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.
Abstract We investigate the topological excitations of rotating spin-1 ferromagnetic Bose–Einstein condensates with spin–orbit coupling (SOC) in an in-plane quadrupole field. Such a system sustains a rich variety of exotic vortex structures due to the spinor order parameter and the interplay among in-plane quadrupole field, SOC, rotation, and interatomic interaction. For the nonrotating case, with the increase of the quadrupole field strength, the system experiences a transition from a coreless polar-core vortex with a bright soliton to a singular polar-core vortex with a density hole. Without rotation but with a fixed quadrupole field, when the SOC strength increases, the system transforms from a central Mermin–Ho vortex into a criss-crossed vortex–antivortex string lattice. For the rotating case, we give a phase diagram with respect to the quadrupole field strength and the SOC strength. It is shown that the rotating system supports four typical quantum phases: vortex necklace, diagonal vortex chain cluster, single diagonal vortex chain, and few vortex states. Furthermore, the system favors novel spin textures and skyrmion excitations including an antiskyrmion, a criss-crossed half-skyrmion–half-antiskyrmion lattice, a skyrmion-meron necklace, a symmetric half-skyrmion lattice, and an asymmetric skyrmion-meron lattice.
We study the topological defects and spin structures of rotating SU(3) spin–orbit-coupled spin F=1 Bose–Einstein condensates (BECs) in an in-plane quadrupole field with ferromagnetic spin interaction, and the BECs is confined by a harmonic trap. Without rotation, as the quadrupole field strength is increased, the spin F=1 BECs with SU(3) spin–orbit coupling (SOC) evolves from the initial Thomas–Fermi phase into the stripe phase; then, it enters a vortex–antivortex cluster state and eventually a polar-core vortex state. In the absence of rotation with the given quadrupole field, the enhancing SU(3) SOC strength can cause a phase transition from a central Mermin–Ho vortex to a vortex–antivortex cluster, subsequently converting to a bending vortex–antivortex chain. In addition, when considering rotation, it is found that this system generates the following five typical quantum phases: a three-vortex-chain cluster structure with mutual angles of approximately 2π3, a tree-fork-like vortex chain cluster, a rotationally symmetric vortex necklace, a diagonal vortex chain cluster, and a density hole vortex cluster. Particularly, the system exhibits unusual topological structures and spin textures, such as a bending half-skyrmion–half-antiskyrmion (meron–antimeron) chain, three half-skyrmion (meron) chains with mutual angles of an approximately 2π3, slightly curved diagonal half-skyrmion (meron) cluster lattice, a skyrmion–half-skyrmion (skyrmion-meron) necklace, and a tree-fork-like half-skyrmion (meron) chain cluster lattice.
Two-component Bose-Einstein condensate offers an ideal platform for investigating many intriguing topological defects due to the interplay between intraspecies and interspecies interactions. The recent realization of spin-orbit coupling in two-component Bose-Einstein condensate, owing to coupling between the spin and the centre-of-mass motion of the atom, provides possibly new opportunities to search for novel quantum states. In particular, the gradient magnetic field in the Bose-Einstein condensate has brought a new way to create topologically nontrivial structures including Dirac monopoles and quantum knots. Previous studies of the gradient magnetic field effect in the Bose-Einstein condensate mainly focused on the three-component case. However, it remains unclear how the gradient magnetic field affects the ground state configuration in the rotating two-component Bose-Einstein condensate with spin-orbit coupling. In this work, by using quasi two-dimensional Gross-Pitaevskii equations, we study the ground state structure of a rotating two-component Bose-Einstein condensate with spin-orbit coupling and gradient magnetic field. We concentrate on the effects of the spin-orbit coupling and the gradient magnetic field on the ground state. The numerical results show that increasing the strength of the spin-orbit coupling can induce a phase transition from skyrmion lattice to skyrmion chain in the presence of the gradient magnetic field. Unlike the study of skyrmion in rotating two-component Bose-Einstein condensate with only spin-orbit coupling, the skyrmion chain can occur under the isotropic spin-orbit coupling when the gradient magnetic field is considered. It is worth noting that the skyrmion chain here is arrayed along the diagonal direction. Next we examine the effect of the gradient magnetic field on spin-orbit coupled two-component Bose-Einstein condensate. For the case of weak spin-orbit coupling and the slow rotation, a phase transition from a single plane-wave to half-skyrmion is found through increasing magnetic field gradient strength. For the case of strong spin-orbit coupling and the fast rotation, the nature of the ground state is shown to support the formation of a hidden vortex as the gradient magnetic field is enhanced. These hidden vortices have no visible cores in density distributions but have phase singularities in phase distributions, which are arrayed along the diagonal direction. This result confirms a new method of creating the hidden vortices in the two-component Bose-Einstein condensate. These topological structures can be detected by using the time-of-flight absorption imaging technique. Our results illustrate that the gradient magnetic field not only provides an opportunity to explore the exotic topological structures in spin-orbit coupled spinor Bose-Einstein condensate, but also is crucial for realizing the phase transitions among different ground states. This work paves the way for the future exploring of topological defect and the corresponding dynamical stability in quantum systems subjected to a gradient magnetic field.
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.
Realization of spinor Bose-Einstein condensate in an optical trap has made it possible to create a variety of topological nontrivial structures, due to the vector character of the order parameter. Recently, artificial spin-orbit coupling in the spinor Bose-Einstein condensate, owing to coupling between the spin and the center-of-mass motion of the atom, provides an unprecedented opportunity to search for novel quantum states. As is well known, the potential well in the Bose-Einstein condensate is adjustable. The toroidal trap is an important model potential because of its simplicity and richness in physics. In particular, the spinor Bose-Einstein condensate under the toroidal trap has brought an ideal platform for studying fascinating properties of a superfluid, such as persistent flow and symmetry-breaking localization. For the case of the spin-orbit-coupled Bose-Einstein condensate, the previous studies of the toroidal trap mainly focused on the two-component or antiferromagnetic case. However, in the presence of a toroidal trap, there remains an open question whether the combined effects of the spin-orbit coupling and rotation can produce previously unknown types of topological excitations in the ferromagnetic Bose-Einstein condensate. In this work, by using quasi two-dimensional Gross-Pitaevskii equations, we study the ground state structure of spin-orbit coupled rotating ferromagnetic Bose-Einstein condensate in the toroidal trap. We concentrate on the effects of the spin-orbit coupling and the rotation on the ground states. The numerical results show that in the presence of a toroidal trap, the ground state structure is displayed as half-skyrmion chain with circular distribution. Adjusting the strength of spin-orbit coupling not only changes the number of half-skyrmion in the system, but also controls the symmetry of half-skyrmion with circular distribution. As the rotation frequency increases, the system undergoes the transitions from the plane wave to the half-skyrmion chain with circular distribution, and eventually developing the half-skyrmion phase of triangular lattice. Next, we examine the effect of spin-independent interaction on spin-orbit coupled rotating spinor Bose-Einstein condensate. As the spin-independent interaction increases, the topological defects in the condensate increase due to the variation of the local magnetic order. We also discuss the influence of well shape on the ground state structure. These topological structures can be detected via the time-of-flight absorption imaging technique. The spin-orbit coupled spinor Bose-Einstein condensate in the toroidal trap is an important quantum platform, which not only opens up a new avenue for exploring the exotic topological structures, but also is crucial for realizing the transitions among different ground states. This work paves the way for futureexploring the topological defects and the corresponding dynamical stability in quantum system subjected to the toroidal trap.
Abstract We investigate the ground-state properties of spin–orbit-coupled pseudo-spin-1/2 dipolar Bose–Einstein condensates in a two-dimensional harmonic trap and an in-plane quadrupole field. The effects of spin–orbit coupling (SOC), dipole–dipole interaction (DDI) and the in-plane quadrupole field on the ground-state structures and spin textures of the system are systematically analyzed and discussed. For fixed SOC and DDI strengths, the system shows a quadrupole stripe phase with a half-quantum vortex, or a quadrupole Thomas–Fermi phase with a half-quantum antivortex for small quadrupole field strength, depending on the ratio between inter- and intraspecies interaction. As the quadrupole field strength enhances, the system realizes a ring mixed phase with a hidden vortex–antivortex cluster rather than an ordinary giant vortex in each component. Of particular interest, when the strengths of DDI and quadrupole field are fixed, strong SOC leads to the formation of criss-crossed vortex string structure. For given SOC and quadrupole field, the system for strong DDI displays a sandwich-like structure, or a special delaminated structure with a prolate antivortex in the spin-up component. In addition, typical spin textures for the ground states of the system are analyzed. It is shown that the system sustains exotic topological structures, such as a hyperbolic spin domain wall, skyrmion-half-antiskyrmion-antiskyrmion lattice, half-skyrmion-skyrmion-half-antiskyrmion lattice, and a drum-shaped antimeron.