Nonlinear analysis of Bose-Einstein Condensates

玻色-爱因斯坦凝聚体的非线性分析

• 批准号: 11640387
• 负责人: WADATI Miki
• 金额: $ 1.66万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640387/
• 关键词: Nonlinear Schrodinger equation; Bose-Einstein condensate; 2-conponent Bosonic atoms; Stability of condensate; Gross-Pitaeyskii equation; Free fall; Toroidal trap; One-dimensional δ-function gas; 2成分ボース原子系; ボース・アインシュタイン凝縮; 凝縮体の崩壊; ザハロフ方程式; ボソン・

1.In the case that the effective interaction between atoms is attractive, the condensate becomes unstable when the number of atoms exceeds some value (critical particle number). Using a new inequality, we discussed rigorously the stability condition of Gross-Pitaevskii equation (Nonlinear Schroedinger equation).2.We analyzed the Bose-Einstein condensate under a troidal trap. In Particular, we clarified the ground state properties the ground state energy decreases and the distribution of atoms is shifted to the central axis.3.For the dispersion relation E〜k^a. we investigated the condition of the condensation. By using the WKB method, we found the relation between the dispersion relation and the power-law of the magnetic trap V〜r^p. When the potential decreases rapidly around r=0, we pointed out that the Bose-Einstein condensation takes place even in one dimension. Also, we showed that the values of a and p change drastically the properties of the transition.4.We analyzed the free tall of the condensate including the atomic interactions. Interference patterns of atomic waves reflect the interactions, which are observable in experiments.5.We developed a statistical mechanics of one-dimensional delta function gas. By the peturbational method, we derived the integral equation of thermal Bethe ansatz equation. This is a fisrt direct proof of the Bethe anasatz.

1.在原子间的有效相互作用是吸引的情况下，当原子数超过某个值（临界粒子数）时，冷凝物就变得不稳定。利用一个新的不等式，我们严格地讨论了Gross-Pitaevskii方程（非线性薛定谔方程）的稳定性条件。我们在一个三线阱下分析了玻色-爱因斯坦凝聚体。特别地，我们澄清了基态性质，基态能量降低，原子分布向中轴偏移。对于色散关系E ~ k^a。我们研究了冷凝的条件。利用WKB方法，我们发现了色散关系与磁阱V ~ r^p幂律之间的关系。当势在r=0附近迅速减小时，我们指出即使在一维中也会发生玻色-爱因斯坦凝聚。此外，我们还证明了a和p的值会极大地改变相变的性质。我们分析了包括原子相互作用在内的冷凝物的自由高度。原子波的干涉图样反映了相互作用，这在实验中是可以观察到的。我们发展了一维函数气体的统计力学。用微动法推导了热贝特安萨兹方程的积分方程。这是贝特人的第一个直接证明。

项目成果

H.Morise et al.: "Dynamics of a wave function for the attractive nonlinear Schrodinger equation under isotropic harmonic confinement potential"Journal of Physical Society of Japan. 70. 3529-3534 (2001)

H.Morise 等人：“各向同性谐波约束势下有吸引力的非线性薛定谔方程的波函数动力学”日本物理学会杂志。

Y.Fujii et al.: "Asymptotic form of the two-point correlation function of the XXZ spin chain"Journal of Physics A. 34. 2657-2657 (2001)

Y.Fujii 等：“XXZ 自旋链的两点相关函数的渐近形式”Journal of Chemistry A. 34. 2657-2657 (2001)

G.Kato: "Graphical representation of the partition function of a one-dimensional delta-function Bose-gas"Journal of Mathematical Physics. 42. 4883-4893 (2001)

G.Kato：“一维 δ 函数玻色气体的配分函数的图形表示”数学物理杂志。

H.Fan et al.: "Quantum cloning mechanics of a d-level system"Physical Review A. 64(09). 064301-1-064301-3 (2001)

H.Fan 等人：“d 级系统的量子克隆力学”Physical Review A. 64(09)。

M.Wadati: "Statistical mechanics of a one-dimensional delta-function Bose gas"Journal of Physical Society of Japan. 70. 1924-1930 (2001)

M.Wadati：“一维δ函数玻色气体的统计力学”日本物理学会杂志。