Dipolar Bose gas with SU(3) spin−orbit coupling held under a toroidal trap

Abstract

We consider a dipolar spin-1 Bose gas with SU(3) spin−orbit coupling trapped in a two-dimensional toroidal trap. Due to the combined effects of SU(3) spin−orbit coupling, dipole−dipole interaction, and spin−exchange interaction, the system exhibits a rich variety of ground-state phases and topological defects, including modified stripe, azimuthal distributed petal and triangular lattice, double-quantum spin vortices, and so on. In particular, by studying the spin texture of such a system, it is found that the formation and transformation between meron and skyrmion topological spin textures can be realized by a choice of dipole−dipole interaction, SU(3) spin−orbit coupling, and spin−exchange interaction. We also give an experimental protocol to observe such novel states within current experimental capacity.

Graphical abstract

Keywords

Bose−Einstein condensate / spin−orbit coupling / dipolar condensate / quantum vortex

1 Introduction

The spin−orbit coupling (SOC) plays a very significant role in several fields of physics, such as the spin Hall effect, topological insulators, and topological superconductors [1-5]. In recent years, experimental realizations of synthetic SOC in Bose−Einstein condensates (BEC) has provided a fresh platform for exploring exotic quantum phenomena and new states of matter [6-11]. In particular, various forms of canonical field couplings have been recently proposed, such as SU(3) SOC [12, 13] and spin tensor momentum couplings [14]. Previous results have been shown that SOC can generate abundant topological defects and completely new quantum phases, such as half-quantum vortex [15], Skyrmion [16], composite soliton [17, 18], topological superfluid phase [19], supersolid phase [20, 21], and lattice phase [22]. The SU(3) SOC with spin operator spanned by the Gell−Mann matrices can more efficiently describe the internal couplings among the three-components [12, 23], and therefore new quantum phases and topological defects are expected to be formed in such a system. Very recently, novel topological defects such as double-quantum spin vortex (DSV) and half-skyrmion (HS) have been observed in the ground state of SU(3) SO coupled BEC [13].

In realistic experiments of ultracold atoms, various external potentials bound to the outside can significantly affect the quantum properties of BEC. A toroidal potential with a nontrivial topology is a novel non-simply connected external potential that can be realized by the formation of a repulsive barrier in the middle of a harmonic magnetic trap by a blue-detuned laser beam [24]. Very recently, such toroidal potential has attracted extensive interests, which provides us a highly controllable platform for the study of fascinating properties of a superfluid, such as persistent flow, symmetry-breaking localization, alternately arranged stripe, and countercircling state ([25-28] and references therein). Typically, in a quasi-two-dimensional (Q2D) annular trap, the stable necklace like states and sustained flow states have been discovered for spin-1/ 2 BEC with SOC [25, 26].

Different from the traditional contact interactions, the long-range and anisotropic character of dipole-dipole interaction (DDI) makes dipolar systems unique and has received widespread attention in both experimental and theoretical studies [29-39]. In particular, with SOC, the DDI also gives rise to exotic states, such as the spontaneous spin vortex [40], quantum quasicrystals [41], higher anisotropy stripe soliton [42], meron spin configuration [43], long-range superfluid order [44], and exotic topological phase [45-49]. Up to now, to our best knowledge, there is little work that has focused on the combined effects of both DDI and SU(3) SOC on the ground state of a spin-1 Bose gas confined in a toroidal trap, which is predicted to present nontrivial ground state phases and topological structures.

In this paper, we consider a dipolar spin-1 Bose gas with SU(3) SOC confined in a Q2D toroidal trap. The ground state phase of such a system is systematically characterized under different spin exchange interactions, and the effects of SU(3) SOC and DDI on the system are discussed. It is found that the ground state phase of such a system shows a rich variety of ground-state phases and topological defects, including modified stripe, azimuthal distributed petal and triangular lattice, double-quantum spin vortices, and so on. More interestingly, the formation and transformation between meron and skyrmion topological spin textures can be realized by a proper choice of the system parameters.

The remainder of the present paper is organized as follows. In Section 2, we introduce the theoretical model of a dipolar spin-1 Bose gas with SU(3) SOC in a Q2D toroidal trap. In Section 3, we present and analyze the topological structures and typical spin textures of the system. Finally, the main results of the paper are summarized in Section 4.

2 Theoretical model

We consider the spin-1 BEC with both DDI and SU(3) SOC trapped in a Q2D toroidal trap. Within the mean-field approximation, the system is described by the Gross−Pitaevskii equation (GPE) [50-52]:

(1)\begin{array}{rl}\mathrm{i}\hslash \frac{\mathrm{\partial }{\psi }_j}{\mathrm{\partial }t}=& (-\frac{{\hslash }^2{\mathrm{\nabla }}^2}{2m}+V(r)+c_{0}n){\psi }_j\\ & +c_2\sum _{j^{\prime }=-1}^1\mathit{F}\cdot {\mathit{f}}_{j{j}^{\prime }}{\psi }_{j^{\prime }}+\sum _{j^{\prime }=-1}^1({\upsilon }_{so}{)}_{j{j}^{\prime }}{\psi }_{j^{\prime }}\\ & +c_{dd}\sum _{j^{\prime }=-1}^1\mathit{b}\cdot {\mathit{f}}_{j{j}^{\prime }}{\psi }_{j^{\prime }},\end{array}

where m being the mass of the atom. The spinor wave function {\psi }_j=( {\psi }_1,{\psi }_0,{\psi }_{-1}{)}^{\mathrm{T}} describes the occupation of three components, and the total particle density n=\sum _{j=1,0,-1}{\psi }_j^{\ast }(\mathit{r}){\psi }_j(\mathit{r}) satisfies N=\int n\mathrm{d}\mathit{r} with N being the total number of atoms. The Q2D geometry can be realized by imposing a very tight trap along the z-direction (i.e., the trap frequency {\omega }_z is so large that the dynamics is frozen into the ground state of the trap). In this case, the external trap can be described as

(2)V(r)={\displaystyle \frac{1}{2}}{I}_{0}m {\omega }_{\mathrm{\perp }}^2 (r- r_0{)}^2,

where r^2=x^2+y^2, I_0 and r_0 characterize the depth and location of the toroidal trap, and {\omega }_{\mathrm{\perp }} is the radial trap frequency of the harmonic potential. The coupling parameters c_0 and c_2 describe the strengths of density−density and spin−exchange interactions, respectively. Here, c_0=4\pi {\hslash }^2 (a_0+2a_2)\gamma /(3m) and c_2=4\pi {\hslash }^2( a_2-a_0)\gamma /(3m) are related to the s-wave scattering lengths in the spin-0 and the spin-2 channels with \gamma =\sqrt{m{\omega }_z/(2\pi \hslash )} being the harmonic trap frequency in the z-direction. \mathit{F} is the spin density, \mathit{F}=(F_x ,F_y, F_z) is defined by F_{\nu }(\mathit{r})={\psi }^{†}{f}_{\nu }\psi, with \mathit{f}=(f_x,f_y,f_z) being the vector of the spin-1 matrices given in the irreducible representation [53, 54].

The SU(3) SOC considered here can be written as {\upsilon }_{so} =\kappa \mathit{\lambda }\cdot \mathit{p} with \kappa being the strength. The Q2D momentum \mathit{p}=(p_x ,p_y), and \lambda = ({\lambda }_x,{\lambda }_y) is expressed in terms of {\lambda }_x ={\lambda }^{(1)}+{\lambda }^{(4)}+{\lambda }^{(6)} and {\lambda }_y={\lambda }^{(2)}- {\lambda }^{(5)}+{\lambda }^{(7)}, with {\lambda }^{(i)}(i=1 ,\dots ,8) being the Gell−Mann matrices [55-58]. In this case, the generators of the SU(3) group can be written as

(3)\begin{array}{r}{\lambda }_x=\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right),{\lambda }_y=\left(\begin{array}{ccc}0& -\mathrm{i}& \mathrm{i}\\ \mathrm{i}& 0& -\mathrm{i}\\ -\mathrm{i}& \mathrm{i}& 0\end{array}\right).\end{array}

The last term on the right-hand side of Eq. (1) descripes the magnetic DDI between atoms with c_{dd}={\mu }_0{g}_F^2{\mu }_B^2\gamma /(4\pi ) being its strength. {\mu }_0, {\mu }_B, and g_F are the vacuum magnetic permeability, Bohr magneton, and the Landé g-factor, respectively. Here \mathit{b} is the effective 2D dipole field defined by b_{\nu }=\int \mathrm{d}{\mathit{r}}_{\perp }^{\prime }\sum _{\nu {\nu }^{\prime }}{Q}_{\nu {\nu }^{\prime }}^{2D}{F}_{{\nu }^{\prime }}(\nu ,{\nu }^{\prime } =x,y) in which Q_{\nu {\nu }^{\prime }}^{2D} is the 2D dipole kernel related to the 3D one. By using the Fourier transform and convolution theorem to express the 2D dipole kernel and the 2D effective dipole field in momentum space [51, 52], we can obtain

(4)\begin{array}{rl}{\tilde{Q}}^{2D}(k_x,k_y)=& -\frac{4\pi }{3}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -2\end{array}\right)\\ & +4\pi \frac{G(\frac{k_{\perp }{a}_z}{\sqrt{2}})}{k_{\perp }^2}\left(\begin{array}{ccc}{k}_x^2& k_x{k}_y& 0\\ k_x{k}_y& k_y^2& 0\\ 0& 0& -k_{\perp }^2\end{array}\right),\end{array}

(5)\begin{array}{l}{b}_{\nu }={\mathcal{F}}^{-1}{\displaystyle \sum _{{\nu }^{\prime }}}{\tilde{Q}}^{2D}(k_x,k_y)\mathcal{F}[F_{{\nu }^{\prime }}],\end{array}

where k_{\perp }=\sqrt{k_x^2+k_y^2}, G(q)=2 q{\mathrm{e}}^{q^2}{\int }_q^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}\mathrm{d}t denoting the complementary error function, and \mathcal{F} represents the Fourier transform operator.

For the sake of numerical calculation, we work in dimensionless parameters via rescaling {\tilde{r}}_{\mathrm{\perp }}=r_{\mathrm{\perp }}/a_0, \tilde{t}={\omega }_{\mathrm{\perp }}t, \tilde{V}=V/(\hslash {\omega }_{\mathrm{\perp }}), {\tilde{\psi }}_j={\psi }_j{a}_0/\sqrt{N}, {\tilde{c}}_0=c_{0}N/(\hslash {\omega }_{\mathrm{\perp }}{a}_0^2), \tilde{k}=k/(\hslash {\omega }_{\mathrm{\perp }}{a}_0), and {\tilde{c}}_{dd}=c_{dd}N/(\hslash {\omega }_{\mathrm{\perp }}{a}_0^2) (the tildes are omitted in following discussion for simplicity) with a_0=\sqrt{\hslash /(m{\omega }_{\mathrm{\perp }})} being the harmonic trap length characterized by {\omega }_{\mathrm{\perp }}. Here we want to note that for a pure dipolar condensate without SOC, we can define a typically dimensionless quantity, {\epsilon }_{dd} ={\mu }_0{\mu }^{2}m/(12\pi {\hslash }^2{a}_s), which is the relative strength between the dipolar and contact interactions, to give a well description of the parameter space of such a system. In the presence of SU(3) SOC, here we do not define such dimensionless ratio, and select typical parameter values to show the possible ground state phases. In addition, according to the rescaling {\tilde{c}}_{dd} =c_{dd}N/(\hslash {\omega }_{\mathrm{\perp }}{a}_0^2), the strength of DDI, can be tuned by altering dimensionless units, the number of atoms, and c_{dd} related to the condensate with different atomic species. In this case, other relevant parameters, such as {\tilde{c}}_0 and {\tilde{c}}_2, also change with the dimensionless unit N/(\hslash {\omega }_{\mathrm{\perp }}{a}_0^2). However, their values can be tuned by the corresponding s-wave scattering lengths, which can be realized by the well-known Feshbach resonance.

The ground state of the system can be obtained numerically by solving the GPE (1) in the imaginary time evolution, and the initial state is prepared with different wave functions [59-61]. To highlight the effects of the SU(3) SOC and DDI, we further fix the location of the toroidal trap r_0=7, the depth of the toroidal trap I_0=1, and density-density interaction strength c_0=100. We would like to point out that the radius of trap will not affect the results qualitatively, similar results can also be observed for other radius. For the toroidal trap with selected parameters, the gas is mainly confined along the edge of the annular trap with \sim 0 density around the center. However, similar to the “ghost” vortex, the phase and spin texture distributions show strong dependence on the ratio between the real and imaginary parts of wave functions, leading to the nonzero phase and spin texture around the center of trapping potential.

3 Results and discussion

3.1 The effects of SU(3) SOC on the ground state structure of the system

In this section, we will perform a detailed study of the effects of SU(3) SOC and DDI on the ground state structure of a dipolar spin-1 Bose gas confined in a two-dimensional annular potential. We begin with the system with fixed DDI strength c_{dd}=10 but varied SU(3) SOC strength, and then move to the case with fixed SU(3) SOC but varied DDI strength. In each case, we first consider the condensate with ferromagnetic spin−exchange interaction and then move to the system with antiferromagnetic spin−exchange interaction.

Fig.1 displays the typical ground-state density and phase distributions of a dipolar spin-1 Bose gas with ferromagnetic spin−exchange interaction, i.e., , for fixed dipolar interaction strength, but for varied SU(3) SOC strength. In the absence of SU(3) SOC, the atoms of each component and its related phase distributions are uniformly distributed along the toroidal trap, and the system maintains rotational symmetry, as shown in Fig.1(a). For small SU(3) SOC strength, as shown in Fig.1(b) for \kappa = 0.5, the system is almost in the miscible phase, it is found that three alternatively arranged density peaks emerge in the {\psi }_1 and {\psi }_{-1} components. From the corresponding phase distribution, it is easy to see that the system is in a amplitude-modulated plane wave state, as shown in the last three columns of Fig.1(b).

With a further increase of \kappa, as shown in Fig.1(c) for \kappa =1, the density distributions of components {\psi }_1 and {\psi }_{-1} form density peaks along the direction of SU(3) SOC, exhibiting spatial antisymmetry within the ring. As \kappa is further increased, the system shows a modified stripe phase along the azimuthal angle. In this case, the total density undergoes separation into six pieces formed by alternatively arranged three components, as shown in Fig.1(d) for \kappa =2. This can be understood by looking at the phase distributions of such a system, it has been shown in previous work that for a spin-1 Bose gas with SU(3) SOC and ferromagnetic interaction, the ground state of such a system occupies one single minimum in the momentum space and corresponds to a threefold-degenerate magnetized phase, and the wave number of the ground-state plane wave increases with the strength of SU(3) SOC [13]. In the present system, the confinement of the two-dimensional annular potential leading to the deformation of plane wave and the formation of three plane waves with different wave vector directions.

We now consider the system with antiferromagnetic spin-exchange interaction, i.e., c_2>0. Fig.2 shows the typical ground-state density and phase distributions of a dipolar spin-1 Bose gas with antiferromagnetic interaction for fixed dipolar interaction, but for varied SU(3) SOC strength. In the absence of SU(3) SOC, the ground state phase of the system is similar to the ferromagnetic condensates, as shown in Fig.2(a). For small \kappa, as shown in Fig.2(b) and (c) for \kappa =0.5,1.0, respectively, it is found that the density distributions of the system form petal, and the three components gradually produce directional stripes along the azimuthal direction, which look like petals. This petal phase is in a sense reminiscent of the stripe phase in spin−orbit coupled BEC, where Rashba SOC spontaneously induces directional density stripes in a homogeneous systems [58, 62]. For strong SU(3) SOC strength, as shown in Fig.2(d) for \kappa =2.0, lattice pattern of density distribution appears in each component. Due to the confinement of the external annular potential, a triangular lattice is formed along the ring for each component, and this density pattern holds for even strong SU(3) SOC strength.

Based on the above results, we can thus conclude that without SU(3) SOC, the system exhibits uniform phase distributions. In the presence of SU(3) SOC, the phase distribution of such a system exhibits untrivial structures, which show strong dependence on the type of spin-exchange interactions. Here we want to note that for the toroidal trap considered in this work, the above stripe and lattice patterns will be distributed along the azimuthal angle direction. Consequently, each petal in the petal phase can achieve the lowest energy by shortening the boundaries. Similar to previous works, the number of petals increase with the strength of SU(3) SOC. In addition, it is interesting to observe that the components {\psi }_1 and {\psi }_{-1} exhibit spatial antisymmetric structure, while component {\psi }_0 possesses spatial rotational symmetry, and the total density of components {\psi }_1 and {\psi }_{-1} is complementary to component {\psi }_0.

3.2 The effects of DDI on the ground state structure of the system

We next turn to the effects of DDI on the ground state of the system. Fig.3 shows the typical ground-state density and phase distributions of a dipolar spin-1 Bose gas with ferromagnetic interaction for fixed SU(3) SOC strength \kappa =0.7, but for varied DDI strength. In the absence of DDI, as shown in Fig.3(a) for c_{dd} =0, the density distribution of each component is almost uniformly distributed along the ring, and the system is in a miscible state with rotational symmetry, and the phase distribution shows that this state is a plane wave occupying a single momentum in the momentum space, which indicates that the time-reversal symmetry is broken. Interestingly, it is found that the system will change from magnetized phase to supersolid-like stripe phase for sufficiently strong DDI in our numerical simulations.

In the presence of DDI, the density distributions of the system are dispersed and staggered on the ring. The component {\psi }_1 is significantly different from other two components, and the component {\psi }_0 coincides with that of component {\psi }_{-1} when rotated by \pi degree. It is interesting to observe that for large enough DDI, the density distribution of component {\psi }_1 forms a double-layer annulus and exhibits a novel layered hexagonal petal structure, as shown in Fig.3(c) for c_{dd}=100. With regard to the phase distribution of the system, it is found that the plane wave is broken and gradually redistributed along the azimuthal angle of the ring potential, showing special petal phase.

Fig.4 shows the typical ground-state density and phase distributions of the system with antiferromagnetic interaction for fixed SU(3) SOC strength \kappa =0.9, but for varied DDI strength. Different from the ferromagnetic condensate shown in Fig.3(a), the system shows phase separation even in the absence of DDI, as shown in Fig.4(a) for c_{dd}=0. If we look at the corresponding phase distributions shown in the fifth to seventh columns, it is easy to see that the phase of each component exhibits triangular lattice structure. The underlying physical mechanism lies in the fact that, for a spin-1 Bose gas with SU(3) SOC and antiferromagnetic interaction, the SU(3) SOC breaks the ordinary phase rule of spinor Bose gases and more than one single-particle minima is occupied.

In the presence of weak DDI strength, the density of component {\psi }_1 presents an interleaving hexagonal distribution, while the density distributions of remaining components tend to be dispersed, as shown in Fig.4(b) for c_{dd}=10. For strong DDI, as shown in Fig.4(c) for c_{dd} =100, the density distribution of such a system is similar to the case of ferromagnetic condensates, but with a more complicated bilayer structure.

More information can be obtained by comparing the phase distributions for fixed SU(3) SOC strength but varied DDI strength. It is not difficult to see that the type of spin−exchange interaction determines the number of minima in the single-particle energy spectrum. The phase distribution of the system show plane wave for ferromagnetic interaction, while triangular lattice structure for antiferromagnetic interaction. An increase in DDI significantly alters the ground-state density distribution, but minimally impacts the ground-state phase distribution.

3.3 Double-quantum spin vortices and skyrmions

To further shed light on the ground-state properties, we next analyze the spin textures of the system. For a spin-1 BEC, the spin density components in the three directions are specifically expressed as [63-67]

(6)\begin{array}{rl}{S}_x=& \frac{1}{\sqrt{2}}\frac{{\psi }_0^{\ast }{\psi }_1+({\psi }_1^{\ast }+{\psi }_{-1}^{\ast }){\psi }_0+{\psi }_0^{\ast }{\psi }_{-1}}{|{\psi }_1{|}^2+|{\psi }_0{|}^2+|{\psi }_{-1}{|}^2},\\ S_y=& \frac{\mathrm{i}}{\sqrt{2}}\frac{{\psi }_0^{\ast }{\psi }_1+({\psi }_{-1}^{\ast }-{\psi }_1^{\ast }){\psi }_0-{\psi }_0^{\ast }{\psi }_{-1}}{|{\psi }_1{|}^2+|{\psi }_0{|}^2+|{\psi }_{-1}{|}^2},\\ S_z=& \frac{{\psi }_1^{\ast }{\psi }_1-{\psi }_{-1}^{\ast }{\psi }_{-1}}{|{\psi }_1{|}^2+|{\psi }_0{|}^2+|{\psi }_{-1}{|}^2},\end{array}

where the spin density vector satisfies |\mathit{S}|=\sqrt{S_x^2+S_y^2+S_z^2}. The transverse magnetization is defined as S_{+}= S_x+\mathrm{i}{S}_y, and the spatial distribution of the topological structure of the system can be well described by the topological charge density

(7)q(x, y)={\displaystyle \frac{1}{4\pi }}\mathit{s}\cdot ({\displaystyle \frac{\mathrm{\partial }\mathit{s}}{\mathrm{\partial }x}}\times {\displaystyle \frac{\mathrm{\partial }\mathit{s}}{\mathrm{\partial }y}}),

with \mathit{s}=\mathit{S}/|\mathit{S}|, and the topological charge is defined as

(8)Q={\displaystyle \int q(x,y)\mathrm{d}x\mathrm{d}y.}

A spin vortex, known as a complex topology induced by symmetry breaking, is characterized by zero net mass current and quantized spin current around an unmagnetized core (in-plane magnetization has a “vortex or antivortex” structure) [51, 53, 68]. Fig.5 shows the typical spatial spin density distributions of a dipolar spin-1 Bose gas with SU(3) SOC. In the absence of DDI, as shown in Fig.5(a) and (b) for the amplitude of the total magnetization |\mathit{S}| and transverse magnetization orientation argS_{+}, we again observe the DSV previously discovered in a homogeneous SU(3) spin−orbit-coupled Bose gas with antiferromagnetic spin−exchange interaction [13]. In present system with both SU(3) SOC and DDI, it is found that the DSV can be formed with a proper choice of DDI and SU(3) SOC even for the ferromagnetic condensate. Typical spin density distributions and transverse magnetization orientation argS_{+} for an ferromagnetic condensate with c_2=-50, c_{dd}=10 and \kappa =1 are shown in Fig.5(c)−(f), from which a stable DSV is observed. More interestingly, the distribution of the longitudinal magnetization forms two inner and outer staggered nested hexagonal structures in Fig.5(e), indicating the twisted arrangement of components {\psi }_1 and {\psi }_{-1}.

Shown in Fig.6 is the typical spin textures of a dipolar spin-1 Bose gas with antiferromagnetic spin−exchange interaction and SU(3) SOC, where the first row represents the spin texture and the second row denotes the local amplifications corresponding to the first row. We first select the same parameters c_2 and c_{dd} as those in Fig.2, but with varied SU(3) SOC strength, as shown in Fig.6(a) and (b). For small SU(3) SOC strength, as shown in Fig.6(a) for \kappa = 0.3, the central region of the spin texture shows meron−antimeron pair structure, which is confirmed by calculating the topological charge Q=\pm 0.5. By increasing the strength of SU(3) SOC to \kappa =0.5, it is interesting to observe that the topological charge converges towards the center, forming a skyrmion in the central region with local topological charge Q=-1, as shown in Fig.6(b).

We next select the same spin–exchange interaction c_2 as that in Fig.4, but with varied SU(3) SOC strength and DDI strength, as shown in Fig.6(c) and (d). Another important finding of our present work is that the transformation between meron and skyrmion topological spin textures can be realized by a choice of DDI, SU(3) SOC, and spin−exchange interaction [69]. In the absence of DDI, a hexagonal lattice consists of alternatively arranged skyrmions and antiskyrmions can be formed with a proper choice of SU(3) SOC and spin−exchange interaction, as shown in Fig.6(c) for c_2=3, and \kappa = 0.3. In addition, in the presence of DDI with same parameter c_{dd} in Fig.2, two sets of alternatively arranged triangular lattices, composed of skyrmion with Q=1 and antiskyrmion with Q=-1, respectively, can be formed by decreasing the spin−exchange interaction based on Fig.6(b), as shown in Fig.6(d) for c_2=3 and \kappa =0.5.

We now provide an experimental protocol for observing the above mentioned ground state phases and topological defects. The physical system can be realized in a spin-1 BEC with ferromagnetic or antiferromagnetic spin-exchange interaction. The SU(3) SOC can be realized by using a similar method of Raman dressing, where three laser beams with different polarizations and frequencies, intersecting at an angle of 2\pi /3 are used for the Raman coupling [8, 58]. In this case, the strength of SU(3) SOC can be controlled by changing the strength of Raman lasers. With regard to the contact interactions, its magnitude, even including its sign, can be achieved by utilizing the well-known Feshbach resonance [70]. Here we want to note that, since the parameters used in our manuscript are dimensionless, and its values can be controlled by changing the dimensionless units, where the number of atoms, the characteristic length of the harmonic trap, and the mass of atoms can be used to change the magnitude of such parameters. In addition, to observe the spin textures, the size of the Bose gas in the z-direction is smaller than the dipole healing length {\xi }_{dd} and the spin healing lengths {\xi }_{sp,i}, and the magnetization-sensitive phase-contrast imaging technique, time-of-flight measurement, or the selective absorption technologies can be used to observe such topological defects. Hence we conclude that the parameters used in this work are within current experimental capacity.

4 Conclusions

In conclusion, we have investigated the ground-state phase of a dipolar spin-1 Bose gas with SU(3) SOC confined in a toroidal trap. The effects of SU(3) SOC, dipole−dipole interaction, and spin−exchange interaction are investigated for ferromagnetic and antiferromagnetic condensates, respectively. Our results show that the ground-state density distributions of the system show strong dependence on the strength of DDI for fixed SU(3) SOC, while its phase distributions is directly related to the spin−exchange interaction. Moreover, a rich variety of ground states and topological defects, such as modified stripe, azimuthal distributed petal and triangular lattice, double-quantum spin vortices, meron−antimeron pair, and skyrmion lattice, can be formed by a proper choice of the system parameters. The results deepen our understanding of spin−orbit coupled phenomena and enrich our new cognition of topological excitation in cold atomic physics.

References

[1] I. Žutić, J. Fabian, and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys. 76(2), 323 (2004)

[2] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Experimental observation of the spin-Hall effect in a two-dimensional spin–orbit coupled semiconductor system, Phys. Rev. Lett. 94(4), 047204 (2005)

[3] S. Murakami, Quantum spin Hall effect and enhanced magnetic response by spin–orbit coupling, Phys. Rev. Lett. 97(23), 236805 (2006)

[4] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82(4), 3045 (2010)

[5] X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83(4), 1057 (2011)

[6] Y. J. Lin, K. Jiménez-García, and I. B. Spielman, Spin–orbit-coupled Bose–Einstein condensates, Nature 471(7336), 83 (2011)

[7] P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Spin–orbit coupled degenerate Fermi gases, Phys. Rev. Lett. 109(9), 095301 (2012)

[8] L. Huang, Z. Meng, P. Wang, P. Peng, S. L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Experimental realization of two-dimensional synthetic spin–orbit coupling in ultracold Fermi gases, Nat. Phys. 12(6), 540 (2016)

[9] J. R. Li,J. Lee,W. Huang,S. Burchesky,B. Shteynas,F. Ç. Top,A. O. Jamison,W. Ketterle, A stripe phase with supersolid properties in spin−orbit-coupled Bose−Einstein condensates, Nature 543(7643), 91 (2017)

[10] J. Zhong, B. Tang, X. Chen, and L. Zhou, Quantum gravimetry going toward real applications, Innovation 3(3), 100230 (2022)

[11] S. C. Guo, Y. M. Xu, R. Cheng, J. S. Zhou, and X. Chen, Thermal Hall effect in insulating quantum materials, Innovation 3(5), 100290 (2022)

[12] R. Barnett,G. R. Boyd,V. Galitski, SU(3) spin–orbit coupling in systems of ultracold atoms, Phys. Rev. Lett. 109(23), 235308 (2012)

[13] W. Han, X. F. Zhang, S. W. Song, H. Saito, W. Zhang, W. M. Liu, and S. G. Zhang, Double-quantum spin vortices in SU(3) spin−orbit-coupled Bose gases, Phys. Rev. A 94(3), 033629 (2016)

[14] K. Sun, C. Qu, Y. Xu, Y. Zhang, and C. Zhang, Interacting spin–orbit-coupled spin-1 Bose–Einstein condensates, Phys. Rev. A 93(2), 023615 (2016)

[15] C. Wu, I. Mondragon-Shem, and X. F. Zhou, Unconventional Bose–Einstein condensations from spin–orbit coupling, Chin. Phys. Lett. 28(9), 097102 (2011)

[16] H. Hu, B. Ramachandhran, H. Pu, and X. J. Liu, Spin–orbit coupled weakly interacting Bose–Einstein condensates in harmonic traps, Phys. Rev. Lett. 108(1), 010402 (2012)

[17] H. Sakaguchi, B. Li, and B. A. Malomed, Creation of two-dimensional composite solitons in spin–orbit-coupled self-attractive Bose–Einstein condensates in free space, Phys. Rev. E 89(3), 032920 (2014)

[18] W. Han, G. Juzeliūnas, W. Zhang, and W. M. Liu, Supersolid with nontrivial topological spin textures in spin–orbit-coupled Bose gases, Phys. Rev. A 91(1), 013607 (2015)

[19] H. Wang, L. H. Wen, H. Yang, C. X. Shi, and J. H. Li, Vortex states and spin textures of rotating spin–orbit-coupled Bose–Einstein condensates in a toroidal trap, J. Phys. At. Mol. Opt. Phys. 50(15), 155301 (2017)

[20] G. I. Martone, Y. Li, L. P. Pitaevskii, and S. Stringari, Anisotropic dynamics of a spin–orbit-coupled Bose–Einstein condensate, Phys. Rev. A 86(6), 063621 (2012)

[21] Z. Wu, L. Zhang, W. Sun, X. T. Xu, B. Z. Wang, S. C. Ji, Y. Deng, S. Chen, X. J. Liu, and J. W. Pan, Realization of two-dimensional spin–orbit coupling for Bose–Einstein condensates, Science 354(6308), 83 (2016)

[22] Y. Xu, Y. Zhang, and B. Wu, Bright solitons in spin–orbit-coupled Bose–Einstein condensates, Phys. Rev. A 87(1), 013614 (2013)

[23] T. Graß, R. W. Chhajlany, C. A. Muschik, and M. Lewenstein, Spiral spin textures of a bosonic Mott insu-9 lator with SU(3) spin–orbit coupling, Phys. Rev. B 90(19), 195127 (2014)

[24] C. Ryu, M. Andersen, P. Cladé, V. Natarajan, K. Helmerson, and W. Phillips, Observation of persistent flow of a Bose–Einstein condensate in a toroidal trap, Phys. Rev. Lett. 99(26), 260401 (2007)

[25] X. F. Zhang, M. Kato, W. Han, S. G. Zhang, and H. Saito, Spin–orbit-coupled Bose–Einstein condensates held under a toroidal trap, Phys. Rev. A 95(3), 033620 (2017)

[26] A. C. White, Y. P. Zhang, and T. Busch, Odd-petal-number states and persistent flows in spin–orbit coupled Bose–Einstein condensates, Phys. Rev. A 95(4), 041604(R) (2017)

[27] J. G. Wang, L. L. Xu, and S. J. Yang, Ground-state phases of the spin–orbit-coupled spin-1 Bose gas in a toroidal trap, Phys. Rev. A 96(3), 033629 (2017)

[28] K. Liu, H. He, C. Wang, Y. Chen, and Y. Zhang, Spin–orbit-coupled spin-1 Bose–Einstein condensates in a toroidal trap: Even-petal-number necklace-like state and persistent flow, Phys. Rev. A 105(1), 013323 (2022)

[29] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Bose–Einstein condensation of chromium, Phys. Rev. Lett. 94(16), 160401 (2005)

[30] M. W. Ray, E. Ruokokoski, K. Tiurev, M. Möttönen, and D. S. Hall, Observation of isolated monopoles in a quantum field, Science 348(6234), 544 (2015)

[31] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, Observation of a dipolar quantum gas with metastable supersolid properties, Phys. Rev. Lett. 122(13), 130405 (2019)

[32] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The physics of dipolar bosonic quantum gases, Rep. Prog. Phys. 72(12), 126401 (2009)

[33] K. Góral, K. Rzążewski, and T. Pfau, Bose–Einstein condensation with magnetic dipole–dipole forces, Phys. Rev. A 61(5), 051601 (2000)

[34] L. Santos, G. V. Shlyapnikov, and M. Lewenstein, Roton–maxon spectrum and stability of trapped dipolar Bose–Einstein condensates, Phys. Rev. Lett. 90(25), 250403 (2003)

[35] F. Malet, T. Kristensen, S. M. Reimann, and G. M. Kavoulakis, Rotational properties of dipolar Bose–Einstein condensates confined in anisotropic harmonic potentials, Phys. Rev. A 83(3), 033628 (2011)

[36] B. Liu, X. Li, L. Yin, and W. V. Liu, Weyl superfluidity in a three-dimensional dipolar fermi gas, Phys. Rev. Lett. 114(4), 045302 (2015)

[37] S. Y. Chä and U. R. Fischer, Probing the scale invariance of the inflationary power spectrum in expanding quasitwo-dimensional dipolar condensates, Phys. Rev. Lett. 118(13), 130404 (2017)

[38] M. O. Borgh, J. Lovegrove, and J. Ruostekoski, Internal structure and stability of vortices in a dipolar spinor Bose–Einstein condensate, Phys. Rev. A 95(5), 053601 (2017)

[39] R. N. Bisset, P. B. Blakie, and S. Stringari, Staticresponse theory and the roton–maxon spectrum of a flattened dipolar Bose–Einstein condensate, Phys. Rev. A 100(1), 013620 (2019)

[40] Y. Deng, J. Cheng, H. Jing, C. P. Sun, and S. Yi, Spin–orbit-coupled dipolar Bose–Einstein condensates, Phys. Rev. Lett. 108(12), 125301 (2012)

[41] S. Gopalakrishnan, I. Martin, and E. A. Demler, Quantum quasicrystals of spin–orbit-coupled dipolar bosons, Phys. Rev. Lett. 111(18), 185304 (2013)

[42] Y. Xu, Y. P. Zhang, and C. Zhang, Bright solitons in a two-dimensional spin–orbit-coupled dipolar Bose–Einstein condensate, Phys. Rev. Lett. 92(1), 013633 (2015)

[43] R. M. Wilson, B. M. Anderson, and C. W. Clark, Meron ground state of Rashba spin–orbit-coupled dipolar bosons, Phys. Rev. Lett. 111(18), 185303 (2013)

[44] G. Chen, J. Ma, and S. T. Jia, Long-range superfluid order in trapped Bose–Einstein condensates with spin–orbit coupling, Phys. Rev. A 86(4), 045601 (2012)

[45] N. Q. Burdick, Y. Tang, and B. L. Lev, Long-lived spin–orbit-coupled degenerate dipolar Fermi gas, Phys. Rev. X 6(3), 031022 (2016)

[46] X. J. Feng, J. X. Li, L. Qin, Y. Y. Zhang, S. Q. Xia, L. Zhou, C. J. Yang, Z. L. Zhu, W. M. Liu, and X. D. Zhao, Itinerant ferromagnetism entrenched by the anisotropy of spin–orbit coupling in a dipolar Fermi gas, Front. Phys. 18(5), 52303 (2023)

[47] N. Su, Q. B. Wang, J. G. Hu, X. H. Su, and L. H. Wen, Topological defects in rotating spin–orbit-coupled dipolar spin-1 Bose−Einstein condensates, J. Phys. At. Mol. Opt. Phys. 53(21), 215301 (2020)

[48] Y. Y. Li, Y. Liu, Z. W. Fan, W. Pang, S. H. Fu, and B. A. Malomed, Two-dimensional dipolar gap solitons in free space with spin–orbit coupling, Phys. Rev. A 95(6), 063613 (2017)

[49] T. Oshima and Y. Kawaguchi, Spin Hall effect in a spinor dipolar Bose–Einstein condensate, Phys. Rev. A 93(5), 053605 (2016)

[50] M. Kato, X. F. Zhang, D. Sasaki, and H. Saito, Twisted spin vortices in a spin-1 Bose–Einstein condensate with Rashba spin–orbit coupling and dipole–dipole interaction, Phys. Rev. A 94(4), 043633 (2016)

[51] Y. Kawaguchi and M. Ueda, Spinor Bose–Einstein condensates, Phys. Rep. 520(5), 253 (2012)

[52] Y. Kawaguchi, H. Saito, K. Kudo, and M. Ueda, Spontaneous magnetic ordering in a ferromagnetic spinor dipolar Bose–Einstein condensate, Phys. Rev. A 82(4), 043627 (2010)

[53] D. M. Stamper-Kurn and M. Ueda, Spinor Bose gases: Symmetries, magnetism, and quantum dynamics, Rev. Mod. Phys. 85(3), 1191 (2013)

[54] D. S. Wang, Y. R. Shi, K. W. Chow, Z. X. Yu, and X. G. Li, Matter-wave solitons in a spin-1 Bose–Einstein condensate with time-modulated external potential and scattering lengths, Eur. Phys. J. D 67(11), 242 (2013)

[55] G. B. Arfken,H. J. Weber,F. E. Harris, Mathematical Methods for Physicists, 7th Ed., Academic Press, New York, 2000

[56] Z. F. Xu, Y. Kawaguchi, L. You, and M. Ueda, Symmetry classification of spin–orbit-coupled spinor Bose–Einstein condensates, Phys. Rev. A 86(3), 033628 (2012)

[57] Z. Lan and P. Öhberg, Raman-dressed spin-1 spin–orbit coupled quantum gas, Phys. Rev. A 89(2), 023630 (2014)

[58] Y. Zhang, L. Mao, and C. Zhang, Mean-field dynamics of spin–orbit coupled Bose–Einstein condensates, Phys. Rev. Lett. 108(3), 035302 (2012)

[59] F. Dalfovo and S. Stringari, Bosons in anisotropic traps: Ground state and vortices, Phys. Rev. A 53(4), 2477 (1996)

[60] M. L. Chiofalo, S. Succi, and M. P. Tosi, Ground state of trapped interacting Bose–Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E 62(5), 7438 (2000)

[61] W. Bao,I. L. Chern,F. Y. Lim, Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose–Einstein condensates, J. Comput. Phys. 219(2), 836 (2006)

[62] C. Wang, C. Gao, C. M. Jian, and H. Zhai, Spin-orbit coupled spinor Bose–Einstein condensates, Phys. Rev. Lett. 105(16), 160403 (2010)

[63] T. Mizushima, N. Kobayashi, and K. Machida, Coreless and singular vortex lattices in rotating spinor Bose–Einstein condensates, Phys. Rev. A 70(4), 043613 (2004)

[64] K. Kasamatsu, M. Tsubota, and M. Ueda, Spin textures in rotating two-component Bose–Einstein condensates, Phys. Rev. A 71(4), 043611 (2005)

[65] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions, Nat. Phys. 7(9), 713 (2011)

[66] S. Banerjee, J. Rowland, O. Erten, and M. Randeria, Enhanced stability of skyrmions in two-dimensional chiral magnets with Rashba spin–orbit coupling, Phys. Rev. X 4(3), 031045 (2014)

[67] B. Dong, Q. Sun, W. M. Liu, A. C. Ji, X. F. Zhang, and S. G. Zhang, Multiply quantized and fractional skyrmions in a binary dipolar Bose–Einstein condensate under rotation, Phys. Rev. A 96(1), 013619 (2017)

[68] H. Saito, Y. Kawaguchi, and M. Ueda, Breaking of chiral symmetry and spontaneous rotation in a spinor Bose–Einstein condensate, Phys. Rev. Lett. 96(6), 065302 (2006)

[69] X. Z. Yu, W. Koshibae, Y. Tokunaga, K. Shibata, Y. Taguchi, N. Nagaosa, and Y. Tokura, Transformation between meron and skyrmion topological spin textures in a chiral magnet, Nature 564(7734), 95 (2018)

[70] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82(2), 1225 (2010)

RIGHTS & PERMISSIONS

Higher Education Press