⇦Chinese Physics Letters, 2016, Vol. 33, No. 11, Article code 110301⇨ Collision Dynamics of Dissipative Matter-Wave Solitons in a Perturbed Optical Lattice * Zheng Zhou(周政)1, Hong-Hua Zhong(钟宏华)2,3**, Bo Zhu(朱博)2, Fa-Xin Xiao(肖发新)1**, Ke Zhu(朱科)1, Jin-Tao Tan(谭金桃)4 Affiliations 1Department of Physics, Hunan Institute of Technology, Hengyang 421002 2Department of Physics, Jishou University, Jishou 416000 3School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082 4Department of physics, Hunan University of Technology, Zhuzhou 412007 Received 30 June 2016 ^*Supported by the National Natural Science Foundation of China under Grant Nos 11547125 and 11465008, the Hunan Provincial Natural Science Foundation under Grant Nos 2015JJ4020 and 2015JJ2114, and the Scientific Research Fund of Hunan Provincial Education Department under Grant No 14A118.
^**Corresponding author. Email: hhzhong115@163.com; xfx123480@126.com Citation Text: Zhou Z, Zhong H H, Zhu B, Xiao F X and Zhu K et al 2016 Chin. Phys. Lett. 33 110301 Abstract We investigate the stability and collision dynamics of dissipative matter-wave solitons formed in a quasi-one-dimensional Bose–Einstein condensate with linear gain and three-body recombination loss perturbed by a weak optical lattice. It is shown that the linear gain can modify the stability of the single dissipative soliton moving in the optical lattice. The collision dynamics of two individual dissipative matter-wave solitons explicitly depend on the linear gain parameter, and they display different dynamical behaviors in both the in-phase and out-of-phase interaction regimes. DOI:10.1088/0256-307X/33/11/110301 PACS:03.75.Lm, 05.45.Yv, 67.85.Hj © 2016 Chinese Physics Society Article Text Solitons are localized nonlinear waves propagating without changing shape over large time intervals, and are sustained by a balance between dispersion and nonlinear interaction. Solitons have been widely investigated in nonlinear optics, high-energy physics, and cold atoms.[1-10] In Bose–Einstein condensates (BECs), matter-wave solitons denote dips (dark) or peaks (bright) in the density profiles of BECs, depending on the nature of interatomic interaction. Since the experimental realization of bright solitons in attractive BECs,[11,12] the dynamics of matter-wave solitons have attracted increasing interest from both theoretical and experimental sides. Due to the potential applications for optical communication and high-precision interferometry,[13,14] the interaction of solitons has been an exciting subject.[15-17] The current research direction in soliton interactions is twofold. The one is nonlocal interactions between solitons in dipolar BECs[18,19] and nonlinearity optics,[20,21] and the other is dissipative soliton interactions in various nonconservative systems.[22-24] In realistic cold atom systems, BEC solitons suffer atomic dissipation due to an inelastic three-body recombination process.[25-28] This indicates that the matter-wave solitons decay and eventually disappear as they propagate. Recently, it has been demonstrated that alimentation of atoms from an external source may compensate for the atomic dissipation so as to form a dynamically stabilized soliton based on the balance between the linear gain and the three-body recombination loss.[29] However, it is focused only on the stable profile of the static soliton. The collision dynamics and stability of the kind of dissipative matter-wave solitons in the presence of an external potential are still unclear. In this Letter, we investigate the stability and collision dynamics of dissipative matter-wave solitons in the presence of a weak optical lattice. By means of the direct approach of soliton perturbation theory based on the multiple-time-scale asymptotic analysis,[31,32] we obtain not only the profile dynamics of a dissipative soliton but also the motion equation of its center of mass in the optical lattice. Furthermore, fixing the three-body recombination loss parameter in a typical experimental value, we numerically show the influence of the linear gain parameter on the stability of the single dissipative soliton moving in the optical lattice. It is found that the stability of a dissipative soliton in the long evolution time can be manipulated by tuning the linear gain parameter solely. Moreover, we explore the collision dynamics of two dissipative matter-wave solitons. We find that their collision dynamics depend on their relative phase difference and the linear gain parameter. We consider a bright dissipative matter-wave soliton formed in a strongly elongated attractive BECs subjected to both the three-body recombination loss of atoms and the linear atomic feeding from an external source in a weakly perturbed optical lattice, and the system can be described by the generalized dimensionless quasi-one-dimensional Gross–Pitaevskii equation[25,29] $$\begin{align} i\frac{\partial u}{\partial t}+\frac{\partial^2u}{\partial x^2}+2|u|^2u=iR(u),~~ \tag {1} \end{align} $$ with the small perturbation term $R(u)$ including weak lattice potential, loss, and gain $$\begin{align} R(u)=-iV_0\cos(kx)u+(\gamma-\xi|u|^4)u.~~ \tag {2} \end{align} $$ In the model, the spatial coordinate $x$ and time $t$ are measured in units of the transverse harmonic oscillator length $l=\sqrt{\hbar/(m\omega)}$ and inverse frequency $\omega^{-1}$, respectively, where $m$ is the atomic mass and $\omega$ is the trapping frequency.[29] The term $\xi$ corresponds to the three-body recombination loss rate, $\gamma$ is related to the constant atomic feeding from an external source, $V_0$ depends on the intensity of the laser forming the lattice, $k$ is the wave vector, and $V_0$, $\gamma$, and $\xi$ are normalized by $\hbar\omega$, so that these quantities become dimensionless.

Fig. 1. (Color online) The dynamically stabilized dissipative matter-wave soliton with a smaller amplitude. (a) The time evolution of the dissipative soliton in weak optical lattice. (b) The profile of the dissipative soliton moving in the optical lattice at different times $t=0$, 900, 4500 and 9000. (c) The position of dissipative solitonic center of mass from original Eq. (1) (line) compared with the result from the soliton perturbation analysis Eq. (14) (point). The parameters are set as $\xi=0.01$, $\gamma=0.0001$, $V_0=-0.01$, and $k=2\pi/256$. The initial soliton solution is taken as $u(x,0)=0.5{\rm sech}[0.5(x-64)]$ with zero initial velocity. Hereafter, all variables and parameters are dimensionless.

In the absence of the perturbation term, i.e., $R(u)=0$, Eq. (1) has a well known single-soliton solution $$\begin{alignat}{1} u(x,t)=2\eta{\rm sech}[2\eta(x-q)]\exp[i(\dot{q}x/2-\phi)],~~ \tag {3} \end{alignat} $$ with the position of the peak of the soliton $q(t)=-4\upsilon t+q_0$ and its phase $\phi(t)=4(\upsilon^2-\eta^2)t+\phi_0$, where the four real parameters $\upsilon$, $\eta$, $q_0$, and $\phi_0$ determine the velocity, height as well as width, initial position, and initial phase of the soliton, respectively. To investigate the dynamics of the dissipative matter-wave soliton in the weak optical lattice by means of the direct approach of the soliton perturbation theory,[31] we set $\epsilon$ as a small positive parameter measuring the weakness of the perturbation. At first, the independent variable $t$ is transformed into multiple-time-scale variables $t_n=\epsilon^nt$, $n=0,1,2,\ldots$, and then the time derivative is replaced by the expansion $$\begin{align} \frac{\partial}{\partial t}=\frac{\partial}{\partial t_0}+\epsilon\frac{\partial}{\partial t_1}+\epsilon^2\frac{\partial}{\partial t_2}+\ldots.~~ \tag {4} \end{align} $$ At the same time, we expand $u$ and $R[u]$ in an asymptotic series $$\begin{align} u=\,&u^{(0)}+\epsilon u^{(1)}+\epsilon^2 u^{(2)}+\ldots,~~ \tag {5} \end{align} $$ $$\begin{align} R[u]=\,&R^{(1)}[u^{(0)}]+\epsilon R^{(2)}[u^{(0)},u^{(1)}]+\ldots.~~ \tag {6} \end{align} $$ Substituting Eqs. (4)-(6) into Eq. (1) and collecting the terms of the same order, then we derive the solution of leading order equation $$\begin{align} u^{(0)}=2\eta e^{-i\theta}{\rm sech} z,~~ \tag {7} \end{align} $$ with $$\begin{align} z=\,&2\eta(x-q),~~\frac{\partial q}{\partial t_0}=-4\upsilon,~~ \tag {8} \end{align} $$ $$\begin{align} \theta=\,&2\upsilon(x-q)+\delta,~~\frac{\partial\delta}{\partial t_0} =-4(\upsilon^2+\eta^2),~~ \tag {9} \end{align} $$ obviously, $$\begin{alignat}{1} R^{(1)}[u^{(0)}]=[-iV_0\cos(kx)+\gamma-\xi|u^{(0)}|^4]u^{(0)}.~~ \tag {10} \end{alignat} $$ Following the results of Ref. [31] and returning to the original time variable, we obtain the time dependence of the soliton parameters $$\begin{align} \frac{d\eta}{dt}=\,&\frac{1}{2}{\rm Re}\int^\infty_{-\infty}R^{(1)}e^{i\theta}{\rm sech} zdz=2\gamma\eta-\frac{256}{15}\xi\eta^5,~~ \tag {11} \end{align} $$ $$\begin{align} \frac{dq}{dt}=\,&\frac{\partial q}{\partial t_0}+\epsilon\frac{\partial q}{\partial t_1}=\frac{\partial q}{\partial t_0}=-4\upsilon,~~ \tag {12} \end{align} $$ $$\begin{align} \frac{d\upsilon}{dt}=\,&-\frac{1}{2}{\rm Im}\int^\infty_{-\infty}R^{(1)}e^{i\theta}{\rm tanh} z {\rm sech} z dz\\ =\,&-\frac{V_0k^2\pi\sin(kq)}{8\eta\sinh(k\pi/4\eta)}.~~ \tag {13} \end{align} $$ Equation (11) indicates that there exists a constant value $\eta_{\rm s}$ after a large interval of time for amplitude of the soliton when the right-hand side of Eq. (11) becomes zero, i.e., $\eta_{\rm s}=(15\gamma/128\xi)^{1/4}$, which is independent of the initial amplitude, and depends only on the ratio of the gain and loss coefficients. This means that by a perturbation procedure that an alimentation of atoms from an external feeding source to the BEC soliton may compensate for the dissipation loss and may lead to a dynamically-stabilized soliton based on the dynamic balance between the linear gain and the three-body recombination loss. Because the three-body recombination loss process is inevitable and the linear gain is added by intentional introduction and can be tuned conveniently with a wide range in the BEC experiment, we fix the three-body recombination loss in a typical experimental value and only change the linear gain parameter throughout this work. Therefore, the amplitude of dissipative soliton can be manipulated by tuning the linear gain parameter solely. The effective mass of the dissipative soliton can be defined as the norm of the dimensionless wave function, i.e., $M=\int^\infty_{-\infty}|u|^2dx=4\eta(t)$, which is proportional to the number of atoms in BECs.[33] Combining Eqs. (12) and (13), if the spatial period of the optical lattice ($\sim k^{-1}$) is always sufficiently larger than the variable spatial width of dissipative soliton ($\sim \eta(t)^{-1}$) during its propagation, we derive the equation of motion for the dissipative matter-wave soliton as follows: $$\begin{align} \frac{d^2q}{dt^2}=2V_0k\sin(kq),~~ \tag {14} \end{align} $$ which describes the dissipative soliton as an effective particle with varying masses. The effective particle approach has been widely used in studying the soliton dynamics of conservative systems.[34,35] According to Eq. (14), we note that the motion equation of the dissipative soliton is not related to the gain and loss coefficients, and only depends on the weak external optical lattice potential configuration in the limit $k/\eta(t)\rightarrow 0$. In other words, both the velocity and position of the dissipative soliton at any given moment are the same for different linear gain parameters on the same initial velocity and initial position. Therefore, under the same initial conditions for the dissipative soliton (including initial mass, velocity and position), the larger the effective mass of dissipative soliton is, the higher energy the dissipative soliton possesses. The analytical results (11) and (14) can be confirmed by the direct numerical simulation of the original Eq. (1) in the following, and the analytical analysis offers valuable guidance to further study the dynamics of dissipative solitons in numerical computation. In the following, to investigate the stability and collision dynamics of the dissipative matter-wave solitons in the weakly perturbed optical lattices, we perform a detailed numerical analysis of Eq. (1) with different dissipation parameters. Throughout the present study, the weak optical lattice potential is fixed at $-0.01\cos(2\pi x/256)$ and the dimensionless three-body recombination loss parameter is fixed at $\xi=0.01$ corresponding to typical experimental values.[29] The stability of the dissipative soliton in the non-conservative systems is of importance.[30] In the first numerical simulation we consider the stability of the mobility of a single dissipative soliton in the optical lattice. The dimensionless gain parameter is selected as $\gamma=0.0001$ and the initial soliton solution is taken as $u(x,0)=0.5{\rm sech}[0.5(x-64)]$ with zero initial velocity at $t=0$. We numerically plot the time evolution of the dissipative soliton and its profile at different times $t=0$, 900, 4500, and 9000 in Figs. 1(a) and 1(b), respectively. Figures 1(a) and 1(b) indicate clearly that the dissipative soliton with a smaller amplitude moving in a lattice site of the optical lattice is stable. Its amplitude reaches a fixed constant value after a certain evolution time, and the fixed constant value for the amplitude of dissipative soliton at large time is $2\eta_{\rm s}=0.37$ through the above soliton perturbation analysis. In Fig. 1(c), it shows the time dependence of the position of dissipative solitonic center of mass, where the result from the soliton perturbation analysis Eq. (14) is confirmed by the direct numerical simulation of the original Eq. (1).

Fig. 2. (Color online) The time evolution of the dissipative soliton with the sufficiently large initial velocity $\upsilon_0=-2$. The other parameters and initial conditions are the same as those in Fig. 1. Here (a) and (b) correspond to the spatiotemporal distribution of $|u|$ and its plan view, respectively.

Fig. 3. (Color online) The unstable dissipative matter-wave soliton with a larger amplitude. The linear gain parameter is set as $\gamma=0.001$, and the other parameters and the initial conditions are the same as those in Fig. 1.

It is worth noting whether the dissipative soliton would perform a trapped oscillation in a potential well or travel across the optical lattice, depending strongly on the initial velocity of dissipative soliton. In Fig. 2, as an example, the same as in Fig. 1 but with the sufficiently large initial velocity $\upsilon_0=-2$, the dissipative soliton can travel across potential barriers of the optical lattice facing in the same direction in a relatively short time. To give the comprehensive study on the role of the linear gain parameter in stability, in Fig. 3 we also give the numerical analysis of Eq. (1) with the gain parameter $\gamma=0.001$ ($2\eta_{\rm s}=0.66$), and the other parameters are the same as those in Fig. 1. In this case, the amplitude of the dissipative soliton reaches a larger value after a transient evolution time, while it is unstable at sufficiently large time. We have repeated these calculations over a wide range of small dissipation parameters. It is shown that the dissipative soliton with a smaller amplitude compared with the initial choice is always robust. However, the dissipative soliton is unstable after a long evolution time when the amplitude $2\eta_{\rm s}$ based on the soliton perturbation analysis exceeds a critical value. This can be intuitively understood that the dissipative soliton possesses a higher energy in the same initial conditions due to the increase of its effective mass, and the dissipative soliton with high energy is unstable when it makes a trapped oscillation in a lattice site.

Fig. 4. (Color online) The interactions between two dissipative matter-wave solitons in both the in-phase and out-of-phase cases: (a) in-phase $\delta=0$ and (b) out-of-phase $\delta=\pi$. The initial condition is taken as $u(x,0)=0.5{\rm sech}[0.5(x-64)]+0.5{\rm sech}[0.5(x+64)]e^{i\delta}$. The linear gain parameter is set as $\gamma=0.00001 (2\eta_{\rm s}=0.208)$, and the other parameters are the same as those in Fig. 1.

Next, we investigate the dynamics of dissipative matter-wave solitonic interactions, and assume that the initial condition is taken as two oppositely moving bright solitons of the form[36,37] $$\begin{alignat}{1} u(x,t)=\,&2\eta_1{\rm sech}[2\eta_1(x+q_1)]e^{i\dot{q}_1x/2}\\ &+2\eta_2{\rm sech}[2\eta_2(x-q_2)]e^{-i\dot{q}_2x/2+i\delta},~~ \tag {15} \end{alignat} $$ with $\delta$ being their relative phase difference. For sufficiently large positive values of $q_1$ and $q_2$, this ansatz (15) approximates a solution comprising two bright solitons located at $q_1$ and $q_2$. In a recent experiment,[15] it has been observed that the interaction between two matter-wave solitons by attractive and repulsive forces depends on their relative phase difference in the conservative system. To investigate the roles of the relative phase difference and the dissipation parameters in collision dynamics, we perform the numerical simulations in both the in-phase $\delta=0$ and the out-of-phase $\delta=\pi$ cases by fixing the three-body recombination loss parameter as $\xi=0.01$ and continuously increasing the linear gain parameter $\gamma$. In Fig. 4, we numerically plot the interactions between two dissipative matter-wave solitons in both the in-phase and out-of-phase cases first with a sufficiently small gain parameter $\gamma=0.00001$ ($2\eta_{\rm s}=0.208$). In the top row of Fig. 4, the amplitudes of the two dissipative solitons reach a fixed value after a transient evolution time. In addition, it is shown that a peak occurs during the collision at the center for $\delta=0$ and a central node for $\delta=\pi$. This indicates that the interaction between two dissipative solitons is attractive for $\delta=0$ and is repulsive for $\delta=\pi$. These collisional features in the dissipative system are analogous to the ones in the conservative system.[15] However, in the in-phase case, the amplitude of the collision of the two dissipative solitons increases as they collide and is no longer changed at large times, whereas in the out-of-phase case, the two dissipative solitons make the same amplitude oscillation during the time evolution, as shown in the bottom row of Fig. 4. As the gain parameter is increased to $\gamma=0.0001$ $(2\eta_{\rm s}=0.37)$, for $\delta=0$ the two dissipative solitons collide in several times, then travel across the lattice potential barrier and eventually move away from each other. For $\delta=\pi$ the two dissipative solitons still remain the same amplitude oscillation at all times in Fig. 5. With the further increase in gain parameter, the amplitude of the dissipative soliton becomes larger than the initial choice. For this case, selecting the gain parameter as $\gamma=0.001$ $(2\eta_{\rm s}=0.66)$, we plot the collision dynamics in Fig. 6. For $\delta=0$ the two dissipative solitons collide once, then they separate rapidly and move away. For $\delta=\pi$ the two dissipative solitons begin to make the same amplitude oscillation. However, after a long evolution time, the two dissipative solitons with the large effective mass become unstable due to the high energy. The results show that the collision dynamics of the two dissipative solitons depend on their relative phase difference. The different dynamical behaviors are displayed in both the in-phase and out-of-phase interaction cases.

Fig. 5. (Color online) The interactions between two dissipative matter-wave solitons. The same as shown in Fig. 4 but with $\gamma=0.0001$: (a) in-phase $\delta=0$ and (b) out-of-phase $\delta=\pi$.

Fig. 6. (Color online) The interactions between two dissipative matter-wave solitons. The same as shown in Fig. 4 but with $\gamma=0.001$: (a) in-phase $\delta=0$ and (b) out-of-phase $\delta=\pi$.

In summary, we have investigated the stability and collision dynamics of the dissipative matter-wave solitons in a weakly perturbed optical lattice. By means of the direct approach of soliton perturbation theory, the dissipative soliton can be treated as an effective particle with varying mass. We obtain the time evolution equations of both the amplitude and the position of dissipative solitonic center of mass. These analytical results can be confirmed by the direct numerical simulation of the original generalized Gross–Pitaevskii equation. In addition, we perform a detailed numerical analysis of the role of the relative phase difference in the interactions between two dissipative matter-wave solitons by fixing the three-body recombination loss parameter in a typical experimental value and continuously increasing the linear gain parameter. The two dissipative solitons display different dynamical behaviors in both the in-phase and out-of-phase interaction regimes. It is worth noting that, as a perturbative method, the effective particle approach has an application condition restricting its use for weak external optical lattice potential and small gain and loss parameters, for which the dissipative soliton can preserve its shape under its propagation. With the increasing dissipation parameters beyond some critical values, one could not obtain a dynamically-stabilized soliton. Our theoretical results may be confirmed under the currently accessible experimental setup.[27] We thank Professor Yan Jia-Ren for his guidance and discussion. References Colloquium : Looking at a soliton through the prism of optical supercontinuumSolitons in nonlinear latticesRaman Self-Frequency Shift of Dissipative Kerr Solitons in an Optical MicroresonatorUltraslow Optical Solitons in a Cold Four-State MediumEnhanced quantum reflection of matter-wave solitonsDynamical evolutions of matter-wave bright solitons in an inverted parabolic potentialGeneration of ring dark solitons by phase engineering and their oscillations in spin-1 Bose-Einstein condensatesBright solitons in spin-orbit-coupled Bose-Einstein condensatesThree-dimensional solitons in two-component Bose—Einstein condensatesNonautonomous dark soliton solutions in two-component Bose—Einstein condensates with a linear time-dependent potentialFormation of a Matter-Wave Bright SolitonFormation and propagation of matter-wave soliton trainsCreation and Detection of a Mesoscopic Gas in a Nonlocal Quantum SuperpositionRealizing bright-matter-wave-soliton collisions with controlled relative phaseCollisions of matter-wave solitonsThe Dynamics of a Bright—Bright Vector Soliton in Bose—Einstein CondensationThe Interaction of Peregrine SolitonsControllable nonlocal interactions between dark solitons in dipolar condensatesInteraction of solitons in one-dimensional dipolar Bose-Einstein condensates and formation of soliton moleculesInteractions of nonlocal dark solitons under competing cubic–quintic nonlinearitiesSolitons shedding from Airy beams and bound states of breathing Airy solitons in nonlocal nonlinear mediaInfluence of gain dynamics on dissipative soliton interaction in the presence of a continuous waveDark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potentialPT-symmetric coupler with a coupling defect: soliton interaction with exceptional pointCollapse and Bose-Einstein Condensation in a Trapped Bose Gas with Negative Scattering LengthEvidence for Efimov quantum states in an ultracold gas of caesium atomsThree-Body Recombination at Vanishing Scattering Lengths in an Ultracold Bose GasEmergence of Chaotic Scattering in Ultracold Er and DyDissipation-managed soliton in a quasi-one-dimensional Bose-Einstein condensateMultipeaked Solitons in Parity-Time Complex Superlattice with Dual PeriodsDirect approach to the study of soliton perturbations of the nonlinear Schrödinger equation and the sine-Gordon equationSecond-order tunneling of two interacting bosons in a driven triple wellDynamics of Matter-Wave Solitons in a Ratchet PotentialSoliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbationsChaotic transport of a matter-wave soliton in a biperiodically driven optical superlatticeBright matter-wave soliton collisions at narrow barriersMean-field analog of the Hong-Ou-Mandel experiment with bright solitons

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