| Bose-Einstein condensates (BEC) have been attractive subjects in recent decades. They not only offer the perfect macroscopic quantum systems to investigate many fundamental problems in quantum mechanics but also have extensively application foregrounds such as in atom laser and quantum computation. In the framework of mean-field theory, the BEC is governed by the Gross-Pitaevskii equation. Based on the Gross-Pitaevskii equation we shall study the chaos, unstable cycle in double-well and chaos control, synchronism in an optical lattice.Our paper is organized as the following six parts. In the first part we shall give a simple introduction to development of optical chaos, chaos in BEC, controlling chaos, anti-control of chaos ,atomic optical, BEC and properties, atomic laser and so on.In part two, we studied the space-time evolution properties of the BEC held in a traveling optical lattice. In this chapter, the dynamic equation of the BEC is deeply investigated and the stability of its stable solution is analyzed. We focus on the features of spatiotemporal chaos of the Bloch-like solution of the system. When the damping is our considerations, we use Lyapunov exponent to present the chaotic region in parametrical space. Features of the transient chaos are studied through numerical method. The transition procession from transient chaos to stationary one has been numerically simulated. In this procession, we find that the final attractors of the transient chaos undergo a series of period-doubling bifurcations. We also simulate the time-series and power spectra.In part three, chaos control and synchronism in BEC are investigated by four methods. Firstly, we suggest a method for eliminating chaos by modulating periodic signals to convert the chaotic state into regular state. As a function of modulation intensity and modulation frequency the maximal Lyapunov exponent is calculated respectively, and the periodic orbits associated with the negative Lyapunov exponent. Secondly, BEC chaos is controlled with the outer period signal parameter modulation. Numerical simulation shows that the chaotic behavior can be well controlled to enter into periodicity by choosing condition of resonance and the best phase matching or intensity. Thirdly, a method of chaos control with linear feedback is presented. By using the method, we propose a scheme of controlling chaotic behavior in a BEC with atomic mirror. The results of the computer simulation show that the chaos into the stables could be realized by adjusting the coefficient of feedback only if the maximal Lyapunov exponent of the system is negative. Fourth, through changing s-wave scattering length by using feshbach resonance, the chaotic behavior can be well controlled to enter into periodicity. Numerical simulation shows that there are different periodic orbits according to different s-wave scattering length only if the maximal Lyapunov exponent of the system is negative. Finally, chaotic synchronization in BEC is investigated.The fourth chapter anti-control of chaos in BEC is put forward by two methods. We suggest a method for generating chaos in BEC by modulating periodic signals to convert the regular states. As a function of modulation intensity and modulation, frequency the maximal Lyapunov exponent is calculated respectively and the chaotic orbits associated with the positive Lyapunov exponent. Finally, we present a method anti-control of chaos in BEC by applying outer periodic signals to convert the periodic state into chaotic state. The periodicity can be well-controlled to enter into chaotic behavior by choosing condition of resonance and best phase matching or intensity phase. Numerical simulation shows that there are chaotic orbits corresponding to different phase matching or intensity only if the maximal Lyapunov exponent is positive.In part five, by using Lyapunov exponent, chaotic time evolutions and the macroscopic quantum self-trapping in BEC are investigated for the particle number density of BEC with three-body interaction. Chaotic attractors, the time series and power spectra are simulated numerically. Properties of chaos are revealed theoretically and numerically in this part. Firstly, we study the chaotic properties in BEC. In the second part, unstable cycle in BEC with three-body interaction is investigated by using Lyapunov exponent. Finally, we study the macroscopic quantum self-trapping in BEC. Double-well trapping in BEC is a simple model. People use it to study many phenomenons.Finally, we briefly summarize our contributions and discuss future work in the last chapter. Here, our main works are involved in chapter's three, four and five.
|
PDF Full Text Request