Bessel Vortices In Spin-orbit-coupled Spin-1 Bose-Einstein Condensates

Bose Einstein condensation(BEC)is a phenomenon in which boson atoms condense into the same state when cooled to near absolute zero.With the deepening of theory and the discovery of experiments,more and more spinor BECs with spin-orbit coupling have been studied.This kind of BEC has many properties,such as vortex,skyrmion and soliton.These play important roles in the spin hall effect,topological insulator,spintronic devices,chiral supersolids,and polarized exciton topological insulator.Therefore,the more accurate the analytical solutions under these models are,the more accurate the systems that understand them will be.The model studied in this paper is the nonlinear Schrodinger equation of spin-orbit coupling,spin-1 and two-dimensional BEC.We first set up the form of the stable state vortex solution of the model,find out the exact solution of the linear part of the model equation,and on this basis,reasonably construct the variational approximate solution under the complete model,and obtain the corresponding energy expression.Then,the quantum phase transition diagram is drawn by using the variational method,and the approximate degree of the variational solution is tested by comparing the numerical solution of the model with the obtained variational solution,and whether the vortex ground state evolves stably with time is studied,and its topological properties are studied.The methods and results we used are as follows:First,when solving the exact solution of the linear part of the model equation and its corresponding energy,we define and explain a generalized momentum operator on the spinorbit coupling term,and use this operator to find the exact solution of the linear part,whose radial funtion is the first kind of Bessel type solution.Then,under the condition that its asymptotic expression satisfies the model equation and orthogonal normalization,the variational approximate solution is constructed by introducing the sech function,and its corresponding energy expression is obtained.Finally,we also discuss the differences and advantages between the Bessel vortex variational solution of our model and the Gaussian variational solution commonly used in the previous articles on BEC.Secondly,in the case of obtaining the approximate analytical solution of the model,we use the simple search method to observe the change of the parameters in the variational solution with the parameters of the nonlinear system,and draw the phase transition diagram of the vortex ground state.Then we use several numerical methods such as imaginary-time propagation and symmetric split-step Fourier algorithm to find the numerical solution.Through it,we find that the approximation degree of the vortex ground state of the model obtained by the variational method is very good.Next,using two methods,linear stability analysis and perturbation dynamics evolution,it is verified that the time evolution of the vortex ground state is very stable.It is also analyzed that the balance between the spin-orbit coupling and the attractive mutual potential energy plays a key role in the prevention of collapse and stability of BEC,which has great potential for experimental implementation.Finally,we derive another expression of the vortex ground state,simply explain its manifold structure,and find that the unit Bloch vector of m=0 vortex ground state ψ0 has a skyrmion structure,and it is proved by drawing and formula calculation that it has a result of a skyrmion number of 1 in the domain within the boundary of r=3.8.