1. Introduction
With the first experimental observation in 1995 [1, 2], the Bose–Einstein condensate (BEC) has been widely studied in both experiment and theory. The collective excitation of matter waves in BECs, such as the bright and dark solitons [3–8], vortex lattices [9–11] and rogue waves [12–14] draw a great deal of interest. The experimental realization of spin–orbit coupling (SOC) in BECs by the NIST group [15] arises a hot research on spin–orbit coupled spin-1/2 BECs. In spin–orbit coupled spin-1/2 BECs, the competition between the intraspecies repulsion and the interspecies repulsion leads to a variety of ground-state phases, such as the plane wave phase, stripe phase and meron with dipole–dipole repulsion [16–20]. The dynamic properties of spin–orbit coupled spin-1/2 BECs are also studied, such as solitons, quantized vortices dynamics and Josephson dynamics [21–29].
In the rotating spin–orbit coupled spin-1/2 BECs, a variety of vortex structures [30–33] are found, such as half-quantum vortex, ring-like structure with domain, giant vortex and circular-hyperbolic skyrmion. The Mermin-Ho vortex, quantum entanglement and quantum knot are observed in the spinor BECs with gradient magnetic field [34–36]. In [37], a tunable SOC is realized in spin-1 BECs by modulating gradient magnetic field in experiment. The SOC strength can be adjusted by changing the momentum impulse from the gradient magnetic field. Recently, the related researches of spin–orbit coupled spin-1/2 and spin-1 BECs in a harmonic trap with gradient magnetic field have been studied [38–41]. In spin-1/2 BECs, the gradient magnetic field and SOC can control the transformation between the single plane-wave phase and half-skyrmion phase. In spin-1 BECs, the influence of SOC and gradient magnetic field on the properties of the monopoles with polar-core vortex. Compared with the previous works [42–45], the monopoles have a long-lived time, therefore, the experimental observation of the monopoles becomes easier.
In this paper, we investigate the ground-state phases of two-dimensional (2D) rotating spin–orbit coupled spin-1/2 BECs loaded in a harmonic plus quartic trap with gradient magnetic field. The competition between gradient magnetic field, SOC and rotation leads to a variety of ground-state phase structures. In the weakly rotation regime, as the increase of gradient magnetic field strength, two types of ground-state phases are found with a fixed SOC, i.e. the unstable phase and the single vortex-line phase. The unstable phase presents the vortex lines structures along the off-diagonal direction. With magnetic field gradient strength increases, the number of vortex lines changes accordingly. As the magnetic field gradient strength increases further, a single vortex line along the diagonal direction, the single vortex-line phase is found. The phase diagram shows that the boundary between the two phases is linear with the relative repulsion λ ≥ 1 and is nonlinear with λ < 1. In the relatively strong rotation regime, in addition to the unstable phase and the single vortex-line phase, the vortex-ring phase is formed for the strong magnetic field gradient and rapid rotation. The vortex-ring phase shows the giant and hidden vortex structures at the center of ring, the strong magnetic field gradient makes the number of the vortices around the ring unchanged (n = 6).
2. Model and Hamiltonian
In our paper, we study the ground-state phases of 2D rotating spin–orbit coupled spin-1/2 BECs loaded in a harmonic plus quartic trap with gradient magnetic field, the expectation value of the Hamiltonian are given as $ \begin{eqnarray}\begin{array}{rcl}E[{\rm{\Psi }}] & \equiv & \langle \widehat{H}\rangle =\displaystyle \int {{\rm{d}}}^{2}{\boldsymbol{r}}{{\rm{\Psi }}}^{\dagger }\left[-\displaystyle \frac{{{\hslash }}^{2}}{2m}{{\rm{\nabla }}}^{2}+V({\boldsymbol{r}})+{\upsilon }_{\mathrm{soc}}-{\rm{\Omega }}{L}_{z}\right.\\ & & \left.+{g}_{F}{\mu }_{B}{\boldsymbol{B}}({\boldsymbol{r}})\cdot {\boldsymbol{\sigma }}\Space{0ex}{3.08ex}{0ex}\right]{\rm{\Psi }}+\displaystyle \int {{\rm{d}}}^{2}{\boldsymbol{r}}({c}_{1}{N}_{\uparrow }^{2}+{c}_{2}{N}_{\downarrow }^{2}\\ & & +{c}_{12}{N}_{\uparrow }{N}_{\downarrow }),\end{array}\end{eqnarray}$ where ${\rm{\Psi }}=({{\rm{\Psi }}}_{\uparrow },{{\rm{\Psi }}}_{\downarrow })$ is the wave function of spin-1/2 BECs, m is the atomic mass and ${\boldsymbol{r}}=(x,y)$ . The densities of spin- $\uparrow $ component ${N}_{\uparrow }=| {{\rm{\Psi }}}_{\uparrow }{| }^{2}$ and spin- $\downarrow $ component ${N}_{\downarrow }=| {{\rm{\Psi }}}_{\downarrow }{| }^{2}$ . The repulsions ${c}_{1}=\tfrac{4\pi {{\hslash }}^{2}{a}_{1}}{m}$ and ${c}_{2}=\tfrac{4\pi {{\hslash }}^{2}{a}_{2}}{m}$ are the intraspecies repulsions, ${c}_{12}=\tfrac{4\pi {{\hslash }}^{2}{a}_{12}}{m}$ is the interspecies repulsion. ${\boldsymbol{\sigma }}$ is the spin density ${\boldsymbol{\sigma }}=({\sigma }_{x},{\sigma }_{y},{\sigma }_{z})=({{\rm{\Psi }}}^{\dagger }{\hat{\sigma }}_{x}{\rm{\Psi }},{{\rm{\Psi }}}^{\dagger }{\hat{\sigma }}_{y}{\rm{\Psi }},{{\rm{\Psi }}}^{\dagger }{\hat{\sigma }}_{z}{\rm{\Psi }})$ with the 2 × 2 spin-1/2 matrices $\widehat{\sigma }$ . The isotropic Rashba SOC is ${\upsilon }_{\mathrm{soc}}\,=-{\rm{i}}{\hslash }\gamma ({\sigma }_{x}{\partial }_{y}-{\sigma }_{y}{\partial }_{x})$ with SOC strength γ. $V({\boldsymbol{r}})=\tfrac{1}{2}m{\omega }_{\perp }^{2}({{\boldsymbol{r}}}^{2}+\beta {{\boldsymbol{r}}}^{4})$ is a harmonic plus quartic trap, here we set β = 0.5. The rotational frequency ω along the z direction with the orbit angular momentum ${L}_{z}=-{\rm{i}}{\hslash }(x{\partial }_{y}-y{\partial }_{x})$ . The parameters gF is Lande factor and μB is Bohr magnetic moment. The gradient magnetic field [46] ${\boldsymbol{B}}({\boldsymbol{r}})=B(x{{\rm{e}}}_{x}-y{{\rm{e}}}_{y})$ , where B is the strength. gF = −1/2 is Lander factor and μB is Bohr magnetic moment.
The time evolution of the mean field is governed by $ \begin{eqnarray}{\rm{i}}{\hslash }\displaystyle \frac{\partial {\rm{\Psi }}}{\partial t}=\displaystyle \frac{\delta E}{\delta {{\rm{\Psi }}}^{* }}.\end{eqnarray}$ The time-dependent Gross–Pitaevskii equations of spin–orbit coupled BECs can be obtained by substituting equation (1 ) into (2 ), $ \begin{eqnarray}\begin{array}{rcl}{\rm{i}}\displaystyle \frac{\partial {{\rm{\Psi }}}_{\uparrow }}{\partial t} & = & \left[-\displaystyle \frac{{{\rm{\nabla }}}^{2}}{2}+V({\boldsymbol{r}})+{\rm{i}}{\rm{\Omega }}(x{\partial }_{y}-y{\partial }_{x})+{c}_{1}| {{\rm{\Psi }}}_{\uparrow }{| }^{2}\right.\\ & & \left.+{c}_{12}| {{\rm{\Psi }}}_{\downarrow }{| }^{2}]{{\rm{\Psi }}}_{\uparrow }+[B(x+{\rm{i}}y)+\gamma (-{\rm{i}}{\partial }_{y}+{\partial }_{x})\Space{0ex}{3.15ex}{0ex}\right]{{\rm{\Psi }}}_{\downarrow },\\ {\rm{i}}\displaystyle \frac{\partial {{\rm{\Psi }}}_{\downarrow }}{\partial t} & = & \left[\Space{0ex}{3.08ex}{0ex}(-\displaystyle \frac{{{\rm{\nabla }}}^{2}}{2}+V({\boldsymbol{r}})+{\rm{i}}{\rm{\Omega }}(x{\partial }_{y}-y{\partial }_{x})\Space{0ex}{3.08ex}{0ex})+{c}_{2}| {{\rm{\Psi }}}_{\downarrow }{| }^{2}\right.\\ & & \left.+{c}_{12}| {{\rm{\Psi }}}_{\uparrow }{| }^{2}]{{\rm{\Psi }}}_{\downarrow }+[B(x-{\rm{i}}y)+\gamma (-{\rm{i}}{\partial }_{y}-{\partial }_{x})\Space{0ex}{3.08ex}{0ex}\right]{{\rm{\Psi }}}_{\uparrow }.\end{array}\end{eqnarray}$
The length (x and y), energy (SOC, rotation and interaction) and magnetic field gradient are measured in units of $\sqrt{{\hslash }/m{\omega }_{\perp }}$ , ℏω⊥ and ${\hslash }{\omega }_{\perp }/({g}_{F}{\mu }_{B}{a}_{h})$ , respectively. Where ${a}_{h}=\sqrt{{\hslash }/m{\omega }_{\perp }}$ is characteristic length of harmonic plus quartic trap. We assume that the intraspecies repulsions c1 = c2 = c and the relative repulsion λ = c12/c in the present paper.
3. Ground-state phases
The ground-state phases of spin-1/2 BECs with isotropic SOC loaded in a harmonic plus quartic trap with gradient magnetic field can be obtained by using the time-splitting Fourier spectral method with the imaginary-time propagation [47, 48], it makes the substitution $t\to {\rm{i}}t$ in equation (3 ), which leads to an exponential decay of the wave function, and a corresponding decay of the eigenstates via. The eigenenergy governs the decay rate, and so the eigenstate with the lowest energy, i.e. the ground state of the system. The spatial and time steps employed in this paper are Δx(y) = 0.05 and ${\rm{\Delta }}t=0.0001$ . The interspecies repulsion c = 1000 and the relative repulsion λ = 1.2 are fixed.
We first study on the ground-state phases of the relatively simple case without SOC (γ = 0). Figure 1 shows the ground-state densities of spin–orbit coupled spin-1/2 BECs with the rotational frequencies ω = 1.0 in (a)–(b) and ω = 1.4 in (c)–(d), respectively. In figure 1(a), the rotation frequency is too small to generate vortex in the harmonic plus quartic trap with magnetic field gradient strength B = 0. The ground-state phase is a plane wave phase, which is similar with the case without rotation. However, the ground-state phase becomes a stable phase with gradient magnetic field B = 10 in figure 1(b). The strength of the center of the gradient magnetic field is zero, which arises the singularity of the topological defect. The densities show a discrete soliton in spin- $\uparrow $ component and a Mermin-Ho vortex in spin- $\downarrow $ component at the center position of trap. As the rotational frequency is increased to ω = 1.4, the vortex lines are shown and they are symmetric about the diagonal direction without magnetic field gradient strength B = 0 in figure 1(c). We investigate the effects of magnetic field gradient on the vortices. The vortices realize the vortex rings with magnetic field gradient strength B = 10 in figure 1(d). A variety of shapes can be formed by the vortices in the inner ring as the magnetic field gradient increases gradually, such as triangular, square and so on.
Figure 1. The ground-state density profiles of rotating spin-1/2 BECs loaded in a harmonic plus quartic trap with gradient magnetic field. The columns in every panel from left to right are the densities of spin- |
When considering SOC, the competition between SOC, rotation and gradient magnetic field leads to the bountiful ground-state phases. Now we study the effects of gradient magnetic field B on the ground-state phases and fix the SOC strength γ = 2. The ground-state phases are shown in figures 2 and 3 with rotational frequencies ω = 1.0 and ω = 1.4, respectively. We find that the vortices arranged radially around the center of the circle without magnetic field gradient B = 0 in figure 2(a), which realize the vortex lattices. As the magnetic field gradient strength increases, the vortex lattices structure evolved into the vortex lines. The vortex lines along the oblique direction, as shown in figures 2(b)–(c). The spin-flip of SOC can generate the vortices, however, the spin is not in the plane. The gradient magnetic field can flip the atomic magnetic moment in the non-plane, it will drive the spin along direction of the gradient magnetic field. The vortices are distributed uniformly along the off-diagonal and diagonal directions. However, the gradient magnetic field inhibits the generation of vortices, which results in the number of vortex lines is unstable, we take the ground-state phase as the unstable phase. As the magnetic field gradient strength increases further, a single vortex line along the diagonal direction is shown in figure 2(d). Though the number of vortices of vortex line decreases, the shape of the ground-state phase keeps unchanged as the gradient magnetic field increases. We take the ground-state phase as the single vortex-line phase.
Figure 2. The ground-state density profiles of rotating spin–orbit coupled spin-1/2 BECs loaded in a harmonic plus quartic trap with gradient magnetic field. The SOC strength γ = 2 and rotation strength ω = 1. The columns in every panel from the left to right are the densities of spin- |
Figure 3. The ground-state density profiles of spin–orbit coupled spin-1/2 BECs in a harmonic plus quartic trap with gradient magnetic field. The SOC strength γ = 2 and rotation strength ω = 1.4. The columns in every panel from the left to right are the densities of spin- |
For the large rotation strength ω = 1.4, the vortices of the ground-state phase arranged radially around the center of the circle without magnetic field gradient B = 0 in figure 3(a). The unstable phases are also shown in figures 3(b)–(e). In the unstable phases regime, with the magnetic field gradient increasing, we find that the number of the vortices along the off-diagonal direction is a constant value in figures 3(c)–(e), which is different from the shown in figure 2. As the magnetic field gradient strength increases further, the strong magnetism of the gradient magnetic field leads to the atoms hard to condensed in the center of the harmonic plus quartic trap. The vortex line evolves into the vortex ring structure. The giant and hidden vortex structures are found at the center of ring. The hidden vortex is only reported in the single-component BEC in [49, 50]. We take the ground-state phase with the vortex ring structure as the vortex-ring phase, as shown in figure 3(f). The strong magnetic field gradient makes the number of the vortices around the ring is a constant value (n = 6).
The phase distributions of the ground-state phase are studied in figure 4. For the small rotation, we can see that the phase distributions from the general vortex structure to the vortex alignment with increasing the magnetic field gradient in figures 4(a)–(b). In figures 4(c)–(d), it is found that the hidden vortex structure at the center of ring with the large rotation. Figure 5 plots the corresponding spin textures of the ground-state phases in figure 4. The spin texture is ${S}_{x}=({{\rm{\Psi }}}_{\uparrow }^{* }{{\rm{\Psi }}}_{\downarrow }+{{\rm{\Psi }}}_{\downarrow }^{* }{{\rm{\Psi }}}_{\uparrow })/| {\rm{\Psi }}{| }^{2}$ , ${S}_{y}=-{\rm{i}}({{\rm{\Psi }}}_{\uparrow }^{* }{{\rm{\Psi }}}_{\downarrow }-{{\rm{\Psi }}}_{\downarrow }^{* }{{\rm{\Psi }}}_{\uparrow })/| {\rm{\Psi }}{| }^{2}$ , ${S}_{z}\,=(| {{\rm{\Psi }}}_{\uparrow }{| }^{2}-| {{\rm{\Psi }}}_{\downarrow }{| }^{2})/| {\rm{\Psi }}{| }^{2}$ . The Skymiron line structures are shown in the spin textures of figures 5(a)–(d).
Figure 5. Panels (a)–(d) are the corresponding spin texture of the ground-state phases in figure 4. The color of the arrows indicates the value of Sz. |
4. Phase diagrams
In order to observe the ground-state phases in experiment easily, we study the phase diagrams of rotating spin–orbit coupled spin-1/2 BECs in a gradient magnetic field. The ground-state phase diagram is spanned by the SOC strength γ and magnetic field gradient strength B with the different relative repulsions λ = 1.2 and λ = 0.6, one can be seen in figure 6. The rotation strength ω = 1. As the increase of gradient magnetic field strength, the BECs experiences a phase transition from the unstable phase to the single vortex-line phase. The unstable phase presents the vortex lines structures along the off-diagonal direction. With magnetic field gradient strength increases, the number of vortex lines changes accordingly (see figures 1(b)–(c)). The single vortex-line phase has a single vortex line along the diagonal direction. The shape of the single vortex-line phase keeps unchanged as the gradient magnetic field increases (see figure 1(d)). The phase diagram shows that the boundaries between the two phases is linear with the relative repulsion λ ≥ 1 (see black dotted line λ = 1.2) and is nonlinear with λ < 1 (see red dotted line λ = 0.6). For the fixed gradient magnetic field, the time reversal symmetry of spin–orbit coupled spin-1/2 BECs is preserved when the relative repulsion λ ≥ 1. The relative repulsion inhibits the formation of the single vortex-line phase.
Figure 6. The ground-state phase diagram is spanned by the SOC strength γ and magnetic field gradient strength B with the different relative repulsions λ = 1.2 and λ = 0.6. The rotation strength ω = 1. Two types of ground-state phases are shown. |
We also discuss the ground-state phase diagram is spanned by rotation strength ω and magnetic field gradient strength B in figure 7. The SOC strength γ = 2 and the relative repulsion λ = 1.2. Three types of the ground-state phases are found. In addition to the unstable phase and the single vortex-line phase, the vortex-ring phase is formed for the strong magnetic field gradient and rapid rotation (ω ≥ 1.3). The vortex-ring phase shows the giant and hidden vortex structures at the center of ring. The strong magnetic field gradient makes the number of the vortices around the ring unchanged (see figure 3(f)).
Figure 7. The ground-state phase diagram is spanned by the rotation strength ω and magnetic field gradient strength B with the SOC strength γ = 2. Three types of ground-state phases are shown. |
5. Summary
In summary, we have investigated the ground-state phases of 2D rotating spin–orbit coupled spin-1/2 BECs loaded in a harmonic plus quartic trap with gradient magnetic field. The competition between gradient magnetic field, SOC and rotation leads to a variety of ground-state phase structures. In the weakly rotation regime, as the increase of gradient magnetic field strength, two types of ground-state phases are found with a fixed SOC, i.e. the unstable phase and the single vortex-line phase. The unstable phase presents the vortex lines structures along the off-diagonal direction. With magnetic field gradient strength increases, the number of vortex lines changes accordingly. As the magnetic field gradient strength increases further, a single vortex line along the diagonal direction, the single vortex-line phase is found. The phase diagram shows that the boundary between the two phases is linear with the relative repulsion λ ≥ 1 and is nonlinear with λ < 1. In the relatively strong rotation regime, in addition to the unstable phase and the single vortex-line phase, the vortex-ring phase is formed for the strong magnetic field gradient and rapid rotation. The vortex-ring phase shows the giant and hidden vortex structures at the center of ring. The strong magnetic field gradient makes the number of the vortices around the ring unchanged (n = 6).