Shaanxi Key Laboratory for Theoretical Physics Frontiers, School of Physics, Northwest University, Xi’an 710069, China
† Corresponding author. E-mail:
zhaolichen3@nwu.edu.cn
Project supported by the National Natural Science Foundation of China (Grant No. 11775176), the Major Basic Research Program of the Natural Science Foundation of Shaanxi Province, China (Grant No. 2018KJXX-094), and the Key Innovative Research Team of Quantum Many-Body Theory and Quantum Control in Shaanxi Province, China (Grant No. 2017KCT-12).
1. IntroductionMulti-component coupled Bose–Einstein condensate (BEC) provides a good platform to study dynamics of vector solitons.[1,2] Many different vector solitons have been obtained in two-component coupled BEC systems, such as bright–bright soliton,[3] bright–dark soliton,[4] dark–bright soliton,[5–7] dark–dark soliton.[8] Moreover, it is possible to find dark–anti-dark soliton in BEC with unequal inter- and intra-species interaction strengths.[9–12] If the superposition of the dark soliton and anti-dark soliton in two components is uniform density, the dark–anti-dark soliton would become the so-called “magnetic soliton”.[9,10] Different from the magnetic properties of these solitons, domain wall (DW) can also exist in two-component BEC with many different settings.[13–16] Furthermore, exact analytical domain wall solutions were derived with both two-body and three-body interactions in a two-component BEC system.[17] The DW solutions usually correspond to kink and anti-kink excitations in each component. With linear coupling effects, an exact DW solution was produced for ratio 3:1 of the inter-species and intra-species interaction coefficients.[18] But the interactions between DWs or their motions were investigated numerically in the most of previously studies. Based on the most of previous studies on vector solitons in two-component BEC, we can know that kink-type excitation can not be obtained with identical inter-species and intra-species interaction strengths.[19–21] Therefore we try to find kink and anti-kink excitations in some integrable cases with different inter-species and intra-species interaction strengths. The integrability could enable us to discuss the interactions between DWs more clearly and conveniently.
We note that pair-transition effects bring many striking localized wave structures in a two-component BEC,[22–26] since the constrain conditions on nonlinear interactions are quite different from the ones in Manakov model.[27] For this integrable case with the pair-transition effects, the inter-species and intra-species interaction strengths can be different. This provides possibilities to investigate nonlinear waves exactly and analytically in the system. Moreover, experiments in BEC systems suggested that pair-particle transition effect became dominant, and single-particle transition could be ignored.[28,29] We therefore try to look for kink and anti-kink excitations in the system. Moreover, more exotic transition dynamics are expected to exist in the pair-transition coupled model,[24,26] compared with transition process for DWs obtained in Refs.[17] and [18].
In this paper, we show that DW excitation can exist in a two-component BEC with pair-transition effects. It is shown that DW excitations can be described exactly by kink and anti-kink excitations in each component. The DW with moving velocities can be described by a generic solution. The DW solutions are given analytically with no three-body interactions, in contrast to the DW reported before.[17] The explicit conditions are for ratio 2:1 of the inter-species and intra-species interaction coefficients, which is different from the ones obtained in BEC with linear inter-conversion between the immiscible components.[18] Furthermore, we derive multi-kink in the system, which can be used to investigate nonlinear interactions between DWs exactly. And in this way, bubble-droplet structure can be obtained. Especially, their interactions demonstrate some striking particle transition dynamics. The striking transition effects make the DW admit quite different collision behavior, in contrast to the collision between solitons or other nonlinear waves. The collisions between kinks induce phase shift, which makes the DW change greatly. With the proper combination of fundamental DWs, collisions can be elastic or inelastic. These characters can be used to manipulate one DW by interacting with other ones.
2. Physical modelOne-dimensional two-component BEC system with particle transition can be described by the Hamiltonian
where
is the particle number operator, the symbol
represents the Hermite conjugation.
and
(
i=1,2) are the intra- and inter-species interactions between atoms.
J1 and
J2 denote single particle transition and pair particle transition coupling strengths,
[30,31] the pair particle transition terms correspond to pair-transition effects in BEC or the four-wave-mixing in optic systems. Recent experimental results in a double-well Bose–Einstein condensate suggested that pair-transition can become dominant with strong interaction between atoms.
[28,29] Therefore, we consider that the case for second-order transition is dominant, namely,
and
. The corresponding dynamic evolution equation can be derived from the Heisenberg equation
for the field operator. Performing the mean field approximation
, we can get the following integrable coupled nonlinear Schrödinger equations (NLSE) in dimensionless form
,
where
q1 and
q2 are functions of the space coordinate
x and time
t, the symbol overbar represents the complex conjugation. Many striking localized wave structures had been given in two-component BEC,
[22–26] as the constrain conditions on nonlinear interactions are quite different from the ones in Manakov model. Since the dark soliton admits kink-like character partly, we will start from investigating dark soliton combing with some other nonlinear waves. Notably, a linear superposition of dark soliton with a nonlinear plane wave can produce kink-like density distribution. This is a key clue to construct DW in BEC with pair-transition effects. With the aid of linear superposition of dark soliton solutions and a general plane wave, we obtain DWs exactly and analytically, this is the main result of this paper. Next, we show the DWs explicitly and investigate the transition behavior during their interaction processes.
3. Analytical domain wall and bubble-droplet solution3.1. Domain wall solutionThe fundamental DW solution can be derived as follows, it is similar to the solutions reported in Refs.[25] and [32]:
where
The real parameter
(
) is the spectral parameter,
and
are the real and imaginary part of
which is root of the characteristic equation.
ci,
ki, and
are the background amplitudes, wave vectors, and frequencies of the dark soliton and plane wave,
only affect the center position of DW. The parameters
ci,
ki, and
satisfy the following formula
, otherwise the solution will not exist.
Based on the solution, we can observe the profiles of DW conveniently. As an example, we show the simplest case for fundamental DW. With
,
, stable domain wall structure can be shown in Fig. 1, the stability of DW is tested numerically with some weak noises, the results indicate that the DW is stable. This DW solution is different from the exact solutions in Refs. [17] and [18] that it is not completely static. The moving DW was investigated numerically by adding a spatial-dependent phase.[17,18] Moreover, many different profiles appear when the spectral parameter or background parameters vary, we can control the velocity, height and gradient of DW by adjusting the value of
, this gives us the opportunity to study the interaction of DWs analytically. When cj or kj is changed and results in
or
, the stable DW and anti-DW will disappear as the backgrounds oscillate periodically. We do not show these cases, since their properties are similar to the one in Fig. 1, just with some additional spatial-temporal periodic behaviors.
3.2. Bubble-droplet statesA striking bubble-droplet structure can be obtained through combining a DW and an anti-DW properly. If the domain wall and the anti-domain wall admit identical velocities, the solution for bubble-droplet structure can be written as:
The implications of parameters are the same as in Subsection 3.1. If only
and
are of unequal values and other parameters are equal respectively, the profile of this solution corresponds to bubble-droplet. The numerical results of this structure and its evolution with no velocity in BEC have been reported before,
[17,18] now we give an exact analytical solution, and bubble-droplet state can be non-static, its velocity can be adjusted by changing the values of
too. As an example, we show one static case in Fig.
2. It is seen that kink–anti-kink structure appears in the two components and they are parallel. The profiles could be considered as a bubble coupled to a droplet, which refers to as bubble-droplet state. When the distance between DWs and anti-DWs are small enough, the cross section of two components can be approximately regarded as a dark soliton and a bright soliton. The dark–bright soliton is similar to dark–bright one obtained in two-component BEC system, but the expressions of them are distinctive, which makes them possess different effective mass.The stability of bubble-droplet state is tested numerically with weak white noises, and its long time evolution is shown in Fig.
3. The results of simulation indicate that bubble-droplet we construct can be stable with weak noises.
If
, the kink and anti-kink will have different velocities. The solution can be used to investigate the interactions between DWs. We note that the interaction processes admit abundant transition behavior between the two components. We can investigate the interactions between DWs and bubble-droplets conveniently through deriving high-order DW solutions.
4. Particles transition during the interaction process between domain wallsWhen the velocities of DW and anti-DW are different, they will collide somewhere and induce new dynamical behavior. From Fig. 4, we can see that parallel and non-parallel situations have completely different properties. In the non-parallel case, stable bubble-droplet state disappears and the backgrounds become asymmetric both in two components. Neither interference behavior nor the tunneling of particles emerges during the collision process, which is different from our previous work on wave properties of solitons.[33,34] However, the parts between kink and anti-kink have different values before and after the collision, this means that the particles are no longer conserved in each component. The particle transition emerges between two components, with the total number of particles in the whole system is constant.
In order to investigate the transition behavior, we numerically calculate the changes in the number of particles of the two components while domain walls interact with each other in Fig. 4. The particle number evolutions for the two components are shown in Fig. 5, which is calculated in the regimes where particle transition happens. It is obvious that there is a transition behavior between the two components, the particles transit in pairs and the total number of them in the whole system is constant. The phase shift induced by the collision can be used to explain why the particle numbers vary.[26]
Then, we further investigate the collision between a DW and a bubble-droplet. Solution for describing the interaction between DW and bubble-droplet can be derived as
where
M and
Mi (
i,
m=1,2) are both 2 × 2 matrices and their elements are
There is a DW and a bubble or a droplet in each component, the interactions of them can be observed analytically and exactly. All these solutions for describing collision between DWs are absent in Refs. [
17] and [
18], which were investigated numerically. As an example, we show one case in Fig.
6, for which DW collides with bubble or droplet in two components. It is seen that the profile of kink state does not vary after collision process, but the bubble and droplet state both vary after the collision. Namely, the bubble state in
q1 component transit to be a droplet state after collision (see the upper picture in Fig.
6), and the process is inverse in the
q2 component (see the lower picture in Fig.
6). The collision with a kink makes the bubble-droplet state change greatly. The collision process also induce particle transition between the two components, and this transition behavior can be calculated directly by the particle number in the bubble state.
Finally, we would like to investigate the collision between two bubble-droplets. The general solution for observing their interaction can be given as
 | |
We investigate the collision between bubble-droplets based on the solution analytically. As an example, we show one case in Fig. 7. The two bubble-droplet states are both kept well after the collision process, but there is a striking transition behavior during the collision process. Two droplet states collide with each other, and there is a state with no particles in the q2 component (see the lower picture in Fig. 7). The particles all transit to q1 component, therefore there is a hump emerging during the collision between two bubbles (see the upper picture in Fig. 7). In fact, the collision process can be also related with the ones in Fig. 6, since a bubble-droplet is constituted of one DW and one anti-DW. Namely, this situation can also be seen as a bubble-droplet colliding successively with a DW and an anti-DW, there are two collisions with DW in this way. The first collision with a DW changes the bubble-droplet to be droplet-bubble state, and the second collision with an anti-DW changes it back to be a bubble-droplet state. Therefore collision with DWs always induce the transition effect in the BEC with pair-particle transition effects discussed here.
If we change the relative velocity between two bubbles (droplets), there will be some different transition behaviors at the center of collision, which reminds us that relative velocity will influence the particle transition. As an example in Fig. 8. It is seen that the particle transition is quite different from the one in Fig. 7. The particle transition directions are inverse with each other for the ones in Fig. 7 and Fig. 8. This comes from the phase shifts induced by DW collisions are different for the two cases. It should be noted that the velocity of DW should be smaller than the velocity of sound on the density background with repulsive interactions between atoms.
5. Conclusion and discussionIn this paper, we give the analytical DW solution in two-component BEC which can be described by coupled NLSEs with pair-transition effects. It is shown that DWs can be described exactly by kink and anti-kink excitations in each component. Bubble-droplet structures can be also obtained from the fundamental DW, and their interactions are investigated analytically. Especially, DW interactions demonstrate some striking particle transition dynamics. In contrast to the collision between solitons or other nonlinear waves, these striking transition effects makes the DW admit quite different collision behavior. The collision between kinks induces some phase shift, which makes the DW profile change greatly. Their collisions can be elastic or inelastic with proper combination of fundamental DWs. For examples, collisions between bubble-droplet states can be elastic, and the collision with one DW is usually inelastic. These characters can be used to manipulate one domain wall or bubble-droplet state by interacting with other DWs.
