# SU(3) Spin–Orbit Coupled Rotating Bose–Einstein Condensate Subject to a Gradient Magnetic Field

^{1}College of Intelligent Manufacturing, Sichuan University of Arts and Science, DaZhou, China^{2}Industry Technology Research Institute of Intelligent Manufacturing, Sichuan University of Arts and Science, DaZhou, China^{3}College of Chemistry and Chemical Engineering, Sichuan University of Arts and Science, DaZhou, China^{4}Department of Physics, Shaanxi University of Science and Technology, Xi’an, China

We consider a harmonically trapped rotating spin-1 Bose–Einstein condensate with SU(3) spin–orbit coupling subject to a gradient magnetic field. The effects of SU(3) spin–orbit coupling, rotation, and gradient magnetic field on the ground-state structure of the system are investigated in detail. Our results show that the interplay among SU(3) spin–orbit coupling, rotation, and gradient magnetic field can result in a variety of ground states, such as a vortex ring and clover-type structure. The numerical results agree well with our variational analysis results.

## 1 Introduction

Spin–orbit coupling (SOC) plays an important role in a variety of physics branches. The realization of SOC in neutral atomic gases has attracted major attention both theoretically and experimentally [1–10]. Previous works have shown that the SOC can induce a variety of topological defects, such as skyrmions and half-quantum vortices and solitons, which are useful for the design and exploration of new functional materials [11–18]. Besides the ground-state structure, the dynamic properties of the system are significantly altered in the presence of SOC due to the close relationship between the spin and motional degrees of freedom in the topological excitations [19–24].

With the development of low-temperature technology, various types of SOC, such as Rashba, Dresselhaus, Weyl, and spin–orbit angular momentum, can be realized in experiments [1, 4, 25, 26]. However, most of previous studies of SOC have focused on the type of SU(2), where the internal states are coupled to their momenta via the SU(2) Pauli matrices. Only few works consider the SU(3) SOC, where the spin operators are spanned by the Gell-Mann matrices, which is more effective in describing the internal couplings among three-component condensates, such as a spin-1 Bose–Einstein condensate (BEC). Recently, Han and his co-authors have considered a homogenous SU(3) SO-coupled Bose gas and obtained the double-quantum spin vortices [27]. On the base of their pioneering research work, Li and Chen have studied the SU(3) SO-coupled BEC confined in a harmonic plus quartic trap [28]. Very recently, the ground states of a harmonically trapped spin-1 BEC with SU(3) BEC affected by the external rotation are investigated in [29], where a clover-type ground-state structure is discovered.

Nowadays, the gradient magnetic field has attracted more and more attention. More specifically, it is found that a variety of topological defects, such as a magnetic monopole and quantum knot, and even the artificial SOC can be realized by controlling the gradient magnetic field [30–34]. Li and his co-authors have investigated the ground state of three-component BEC in the gradient magnetic field and obtained the central Mermin–Ho vortex, magnetic monopole, and symmetry vortex lattices [35, 36]. They have also investigated the ground state of SU(2) SO-coupled BEC in the magnetic field and found that the skyrmion chain can induced by the isotropic SU(2) SOC [37]. As far as we know, there is little study on an SU(3) SO-coupled BEC subject to both rotation and gradient magnetic field, which is we attempt to do. In this work, we consider an SU(3) SO-coupled rotating BEC subject to a gradient magnetic field and show that the system has a rich variety of ground states. Here, we want to note that, in real experiments, it is difficult to realize a rotating BEC with fixed SOC [38–40]. To do so, we can rotate the lasers creating SOC for an isotropic trap or rotate both the lasers creating SOC and an anisotropic trap, which leads to an effective time-independent Hamiltonian, *H*_{eff} = *H* − Ω*L*_{z}, describing the system in a rotating frame of Ref. [41].

The rest of this paper is organized as follows. In Section 2, we formulate the theoretical model describing an SU(3) SO-coupled BEC subject to both rotation and gradient magnetic field and briefly introduce the numerical method. The numerical results and its corresponding theoretical analysis through the variational approach are presented in Section 3. Finally, in Section 4, the main results of the work are summarized.

## 2 Model and Method

To begin with, we consider a quasi-two-dimensional (Q2D) spin-1 BEC with SU(3) SOC [27], which is confined in a controllable magnetic field [36]. Under the mean-field approximation, the Hamiltonian of such a system can be written as [42–44]

where **r** = (*x*, *y*) denotes the spinor order parameters of the spin-1 BEC with normalization *∫*Ψ^{+}Ψ*d*^{2}*r* = *N*. *M* is the atomic mass, *n* = *∑*_{m}*n*_{m} with *n*_{m} = |*ψ*_{m}(*r*)|^{2}, and *m* = 0, ±1 is defined as the condensate density. The Q2D geometry can be realized by imposing a strong harmonic potential *ω*_{z} ≫ *ω*_{⊥}. In this case, the external trapping potential can be written as *ω*_{⊥} being the radial trapping frequencies. For the interaction terms, the coupling parameters are given by *a*_{0,2} are two-body *s*-wave scattering lengths for total spins 0 and 2, respectively. *L*_{z} = − *iℏ*(*x∂*_{y} − *y∂*_{x}) is the *z*-component of the angular momentum, and Ω is the angular frequency of the external rotation. The vector of spin-1 matrices is defined by *F*_{x}, *F*_{y}, and *F*_{z} are the 3 × 3 Pauli spin-1 matrices.

The SU(3) SOC considered in the present work can be written as *ν*_{so} = *κλ* ⋅ *p*, where *κ* is the strength of SU(3) SOC and *λ*_{x} = *λ*^{(1)} + *λ*^{(4)} + *λ*^{(6)} and *λ*_{y} = *λ*^{(2)} − *λ*^{(5)} + *λ*^{(7)}, with *λ*^{(i)}(*i* = 1, 2, …8) being the Gell-Mann matrices. In this case, the generators of the SU(3) group can be written as [45]

Note that the SU(3) SOC in the Hamiltonian involves all the pairwise couplings between three states. This is distinct from the SU(2) case, where the state of *ψ*_{1}(*r*) and *ψ*_{−1}(*r*) cannot be coupled directly. In real experiments, the Hamiltonian with SU(3) SOC can be realized using a similar method of Raman dressing as in the SU(2) case, where three laser beams with different polarizations and frequencies, intersecting at an angle of 2*π*/3, are used for the Raman coupling [27]. The external magnetic field is given by *B*_{1} = *B* is the magnetic field gradient in the 2D plane, and we focus on the case of axial bias field *B*_{2} = 0. The Lande factor *μ*_{B} is the Bohr magneton. The ground state and dynamics of the system can be described by the following dimensionless coupled Gross–Pitaevskii equations:

where the length and time units are chosen as *ω*_{⊥}, *κ* = *κ*/(*ℏω*_{⊥}*l*_{⊥}), *B* = *B*g_{F}*μ*_{B}*l*_{⊥}/(*ℏω*_{⊥}), and

## 3 Results and Discussions

In what follows, we will perform a detailed numerical study of the effects of system parameters on the ground-state structure. The richness of the present system lies in the large number of free parameters, which include the spin-dependent and spin-independent contact interactions, SU(3) SOC, rotation, and gradient field. To highlight the effects of the SOC, rotation, and gradient field, we focus on the antiferromagnetic condensate with *c*_{2} > 0 and fix the contact coupling parameters *c*_{0} = 100 and *c*_{2} = 3.

We first consider the system without SOC and fix the rotation frequency as Ω = 0.7. The typical density and phase distributions of such a system are shown in Figure 1 for different gradient magnetic fields, where only component 1 is shown as the other two components show similar behavior. In the absence of both gradient field (*B* = 0) and SU(3) SOC (*κ* = 0), the system is located at the center of the external harmonic potential, and no other lumps are formed. In this case, discrete vortices are formed due to the external rotation, but no vortex lattice can be formed due to the small contact interactions, as shown in Figure 1A. In the presence of a gradient magnetic field, the density distribution of the system shows a ring structure, and the size of center hole increases with the strength of the gradient magnetic field, as shown in Figures 1B,C for *B* = 1 and *B* = 2, respectively. In these cases, a vortex ring is formed along the ring direction (see Figure 1F for the phase distribution of Figure 1C), and the number of vortex also increases with the gradient magnetic field. Furthermore, if we increase the strength of gradient magnetic to *B* = 3 (see Figure 1D), most of vortices begin to move into the central hole, and the vortex ring is gradually destroyed. Upon increasing the gradient field to *B* = 4, all the vortices move to the central hole and form a giant vortex at the center of the trapping potential. In this case, no visible vortex remains, as shown in Figure 1E. Comparing Figures 1A–E, it is found that the radius of the ring increases with the gradient magnetic field.

**FIGURE 1**. Typical ground-state structures of a rotating BEC as a function of the gradient magnetic field in the absence of SU(3) spin–orbit coupling. **(A)**–**(E)** Density distributions of component 1 for different gradient magnetic fields: **(A)** *B* = 0, **(B)** *B* = 1, **(C)** *B* = 2, **(D)** *B* = 3, and **(E)** *B* = 4. **(F)** The corresponding phase distribution of **(C)**. Other parameters are fixed as *c*_{0} = 100, *c*_{2} = 3, Ω = 0.7, and *κ* = 0, and the scale of each figure is [−12.8,12.8] in units of

Now, we turn our attention to the combined effects of SU(3) SOC and gradient magnetic on the ground-state structure of the system and thus set Ω = 0. Figure 2 shows the typical density and phase distributions of the system for the varying SU(3) SOC and gradient magnetic field. Previous studies on the SU(3) SOC Bose gases have shown that the triangular lattice structure is energy favorable for both homogeneous and confined systems [27, 29]. In the present system, we find a similar triangular lattice structure in the absence of the gradient magnetic field (*B* = 0), as shown in Figure 2A. In the presence of a gradient magnetic field, it is found that, with the increasing strength of the gradient magnetic field, the triangular lattice structure is gradually destroyed. In addition, the system evolves into three parts in space and eventually into a clover-type structure, as shown in Figures 2B–D for *B* = 1, 2, 3, respectively. Here, we want to note that a similar clover-type structure has been previously discovered in an SU(3) SO-coupled Bose gas with rotation [29]. However, the clover-type structure found in the present work is induced by the combined effects of both SU(3) SOC and a gradient magnetic field, which give us another way to realize the clover-type structure in a non-rotating system. It is interesting to observe that the distance among such three space parts of the clover-type structure is unchanged if we increase the strength of SU(3) SOC but fix the strength of the gradient magnetic field, as shown in Figures 2D,E for *B* = 3, *κ* = 3 and *B* = 3, *κ* = 5, respectively. If we look at the phase distribution of the clover-type structure, as shown in Figure 2F for Figure 2D, it is easy to find that the local lump is in the plane wave phase, which is consistent with our previous results on the SU(3) SO-coupled system [27].

**FIGURE 2**. Typical ground-state structures of an SU(3) spin–orbit-coupled BEC as a function of SU(3) spin–orbit coupling and gradient magnetic field in the absence of rotation. **(A)**–**(D)** Density distributions of component 1 for fixed SU(3) spin–orbit coupling *κ* = 3 and for different gradient magnetic fields: **(A)** *B* = 0, **(B)** *B* = 1, **(C)** *B* = 2, and **(D)** *B* = 3. **(E)** Density distribution of the system for *B* = 3 and *κ* = 5. **(F)** The corresponding phase distribution of **(D)**. Other parameters are fixed as *c*_{0} = 100, *c*_{2} = 3, and Ω = 0, and the scale of each figure is [-12.8,12.8] in units of

Finally, we move to the combined effects of SU(3) SOC, rotation, and the gradient magnetic field on the ground-state structure of the system. To highlight the effect of SU(3) SOC, we further fix Ω = 0.7. Figure 3 shows the typical ground-state structures of a rotating BEC as a function of the SU(3) SOC and gradient magnetic field. In the absence of SU(3) SOC, the system shows a ring structure with a vortex ring (see Figure 3A for *B* = 2 and *κ* = 0), which is consistent with that reported in Figure 1C. In the presence of SU(3) SOC, it is found that the annular structure is destroyed even for a small SU(3) SOC, as shown in Figure 3B for *κ* = 0.5 and *B* = 2, where two asymmetry lumps with visible vortices are formed. Further increasing the strength of SU(3) SOC, this tendency is becoming more and more obvious, and eventually, the system evolves into a single lump, as shown in Figures 3C,D, and F (the corresponding phase distribution of (Figure 3D)) for *κ* = 1 and *κ* = 2, respectively. This is different from the phase diagram in [38], where the density distributions show some symmetry and the center of mass of the system is still around the center of the trap (which also can be seen from their momentum distributions). For the present case, the symmetry of the system is further broken and the center of mass of the system deviates from the center of the trap. Actually, such difference can be understood by the fact that the SU(3) SO-coupled system has three discrete minima in the single-particle energy spectrum, and the number and weight of such three minima are selected for different strengths of SOC. Similar to the former case, it is interesting to observe that the distance between the lump and the center of trapping potential increases with the gradient magnetic field (see Figure 3E for *B* = 1 and *κ* = 2). Here, we want to note that we have also calculated other rotation frequencies, and the results show similar behavior.

**FIGURE 3**. Typical ground-state structures of a rotating BEC as a function of the SU(3) spin–orbit coupling and gradient magnetic field. **(A)**–**(D)** Density distributions of component 1 for a fixed gradient magnetic field *B* = 2 and for different SU(3) spin–orbit coupling: **(A)** *κ* = 0, **(B)** *κ* = 0.5, **(C)** *κ* = 1, and **(D)** *κ* = 2. **(E)** Density distribution of the system for *κ* = 2 and *B* = 1. **(F)** The corresponding phase distribution of **(D)**. Other parameters are fixed as *c*_{0} = 100, *c*_{2} = 3, and Ω = 0.7, and the scale of each figure is [−12.8,12.8] in units of

To give a clearer understanding of the above results, we employ the variational method and analytically calculate the possible ground state. We begin with the single-particle energy spectrum of the system, which can be obtained by diagonalizing the kinetic energy and SOC terms. It is found that there exist three discrete minima residing on the vertices of an equilateral triangle in the momentum space. As discussed in [27], a threefold-degenerate plane wave ground state is selected as the ground state for a ferromagnetic condensate with SU(3) SOC, while three discrete minima with unequal (equal) weights are selected for the antiferromagnetic case. Consequently, the variational wave function can be written as Ψ = Ψ_{1} + Ψ_{2} + Ψ_{3}, where

where Ψ_{1}, Ψ_{2}, and Ψ_{3} are the wave functions corresponding to the three minima of the single-particle spectrum, ( −2*κ*, 0),

It is found that the minimum energies are equal to *x*-axis, the second quadrant and the third quadrant. In addition, we find that these positions are only related to the gradient magnetic field *B* and show independence with SU(3) SOC *κ*. All the results are consistent with those reported in Figure 2. When the rotation is included, we have

In this case, the second equation has the minimum energy *B*, which are consistent with those reported in Figure 3.

## 4 Conclusion

We have investigated the ground-state structure of a harmonically trapped rotating spin-1 BEC with SU(3) SOC subject to a gradient magnetic field. In the absence of SU(3) SOC, the system shows an annular structure, where a vortex ring is formed. In the absence of rotation, it is found that the clover-type structure discovered in the previous work can also be induced by the combined effects of SU(3) SOC and gradient magnetic field, and their distance shows sole dependence on the gradient magnetic field. When the rotation is included, we found that only one lump is formed in the three quadrants, and the distance between the lump and the center of trapping potential increases with the gradient magnetic field. Finally, we have employed the variational method and analytically calculated the possible ground state, which agrees well with our numerical simulations.

## Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

## Author Contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

## Funding

This work was supported by the National Natural Science Foundation of China under Grant Nos. 12075163, 12175129, 11775253, 12175027, and 11875010; the Dazhou Science and Technology Bureau under Grant No. 2019CYR0001; and the Natural Science Foundation of Chongqing under Grant No. cstc2019jcyj-msxmX0217.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s Note

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Keywords: Bose–Einstein condensate, SU(3) spin–orbit coupling, gradient magnetic field, rotation, GP equation

Citation: Chen G-P, Tu P, Qiao C-B, Zhu J-X, Jia Q and Zhang X-F (2021) SU(3) Spin–Orbit Coupled Rotating Bose–Einstein Condensate Subject to a Gradient Magnetic Field. *Front. Phys.* 9:768799. doi: 10.3389/fphy.2021.768799

Received: 01 September 2021; Accepted: 29 September 2021;

Published: 02 November 2021.

Edited by:

Zhaoxin Liang, Zhejiang Normal University, ChinaReviewed by:

Qing Sun, Capital Normal University, ChinaWei Yi, University of Science and Technology of China, China

Xiang-Fa Zhou, University of Science and Technology of China, China

Copyright © 2021 Chen, Tu, Qiao, Zhu, Jia and Zhang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Guang-Ping Chen, chengp205@126.com; Xiao-Fei Zhang, xfzhang_physics@163.com