Dark Solitons with Majorana Fermions in Spin-Orbit-Coupled Fermi Gases

Abstract

We show that a single dark soliton can exist in a spin-orbit-coupled Fermi gas with a high spin imbalance, where spin-orbit coupling favors uniform superfluids over non-uniform Fulde-FerrellLarkin-Ovchinnikov states, leading to dark soliton excitations in highly imbalanced gases. Above a critical spin imbalance, two topological Majorana fermions (MFs) without interactions can coexist inside a dark soliton, paving a way for manipulating MFs through controlling solitons. At the topological transition point, the atom density contrast across the soliton suddenly vanishes, suggesting a signature for identifying topological solitons. Solitons, topological defects arising from the interplay between dispersion and nonlinearity of underlying systems, are significant for many different physical branches [1][2][3][4]. The realization of cold atomic superfluids provides a clean and controllable platform for exploring soliton physics. In cold atomic gases, dark solitons represent quantum excitations of a superfluid with the superfluid order parameter vanishing at the soliton center in conjunction with a phase jump across the soliton. Dark solitons have been extensively investigated in cold atoms [5][6][7][8][9][10][11][12][13]. In particular, dark solitons have recently been experimentally observed in strongly interacting spin balanced Fermi gases [14], where the Cooper pairing wave-function has a phase jump across the soliton [15][16][17][18][19][20][21]. However, dark solitons in the presence of a large spin imbalance have not been well explored.With a large spin imbalance, the ground state of the superfluid is theoretically predicted to be the spatially non-uniform Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase [22] with finite momentum pairing in 1D and quasi-1D [23][24][25][26][27][28], which is partially verified by the experiment [29]. This spatially non-uniform phase does not support dark soliton excitations that usually occur in BCS-type uniform superfluids. On the other hand, uniform superfluids can exist in large spin imbalanced Fermi gases [30] in the presence of spin-orbit (SO) coupling. Since SO coupling for cold atoms has been experimentally generated recently for both bosons and fermions [31][32][33][34][35][36], a natural question is whether such SO coupled superfluids with large spin imbalances can also support dark solitons.More interestingly, it is well known that defects (vortices, edges, etc.) in SO coupled fermionic superfluids with large spin imbalances can accommodate Majorana fermions (MFs) [37][38][39][40][41][42] In this Letter, we address these two important questions by studying dark solitons in degenerate Fermi gases (DFGs) trapped in 1D harmonic potentials with the experimentally already realized SO coupling and spin imbalances. Here the spin imbalance is equivalent to a Zeeman field. In the absence of SO coupling, the FFLO state [22] with an oscillating order parameter amplitude is the ground state [24] with a large Zeeman field, which cannot support dark solitons. With SO coupling, we find (i...

Traveling Majorana Solitons in a Low-Dimensional Spin-Orbit-Coupled Fermi Superfluid

“…2. Already at zero velocity, the ABS splits into two branches due to the combined effects of spin-orbit coupling and effective Zeeman field [26,27]. With increasing the soliton velocity, the energy of the upper ABS gradually increases and merges into the quasi-particle scattering continuum at v h ≃ 0.22v F , which is much smaller than the pair-breaking velocity v pb ≃ 0.45v F .…”

“…Traveling Majorana solitons with fixed π phase step, if experimentally observed to oscillate inside a Fermi cloud, would be a smoking-gun proof of the existence of long-sought topological superfluids. We note that stationary dark solitons with Majorana fermions in a spin-orbit coupled Fermi gas were recently predicted [26,27]. However, the crucial issue raised in any practical manipulations, i.e., the fate of these solitons at a finite velocity of motion, was not addressed.…”

“…Majorana solitons can be created by imprinting a sharp phase jump [13,15]. The observation of their oscillations seems to be an experimental challenge, as the density profile of Majorana solitons remains flat [26,27]. One then has to measure the local pairing order parameter.…”

Bright solitons in a two-dimensional spin-orbit-coupled dipolar Bose-Einstein condensate

“…This achievement has ignited tremendous interest in this field because of the dramatic change in the single particle dispersion (induced by spin-orbit coupling) which in conjunction with the interaction leads to many exotic superfluids [35][36][37][38][39][40][41][42][43][44][45](also see [46][47][48][49][50][51][52][53] for review). Such change in dispersion also results in exotic solitons even when the interaction is contact (without dipole-dipole interactions), including 1D bright solitons [54][55][56][57][58][59][60] for a BEC with attractive contact interactions, 1D dark [61,62] and gap solitons [63][64][65] for a BEC with repulsive contact interactions, as well as 1D dark solitons for Fermi superfluids [66,67]. These solitons exhibit unique features that are absent without spin-orbit coupling, for instance, the plane wave profile with a spatially varying phase and the stripe profile with a spatially oscillating density for BECs, as well as the presence of Majorana fermions inside a soliton for Fermi superfluids.…”

Topological Fulde–Ferrell superfluids of a spin–orbit coupled Fermi gas

“…These intriguing ground states are mainly due to the appearance of a double well structure of the underlying single particle spectrum. In Fermi gases, the ground state of superfluids with SO coupling can become topological in the presence of out-of-plane Zeeman fields and such topological superfluids accommodate Majorana fermions in low dimensions 80,81,82,83,84,85,86,87,88 In this review, we will present some essential aspects of topological FF superfluids. In Sec.…”