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 †

Majorana fermions (MFs), quantum particles that are their own antiparticles, are not only of fundamental importance in elementary particle physics and dark matter, but also building blocks for fault-tolerant quantum computation. Recently MFs have been intensively studied in solid state and cold atomic systems. These studies are generally based on superconducting pairing with zero total momentum. On the other hand, finite total momentum Cooper pairings, known as Fulde-Ferrell (FF) Larkin-Ovchinnikov (LO) states, were widely studied in many branches of physics. However, whether FF and LO superconductors can support MFs has not been explored. Here we show that MFs can exist in certain types of gapped FF states, yielding a new quantum matter: topological FF superfluids/superconductors. We demonstrate the existence of such topological FF superfluids and the associated MFs using spin-orbitcoupled degenerate Fermi gases and derive their parameter regions. The implementation of topological FF superconductors in semiconductor/superconductor heterostructures is also discussed.

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.

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.

We study solitary waves of polarization (magnetic solitons) in a two-component Bose gas with slightly unequal repulsive intra-and interspin interactions. In experimentally relevant conditions we obtain an analytical solution which reveals that the width and the velocity of magnetic solitons are explicitly related to the spin healing length and the spin sound velocity of the Bose mixture, respectively. We calculate the profiles, the energy and the effective mass of the solitons in the absence of external fields and investigate their oscillation in a harmonic trap where the oscillation period is calculated as a function of the oscillation amplitude. The stability of magnetic solitons in two dimensions and the conditions for their experimental observation are also briefly discussed. and so on. Because of the interplay of nonlinearity and dispersion, solitons can move in their medium without loosing their shape and thus have important application in information processing. Among various physical systems, ultracold atomic gases provide a prominent platform for the investigation of solitons which can be engineered by phase imprinting, density imprinting, quantum quenches, etc. Soon after the realization of BoseEinstein condensation, dark and bight solitons characterized by density notches and density bumps have been actually observed in repulsive [4,5] and attractive [6] interacting Bose gases, respectively.