Dynamic evolution of vortex solitons for coupled Bose-Einstein condensates in harmonic potential trap

We studied the evolution of vortex solitons in two-component coupled Bose-Einstein condensates trapped in a harmonic potential. Using a two-dimensional coupled Gross-Pitaevskii equation model and a variational method, we theoretically derived the vortex soliton solution. Under an appropriate parametric setting, the derived vortex soliton radius was found to oscillate periodically. The derived quasi-stable states with typical nonlinear features are pictorially demonstrated and can be used to guide relevant experimental observations of vortex soliton phenomena in coupled ultracold atomic systems.


I. INTRODUCTION
During the past two decades, there has been a strong research concentration on Bose-Einstein condensate (BEC)-related studies and nonlinear phenomena investigation has been a hot topic.Up to now, tremendous progress has been made on studies of typical nonlinear features such as solitons and vortices, and stability that has been predicted 1 in competing nonlinear media is a concern.It has been shown that collapse can be avoided with the influence of the nonlocality of the nonlinearity 2 and that modulational instability 3 can be prevented if the degree of nonlinearity is confined within a certain threshold.One particular nonlinear feature, e.g., white solitons, can undergo periodic oscillation and has been proved to be stable. 4Moreover, in self-focus media, the propagation of vortices stays relatively stable as a result of the nonlocal nonlinearity. 5Better control of the physical situation can be achieved in coupled ultracold atomic gases such as multi-component BECs 6 or BECs with impurities.It has been confirmed that the instability of vortexes will be eliminated significantly by the nonlocal nonlinear response and can form quasi-stable rotating or breathing states. 7n this study, we carried out a theoretical investigation of the vortex dynamics of a two-component coupled BEC system.Using a two-dimensional coupled Gross-Pitaevskii equation (GPE) model [8][9][10][11][12][13][14][15][16] and a variational method, 17,18 we derived the analytical vortex soliton solution for the dilute component and determined that, under a certain parametric setting, the system's distribution width, which is a time-dependent parametric function, oscillates periodically such that the vortex ring showing up in the system expands and contracts alternately and stays in a quasi-stable dynamic state.The key information obtained in this theoretical study, e.g., the vortex oscillation period, can be used to guide relevant experimental studies observing the vortex dynamics in coupled ultracold atomic systems.
The rest of the paper is organized as follows.Section II describes the GPE model and the procedural details of the variational approach adopted toward the analytical solution derivation.Section III gives the results of the analysis and the discussion.Finally, Section IV gives the concluding remarks.a Electronic mail: wangying@just.edu.cnb Electronic mail: w.wang@hw.ac.uk 2158-3226/2017/7(10)/105209/6

II. THE GPE MODEL AND THE VARIATIONAL DERIVATION OF THE VORTEX SOLITON SOLUTION
The realization of Bose-Einstein condensation in traps raised the interesting topic regarding investigating multi-component BEC.Such kind of BEC was first realized with atoms occupying different hyperfine states, and later with different kind of atoms. 19Under the mean field theory framework, we investigated the vortex soliton dynamics for a coupled BEC atomic system according to the following coupled Gross-Pitaevskii equation (GPE): 17,20 where m a and m b are the masses of the dilute and dense component atoms respectively in the twocomponent system.Ω a is the external harmonic trapping potential strength in which the ) is the external trapping potential strength limiting the m b component, g describes the nonlinear interaction strength inside the dense component, and χ and ϕ are the wave functions of the m a and m b components, respectively.Meanwhile, the m a component experiences a type of inter-component interaction described by the parameter ηg, where the parameter η denotes the relative strength of the coupled interaction.To study the dynamic evolution of the coupled BEC, we utilize a variational method to solve Eqs. ( 1) and ( 2).First, we decouple the two equations by adopting the Tomas-Fermi approximation for Eq. ( 2) as 17 To solve Eq. ( 1), we assume the following 2D vortex soliton (with vorticity S = 1) ansatz for χ(r): 18,21 where σ(t) is a time-dependent parametric function describing the system distribution width, γ(t) is a function of t contributing to the phase, and C 0 is the normalization constant.Substituting the ansatz (3) into Eq.( 1) and solving the imaginary part of Eq. ( 1), we obtain the following explicit expression for γ(t): Next, we consider the action S = Ldrdt, where the Lagrangian density for decoupled Eq. ( 1) is Substituting the ansatz (3) into L and integrating the Lagrangian density over the spatial variables, we obtain the Lagrangian depending on σ(t) as follows: Substituting Eq. ( 4) into the Lagrangian above, we re-express the Lagrangian using σ(t) and σ(t) as follows: Now, we apply the Euler-Lagrangian equation ), where a 0 is the initial s-wave scattering length.
and then we reach the equation for the width σ(t) as follows: where Fig. 1 shows the potential function curve for three different nonlinear interaction constants: η = 0.0, 0.01, 0.1.Since the weak interaction g is very small, we can neglect its impact on the analytical derivation at the first instance.Therefore, without Eq.( 12), Eq. ( 9) is solvable with the following analytical solution: where where σ 0 is the original distribution width at t≤ 0.

III. ANALYSIS AND DISCUSSION
Here, we investigate the amplitude of the wave function (3) with the ansatz of the vortex soliton format as follows: To capture the dominant feature of the dynamic evolution of the wave function, we calculate the vortex soliton peak position by taking the derivative of Eq. ( 17) with respect to r: Setting Eq. ( 18) to zero, we get the formula for the peak position as follows: which means that the vortex soliton's peak position is time varying since σ(t) is a periodic function with period 2π/ω.This means that the vortex soliton's radius oscillates between the maximum value √ B − A and the minimum value √ B + A. Fig. 2 shows the periodic variation of r max at time t.Fig. 3 shows eight pictorial plots of the vortex soliton shape at eight timing positions.For comparison purpose, the vortex soliton plots based on the numerical solution of the coupled Eqs. ( 1) and (2) at four timing positions are shown in Fig. 4, where the numerical calculation starts from the initial wave function Ar exp(− r 2 2σ 0 + iθ) (A is normalization constant) and is based on the coupled partial differential equations solving module in the software package of Mathematica.We can see similar evolutionary trend for the vortex soliton shape.
Here, it is necessary to comment on the occurrence of this dynamic stability.The oscillation amplitude of σ(t) around the minimum position of V (σ) should not be too large and is confined inside the trapping region of V (σ)'s "potential" curve.Moreover, our pure analytical derivation is based on the assumption that the nonlinear interaction strength constants g and η are small.A larger g or η value means a significant deviation of the dynamic quantities such as the oscillation period and the stable trapping range of V (σ); from Eqs. ( 9), ( 10), (11), and ( 12), however, we can see that the existence of σ(t)'s quasi-stable oscillation feature will not be affected provided that η 2 g 8 < m a Ω 2 a .The qualitative property (i.e., the oscillation of the system's distribution width σ(t)) is a particular feature of our theoretical treatment presented here, which can be used to guide relevant experimental observations of vortex soliton phenomena in coupled ultracold atomic systems.

IV. CONCLUSION
In this study, using a the two-dimensional coupled GPE model and a variational method, we investigated the vortex soliton dynamics for a coupled two-component BEC system.Through the vortex soliton ansatz of the wave function for the dilute component, we finally derived the analytical vortex soliton solution, in which, under an appropriate parametric setting (i.e., σ(t) lies in the trapping region of the potential function V (σ)), the vortex soliton's radius oscillates periodically and, thus, exists in a quasi-stable dynamic state.The theoretical results obtained here can be used to guide relevant experimental observations of vortex soliton phenomena in coupled ultracold atomic systems.