In the light of the ϕ-mapping topological current theory, the Mermin–Ho vortices and the monopoles in three-component spinor BEC are studied. It is pointed out that these two topological structures respectively inhere in two-dimensional and three-dimensional topological currents which can be derived from the same field tensor k μν , and both these topological structures are characterized by the ϕ-mapping topological numbers—Hopf indices and Brouwer degrees. Furthermore, the spatial bifurcation of Mermin–Ho vortices and the generation and annihilation of monopoles are also discussed.In the light of the ϕ-mapping topological current theory, the Mermin–Ho vortices and the monopoles in three-component spinor BEC are studied. It is pointed out that these two topological structures respectively inhere in two-dimensional and three-dimensional topological currents which can be derived from the same field tensor k μν , and both these topological structures are characterized by the ϕ-mapping topological numbers—Hopf indices and Brouwer degrees. Furthermore, the spatial bifurcation of Mermin–Ho vortices and the generation and annihilation of monopoles are also discussed.

Journal of Physics A Mathematical and GeneralDec 17, 200218 citations

Knotlike vortex lines in spinor BEC

Yi-Shi Duan, Xin Liu, Jie Yang

Annals of PhysicsFeb 4, 20043 citations

Coreless and singular vortex lattices in rotating spinor Bose-Einstein condensates

Takeshi Mizushima, Naoko Kobayashi, Kazushige Machida

We theoretically investigate vortex-lattice phases of rotating spinor Bose-Einstein condensates (BEC's) with a ferromagnetic spin interaction by numerically solving the Gross-Pitaevskii equation. The spinor BEC under slow rotation can sustain a rich variety of exotic vortices due to the multicomponent order parameters, such as Mermin-Ho and Anderson-Toulouse coreless vortices (the two-dimensional Skyrmion and meron) and nonaxisymmetric vortices with shifting vortex cores. Here, we present the spin texture of various vortex-lattice states at higher rotation rates and in the presence of an external magnetic field. In addition, the vortex phase diagram is constructed in the plane of the total magnetization $M$ and the external rotation frequency $\ensuremath{\Omega}$ by comparing the free energies of possible vortices. It is shown that the vortex phase diagrams in the $M\text{\ensuremath{-}}\ensuremath{\Omega}$ plane may be divided into two categories: (i) the coreless vortex lattice formed by the several types of Mermin-Ho vortices and (ii) the vortex lattice filling in the cores with the pure polar (antiferromagnetic) state. In particular, it is found that the type-(ii) state forms composite lattices of coreless and polar-core vortices. The difference between type (i) and type (ii) results from the existence of the singularity of the spin textures, which may be experimentally confirmed by the spin imaging within polarized light recently proposed by Carusotto and Mueller. We also discuss the stability of triangular and square lattice states for rapidly rotating condensates.We theoretically investigate vortex-lattice phases of rotating spinor Bose-Einstein condensates (BEC's) with a ferromagnetic spin interaction by numerically solving the Gross-Pitaevskii equation. The spinor BEC under slow rotation can sustain a rich variety of exotic vortices due to the multicomponent order parameters, such as Mermin-Ho and Anderson-Toulouse coreless vortices (the two-dimensional Skyrmion and meron) and nonaxisymmetric vortices with shifting vortex cores. Here, we present the spin texture of various vortex-lattice states at higher rotation rates and in the presence of an external magnetic field. In addition, the vortex phase diagram is constructed in the plane of the total magnetization $M$ and the external rotation frequency $\ensuremath{\Omega}$ by comparing the free energies of possible vortices. It is shown that the vortex phase diagrams in the $M\text{\ensuremath{-}}\ensuremath{\Omega}$ plane may be divided into two categories: (i) the coreless vortex lattice formed by the several types of Mermin-Ho vortices and (ii) the vortex lattice filling in the cores with the pure polar (antiferromagnetic) state. In particular, it is found that the type-(ii) state forms composite lattices of coreless and polar-core vortices. The difference between type (i) and type (ii) results from the existence of the singularity of the spin textures, which may be experimentally confirmed by the spin imaging within polarized light recently proposed by Carusotto and Mueller. We also discuss the stability of triangular and square lattice states for rapidly rotating condensates.

arXiv:cond-mat/0407684Physical Review AOct 12, 200492 citations

Ground state of a rotating Bose-Einstein condensate with in-plane quadrupole field

Liu Jing-Si, Li Ji, Liu Wu-Ming

Compared with the scalar Bose-Einstein condensate, the spinor Bose-Einstein condensate, in which internal degrees of freedom are essentially free, has aroused the great interest in the study of topological excitations. In particular, the spinor Bose-Einstein condensate with rotation provides a new opportunity for studying novel quantum states including a coreless vortex and vortex lattice. To date, in the presence of rotation, a great many of studies on the topological excitations have focused on the Bose-Einstein condensate system with the uniform Zeeman field or without external magnetic field. However, the ground state structure of a rotating Bose-Einstein condensate in the presence of in-plane gradient-magnetic-field remains an open question. In this work, by using the imaginary-time propagation method, we study the ground state structure of a rotating Bose-Einstein condensate with in-plane quadrupole field. We first examine the effect of in-plane quadrupole field on trapped spinor Bose-Einstein condensate. The numerical results show that Mermin-Ho vortex can be induced only by the cooperation between quadrupole field and rotation. When magnetic field gradient is increased, the vortices around Mermin-Ho vortex display the symmetrical arrangement. For an even larger magnetic field gradient strength, the system only presents the Mermin-Ho vortex because the in-plane quadrupole field can prevent the vortices around Mermin-Ho vortex from occurring. Next, we examine the effect of the rotation on trapped spinor Bose-Einstein condensate. A phase transition from a polar-core vortex to a Mermin-Ho vortex is found through applying a rotational potential, which is caused by the cooperation between the in-plane quadrupole field and the rotation. We further study the combined effects of spin exchange interaction and density-density interaction. The results confirm that in the presence of the quadrupole field both spin exchange interaction and density-density interaction, acting as controllable parameters, can control the number of the vortices around Mermin-Ho vortex. The corresponding number of the vortices shows step behavior with increasing the ratio between spin exchange interaction and density-density interaction, which behaves as hexagon, pentagon, square and triangle. It is found that two types of topology structures, i.e., the hyperbolic meron and half-skyrmion, can occur in the present system. These vortex structures can be realized via time-of-flight absorption imaging technique. Our results not only provide an opportunity to investigate the exotic vortex structures and the corresponding phase transitions in a controlled platform, but also lay the foundation for the study of topological defect subjected to gauge field and dipolar interaction in future.Compared with the scalar Bose-Einstein condensate, the spinor Bose-Einstein condensate, in which internal degrees of freedom are essentially free, has aroused the great interest in the study of topological excitations. In particular, the spinor Bose-Einstein condensate with rotation provides a new opportunity for studying novel quantum states including a coreless vortex and vortex lattice. To date, in the presence of rotation, a great many of studies on the topological excitations have focused on the Bose-Einstein condensate system with the uniform Zeeman field or without external magnetic field. However, the ground state structure of a rotating Bose-Einstein condensate in the presence of in-plane gradient-magnetic-field remains an open question. In this work, by using the imaginary-time propagation method, we study the ground state structure of a rotating Bose-Einstein condensate with in-plane quadrupole field. We first examine the effect of in-plane quadrupole field on trapped spinor Bose-Einstein condensate. The numerical results show that Mermin-Ho vortex can be induced only by the cooperation between quadrupole field and rotation. When magnetic field gradient is increased, the vortices around Mermin-Ho vortex display the symmetrical arrangement. For an even larger magnetic field gradient strength, the system only presents the Mermin-Ho vortex because the in-plane quadrupole field can prevent the vortices around Mermin-Ho vortex from occurring. Next, we examine the effect of the rotation on trapped spinor Bose-Einstein condensate. A phase transition from a polar-core vortex to a Mermin-Ho vortex is found through applying a rotational potential, which is caused by the cooperation between the in-plane quadrupole field and the rotation. We further study the combined effects of spin exchange interaction and density-density interaction. The results confirm that in the presence of the quadrupole field both spin exchange interaction and density-density interaction, acting as controllable parameters, can control the number of the vortices around Mermin-Ho vortex. The corresponding number of the vortices shows step behavior with increasing the ratio between spin exchange interaction and density-density interaction, which behaves as hexagon, pentagon, square and triangle. It is found that two types of topology structures, i.e., the hyperbolic meron and half-skyrmion, can occur in the present system. These vortex structures can be realized via time-of-flight absorption imaging technique. Our results not only provide an opportunity to investigate the exotic vortex structures and the corresponding phase transitions in a controlled platform, but also lay the foundation for the study of topological defect subjected to gauge field and dipolar interaction in future.

Acta Physica SinicaJan 1, 20178 citations

M

and the external rotation frequency $\ensuremath{\Omega}$ by comparing the free energies of possible vortices. It is shown that the vortex phase diagrams in the $M\text{\ensuremath{-}}\ensuremath{\Omega}$ plane may be divided into two categories: (i) the coreless vortex lattice formed by the several types of Mermin-Ho vortices and (ii) the vortex lattice filling in the cores with the pure polar (antiferromagnetic) state. In particular, it is found that the type-(ii) state forms composite lattices of coreless and polar-core vortices. The difference between type (i) and type (ii) results from the existence of the singularity of the spin textures, which may be experimentally confirmed by the spin imaging within polarized light recently proposed by Carusotto and Mueller. We also discuss the stability of triangular and square lattice states for rapidly rotating condensates.