Dynamics of solitons in ultracold Bose gases with tunable interactions

Abstract

We review our recent theoretical advances in the dynamics of soliton in ultracold atomic gases with different types of interactions, including spin−orbit coupling and nonlocal Rydberg interactions. By using the variational approximation and the numerical simulation of coupled Gross−Pitaevskii equations, the stability and dynamics of soliton are investigated in full parameter space. Both the analytical and numerical results show that the stability and dynamics of solitons, including both bright and ring dark solitons, show strong dependence on the strength of spin−orbit coupling or the Rydberg interaction. Our results open up alternate ways in the quantum control of soliton in ultracold atom system.

Graphical abstract

Keywords

Bose−Einstein condensate / spin−orbit coupling / Rydberg interaction / soliton / ring dark soliton

1 Introduction

Ultracold atomic gas (known as Bose−Einstein condensate (BEC)) with macroscopic quantum properties and highly tunable spatiality, has become an ideal platform for the study of fundamental problems in a variety of physical fields, such as condensed matter physics, quantum information, quantum computation, and nonlinear dynamics [1−6]. Recently, the experimental realization of synthetic spin−orbit coupling (SOC) in ultracold atomic system has drawn considerable attentions in several fields of physics, since it opens up a new avenue to study some uncharted territory, which are central to many exotic phenomena such as topological insulators, quantum Hall effect, and superconductivity [7−17].

The presence of SOC enhance the close relationship between the spin and motional degrees of freedom in the topological excitations [18]. Previous studies have been shown that SOC can induce a variety of generate topological defects, such as soliton, vortex, skyrmions, monopoles, and knots, and completely new quantum phases [19−27]. On the other hand, the experimental realization of Rydberg dressing technology also provide us a significant platform for investigating novel states of matter and its related dynamics, where the presence of the nonlocal Rydberg interactions can induce roton excitation, leading to spontaneous breaking of translational symmetry of the system [28−35]. Combined with such two controllable parameters, it is expected that not only the static but also the dynamic properties of such topological excitations are significantly different form the system without SOC.

In this article, we review our recent theoretical advances in the dynamics of soliton in Bose gas with different types of interactions. In Section 2, we introduce the theoretical model describing a Bose gas with SOC or Rydberg interaction, and the numerical methods. In Section 3, the formation, stability, and dynamics of bright and ring dark soliton (RDS) in spin−orbit coupled or Rydberg-dressed Bose gas are investigated. Finally, we summarize in Section 4.

2 Theoretical model

The model we consider here is a two-component BEC with SOC and nonlocal Rydberg interactions confined in a quasi-two-dimensional (Q2D) harmonic potential, which can be realized by adding a very strong harmonic confinement along the axial direction. Within the mean-field approach, the Gross−Pitaevskii (GP) energy functional of such a system can be written as \mathcal{H}={\mathcal{H}}_0+{\mathcal{H}}_{\mathrm{i}\mathrm{n}\mathrm{t}}, with

(1)\begin{array}{rl}{\mathcal{H}}_0=& \int {\mathrm{\Psi }}^{\ast }(\mathit{r})[-\frac{{\hslash }^2{\mathrm{\nabla }}^2}{2m}+{\mathcal{V}}_{so}+V_{\mathrm{e}\mathrm{x}\mathrm{t}}(\mathit{r})]\mathrm{\Psi }(\mathit{r}){\mathrm{d}}^2\mathit{r},\\ {\mathcal{H}}_{\mathrm{i}\mathrm{n}\mathrm{t}}=& \frac{1}{2}\int {\mathrm{d}}^2\mathit{r}\sum _{i,j=\uparrow ,\downarrow }{g}_{ij}{\psi }_i^{\ast }(\mathit{r}){\psi }_j^{\ast }(\mathit{r}){\psi }_j(\mathit{r}){\psi }_i(\mathit{r})\\ & +\frac{1}{2}\int {\mathrm{d}}^2\mathit{r}{\mathrm{d}}^2{\mathit{r}}^{\prime }\sum _{i,j=\uparrow ,\downarrow }{\psi }_i^{\ast }(\mathit{r}){\psi }_j^{\ast }({\mathit{r}}^{\prime })U_{ij}(\mathit{r}-{\mathit{r}}^{\prime }){\psi }_j({\mathit{r}}^{\prime }){\psi }_i(\mathit{r}).\end{array}

Here m is the atomic mass, and \mathit{r}=(x,y). \mathrm{\Psi }=({\psi }_{\uparrow },{\psi }_{\downarrow }{)}^{\mathrm{T}} denotes spinor order parameter with pseudospin state \uparrow and \downarrow, satisfying the normalization condition \int {\mathrm{d}}^2\mathit{r}{\mathrm{\Psi }}^{†}\mathrm{\Psi }=N with N being the total particle number. The SOC considered here is the Rashba-type, and can be written as {\mathcal{V}}_{\mathrm{s}\mathrm{o}}=-\mathrm{i}\hslash \kappa ({\sigma }_x{\mathrm{\partial }}_x+{\sigma }_y{\mathrm{\partial }}_y) with \kappa being the strength and {\sigma }_{x,y} being the Pauli matrices. The Q2D external potential considered here is a harmonic potential, which can be written as V_{\mathrm{e}\mathrm{x}\mathrm{t}}(\mathit{r})=m({\omega }_x^2{x}^2+{\omega }_y^2{y}^2)/2 ({\omega }_x={\omega }_y\equiv {\omega }_{\mathrm{\perp }} for isotropic harmonic potential, \mathrm{\Omega }={\omega }_{\mathrm{\perp }}/{\omega }_z denotes the aspect ratio). In this case, the effective intra- and inter-component contact interaction parameters can be written as g_{\uparrow \uparrow (\downarrow \downarrow )}=2\sqrt{2\pi }{\hslash }^2{a}_{\uparrow \uparrow (\downarrow \downarrow )}/(m{a}_z) and g_{\uparrow \downarrow }=g_{\downarrow \uparrow }=2\sqrt{2\pi }{\hslash }^2{a}_{\uparrow \downarrow }/(m{a}_z), where a_{\uparrow \uparrow (\downarrow \downarrow )} and a_{\uparrow \downarrow } are the corresponding s-wave scattering lengths. The nonlocal Rydberg interaction potential can be written as U_{ij}(\mathit{r})=C_6^{(ij)}/(R_c^6+|\mathit{r}{|}^6) with C_6^{(ij)} and R_c being the interaction strength and Rydberg blockade radius, respectively [36, 37].

The dynamics of the system, including the bright and ring dark solitons discussed below, can be obtained numerically by using the the time-splitting Fourier spectral method [38−41]. In addition, the alternating-direction implicit method is also employed, and the obtained results are cross-checked and agree well with each other. Furthermore, to simplify the calculation, we will work in natural units by setting \hslash =m=1.

3 Dynamics of soliton

In what follows, we will perform a detailed study of the effects of SOC and nonlocal Rydberg interactions on the formation, stability, and dynamics of a variety of soliton, with an emphasis on the bright and ring dark solitons. The richness of the system considered in this work lies in the large number of free parameters, such as contact interactions, SOC, and the nonlocal Rydberg interactions. To highlight the effects of SOC and nonlocal Rydberg interactions on the dynamics of solitons, we further simplify the situation by considering the equal case with g_{\uparrow \uparrow }=g_{\downarrow \downarrow }\equiv g, and C_6^{(\uparrow \uparrow )}=C_6^{(\downarrow \downarrow )}=C_6^{(\uparrow \downarrow )}\equiv C_6.

3.1 Dynamics of bright soliton in a trapless spin−orbit coupled Bose gas

As is well known, in high-dimensional free space, bright soliton is usually unstable and will be subject to collapse. To obtain stable bright soliton, different physical mechanisms have been elaborated [42−46]. Typically, a rapid oscillation of the strength of contact interaction has been proposed for ultracold atomic systems [42]. Here we wonder whether matter-wave bright solitons can be stabilized in 2D free space by a proper choice of ramp scheme and SOC?

The static and dynamic properties of such a system can be well described by the following dimensionless GP equations:

(2)\begin{array}{rl}& \mathrm{i}\frac{\mathrm{\partial }{\psi }_{\uparrow }}{\mathrm{\partial }t}=\left(-\frac{{\mathrm{\nabla }}^2}{2}+{\mathcal{V}}_{\mathrm{s}\mathrm{o}}+V_{\mathrm{e}\mathrm{x}\mathrm{t}}(r)+g(|{\psi }_{\uparrow }{|}^2+|{\psi }_{\downarrow }{|}^2)\right){\psi }_{\uparrow },\\ & \mathrm{i}\frac{\mathrm{\partial }{\psi }_{\downarrow }}{\mathrm{\partial }t}=\left(-\frac{{\mathrm{\nabla }}^2}{2}+{\mathcal{V}}_{\mathrm{s}\mathrm{o}}+V_{\mathrm{e}\mathrm{x}\mathrm{t}}(r)+g(|{\psi }_{\downarrow }{|}^2+|{\psi }_{\uparrow }{|}^2)\right){\psi }_{\downarrow }.\end{array}

Here a periodic modulation of SOC can be obtained by making the strength of SOC oscillate at frequency \omega. To avoid nonadiabatic disturbances, we gradually switch on the periodically varying SOC and simultaneously turn off the radial confinement potential as [42, 47]

(3)\kappa (t)=f(t)({\kappa }_0+{\kappa }_1\sin (\omega t)),

(4){\omega }_{\mathrm{\perp }}^2(t)=1-f(t),

where f(t) is a ramp function given by

(5)f(t)=\{\begin{array}{ll}t/T,& 0\le t\le T;\\ 1,& t>T.\end{array}

We gradually switch off the trap and increase the strength of SOC according to the liner ramp function given by Eq. (5) with T=10. The initial wavefunctions are chosen as a superposition of stabilized Townes solitons, which can be written as {\psi }_{\uparrow ,\downarrow }={\mathbf{\Phi }}_{S_{\uparrow ,\downarrow }}\equiv {\alpha }_{\uparrow ,\downarrow }{\mathbf{\Phi }}_S. Here the superposition coefficients are real constants and satisfying the normalization condition {\alpha }_{\uparrow }^2+{\alpha }_{\downarrow }^2=1 [48, 49].

Fig.1 shows the time evolutions of the system for both fixed (a) \kappa (t)=0.5f(t) and (b) \kappa (t)=0.6f(t), and periodically varying SOC (c) \kappa =f(t)(0.6+0.2\sin (30t)). From a small strength of fixed SOC, such as \kappa (t)=0.5f(t) shown in Fig.1(a), it is easy to see that within the period of ramp function, the peak densities decrease with time [see Fig.2(a)], accompanied by a rapid oscillation. After the trapping potential is switched off, the peak densities exhibit slow oscillations with time, but almost keep constants. A increases of the strength of fixed SOC, such as \kappa (t)=0.6f(t) shown in Fig.1(b), it is found that dynamically stabilized vector bright solitons can be realized after the external potential is switched off (by comparing the total density distributions). Moreover, it is easy to see that the oscillation frequencies of the peak densities increases with the strength of SOC.

Shown in Fig.1(c) is the time evolutions of the system for a periodically varying SOC \kappa =f(t)(0.6+0.2\sin (30t)). Similar to the cases of fixed SOC, dynamically stabilized vector bright solitons can be formed as long as the oscillating term is smaller than the fixed term, which can be seen from the time evolution of peak density shown in Fig.2(c).

Based on the above results, we can thus conclude that the presence of SOC balances the attractive contact interactions, resulting a new stability mechanism, which can be used to realize dynamically stabilized vector bright solitons with a proper choice of the ramp scheme.

3.2 Dynamics of ring dark soliton in spin−orbit coupled Bose gas

In what follows, we turn our attention to the dynamics of RDS. Compared with other types of dark soliton in 2D trapping poential, RDS exhibits high stability, rich dynamics and special symmetry, leading to the formation of a variety of vortex structure and patterns [50−54]. Previous studies mainly focus on the effects of contact interactions on the dynamics of RDS in ultracold Bose gases [55−70].

Different from the RDS in nonlinear optics, where the background is a homogeneous, the background of the BEC system should be described by the Thomas−Fermi approximation, it is hard to obtain the analytical solution of the RDS. Since we focus on the effects of a variety of interactions on the dynamics of RDS, we prepare the initial wave functions, where each component contains a single RDS with different initial radii [50, 54], as

(6)\begin{array}{rl}{\psi }_i(x,y,0)=& (1-\frac{{\mathrm{\Omega }}^2{r}^2}{4})\\ & \times [\cos {\varphi }_i(0)\tanh Z(r_1)+\mathrm{i}\sin {\varphi }_i(0)]{\mathrm{e}}^{\mathrm{i}\kappa x},\end{array}

where i=(\uparrow ,\downarrow ), \mathrm{\Omega }={\omega }_{\mathrm{\perp }}/{\omega }_z denotes the aspect ratio of the external harmonic potential, Z(r_1)=(r_1-R_{i0})\cos {\varphi }_i(0) with r_1=\sqrt{(1-e_c^2)x^2+y^2} and e_c being the eccentricity of the ring. R_{i0} is the initial radius and \mathrm{\Delta }R=R_{20}-R_{10} denotes its radius difference. \cos {\varphi }_i(0) is the depth of the input soliton. The input soliton is the so-called stationary black RDS for \cos {\varphi }_i(0)=1, while a gray RDS for \cos {\varphi }_i(0)\ne 1 (which will exhibit exotic decay dynamics).

Different from the initial wave functions used for Bose gas with only contact interaction, a plane wave background {\mathrm{e}}^{\mathrm{i}\kappa x} is added in Eq. (6). This can be understood by the fact that, in the presence of SOC, the ground state is a plane wave state for , and stripe state for g_{\uparrow \downarrow }/g>1. To investigate the dynamics of RDS, we will focus on , and thus a plane-wave background is added as the last term in Eq. (6).

To highlight the effects of interactions on the dynamics of RDS, we further simplify the situation by fixing aspect ratio of the external harmonic potential \mathrm{\Omega }=0.028, the initial radius R_{10}=27.9 and R_{20}=28.9 with \mathrm{\Delta }R=1. More interestingly, it is found that for the parameters considered here, there exists a critical value of the SOC, {\kappa }_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.03, above which the system becomes unstable in a very short time interval. To highlight the effect of the SOC and avoid this instability, we set \kappa =0.01 throughout this paper.

The following dynamics of two RDSs with the circular symmetry in a spin-orbit coupled BEC is shown in Fig.3, where the corresponding one-dimensional density distributions are given simultaneously in Fig.4. It is interesting to observe that the dynamical behavior here is different from the case without SOC, where the RDSs shrinked but without oscillating. In the presence of SOC, the radius of RDSs first increase to a maximum [Fig.3(b) and Fig.4(c) for t=60] and then gradually decrease to its minimum value [Fig.3(d) and Fig.4(d) for t=138].

More interestingly, the radii of RDSs maintain circular symmetry until they are minimized at t=138, which indicating that the presence of the SOC increases the lifetime of RDSs (here the lifetime of the RDS is defined as the time interval between the start and the time point when snake instability occurs).

We thus conclude that the presence of the SOC not only increases the lifetime of RDS, but also changes their attenuation kinetics [71].

3.3 Dynamics of ring dark soliton in Rydberg-dressed Bose gas

So far, we have focused on the effects of SOC on the dynamical properties of soliton. Different from the short range contact interaction and SOC, the nonlocal Rydberg interactions, which is long-rang and isotropic, can induce roton excitation, and thus leading to the spontaneous breaking of translational symmetry of the system. Due to the close relationship between the dynamics of vortices and the spontaneous symmetry breaking, it is of particular interest to investigate the dynamics of RDS and its related vortices in the presence of nonlocal Rydberg interaction.

The model we consider here is a two-component Bose gas with nonlocal Rydberg interactions. The dynamics of such a system can be well described by the following coupled GP equations [72],

(7)\begin{array}{rl}\mathrm{i}\frac{\mathrm{\partial }{\psi }_1}{\mathrm{\partial }t}=& (-\frac{1}{2}{\mathrm{\nabla }}^2+\frac{1}{2}{\mathrm{\Omega }}^2{r}^2+g_{11}|{\psi }_1{|}^2+g_{12}|{\psi }_2{|}^2\\ & +A_{11}|{\psi }_1{|}^2+A_{12}|{\psi }_2{|}^2){\psi }_1,\\ \mathrm{i}\frac{\mathrm{\partial }{\psi }_2}{\mathrm{\partial }t}=& (-\frac{1}{2}{\mathrm{\nabla }}^2+\frac{1}{2}{\mathrm{\Omega }}^2{r}^2+g_{22}|{\psi }_2{|}^2+g_{21}|{\psi }_1{|}^2\\ & +A_{22}|{\psi }_2{|}^2+A_{21}|{\psi }_1{|}^2){\psi }_2,\end{array}

where the Q2D Rydberg interactions are rescaled as C_6^{(ii)}/R_c^4 and C_6^{(ij)}/R_c^4, and we denote the Rydberg interactions as A_{ii} and A_{ij} for simplicity.

Previous studies have been shown that the eccentricity plays an important roles in determining the stability and dynamics of the system system. For system with contact interactions, the distortion of the ring shape can be realized by changing the eccentricity, leading to the symmetry breaking [56]. To highlight the effects of the nonlocal Rydberg interaction, we further fix e_c=0.3 (which is below the critical e_{\mathrm{m}\mathrm{a}\mathrm{x}}) and \cos {\varphi }_1(0)=\cos {\varphi }_2(0)=0.75 to suppress the initial shrink of the RDSs, and the collisions of the RDSs and the following dynamics of vortices, respectively. Finally, the initial wave functions are chosen as Eq. (6) but without the plane wave background {\mathrm{e}}^{\mathrm{i}\kappa x}.

Fig.5 and Fig.6 show the time evolution of RDS in component 1 and the vortices formed after its collapse for a small value of nonlocal Rydberg interaction A=0.01. It is easy to see that the radius of RDS first increases to the maximum value, and then gradually decreases to its minimum one [Fig.5(b) for t=60 and Fig.5(c) for t=124]. In this case, the RDS first expand toward to the low-density region under the effects of quantum pressure, and then return back when it reaches at the Thomas−Fermi (TF) radius until it deforms and break into several lumps, where a necklace-like structure is formed, as shown in Fig.5(d) for t=129.

Here we note that the above dynamics can be understood if we look at the analytical solution of RDS. According to Eq. (6), the oscillate region of RDS is ranging from R_{\mathrm{m}\mathrm{i}\mathrm{n}}=8 to R_{\mathrm{m}\mathrm{a}\mathrm{x}}=58, where R_{\mathrm{m}\mathrm{a}\mathrm{x}} is the rim of the condensate and the so-called TF radius. In addition, such deformation is induced by the nonlocal Rydberg interaction, leading to the continuous-to-discrete rotational symmetry breaking, which is different from previous cases without nonlocal Rydberg interaction. Here we chose the eccentricity e_c=0.3, which is below the critical e_{\mathrm{m}\mathrm{a}\mathrm{x}}, the initial RDSs no longer oscillate periodically and can survive for a longer lifetime, as shown in Fig.5(d).

After that, the lumps gradually evolve into several vortex pairs, and the vortex pairs and lumps continuously recombine to each other. In this case, it is interesting to find that the new formed vortex pairs can be classified into two classes: one can survive for a longer lifetime (created at the center of the condensate and marked by solid circles in Fig.5 and Fig.6), and the other one is generated symmetrically from evolution of lumps along the x- and y-axis or the edge of the condensate (marked by hollow circles in Fig.5 and Fig.6).

Since the dynamics of the first class of vortex pairs is similar to previous results, here we focus on the second class one. It is easy to see that the first two vortex pairs are generated by residual RDS along x-axis, and then the other two are generated by the lumps moving along \pm y-axis, as shown in Fig.5(g) and (h). After that, the four pairs of vortex pair continue to move along the periphery of the condensate, during which the vortex and antivortex within one vortex pair start to move apart from each other.

The first recombination of vortex pair occurs after one eighth circle, where the eight separated vortices and antivortices approach in pair and form four new vortex pair arranged in \times-shape, as shown in Fig.5(i). The new formed vortex pairs continue to move along the periphery of the condensate symmetrically until reaching the edge of \pm y-axis, as shown in Fig.5(j) and Fig.6(a), where we observe the second recombination of vortex pairs. After that, the two new vortex pair, which are formed by vortex from clockwise direction and antivortex from counterclockwise direction, move toward the center of condensate along the \pm y-axis. The third recombination of vortex pair occurs at the remaining four vortices, which return back along their original path and form vortex pairs at both ends of the \pm x-axis.

From Fig.6(f) and (g), we can observe the collision of two vortex pair of the second class occurs along x-axis, while the locations of two vortex pair of the first class just change a little. This indicating that the velocity of the second class is faster than the one of the first class, leading to the shorter lifetime of the second class of vortex pair. In order to give a clear description of the exotic dynamics of the second class of vortex pair, the schematic illustration of its dynamics is shown in Fig.7, where the red and blue solid circles indicate the vortices with different circulation, while the red and blue dashed circles indicate the key point of the motion trajectory.

Another interesting finding is the number of the second class of vortex pair decreases with the strength of Rydberg interaction, as shown in Fig.8 for A=0.08, where no pair of vortex pair is generated from the lumps. We thus conclude that there exists a critical value of Rydberg interaction, below which the number of vortex pair decreases with Rydberg interaction, while no visible vortices generated for value above such critical one. Furthermore, our numerical results show although the maximum radius of RDS increases with the strength of Rydberg interaction, its lifetime decreases with the Rydberg interaction, as shown in Fig.9. More interestingly, different from the cases with only short-range interactions, in the presence of long-range and isotropic interaction, there will be no visible vortices generated from the collapse of RDS for sufficiently strong Rydberg interaction.

We have also carried out the simulations for A=0. The obtained numerical results show that the dynamics of the first class of vortex dipole exhibits qualitative similarity to the system with Rydberg interaction, while the early dynamics of the second class is almost similar to system with Rydberg interaction.

3.4 Self-structured pattern formation of ring dark soliton in Rydberg-dressed Bose gas

In Ref. [73], we have investigated the dynamics and pattern formation of ring dark solitons in a two-dimensional binary BECs with tunable contact interactions, where the modulation frequency of the inter-component interaction is resonant or nonresonant with the one of the trapping potential. For the physical system considered here, in the case of a large detuning between the driving fields and the atoms, a time-periodic modulation of the Rydberg-dressing soft-core potential can be realized by periodically tuning the Rabi frequency of the driving field. Hence it is desirable to investigate the effects of tunable Rydberg interactions on the dynamic properties of RDS in a binary BEC.

We begin with the same model as Eq. (7) with initial wave function Eq. (6) without the plane wave background {\mathrm{e}}^{\mathrm{i}\kappa x}. Since we focus on the periodic varying of Rydberg interactions, both the intra- and inter-component interactions can be chosen. As the first attemptation, we fix A_{11}=A_{22}\equiv A, g_{11}=g_{22}=g_{12}=g_{21}\equiv 1, which is most closely related to the {}^{87}Rb system, and focus on the periodic modulation of the inter-component Rydberg interactions as [74]

(8)A_{ij}(t)=0.08+k\sin (\omega t).

Here k is the modulation amplitude, satisfying 0\le k\le 1. \omega is the modulation frequency, resulting in a modulation period of 2\pi /\omega. Since there is a large number of free parameters, we further fix k=0.02 and \omega =\mathrm{\Omega } (the modulation frequency of the inter-component Rydberg interaction \omega resonates with the natural frequency of the trapping potential \mathrm{\Omega }). It is easy to see that the parameter region of the periodic modulation of inter-component Rydberg interaction is [0.06, 0.1].

As we discussed before, the distortion of the ring shape of RDS can be realized by changing the eccentricity, which is used to induce snake instability. Here a periodic modulation of interaction is considered, we thus set eccentricity e_c=0 and focus on the dynamics of two black RDSs (\cos {\varphi }_1(0)=\cos {\varphi }_2(0)=1). Our results show that the presence of periodic modulation of Rydberg interaction results in rich nonequilibrium dynamics of the RDS and vortices after its collapse, where different types of vortex necklace, vortex dipole, and self-structured pattern formation are observed.

Our main results are summarized in Fig.10, where the typical “phase diagram” of nonequilibrium dynamics of the system as a function of intra-component Rydberg interaction is depicted. In order to give a more clear description and classification of the dynamical behavior, three different parameter regions, i.e., , A\in [A_{ij}^{\mathrm{m}\mathrm{i}\mathrm{n}},A_{ij}^{\mathrm{m}\mathrm{a}\mathrm{x}}], and A>A_{ij}^{\mathrm{m}\mathrm{a}\mathrm{x}}, will be discussed, respectively.

We begin with the region , where the intra-component Rydberg interaction is smaller than the minimal value of inter-component one. Fig.11 shows the typical time evolution of two black RDSs for Rydberg interaction A=0.01, where the top row is for component 1 and the bottom one is for component 2. Our results show that in the presence of periodic modulation of the inter-component Rydberg interaction, the initial RDS firstly evolve into a series of concentric black RDSs (the number of concentric black RDSs generated by the initial RDS decreases with intra-component Rydberg interaction), and then into a variety of self-structured pattern formation, where the patterns of two components are complementary. Here we want to note that the initial circular symmetry of the system (eccentricity e_c=0) is gradually breaking due to the presence of long-rang and isotropic Rydberg interaction, as shown in Fig.11(d)−(f) for t=140,270,350, respectively.

Fig.12 and Fig.13 show the time evolution of RDSs for Rydberg interaction A=0.08, where the intra-component Rydberg interaction is in the region A\in [A_{ij}^{\mathrm{m}\mathrm{i}\mathrm{n}},A_{ij}^{\mathrm{m}\mathrm{a}\mathrm{x}}]. In this case, the initial RDS no longer evolves into new RDSs, it first perform radial oscillation, during which the snake instability sets in, and then the distorted RDSs develop into a necklace structure, where 20 vortex pairs arranged themselves along the ring, as shown in Fig.12(c) for t=92 [54]. With time going on, the adjoining vortices keep recombining repeatedly. It is interesting to see that the quartets of pairs successively expel off the necklace in the + shape direction, move inward to the center of the condensate and form a mini RDS at the center of the condensate, as shown in Fig.12(e) for t=194. After that, the mini RDS splits into lumps, which return outward along the same way in the cross direction. With the development of instability, such lumps split into vortices, which either moving outward to the edge of the condensate or recombining with lumps coming from the center into a arc-shaped lump solitons, as shown in Fig.13(h)−(j) for t=205,215,217, respectively [75]. Finally, the dynamic of the remained vortex pairs become chaotic, as shown in Fig.13(l) for t=500. Here we want to note that within this parameter region, there is no new RDS and self-structured pattern formation.

Finally, we move to the region with A>A_{ij}^{\mathrm{m}\mathrm{a}\mathrm{x}}. Fig.14 and Fig.15 show the typical time evolution of RDSs for A=0.12. It is easy to see that the earlier dynamics of the RDS is similar to the region with A\in [A_{ij}^{\mathrm{m}\mathrm{i}\mathrm{n}},A_{ij}^{\mathrm{m}\mathrm{a}\mathrm{x}}], where vortex pairs form a necklace structure along the ring, as shown in Fig.14(c). However, it is interesting to see that the following dynamics of the system exhibits both vortices dynamics and self-structured patterns formation, as shown in Fig.15.

We thus conclude that the dynamics of the RDS shows strong dependence on the periodic modulation of the Rydberg interactions. Within the region , the dynamics of RDS exhibits self-structure patterns formation after its collapse. For the region with A\in [A_{ij}^{\mathrm{m}\mathrm{i}\mathrm{n}},A_{ij}^{\mathrm{m}\mathrm{a}\mathrm{x}}], the dynamics of RDS is similar to the system with fixed contact interactions, where no self-structure patterns formation can be observed. For A>A_{ij}^{\mathrm{m}\mathrm{a}\mathrm{x}}, the system exhibits rich dynamics behavior, where exotic vortex structure and self-structure patterns are observed.

As discussed in previous works, one of the key effects of periodic modulation is to extend the lifetime of RDS. Fig.16 illustrates the scatter and fitting curve of the lifetime of RDSs as a function of the intra-component Rydberg interaction, where the red dots represent the values obtained from our numerical results and the blue line is a curve fitted from the true value. It is easy to see that the lifetime of RDS decreases as the intra-component Rydberg interactions is close to the inter-component one, which provides us another way to extend the lifetime of RDS by tunable Rydberg interaction [76].

Last but not least, we want to provide an experimental protocol for observing such exotic dynamics. A two component system of BECs can be realized by selecting |F=1,m_f=-1⟩, and |2,1⟩ spin states of {}^{87}Rb, where the Rydberg interaction can be realized by using the Rydberg dressing technique and its strength (including a periodic modulation) can be realized by tuning the two photon Rabi frequency and detuning [77, 78]. With regard to the periodical modulation of SOC, it is actually difficult to realized a time-oscillating SOC by using the Raman coupling. With the progress of new experimental scheme, we hope that experimental physicists will be interested in the realization of time-oscillating SOC.

4 Conclusions

In summary, we have examined the controllable dynamics of matter wave solitons, including the bright and RDSs, in two-component Bose gas with either SOC or nonlocal Rydberg interaction. We first present a proper ramp scheme to realize dynamically stabilized vector bright solitons in a trapless spin−orbit coupled Bose gas, and then discuss the effects of SOC on the dynamics of RDS, where the presence of the SOC not only increases the lifetime of RDS, but also changes their attenuation kinetics. For the system with Rydberg interaction, the presence of nonlocal Rydberg interaction induces the deformation of the RDS, leading to the continuous-to-discrete rotational symmetry breaking. In this case, the dynamics of the followed vortices can be classified into two classes, whose number, dynamics, and lifetime show strong dependence on the strength of Rydberg interaction. Finally, the nonequilibrium dynamics of RDS in a two-component Bose gas with tunable Rydberg interactions has been investigated, where three distinct parameter regions are identified, showing strong dependence on the tunable Rydberg interaction. We hope that these results can stimulate studies of soliton dynamics in ultracold atoms and optics in general and further studies of other topological excitations in particular.

Owing to the recent developments in the experimental implementation of different types of SOC, a natural extension of current research is to investigate the effects of different types of SOC, such as NIST, Weyl, Raman lattice, spin−tensor−momentum coupling, and SU(3) types of SOC, on the stability and dynamics of a variety of topological defects. On the other hand, another type of long range interaction, that is the dipole−dipole interaction, can also be included. Due to the anisotropic nature of the dipole−dipole interaction, the stability, and especially the dynamics of different types of topological defects will shows totally different dynamical behaviors.

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