Spin-Orbit Coupling And Magnetism In Optical Lattices

In this thesis,we propose the experimental realization of one dimensional,two dimensional and three dimensional spin-orbit coupling.Moreover,we also propose the experimental detection of the topological phenomenon induced by the spin-orbit-coupling.Theoretically,we minaly discuss the topological phases and current affected by the spin-orbit-coupling.We also study the novel magnetic phases and the phase transition induced by the spin-orbit-coupling in the spin-1/2 system.In the spin-1 system,the spin-tensor-momentum coupling can induce novel spin-tensor magnetism.The main results are as following,1.In this work,we propose the quantum simulation of symmetry-protected topological states for interacting fermions using alkaline-earth-like atoms ¹⁷³Yb in quasi-one-dimensional optical lattice.The alkaline-earth-like atoms have a long-life excited state ³P₀,which is called as clock state.The Hamiltonian of the system has the U?1? particle-number-conservation and the chiral symmetries,which form an antiunitary group [U?1?×Z₂^T].The symmetry-protected topological phase has fermionic edge states and the corresponding topological invariant is Z₄.The above symmetry-protected topological phases is in clear contrast to existing proposals of realizing symmetry-protected topological phases with bosonic edge modes.The Bose symmetry-protected topological phases are protected by the SU?N?/Z_N symmetry and can be realized in pure bosonic systems.The interacting fermionic symmetry-protected topological phase is also fundamentally different from noninteracting fermionic symmetry-protected topological phases,due to the distinct topological invariants and classifications.We numerically calculate the ground state of the system to plot the phase diagram and propose the topological phase transitions drived by the spin-exchange interaction.The symmetry-protected topological phases can be probed by detecting the local occupation of the clock states at the edges.2.We study the ground-state current properties of interacting fermions in a one-dimensional optical lattice clock with spin-orbit coupling.Adopting the concept of synthetic dimensions,we explicitly consider the hyperfine spin states in the clockstate manifolds and map the system to multiple two-leg ladders combine with a real dimensional with uniform magnetic flux penetrating the plaquettes of each ladder.We numerically investigate the effect of spin-exchange interactions on the many-body ground state properties such as the current flow and the density-density correlations by using the density matrix renormalization group.Our numerical results reveal a rich phase diagram,where the interactions drastically modify the Meissner and the vortex states in the noninteracting case.In particular,we show the existence of an exotic interaction-induced vortex state,where spin currents emerge between different ladders together with spin-density-wave order in the ground state.The spin-exchange and density-density interactions in the clock-state manifolds can be readily tuned by external magnetic field through the orbital Feshbach resonance,or by transverse trapping frequencies through the confinement-induced resonance.Our results have interesting implications for future experiments.3.We propose a scheme to realize two-dimensional Rashba spin-orbit coupling and three-dimensional Weyl types of spin-orbit coupling in a three-dimensional optical lattice clock and explore their topological phases.The three-dimensional optical lattice potential is generated by magic wavelength lasers,two states ?g? and ?e? suffer the same lattice potential.The clock laser that couples two states ?g? and ?e? is coherently splitted into four beams propagating along different directions and their interference generates the SOC.3D Weyl ?2D Rashba?types of SOC are realized when the wave vectors of the clock laser do?not?possess z-component.The three-dimensional Weyl types of SOC can induce a rich phase diagram,which contains topological phases with various numbers of Weyl points as well as a fully gapped three-dimensional Chern insulator phase.The Weyl points have two types,type-I and type-II.The Weyl points must appear on the pair.Different Weyl points in a pair have opposite topological charges,and are connected by gapless Fermi arcs on the surface.The two-dimensional Rashba of spin-orbit coupling induces topological bands.The three-dimensional Weyl type of spin-orbit coupling induces a topological surface around the weyl point.The spin-textures of can be detected using suitably designed spectroscopic sequences.Three spectroscopic sequences are proposed to accurately measure the spin-textures using a combination of Rabi spectroscopy and time-of-flight images.4.We studied the magnetic proprieties of the ground state in the presence of the spin-orbit-coupling and the repulsive on-site interaction.We find that spin-orbit-coupling can drive the phase transition from spin-rotating antiferromagnetic Mott insulator phase to spin-rotating ferromagnetic Mott insulator phase.The above phases can be identified by means of the spin-correlation function and the spin-structure factor S?k?.We plot the phase diagram of the ground state in the plane of U and ?.There is a second-order quantum phase transition between the spin-rotating ferromagnetic Mott insulator and the spin-orbit-coupling-induced metallic phase.The spin-orbit-coupling-induced metallic phase is predicted at the strong spin-orbit-coupling.The spin-rotating means the spin orientations of the nearest-neighbor sites are not parallel or antiparallel but have an intersection angle ???0.??.In the spin-rotating antiferromagnetic Mott insulator,the corresponding angle are ?/2<?<?,and the quasi-long-range spin-correlation function decays as a power law and changes the sign with a period 2<T<4.Whereas for the spin-rotating ferromagnetic Mott insulator with ?<?/ 2,the spin-correlation function also decays as a power law,but spiral in real space with period T>4.Finally,we find that the momentum k_peak can also be affected dramatically by spin-orbit-coupling,which is equal to the corresponding momentum at the peak of the spin-structure factor.The analytical expression of this momentum peakk with respect to the spin-orbit-coupling strength is also derived,which suggests that the predicted spin-rotating ferromagnetic (k_peak<?/2) and antiferromagnetic ??/2?<k_peak<?) correlations can be detected experimentally by measuring the spin-structure factor S?k? via time-of-flight imaging.5.The quantum magnetism characterization of a large spin ??1? system naturally involves not only spin-vectors but also spin-tensors.While certain types of spin-vector?e.g.,ferromagnetic,spiral?and spin-tensor?e.g.,nematic in frustrated lattices?orders have been investigated separately,but the coexistence and correlation between them have not been well explored.Here we propose and characterize a novel quantum spiral spin-tensor magnetic order on a spin-1 Heisenberg chain as well as a spiral spin-tensor Zeeman field.The above theoretical model can be realized using a Raman-dressed cold atom optical lattice in experiment.We plot the phase diagram of ground state through density-matrix renormalization group numerical calculation.We characterize the magnetism of the ground state by calculating the coexistence of spin-vector and spin-tensor local orders as well as spin-vector and spin-tensor correlation functions.Our studies may open a door for exploring novel magnetic orders and studing spin-tensor electronics/atomtronics in large-spin systems.