Quantized Vortex States and Dynamics in Bose-einstein Condensates

In this thesis, we analytically and numerically study quantized vortex states aswell as their dynamics in rotating Bose-Einstein condensate at extremely low tem-perature.We present an efficient method–gradient flow with discrete normalization tofind the stationary solutions of the Gross-Pitaevskii equation and coupled Gross-Pitaevskii equations. We give a mathematical justification for the correctness ofthe method. Based on these solutions, we obtain the equilibrium properties such asquantized vortex states of trapped Bose-Einstein condensated gases under rotationat extremely low temperature.We also present a new, efficient and spectrally accurate method–time-splittingspectral method—to numerically solve the Gross-Pitaevskii equation, coupled Gross-Pitaevskii equations and generalized Gross-Pitaevskii equations. Based on the time-dependent solutions of these equations, we numerically obtain the dynamics ofquantized vortex states in rotating one-component Bose-Einstein condensate, two-component Bose-Einstein condensates and spinor Bose-Einstein condensates at ex-tremely low temperature.