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Acta Physica Sinica

Effects of linear Zeeman splitting on the dynamics of bright solitons in spin-orbit coupled Bose-Einstein condensates

Lin Wen, Yi Liang, Jing Zhou, Peng Yu, Lei Xia, Lianbin Niu, Xiao-Fei Zhang

Published Jan 1, 2019• 12 citations
Publisher

Abstract

Solitons as self-supported solitary waves are one of the most fundamental objects in nonlinear science. With the realization of Bose-Einstein condensate, matter-wave solitons have aroused enormous interest due to their potential applications in atomic transport and atomic interferometer. In recent years, the artificial spin-orbit coupling has been realized in ultracold atoms, thus providing a new platform to study the nonlinear matter wave solitons under a gauge field, and a variety of novel soliton phenomena have been successively predicted. In this paper, we analyze the effects of linear Zeeman splitting on the dynamics of bright-bright solitons in spin-orbit coupled two-component Bose-Einstein condensate, via the variational approximation and the numerical simulation of Gross-Pitaevskii (GP) equations. For the SU(2) spin-rotational invariant attractive atomic interaction in a uniform case without external trap, we take a hyperbolic secant function as the variational Ansatz for bright soliton in variational approximation, and derive the Euler-Lagrange equations describing the evolution of the Ansatz parameters. By solving the time-independent Euler-Lagrange equations, we find two stationary solitons each with a finite momentum for a weak spin-orbit coupling due to the linear Zeeman splitting. Linearizing the Euler-Lagrange equations around these stationary solitons, we further obtain a zero-energy Goldstone mode and an oscillation mode with frequency related to linear Zeeman splitting: the former indicates that the continuous translational symmetry of the stationary solitons will be broken under a perturbation, and the later shows that the stationary solitons will oscillate under a perturbation. Furthermore, by solving the time-dependent Euler-Lagrange equations, we also obtain the exact full dynamical solutions of Ansatz parameters, and observe that the linear Zeeman splitting affects the period and velocity of soliton's oscillation and linear motion, which may provide a new method to control the dynamics of solitons. All the variational calculations are also confirmed directly by the numerical simulation of GP equations.

DOI
10.7498/aps.68.20182013

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Solitons as self-supported solitary waves are one of the most fundamental objects in nonlinear science. With the realization of Bose-Einstein condensate, matter-wave solitons have aroused enormous interest due to their potential applications in atomic transport and atomic interferometer. In recent years, the artificial spin-orbit coupling has been realized in ultracold atoms, thus providing a new platform to study the nonlinear matter wave solitons under a gauge field, and a variety of novel soliton phenomena have been successively predicted. In this paper, we analyze the effects of linear Zeeman splitting on the dynamics of bright-bright solitons in spin-orbit coupled two-component Bose-Einstein condensate, via the variational approximation and the numerical simulation of Gross-Pitaevskii (GP) equations. For the SU(2) spin-rotational invariant attractive atomic interaction in a uniform case without external trap, we take a hyperbolic secant function as the variational Ansatz for bright soliton in variational approximation, and derive the Euler-Lagrange equations describing the evolution of the Ansatz parameters. By solving the time-independent Euler-Lagrange equations, we find two stationary solitons each with a finite momentum for a weak spin-orbit coupling due to the linear Zeeman splitting. Linearizing the Euler-Lagrange equations around these stationary solitons, we further obtain a zero-energy Goldstone mode and an oscillation mode with frequency related to linear Zeeman splitting: the former indicates that the continuous translational symmetry of the stationary solitons will be broken under a perturbation, and the later shows that the stationary solitons will oscillate under a perturbation. Furthermore, by solving the time-dependent Euler-Lagrange equations, we also obtain the exact full dynamical solutions of Ansatz parameters, and observe that the linear Zeeman splitting affects the period and velocity of soliton's oscillation and linear motion, which may provide a new method to control the dynamics of solitons. All the variational calculations are also confirmed directly by the numerical simulation of GP equations.
We have investigated the dynamics of bright solitons in a spin–orbit coupled spin-1 Bose–Einstein condensate analytically and numerically. By using the hyperbolic sine function as the trial function to describe a plane wave bright soliton with a single finite momentum, we have derived the motion equations of soliton’s spin and center of mass, and obtained its exact analytical solutions. Our results show that the spin–orbit coupling couples the soliton’s spin with its center-of-mass motion, the spin oscillations induced by the exchange of atoms between components result in the periodical oscillation of center-of-mass, and the motion of center of mass of soliton can be viewed as a superposition of periodical and linear motions. Our analytical results have also been confirmed by the direct numerical simulations of Gross–Pitaevskii equations.
We analyze the dynamics of two-component vector solitons, namely, dark-bright solitons, via the variational approximation in Bose-Einstein condensates. The system is described by a vector nonlinear Schr\"odinger equation appropriate to multicomponent Bose-Einstein condensates. The variational approximation is based on a hyperbolic tangent (hyperbolic secant) for the dark (bright) component, which leads to a system of coupled ordinary differential equations for the evolution of the ansatz parameters. We obtain the oscillation dynamics of two-component dark-bright solitons. Analytical calculations are performed for same-width components in the vector soliton, and numerical calculations extend the results to arbitrary widths. We calculate the binding energy of the system and find it to be proportional to the intercomponent coupling interaction and numerically demonstrate the breakup or unbinding of a dark-bright soliton. Our calculations explore observable eigenmodes, namely, the internal oscillation eigenmode and the Goldstone eigenmode. We find analytically that the number of atoms in the bright component is required to be less than the number of atoms displaced by the dark soliton in the other component in order to find the internal oscillation eigenmode of the vector soliton and support the existence of the dark-bright soliton. This outcome is confirmed by numerical results. Numerically, we find that the oscillation frequency is amplitude independent. For dark-bright solitons in 87Rb^{87}\mathrm{Rb}87Rb we find that the oscillation frequency range is 90 to 405 Hz and therefore observable in multicomponent Bose-Einstein condensate experiments.
We investigate the matter-wave solitons in a spin–orbit-coupled spin-1 Bose–Einstein condensate using a multiscale perturbation method. Beginning with the one-dimensional spin–orbit-coupled threecomponent Gross–Pitaevskii equations, we derive a single nonlinear Schrödinger equation, which allows determination of the analytical soliton solutions of the system. Stationary and moving solitons in the system are derived. In particular, a parameter space for different existing soliton types is provided. It is shown that there exist only dark or bright solitons when the spin–orbit coupling is weak, with the solitons depending on the atomic interactions. However, when the spin–orbit coupling is strong, both dark and bright solitons exist, being determined by the Raman coupling. Our analytical solutions are confirmed by direct numerical simulations.
Abstract We explore the existence and stability domains of stationary and moving bright solitons in spin–orbit-coupled spin-1 Bose–Einstein condensates in an external Zeeman field. Two families of bell-shape bright solitons (plane-wave (PW) phases) and two families of stripe-shape bright solitons (standing-wave (SW) phases) are obtained. We find a relation between the existence of these bright solitons and the single-particle energy spectrum, as well as the requirements of their existence for atomic interactions. The stability of four families of bright solitons is systematically analyzed using the linear stability analysis method. For two families of PW bright solitons, they are unstable only when the strength of the Zeeman field is above a critical value. The critical value is affected by the spin–orbit coupling and hardly affected by atomic interactions. When the velocities of solitons are zero (nonzero), the critical values of these two families are equal (unequal). For two families of SW bright solitons, their stability domains are identical in the absence of the Zeeman field, and they are unstable when the ferromagnetic interaction is strong. In the presence of the Zeeman field, their stability domains are complementary and complex, and one family of SW bright solitons can exist stably in the area with stronger ferromagnetic interaction. Furthermore, we find the collisions of stable bright solitons with different velocities can generate intriguing dynamics.
We analytically and numerically study the different types of solitary wave in the two-component helicoidal spin-orbit coupled Bose-Einstein condensates (BECs). Adopting the multiscale perturbation method, we derive the analytical bright and dark solitary wave solutions of the system, and the stationary and moving bright (dark) solitary waves are obtained. The effects of spin-orbit coupling, the helicoidal gauge potential, the momentum, the Zeeman splitting, and the atomic interactions on the solitary wave types are discussed, and it is found that the coupling of these physical parameters can manipulate different types of solitary waves in the system. The results indicate that the helicoidal gauge potential breaks the symmetric properties of the energy band of the system and adjusts the energy band structure, thus further effecting the solitary wave properties, i.e., stationary or moving solitary wave, bright, or dark solitary wave. Correspondingly, the analytical predictions for exciting stationary or moving bright (dark) solitary wave in parameter space are obtained. In particular, the helicoidal gauge potential changes the solitary wave types drastically for the weak spin-orbit coupling, i.e., in the absence of the helicoidal gauge potential, only dark (bright) solitary wave solutions exist in the system with repulsive (attractive) atomic interaction; however, in the presence of the helicoidal gauge potential, both dark and bright solitary waves can exist in the system regardless of whether the atomic interaction is repulsive or attractive. In addition, we investigate the stability of solitary waves and obtain the stability regions of different types of solitary waves by applying the linear stability analysis. The dynamic evolution results of the solitary waves by the direct numerical simulation not only validate the linear stability analysis but also confirm the analytical prediction of the solitary waves. Finally, the collision effects between solitary waves are also presented by the numerical simulation. It is shown that the interactions between solitary waves in the system have both elastic and inelastic collisions, which are closely related to the position of solitary wave states in the linear energy band. Our results provide a potential way to adjust the types of solitary waves in BECs with helicoidal gauge potential.
The existence and dynamics of solitons in quasi-one-dimensional Bose-Einstein condensates with spin-orbit coupling (SOC) and attractive two-body interactions are described for two coupled atomic pseudospin components with slowly and rapidly varying time-dependent Raman frequency. By varying the Raman frequency linearly in time, it was shown that ordinary nonlinear Schr\"odinger-type bright solitons can be converted to striped bright solitons, and vice versa. The internal Josephson oscillations between atom number of the coupled soliton components and the corresponding center-of-mass motion are studied for different parameter configurations. In this case, a mechanism to control the soliton parameters is proposed by considering parametric resonances, which can emerge when using time-varying Raman frequencies. Full-numerical simulations confirm variational analysis predictions when applied to the region where regular solitons are expected. In the limit of high frequencies, the system is described by a time-averaged Gross-Pitaevskii formalism with renormalized nonlinear and SOC parameters and modified phase-dependent nonlinearities. By comparing full-numerical simulations with averaged results, we have also studied the lower limits for the frequency of Raman oscillations in order to obtain stable soliton solutions.