自旋轨道耦合BEC系统的混沌特性研究

In the project we shall study the classical chaotic solutions and their bifurcation diagrams, Lorenz attractors, Lyapunov indexes, Poincare sections, time series and power spectra for the spin-orbit coupling Bose-Einstein condensates (BECs). At the same time, we shall calculate numerically the energy spectra and macro wavefunctions for the spin-orbit coupled BECs with high precision. And then we shall make some statistical analyses on the calculated energy spectra using a random matrix quantum statistical theory to arrive the corresponding statistical properties of the nearest neighbor level pacing distributions, the Δ3 statistics-the stiffness of spectra and to discuss the changes of level curvature distributions with different confined parameteres. We shall summarize the common chaotic characteristics for the Rashba and Dresselhaus spin-orbit coupled BECs with static and dynamic intensity, and for spin-orbit coupled BECs under the driving of periodic kickors and those of harmonic oscillators, We would explain better the chaotic characteristics of spin-orbit coupled BECs in theory in order to provide some useful theoretical results for the researched on atomic lasers, quantum computations and quantum informations, chip technologies, precision measurements and nanotechnology applications and to provide a certain theoretical guidances to the chaotic control of coupled BECs.

探讨自旋轨道耦合BEC的经典混沌解以及其分岔图、吸引子、Lyapunov指数、庞加莱截面图、时间序列和功率谱等；高精度数值计算自旋轨道耦合BEC的能谱和宏观波函数，利用随机矩阵提出量子统计理论对所得到的能谱进行统计分析，得到相应的最近能级间距分布图象和相应Δ3统计特性以及能级曲率分布随参数的变化；从而得到静态和动态耦合强度下、周期打子和谐振子驱动下的Rashba和 Dresselhaus 自旋轨道耦合BEC中参数诱导量子混沌现象的一般规律，以期理论上进一步描述自旋轨道耦合BEC的混沌特性，解释其混沌现象，为自旋轨道耦合BEC在原子激光、量子信息与量子计算、芯片技术、精密测量和纳米技术等领域的应用提供有益的理论依据，为自旋轨道耦合BEC的混沌控制提供一定的理论指导。

本项目研究了自旋轨道耦合BEC原子气的动力学性质，探讨了局域化波包到达平衡态的演化过程，计算了相关体系的经典混沌解、经典轨迹、庞加莱截面图、时间序列和功率谱等；高精度地计算了自旋轨道耦合BEC的能谱和宏观波函数，利用随机矩阵提出量子统计理论对所得到的能谱进行统计分析，给出同等强度的Rashba和Dresselhuas自旋轨道耦合作用下有限双光子失谐对量子三临界点和相变的效应。总结了Rashba和 Dresselhaus 自旋轨道耦合BEC中参数诱导量子混沌现象的一般规律，以期理论上进一步描述自旋轨道耦合BEC的混沌特性，解释其混沌现象，为自旋轨道耦合BEC在原子激光、量子信息与量子计算、芯片技术、精密测量和纳米技术等领域的应用提供有益的理论依据，为自旋轨道耦合BEC的混沌控制提供一定的理论指导。