Study On The Dyinamics And Quarntum Entanglenrment Of Many Body System Based On The Majorana Representation

With the birth and rapid development of quantum mechanics,quantum and its correspond theory has not only shown its importance on explaining the physics world,but also played an important role in the progress of modern society.On one hand,as the development of information science and material science,how to improve the efficiency and quality in information processing and design new type of material by the theory of quantum mechanics has become an important subject in modern technology progress.On the other hand,the research on the physic system of realizing quantum computation has made a significant progress.One of the most promising program of quantum computation is the cold atom system.In this thesis,we choose the distributions and trajectories of the Majorana points in the Majorana representation as an intuitive method.The properties,such as dynamic evolution and quantum entanglement,of many body quantum system has been studied.It will deepen the understanding of the many-body cold-atom system and its symmetry-dependent physical mechanics.This thesis consists of five chapters,and our main works are given in Chapter 3 and 4.In the first and second chapter,an introducing of the research background and related background knowledge is made,including the proposals and developments of quantum mechanics,condensed matter physics and quantum entanglement.Furthermore,the basic knowledge of Schwinger's angular momentum-boson model,the Majorana representation,the double-well potential model and quantum entanglement are introduced in detail.In chapter 3,by using Majorana representation,we study the dynamical evolution of the nonlinear two-mode boson system both in the mean-field and the second quantized models.The motion equations of the Majorana points has been derived and the difference between the two models is analyzed and presented by the star equations.The quantum state evolution in the second-quantized model as well as its relation and difference to mean-field model has been studied.Furthermore,by using the fidelity between the states of the two models and the correlation between stars,we also discussed the quantum state evolution and corresponding self-trapping phenomenon with different boson numbers and nonlinear strength.In chapter 4,we study the two and three qubits quantum entanglement through the distances between the Majorana points,analyze the symmetries of several typical entangled states,and discuss the relation between the Majorana points' distances and the entanglement classification as well as the measurement.Besides,we also generalize the Majorana representation for two qubits to the situation of any generic state,and have some preliminary results.Finally,we give a summary and prospect of this paper.