Topological Excitations In Spinor Bose–Einstein Condensates

The experimental realization of Bose-Einstein Condensation(BEC)has been promoting the rapid development in field of ultra-cold atom and molecular physics.Due to its macroscopic quantum coherent properties and unprecedented artificial controllability,not only has BEC been the basic testing ground for the quantum many-body physics,but also it provides a remarkably versatile experimental platform for simulating some analogues physical phenomena in the condensed matter physics and nonlinear science.In recent years,with the continuous progress in associated experimental techniques,the topics of exploring the novel quantum states and its peculiar properties in BEC have been the international research hotspot.This thesis will focus on the topological excitations in spinor Bose–Einstein condensates,including the mean-field theory of spinor Bose-Einstein condensate,symmetry classification of order parameter manifold,classification of topological excitations and the numerical research for some kinds of topological excitations.The first chapter devotes to the foundational theory in BEC.First,We give an overview of history about the development of the research for BoseEinstein Condensation.we introduce the foundational theories of ideal BoseEinstein gas and interacting Bose-Einstein gas.Later,we introduce the two-body scattering theory,and then,derive the important physical parameter of scattering length,explain its physical meaning,and introduce Feshbach resonance,a kind of Common methods of tuning scattering length.In the second chapter,we first introduce second quantization method of interacting spinor Bose-Einstein Condensation,and derive the interacting Hamiltonian of spinor BEC with spin-1,spin-2,and spin-3 respectively.And the,using the mean-field theory to obtain the Gross-Pitaevskii equation of spinor Bose-Einstein Condensation with spin-1 and spin-2.Later,applying the GrossPitaevskii equation of spinor BEC to research the possible ground sates of system with spin-1 and spin-2,and obtain their phase diagrams for some parameters.In the third chapter,we first apply theory of group theory and differentiable manifold to the BEC order parameter space,and introduce conception of order parameter manifold.After that,we give the procedure to find ground sates using symmetry analysis.By this method,we classify the possible ground state of systems with spin-1 and spin-2.In the end,we simply introduce the Homotopy theory for classification of topological excitations,and then analysis possible kind of topological excitations in spinor order parameter.In the fourth chapter,first,by exploring the spin-orbit-coupled rotating pseudo-spin-1/2 Bose-Einstein condensates with the stochastic projected GrossPitaevskii equations,we prove the existence of the circular-hyperbolic skyrmion.And then,we investigate the half-skyrmion excitations induced by spin-orbit coupling in the rotating and rapidly quenched spin-1 Bose-Einstein condensates.Our results show that the half-skyrmion excitation depends on the combination of spin-orbit coupling and rotation,and it originates from a dipole structure of spin which is always embedded in three vortices constructed by each condensate component respectively.In the end,by using the imaginary-time propagation method,we study the ground state structure of a rotating Bose-Einstein condensate with in-plane quadrupole field.The numerical results show that,Mermin-Ho vortex can be induced only by the cooperation between quadrupole field and rotation.It is found that two types of topology structures,the hyperbolic meron and half-skyrmion,can occur in the present system.