1. Introduction
The formation of a steady state structure of Bose–Einstein condensates (BECs) in optical lattices (OL) is a consistently employed method. The OL used are either periodic or quasi-periodic, such as those described by $\cos $ or $\sin $, the properties of BECs in a two-dimensional (2D) quasi-periodic OL with eightfold rotational symmetry was investigated by numerically solving the Gross–Pitaevskii equation (GPE) [1]. The out-of-equilibrium dynamics of a dilute, lattice-confined Bose–Fermi mixture initialized in a highly excited state consisting of boson-fermion pairs (doublons) occupying single lattice sites was studied [2]. A BEC was experimentally investigated, and placed in a one-dimensional (1D) OL whose phase or amplitude is modulated in a frequency range resonant with the first bands of the band structure [3].
For nonperiodic type OL, as exemplified by the Bessel optical lattices (BL), which vary spatially non-periodically and have a completely different structure depending on the order of the lattice, the existence conditions for new surface crescent, dipole, tripole, and quadrupole solitons formed at the interface of a focusing photorefractive medium and a medium imprinted with a BL were found [4]. A scheme is proposed to generate stable spatiotemporal solitons in a cold Rydberg atomic system with a BL potential, by utilizing electromagnetically induced transparency [5]. The nature of solitons in BL has also been widely studied [6–9].
For BECs in BL, this research has studied the steady state structure of BECs and their properties in a single BL, which makes the influence of the BL formation potential field on the structure of BECs very monotonous. In order to make BLs present rich structures, in our work, we firstly propose the concept of a superposed Bessel optical lattices (SBL), which makes it possible to form a rich steady state structure and manipulate the steady state structure in it.
In addition to considering the effect of potential field, a lot of interest has been recently drawn to the possibility of implementing artificially engineered spin-orbit coupling (SOC) in the spinor BECs [10]. Under the action of the competing self- and cross-interaction, the matter-wave solitons are investigated in spinor BECs, stable 2D two-component solitons of the mixed-mode type are found if the cross-interaction between the components is attractive, while the self-interaction is repulsive in each component. Stable solitons of the semi-vortex type are formed in the opposite case [11]. Gap solitons can be created in free nearly 2D space in dipolar spinor BECs with the SOC, subject to tight confinement, with size a⊥, in the third direction [12]. For quasi-2D patterns, with lateral sizes l ≫ a⊥, the kinetic-energy terms in the respective spinor. The behavior of the formation of stable bright solitons due to the presence of spin-orbit interactions is investigated in bipolar BECs of quasi-1D 164Dy atoms [13]. The shape of the anisotropic vortex solitons with respect to the in-plane polarization of the atomic dipole moments in the underlying BEC may be effectively controlled by the strength Ω of the Zeeman splitting [14]. A transition from the horizontal to vertical shape with the increase of Ω is found numerically and single-particle dynamics in SOC BECs have been studied in experiments exploring Landau–Zener tunneling and Zitterberg physics [15].
In addition to the effects of SOC interactions in atomic BECs on the system behavior, the effects of SOC interactions in exciton polarization exciton BECs on the formation of the system steady state structure have also been investigated [16, 17]. SOC between the two polarization components generated by the TE-TM energy splitting of a hole-photon acts in conjunction with Zeeman splitting to lift the degeneracy between vortex solitons of the opposite topological charge and to make them have different density profiles at fixed energies. Two-dimensional ground states and vortex solitons in polaronic condensates with SOC and Zeeman splitting evolving in microcavity column arrays was studied [18]. In addition, the exciton polaritons in the micropillar two-dimensional Lieb lattice are studied in detail [19].
In summary, the formation of steady state structures in BECs may be affected by a number of factors, but the two most intriguing aspects are the role of the potential field and the interactions between particles. In this paper, based on the introduction of a superposed BL potential field, we study the formation of steady state structures in atomic BECs in the presence of SOC interactions, and realize the manipulation of the steady state structures by adjusting the parameters of the SBL and SOC interaction.
2. Model and superposed Bessel lattice
Under the mean-field approximation, the evolution of the spinor wave functions of the BECs, ψ = (ψ1, ψ2), is governed by the coupled Gross–Pitaevskii equations [20–25],
$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}\displaystyle \frac{\partial {\psi }_{1}}{\partial t} & = & \left[-\displaystyle \frac{1}{2}\left(\displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}+\displaystyle \frac{{\partial }^{2}}{\partial {y}^{2}}\right)+{V}_{1}(x,y)\right.\\ & & \left.+{g}_{1}| {\psi }_{1}{| }^{2}+{g}_{12}| {\psi }_{2}{| }^{2}-{\rm{\Omega }}{\hat{L}}_{z}\right]{\psi }_{1}+{\hat{\partial }}_{-}{\psi }_{2},\\ {\rm{i}}\displaystyle \frac{\partial {\psi }_{2}}{\partial t} & = & \left[-\displaystyle \frac{1}{2}\left(\displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}+\displaystyle \frac{{\partial }^{2}}{\partial {y}^{2}}\right)+{V}_{2}(x,y)\right.\\ & & \left.+{g}_{2}| {\psi }_{2}{| }^{2}+{g}_{12}| {\psi }_{1}{| }^{2}-{\rm{\Omega }}{\hat{L}}_{z}\right]{\psi }_{2}-{\hat{\partial }}_{+}{\psi }_{1},\end{array}\end{eqnarray}$
where g1,2 is the strength of the interaction within the components, g12 is the strength of the interaction between the components. ${\hat{\partial }}_{\pm }={\alpha }_{x}\partial /\partial x\pm {\rm{i}}{\alpha }_{y}\partial /\partial y$, αx,y denotes the SOC interaction strength in the x and y directions, respectively. ${\hat{L}}_{z}=-{\rm{i}}(x{\partial }_{y}-y{\partial }_{x})$ is the angular momentum operator and Ω is the angular velocity.The potential energy term V1,2(x, y) in the equation plays a crucial role in the formation of the steady state structure. The choice of a SBL is our point of innovation. The optical lattices used to trap atoms so that they can form BECs, which take the form of periodic forms, e.g. $\cos $ or $\sin $ forms, and nonperiodic forms, and the BL is a typical nonperiodic light lattice. Since the structure of BL is described by the Bessel function, its basic form is $-{V}_{0}{J}_{n}^{2}(\sqrt{2b}r)$, where the minus sign is for the formation of the potential well structure to constrain atoms for BECs formation, V0 is the depth of the potential well, ${J}_{n}(\sqrt{2b}r)$ is the n-order Bessel function with space variable r, and b is the scaling factor of the Bessel function.
In order to identify the structure of the SBL, for simplicity, let us analyze the structure of the BL first. Figure 1 gives the curves of the basic form BL $-{V}_{0}{J}_{n}^{2}(\sqrt{2b}r)$ with respect to r, which correspond to the orders n = 0, 1, 2, respectively. It is clear that when the curve is rotated around the axis perpendicular to the r direction, a potential well structure is formed at the extremes of the function, with the difference that a potential well structure is formed in the center of the BL when n = 0, and a toroidal potential well structure is formed when n = 1,2.
Figure 1. Basic form BL $-{V}_{0}{J}_{n}^{2}(\sqrt{2b}r)$ curve with r, V0 = 1, b = 0.5. |
The SBL we propose is a linear superposition of m BLs with different center positions or orders, and is expressed as
$\begin{eqnarray}V(x,y)=\displaystyle \sum _{i=1}^{m}-{V}_{0i}{J}_{{n}_{i}}^{2}\left(\sqrt{2{b}_{i}}\sqrt{{\left(x-{\tilde{x}}_{i}\right)}^{2}+{\left(y-{\tilde{y}}_{i}\right)}^{2}}\right),\end{eqnarray}$
where the ith (i = 1, 2 ⋯ , m) BL is described as $\begin{eqnarray*}-{V}_{0i}{J}_{{n}_{i}}^{2}\left(\sqrt{2{b}_{i}}\sqrt{{\left(x-{\tilde{x}}_{i}\right)}^{2}+{\left(y-{\tilde{y}}_{i}\right)}^{2}}\right),\end{eqnarray*}$
ni denotes the order of the BL, V0i denotes the intensity of the BL, bi denotes the scaling factor of the BL, $({\tilde{x}}_{i},{\tilde{y}}_{i})$ is the center coordinate of the ith BL, by adjusting the value of $({\tilde{x}}_{i},{\tilde{y}}_{i})$, the BL can be adjusted. Therefore, the structure of the superposition potential field can be completely adjusted by adjusting the parameter set $(m,{n}_{i},{V}_{0i},{b}_{i},{\tilde{x}}_{i},{\tilde{y}}_{i})$.A typical SBL structure is shown in figure 2, which represents the SBL formed by the BLs with different center positions $({\tilde{x}}_{i},{\tilde{y}}_{i})$ and orders ni, respectively. The red region, which has the smallest potential energy, denotes the region of potential wells, which have a binding effect on atoms in atomic BECs. In general, the atoms in the BECs are distributed in the potential wells, so the distribution of the potential wells have an important influence on the steady state structure of the BECs. SBLs with different potential well distributions can form different steady state structures in them.
Figure 2. SBLs formed by BLs with different center positions $({\tilde{x}}_{i},{\tilde{y}}_{i})$ and order ni, the red region has the smallest potential energy, denoting the potential well region, the blue region has the largest potential energy, denoting the potential barrier region, and the other regions have potentials as shown in colormap. (a) SBL formed by two zeroth-order BLs with center coordinates (−2, 0), (2, 0). (b) SBL formed by two first-order BLs with center coordinates (−2, 0), (2, 0). (c) SBL formed by four zeroth-order BLs with center coordinates (−2, 2), ( − 2, − 2), (2, 2), (2, − 2). (d) SBL formed by four first-order BLs with center coordinates (−2, 2), ( − 2, − 2), (2, 2), (2, − 2). The intensity of the BL in the SBL is V0i = 20, and the direction of the axes of the right-angled coordinate system is shown by the arrows in (a). |
Figure 2(a) is the superposition potential field formed by two zeroth-order BLs, in which the potential wells are symmetrically distributed about the separation of the y axis, (b) is the superposition potential field formed by two first-order BLs, in which the potential wells are distributed in the shape of a spindle, (c) is the superposition potential field formed by the superposition of four zeroth-order BLs, in which the potential wells are symmetrically distributed about the center of the potential field, and (d) is the superposition potential field formed by four first-order BLs, in which the potential wells are distributed in the center of the potential field. Compared with the superposition potential field formed by two BLs, the superposition potential field formed by four BLs has a more complex spatial distribution. The BL scaling factor and BL center coordinate plays a decisive role in the distribution of the potential field, which can be adjusted to obtain a richer potential field distribution.
3. Steady state structures
In order to study the steady state structure in the SBL more deeply, we first investigate the steady state structure in a single BL potential well. By numerically solving the GPE (1 ), we can obtain the corresponding steady state structure of the system and analyze the dynamics of the steady state structure.
3.1. Steady state structures formed in BL
When discussing the effect of potential energy on the steady state structure, it is generally believed that the two components feel the same potential energy V1 = V2, so that the two components have the same density distribution and form a symmetric structure, i.e. ∣ψ1∣2 = ∣ψ2∣2. Of course, it is not ruled out that when the interactions between the components are sufficiently large, a phase separation phenomenon occurs, so that the density distributions of the two components will be different ∣ψ1∣2 ≠ ∣ψ2∣2, an asymmetric structure is usually formed.
In search of alternative means of forming asymmetric structures, we choose two components to feel different potential energy interactions, i.e. V1 ≠ V2, and this purpose can be achieved by the different center positions of the two BLs. For theoretical investigation, it can be assumed that each of the components are independently subject to an external trapping potential [26]. For experimental investigation, since Bessel lattices are achieved by a non-diffracting beam [27, 28], and BEC atoms can be trapped in the minima of the beam [29], the beam centers of the Bessel lattices perceived by the two atoms can be made to coincide. Therefore, we conceive an experimental scheme in which two beams are used to achieve V1 and V2, respectively. Firstly, the centers of the two beams are made to coincide, and secondly, one of the beams is moved very slowly so that the centers of the two beams are displaced relative to each other, thus subjecting the two parts of the atoms to different potential.
Figure 3 gives the patterns of steady state structures formed in BLs of different orders and center positions. It is clear that the typical patterns of steady state structures formed are Gaussian, toroidal, and vortex superposition modes, and the numbers in the figure denote the center coordinates of the two BLs. In (a), the steady state modes formed in the zeroth-order BL with the center position distributed diagonally along the coordinate system. One of the two important features is that Gaussian modes are formed in the zeroth-order BL, and the other is that the position of the BL potential wells determines the position of the Gaussian modes, which realizes the manipulation of the steady state structure by the BL.
Figure 3. We choose the centers of V1 and V2 not to coincide so that V1 ≠ V2, which leads to ψ1 ≠ ψ2. The steady state structure in a single BL (m = 1) potential well with different parameters in the absence of SOC interactions is considered, with the red region having the highest density, the blue region having a density of 0, and the white region having a density in between, and the solid green line denoting the contour plot of the potential well. (a) Gaussian-type steady state structure formed by two components in two zeroth-order BLs with different centers, ${V}_{1}=-20{J}_{0}^{2}\left(\sqrt{0.2}\sqrt{{\left(x+2\right)}^{2}+{\left(y\mp 2\right)}^{2}}\right)$, ${V}_{2}\,=-20{J}_{0}^{2}\left(\sqrt{0.2}\sqrt{{\left(x-2\right)}^{2}+{\left(y\pm 2\right)}^{2}}\right)$. (b) Toroidal steady state structure formed by two components in two first-order BLs with different centers, ${V}_{1}=-20{J}_{1}^{2}\left(\sqrt{0.2}\sqrt{{\left(x+2\right)}^{2}+{\left(y\mp 2\right)}^{2}}\right)$, ${V}_{2}\,=-20{J}_{1}^{2}\left(\sqrt{0.2}\sqrt{{\left(x-2\right)}^{2}+{\left(y\pm 2\right)}^{2}}\right)$. (c) Toroidal and Gaussian steady state structures formed by two components in zeroth-order and first-order BLs with different centers, respectively, ${V}_{1}\,=-20{J}_{1,0}^{2}\left(\sqrt{0.2}\sqrt{{\left(x+2\right)}^{2}+{\left(y-2\right)}^{2}}\right)$, ${V}_{2}=-20{J}_{0,1}^{2}\left(\sqrt{0.2}\sqrt{{\left(x-2\right)}^{2}+{\left(y-2\right)}^{2}}\right)$. (d) The first column on the left depicts the vortex superposition structure of the two components in a first-order BL with overlapping centers (${V}_{\mathrm{1,2}}=-20{J}_{1}^{2}\left(\sqrt{0.2}\sqrt{{x}^{2}+{y}^{2}}\right)$), and the second column shows the vortex superposition structure of the two components in a second-order BL with overlapping centers (${V}_{\mathrm{1,2}}=-20{J}_{2}^{2}\left(\sqrt{0.2}\sqrt{{x}^{2}+{y}^{2}}\right)$). The two columns on the right show the vortex superposition structure formed when the first component is in the first-order BL and the second component is in the second-order BL, ${V}_{\mathrm{1,2}}=-20{J}_{1,2}^{2}\left(\sqrt{0.2}\sqrt{{x}^{2}+{y}^{2}}\right)$. The other parameters are Ω = 0, V0 = 20, αx = 0, αy = 0, b = 0.1, g1 = 10.0, g2 = 10, g12 = 5. |
In figure 3(b), the steady state structure pattern formed in the first-order BL with diagonal distribution of the center position is given. It can be easily seen that all the atoms are uniformly distributed in the first loop potential well of the first-order single BL potential field, which forms a rare toroidal pattern, and the two BLs are able to manipulate the toroidal pattern accurately.
In figure 3(c), we increase the complexity a little bit so that the two components feel different BLs with different center positions and different orders, e.g. the first component is in the first-order BL on the lower left and the second component is in the zeroth-order BL on the upper right, or vice versa. Easily recognizable, the two orders of BL form a toroidal mode and a Gaussian mode, respectively, and the positions of the two modes are completely constrained by the position of the BL. There is also a detail to be taken into account, as the presence of the two different modes causes small depressions in the toroidal mode, which break the complete symmetry of the toroidal mode.
In figure 3(d), we give the positive and negative vortex superposition mode [23, 30, 31], the distribution of this mode splits along the angular direction, and its formation is much more difficult, it is difficult to form such a stable mode in two BLs when they have different center positions, i.e. when they are not concentric. Therefore, to study the vortex superposition mode, we choose concentric BLs, and the first column represents the symmetric (∣ψ1∣2 = ∣ψ2∣2) vortex superposition state with an angular momentum quantum number of 6 formed by the two components in concentric BLs of the first-order. The second column represents the symmetric vortex superposition states with an angular momentum quantum number of 2 formed in a concentric second-order BL with both components. It can be contrasted that the angular momentum quantum numbers of the modes in the first-order BL are larger than those of the modes in the second-order BL, i.e. the formation of a vortex superposition state with a larger angular momentum quantum number requires a larger potential well depth. In order to form vortex superposition modes in which the two components feel different BL effects, which cannot be achieved by BLs with different center positions, we chose concentric first-order and second-order BLs, as shown in the two columns on the right, in which vortex superposition modes with the same angular momentum quantum number and different single radii were formed. It should be noted that the positions of the vortex superposition modes are also completely determined by BL to realize the BL's manipulation of the steady state modes.
3.2. Steady state structures formed in SBL
It is extremely difficult to deform the steady state structure in BECs, but by introducing the BL superposition potential field structure, we cannot only form the steady state structure in it, but also flexibly adjust the distribution of the steady state structure by the parameters of the potential field structure. In this section, we discuss the steady state structure pattern formed in the superposition potential field structure shown in figure 2, in which the typical steady state structure is formed as shown in figure 4, and it is easy to see that the steady state structure pattern of the system is completely dependent on the layout of the potential field, and the distribution of the density is the largest in the region of the potential field's potential wells, and in the other regions of the potential field, the density distribution is almost zero.
Figure 4. The steady state structure formed in the superposition potential well formed in figure 2. |
It should be noted that the last column is the steady state structure pattern formed in the superposition of four first-order BLs, and the atoms are trapped in the central potential well because the central potential well is deeper compared with the surrounding four potential wells, forming a Gaussian-type steady state structure pattern. Since the superposition potential field structure can be flexibly adjusted by its parameters, and the BECs steady state density distribution is completely dependent on the potential field structure, the steady state structure of the BECs can be adjusted by the superposition potential field, thus realizing the structural manipulation of the BECs.
4. Interference process of BECs in a SBL
The occurrence of interference between the internal parts of the BECs is a fundamental manifestation of the quantum macroscopic volatility of the BECs. In order to verify this interference phenomenon, we let the steady state modes (figure 4(a) and (c)) of the separated states formed in the two or four superposition BLs with n = 0 freely diffuse so that the parts meet each other in order to produce the interference phenomenon. Free diffusion means that after the formation of the steady state mode, we remove the superposition potential field, i.e. V1 = V2 = 0, and let the components keep their centers fixed to start diffusion and meet. The diffusion rate of each part is determined by the nonlinear interaction coefficients gj of the atoms inside the component, and the BECs diffuse faster when there are strong repulsive interactions between the atoms, while on the other hand, weak repulsive interactions between the atoms, or even the presence of attractive interactions, will inhibit the diffusion tendency.
The free diffusion interference process of the bipolar and quadrupolar separated steady state modes is shown in figure 5 and figure 6. Figure 5 gives the free diffusion (Vj = 0) and interference process of the separated steady state structure shown in figure 4(a) under the effect of different gj (gj = 10, the first row, gj = 100, the second row). It can be seen that, when there is a weak repulsion between atoms (gj = 10), the free diffusion of BEC wave packets is slower, due to the small radius of the BEC wave packets, they are far away from each other at t = 1, t = 1.5, the wave packets continue to diffuse, close to each other, and when t = 2.0, they meet and produce interference, and when t = 2.3, a more pronounced interference is produced. When t = 2.3, the interference is more pronounced.
Figure 5. The free diffusion (Vj = 0) and interference processes of the steady state modes under different gj (gj = 10 (top), gj = 100 (bottom)) depicted by figure 4(a) and (c). |
Figure 6. The interference process of free diffusion for the formation of a steady state structure in a complex potential field, gj = 10, is shown in figure 2(c). |
When there is a strong repulsion between atoms (gj = 100), the free diffusion of atomic wave packets is faster, and the radius of the BEC wave packets is larger, at t = 1, the wave packets meet each other, resulting in interference. With the increase of time, the interference is more and more obvious, and there are dark and light interference fringes. The phenomenon and the classical wave dynamics in the interference of the wave is completely consistent with the mechanism, in addition, the distribution of the number of atoms in the interference fringes is not the same, although the interference is not the same. Additionally, the distribution of atoms in the interference fringes is not uniform, and the distribution of atoms in the center fringe is larger.
The interference process of the free diffusion of the steady state structure in the complex potential field shown in figure 4(c) is shown in figure 6. It can be seen that the center of each part of the quantum BECs coincides with the center of the potential well, and after the superposition potential field is removed, each part is centered on the potential well and freely diffuses, t = 0.7, the BECs start to diffuse, t = 1.0, each part of the BECs diffuses to just touch each other, and after further diffusion, at t = 1.5, each part meets and interferes with the distribution of atoms on the coordinate axes. When t = 2.3, complete interference occurs, and the atoms in the BECs are distributed in a dotted pattern, with more atoms in the center dots and at the dots on the coordinate axes. The interference patterns in the two-BL superposition potential field are completely different from those in the four-BL superposition potential field, with the former showing bright and dark interference fringes, and the latter showing interfering dots.
5. Vortex state in BLs
In this section, we discuss the structural features of BECs in a superposed BL potential field in the rotating case. In general, the rotating potential well method, the laser stirring method, and the phase imprinting method can be used to produce a rotational effect on the BECs, which is equivalent to the description of the structure of the BECs in a rotating coordinate system. In the theoretical calculations, it is necessary to add Coriolis terms to the GPEs in the laboratory coordinate system to obtain the dynamic equations in the rotating coordinate system in order to describe the dynamic behavior of the BECs in the rotating coordinate system.
5.1. Vortex array in Gaussian steady state
The basic feature of a rotating BEC is the formation of vortices or vortex lattices in it, which is a manifestation of the superfluid properties of the BECs. We choose the BL parameter as V0 = 100, n = 0, b = 0.01, which means the center of the potential field is a deep potential well with a large width, so that in the process of rotation, the potential well can effectively bind the atoms in the central potential well. It is easier to form the vortex state, and the formation of the vortex state is shown in figure 7. A vortex state is a singularity in the phase distribution of a condensed state, which cannot form a condensed state in a condensate when the angular rate of rotation is small, but can form a vortex state when the angular rate of rotation reaches a certain critical value.
Figure 7. Variation of vortex array states with angular rate Ω in a single potential well without SOC interaction (α = 0), other parameters V0 = 100, b = 0.01, n = 0, g1 = 100, g2 = 100, g12 = 50. |
The vortex manifests itself as a phase singularity in the condensed wave function, and the vortex states formed at different rotation rates are shown in figure 7. The vortex state cannot be formed in the condensate when the rotation rate is small. When the angular rate of rotation reaches a critical value, vortex states begin to form in the condensate. For example, Ω = 0.7, a single vortex state is formed in the center of the condensate, and when the rotational angular rate increases, e.g. Ω = 0.85, a vortex lattice is formed in which the outer vortices are symmetrically distributed about the center vortex and triangularly distributed with the center vortex as the apex. When the angular rate increases, e.g. Ω = 0.9, more vortices appear in the vortex lattice, and the layout of vortices in the vortex lattice remains unchanged.
In order to study the effect of the SOC interaction on the vortex states, we let the two components feel the common zeroth-order BL potential well. It is reasonable that the density distributions of the two components should be the same when they are subjected to the same potential, but when there is a SOC interaction, the density distributions of the two components are no longer the same, and asymmetric vortex lattice structure patterns are formed.
The effect of the SOC interaction on the vortex mode is shown in figure 8, as shown in (a), in the absence of SOC interactions, the two components form a symmetric single vortex mode. In (b), when there is a SOC interaction, e.g. α = 0.3, the circular symmetry presented by the BECs density distribution is broken, and the two components form an asymmetric structural pattern, with an elliptical density distribution for the first component and an approximate circular density distribution for the second component. The centers of the two vortex states no longer coincide, and the number of small vortices contained in the two vortex states is different. The first component shows a single vortex structure pattern, and the second component shows a double vortex structure pattern. It should be pointed out that, compared with the figure in 8(a), the density distribution at the vortexes (in fact, the vacancies in the density distribution) is no longer a regular circle.
Figure 8. Effect of SOC interactions on vortex superposition states, variation of per-component density distribution with SOC interaction intensity, other parameters V0 = 30, b = 0.01, n = 0, Ω = 0.5, g1 = 100, g2 = 100, g12 = 50. |
In figure 8(c), when the SOC interaction strength continues to increase, e.g. α = 0.7, the density distribution of the first component is an overall rounded triangle, which contains two small vortices, and its density distribution is irregularly elliptical. The second component has a nearly circular density distribution with three small vortices.
In figure 8(d), when the SOC interaction increases to a large level, e.g. α = 0.9, the first component density distribution is still approximately rounded triangular, but its edges become blurred and it contains three small vortices. The density distribution at the vortexes penetrates to the edge of the BECs, and the second component is still approximately circular and it contains four small vortices, one in the center of the BECs, and the other three are symmetrically located. The second component is still approximately circular, containing four small vortices, one in the center of the BECs and three symmetrically distributed at the edge of the BECs.
In summary, the SOC interaction not only breaks the symmetry between the components, so that the density distributions of the two components are no longer the same, but also breaks the symmetry within the components, so that the spatial distribution of the BECs is no longer a regular pattern.
5.2. Vortex states in the toroidal structure
In order to further investigate the effect of SOC interactions on the vortex states, we also choose the annular vortex states in the first-order annular BL for analysis. It should be noted that the cyclic vortex state is different from the aforementioned cyclic steady state. The vortex state carries angular momentum and has a singularity on the phase diagram, and the phase varies periodically around the singularity, with the number of periodic variations equal to the angular momentum quantum number, while the cyclic steady state carries no angular momentum.
Figure 9 gives the variation of the toroidal vortex state formed in the first-order toroidal BL with the strength of the SOC interaction for a BEC with a rotation rate of Ω = 0.570. The phase variation is an important sign in describing the vortex states, and we give not only the density distribution of each component, but also the corresponding phase distribution.
Figure 9. The effect of SOC interaction on the toroidal vortex state. The left two columns correspond to the changes of density distribution and phase distribution of the first component with the strength of SOC interaction, the right two columns correspond to the changes of density distribution and phase distribution of the second component with the SOC interaction, and the other parameters are V0 = 100, b = 0.1, n = 1, Ω = 0.570, g1 = 500.0, g2 = 500, g12 = 100. |
In figure 9(a), the symmetric toroidal vortex state is formed in the BEC when the SOC interaction strength is 0. Its density distribution is the same as that of the toroidal steady state, but the phase diagram shows that there is a singularity, and the phase varies periodically around the singularity, which can be seen to carry an angular momentum quantum number of 3.
In figure 9(b), when there is a SOC interaction, αx = αy = 1, the density distribution is the same as that of the toroidal steady state, the SOC interactions in the x and y directions are equal, the two components still form a symmetric structure, and the density distribution of the toroidal vortex state is the same as that in (a), but the angular momentum is greatly increased from the phase diagram, with an angular momentum quantum number of 9. That is to say, the symmetric SOC interactions do not change the density distribution of the toroidal vortex state, but they do input angular momentum to the system. This means that the symmetric SOC interaction does not change the density distribution of the toroidal vortex state, but inputs angular momentum to the system, increasing the angular momentum quantum number by 6.
In figure 9(c) and (d), the density and phase distributions of the system under asymmetric SOC interactions in the x and y directions is depicted. It can be seen that small vortices appear on the edges of the annular large vortex, and the missing densities of the two components are not perfectly symmetric and appear complementary, i.e. where there is a missing component in the first one, there is no missing component in the second one, or vice versa. The symmetry of the phase diagrams is completely broken, with the corresponding phase singularities appearing at the small vortices. In (d), the SOC interaction strengths in the two directions are interchanged with respect to (c), which corresponds to a spatial rotation of π/2 in (c).
6. Summary
In this paper, we first constructed the SBL, which has a richer structure than a single BL, and we can flexibly manipulate the structure of the BECs placed in it. For example, we can realize the coexistence of Gaussian and toroidal structures, and we can easily form vortex superposition structures in it. More importantly, we can realize the dot matrix structure of BECs by adjusting the structural parameters of SBL. The steady state structure formed in the SBL produces an obvious interference effect when the potential field is removed, verifying the macroscopic quantum interference effect. In addition, we analyze that the SOC interactions can be transformed into the angular momentum of the vortex state through the modulation of the vortex state by the SOC interactions.