摘要
在当前的冷原子研究领域中,光晶格中玻色爱因斯坦凝聚体(BEC)动力学特性的研究已成为其中的热点。大量的理论和实验研究表明,原子间相互作用对光晶格中BEC的动力学性质有重要的影响,如它对BEC的能带结构、隧穿特性、稳定性、自俘获有着显著的影响。光晶格中BEC的自俘获现象属于非线性量子现象,与冷原子系统的稳定性以及量子相变有关。然而,目前大部分研究仅仅局限于考虑原子间两体相互作用下光晶格中BEC的动力学特性。我们知道要研究真实的动力学特性,原子间三体相互作用是不能忽略的。因此,本文一方面,研究同时考虑原子两体相互作用和三体相互作用时,三体相互作用对光晶格中BEC的能带特征、动力学稳定性和自俘获的影响；另一方面,研究FS光晶格中,两个谐波晶格的相对相位对BEC的能带结构、隧穿特性及动力学稳定性的影响。论文包括六章：
第一章简单介绍了BEC的研究历史、实现BEC的相关技术以及BEC的应用前景和研究意义。
第二章呈现了BEC的一些基本理论知识。
第三章研究了两体和三体相互作用下BEC在一维光晶格中的动力学行为。运用两模近似,对描述一维光晶格的BEC的耦合GP方程做了一些数值上的分析,讨论了三体相互作用λ对体系定态解和稳定性的影响。发现λ不仅改变了体系的分叉点而且对体系的宏观量子自囚禁有极大的影响。另外对能极差γ加上一个周期调制γ=A0+A1 sin(ωt)后,光晶格中的BEC表现出更加丰富的动力学特性。特别地讨论了,随着γ的变化,相对粒子布局数将从周期振动经倍周期分岔进入混沌。
第四章运用一个非线性三模模型研究了FS光晶格中BEC的能带结构。在线性情况下,晶格间的相对相位强烈地影响着系统的能带结构。在非线性情况下,当原子间的相互作用达到四个临界值时,系统的能带在体系的中间能级上出现了非常奇异的碗状结构。为此我们研究了体系本征态的稳定性。
第五章在平均场框架下,用非线性三模模型分析了BEC在加速FS光晶格中的能带结构和隧穿动力学。在线性情形下,能带结构和两个谐波晶格的相对相位有很大的关联。非线性情况下,随着相互作用的变化BEC的能带有很奇特的变化。由于能级上拓扑结构的扭曲,隧穿动力学将有明显的变化。对于弱非线性,能带结构和线性情况相似,同时隧穿几率非零。强非线性时,中间能级有明显的绝热隧穿,也就是说在中间能级,隧穿率作为扫描率的α的函数在中间能级呈现了不规则的振荡。
在本文的最后,对我们的工作作了总结,并对将来的研究作了展望。
Dynamical behaviors of Bose-Einstein condensate (BEC) in optical lattices have attracted enormous attention both experimentally and theo-retically. A lot of research shows the atoms interactions play a important role for dynamic properties of BEC in optical lattices. The energy band structure, tunneling dynamics, the stabilities, self-trapped states of BEC are affected dramatically by atoms interactions. The self-trapped phenom-ena of BEC in optical lattice belong to the fields of nonlinear quantum, it is relevant to the stabilities of cold atoms system and quantum phase tran-sition. However, most present studies are only focused on considering the two-body interaction for dynamic behaviors of BEC in optical lattice. In fact, The three-body interaction can not be ignored for the real behaviors of BEC. It have influenced dramatically for Dynamic behaviors of BEC. So, one aim of this article is to study the dynamic behavior of BEC in op-tical lattices with two-body and three-body interactions. The three-body interaction have important affect on the stabilities and the macroscopic quantum self-trapping of the system. On the other hand, the aim of this article is to investigate the influence of the relative phase of the two lat-tice harmonics on the band structure, the tunneling dynamics of BEC in Fourier-Synthesized optical. The major result are listed as follows:
ChapterⅠ:The history of research, the realization technique, and application prospect and research meaning of BEC are reviewed briefly.
ChapterⅡ:We present on the basic theory of BEC.
ChapterⅢ:We study the dynamical behaviors of BEC with two- and three-atom interactions in optical lattices with analytical and numeri-cal methods. It is found that the steady-state relative population appears tuning-fork bifurcation when the system parameters are changed to certain critical values. In particular, the existence of three-body interaction not only transforms the bifurcation point of the system but also affects greatly on the macroscopic quantum self-trapping behaviors associated with the critically stable steady-state solution. In addition, we also investigated the influence of the initial conditions, three-body interaction, and the energy bias on the macroscopic quantum self-trapping. Finally, by applying the periodic modulation on the energy bias, we observed that the relative pop-ulation oscillation exhibits a process from order to chaos, via a series of period-doubling bifurcations.
Chapter IV:By employing a nonlinear three-mode model, the band structure of BEC in Fourier-Synthesized optical lattices is studied, where the nonlinearity is coming from the mean field treatment of interaction be-tween atoms. In linear case, we obtain the band structure for certain poten-tial values. It is demonstrated that the energy band structure is strongly dependent on the value of relative phase of the two lattice harmonics. In the nonlinear case, we show that the eigenenergies as the functions of the quasi-momentum have a novel bowl structure in the middle energy level . We find that there exist four critical values of interaction strength at which the band structure will undergo interesting changes. Furthermore, the stability of eigenstate is also investigated.
Chapter V:By applying the mean field theory, we study the band structure and tunneling dynamics of BEC in accelerate Fourier-Synthesized optical lattices. It is found that the band structure undergo interesting changes. Because of these topological distortions on the energy levels, we expect that the tunneling dynamics appear dramatic change. The tunnel-ing probability is nonzero when the nonlinear term is very small and the energy levels keep the same topological structure as that in the linear case. Especially, the tunneling probability as a function of sweeping rate shows an irregular oscillation for the state on the middle level.
At last, we make a conclusion and outlook of this thesis.
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