support Majorana fermions with non-Abelian exchange statistics in one dimension (1D). However, as shown in recent theory and experiments for 1D spin-imbalanced Fermi gases, the Zeeman field can lead to the long-sought Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids with nonzero center-of-mass momentum Cooper pairings, in contrast to the zero center-of-mass momentum pairings in Majorana superfluids. Therefore a natural question to ask is which phase, FFLO or Majorana superfluids, will survive in SO-coupled Fermi gases in the presence of a large out-of-plane Zeeman field. In this paper, we address this question by studying the mean-field quantum phases of 1D (quasi-1D) SO-coupled fermionic cold-atom optical lattices.
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids, Cooper pairings with finite momentum, and Majorana fermions (MFs), quasiparticles with non-Abelian exchange statistics, are two topics under intensive investigation in the past several decades, but unambiguous experimental evidence for them has not been found yet. Here we show that the recent experimentally realized shaken optical lattice provides a pathway to realize FFLO superfluids and MFs. By tuning the shaking frequency and amplitude, various couplings between the sss and ppp orbitals of the lattice (called the pseudospins) can be generated. We show that the combination of the sss- and ppp-band dispersion inversion, the engineered pseudospin coupling, and the on-site attractive interaction naturally allows the observation of FFLO superfluids as well as MFs in different parameter regions.
We study the phase diagram of a two-dimensional ultracold Fermi gas with the synthetic spin-orbit coupling (SOC) that has recently been realized at the National Institute of Standards and Technology (NIST). Because of the coexistence of anisotropic SOC and effective Zeeman fields in the NIST scheme, the system shows a rich structure of phase separation involving exotic gapless superfluid states and Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing states with different center-of-mass momenta. In particular, we characterize the stability region of FFLO states and demonstrate their unique features under SOC. We then show that the effective transverse Zeeman field in the NIST scheme can qualitatively change the landscape of the thermodynamic potential which leads to intriguing effects such as the disappearance of pairing instability, the competition between different FFLO states, and the stabilization of a fully gapped FFLO state. These interesting features may be probed, for example, by measuring the in situ density profiles or by the momentum-resolved radio-frequency spectroscopy.