1. Introduction

After the first experimental realization of Bose-Einstein condensates (BECs) in dilute alkali metal vapors^[1], their study has experienced enormous experimental and theoretical advancements. The observation of the bright solitons and the quantum interference phenomena between two coupled BEC have yielded a stunning new demonstration of the wave-like behavior of atoms and provided us with the perspective of getting a better understanding of the complicated behavior of quantum many-body systems and with the hope of realizing novel concrete applications of quantum mechanics, such as atom interferometers and atom lasers^[2].

Currently, the entire experiment of the BEC is carried out at ultralow temperatures and takes place in an electromagnetic trap of the order of millimetres or smaller. In most experiments, a harmonic potential, or very nearly so, is employed, but a wide variety of potentials can now be constructed experimentally. In recent experiments BECs have been successfully loaded in optical lattices^[3](OL) and, ever since, this field has attracted considerable attention. A periodic OL potential is created along the waveguide axis (x axis) by interference patterns from multiple laser beams^[4]. It can be modeled byV^opt (x)=V ₀ cos(kx). The longitudinal envelope of the OL potential is determined by the profile of the laser beam waist and the lattice wave-number k can be tuned by the geometry of laser beams. The great flexibility of an OL potential arises from the fact that the above parameters can be tuned experimentally, providing precise control over the shape and time-variation of the external potential. OL potentials are, therefore, of particular interest to a number of BEC applications—ranging from matter-wave optics to precision measurements and quantum information processing.

Using highly anisotropic traps, it is possible to produce quasi-one-dimensional (1D) BECs (cigar-shaped). The 1D regime is valid when the transverse dimensions of the condensate are of the order of or less than the atomic interaction length, and both are much smaller than its longitudinal dimension. Thus the condensate has the form of an ellipsoid stretched along one of its major axes. In this regime the BEC remains phase coherent. For shorter times the nonlinear Schrodinger equation (NLSE), also called Gross-Pitaevskii equation(GPE), provides an accurate description of the BEC soliton dynamics.^{[5, 6]}

If more than one distinct condensate is trapped in the confining potential there is a macroscopic quantum tunneling between the two BECs, and the GPE is no longer sufficient. In that case, the wave-function for each BEC satisfies its own GPE coupled to the others by nonlinear mean-field interactions. Thus, in the 1D regime, the BEC dynamics is governed by a set of coupled NLSEs with an external potential. Notice that there are two different types of atomic tunneling between coupled BECs, viz. external tunneling and internal tunneling^{[7, 8]}. The coherent atomic tunneling between the two BECs confined in a double-well potential and the macroscopic quantum self-trapping phenomenon have been theoretically predicted and examined in Refs. [9–11]. Williams et al. have proposed an experiment to use the binary mixture of states in order to observe the internal tunneling among BECs confined in a single-well potential^[12]. With the proceeding of the theoretical investigations, related experimental investigations have been also reported. In Ref. [13], the BEC atoms are confined in an array of optical traps in a gravitational field, and the tunneling among the BECs confined in the multiwell potential has been observed by Anderson and Kesevich. Hall and co-workers have achieved the macroscopic quantum coherence of the two BECs in different hyperfine spin states of ⁸⁷Rb atoms, and the relative phase of the two BECs has been measured^[14]. Despite the aforementioned experimental and theoretical investigations, the analytical form of the coupled BEC solitons in an OL potentials and the domain of their existence were very few addressed in the previous publication. The aims of the present paper are to analytically discuss the coherent evolution of two coupled BEC solitons confined in an OL with external tunneling, and some novel results are obtained.

2. Formalism

To consider the dynamics of BECs trapped in an OL, we start with a cigar-shaped condensate. If considering only two-body, mean-field interactions, a dilute Bose-Einstein gas dynamics can be modeled using an effective 1D NLSE in the following dimensionless form^[15]

where u≡u(x, t) is the mean-field BEC wave-function, and the nonlinearity coefficient g=±1 accounts for repulsive(+) and attractive(-) interatomic interactions, respectively, and relates to the s-wave scattering length as. In this paper, we study the coherent evolution of bright solitons (for g=-1). V(x) is the normalized external confining potential. It is convenient to decompose the potential V(x) as follows

where the term V_con describes the magnetic trap(MT). The periodic potential, V_opt , is formed by an OL, k is the wave-number and V ₀ is the lattice depth measured in units of the lattice recoil energy, E_rec . The parameter Ω = ω_x/ω_⊥, where ω_x and ω _⊥ are the confining frequencies of MT in the axial and transverse directions. In Eq. (1), t is the dimensionless time measured in terms of ${\omega }_{\perp }^{-1}$ and x is the longitudinal spatial variable measured in units of the transverse harmonic-oscillator length, a _⊥ (h/mω _⊥)^1/2. Accordingly, |u|² corresponds to the rescaled population density of the condensate measured in units of mω _⊥/4πħ|a_s |, and m is the atomic mass. In practice, the MT has a flat central portion supporting several hundreds of periods of the periodic potential. The whole potential is sufficient to satisfy the desired near-periodic potential. It is also possible to load the condensate onto V(x) and then adiabatically remove V_con , leaving only the periodic potential. In this work, we let V_con = 0 and focus on optical lattices, viz.

For the wave-function u of two well-separated BECs we can write u=u ₁+u ₂ and assuming mutual coupling only through intensity overlap in the |u|² term we find the following coupled equations

where u_i (i=1,2) is the wave-function of the two BECs respectively and u_i is normalized to the number of particles in the i-th BECs. N=N ₁+N ₂=${\int }_{-\infty }^{+\infty }$|u ₁|² dx+∫+∞-∞|u ²|² dx is the rescaled total number of atoms(a conserved quantity). In Eq. (4), we have neglected the fast varying term $u_{3-i}^2$ $u_i^{*}$ (i=1,2), where ^* denotes the complex conjugate. Compared with Ref. [7, 16], Eq. (4) is a specific case in which the two BECs are symmetrical and there is no linear coupling.

3. Computation and discussion

In the absence of any potential and mutual coupling, Eq. (4) possesses the stationary bright soliton solutions in the general form of

The parameter A_i represents the amplitude (and inverse width), x _0i and ẋ_oi = dx _0i/dt represent the location and the velocity of the soliton center, and ϕ_i is the phase. During the evolution of the two coupled BEC solitons, we assume that the wave-function ui retains the Sech-shape given by Eq. (5), but A_i ,x _0i,ϕ_i are the functions of time as a result of the weak interaction. The condition of the normalization is N=${\int }_{-\infty }^{+\infty }$|u ₁|² dx + ${\int }_{-\infty }^{+\infty }$|u ₁|² dx = 2(A ₁+A ₂).

The averaged Lagrangian of Eq. (4) can be obtained with the usual variational approach^[17–19]

where

Using $\frac{\partial L\left(t\right)}{\partial \sigma }-\frac{d}{dt}\left(\frac{\partial L\left(t\right)}{\partial \dot{\sigma }}\right)=0,(\sigma ={\varphi }_1,{\varphi }_2,A_1,A_2,x_{01},x_{02}),$ then

Equations (8a)~(8c) determine the evolution characteristics of the two coupled BEC solitons. Although from Eq. (8a) we infer that Ai is still a constant, Eq. (8c) can be regarded as an equation determining the phase ϕ _i and the amplitude A_i once the other parameters are known so that the equations are self-consistent. We assume that the system of the BEC solitons is symmetric, A ₁ = A ₂ = A, x ₀₁ = Δ/2 and x ₀₂ = -Δ/2, viz. Δ = x ₀₁ - x ₀₂ is the separation between the BEC solitons. From Eq. (8b), we can obtain

According to Eq. (7), then

An anharmonic effective potential is derived using the mechanical analogy,

where, $V_{\mathit{eff}1}=\frac{8}{3A}\frac{k\pi V_0}{\sinh \left(\frac{k\pi }{2A}\right)}\cos \left(\frac{k\Delta }{2}\right)$ is the effective potential resulting from the OL, and $V_{\mathit{eff}2}=\frac{8}{3A}{L}_{12}=-\frac{64}{3}{A}^2\left[\frac{-1}{{\sinh }^2\left(A\Delta \right)}+\frac{\cosh \left(A\Delta \right)}{{\sinh }^3\left(A\Delta \right)}\ln \frac{\cosh \left(A\Delta \right)+1+\sinh \left(A\Delta \right)}{\cosh \left(A\Delta \right)+1-\sinh \left(A\Delta \right)}\right]$ arises from the weak mutual interaction of two BEC solitons. From Eq. (9), we can conclude that this interaction will lead to changes in the BEC soliton dynamic characteristics.

 

Fig. 1 The effective potential V _eff2 resulting from the weak mutual interaction of two BEC solitons versus the separation Δ of the two BEC solitons

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By numerical simulations, V _eff2 versus the separation Δ is plotted in Fig. 1 which indicates that the interaction between the two coupled BEC solitons forms a potential well. The separation within which the two solitons can interact is finite, and the effective potential is approximately equal to zero beyond the special separation (see Fig. 1). This is because that the wave-functions of the two BEC solitons do not overlap in that case. We can also conclude that the interaction of two BEC solitons is attractive, and that the solitons will move down the potential slope towards x=0, an equilibrium point, and oscillate around the equilibrium point. Beyond the potential field, the two solitons are in neutral equilibrium. From Fig. 1, we find that the width and depth of the effective potential well are very sensitive to the amplitude A of a soliton. For example, the effective potential well becomes deeper and narrower as A increases. The separation where the two solitons can interact becomes smaller. This is because that the amplitude A reflects the number of atoms in a BEC soliton, and the bigger the number of atoms is, the greater the ability of the self-focusing is. On the other hand, in the absence of mutual coupling, the OL leads to a periodical potential field V _eff1. The periodical potential is relative to the parameters of the OL and solitons. At the same time, the wave-number k has a greater effect on the amplitude of the effective potential V _eff1. The larger the wave-number is, the smaller the amplitude becomes.

In Fig. 2, we plot the effective potential Veff versus the separationΔaccording to Eq. (11) when both the coupling effects between the two BEC solitons and the OL are present to the effective potential. The parameters of the system are A=1, and V ₀=0.5. The solid, dashed and dotted lines correspond to the wave-number k=0.1, 0.5, 1.5 respectively. We find that the wave-number k plays an important role. It deforms the shape of the effective potential, and changes the dynamic characteristics of the system.

 

Fig. 2 The effective potential Veff versus the separation Δof the two coupled BEC solitons

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In the case of long-period OL, the effective potential of the OL compensates that of the two mutually coupled solitons. V _eff1 is dominant and its shape is fluted with V _eff2 around the point x=0. Apart from the flute, the whole effective potential is similar to the case without mutual coupling. The system is in a stable equilibrium at the minimum of the effective potential or in a vibration around the stable equilibrium point.

In the case of intermediate OL period, V _eff2 is dominant within the separation where the two solitons can interact because the amplitude of V _eff1 is smaller and two small-peak values appear whose amplitude depends on the amplitude of the OL and the slope of V _eff2. When the peaks of the effective potential are low, the BEC solitons may cross over the peaks and meet at the bottom of the effective potential if the solitons possess larger initial energy. In that situation, the separation within which the two solitons can meet by mutual attraction becomes much wider. We notice that the OL potential will be considered as a perturbation parameter if the OL period is much smaller than the width of the BEC soliton. Then the variational approach is non-effective and the perturbation theory will be employed^[20]. Hence our investigations are valid when the OL is assumed to be smooth and slowly varying on the soliton scale.

In order to get an understanding of the novel phenomena, in Fig. 3, we show the motion of the two coupled solitons by a direct numerical integration of Eq. (4). In our numerical simulations, the initial input pulse is u_i (t=0)=Asech(x - x _0i) and the other parameters are k=0.5, and A=1 for Fig. 3(a) and A=2 for Fig. 3(b, c, d). The initial locations of the motionless BEC soliton center x _0i (i=1, 2) and the lattice depth of the OL V ₀ are indicated in the figures. When Δ is smaller, the two solitons will meet at x=0 because of the attractive interaction as shown in Fig. 3(a). In the case of a larger Δ, the interaction between the two solitons is sufficiently small, and each of the two solitons evolves independently. When we initialize the solitons at the respective minimum of the potential, the solitons remain steady and are in stable equilibrium, which is indicated in Fig. 3(b). If the initial position of the soliton center is around the bottom of the potential, the corresponding soliton will vibrate around the minimum, and the frequency of the vibration is relative to V ₀ and A (see Fig. 3(c) and (d)).

 

Fig. 3 The two coupled BEC soliton density evolve in an OL. The initial locations of the motionless BEC soliton center x _0i (i=1, 2) , the amplitude A and the lattice depth of the OL V ₀ are: (a) x _01,2=±1, A=1 and V ₀=0.5, (b) x _01,2=±6.28, A=2 and V ₀=0.5, (c) x _01,2=±8, A=2 and V ₀=0.5, (d) x _01,2=±8, A=2 and V ₀=1.5.

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Fig. 4 The two coupled BEC soliton density evolve in a slowly tuning OL. The OL drags the solitons and takes them to another location. The center of the solitons is initialized at the respective effective potential minimum.

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From the above investigation, we can conclude that an OL may capture a coupled BEC soliton. On the other hand, the OL potential can be tuned by changing the lattice wave-number k. Hence, we can change the separation of the two coupled BEC solitons by slowly changing the OL, as long as the OL tuning is slow enough to ensure adiabatic change. Fig. 4 shows that a slowly tuning OL drags two BEC solitons and takes them to another position. The initial input pulse is u_i (t=0)=Asech(x-x _0i), the parameters of the system are selected as A=2, V ₀=0.5, and k=1.5-0.01t, and the initial location of the motionless BEC soliton center is x _0i=±6.28. We notice that a moving localized attractive impurity or a moving OL (a total OL is moved) may drag a stationary soliton^{[20, 21]}, but here our aim is to drag the soliton by tuning the wave-number k of an OL. Apparently, this issue is important for applications of the BEC.

4. Conclusions

The characteristics of two coupled BEC solitons trapped in an OL are investigated with the variational approach and direct numerical simulations of the GPE. Our investigation shows that an OL may be used not only to sustain two coupled solitons but also to manipulate them at will. Its effect depends on the initial location of the BEC soliton, the lattice amplitude and wave-number, and the amplitude of the coupled BEC solitons. Furthermore, the interaction between two nonlinear coupled BEC solitons is the attractive effect within some separation. Either in view of understanding their dynamic evolution or in view of their practical application, these properties are important.