Main TextIn monolayer transition metal dichalcogenides (TMDs), there is a valley pseudospin 1/2 which describes the two inequivalent but energy degenerate band edges (the ±K valleys) at the corners of the hexagonal Brillouin zone 1 . With broken inversion symmetry, electrons in the two valleys can have finite orbital contributions to their magnetic moments which are equal in magnitude but opposite in sign by time reversal symmetry. This orbital magnetic moment is thus linked to the valley pseudospin in the same way that the bare magnetic moment ( S) is linked to the real spin S, where is the Bohr magneton and is the Lande -factor. The orbital magnetic moment in turn has two parts: a contribution from the parent atomic orbitals, and a "valley magnetic moment" contribution from the lattice structure 1 (Fig. 1a, [18][19][20][21][22][23] , are subject to a momentum-dependent gauge field arising from electron-hole exchange, or valley-orbit coupling, which at zero magnetic field is predicted to result in massless and massive dispersion respectively within the light cone 24 . This implies the possibility of controlling excitonic valley pseudospin via the Zeeman effect in an external magnetic field.Our measurements of polarization-resolved photoluminescence (PL) in a perpendicular magnetic field are performed on mechanically exfoliated WSe2 monolayers. We have obtained consistent results from many samples. The data presented here are all taken from one sample at a temperature of 30 K. In order to resolve the splitting between the +K and -K valley excitons, which is significantly smaller than the exciton linewidth (~10 meV), we both excite and detect with a single helicity of light. In this way we address one valley at a time, and the splitting can be determined by comparing the peak positions for different polarizations. The splitting in the applied magnetic field breaks the valley degeneracy, enabling control of the valley polarization. To investigate this we measure the degree of PL polarization for both helicities of incident circular polarization. Figure 2a shows PL for σ -excitation with σ -(red) and σ + (orange) detection at a field of -7 T. The suppression of the σ + signal relative to the co-polarized σ -peak is a signature of optically pumped valley polarization 7-10 . The degree of exciton valley polarization is clearly larger for σ + excitation than for − (Fig. 2b). On the other hand, when the magnetic field is reversed to +7 T (Figs. 2c and d) the polarization becomes larger for − . This observation implies that, while the sign of the valley polarization is determined by the helicity of the excitation laser, its magnitude depends on the relationship between the helicity and the magnetic field direction. Figure 2e shows the degree of PL polarization for both σ + (blue) and σ -(red) excitation as a function of B between -7 T and +7 T for the neutral exciton peak. It is linear in B with a negative (positive) slope. This "X" pattern implies that the valley Zeeman splitting induces an asymmetry in the intervalle...

Two-dimensional (p(x)+ip(y)) superfluids or superconductors offer a playground for studying intriguing physics such as quantum teleportation, non-Abelian statistics, and topological quantum computation. Creating such a superfluid in cold fermionic atom optical traps using p-wave Feshbach resonance is turning out to be challenging. Here we propose a method to create a p(x)+ip(y) superfluid directly from an s-wave interaction making use of a topological Berry phase, which can be artificially generated. We discuss ways to detect the spontaneous Hall mass current, which acts as a diagnostic for the chiral p-wave superfluid.

Spin-orbit coupled ultra-cold atoms provide an intriguing new avenue for the study of rich spin dynamics in superfluids. In this Letter, we observe Zitterbewegung, the simultaneous velocity (thus position) and spin oscillations, of neutral atoms between two spin-orbit coupled bands in a Bose-Einstein condensate (BEC) through sudden quantum quenches of the Hamiltonian. The observed Zitterbewegung oscillations are perfect on a short time scale but gradually damp out on a long time scale, followed by sudden and strong heating of the BEC. As an application, we also demonstrate how Zitterbewegung oscillations can be exploited to populate the upper spin-orbit band, and observe a subsequent dipole motion. Our experimental results are corroborated by a theoretical and numerical analysis and showcase the great flexibility that ultra-cold atoms provide for investigating rich spin dynamics in superfluids.PACS numbers: 67.85. De, 03.75.Kk, 67.85.Fg Introduction.-The Zitterbewegung (ZB) oscillation, first predicted by Schrödinger in 1930 [1] for relativistic Dirac electrons, describes the fast oscillation or trembling motion of electrons arising from the interference between particle and hole components of Dirac spinors. Although fundamentally important, the ZB oscillation is difficult to observe in real particles. In the past eight decades, analogs of the ZB oscillation have been predicted to exist in various physical systems [2-8], ranging from solid state (e.g., semiconductor quantum wells) to trapped cold atoms, but experimentally a ZB analog has only recently been observed using trapped ions as a quantum emulator of the Dirac equation [9]. A crucial ingredient for the ZB oscillation is the coupling between spin and linear momentum of particles, leading to simultaneous velocity and position oscillations accompanying the spin oscillation, which distinguishes ZB from Rabi oscillations where spin oscillations between two bands do not induce velocity and position oscillations. * These authors contributed equally to this work †

We investigate the BCS-BEC crossover in three-dimensional degenerate Fermi gases in the presence of spin-orbit coupling (SOC) and Zeeman field. We show that the superfluid order parameter destroyed by a large Zeeman field can be restored by the SOC. With increasing strengths of the Zeeman field, there is a series of topological quantum phase transitions from a nontopological superfluid state with fully gapped fermionic spectrum to a topological superfluid state with four topologically protected Fermi points (i.e., nodes in the quasiparticle excitation gap) and then to a second topological superfluid state with only two Fermi points. The quasiparticle excitations near the Fermi points realize the long-sought low-temperature analog of Weyl fermions of particle physics. We show that the topological phase transitions can be probed using the experimentally realized momentum-resolved photoemission spectroscopy.

We propose to use the recently predicted two-dimensional "weak-pairing" px + ipy superfluid state of fermionic cold atoms as a platform for topological quantum computation. In the core of a vortex, this state supports a zero-energy Majorana mode, which moves to finite energy in the corresponding topologically trivial "strong-pairing" state. By braiding vortices in the "weak-pairing" state, unitary quantum gates can be applied to the Hilbert space of Majorana zero modes. For readout of the topological qubits, we propose realistic schemes suitable for atomic superfluids.

Spin-orbit coupling (SOC), the interaction between the spin and momentum of a quantum particle, is crucial for many important condensed matter phenomena. The recent experimental realization of SOC in neutral bosonic cold atoms provides a new and ideal platform for investigating spin-orbit coupled quantum many-body physics. In this Letter, we derive a generic Gross-Pitaevskii equation as the starting point for the study of many-body dynamics in spin-orbit coupled Bose-Einstein condensates. We show that different laser setups for realizing the same SOC may lead to different mean-field dynamics. Various ground state phases (stripe, phase separation, etc.) of the condensate are found in different parameter regions. A new oscillation period induced by the SOC, similar to the Zitterbewegung oscillation, is found in the center-of-mass motion of the condensate.

We demonstrate that a Weyl point, widely examined in 3D Weyl semimetals and superfluids, can develop a pair of non-degenerate gapless spheres. Such a bouquet of two spheres is characterized by three distinct topological invariants of manifolds with full energy gaps, i.e., the Chern number of a 0D point inside one developed sphere, the winding number of a 1D loop around the original Weyl point, and the Chern number of a 2D surface enclosing the whole bouquet. We show that such structured Weyl points can be realized in the superfluid quasiparticle spectrum of a 3D degenerate Fermi gas subject to spin-orbit couplings and Zeeman fields, which supports Fulde-Ferrell superfluids as the ground state. [17,18]. A Weyl point can also be regarded as a monopole in 3D momentum space that exhibits an integer topological charge, i.e., the quantized first Chern number of a surface enclosing the singularity. Weyl points were also suggested to exist in the quasiparticle spectrum of superfluid 3 He A phase [2]. In contrast to traditional fully gapped superfluids, the Weyl superfluids bear doubly degenerate nodes pinned to zero energy, around which the quasiparticle energies disperse linearly in all directions. Most recently, the existence of such Weyl nodes has also been generalized to various coldatom superfluids and solid-state superconductors [19][20][21][22][23][24][25][26][27][28].In this Letter, we investigate whether a Weyl point can develop a nontrivial structure at zero energy and whether there exist any topological property protecting the developed structure. (i) We first consider a toy model to examine the possibility for a Weyl point to develop a gapless structure. Mathematically, the structured Weyl point can be viewed as a bouquet of two spheres (or wedge of two spheres) [29], which is a new class of topological state that has not been explored previously. Amazingly, the zero-energy bouquet is characterized by three distinct topological invariants: the first Chern number of a surface enclosing the whole bouquet, the zeroth Chern numbers of the interiors of the two spheres, and the winding number of a loop enclosing the touching point in the plane of symmetry. (ii) We further show that the structured Weyl points can be physically realized in the quasiparticle excitation spectrum of a 3D spin-orbit-coupled (SOC) fermionic cold-atom superfluid with the FuldeFerrell (FF) ground state. FF superfluids [30][31][32][33][34][35][36][37][38][39][40][41] have been studied recently in SOC degenerate Fermi gases subject to Zeeman fields, which yield asymmetries of the Fermi surface and induce the FF Cooper pairing with nonzero total momenta. We obtain a rich phase diagram in the gapless region of 3D FF superfluids, where not only the featureless Weyl points [19][20][21][22][23][24][25][26][27][28] but also the structured Weyl points emerge. Note that the featureless Weyl points have been well studied in SOC FF superfluids [24], and here we focus only on the novel structured Weyl points. (iii) We also discuss how the structured W...

We consider a trapped atomic system in the presence of spatially varying laser fields. The laser-atom interaction generates a pseudospin degree of freedom (referred to simply as spin) and leads to an effective spin-orbit coupling for the fermions in the trap. Reflections of the fermions from the trap boundaries provide a physical mechanism for effective momentum relaxation and nontrivial spin dynamics due to the emergent spin-orbit coupling. We explicitly consider evolution of an initially spin-polarized Fermi gas in a two-dimensional harmonic trap and derive nonequilibrium behavior of the spin polarization. It shows periodic echoes with a frequency equal to the harmonic trapping frequency. Perturbations, such as an asymmetry of the trap, lead to the suppression of the spin echo amplitudes. We discuss a possible experimental setup to observe spin dynamics and provide numerical estimates of relevant parameters.