【作者】 吴雷；

【导师】 张解放；

【作者基本信息】 浙江师范大学 ， 非线性物理， 2007， 硕士

【摘要】 玻色—爱因斯坦凝聚是一种崭新而奇特的物质状态,孤立波是一种可以稳定传播的物质存在形态。在玻色—爱因斯坦凝聚中研究孤立波的激发和稳定传播等问题是近几年来非线性科学研究的一个热点。本文研究在时变散射长度和时变谐振子势共同作用下玻色—爱因斯坦凝聚体的动力学演化问题,其中包括波函数的精确自相似演化及其可能的应用、一维几何边界条件下亮孤立波的传播特性,以及平面波背景上周期微扰的调制不稳定性分析等。本文的主要内容如下:在论文第一章,我们首先简单介绍无相互作用玻色气体的凝聚特性,并讨论在非热力学极限下低维系统出现玻色—爱因斯坦凝聚的可能性。然后介绍当玻色了之间存在相互作用时描述凝聚体动力学演化的Gross-Pitaevskii方程,并简要介绍实验上散射长度和谐振子势的产生和控制问题,为下文的研究奠定基础。在论文第二章,我们着重研究凝聚体定态波函数在时变散射长度和时变谐振子势共同作用下的演化问题。我们发现当散射长度和谐振子势满足一定的关系时,凝聚体定态波函数将满足精确的自相似演化规律。基于这个发现,我们提出了利用波函数精确自相似膨胀的性质来提高凝集体定态轮廓测量精度的实验方案。更进一步,我们提出了利用波函数精确自相似压缩的性质来提高物质波的局部密度来检验Gross-Pitaevskii方程有效性的实验方案。特别地,在高维情况下,我们还得到了涡旋的精确自相似演化条件。进一步的分析发现在自相似演化过程中,涡旋的拓扑指数保持不变。在论文第三章,作为一维定态波函数精确自相似演化的特例,我们把含有时变散射长度和时变谐振子势的一维Gross-Pitaevskii方程转化成了标准的非线性薛定谔方程。当粒子间存在吸引相互作用时,我们得到了亮孤立波解:该孤立波既可以在零背景上传播,也可以镶嵌在平面波背景中传播。进一步的分析发现,孤立波的运动完全由谐振子势控制,而零背景上孤立波的振幅则完全由散射长度控制。对于平面波背景上的波,我们通过解析的方法对该背景上的微扰的调制不稳定性问题进行了分析。最后,当时变散射长度和时变谐振子势近似满足可积条件时,我们通过孤子微扰理论得到了亮孤立波的近似解,并与数值模拟进行了对比。同理,当粒子间存在排斥相互作用时,我们可以得到暗孤立子解。为简单见,在文中不再赘述。在论文第四章,作为对调制不稳定分析的一个扩展,我们讨论了考虑玻色子之间的三体吸引相互作用后平面波背景上周期微扰的调制不稳定性问题。解析和数值模拟显示,谐振子势既可以使本来调制稳定的波变得调制不稳定,也可以使本来调制不稳定的波变得调制稳定。最后是对全文工作的总结和展望。更多还原

【Abstract】 Observation of Bose-Einstein condensates (BEC) in ultracold gases of weakly interacting alkali metals has opened up a new possibility for studying the quantum phenomena in macroscopic scales. In particular, the experimental realization of solitons and vortices in BEC has stimulated intensive studies of the nonlinear excitations of matter waves. In this thesis, we investigate the evolution dynamics of BEC matter waves in the presence of time-dependent scattering length and time-dependent harmonic potential, including the exact self-similar evolution of BEC matter waves and their potential applications, the propagation properties of one-dimensional BEC bright solitons and the modulation instabilities on plane waves. This thesis is organized as follows:In Chapter 1, we firstly introduce the basic concept of BEC in ideal gases, and discuss the possibility of BEC in lower dimensional geometry. Then we introduce the Gross-Pitaevskii equation which governs the dynamic of BEC when the interatomic interaction is turned on, and discuss how to control the interatomic interaction and external harmonic potential in experiment.In Chapter 2, we discover that the profile of BEC stationary state can evolve exact self-similarly when the time-dependent interatomic interaction and harmonic potential satisfy a certain condition both for one-, two- and three-dimensional geometry. Based on this discovery, the accuracy in measuring the stationary-state profile is improved when the direct measurement is not available. Furthermore, we also propose a feasible experiment to squeeze the matter wave into very high local matter density to check the validity of three-dimensional Gross-Pitaevskii equation. Specifically, we also find the self-similar condition for vortex in higher dimensional geometry. Further analysis reveals that the topological charge of vortex does not change during the self-similar evolution.In Chapter 3, we present the integrable condition for one-dimensional Gross-Pitaevskii equation and obtain the exact analytical solution which describes the modulation instability and the propagation of bright solitary wave on a continuous wave background. Moreover, we also find the approximate bright solitary wave solutions under the near-integrable conditions. Both of these solutions show that, the amplitude of bright solitary wave with zero boundary condition depends on the scattering length while its motion depends on the external potential. Dark soliton solution can also found in the similar way, but for simplicity, we are not going to discuss it.In Chapter 4, we investigate the modulation instability of Bose-Einstein condensate with three-body interatomic interaction and external harmonic trapping potential. Both of our analytical and numerical results show that, the external potential will either cause the excitation of modulation instable modes or restrain the modulation instable modes from growing.Conclusions and outlook are given in Chapter 5.更多还原