We present an overview of our recent theoretical studies on the quantum phenomena of the spin-1 Bose-Einstein condensates, including the phase diagram, soliton solutions and the formation of the topological spin textures. A brief exploration of the effects of spin-orbit coupling on the ground-state properties is given. We put forward proposals by using the transmission spectra of an optical cavity to probe the quantum ground states: the ferromagnetic and polar phases. Quasi-one-dimension solitons and ring dark solitons are studied. It is predicted that characteristics of the magnetic solitons in optical lattice can be tuned by controlling the long-range light-induced and static magnetic dipoledipole interactions; solutions of single-component magnetic and single-, two-, three-components polar solitons are found; ring dark solitons in spin-1 condensates are predicted to live longer lifetimes than that in their scalar counterparts. In the formation of spin textures, we have considered the theoretical model of a rapidly quenched and fast rotating trapped spin-1 Bose-Einstein condensate, whose dynamics can be studied by solving the stochastic projected Gross-Pitaevskii equations. Spontaneous generation of nontrivial topological defects, such as the hexagonal lattice skyrmions and square lattice of half-quantized vortices was predicted. In particular, crystallization of merons (half skyrmions) can be generated in the presence of spin-orbit coupling.
We present an overview of our recent theoretical studies on the quantum phenomena of the spin-1 Bose-Einstein condensates, including the phase diagram, soliton solutions and the formation of the topological spin textures. A brief exploration of the effects of spin-orbit coupling on the ground-state properties is given. We put forward proposals by using the transmission spectra of an optical cavity to probe the quantum ground states: the ferromagnetic and polar phases. Quasi-one-dimension solitons and ring dark solitons are studied. It is predicted that characteristics of the magnetic solitons in optical lattice can be tuned by controlling the long-range light-induced and static magnetic dipoledipole interactions; solutions of single-component magnetic and single-, two-, three-components polar solitons are found; ring dark solitons in spin-1 condensates are predicted to live longer lifetimes than that in their scalar counterparts. In the formation of spin textures, we have considered the theoretical model of a rapidly quenched and fast rotating trapped spin-1 Bose-Einstein condensate, whose dynamics can be studied by solving the stochastic projected Gross-Pitaevskii equations. Spontaneous generation of nontrivial topological defects, such as the hexagonal lattice skyrmions and square lattice of half-quantized vortices was predicted. In particular, crystallization of merons (half skyrmions) can be generated in the presence of spin-orbit coupling.
Abstract We investigate the ground-state properties of spin–orbit (SO)-coupled Bose–Einstein condensates confined in a spin-dependent asymmetric potential composed of harmonic and quartic contributions. The combined effect of the two potentials gives rise to an effective spatially inhomogeneous Zeeman field along the z -direction, and the complex competition among SO coupling, interatomic interactions, and this effective field leads to the emergence of nontrivial topological phases and complex spin textures in the system. For weak interatomic interactions, as the SO coupling strength increases, the system undergoes a series of transitions between ground-state phases: from a spin-polarized state, to an Anderson–Toulouse (AT) vortex state, a high-order radial dark soliton state, and finally to a domain structure. For strong interatomic interactions, the condensate evolves from a spin-polarized state to an AT vortex state, and then to a spatially coexisting state of AT and Mermin–Ho vortices, which corresponds to a novel topological excitation in pseudospin space: a skyrmion-dominated texture embedded with circular-hyperbolic antimeron pairs, realizing the coexistence and nesting of multiple topological defects. For sufficiently strong SO coupling, both components eventually evolve into similar multiring-shaped higher-order radial dark soliton states, with the number of soliton rings increasing as the SO coupling strength increases. Our results enrich the ground-state phases and topological excitations of cold atomic systems and provide new insights for exploring and manipulating nontrivial topological excitations in such systems.
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.
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 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.
We consider a spin-1 ferromagnetic Bose–Einstein condensate with Rashba spin–orbit coupling (SOC) and dipole–dipole interaction (DDI) in a double-well potential. For the nonrotating case, the ground state of the system is obtained by using the imaginary time propagation method, and a ground-state phase diagram is given with respect to the SOC strength and the DDI strength. It is shown that the nonrotating system sustains rich exotic ground-state phases including the ox-horn phase with hidden vortices and antivortices, density droplet phase with vortex–antivortex pairs, modulated stripe phase with vortex–antivortex chains, checkerboard phase with visible vortex–antivortex chains and hidden vortex–antivortex pairs, and the triangular vortex–antivortex lattice phase. For the rotating case, the dynamics of the system is investigated by using a phenomenological dissipation model. The effects of SOC, DDI and rotation on the unique dynamic behaviors of the rotating system and the final steady-state structures are revealed and analyzed. In particular, the rotating system supports fascinating novel spin textures and skyrmion excitations, and experiences a series of topological structure transitions such as the transitions from an interlaced skyrmion–antiskyrmion lattice pair to a skyrmion–meron lattice pair and then to a meron lattice pair.