Deformation scheme of nonlinear Rosen–Zener tunneling for Bose–Einstein condensates in a triple-well potential

We investigate Rosen–Zener tunneling for Bose–Einstein condensates in a triple-well potential within the framework of mean-field treatment. Firstly, we exactly calculate tunneling dynamics for triple-well in the linear case, where the population evolution of each well is robust and all atoms are finally trapped in the single well that is initially uploaded. In this case, tunneling dynamics are symmetrical when all atoms are populated in the first well and third well. However, as the nonlinear interaction is introduced, tunneling dynamics will be significantly changed. On the one hand, the symmetry will be broken, some atoms will not be confined to the starting well. On the other hand, nonlinear Josephson oscillation will be presented within a fixing interval. When the interaction exceeds this interval, the self-trapping solution emerges.

In a mean-field approximation, the dynamics of BEC trapped in a finite-sized-well potential is described by the Gross-Pitaevskii equation (GPE) or nonlinear Schrödinger equation (NLSE) [18], where the nonlinearity naturally arises from the onsite interactions. Numerous studies have shown that this interaction will significantly affect the tunneling dynamics between the wells [7][8][9][10][11][12][13]. For example, in the double-well potential, the RZ tunneling exhibits Josephson oscillation with a weak interaction [4]. When the atomic interaction is strong enough, the Josephson oscillation between two wells is completely blocked and all atoms are trapped in one well, which is termed as self-trapping phenomenon. The RZ model also has been extended to two-component BEC and interferometry of an interacting BEC trapped in H. Cao (&) School of Material Science and Engineering, Chongqing Jiaotong University, Chongqing 400074, China e-mail: ch198772ch@163.com double-well potential with a periodically modulating barrier [19,20]. The other nonlinear models in the double-well potentia also exhibit interesting phenomena. In the nonlinear LZ tunneling, there exists a nonzero tunneling probability in the adiabatic limit for large enough interaction strength [1]. For the nonlinear DK transition, the nonlinearity leads to the asymmetry of transition probability in different initial states [5].
On the other hand, the addition of a third well to a double-well system suggests that an even richer dynamics should be observed, which will be more complicated and no longer integrable. The LZ model has been extended to BECs trapped in a triple-well and investigated its tunneling dynamics with the existence of nonlinearity, which shows many striking properties distinguished from that of the double-well case [2,7,8]. For a very small nonlinear parameter, the energy levels still retain the same topological structure as its linear counterpart, the adiabaticity breaks down and manifests the presence of a nonzero tunneling probability. At the same time, the nonlinear LZ tunneling is irregular showing an unresolved sensitivity to the sweeping rate, a phenomenon attributed to the presence of the chaotic state. Recently, nonlinear LZ dynamics of Rydberg-dressed Bose-Einstein condensates have been investigated in a triple-well potential, in which long-range soft-core interaction gives rise to strong nearest-and next-nearest-neighbor interactions in the triple-well system [21]. The longrange soft-core interaction induces a far more controllable manner than BECs with onsite interaction. However, the RZ model has not been extended to BECs trapped in triple-well.
In this paper, we extend the Rosen-Zener model to BECs trapped in triple-well potential and examine how nonlinearity affects the tunneling dynamics in this system. Over the years, the study of deformation schemes for the RZ model has also been uninterrupted. In addition to the initially proposed hyperbolic tangent type [22], other forms include Gaussian [23], exponential [24], and hyperbolic tangent [25]. We propose a special hyperbolic-secant model distinguished from that of the original model, in our model, the Hamiltonian is only constructed by a single parameter. We give its analytic solution in a linear case, and examine tunneling dynamics in the presence of nonlinearity.

The model and analytic solution in linear case
We investigate the dynamics of a BEC in a triple-well system under mean-field approximation, the wave function j WðtÞi of the system is described by in which jw i iði ¼ 1; 2; 3Þ are three spatial modes localized in the first, second, and third wells, satisfying jw 1 i ¼ ½1; 0; 0 T , jw 2 i ¼¼ ½0; 1; 0 T and jw 3 i ¼ ½0; 0; 1 T . c i ðtÞði ¼ 1; 2; 3Þ are the probability amplitudes of three modes, which are fixed as j c 1 ðtÞ j 2 þ j c 2 ðtÞ j 2 þ j c 3 ðtÞ j 2 ¼ 1. The triple-well system described by the dimensionless Schördinger equation [26][27][28] i d dt with the nonlinear Hamiltonian where g is the nonlinear parameter describing interaction strength between atoms. When g [ 0, the interaction is repulsive. K is the tunneling rate between nearest neighbor traps, and c is a bias field to create a potential height difference between neighboring traps. A schematic representation of our model is shown in Fig. 1. If the potential wells are linearly biased and the tunneling rate keeps constant, nonlinear Landau-Zener (LZ) tunneling will be presented in this system. Landau-Zener dynamics has produced a vast field of research and is still lucrative in terms of its modernday applications [7][8][9][10]21]. In contrast to the LZ model, in the RZ model, the bias of each well keeps a constant and the tunneling rate is non-linearly variable [22][23][24][25]. In our paper, we will investigate Rosen-Zener tunneling in BECs trapped in triple-well, so setting the tunneling rate in the form of there is just one adjustment parameter c, which is the same as a bias field. When g ¼ 0, this kind of RZ model is precisely in a linear system, and its dynamics can be exactly solved. To continue, we rewrite the Hamiltonian of Eq. 3 as where J i ði ¼ x; y; zÞ denotes the general angular-momentum operator with quantum number j ¼ 1, and its components satisfying ½J i ; J j ¼ ie ijk J k . Differing from special function (original) method, here we invoke a dynamical invariant [29,30] to resolve the dynamics of the above system IðtÞ ¼U y ðtÞJ z UðtÞ ¼ sin hðtÞ cos uðtÞJ x þ sin hðtÞ sin uðtÞJ y þ cos hðtÞJ z ; in which jmiðm ¼ 0; AE1Þ represents the eigenstate of J z . For triple-well system, jmi can be regarded as spatial modes of each well, satisfied j1i ¼ jw 1 The dynamical basis of the BEC system in a linear case can be formulated as j WðtÞi ¼ P m k m e ia m j / m ðtÞi, k m is constant and the so-called LR phase a m is given by Correspondingly, we can compute how the atom is populated in each well, i.e., P i ðtÞ ¼j c i j 2 ¼ jhw i j WðtÞij 2 ði ¼ 1; 2; 3Þ. We have discussed that the populations of three wells vary with time by manipulating the initial states, and it has been depicted in Fig. 2. As shown in Fig. 2, the tunneling dynamics are sensitive to the choice of the initial mode. When all atoms are initially populated in the first well and third well (see Fig. 2a and Fig. 2c), tunneling dynamics are symmetrical. As time goes on, the population of the starting well (well 1 or well 3) is firstly reduced, then it will increase when a fixed value is reached. The populations of the other two wells increase first and then decrease. Finally, all atoms are located in the initial well (well 1 or well 3). Unlike the previous two cases, when the BEC is initially prepared in well 2, the populations in well 1 and well 3 are changed in the same way, increasing first and then decreasing. Finally, all atoms are located in well 2. These three cases share a similar feature in that population evolution is robust and smooth and all atoms are finally trapped in the single well where it is initially uploaded.
3 Nonlinear RZ tunneling in triple-well system

Numerical Results
In this section, we will study the dynamics of the nonlinear interaction (g 6 ¼ 0) BEC for different initial states. The analytical solution in the above section is not available when the nonlinear interaction is introduced in the Hamiltonian (3), we therefore exploit a fourth or fifth-order Runge-Kutta algorithm to trace the tunneling dynamics numerically. The similar numerical methods are widely used to solve the dynamics of complex system [31][32][33][34][35].
Then we depict the contour plots for population of three wells with different initial states in Fig. 3. To investigate the interplay between g=c and initial state, we choose the population changing as a function of g=c and time (evolving from À5=c to 5=c). Among them, Fig. 3a, b, and c stands for population evolution in three wells when all atoms are initially populated in well 1. Figure 3d, e, and f show the population evolution when all atoms are initially filled in well 2. Figure 3g, h and i represent the population evolution of each well when all atoms are initially filled in well 3.
It is clear that the non-linear interaction affects the populations of each well differently under the different initial states. When all atoms are initially in the first well, i.e., P 1 ðÀ5=cÞ ¼ 1 [ see Fig. 3a, b and c], as time goes on, the population evolution is slightly modified from the linear counterpart, which is only affected to a small extent by non-linear interactions at g=c\6. Differing from the linear case, some atoms do not finally get confined to the starting well. When g=c [ 6, the populations in the three wells will not be altered. For all atoms initially located in well 2 [ Fig. 3d, e, and f], the population in two of these wells was significantly affected by nonlinear interactions. When g=c ranges from 2 to 6, it undergoes rapid oscillations in well 1 and well 2, but only in the region where time is greater than 0. Finally, the atoms were Fig. 2 Time evolution of population in each well P i ðtÞ for different initial states. a starting from the first well (P 1 ðtÞ ¼ 1; P 2 ðtÞ ¼ 0; P 3 ðtÞ ¼ 0), b starting from the second well (P 1 ðtÞ ¼ 0; P 2 ðtÞ ¼ 1; P 3 ðtÞ ¼ 0), Rosen-Zener starting from the third well (P 1 ðtÞ ¼ 0; P 2 ðtÞ ¼ 0; P 3 ðtÞ ¼ 1) not bounded in one well. The population in well 3 has a small impact compared to the other two well. Differing from the above two cases, when all atoms are initially located in well 3 [ Fig. 3g, h and i], the population evolution of the BECs in all wells has dramatically changed due to the existence of the nonlinear interaction. Oscillations behavior occurs in every well as the values of g=c ranges from 2 to 6. As a result, the population will be affected by nonlinearity no matter how the initial state is. There exists a numerical interval for the nonlinearity where the population varies drastically.

Adiabatic eigenspectra
To further explain these phenomena, we will introduce the eigenenergy level structure of the system. Ignoring a total phase, the dynamics of the above three-well system can be depicted by a classical Hamiltonian of two-degree freedom. The amplitudes of each site can be expressed in terms of the total density and a phase factor as c i ¼ ffiffiffiffi ffi P i p e ia i [7,8,36]. The relative phase factors can be defined as b 1 ¼ a 2 À a 1 and b 3 ¼ a 2 À a 3 . Using P 1 þ P 2 þ P 3 ¼ 1, the Hamiltonian Eq.(3) can be expressed similarly to a classical Josephson Hamiltonian of the form [36] and the dynamics are given by the conjugate equations The fixed point or minimum energy point of the classical Hamiltonian system (11) corresponds to the eigenstate of the quantum system. To derive the analytical expressions of these fixed points is difficult, but we can obtain them numerically and then plot the eigenenergies as the function of the time in Fig. 4. In Fig. 4, we note that it is different from the earlier reference [18,19], there is no looping new feature in our figures. As the intensity of nonlinearity increases, the energy levels move in the vertical direction. For the case of weak nonlinearity (g=c ¼ 0:5), the energy levels shift slightly compared to the linear case (see Fig. 4a). With an increase in nonlinearity (g=c ¼ 2), the upper level moves up a greater distance and the structure has also been changed, the middle level and lower level move with it. When the interaction is stronger (g=c ¼ 5), the structures of the three levels have been modified. Interestingly, the part of the upper and lower energy levels are combined in some intervals. For still stronger interaction (g=c ¼ 8 and 10), the energy levels will not be changed. The effect of non-linearity on energy levels is similar to the effect on population (Fig. 3). When the value of nonlinearity intensity is ranged from 2 to 6, the effect on the energy levels and population is greater. There exists a critical value beyond which the energy level and population will not be changed. In our paper, this value is close to 6.

Transition to Self-trapping
In the presence of nonlinear interactions, the meanfield equations are solved numerically with a given set of parameters and initial conditions. The population in Fig. 3 indicates the emergence of self-trapping when the nonlinear interaction is strong. To characterize dynamics in the time limit, we calculate the time averaged populations in individual wells through [8,21] where s is the interval of time. We plot the averaged populations in their dependence of the interaction strength with initial conditions P 1 ¼ 1, P 2 ¼ 1, In Fig. 5a, the initial state is given by ½1; 0; 0 T (P 1 ¼ 1). When g=c ¼ 0 (linear case), the average populations are obtained with the help of Eqs. (8) and (9), hP 1 i % 0:84; hP 2 i % 0:13; hP 2 i % 0:03: The majority of the particles are found in the left well for the linear case. With an increase in the nonlinearity, the averaged population in well one hP 1 i decreases at first, and then increases. When the interaction parameter is larger than 4, the averaged population hP 1 i jumps up and tends to unite soon after, it is fully localized in the well 1. In this case, the averaged population of well two and three keeps zero. Here we see a smooth transition from the initial densities toward the self-trapping regime. In Fig. 5b, the initial condition is ½0; 1; 0 T (P 2 ¼ 1). Similarly, the averaged populations are readily obtained for the linear case, in which the majority of the particles are found in well two. With an increase in the nonlinearity, the averaged population hP 2 i increases smoothly at the beginning, then passes a turbulence interval [2,6]. The selftrapping emerges such that the BECs localize in well two when the interaction parameter is larger than 6. For Fig. 5c, the initial state is given by ½0; 0; 1 T (P 3 ¼ 1). When g=c ¼ 0, the averaged populations has some correspondence to that of Fig. 4a due to behind symmetry, hP 1 i % 0:03; hP 2 i % 0:13; hP 3 i % 0:84: With an increase in the nonlinearity, the averaged population in three wells have a turbulence when the interval of interaction ranges from 2 to 6.3. The averaged population in well three hP 3 i decreases monotonically and increases smoothly to unite. When the interaction parameter is larger than 6.3, the population tends to fully localize in well three.
Comparing the above three cases, we find that the smooth transition of the BEC to self-trapping is broken by different initial states, and the transition to selftrapping of the BEC in triple-well systems can be effectively controlled by different initial states as the nonlinear interaction is strong. We note that selftrapping phenomenon also exists in LZ model [7,8,21]. In this model, the averaged population will encounter turbulent interval as the nonlinearity increased, which represents chaotic motions. Differing from LZ model, transition to self-trapping is smooth in our model, indicating no chaotic behavior.
In BEC experiment, quantum particle can be transported between the wells and the interaction strength g between atoms can be controlled by Feshbach resonance [37,38]. Based on this, the related reference have simulated 7 Li soliton experiment [37][38][39][40], which shows that the adiabatic transport of a BEC containing 2000 7 Li atoms can be achieved over 20 lm within an ambient harmonic trap [39,40], and it contains thousands of interacting atoms distinguished from the previous single particle cases [41,42]. Our model is also applicable to 2000 7 Li soliton experiment. In this case, the tunneling rate K can be controlled by magnetic fields. Figure 3 stands for the 2000 7 Li atoms occupation in each well at any time. Consequently, we can adjust different initial states to get the suitable 7 Li atom occupation for three wells. Compared to their results, our research can be used to implement the self-trapping phenomenon of 7 Li atom, but they are more interested in the transport between well 1 and well 3.

Conclusion
In conclusion, the RZ tunneling dynamics are analyzed for two important cases: linearity and nonlinearity, which exhibit different dynamical behaviors. In the linear case, all atoms are finally trapped in the single well where it is initially uploaded. However, as the non-linearity increases, this phenomenon will be broken. When the value of non-linearity intensity is in the range of 2 to 6, both the energy level and the population have been changed dramatically. Differing from the previous reference [1], our energy levels do not appear topological structures, only moving in the vertical direction, which is similar to our previous results for the two levels [6]. At the same time, there exists a critical value for nonlinearity beyond which the energy level and population will not be changed, known as self-trapping, which can be effectively controlled by different initial states. In the future, we hope our theoretical discussion will stimulate experiments in optical traps.
Funding The authors have not disclosed any funding.
Data Availibility Statement All data generated or analyzed during this study are included in this article.

Declarations
Conflicts of interest There is no conflict of interest.