Cold atoms with laser-induced spin-orbit (SO) interactions provide promising platforms to explore novel quantum physics, in particular the exotic topological phases, beyond natural conditions of solids. The past several years have witnessed important progresses in both theory and experiment in the study of SO coupling and novel quantum states for ultracold atoms. Here we review the physics of the SO coupled quantum gases, focusing on the latest theoretical and experimental progresses of realizing SO couplings beyond one-dimension (1D), and the further investigation of novel topological quantum phases in such systems, including the topological insulating phases and topological superfluids. A pedagogical introduction to the SO coupling for ultracold atoms and topological quantum phases is presented. We show that the so-called optical Raman lattice schemes, which combine the creation of the conventional optical lattice and Raman lattice with topological stability, can provide minimal methods with high experimental feasibility to realize 1D to 3D SO couplings. The optical Raman lattices exhibit novel intrinsic symmetries, which enable the natural realization of topological phases belonging to different symmetry classes, with the topology being detectable through minimal measurement strategies. We introduce how the non-Abelian Majorana modes emerge in the SO coupled superfluid phases which can be topologically nontrivial or trivial, for which a few fundamental theorems are presented and discussed. The experimental schemes for achieving non-Abelian superfluid phases are given. Finally, we point out the future important issues in this rapidly growing research field.
A Majorana zero bound mode exists in the vortex core of a chiral
p+ip superconductor or superfluid, which can be driven from an
s-wave pairing state by two-dimensional spin-orbit coupling. We propose here a novel scheme based on realistic cold atom platforms to generate two-dimensional spin-orbit interactions in a blue-detuned square optical lattice, and predict both the quantum anomalous Hall effect and chiral topological superfluid phase in the experimentally accessible parameter regimes. This work may open a new direction with experimental feasibility to observe non-Abelian topological orders in cold atom systems.
The experimental realizations of spin-orbit coupling (SOC) in cold atom gases put topological superfluids (TSs) under the spotlight. The topological protection of suggested superfluid models comes from the local antiunitary symmetries of ten Altland-Zirnbauer symmetry classes. In this paper we theoretically study a two-dimensional BDI model describing spin-orbit-coupled bilayer fermionic gases with
s-wave interaction to realize mirror-symmetry- (nonlocal and unitary) protected gapped TS, beyond the paradigms suggested before. This gapped phase not only shows linearly dispersive Majorana zero modes on mirror invariant one-dimensional boundaries, but also shows Majorana corner states to realize the second-order TS (TS2) with
s-wave pairing, which is explicitly different from the nonconventional pairings in the suggested systems to realize TS2. Generally, engineering the single-particle band structures should be easier than directly engineering the complicated pairings. Considering the tremendous progress in manipulating cold atom systems, our work provides a possible route (by engineering the single-particle physics, such as SOC) to realize mirror-symmetry-protected gapped TS and TS2 in cold fermionic systems in the future.
Recent advanced experimental implementations of optical lattices with highly tunable geometry open up new regimes for exploring quantum many-body states of matter that had not been accessible previously. Here we report that a topological fermionic superfluid with higher Chern number emerges spontaneously from
s-wave spin-singlet pairing in an orbital optical lattice when its geometry is tuned to explicitly break reflection symmetry. Qualitatively distinct from the conventional scheme that relies on higher partial-wave pairing, the crucial ingredient of our model is topology originating from mixing higher Wannier orbitals. It leads to unexpected changes in the topological band structure at the single-particle level, i.e., the bands are transformed from possessing two flux-$\ensuremath{\pi}$ Dirac points into a single quadratic touching point with flux $2\ensuremath{\pi}$. Based on such engineered single-particle bands, spin-singlet pairing of ultracold fermions arising from standard
s-wave attractive interaction is found to induce higher Chern number (Chern number of 2) and topologically protected chiral edge modes, all occurring at a higher critical temperatures in relative scales, potentially circumventing one of the major obstacle for its realization in ultracold gases.
Topological superfluid, new quantum matter that possesses gapless exotic excitations known as Majorana fermions, has attracted extensive attention recently. These excitations, which can encode topological qubits, could be crucial ingredients for fault-tolerant quantum computation. However, creating and manipulating multiple Majorana fermions remain an ongoing challenge. Loading a topologically protected system in multi-layer structures would be a natural and simple way to achieve this goal. Here we investigate the system of bilayer Fermi gases with spin-orbit coupling and show that the topological condition is significantly influenced by the inter-layer tunneling, yielding two novel topological phases, which support more Majorana Fermions. We demonstrate the existence of such novel topological phases and associated multiple Majorana fermions using bilayer Fermi gases trapped inside a harmonic potential. This research pave a new way for generating multiple Majorana fermions and would be a significant step towards topological quantum computation.
We propose a theoretical idea to reach the
p-wave superfluid phase in an ultracold Fermi gas. The key of our idea is that the pairing symmetry of a Fermi superfluid is fully dominated by the symmetry of the superfluid order parameter, which is essentially given by the product of a pair amplitude and a pairing interaction. Noting this, in our proposal, we first prepare a
p-wave pair amplitude not by using a
p-wave interaction, but by using the phenomenon that a
p-wave pair amplitude is induced in an
s-wave superfluid Fermi gas with antisymmetric spin-orbit interaction. In this case, although the system is still in the
s-wave superfluid state with the
s-wave superfluid order parameter, when one suddenly replaces the
s-wave interaction by an appropriate
p-wave one (which is possible in cold Fermi gases by using a Feshbach resonance technique), the product of the
p-wave interaction and the
p-wave pair amplitude that has already been prepared in the spin-orbit-coupled
s-wave superfluid state immediately gives a finite
p-wave superfluid order parameter. Thus, at least just after this manipulation, the system is in the
p-wave superfluid state, being characterized by the artificially produced
p-wave superfluid parameter. In this paper, to assess our idea, we evaluate the
p-wave pair amplitude in a spin-orbit-coupled
s-wave superfluid Fermi gas at
T=0. We determine the region where a large
p-wave pair amplitude is obtained in the phase diagram with respect to the strengths of the
s-wave pairing interaction and the spin-orbit coupling. We also discuss the accessibility of this optimal region on the viewpoint of the superfluid phase transition temperature. Since the achievement of a
p-wave superfluid Fermi gas is one of the most crucial issues in cold-atom physics, our proposal would be useful for this exciting challenge.