The Matter-wave Solitons In Spin-orbit Coupled Bose-einstein Condensates

In recent years,Bose-Einstein condensate(BEC)has become an important platform for the study of macroscopic quantum phenomena.Especially,the s-tudy of nonlinear collective excitations in this system has aroused great interest,such as the Anderson localization of matter waves,production of bright and dark solitons,generation of vortex-antivortex dipoles,emulation of gauge fields and spin-orbit coupling,etc.Recently,the realization of the artificial spin-orbit cou-pling in a neutral atomic BECs has stimulated lots of works.Particularly,the studies for vortex structures in rotating spin-orbit coupled BECs and trapped 2D atomic BECs with spin-independent interactions in the presence of the isotropic spin-orbit coupling are very popular.Further,the people’s research enthusiasm for matter-wave soliton has been growing.There are a variety of soliton struc-tures in this system,such as dark solitons,bright solitons and strip solitons,etc.Soliton has a unique feature that its shape does not change in the process of propagation.This paper mainly studies the solitons and the properties of soliton in spin-orbit coupled BECs.Firstly,a brief introduction to the spin-orbit coupled BECs is made,in-cluding the realization of spin-orbit coupling in experiment and some nonlinear excitation structures in this system.Then we introduce some backgrounds and present research situations of the matter-wave soliton,which includes solitons and the properties of soliton in spin-1/2 and spin-1 systems.In the second chapter,we study the soliton in the spin-orbit coupled spin-1/2 BECs system.Starting from the Gross-Pitaevskii(GP)equation describing the two-component spin-orbit coupled BECs,we get a KdV(Korteweg-de Vries)-like equation by using a perturbation method.In order to obtain the solution of this equation,we use the perturbation method for the second time and assume that the Raman coupling is much smaller than the spin-orbit coupling,and finally the analytical solution of the KdV-like equation is obtained.Further,we find that different types of moving solitons exist,such as dark-dark solitons,bright-bright solitons,and dark-bright solitons.The types of solitons mainly depend on the type of interactions(attractive interaction and repulsive interaction).In general case,for repulsive interaction,there only exists dark soliton,while for attractive interaction,there is bright soliton.However,due to the effects of spin-orbit coupling,we find a novel phenomenon that when the inter-and intra-species interactions are attractive we are capable of getting dark solitons.And with the Raman coupling strength changing the type of soliton will change.In the third chapter,we discuss the matter-wave solitons in spin-orbit cou-pled spin-1 BECs.Starting from the three-component coupled GP equations that describe this system,a nonlinear Schr¨odinger equation is obtained by using multi-scale perturbation method.This equation has soliton solutions,and ac-cording to this equation we not only get the energy spectrum structures of the system but also the parameter regions for different types of solitons.Further,the research reveals that there only exist positive mass bright and dark solitons in the case of weak spin-orbit coupling and strong Raman coupling,while in the case of strong spin-orbit coupling and weak Raman coupling there exist both positive mass dark and bright solitons and negative mass dark and bright soli-tons.The types of solitons are determined by the signs of dispersion coefficient and nonlinear coefficient.When the two coefficients have same signs,there exists dark soliton,and when they are opposite,the system has bright soliton solutions.Then we verify our analytical stationary and moving soliton solutions by mean-s of the numerical simulations.In detail,we adopt fourth-order Runge-Kutta method to solve the GP equations with our analytical solutions as initial wave functions.It is found that the solitons can keep their shapes for a very long time.This shows that our analytical results are very correct.Finally,a brief summary of these two tasks and prospects in this area are given.