Ground State Phases And Spin Orders Of The Weyl Coupled Spinor Bose Gases

Spin-orbit coupling occurs extensively in many physical systems, it links a microscopic particle’s quantum-mechanical spin to its spatial motion. It is responsible for the fine structure of the spectra in the field of atomic and molecular physics; For solid-state matter, it depicts the crystal’s electric field felt when the electrons which carries spin move through; it also plays crucial roles in the spin-Hall effect, topological insulators and topological superconductors, etc. However, for the cold-atom gases, owing to the atoms are electric neutral, the spin-orbit coupling is not a intrinsic interaction. In the years 2009 to 2011, thanks to the Raman coupling tecniques, a series of experiments carried out by the NIST group engeering the synthetic vector potential, magnetic field, electric field, and non-Abelian gauge fields, i.e. the spin-orbit coupling in successive. From then, the studies of the spin-orbit coupled cold-atom gases became a prolonged research interest in the cold-atom community.The spin-orbit coupling realized experimentally by NIST group is a one-dimensional configuration which couples the spins of the cold atoms to their momenta in one spatial direction. It is equiverlent to the equally weighted combination of the well-known Rashba-and Dresselhaus-type spin-orbit coupling in form. In the last two years, the field still attracts experimental interests, and persistent experimental progresses were reported, including the Raman spin-orbit coupled spin-1 Bose gases, two-dimensional spin-orbit coupled ultra-cold Fermi gases, and two-dimensional spin-orbit coupled Bose gases in optical lattice. The experimental schemes for the realization of the most symmetrical three-dimensional spin-orbit coupling, i.e. the Weyl spin-orbit coupling configuration also were proposed. All these experimental efforts make the spin-orbit coupled cold atom gases a persistent research interests in recent years.The theoretical and experimental works both show that some novel quantum phases occur in the spin-orbit coupled Bose gases. There are plane wave phase, stripe superfluid phase, and zero-momentum phase for NIST spin-orbit coupled pseudospin-1/2 and spin-1 Bose gases, for the latter configuration, the spatially modulated nematic orders also appear in the stripe phase. These ground state phases also appear in the Rashba coupled gases. For Weyl spin-orbit coupled pseudospin-1/2 Bose gases, the plane wave and the stripe ground states occur again in the homogeneous systems. When a tight external confiment presents, the various novel half-quantum vortex phases and spontaneously localizesd lattice phases will appear as the ground states.In this thesis, we study the Weyl spin-orbit coupled pseudospin-1/2 and spin-1 Bose gases respectively. These works include two parts.Firstly, we present a mean-field study of the ground state of Weyl spin-orbit coupled pseudospin-1/2 Bose gases.We present a variational study of pseudospin-1/2 Bose gases in a harmonic trap with weak 3D spin-orbit coupling of Weyl type. We notice that the Weyl coupling breaks the parity symmetry, the harmonic oscillator states with different parities will interact with each other, leaving only the total angular momentum a constant of motion, which inspires us to approximate the single particle state as the 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 with opposite magnetic quantum numbers. The variational results essentially reproduce the 3D skyrmion-like ground state recently identified by Kawakami et. al. [Phys. Rev. Lett.109,015301 (2012)] 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:one phase has parity symmetric and axisymmetric density distributions, while the other phase 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 topology of the 3D magnetic skyrmion-like spin texture is unaffected.Secondly, we present the mean-field study of the ground state of Weyl spin-orbit coupled spin-1 Bose gases.We present a variational study of the spin-1 Bose gases in a harmonic trap with three-dimensional spin-orbit coupling of Weyl type, and obtain a phase diagram. Two ground state phases, namely the magnetic and the nematic phases, are identified depending on the spin-independent and the spin-dependent interactions. Unlike the non-spin-orbit-coupled spin-1 Bose-Einstein condensate for which the phase boundary between the magnetic and the nematic phase lies exactly at zero spin-dependent interaction, the boundary is modified by the spin-orbit-coupling here. We also find the magnetic phase is featured with phase-separated density distributions,3D skyrmion-like spin textures and competing magnetic and biaxial nematic orders, while the nematic phase is featured with miscible density distributions, zero magnetization and spatially modulated uniaxial nematic order.