1. Introduction
The experimental observation of BECs in trapped dilute gases [1-3] has aroused intense research enthusiasm over the dynamical properties of BECs due to new opportunities to study novel quantum phenomena in various physical fields with nonlinear many-body interactions. As is well known, BECs are just such systems with nonlinear many-body interactions. It is precisely because of the presence of nonlinear many-body interactions that BECs exhibit rich and important dynamical properties. The most important dynamical properties of BECs include chaos [4-21], soliton [22-30], Josephson oscillation [5-7, 31-37], phase transition [38-43] and macroscopic quantum self-trapping [44-52]. Although a large number of achievements have been reported, there are still many open problems about BEC dynamics awaiting clarification.
The appropriate mathematical tool to describe BEC dynamics is the Gross-Pitaevskii equation (GPE) providing an appropriate starting point to investigate the underlying many-body system on the mean-field level [53]. The nonlinear term in the GPE represents the nonlinear interatomic interaction exerting an important impact on BEC behaviors [53]. The s-wave scattering length is decisive for the nonlinear interatomic interaction. Experimental research has demonstrated that a space-dependent s-wave scattering length can be achieved by a (longitudinally) changing transversal confinement or an inhomogeneity of the external magnetic field in the vicinity of a Feshbach resonance [54-57]. In this paper, we will consider a space-dependent s-wave scattering length. Owing to this, the nonlinear parameter in the GPE is space-dependent too.
BECs possess rich and complex spatial structures [53]. The main purpose of this paper is to investigate the spatial structure of a BEC in the frame of the GPE. No doubt the dynamical properties of BECs are significantly affected by the spatial structures of BECs, namely, the special distribution of BEC atoms. The research on the spatial structures of BECs is of great significance to the stability and control of BECs. BECs in different trap potentials may exhibit some different dynamical behaviors, in other words, the trap configuration can be of important influence on the dynamical behaviors of BECs. In this work, the considered BEC system is loaded in a combined trap consisting of a harmonic and an optical lattice trap. Such a combined trap can be obtained by superimposing the periodic optical potential of a far-detuned standing wave on the harmonic potential of a magnetic trap [58, 59]. The remarkable tunability and dynamical control of harmonic and optical lattice potentials offer great convenience to investigate quantum mechanical behaviors of BECs. BECs in a combined trap consisting of harmonic and optical lattice traps are promising candidates to study versatile quantum phenomena in external physical fields. Theoretical analyses of this work reveal that there is a space-dependent atomic current in the considered BEC system. Undoubtedly, the atomic current will play an important role in determining the spatial structure of the BEC.
The remainder of this manuscript is organized as follows. In section 2 the theoretical analyses are performed. In section 3 we present the numerical analyses of the spatial structure of the BEC system. In section 4 a brief summary is given.
2. Theoretical analyses
At sufficiently low temperatures and in the mean-field approximation, the macroscopic quantum wave function ψ(x, t) describing the dynamical evolution of a quasi-one-dimensional (1D) BEC system satisfies the time-dependent mean-field Gross-Pitaevskii equation
$\begin{eqnarray}{\rm{i}}{\hslash }\displaystyle \frac{\partial \psi }{\partial t}=-\displaystyle \frac{{{\hslash }}^{2}}{2m}\displaystyle \frac{{\partial }^{2}\psi }{\partial {x}^{2}}+{V}_{\mathrm{ext}}(x)\psi +g{\left|\psi \right|}^{2}\psi ,\end{eqnarray}$
here ℏ is the Planck constant, ${V}_{\mathrm{ext}}(x)\,=V\cos (2{kx})+\tfrac{1}{2}m{\omega }^{2}{x}^{2}$ is the combined trap potential with an optical lattice potential $V\cos (2{kx})$ and a harmonic potential $\tfrac{1}{2}m{\omega }^{2}{x}^{2}$, V is the intensity of the optical lattice potential, k is the wave number of the optical lattice, m is the atomic mass, ω is the frequency of the harmonic potential, g characterizes the nonlinear coefficient corresponding to the two-body interactions proportional to the s-wave scattering length as [60]. In experiments the sign and value of as can be conveniently changed according to demand through the Feshbach resonance technique [54-56]. as > 0 indicates a repulsive interaction and as < 0 corresponds to the attractive case.In the presence of a space-dependent s-wave scattering length, we have a space-dependent nonlinear parameter as below [56]
$\begin{eqnarray}g(x)={g}_{0}+{g}_{1}{\sin }^{2}(\kappa x),\end{eqnarray}$
where g0 is the nonlinear parameter in the absence of modulation and g1 is the amplitude of the modulation. κ is the wave number of the modulating laser.In order to simplify equation (1 ), we rescale the wave function and introduce dimensionless parameters as follows1 ) can be expressed as the following dimensionless form7 ) is a small correction to the fundamental one, $\cos (2\kappa x)$. Thus, to order O(ϵ), the effective potential (7 ) is a modified lattice rather than a superlattice.
$\begin{eqnarray}\begin{array}{rcl}\,\,\,\,\tilde{\psi } & = & \displaystyle \frac{\psi }{\sqrt{n}},\,\,\tilde{t}=\displaystyle \frac{{\hslash }}{m}{\left[k\right]}^{2}t,\\ \tilde{x} & = & [k]x,\,\,\tilde{\omega }=\displaystyle \frac{m}{{\left[k\right]}^{2}{\hslash }}\omega ,\\ \,\,\tilde{V} & = & \displaystyle \frac{m}{{\left[k\right]}^{2}{{\hslash }}^{2}}V,\,\,\tilde{\kappa }=\displaystyle \frac{\kappa }{[\kappa ]},\\ \tilde{k} & = & \displaystyle \frac{k}{[k]},\,\,{\tilde{g}}_{\mathrm{0,1}}=\displaystyle \frac{{mn}}{{\left[k\right]}^{2}{{\hslash }}^{2}}{g}_{\mathrm{0,1}},\end{array}\end{eqnarray}$
with n being the number density of BEC atoms. It may not result in misunderstanding to remove ‘∼' from the above variables. So equation ( $\begin{eqnarray}\begin{array}{l}{\rm{i}}\displaystyle \frac{\partial \psi }{\partial t}=-\displaystyle \frac{1}{2}\displaystyle \frac{{\partial }^{2}\psi }{\partial {x}^{2}}+\left[V\cos (2{kx})+\displaystyle \frac{1}{2}{\omega }^{2}{x}^{2}\right]\psi \\ \quad +[{g}_{0}+{g}_{1}{\sin }^{2}(\kappa x)]{\left|\psi \right|}^{2}\psi .\end{array}\end{eqnarray}$
Considering the solution of the above equation bear the form of $\psi =\phi /\sqrt{g(x)}$ and g(x) ≠ 0, we have [56] $\begin{eqnarray}\begin{array}{l}{\rm{i}}{\phi }_{t}=-\displaystyle \frac{1}{2}{\phi }_{{xx}}+{\left|\phi \right|}^{2}\phi +\left[\Space{0ex}{3.0ex}{0ex}V\cos (2{kx})\right.\\ \quad \left.+\displaystyle \frac{1}{2}{\omega }^{2}{x}^{2}\right]\phi +{\hat{V}}_{\mathrm{eff}}(x)\phi ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\hat{V}}_{\mathrm{eff}}(x)=\displaystyle \frac{1}{2\sqrt{g}}\displaystyle \frac{{\partial }^{2}\sqrt{g}}{\partial {x}^{2}}\\ \quad -\displaystyle \frac{1}{g}{\left(\displaystyle \frac{\partial \sqrt{g}}{\partial x}\right)}^{2}+\displaystyle \frac{1}{\sqrt{g}}\displaystyle \frac{\partial \sqrt{g}}{\partial x}\displaystyle \frac{\partial }{\partial x}.\end{array}\end{eqnarray}$
In this work we consider the situation with g0 ≫ g1 and set a new parameter ϵ = κg1/g0 ≪ 1, and then ${\hat{V}}_{\mathrm{eff}}(x)$ can be directly expressed as $\begin{eqnarray}\begin{array}{l}{\hat{V}}_{\mathrm{eff}}(x)=-\displaystyle \frac{3{\varepsilon }^{2}}{16}+\displaystyle \frac{\kappa \varepsilon \cos (2\kappa x)}{2}\\ \quad +\displaystyle \frac{3{\varepsilon }^{2}\cos (4\kappa x)}{16}+\displaystyle \frac{\varepsilon \sin (2\kappa x)}{2}\displaystyle \frac{\partial }{\partial x}.\end{array}\end{eqnarray}$
As has been pointed out in [56], the purpose of introducing the small quantity ϵ is to evince the fact that the $\cos (4\kappa x)$ harmonic in equation (Usually, the general solution of equation (5 ) can be selected as8 ) to the nonlinear equation (5 ) and separating the real and imaginary parts of the generated equation yield10 ) over the spatial coordinate x leads to11 ), we can obtain ${\rm{d}}\theta /{\rm{d}}x={\rm{\Gamma }}{{\rm{e}}}^{-\varepsilon \cos (2\kappa x)/2\kappa }/{R}^{2}(x)$. And then inserting it into equation (9 ) leads to
$\begin{eqnarray}\phi (x,t)=R(x)\exp ({\rm{i}}[\theta (x)-\mu t]).\end{eqnarray}$
Applying solution ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{{\rm{d}}}^{2}R(x)}{{\rm{d}}{x}^{2}}-R(x){\left(\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}x}\right)}^{2}+2\mu R(x)-2{R}^{3}(x)\\ \quad -2[V\cos (2{kx})+\displaystyle \frac{1}{2}{\omega }^{2}{x}^{2}]R(x)\\ \quad -[\kappa \varepsilon \cos (2\kappa x)+\displaystyle \frac{3{\varepsilon }^{2}}{8}\cos (4\kappa x)]R(x)\\ \quad -\varepsilon \sin (2\kappa x)\displaystyle \frac{{\rm{d}}R(x)}{{\rm{d}}x}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}2\displaystyle \frac{{\rm{d}}R(x)}{{\rm{d}}x}\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}x}+R(x)\displaystyle \frac{{{\rm{d}}}^{2}\theta }{{\rm{d}}{x}^{2}}=\varepsilon \sin (2\kappa x)R(x)\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}x}.\end{eqnarray}$
Performing the integration in equation ( $\begin{eqnarray}\begin{array}{rcl}J(x) & = & {R}^{2}\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}x}={\rm{\Gamma }}{{\rm{e}}}^{-\varepsilon \cos (2\kappa x)/(2\kappa )},\\ {\rm{\Gamma }} & = & {R}^{2}({x}_{0}){\theta }_{x}({x}_{0}){{\rm{e}}}^{\varepsilon \cos (2\kappa {x}_{0})/(2\kappa )}.\end{array}\end{eqnarray}$
Here Γ is an integral constant related to the initial and boundary conditions, θx(x0) is the value of the derivative of θ with respect to x at x0. According to equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{{\rm{d}}}^{2}R(x)}{{\rm{d}}{x}^{2}}+2\mu R(x)-2{R}^{3}(x)-\displaystyle \frac{{{\rm{\Gamma }}}^{2}{{\rm{e}}}^{-\tfrac{\varepsilon \cos (2\kappa x)}{\kappa }}}{{R}^{3}(x)}\\ \quad -2\left[V\cos (2{kx})+\displaystyle \frac{1}{2}{\omega }^{2}{x}^{2}\right]R(x)\\ \quad -[\kappa \varepsilon \cos (2\kappa x)+\displaystyle \frac{3}{8}{\varepsilon }^{2}\cos (4\kappa x)]R(x)\\ \quad -\varepsilon \sin (2\kappa x)\displaystyle \frac{{\rm{d}}R(x)}{{\rm{d}}x}=0.\end{array}\end{eqnarray}$
Obviously, the above equation is a very complex nonlinear one describing a complex nonlinear system. Such a nonlinear system can exhibit rich nonlinear behaviors. Chaos is one of the most typical and important nonlinear behaviors. Sometimes chaos can be useful, but in other situations, it may be a destructive factor [61]. As a result, how to result in and suppress chaos are always hot topics. Here in this work, we investigate the regular and chaotic spatial structures of a BEC in a combined trap. What should be pointed out is that both κ and k are laser wave numbers, so they have the same unit, namely [k] = [κ].As long as the phase θ is not a constant but a space-dependent function, the derivative dθ/dx in equation (11 ) will not be zero. In this situation J(x) ≠ 0 and it announces there exists a nonlinear space-dependent atomic current in the BEC, which means the parameter Γ in equation (11 ) is a non-zero one. Γ will undoubtedly play an important role in determining the intensity of the space-dependent atomic current. We plot the space-dependent atomic current in figure 1 for ϵ = 0.001, κ = 0.5, Γ = 1.2. From figure 1 one can see that the current oscillates periodically with space. For a small ϵ the atomic current varies cosinusoidally with space and Γ can be treated as the intensity of the atomic current because the average value of ${{\rm{e}}}^{-\varepsilon \cos (2\kappa x)/2\kappa }$ in a period is about one. What should be pointed out is that strong enough nonlinear currents may cause complex and even chaotic behaviors in nonlinear systems [61].
Figure 1. The space-dependent atomic current with ϵ = 0.001, κ = 0.5, Γ = 1.2. |
3. Numerical analyses
In order to explore the spatial structure of the BEC (namely the spatial distribution of BEC atoms) with current (11 ) in the combined trap, we will numerically solve equation (12 ) in the range 0 < x < 50 and present the findings with the software package ‘Mathematica' 12.0. Undoubtedly, the atomic current can exert an important influence on the spatial structure of the BEC and even lead the BEC into a complex spatial structure.
Firstly, we consider the cases with and without the atomic current. In experiments, the initial and boundary conditions cannot be determined exactly due to the fluctuation of the atomic thermal cloud. Thus in this paper, we choose the possible values of R(0) and $\dot{R}(0)$ in our numerical calculations. Keeping ϵ, κ, and Γ the same as in figure 1 and setting the other parameters as μ = 1.8, V = 5.0, k = 7.5, ω = 0.01, R(0) = 1, $\dot{R}(0)=0$, we plot the phase space orbit in ($R,\,\dot{R}$) plane in figure 2(a) and the corresponding spatial evolution curve of R(x) in figure 2(b). From figure 2(a) we can observe a typical chaotic attractor whose phase orbit evolutes with the coordinate in a finite region in the phase space and displays obvious confusion, which is one of the essential characteristics of chaos. Figure 2(b) shows a typical chaotic spatial evolution curve of R(x). The chaotic attractor in figure 2(a) and the spatial evolution curve of R(x) in figure 2(b) denote that the BEC is in a chaotic spatial structure. As has been illuminated above, the value of Γ for figures 2(a) and (b) is 1.2, implying the existence of an atomic current in the BEC. For comparison, we consider the case with θ being a constant and the other parameters being kept the same as figure 2(a), which will result in $J(x)={R}^{2}{\rm{d}}\theta /{\rm{d}}x={\rm{\Gamma }}{{\rm{e}}}^{-\varepsilon \cos (2\kappa x)/2\kappa }=0$, meaning the disappearance of the atomic current and Γ = 0. The corresponding phase portrait in figure 2(c) and spatial evolution curve of R(x) in figure 2(d) indicate that the degree and region of chaos are significantly reduced although the system remains in a chaotic state. This manifests the reduction in the degree and region of the spatial chaos of the BEC system. The above numerical calculation results show that the spatial structures of the BEC system may be very different with and without the atomic current. The two chaotic attractors localized in finite regions in figures 2(a) and (c) announce the boundedness of the two chaotic solutions of equation (12 ) corresponding to the two chaotic attractors.
Figure 2. (a), (c) and (e) are plots of the orbits in the equivalent phase space $(R,\,\dot{R})$ with μ = 1.8, V = 5.0, ϵ = 0.001, κ = 0.5, k = 7.5, ω = 0.01. (a), (b), (c) and (d): $\,R(0)=1,\,\dot{R}(0)=0;$ (e) and (f): $\,R(0)=1.04,\,\dot{R}(0)=0.01$. (a), (b), (e) and (f): Γ = 1.2; (c) and (d): Γ = 0. (b), (d) and (f) are the spatial evolution curves of R(x) corresponding to (a), (c) and (e), respectively. |
In fact, chaotic behaviors in nonlinear physical systems are very sensitive to the initial conditions. Very small changes in the initial conditions may significantly change the chaotic attractor of the system or lead the chaotic attractor to lose its boundedness [61]. The sensitivity of chaos to the initial conditions is known as the butterfly effect in chaos theory, which is also a distinct characteristic of chaos [61]. Next, we will demonstrate this kind of sensitivity through numerical calculations. Changing the values of R(0) and $\dot{R}(0)$ to 1.04 and 0.01 respectively and keeping the other parameters the same as in figure 2(a), we plot a phase portrait in figure 2(e) and the corresponding spatial evolution curve of R(x) in figure 2(f). In the process of plotting figure 2(e) we find that the phase orbit evolutes in a finite region in the early stage, but it extends far away in the late stage. This can also be judged from figure 2(f). The corresponding spatial evolution curve of R(x) in figure 2(f) indicates that R(x) tends to a very big value when x > 45, which will inevitably lead to a high atomic density. That means the system will lose its stability. Under this circumstance, the BEC internal structure will become very disrupted, and solitons and vortices may be created in the system [62, 63]. For attractive BECs an increasing atomic density can cause collapse when the atomic density exceeds a critical value [53]. The above findings show that very small changes in the initial conditions have significantly changed the chaotic spatial structure of the BEC and even force the BEC to lose its boundedness.
The term related to the chemical potential μ in equation (12) is a linear one. Usually, linear terms in nonlinear physical systems can effectively suppress system chaos when it is strong enough [61]. Only increasing the chemical potential and keeping the other parameters the same as in figure 2(a), we plot the phase portraits in figures 3(a)-(c). When the chemical potential μ increases to 3, the system is still in a chaotic state, but the chaos degree is visibly reduced, as can be seen in figure 3(a). When the chemical potential μ further increases to 31, figure 3(b) shows that although the system is still in a complex and even chaotic state, some periodic orbits can be seen in the phase space. This means the chaos degree is further reduced. And finally, when the chemical potential μ reaches 300, only one periodic orbit appears in the phase space of figure 3(c), which demonstrates that the system is in a single-periodic spatial structure and BEC atoms are in a single-periodic spatial distribution. The evolution of the phase portraits in figure 3 indicates that in the presence of the atomic current, a large chemical potential is required to effectively suppress the chaotic spatial structure of the BEC.
Figure 3. Plots of the orbits in the equivalent phase space $(R,\,\dot{R})$ with V = 5.0, ϵ = 0.001, κ = 0.5, Γ = 1.2, k = 7.5, ω = 0.01, $R(0)=1,\,\dot{R}(0)=0$ and different values of μ. |
As has been pointed out, Γ can be treated as the intensity of the atomic current for a small ϵ. In plotting figures 2 and 3 we have kept ϵ as a small quantity. In our study, we find that keeping the other parameters the same as in figure 3(c) and continuously increasing Γ will gradually destroy the single-periodicity of the system. When the value of Γ exceeds 10, the single-periodicity of the system will be destroyed. And for 10 < Γ < 23 the system is in a series of quasi-periodic states. At last, when Γ arrives at 24, a clear chaotic attractor appears in the phase space, as can be seen in figure 4(a). The corresponding spatial evolution curve of R(x) shown in figure 4(b) also indicates that R(x) undergoes a chaotic spatial evolution. Figure 4 demonstrates that the regular spatial structure of the BEC has been destroyed by a strong enough atomic current and the system has stepped into a state with a chaotic spatial structure via a quasi-periodic route. What is surprising is that further increasing the value of Γ can eliminate the chaotic spatial structure and allow the system to always stay in a series of single-periodic states even for a very large Γ, denoting that large nonlinear currents cannot necessarily destabilize the system and result in a complex spatial structure. As an example, we plot the single-periodic phase orbit and corresponding spatial evolution curve of R(x) for Γ = 100 in figures 5(a) and (b), respectively. The other parameters are the same as in figure 4. Figure 5 shows that even the value of Γ reaches 100, the BEC system remains in a single-periodic spatial structure. Figures 4 and 5 illustrate that, in the presence of a large chemical potential, a strong enough atomic current is necessary to make the system lose its periodic spatial structure; however, when the atomic current intensity exceeds a critical value, the system chaos will be eliminated and the system will always be in single-periodic spatial structures.
Figure 4. The phase portrait in $(R,\,\dot{R})$ plane and corresponding spatial evolution curve of R(x) for Γ = 24. The other parameters are the same as in figure 3(c). |
Figure 5. The phase portrait in $(R,\,\dot{R})$ plane and corresponding spatial evolution curve of R(x) for Γ = 100. The other parameters are the same as in figure 3(c). |
Now, we study the sensitivity of the spatial structure of the BEC to the intensity of the atomic current. In our numerical simulations we find that when the atomic current is very weak, the spatial structure of the BEC will exhibit a strong sensitivity to the intensity of the atomic current. In order to illustrate this problem, we set the parameters as μ = 1.8, V = 5.0, ϵ = 0.001, κ = 0.5, k = 7.5, ω = 0.01, R(0) = 1, $\dot{R}(0)=0$ and plot the phase portraits in figure 6 for small values of Γ. From figure 6(a) with Γ = 1.11 × 10−5 we observe a particular phase space diagram with a very simple right half but a complex left half showing clear chaotic features. When Γ increases a little to 1.12 × 10−5, a dramatic change occurs in the phase space and the phase orbits only appear in the area of R(x) > 0, see figure 6(b). The evolution of the phase portraits in figure 6 illustrates that when the atomic current is very weak, a very small change in the atomic current intensity can dramatically change the spatial distribution of BEC atoms.
Figure 6. Plots of the orbits in the equivalent phase space $(R,\,\dot{R})$ with μ = 1.8, V = 5.0, ϵ = 0.001, κ = 0.5, k = 7.5, ω = 0.01, R(0) = 1, $\dot{R}(0)=0$ and different values of Γ. |
In this paragraph, we explore the effects of the combined trap parameters on the spatial structure of the BEC. Starting from figure 3(c), we find that continuously increasing the frequency ω of the harmonic potential, the amplitude V or the wave number k of the optical lattice respectively can gradually put the system into chaotic states despite the large chemical potential related to the linear item usually suppressing chaos in nonlinear systems. Hence, there is no doubt that both the harmonic and optical lattice components of the combined trap have important effects on the spatial distribution of BEC atoms. We take the frequency ω of the harmonic potential as the control parameter to illustrate the process from a single-periodic state to a chaotic one. When ω ≥ 0.15, the system obviously loses its spatial single-periodicity. The phase portrait for ω = 0.15 shows that the spatial single-periodicity of the system has been identifiably destroyed, as can be seen in figure 7(a). When ω is increased to 0.35, the phase portrait in figure 7(b) demonstrates clear chaotic features, implying that the system is in a chaotic state and BEC atoms are in a chaotic spatial distribution. And when ω is further increased to 0.476, the chaotic region is further enlarged and the chaotic features become more obvious, see figure 7(c). This denotes that the system is in a more chaotic state and BEC atoms are in a very complex spatial distribution. In order to suppress the chaos in figure 7(c), it is revealed that the chemical potential must exceed 6000. Figure 8 shows the phase portrait for μ = 6000 and the other parameters are the same as figure 7(c). It can be seen in figure 8 that the chaos disappears completely and the system enters a single-periodic state, indicating that the spatial chaos in the BEC under some system conditions requires a very large chemical potential to suppress.
Figure 7. Plots of the orbits in the equivalent phase space $(R,\,\dot{R})$ with μ = 300, V = 5.0, ϵ = 0.001, κ = 0.5, Γ = 1.2, k = 7.5, R(0) = 1, $\dot{R}(0)=0$ and different values of ω. |
Figure 8. The phase portrait in $(R,\,\dot{R})$ plane for μ = 6000 and the other parameters are the same as figure 7(c). |
4. Conclusion
We have investigated the regular and chaotic spatial structures of a BEC with a space-dependent s-wave scattering length in a combined trap. For a BEC with a space-dependent phase, there is a space-dependent atomic current in the system. The atomic current can be adjusted with the Feshbach resonance technique and its intensity can be determined by the initial and boundary conditions. Research findings demonstrate that the existence of the atomic current can greatly affect the spatial structure of the BEC system. The spatial structures of the BEC with and without the atomic current are very different. A large chemical potential is required to suppress the chaotic spatial structure of the BEC system. Despite the existence of a large chemical potential, as the intensity of the atomic current increases, the system moves from a periodic state to a chaotic one via a quasi-periodic route. It is often assumed that strong enough currents may cause complex and even chaotic behaviors in nonlinear systems. However, in the presence of a large chemical potential, a strong atomic current unexpectedly suppresses the chaotic spatial structure of the BEC and always keeps the system in a series of single-periodic states as the atomic current gets stronger. Continuously adjusting the parameters of the combined trap can gradually put the BEC into chaotic spatial structures.