Nonlinear Phenomena and their Controls in Bose-Einstein Condensates

为了阐明超冷中性玻色原子玻色-爱因斯坦凝聚体的静态和动态性质，我们研究了以下问题.对于一维量子可积粒子系统，准动量分布和激发能谱分别由Lieb-Liniger（LL）积分方程和Yang-Yang（YY）积分方程描述。这些积分方程通常采用数值方法求解，而解析方法的研究较少。特别是，弱耦合的情况下，已知是一个困难的问题。本文应用幂级数展开方法分析了自旋为1/2的费米子系统，其中两种费米子（自旋向上和自旋向下）通过吸引的δ函数势相互作用。在强耦合区，存在自旋对态。这意味着在超冷气体中存在BCS态.物质波在F=1旋量BEC凝聚体中的传播在光阱中，实现了具有内自由度的凝聚体。结果表明，当原子间相互作用势和自旋交换相互作用的大小相同时，三分量Gross-Pitaevskii（GP）方程是可积的。通过逆散射方法，得到了N孤子解。对于吸引情形，存在亮孤子，并将亮孤子分为极性孤子和铁磁孤子。对于排斥的情况，类似的结果保持（在准备中）。对于一般的两个耦合常数，物质波的传播进行了研究。由于今年是计划的最后一年，我们将分析扩展到非对易时空中的孤子方程、一维排斥过程中的输运、磁场中两个自旋的几何相位和量子纠缠等课题。

To clarify the static and dynamic properties of the Bose-Einstein condensates of ultra-cold neutral bosonic atoms, the following subjects were investigated.1. Quantum delta-function gasesFor one-dimensional quantum integrable particle systems, quasi-momentum distribution and excitation energy spectrum are described by Lieb-Liniger (LL) integral equation and Yang-Yang (YY) integral equation, respectively. Those integral equations are usually solved numerically, but the analytic work are few. In particular, the weak coupling case is known to be a hard problem. We apply the power-series expansion method for analysis of spin-1/2 fermion system where two kinds of fermions (spin up and spin down) interact through the attractive delta-function potential. In the strong coupling region, spin-pair states are shown to exist. This implies the occurrence of BCS state in ultra-cold gases.2. Matter-wave propagations in F=1 spinor BEC condensate.In optical trap, the condensates with internal degrees of freedom are realized. It was found that, when the magnitudes of inter-atomic potential and spin-exchange interaction are same, three-component Gross-Pitaevskii (GP) equation is integrable. Through the inverse scattering method, N-soliton solutions are obtained. For the attractive case, bright solitons exist and those are classified into polar soliton and ferromagnetic soliton. For the repulsive case, the similar results hold (in preparation). For general two coupling constants, matter-wave propagations are investigated. From the analysis of plane-wave solutions, we can show the existence of polar soliton and ferromagnetic soliton.Since this year is the final year of the project, we extended analyses to the subjects such as soliton equations in non-commutative space-time, transports in one-dimensional exclusion processes and, geometric phases and quantum entanglements of two spins in a magnetic field.