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Chinese Physics B

Weakly interacting spinor Bose–Einstein condensates with three-dimensional spin–orbit coupling

Shu-Wei Song, Rui Sun, Hong Jin Zhao, Xuan Wang, Bao-Zhong Han

Published Apr 1, 2016• 3 citations
Publisher

Abstract

Starting from the Hamiltonian of the second quantization form, the weakly interacting Bose–Einstein condensate with spin–orbit coupling of Weyl type is investigated. It is found that the SU (2) nonsymmetric term, i.e., the spin-dependent interaction, can lift the degeneracy of the ground states with respect to the z component of the total angular momentum J z , casting the ground condensate state into a configuration of zero J z . This ground state density profile can also be affirmed by minimizing the full Gross–Pitaevskii energy functional. The spin texture of the zero J z state indicates that it is a knot structure, whose fundamental group is π 3 ( M ) ≅ π 3 ( S 2 ) = Z .

DOI
10.1088/1674-1056/25/4/040305

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Starting from the Hamiltonian of the second quantization form, the weakly interacting Bose–Einstein condensate with spin–orbit coupling of Weyl type is investigated. It is found that the SU (2) nonsymmetric term, i.e., the spin-dependent interaction, can lift the degeneracy of the ground states with respect to the z component of the total angular momentum J z , casting the ground condensate state into a configuration of zero J z . This ground state density profile can also be affirmed by minimizing the full Gross–Pitaevskii energy functional. The spin texture of the zero J z state indicates that it is a knot structure, whose fundamental group is π 3 ( M ) ≅ π 3 ( S 2 ) = Z .
We present a variational study of the spin-1 Bose gases in a harmonic trap with three-dimensional spin–orbit (SO) coupling of Weyl type. For weak SO coupling, we treat the single-particle ground states as the form of perturbational harmonic oscillator states in the lowest total angular momentum manifold with j = 1, m j = 1, 0, −1. When the two-body interaction is considered, we set the trail order parameter as the superposition of three degenerate single-particle ground-states and the weight coefficients are determined by minimizing the energy functional. 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-SO-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 SO-coupling. We 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. The emergence of higher spin order creates new opportunities for exploring spin-tensor-related physics in SO coupled superfluid.
We consider a SU(3) spin–orbit coupled Bose–Einstein condensate confined in a harmonic plus quartic trap. The ground-state wave functions of such a system are obtained by minimizing the Gross–Pitaevskii energy functional, and the effects of the spin-dependent interaction and spin–orbit coupling are investigated in detail. For the case of ferromagnetic spin interaction, the SU(3) spin–orbit coupling induces a threefold-degenerate plane wave ground state with nontrivial spin texture. For the case of antiferromagnetic spin interaction, the system shows phase separation for weak SU(3) spin–orbit coupling, where three discrete minima with unequal weights in momentum space are selected, while hexagonal honeycomb lattice structure for strong SU(3) SOC, where three discrete minima with equal weights are selected.
Abstract We investigate the ground state phases of the two-component Bose–Einstein condensates with three-dimensional Weyl-type spin–orbit coupling (SOC) loaded in a cubic optical lattice. The single-particle lowest energy band exhibits one, two or four pairs of degenerate ground states depending on the configuration of the SOC. We find two types of the supersolid (SS) phases with the isotropic SOC, and the SS phases are signaled by the momentum peaks distribution. We also discuss the spin textures of the SS phases, the spin vectors consist alternatively of two sets of continuous textures at the even and the odd lattice sites.
The recent experimental creation of knotted solitons in spin-1 Bose-Einstein condensates opens an interesting avenue for exploring the stability and dynamics of knot solitons. Knotted and helical vortices can be accommodated in spin-orbit-coupled antiferromagnetic spin-1 Bose-Einstein condensates. We demonstrate the stability of these topological structures via numerical simulations of the corresponding Gross-Pitaevskii equation. By changing the strength of the spin-orbit coupling, we find that the ground state undergoes a change from a knotted spin texture to a helical spin texture. Our results reveal that these types of topological object are experimentally realizable in spin-1 Bose-Einstein condensates.
In this paper, we investigate the dynamic properties of spin–orbit coupling spin-1 ferromagnetic Bose–Einstein condensates with different trap geometry. Our results are obtained in terms of the three-component Gross–Pitaevskii equation of mean-field theory. Two kinds of trap potential are discussed: isotropic and anisotropic. It is shown that the spin-exchange dynamics are greatly influenced by trap geometry. For the latter with weak spin–orbit coupling strength, we find that the three-component oscillation accelerates and some small difference emerges between component m=1 and m=-1. With the increase in spin–orbit coupling strength, the three components reach almost the same population. In addition, the kinetic energy of the system changes within a small scope for strong spin–orbit coupling, as opposed to a constant value in an isotropic trap. The density distributions display that the stripe phase appears with the increase in spin–orbit coupling strength. The method of generating stripe structure is different from the ground state of ferromagnetic condensate. For isotropic trap, the spatial separation of top and bottom spin–orbit condensates in component m=1 and m=-1 occurs at weak spin–orbit coupling, and square lattice appears at strong spin–orbit coupling.
The fragmentation of spin-orbit-coupled spin-1 Bose gas with a weak interaction in external harmonic trap is explored by both exact diagonalization and mean-field theory. This fragmentation tendency, which originates from the total angular momentum conservation, is affected obviously by the spin-orbit-coupling strength and the spin-dependent interaction. Strong spin-orbit interaction raises the inverse participation ratio, which describes the number of significantly occupied single-particle states. As the spin-dependent interaction changes from antiferromagnetic to ferromagnetic, the peak values in the inverse participation ratio become lower. Without the confinement of the appointed total angular momentum, the condensate chooses a zero or finite total angular momentum ground state, which is determined by both the interaction and the spin-orbit-coupling strength.