Chuanwei Zhang

Magnetic control of valley pseudospin in monolayer WSe2

Supplementary Text S1. Computing the valley Zeeman effect in multiple samples. The peak splitting due to the valley Zeeman effect is small compared to the width of the photoluminescence (PL) peaks so care must be taken in determining the Zeeman splitting. The peaks are slightly asymmetric, with shape varying somewhat with magnetic field, and do not conform well to a Gaussian or Lorentzian peak shape. We use two methods to determine the peak position and hence the Zeeman splitting, both of which make no assumptions about the peak shape. As shown in Figure 1c, they agree very well. The first, “max point”, simply finds the 15 points in each spectrum with the most counts and assigns the peak position to the median value of these points. This method is insensitive to the trion peak, which is too far away to influence these points; however, it is more sensitive to noise as it only considers a few points. The second method, “weighted average”, computes the “center of mass” of the peak, , where is the PL spectral density and E is photon energy. In this method the effect of noise is greatly reduced because it makes use of all the several hundred points that make up the spectrum, but on the other hand it is more sensitive to the trion peak, which will tend to over-weight the low-energy side of the peak. However, since the valley exciton Zeeman splitting is small and we are interested in the difference between the σ+ and σpeaks, the weak trion effects on both peaks tend to balance each other out. The data in Fig. 1c and Fig. S1a are from two different samples. We can see that the splitting as a function of magnetic field obtained by these two different methods has little difference. Eight samples were measured and all were observed to have a splitting linear in the applied field. In Figure S1b we plot the fitted slope of the splittings from all the samples, in units of Bohr magnetons. The data presented in the main text is from the fifth sample. We see that the data can be split into two groups with mean values of 1.57 μB and 2.86 μB. The origin of this bimodal distribution in the g-factors is unclear due to the lack of understanding of what external factors can affect the g-factors in these new materials. Future studies will be necessary to quantitatively determine the effect of variables such as strain, doping, and substrate on the magnetic properties. However, all samples show similar behavior in their valley polarization as a function of the applied field (i.e., the “X” and “V” patterns).

px+ipy superfluid from s-wave interactions of fermionic cold atoms.

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.

BCS-BEC crossover and topological phase transition in 3D spin-orbit coupled degenerate fermi gases

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.

Observation of Zitterbewegung in a spin-orbit-coupled Bose-Einstein condensate

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.

Structured Weyl Points in Spin-Orbit Coupled Fermionic Superfluids

We demonstrate that a Weyl point, widely examined in 3D Weyl semimetals and superfluids, can develop a pair of nondegenerate 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.

Mean Field Dynamics of Spin-Orbit Coupled Bose-Einstein Condensates

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.

Dicke-type phase transition in a spin-orbit-coupled Bose–Einstein condensate

Spin-orbit-coupled Bose-Einstein condensates (BECs) provide a powerful tool to investigate interesting gauge field-related phenomena. Here we study the ground state properties of such a system and show that it can be mapped to the well-known Dicke model in quantum optics, which describes the interactions between an ensemble of atoms and an optical field. A central prediction of the Dicke model is a quantum phase transition between a superradiant phase and a normal phase. We detect this transition in a spin-orbit-coupled BEC by measuring various physical quantities across the phase transition. These quantities include the spin polarization, the relative occupation of the nearly degenerate single-particle states, the quantity analogous to the photon field occupation and the period of a collective oscillation (quadrupole mode). The applicability of the Dicke model to spin-orbit-coupled BECs may lead to interesting applications in quantum optics and quantum information science.

Tunable spin-orbit coupling via strong driving in ultracold-atom systems

Spin-orbit coupling is an essential ingredient in topological materials, conventional and quantum-gas-based alike. Engineered spin-orbit coupling in ultracold-atom systems-unique in their experimental control and measurement opportunities-provides a major opportunity to investigate and understand topological phenomena. Here we experimentally demonstrate and theoretically analyze a technique for controlling spin-orbit coupling in a two-component Bose-Einstein condensate using amplitude-modulated Raman coupling.

Testable Signatures of Quantum Nonlocality in a Two-Dimensional Chiral p-Wave Superconductor

A class of topological excitations-the odd-winding number vortices-in a spinless 2D chiral p-wave (px+ipy) superconductor traps Majorana fermion states in the vortex cores. For a dilute gas of such vortices, the lowest energy fermionic eigenstates are intrinsically nonlocal. We predict two testable signatures of this unusual quantum nonlocality in quasiparticle tunneling experiments. We discuss why the associated teleportationlike phenomenon does not imply the violation of causality.

Nonequilibrium Spin Dynamics in a Trapped Fermi Gas with Effective Spin-Orbit Interactions

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.