Dynamics Of A Bose-Einstein Condensate With Modulation Potential Barrier

The realization of Bose-Einstein condensation(BEC)in alkali metal atomic gases is one of the most significant achievements in condensed matter physics over the past two decades.BEC systems are highly controllable and have been used as a platform to study various quantum phenomena both theoretically and experimentally,including the dynamics of vortices which is an important topic in nonlinear physics.This paper studies dynamics of a vortex ring(VR)crossing a coaxial obstacle with periodically modified spherical surface and discusses the dynamical excitation when a spherical obstacle moves in a narrow elongated potential well in a two-component Bose-Einstein condensate with spin-orbit coupling(SOC).This paper is organized as follows.In chapter 1,we first introduce the research background of BEC theory and experiments,the basic properties of a VR in uniform BEC systems,and the dynamics of an elliptical VR.Then we review the time evolution and scattering interactions of vortex lines or VRs and the interesting reconnections of vortex lines,as well as the dynamics of a VR crossing a coaxial sphere.Finally,the chapter presents the research background of spin density waves induced by moving potential barriers.In chapter 2,we first introduce the main numerical methods,including mean field theory,and the Gross-Pitaevskii(GP)equation and the fourth-order Runge-Kutta method.Then we briefly introduce the Bogoliubov excitation theory,In chapter 3,we investigate dynamics of a VR encountering a coaxial obstacle with periodically modified spherical surface.As the VR moves towards the obstacle,its radius would expand,and then the circular VR is deformed to be gear shaped when passing over the obstacle.When the VR is moving away from the obstacle,the result depends on the wave number of the periodic distortion and the distortion amplitude and it can be broadly classified into four regimes as the distortion amplitude increases.If the amplitude is finite,this gear-shaped VR can evolve into a helical Kelvin wave state.Then for an intermediate distortion of the obstacle,the expanded VR can split into several smaller VRs.These split VRs would travel some distances forwards and then reconnect with each other,forming one new forward-moving Kelvin wave,or two Kelvin waves of different sizes,with the bigger one moving forwards and the smaller one moving backwards.If the distortion amplitude is very large,the gear-shaped VR would be inclined to form one complete Kelvin-wave VR.We give a qualitative phase diagram as a function of the distortion amplitude.In Chapter 4,we study dynamical excitation of a spin-1/2 BEC system with SOC in a narrow trap.The energy spectra of the ground state and density distributions for different system sizes are obtained by solving the GP equation.Then the excitation spectrum is calculated using the Bogoliubov theory.Finally we investigate the dynamics of a spherical obstacle moving in the elongated box trap.In a wider trap,vortex pairs are released when an obstacle moving in the plane-wave direction.In the narrower trap,a long density belt accompanied by density islands are found ahead the obstacle,as well as a decay of density behind the obstacle.In Chapter 5,we provide a summary and some prospects for further research.