基于变量重构的BEC涡旋陀螺涡动模态稳定性研究

Aimed at the quantitative stability analysis and high-stability control issues of the whirling modes for a Bose-Einstein condensate (BEC) vortex gyroscope, the quantitative stability analysis and control for the nutation and precession modes of the superposition BEC vortex coupling system under the three dimensional annular potential well has been studied based on variable reconstruction method. Firstly, the superposition-BEC vortex coupling dynamical model is built, and the 2 nonlinear equations about the nonlinear wave functions are converted into the single variable linear equation about the complex-variable disturbance by variable reconstruction. The stability equivalence of the two systems before and after variable reconstruction has been proven, and the inherent relationships between the distribution of the closed-loop poles of the complex-variable system as well as the different whirling modes are revealed, yielding the mathematical token method of differentiating the nutation and precession modes for BEC vortex gyroscopes. Secondly, the necessary and sufficient conditions of the critical stability for nutation and precession are studied, revealing analytically the effecting law of the angular moment, potential well parameters and other outer disturbance factors on the relative stability of the nutation and precession modes. The concept of complex-variable frequency characteristics is proposed, and the nominal and robust stability criteria of nutation and precession modes for BEC vortex gyroscopes are developed. Finally, to realize the high-stability control for BEC vortex gyroscopes, a closed-loop control scheme based on interference-pattern space and time information and potential well control is presented. According to the presented stability criteria, the controller parameters have been further designed quantificationally and optimized, and experiments will be performed to demonstrate the effectiveness and superiority of the proposed methods.

针对BEC涡旋陀螺涡动模态稳定性分析和高稳定度控制的难题，本项目探索基于变量重构的三维环状势阱下叠加态BEC涡旋耦合系统的章动和进动稳定性分析理论与高稳定度控制方法。在建立叠加态BEC涡旋耦合动力学模型的基础上，引入变量重构的思想，将基于两个波函数的非线性GP方程组转化为一个复变量扰动的单变量线性方程；证明变量重构前后系统稳定的等价性，研究复变量系统极点分布与章动、进动间的映射关系，提出区分BEC涡旋陀螺章动和进动模态的数学表征方法；其次推导章动和进动临界稳定的充要条件，明晰角动量大小和势阱参数及外部扰动等因素对章动和进动相对稳定性的影响规律，提出量子系统复变量频率特性概念，建立BEC涡旋陀螺不同涡动模态的标称和鲁棒稳定判据；最后，提出基于干涉图样时空信息和势阱控制相结合的闭环测控方案，利用提出的章动和进动稳定判据，对系统的参数进行定量和优化设计，并进行实验验证。

针对BEC涡旋陀螺章动和进动稳定性分析和高稳定度控制的难题，本项目首先从平均场近似的Gross-Pitaevskii方程出发，在量子陀螺圆周对称近似边界条件下，对激子极化激元叠加态物质波的局域时空演化过程建模。.其次，研究了处于无序势中的二维激子极化激元凝聚正反涡旋叠加态在载体旋转情形下的稳定性和动力学特性，模拟了量子涡旋陀螺在实际应用中的场景，并初步验证了圆周对称微腔中激子极化激元BEC涡旋体系陀螺效应的存在。通过引入具有时间分量的旋转坐标系，对Gross-Pitaevskii方程进行重构，得到具有科里奥利项的刻画旋转状态的Gross-Pitaevskii方程。进而用实时演化方法研究了局域粒子数和半导体微腔旋转角速率之间的关系。研究了激发区域旋转速率与载体旋转速率的关系，并给出了载体旋转对涡旋叠加态相位稳定性的影响。.随后，对陀螺系统稳定性首次进行了专门、系统的研究。微腔中的激子极化激元BEC体系是一个多体泵浦-耗散动平衡体系，具有复杂的稳定性条件，本文选取了两个最主要的非固有噪声因素——泵浦光噪声和材料无序噪声作为主要研究对象，同时进行了非含时、含时两种维度的分析，利用非含时方法对大量边界条件进行快速筛选、利用含时方法对噪声干扰进行深度分析。准确分析了泵浦噪声、材料无序噪对激子极化激元BEC涡旋体系的时间稳定性和空间稳定性产生的干扰，初步发现了泵浦噪声对激子极化激元BEC体系产生的四种效应和材料无序噪声对激子极化激元BEC体系产生的反置特性，形成了一系列指导量子涡旋陀螺工程样机实现的重要理论参考。.最后，设计了和搭建了机物微腔中激子极化激元的实验研究的光路，并对微腔中激子极化激元进行了实验研究。实现了高斯光、涡旋光泵浦作用下的激子极化激元的激射和凝聚，并在此基础上对激射区域分布与泵浦光以及材料不均匀等因素的关联进行了分析，与理论计算的结果进行了对比。