Spatiotemporal Bloch States Of A High-frequency Driven Bose-Einstein Condensate With Spin-orbit Coupling

Spin-orbit coupling(SOC)describes the interaction between the spin and orbital motion of a quantum particle,is crucial for many important condensed matter phenomena.In recent years,researchers have realized and managed SOC in ultracold atomic systems through external laser fields.In particular,the twocomponent BEC system with SOC trapped in the optical lattice has also been widely studied.In the mean-field theory,as a many-body system,a BEC governed by Gross-Pitaevskii equation(GPE)provides an important basis for studying the corresponding physical properties.The exact solutions and analytically perturbed solutions to the GPE can be used to discuss many physical properties such as the macroscopic quantum(or semi-classical)chaos,BEC stability,the superfluid velocity and flow density,and the generation of solitons.However,the GPE with nonlinear interactions is not easy to solve in general and the BEC system with SOC is a more complex nonlinear system which poses a challenge to finding the exact solutions.Here we are motivated in generation of the exact solutions to such a quasistationary system with SOC and the physical properties of BEC,by using a high-frequency external field to adjust and reconstruct the system parameters.This paper is divided into four chapters,the first chapter is the introduction part,the second and third chapters are the author's main research work,and the fourth chapter is the full text summary.The main contents are as follows:In the first chapter,we briefly introduce the history and basic theory of atomic BEC studies,as well as the mean field theory of BEC and the GPE.Then we briefly introduce the SOC,Ultracold atomic systems and several different types of spinorbit coupling in Bose-Einstein condensate.Finally,we introduce the quantum entanglement.In the second chapter,we consider a SO-coupled BEC driven by a highfrequency field and confined in an optical lattice through which we study the spatiotemporal Bloch states and the associating physical properties.First,based on the RWA,such a driven system is approximated by a quasistationary one and the relating exact spatiotemporal Bloch states to the latter system are obtained.We analytically show that the SOC strength can be adjusted by the high-frequency field such that the existence conditions of the exact particular solutions are met.Meanwhile we find that areas of the parameter regions on the SOC-vs latticestrength plane for the existence of the exact solution enlarge with the increase in Rabi coupling strength.Then,we show the periodic distributions of the atomic number density.Furthermore,we analytically and numerically demonstrate the several novel features related to the exact states:(1)the SOC leads to that the spatiotemporal Bloch states become the spin-motion entangled states;(2)the SOC influences the population imbalance between two BEC components;(3)SOC can be applied to manipulate the stable atomic flow,which is conducive to quantum transport of the BEC for different application purposes.In the third chapter,We investigated the instability of the stationary periodic solution to BEC trapped in optical lattices.We give the Bloch solution of the series form to the Bogoliubov-de Gennes equation and discuss the condition of series index n =-1,0,1 and n =-2,-1,0,1,2.By taking different parameters separately,we obtained the different areas of the instability on the chemical potential-vs wave vector plane.From the results,we can see that when the lattice depth decreases,the amplitude of the periodic solution increases,The corresponding system instability area will be increase.Conversely,the instability region of the system decreases as the lattice depth increases and the amplitude of the periodic solution decreases.Therefore,the results allow us to control the instability regions of the zero-order periodic solution by changing different parameters,then,we can get a stable BEC system with application value.In chapter four,we give a summary to the whole paper,and give a prospectto the instability and the other physical properties of BEC which trapped in the optical lattice.