Abstract:
Phases of matter are conventionally characterized by order parameters describing the type and degree of order in a system. For example, crystals consist of spatially ordered arrays of atoms, an order that is lost as the crystal melts. Like- wise in ferromagnets, the magnetic moments of the constituent particles align only below the Curie temperature, TC. These two examples reflect two classes of phase transitions: the melting of a crystal is a first-order phase transition (the crystalline order vanishes abruptly) and the onset of magnetism is a second- order phase transition (the magnetization increases continuously from zero as the temperature falls below TC). Such magnetism is robust in systems with localized magnetic particles, and yet rare in model itinerant systems where the particles are free to move about. Here for the first time, we explore the itinerant magnetic phases present in a spin-1 spin-orbit coupled atomic Bose gas; in this system, itinerant ferromagnetic order is stabilized by the spin-orbit coupling, vanishing in its absence. We first located a second-order phase transition that continuously stiffens until, at a tricritical point, it transforms into a first- order transition (with observed width as small as h x 4 Hz). We then studied the long-lived metastable states associated with the first-order transition. These measurements are all in agreement with theory.

Abstract:
We develop a field integral formalism to study spin-orbit-coupled (SOC) Bose gases with arbitrary interspecies interaction. We identify various features arising from the interplay of SOC and interspecies interaction, including a roton minimum in the excitation spectrum and dual effects of SOC on ground-state energies depending on interspecies interactions. Counterintuitively, we find that at low interspecies interaction the SOC stabilizes the system by suppressing the quantum depletion. We show that the static structure factor is immune to the SOC in the phase space where time-reversal symmetry is preserved. Furthermore, we present an alternate way of studying phase fluctuations of the system.

Abstract:
Spin-orbit coupling is predicted to have dramatic effects on thermal properties of a two-component atomic Bose gas. We show that in three spatial dimensions it lowers the critical temperature of condensation and enhances thermal depletion of the condensate fraction. In two dimensions we show that spin-orbit coupling destroys superfluidity at any finite temperature, modifying dramatically the cerebrated Berezinskii-Kosterlitz-Thouless scenario. We explain this by the increase of the number of low energy states induced by spin-orbit coupling, enhancing the role of quantum fluctuations.

Abstract:
We theoretically study an interacting few-body system of Rashba spin-orbit coupled two-component Bose gases confined in a harmonic trapping potential. We solve the interacting Hamiltonian at large Rashba coupling strengths using Exact Diagonalization scheme, and obtain the ground state phase diagram for a range of interatomic interactions and particle numbers. At small particle numbers, we observe that the bosons condense to an array of topological states with n+1/2 quantum angular momentum vortex configurations, where n = 0, 1, 2, 3... At large particle numbers, we observe two distinct regimes: at weaker interaction strengths, we obtain ground states with topological and symmetry properties that are consistent with mean-field theory computations; at stronger interaction strengths, we report the emergence of strongly correlated ground states.

Abstract:
We study the zero-temperature phase diagram of a spin-orbit-coupled Bose-Einstein condensate of spin $1$, with equally weighted Rashba and Dresselhaus couplings. Depending on the antiferromagnetic or ferromagnetic nature of the interaction, we find two kinds of striped phases with qualitatively different behaviors in the modulations of the density profiles. Phase transitions to the zero-momentum and the plane-wave phases can be induced in experiments by varying the Raman coupling strength and the quadratic Zeeman field. The properties of these transitions are investigated in detail, and the emergence of tricritical points, which are the direct consequence of the spin-dependent interactions, is explicitly discussed.

Abstract:
We present a variational study of pseudo-spin $1/2$ Bose gases in a harmonic trap with weak 3D spin-orbit coupling of $\bmsigma\cdot\mathbf{p}$ type. This spin-orbit coupling mixes states with different parities, which inspires us to approximate the single particle state with the eigenstates of the total angular momentum, i.e. superposition of harmonic $s$-wave and $p$-wave states. As the time reversal symmetry is protected by two-body interaction, we set the variational order parameter as the combination of two mutually time reversal symmetric eigenstates of the total angular momentum. The variational results essentially reproduce the 3D skyrmion-like ground state recently identified by Kawakami {\it et al.}. We show that these skyrmion-like ground states emerging in this model are primarily caused by $p$ wave spatial mode involving in the variational order parameter that drives two spin components spatially separated. We find the ground state of this system falls into two phases with different density distribution symmetries depending on the relative magnitude of intraspecies and interspecies interaction: Phase I has parity symmetric and axisymmetric density distributions, while Phase II is featured with special joint symmetries of discrete rotational and time reversal symmetry. With the increasing interaction strength the transition occurs between two phases with distinct density distributions, while the topological 3D skyrmion-like spin texture is symmetry protected.

Abstract:
We investigate the stability of supercurrents in a Bose-Einstein condensate with one-dimensional spin-orbit and Raman couplings. The consequence of the lack of Galilean invariance is explicitly discussed. We show that in the plane-wave phase, characterized by a uniform density, the supercurrent state can become dynamically unstable, the instability being associated with the occurrence of a complex sound velocity, in a region where the effective mass is negative. We also discuss the emergence of energetic instability in these supercurrent states. We argue that both the dynamical and the energetic instabilities in these systems can be generated experimentally through excitation of the collective dipole oscillation.

Abstract:
Supersolid is a long-sought exotic phase of matter, which is characterized by the coexistence of a diagonal long-range order of solid and an off-diagonal long-range order of superfluid. Possible candidates to realize such a phase have been previously considered, including hard-core bosons with long-range interaction and soft-core bosons. Here we demonstrate that an ultracold atomic condensate of hard-core bosons with contact interaction can establish a supersolid phase when simultaneously subjected to spin-orbit coupling and a spin-dependent periodic potential. This supersolid phase is accompanied by topologically nontrivial spin textures, and is signaled by the separation of momentum distribution peaks, which can be detected via time-of-flight measurements. We also discuss possibilities to produce and observe the supersolid phase for realistic experimental situations.

Abstract:
The chiral magnetic and chiral separation effects---quantum-anomaly-induced electric current and axial current along an external magnetic field in parity-odd quark-gluon plasma---have received intense studies in the community of heavy-ion collision physics. We show that analogous effects occur in rotating trapped Fermi gases with Weyl-Zeeman spin-orbit coupling where the rotation plays the role of an external magnetic field. These effects can induce a mass quadrupole in the atomic cloud along the rotation axis which may be tested in future experiments. Similar effects also exist in rotating trapped Bose gases with Weyl-Zeeman spin orbit coupling. Our results suggest that the spin-orbit coupled atomic gases are potential simulators of the chiral magnetic and separation effects.

Abstract:
The realization of artificial gauge fields and spin-orbit coupling for ultra-cold quantum gases promises new insight into paradigm solid state systems. Here we experimentally probe the dispersion relation of a spin-orbit coupled Bose-Einstein condensate loaded into a translating optical lattice by observing its dynamical stability, and develop an effective band structure that provides a theoretical understanding of the locations of the band edges. This system presents exciting new opportunities for engineering condensed-matter analogs using the flexible toolbox of ultra-cold quantum gases.