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Results in Physics

Stationary solutions and dynamical evolution of the nonlocal Gross–Pitaevskii equation in spin-2 Bose–Einstein condensates

Hongmei Li, Peng Li, Xuefei Wu

Published Jan 21, 2023• 2 citations
Publisher

Abstract

In this paper, we investigate the nonlocal Gross–Pitaevskii equation which describes the phenomena of Bose–Einstein condensates under the mean field approximation. Using mean-field theory and adopting separation of variables method, the five-component Gross–Pitaevskii spin–orbit-coupled spin-2 Bose–Einstein condensate model can be reduced to five-component ordinary differential equation, then twelve families of solutions are obtained by the combination of some Jacobi elliptic functions. In particular, when the modulus of Jacobi elliptic function tends to one, a series of bright soliton and dark soliton and their combined solution are obtained. These periodic and hyperbolic soliton solutions are analyzed in detail, the nature and images will be of practical application to provide theoretical guidance for spin systems.

DOI
10.1016/j.rinp.2023.106230

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We investigate the different types of matter-wave solitons in spin-orbit-coupled spin-2 spinor Bose-Einstein condensates. Using mean-field theory and adopting the multiscale perturbation method, the original five-component Gross-Pitaevskii spin-orbit-coupled spin-2 spinor Bose-Einstein condensate model can be reduced to a single effective nonlinear Schrödinger equation, which allows us to find analytical soliton solutions of this system. In this way, for different regimes of the spin-orbit coupling, Raman coupling, and interatomic interactions, we find approximate bright and dark soliton solutions. Particularly, the type of solitons depends on the dispersion properties of the system. When the lowest-energy band has a single-well structure, we find there only exist positive mass bright or dark solitons due to the dispersion coefficient of effective nonlinear Shrödinger equation always positive. However, when the lowest-energy band has a double-well structure, there will appear positive (negative) mass bright or dark solitons because the sign of the dispersion coefficient can be positive (negative) under different momentum. We employ direct numerical simulation of the original five-component Gross-Pitaevskii equations to confirm the analytical results.
In this paper, we investigate a five-component Gross–Pitaevskii equation, which is demonstrated to describe the dynamics of an F=2 spinor Bose–Einstein condensate in one dimension. By employing the Hirota method with an auxiliary function, we obtain the explicit bright one- and two-soliton solutions for the equation via symbolic computation. With the choice of polarization parameter and spin density, the one-soliton solutions are divided into four types: one-peak solitons in the ferromagnetic and cyclic states and one- and two-peak solitons in the polar states. For the former two, solitons share the similar shape of one peak in all components. Solitons in the polar states have the one- or two-peak profiles, and the separated distance between two peaks is inversely proportional to the value of polarization parameter. Based on the asymptotic analysis, we analyze the collisions between two solitons in the same and different states.
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 By adopting a mean-field Gross–Pitaevskii description of spinor polariton Bose–Einstein condensates under nonresonant pumping, we investigate the static and dynamical properties of dark–bright (DB) solitons. We derive the equation of motion for the center of mass of the DB solitons’ center analytically by using the Hamiltonian approach. The resulting equation captures how the combination of the open-dissipative character and the spin degrees of freedom of a polariton Bose–Einstein condensate affects the properties of DB solitons, i.e. DB solitons relax by blending with the background at a finite time. In this case, we also determine the lifetime of the DB solitons. Further numerical solutions of the modified dissipative two-component Gross–Pitaevskii equations are in excellent agreement with the analytical results. In the presence of the Langevin noise, we demonstrate that the DB solitons can still propagate for a long time, which is sufficient for their experimental observations within current facilities.
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 report new matter-wave solutions of the one-dimensional spin-1 Bose-Einstein condensate system by combining global spin-rotation states and similarity transformation. Dynamical behaviors of non-stationary global spin-rotation states derived from the SU(2) spin-rotation symmetry are discussed, which exhibit temporal periodicity. We derive generalized bright-dark mixed solitons and new rogue wave solutions and reveal the relations between Euler angles in spin-rotation symmetry and parameters in ferromagnetic and polar solitons. In the modulated spin-1 Bose-Einstein condensate system, new solutions are derived and graphically illustrated for different types of modulations. Moreover, numerical simulations are performed to investigate the stability of some obtained solutions for chosen parameters.