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.
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.
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 investigate the ground-state phases and spin textures of spin–orbit-coupled dipolar pseudo-spin-1/2 Bose–Einstein condensates in a rotating two-dimensional toroidal potential. The combined effects of dipole–dipole interaction (DDI), spin–orbit coupling (SOC), rotation, and interatomic interactions on the ground-state structures and topological defects of the system are analyzed systematically. For fixed SOC strength and rotation frequency, we provide a set of phase diagrams as a function of the DDI strength and the ratio between inter- and intra-species interactions. The system can show rich quantum phases including a half-quantum vortex, symmetrical (asymmetrical) phase with quantum droplets (QDs), asymmetrical segregated phase with hidden vortices (ASH phase), annular condensates with giant vortices, triangular (square) vortex lattice with QDs, and criss-cross vortex string lattice, depending on the competition between DDI and contact interaction. For given DDI strength and rotation frequency, the increase of the SOC strength leads to a structural phase transition from an ASH phase to a tetragonal vortex lattice then to a pentagonal vortex lattice and finally to a vortex necklace, which is also demonstrated by the momentum distributions. Without rotation, the interplay of DDI and SOC may result in the formation of a unique trumpet-shaped Bloch domain wall. In addition, the rotation effect is discussed. Furthermore, the system supports exotic topological excitations, such as a half-skyrmion (meron) string, triangular skyrmion lattice, skyrmion–half-skyrmion lattice, skyrmion–meron cluster, skyrmion–meron layered necklace, skyrmion–giant-skyrmion necklace lattice, and half-skyrmion–half-antiskyrmion necklace.
The ground-state properties of a spin-orbit-coupled rotating Bose-Einstein condensate with a gradient magnetic field in a toroidal trap are studied. When gradient magnetic fields are taken into account, the results are quite novel in that, in the spin-orbit-coupled nonrotating Bose-Einstein condensate with small gradient magnetic field intensity, the ferromagnetic system presents a plane-wave phase and satisfies the translational symmetry, and the antiferromagnetic system presents a stripe phase. With the increase of the magnetic field intensity, the translational symmetry of the ground state of the ferromagnetic system is broken, and the phase change tends to be similar in both ferromagnetic and antiferromagnetic systems. Considering the spin-orbit-coupled rotating Bose-Einstein condensate, the system gradually transforms from the nonsingle half-skyrmion chain to the single half-skyrmion chain with the increase of the gradient magnetic field, while the single half-skyrmion chain transforms to the nonsingle half-skyrmion chain with the increase of spin-orbit coupling. In addition, we obtain the phase diagram as a function of the gradient magnetic field and the spin-orbit-coupling intensity.
We investigate the ground-state structures of dipolar spin-orbit-coupled Bose-Einstein condensates in a toroidal trap. Combined effects of dipole-dipole interaction (DDI) and spin-orbit coupling (SOC) on the ground states of the system are discussed. A ground-state phase diagram is obtained as a function of the SOC and DDI strengths. As two new degrees of freedom, the DDI and SOC can be used to obtain the desired ground-state phases and to control the phase transition between various ground states. In particular, the system displays exotic topological structures and spin textures, such as half-quantum vortex, vortex string, vortex necklace, complex vortex lattice including giant vortex and hidden antivortex chains, different skyrmions, meron (half-skyrmion)-antimeron (half-antiskyrmion) necklace, and composite meron-antimeron lattice.
Abstract The ground-state properties of a spin–orbit-coupled spin-1 Bose–Einstein condensate (BEC) in a toroidal trap are investigated. The exotic ground states, such as the necklace state, the persistent flow and even the vortex state, for the three components in the system are induced by adjustable external parameters. It is found that the petal number of the necklace states is increased with increasing the intensity of the spin–orbit coupling (SOC), and the anisotropy of the SOC can be used to control the structure of the ground states. The rotation of the condensate induced by the external field makes the densities of the three components asymmetric and tends to transform the necklace state to the persistent flow. The radius of the toroidal potential is another degree of freedom for manipulating the necklace state. In addition, the transition between the necklace state and the vortex, intermediated by the persistent flow, can be controlled by the ratio of the density-density and spin-exchange interactions. All of above results enrich our knowledge about the properties of the ground states of spin-1 BEC trapped in the toroidal trap.
The spinor Bose-Einstein condensate (BEC) provides an ideal platform for observing and manipulating topological structures, which arise from the spin degrees of freedom and the superfluid nature of the gas. Artificial helicoidal spin-orbit coupling (SOC) in the spinor BEC, owing to the spatially varying gauge potential and the more flexible adjustability, provides possibly an unprecedented opportunity to search for novel quantum states. The previous studies of the BEC with helicoidal SOC mainly focus on the two-component case. However, there are few reports on the studies of helicoidal SOC in three-component BEC. Especially considering one-dimensional three-component BEC, whether the helicoidal SOC can generate previously unknown types of topological excitations and phase diagrams is still an unsolved problem. In this work, by solving quasi one-dimensional Gross-Pitaevskii equations, we study the ground state structure of one-dimensional helicoidal spin-orbit coupled three-component BEC. The numerical results show that the helicoidal SOC can induce a phase separation among the components in ferromagnetic BEC. Through numerical calculations of the system, a phase diagram is obtained as a function of the helicoidal SOC strength and gauge potential, which shows the critical conditions for phase separation and phase miscibility in ferromagnetic BEC. Meanwhile, we also study the influences of the helicoidal SOC and the gauge potential on the antiferromagnetic BEC ground state. The numerical results show that the helicoidal SOC is beneficial for the miscibility in antiferromagnetic BEC. When the helicoidal SOC strength or gauge potential increases, the ground state of antiferromagnetic BEC exhibits a stripe soliton structure. Adjusting the strength of helicoidal SOC or gauge potential can control the transitions between a plane-wave soliton and a stripe soliton. In addition, we show the changes of the particle number density maximum and the number of peaks of stripe solitons for adjusting the helicoidal SOC strength or gauge potential. Our results show that helicoidal spin-orbit coupled BEC not only provides a controlled platform for investigating the exotic topological structures, but also is crucial for the transitions between different ground states. This work paves the way for exploring the topological defect and the corresponding dynamical stability in quantum systems subjected to the helicoidal SOC in future.