It is to be noted that ψ 1 2 , ψ 0 2 , and ψ − 1 2 (the left three columns) are the density distributions of three components m _F = 1, m _F = 0, and m _F − 1, and the corresponding phase distributions are given by θ ₁ = arg  ψ ₁, θ ₀ = arg  ψ ₀, and θ ₋₁ = arg  ψ ₋₁ (the right three columns), respectively. In Figure 1A, we give the initial quantum state of spin-1 BECs with SOC strength k = 2 loaded in an anharmonic trap. The system supports an approximately disk-shaped condensate without any topological defects in each component; therefore, it presents a plane wave phase [38]. Figures 1B–G show the typical dynamics of the component densities and phases with the anharmonic trap beginning to rotate suddenly with Ω = 2, when the system has been prepared to the initial quantum state. It is shown that during the early evolution period (see Figure 1B), there is a slightly turbulent oscillation in BECs, where the approximately disk-shaped densities are broken and begin to deform. Moreover, ghost vortices are formed in the boundary regions of the BECs due to the surface wave excitations [24, 27]. Then, the system reaches the minimum in the density distribution, and complex turbulent oscillations appear in the phase distributions (Figure 1C), which results in the surface waves propagating along the surfaces. When t = 6, the ghost vortex [24, 39] enters the central region of the external potential and develops into visible vortices which present irregular distributions (Figure 1D). With further time evolution, the component density distributions gradually become regular and the phase defects gradually gather to trap the center (Figures 1E,F). Eventually, the system develops into two vortex necklace structures along the azimuthal direction with an irregular density hole. In addition, the vortices in the density hole in the three components differ by one quantum number in turn (see Figure 1G). One of the reasons for the abovementioned density structures in Figure 1G is the presence of the anharmonic trap. A similar necklace structure has been found in the literature [20, 27, 40], and in the first Ref, ground states are studied in an unharmonic trap, and the latter two are dynamics in a quasicrystalline optical lattice and harmonic trap, respectively. In addition, we know that the shape of the external potential plays an important role in the dynamic properties of the BECs; for example, the system confined in a toroidal trap eventually evolved into a density hole with giant vortex structures.

Next, we study the dynamics of rotating spin-orbit-coupled spin-1 BECs with an in-plane gradient magnetic field in an anharmonic trap. In Figure 2A, we show the initial quantum state of spin-1 BECs with SOC strength k = 2 and in-plane quadrupole field strength B = 2. The vortices occur only in the central regions of the components m _F = 1 and m _F = −1, which form a polar-core vortex [41] with winding combination − 1,0,1 , as shown in Figure 2A. Essentially, the polar-core vortices in Figure 2A are coreless vortices because there is no phase defect in the total density of the system. Our numerical simulation shows that the formation of the central polar-core vortex is relevant to the interplay between the quadrupole field and the SOC. The combined effects of the quadrupole field and the SOC generate a special saddle point structure, where the in-plane magnetization of the spin occurs in the particular magnetic field and the amplitude of the total magnetization F at the saddle point is zero. To satisfy the conservation of angular momentum, the two vortices at the center of components m _F = 1 and m _F = −1 must rotate backward, so they have opposite winding numbers. The typical dynamics of the component densities and phases after the anharmonic trap begins to rotate suddenly with Ω = 2 and are given in Figures 1B–G. As seen from the phase distributions of Figure 2B, there are slightly turbulent oscillations and some ghost vortices. At the same time, the components m _F = ± 1 obey anti-symmetric with a diagonal line, and the component m _F = 0 exists in two vortex dipoles (vortex–antivortex pairs). Then, the component densities are elongated along the diagonal line and are separated (Figure 2C). In addition, the central polar-core vortex and a small number of ghost vortices outside the atomic cloud (last three columns of Figure 2B) transform into complex turbulent oscillations in the phase distribution (last three columns of Figure 2C). When t = 1.3, the system becomes fragments and reaches the minimum in the density distributions, and the corresponding distributions present more complicated turbulence oscillations, which makes the boundary of the BECs unstable and induces surface waves propagating along the surfaces. With further time evolution, the condensates gradually split into two obvious parts about a diagonal line and the center of the system appears to dissipate (see Figures 2E,F). The visible vortices of the system eventually link up with each other along the principal diagonal, that is, a diagonal vortex chain, with the remaining vortices distributed symmetrically as far as possible on both sides of the diagonal line (Figure 2G). Physically, here the vortex chain is a result of the combined effect of SOC, in-plane quadrupole field, and rotation. The magnetic field gradient causes a magnetic force so that the condensate is difficult to concentrate at the center and is divided into two parts, and on the other hand, due to the existence of rotation, the competition among the magnetic force, SOC, and the rotation, a vortex chain appears. The vortex chain structure has been observed in the literature [42, 43], where only ground states with in-plane gradient magnetic fields are considered. The abovementioned research shows that in the dynamics, the combined effect of gradient magnetic fields, SOC, and rotation play a key role in the generation of exotic topological defects. On the experimental side, searching for novel quantum phases in this highly tunable system is still an ongoing work. Our findings will stimulate further research in searching for various novel states in spin-orbit coupled BECs subject to effective gradient fields. In addition, tunable atomic spin-orbit coupling synthesized with a modulating gradient magnetic field has been realized [44–46]

The skyrmion is a type of topological soliton, which was originally suggested in nuclear physics by Skrme to elucidate baryons as a quasiparticle excitation with spins pointing in all directions to wrap a sphere [47]. A skyrmion can be viewed as the reverse of the local spin, and the spin-vector sweeps the whole unit sphere; therefore, its topological charge is Q = 1. As shown Figure 3 and Figure 4, we display the typical transitions of the topological charge density and spin texture. Displayed in Figures 3A,B are the topological charge density and the spin texture, and the corresponding density and phase distributions (t = 6) are given in Figure 1D. Figures 3C–G represent the typical local amplifications of the spin texture in Figure 3B, and Figure 3H shows the topological charge density of t = 11 (Figure 1E). Considering the limited resolution of the texture, in Figure 3I, we only show the spin texture in the core region, and the typical local amplifications of the full spin texture are given in Figures 3J–N. Our numerical calculation shows that Figures 3C–E and Figures 3J–M denote half-antiskyrmions [21, 48] with topological charge Q = −0.5, and Figures 3F,G and Figure 3N present antiskyrmions with topological charge Q = −1, which indicates that the spin texture in the process of evolution comprises different types of half-antiskyrmions (antimerons) and antiskyrmions. Due to the transitional spin texture being unstable, we cannot determine exactly what the topological defect is. Shown in Figures 3O,P are the topological density and spin texture of the steady state, and the local amplifications are given in Figures 3Q–U, Our calculation results demonstrate that the topological defect of the central region in the green spot is a hyperbolic antiskyrmion with unit topological charge Q = −1. The hyperbolic skyrmion and hyperbolic half-skyrmions have been reported in the literature [21, 47–49]. The topological defect in Figures 3Q–T is half-antiskyrmion (antimeron) with topological charge Q = −0.5, where different colors of dots represent half-antiskyrmions with different shapes. As seen from Figure 3O, there are seven half-skyrmions around the trap center, which develop into a half-antiskyrmion necklace structure along the azimuthal direction, and the external region of the cloud is occupied by a larger half-antiskyrmion necklace comprising eleven half-antiskyrmions, that is, the half-antiskyrmion necklace consists of two concentric annular half-antiskyrmions. In short, the spin texture undergoes a transition from an irregular antimeron and antiskyrmion structures to a double half-antiskyrmion necklace structure.

Figure 4A shows the spin structure corresponding to Figure 2A, and it indicates that the spin of the initial state is fully magnetized into the plane and whose arrangement resembles the configuration of a quadrupole magnetic field. The center of the quadrupole magnetic field is a saddle point with the magnetic field strength being 0, which has no magnetization. However, due to the continuity of the wave function, the spin function S (r) in the center point must be continuous. If S (0) has a certain magnitude, it must point in a certain direction. However, due to the symmetry of the system, no matter S (0) choice, any direction will break the continuity of S (r) at the center point. So, S (0) can only take 0 to satisfy the continuity condition. Therefore, the components m _F = 1 and m _F = −1 form a central vortex core. To satisfy the conservation of angular momentum, the two vortices at the center of F = 1 , m F = 1 and F = 1 , m F = − 1 components must rotate in opposite directions, which forms central polar-core vortex (see Figure 2A). Figures 4B,C exhibit the topological charge density and spin texture of t = 15, which corresponds to Figure 2F. From Figures 4D–J, there are seven types of half-antiskyrmions with respective local topological charge Q = −0.5, and they seem to be forming a half-antiskyrmion chain along the diagonal. As shown in Figures 4K,L are the topological charge density and spin texture of the steady state, and the local amplifications are given in Figures 4M–R. The topological structure of the leading diagonals is a chain comprising different kinds of half-antiskyrmions, whose respective local topological charge Q = −0.5. Moreover, the local topological charge in each red spot is Q = −0.5, which indicates that the topological structures on both sides of the leading diagonals are a symmetric half-antiskyrmion (antimeron) lattice with it. In a word, the spin texture transforms from a structure similar to a quadrupole magnetic field to a symmetric half-antiskyrmion lattice with respect to the diagonal of the half skyrmion chain. Essentially, the half-antiskyrmions (antiskyrmions) of different shapes in Figure 3 and Figure 4 are caused by the complicated interplay among the SOC, rotation, in-plane gradient magnetic field, and dissipation.

The dynamic process can also be characterized by the time evolution of the average angular momentum per atom L z , where the dependence of L z on Ω, k, and B is displayed in Figure 5. Take the blue curve as an example, when the system suddenly begins to rotate with Ω = 2, L z increases rapidly with the time evolution (0 ≤ t ≤ 15), and the gradually (15 ≤ t ≤ 100) approaches a maximum value (equilibrium value). Essentially, the combined effects of the continuous input of angular momentum, quantum fluid nature, SOC, anharmonic trap, and the dissipation result in the formation of a steady vortex structure in the three components. Next, we take the green curve as an example, and as the system is rotated with Ω = 2, L z increases rapidly (0 ≤ t ≤ 5) and suddenly a small (5 ≤ t ≤ 8) decrease occurs and finally tends to be an equilibrium value. Compared with the former, we find that the final L z is larger for the latter. That is, the combined effects of the quantum fluid nature, SOC, in-plane gradient magnetic field, and dissipation enhance the creation of vortices on the diagonal.