Exact Analysis of Bose-Einstein Condensates and its Applications

1. 2004年，我们发现了描述三组分玻色-爱因斯坦凝聚体动力学的Gross-Pitaevskii （GP）方程耦合常数的可积条件。用逆散射法对系统进行了分析，阐明了孤子的碰撞特性。进一步，对于一般耦合常数，通过painlevel检验研究了三分量GP方程的可积结构。证明了可积条件存在两种情况；一个是三分量Manakov方程，另一个是我们的方程。前者只包含密度-密度相互作用，而后者描述密度-密度和自旋-自旋相互作用。凝析油的自旋动力学已成为研究最活跃的学科之一。对于一维中delta函数相互作用的自旋1/2费米气体，我们分析了系统的Yang-Yang积分方程，得到了系统的基态能量。作为一种解法，我们采用了2002年M.Wadati对Lieb-Liniger积分方程引入的级数展开…More方法。其中，具有挑战性的方面是处理吸引相互作用和包括外部磁场。我们成功地分析了系统的弱相互作用和强相互作用。与常识相反，弱耦合情况是非常困难和微妙的。计算了磁化强度与耦合常数和磁场的关系，并对相进行了分类。存在三个阶段；完全配对的非极化相，完全极化的无配对相以及以上两相的共存。证明了三个相之间的量子跃迁。非线性光学中的自诱导透明（SIT）在20世纪60年代得到了广泛的研究，为建立孤子的概念提供了有用的信息。电致透明作为一种自然而重要的发展趋势，引起了人们的广泛关注。通过对两束激光与三能级原子相互作用强度的考察，发现了系统的可积条件。通过Backlund变换，得到了孤子解，并详细分析了碰撞特性。多分量孤子系统通常会发生各种碰撞，如常规弹性碰撞、能量交换、脉冲压缩等。EIT可以与多组分玻色-爱因斯坦凝聚相联系和结合。提出了它在量子信息处理中的应用。在量子力学中，人们一直认为，要具有实值能量，能量算符必须是厄米算子（自共轭）。然而，实际能量并不一定是由于能量算符的密闭性。通过使用孤子理论的一个公式，我们明确地证明了非厄米特势是以系统的方式构造的。这解释了奇偶时间对称势理论中特别引入势的原因。少

1. In 2004, we discovered an integrable condition of the coupling constants for the Gross-Pitaevskii (GP) equation which describes the dynamics of 3-component Bose- Einstein condensates. We analyzed the system by the inverse scattering method and clarified the collision properties of solitons. Further, for generic coupling constants, integrable structure of 3-component GP equation was investigated by the Painleve test. It was proved that there exist 2 cases for integrable conditions; one is the 3-component Manakov equation and the other is our equation. The former contains only density -density interaction, while the latter describes both density-density and spin-spin interactions. Spin dynamics of the condensates has become one of the most actively studied subjects.2. For the delta-function interacting spin1/2 Fermi gas in one-dimension, we analyzed the Yang-Yang integral equation to obtain the ground state energy of the system. As a method of solution, we employed a series expansion … More method which was introduced for the Lieb-Liniger integral equation by M.Wadati in 2002. Among many, challenging aspects are to treat attractive interactions and to include external magnetic field. We succeeded in analying the system both in weak and strong interactions. Contrary to the common sense, the weak coupling case is known to be difficult and subtle. And, we calculated the magnetization as a function of coupling constant and the magnetic field, and classified the phases. There exist three phases ; completely paired non-polarized phase, completely polarized phase without pairs and the coexistence of the above 2 phases. Quantum transitions among the three phases are proved.3. Self-induced transparency (SIT) in nonlinear optics was studied extensively in 1960's and gave useful information for establishing the concept of solitons. As a natural and nontrivial development, electromagnetically-induced transparency (EIT) has attracted much attention. Examining the interaction strengths among two laser lights and 3-level atoms, we found integrable condition for the system. By the Backlund transformation, soliton solutions are obtained and the collision properties are analyzed in detail. Common to the multi-component soliton systems, there occur a variety of collisions, conventional elastic collision, energy exchange, pulse compression etc. EIT can be related and combined to the multi-component Bose-Einstein condensations. Its application to quantum information processing has been suggested.4. In quantum mechanics, it has been thought that to have real valued energies the energy operator has to be hermitian (self-conjugate). However, real energies are not necessarily due to the hermicity of the energy operator. By using a formulation of the soliton theory, we have shown explicitly that non-hermitian potentials are constructed in a systematic manner. This explains the adhoc introduction of potentials in the theory of parity-time symmetric potentials. Less