Manipulation Of Bose-Einstein Condensate By Magnetic Field And Spin-Orbit Coupling

This article models the Bose-Einstein condensate(BEC)using the Gross-Pitaevskii(GP)equation under the mean-field approximation and investigates the ground state and dynamical evolution characteristics of the system under the magnetic field and spin-orbit coupling using partial analytical and numerical simulation methods.Chapter 1 mainly introduces the theoretical predictions,experimental implementation,operation methods,and some applications of BEC.Chapter 2 introduces the analytical and numerical methods used in this article,namely the factorization method and the neural network method.The main content of this article is divided into two parts,which are covered in chapters 3 to 7.chapters 3,4,and 5 mainly study the ground state of BEC systems under the influence of spin-orbit coupling and magnetic fields.Chapters 6 and 7,on the other hand,investigate the dynamic evolution and formation of topological structures in BEC condensates under magnetic fields.In chapter 3,we propose a method to achieve controllable energy level inversion in a one-dimensional Bose-Einstein condensate.The linearized equation can be solved exactly,and the results show that any excited state can be transformed into the ground state by adjusting the strength of the spin-orbit coupling and the magnetic field gradient.Numerical solutions of the nonlinear system show that the results are consistent with the linear case under repulsive interactions between components.However,under attractive interactions,superposition states and boundary states are generated.In chapter 4,we will discuss the stationary solutions of two-dimensional Bose-Einstein condensates under the combined effects of Zeeman splitting and spin-orbit coupling.By introducing a generalized momentum operator,the linearized system can be solved exactly.The exact solutions are Bessel vortices and modified Bessel vortices under weak and strong Zeeman splitting,respectively.The ground state of the nonlinear system can be constructed with the help of neural networks.As the cross-interaction increases,the ground state of the system changes from vortex states to mixed states.We also analyze the spin vector distribution of the states,and the spin vector distribution corresponding to Bessel vortices has a Neel-type skyrmion structure,and the corresponding topological number can be analytically expressed.In contrast,modified Bessel vortices and mixed states cannot form skyrmions.In chapter 5,we extend the system from spin 1/2 to spin 1 and discuss the stationary vortex solutions of two-dimensional Bose-Einstein condensates with spin-orbit coupling.Similar to the spin 1/2 system,the solutions of the spin 1 system are still Bessel vortices.We use the variational method to give an approximate analytical expression that can match the numerical results well.Using the variational results,we also find that the ground state of the system corresponds to Bessel vortices with different quantum numbers under different interaction conditions.In addition,we analyze the topological properties of the ground state and perform stability analysis using the variational results.In chapter 6,we propose a new scheme for generating three-dimensional skyrmions in a Bose-Einstein condensate with a spin-1 ferromagnetic spin structure by manipulating multipole magnetic fields and a pair of counter-propagating laser beams.The results show that topological number Q = 2 three-dimensional skyrmions can be created by hexapole magnetic fields and laser beams.Moreover,vortex ring and knot structures are found in skyrmions.In our model,the topological number can be calculated analytically,which means that this method can be extended to create skyrmions with any topological number.As an example,topological number Q = 3,4 three-dimensional skyrmions are also demonstrated,and they can be distinguished by density distributions with specific quantization axes.These topological structures in spin-1 ferromagnetic Bose-Einstein condensates are expected to be realized in experiments.In chapter 7,we use the magnetic field constructed in chapter 6 to generate magnetic monopoles at the zero point of the magnetic field.Since the constructed magnetic field is periodic,multiple magnetic field zero points can be generated.We use two magnetic field zero points to produce a pair of positive and negative magnetic monopoles.By slowly adjusting the bias magnetic field,the two magnetic monopoles are brought close together to interact with each other.When the two magnetic monopoles completely overlap,the magnetic monopoles disappear,but a solitary Dirac string remains.