| Bose-Einstein condensation(BEC)is a common physical phenomenon that involves many fields in physics.Under the inspiration of Bose,in 1924 Einstein predicted the phenomenon of condensation for an ideal gas of neutral atoms obeyed Bose-statistics when the temperature is low enough.Because the conditions of realizing BEC are very harsh,the BEC of near-perfect gas of alkali atom were realized experimentally through laser-cooling,magnetic trap and evaporative cooling technology until 1995.BEC is a macroscopiccal quantum system that is composed of thousands or even millions of atoms which completely lost all the characteristics of their isolated particles atoms.The quantum phenomenon is brought in macroscopic scale by BEC,so its matter waves can be described by a single macro wave function.With the realization of BEC in the experiment,the research of BEC has gradually become a hot subject.The optical lattice that was formed by laser coherent superposition is a very convenient and useful tool to manipulate precisely BEC.It is also widely used in the study of BEC.The combination of optical lattice and BEC opens up many new research directions for us.As a macroscopic system of quantum mechanics,BEC has been widely used in atomic clock,atom laser,quantum computation,quantum information processing and other fields.BEC is a typical nonlinear system,and chaos in BEC had confirmed.In view of chaotic influence on the stability of the BEC,the research of chaos control and synchronization is very meaningful and valuable work.The study of BEC is of great significance for the rational use of condensed matter and the development of new materials.Based on the Gross-Pitaevskii(G-P)form of weakly interacting Bose theory,this dissertation takes G-P equation in the framework of mean field theory as the main model.We mainly study the stability,chaos control and synchronization of BEC system loaded into a moving optical lattice potential with damping effect.In addition,the chaotic dynamic behavior,chaos control and chaos synchronization of BEC in tilted optical lattice are demonstrated numerically.The content of the dissertation is listed as follow:1.Research history of chaos theory,chaos control and chaos synchronization in nonlinear systems are introduced systematically.The properties of BEC system and the research progress of chaos in BEC are briefly introduced.2.The stability and the chaotic behavior of BEC in a moving optical lattice potential with attractive interaction are studied.Through theoretical and numerical analysis,the parameters ranges of the chaotic region of the BEC system are estimated.The maximum Lyapunov exponent of the BEC system with different parameters,bifurcation diagram,chaotic attractor,time series and frequency power spectra are shown through the numerical simulation,and the chaotic characteristics of the BEC are further clarified.3.Four new methods to realize chaos control of BEC system are proposed,which include constant bias,periodic force,wavelet function and linear feedback control.The maximum Lyapunov exponent of BEC system is calculated by numerical simulation with different control parameters.According to Lyapunov stability theory,the chaotic motions can be converted into periodic motions only if the maximum Lyapunov exponent of the system is negative.Meanwhile,the attractor and the corresponding time series diagram of the different periodic orbits with different values of the control parameters are given.Numerical simulations are also given to demonstrate the feasibility of the proposed control strategy.4.Chaos synchronization and anti-synchronization of two BEC systems are studied.On the basis of Lyapunov stability theory,linear stability and Routh Hurwitz criterion,the process of chaotic synchronization and anti-synchronization of BEC system is theoretically discussed by using sinusoidal coupling,linear coupling,hyperbolic sine coupling and active control method.The chaos synchronization criterions of different methods are obtained.In addition,the relation between chaos synchronization time and control parameters is analyzed.Simulation results of chaos synchronization and anti-synchronization are provided to demonstrate the effectiveness of the proposed methods.5.The dynamic equation of the BEC system in a 1-D tilted optical lattice is solved numerically,and the chaotic characteristics of the system are illustrated by the attractor and time series.The influence of amplitudes,tilt factors of the optical lattice potential and initial conditions on the chaotic motion of the system is discussed.The effect of chaos control is revealzed by using the method of constant migration,and feasible methods of realizing chaos control are found.Finally,we give a summary and an induction of this research work and give a outlook of the applications of the BEC system.The dissertation provides a new direction for future research. |
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