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Exact Analysis of Bose-Einstein Condensates and its Applications

玻色-爱因斯坦凝聚体的精确分析及其应用

基本信息

批准号：
18540368
负责人：
WADATI Miki
金额：
$ 2.66万
依托单位：
Tokyo University of Science (2007-2009)The University of Tokyo (2006)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2009
项目状态：
已结题

项目摘要

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

1. 2004 年，我们发现了 Gross-Pitaevskii (GP) 方程的耦合常数的可积条件，该方程描述了三分量玻色-爱因斯坦凝聚体的动力学。我们通过逆散射方法对系统进行了分析，阐明了孤子的碰撞特性。此外，对于通用耦合常数，通过 Painleve 检验研究了 3 分量 GP 方程的可积结构。证明了可积条件存在2种情况；一个是三分量马纳科夫方程，另一个是我们的方程。前者仅包含密度-密度相互作用，而后者描述了密度-密度和自旋-自旋相互作用。凝聚态自旋动力学已成为研究最活跃的课题之一。 2．对于一维的δ函数相互作用的自旋1/2费米气体，我们分析了Yang-Yang积分方程以获得系统的基态能量。作为求解方法，我们采用了 M.Wadati 于 2002 年为 Lieb-Liniger 积分方程引入的级数展开方法。其中，具有挑战性的方面是处理吸引相互作用并包括外部磁场。我们成功地分析了弱相互作用和强相互作用下的系统。与常识相反，弱耦合情况众所周知是困难且微妙的。并且，我们计算了磁化强度作为耦合常数和磁场的函数，并对相位进行了分类。存在三个阶段；完全配对的非极化相、无配对的完全极化相以及上述两相的共存。证明了三相间的量子跃迁； 3．非线性光学中的自致透明 (SIT) 在 20 世纪 60 年代得到了广泛的研究，并为建立孤子的概念提供了有用的信息。作为一种自然而重要的发展，电磁感应透明（EIT）引起了广泛的关注。通过检查两个激光和三能级原子之间的相互作用强度，我们发现了系统的可积条件。通过贝克兰德变换，得到了孤子解，并详细分析了碰撞特性。多分量孤子系统所共有的，会发生多种碰撞，常规弹性碰撞、能量交换、脉冲压缩等。EIT可以与多分量玻色-爱因斯坦凝聚相关联。提出了其在量子信息处理中的应用。 4．在量子力学中，人们认为要具有实值能量，能量算子必须是埃尔密（自共轭）的。然而，真正的能量并不一定是由于能量算子的封闭性造成的。通过使用孤子理论的公式，我们明确地表明非厄米势是以系统的方式构造的。这解释了宇称时间对称势理论中临时引入势的原因。较少的