The internal degrees of freedom provided by ultracold atoms provide a route for realizing higher dimensional physics in systems with limited spatial dimensions. Nonspatial degrees of freedom in these systems are dubbed ``synthetic dimensions.'' This connection is useful from an experimental standpoint but complicated by the fact that interactions alter the condensate ground state. Here we use the Gross-Pitaevskii equation to study the ground-state properties of a spin-1 Bose gas under the combined influence of an optical lattice, spatially varying spin-orbit coupling, and interactions at the mean-field level. The associated phases depend on the sign of the spin-dependent interaction parameter and the strength of the spin-orbit field. We find ``charge''- and spin-density-wave phases which are directly related to helical spin order in real space and affect the behavior of edge currents in the synthetic dimension. We determine the resulting phase diagram as a function of the spin-orbit coupling and spin-dependent interaction strength, considering both attractive (ferromagnetic) and repulsive (polar) spin-dependent interactions, and we provide a direct comparison of our results with the noninteracting case. Our findings are applicable to current and future experiments, specifically with $^{87}\mathrm{Rb}, ^{7}\mathrm{Li}, ^{41}\mathrm{K}$, and $^{23}\mathrm{Na}$.
The internal degrees of freedom provided by ultracold atoms provide a route for realizing higher dimensional physics in systems with limited spatial dimensions. Nonspatial degrees of freedom in these systems are dubbed synthetic dimensions.'' This connection is useful from an experimental standpoint but complicated by the fact that interactions alter the condensate ground state. Here we use the Gross-Pitaevskii equation to study the ground-state properties of a spin-1 Bose gas under the combined influence of an optical lattice, spatially varying spin-orbit coupling, and interactions at the mean-field level. The associated phases depend on the sign of the spin-dependent interaction parameter and the strength of the spin-orbit field. We find charge''- and spin-density-wave phases which are directly related to helical spin order in real space and affect the behavior of edge currents in the synthetic dimension. We determine the resulting phase diagram as a function of the spin-orbit coupling and spin-dependent interaction strength, considering both attractive (ferromagnetic) and repulsive (polar) spin-dependent interactions, and we provide a direct comparison of our results with the noninteracting case. Our findings are applicable to current and future experiments, specifically with 87Rb,7Li,41K, and 23Na.
Since the realization of Bose-Einstein condensation in 1995, experimentalists in ultracold atomic gas community have developed a set of entirely new methods to study quantum gases. In ultracold atomic gases, implementation of synthetic gauge fields, the spin-orbit coupling, optical lattices and Feshbach resonances, provides a clean and highly controllable platform to tune the parameters of the system, which are typically fixed for real materials. Motivated by the rapid developments of experimental progresses, I investigate the ultracold atomic gases with synthetic gauge fields. \n \nIn this thesis, a bosonic gas in optical lattices in the presence of an effective staggered magnetic flux will be studied, using a generalized Bose-Hubbard model. Phase transition from the Mott insulating phase to the superfluid phase will be discussed. Novel superfluid phases including the stripe phase and the plane wave phase will be discussed. Furthermore, the existence of a tricritical point in the phase diagram is predicted. \n \nTo study Fermi systems with synthetic gauge fields, a spin-1=2 Fermi gas with Rashba spin-orbit coupling is investigated by studying the structure of vortex line through the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation (BCSBEC) crossover. Due to the spin-orbit coupling, the s-wave interacting Fermi gas behaves like an effective p-wave paired system. A discussion about the vortex line structure will be provided by calculating the spatial distribution of the gap function, the density, the particle current, and also the elementary excitations. It has been found that the most robust superfluid occurs around the unitary point, indicated by the smallest healing length and the largest critical current. \n \nFinally, motivated by the recent progress of p-wave resonantly Fermi gas in experiment, an extension of Nozières-Schmitt-Rink method to the p-wave case will be given. An investigation of the normal state properties will be given by calculating the equation of state and the p-wave contacts, which have been measured by the experiment. Furthermore, the superfluid transition temperature with realistic parameters appropriate for experiment is calculated and is found to be within experimental reach.
We theoretically establish the mean-field phase diagram of a homogeneous spin-1, spin-orbit-coupled Bose gas as a function of the spin-dependent interaction parameter, the Raman coupling strength, and the quadratic Zeeman shift. We find that the interplay between spin-orbit coupling and spin-dependent interactions leads to the occurrence of ferromagnetic or ferronematic phases which also break translational symmetry. For weak Raman coupling, increasing attractive spin-dependent interactions (as in 87Rb or 7Li) induces a transition from a uniform to a stripe XY ferromagnet (with no nematic order). For repulsive spin-dependent interactions (as in 23Na), however, we find a transition from an XY spin spiral phase ($\ensuremath{\langle}{S}_{z}\ensuremath{\rangle}=0$ and uniform total density) with uniaxial nematic order to a biaxial ferronematic, where the total density, spin vector, and nematic director oscillate in real space. We investigate the stability of these phases against the quadratic Zeeman effect, which generally tends to favor uniform phases with either ferromagnetic or nematic order but not both. We discuss the relevance of our results to ongoing experiments on spin-orbit-coupled, spinor Bose gases.
We study ultracold bosonic atoms with synthetic three-dimensional spin-orbit (SO) coupling in a cubic optical lattice. In the superfluidity phase, the lowest energy band exhibits one, two, or four pairs of degenerate single-particle ground states depending on the SO-coupling strengths, which can give rise to condensate states with spin stripes for weak atomic interactions. In the deep Mott-insulator regime, the effective spin Hamiltonian of the system combines three-dimensional Heisenberg exchange interactions, anisotropy interactions, and Dzyaloshinskii-Moriya interactions. Based on Monte Carlo simulations, we numerically demonstrate that the resulting Hamiltonian with an additional Zeeman field has a rich phase diagram with spiral, stripe, vortex crystal, and especially Skyrmion crystal spin textures in each xy-plane layer. The obtained Skyrmion crystals can be tunable with square and hexagonal symmetries in a columnar manner along the z axis, and moreover are stable against the interlayer spin-spin interactions in a large parameter region.
The dissertation explores the effects of synthetic spin-orbit coupling on the behaviour of quantum gases in several different contexts. We first study realistic methods to create vortices in spin-orbit-coupled (SOC) Bose-Einstein condensates (BEC). We propose two different methods to induce thermodynamically stable static vortex configurations: (1) to rotate both the Raman lasers and the anisotropic trap; and (2) to impose a synthetic Abelian field on top of synthetic spin-orbit interactions. We solve the Gross-Pitaevskii equation for several experimentally relevant regimes and find new interesting effects such as spatial separation of left- and right-moving spin-orbit-coupled condensates, and the appearance of unusual vortex arrangements. Next we consider cold atoms in an optical lattice with synthetic SOC in the Mott-insulator regime. We calculate the parameters of the corresponding tight-binding model and derive the low-energy spin Hamiltonian which is a combination of Heisenberg model, quantum compass model and Dzyaloshinskii-Moriya interaction. We find that the Hamiltonian supports a rich classical phase diagram with collinear, spiral and vortex phases. Next we study the time evolution of the magnetization in a Rashba spin-orbit-coupled Fermi gas, starting from a fully-polarized initial state. We model the dynamics using a Boltzmann equation, which we solve in the Hartree-Fock approximation. The resulting non-linear system of equations gives rise to three distinct dynamical regimes controlled by the ratio of interaction and spin-orbit-coupling strength λ: for small λ, the magnetization decays to zero. For intermediate λ, it displays undamped oscillations about zero and for large λ, a partially magnetized state is dynamically stabilized. Motivated by an interesting stripe phase which appears in BEC with SOC [Li et al., Phys. Rev. Lett. 108, 225301 (2011)], we study the finite-temperature phase diagram of a pseudospin-1/2 Bose gas with contact interactions. We show that strong inter-spin interactions can lead to the appearance of magnetically ordered phases at temperatures above the superfluid transition. For the case of inter-spin attraction, we also discuss the possibility of a bosonic analogue of the Cooper-paired phase, however this state is not energetically favourable. We extend our calculations to a spin-orbit-coupled Bose gas to investigate the possibility of stripe ordering in the normal phase. However, within our approximations, we do not find an instability towards stripe formation. Finally, we consider a two-dimensional Bose gas at zero temperature with an underlying quartic single-particle dispersion in one spatial direction. This Hamiltonian can be realized using the NIST scheme of spin-orbit coupling [Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature 471, 83 (2011)], or using the shaken lattice scheme of Parker et al. [C. V. Parker, L.-C. Ha and C. Chin, Nature Physics 9, 769 (2013)]. By numerically comparing energies of various trial wave-functions, we show that, at low densities, the ground state is strongly correlated, in contrast to a typical mean-field BEC. The trial wave-function with the lowest energy is of Jastrow-type and it describes a state with finite, but strongly reduced, condensate fraction.
Artificial synthetic gauge field and spin-orbit coupling has been extensively studied following their experimental realization in ultracold atomic systems. Thanks for the versatile controllability, such systems not only provide possibilities to simulate and study important models in multidisciplinary fields of physics, but also work as an excellent platform to engineer novel states of matter and quantum phenomena. This paper reviews some recent progresses on the study of ultracold atomic systems with spin-orbit coupling, focusing on the effects induced by dissipation, novel interaction forms, large symmetry of spins, and long-range interactions. The investigation in these aspects is closely related to the characteristics of ultracold atomic systems, hence can bring new inspirations and perspectives on the understanding of spin-orbit coupling. In this review, we firstly investigate the appearance of a topological superradiant state in a quasi-one-dimensional Fermi gas with cavity-assisted Raman process. A cavity-assisted spin-orbit coupling and a bulk gap opening at half filling will be induced by the superradiant light generated in the transversely driven cavity mode. The topological superradiant state and the corresponding topological phase transition in the system can be driven by this mechanism. Then, symmetry-protected topological states of interacting fermions will be introduced in a quasi-one-dimensional cold gas of alkaline-earth-like atoms. Raman-assisted spin-orbit couplings in the clock states, together with the spin-exchange interactions in the clock-state manifolds will give rise to symmetry-protected topological states for interacting fermions, by taking advantage of the separation of orbital and nuclear-spin degrees of freedom in these alkaline-earth-like atoms. Furthermore, we show that an exotic topological defect, double-quantum spin vortices, which are characterized by doubly quantized circulating spin currents and unmagnetized filled cores, can exist in the ground states of SU(3) spin-orbit-coupled Bose-Einstein condensates. It is found that the combined effects of SU(3) spin-orbit coupling and spin-exchange interaction determine the ground-state phase diagram. Finally, we demonstrate that spin-orbit coupling and soft-core long-range interaction can induce an exotic supersolid phase of Bose gas, with the emergence of spontaneous circulating particle current. This implies that a finite angular momentum can be generated with neither external rotation nor synthetic magnetic field, and the direction of the angular momentum can be altered by adjusting the strength of spin-orbit coupling or interatomic interaction.
We study the interplay between large-spin, spin-orbit coupling, and superfluidity for bosons in a two-dimensional optical lattice, focusing on the spin-1 spin-orbit-coupled system recently realized at the Joint Quantum Institute [Campbell et al., arXiv:1501.05984]. We find a rich quantum phase diagram where, in addition to the conventional phases---superfluid and insulator---contained in the spin-1 Bose-Hubbard model, there are new lattice symmetry breaking phases. For weak interactions, the interplay between two length scales, the lattice momentum and the spin-orbit wave vector, induce a phase transition from a uniform superfluid to a phase where bosons simultaneously condense at the center and edge of the Brillouin zone at a nonzero spin-orbit strength. This state is characterized by spin-density-wave order, which arises from the spin-1 nature of the system. Interactions suppress spin-density-wave order, and favor a superfluid only at the Brillouin zone edge. This state has spatially oscillating mean-field order parameters, but a homogeneous density. We show that the spin-density-wave superfluid phase survives in a two-dimensional harmonic trap, and thus establish that our results are directly applicable to experiments on 87Rb,7Li, and 41K.