Multidimensional Tunnelling Effect and Complexified Classical Dynamics

多维隧道效应和复杂的经典动力学

基本信息

批准号：
13640410
负责人：
IKEDA Kensuke
金额：
$ 2.56万
依托单位：
Ritsumeikan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

Multi-dimensionality of the systems radically influences tunnelling phenomena. In particular, if the system is classically non-integrable, complicated tunneling phenomenona, which are due to the presence of chaotic set and are called chaotic tunnelling, are observed. The fundamental mechanism of chaotic tunnelling is investigated by using classical dynamics extended to the fully complex domain, i.e. the complex semiclassical method.(1) Tunnelling in the presence of chaos is investigated for a class of quantum map system. Extensive numerical studies reveal that the tunnelling trajectories dominantly contributing the dynamical tunnelling process form a very limited class of sets in the initial manifold, which is called "Laputa chains" from their characteristic shape. Mathematical significance of such a set is investigated applying the results of hormorphic dynamical theory to numerically clarified natures. The main results is that the closure of Laputa chain is bounded by two sets, namel … More y, the Jula set J^+, and the filled-in Julia set K^+, from below and above, respectively. It is further conjectured that K^+=J^+. If this is the case, the closure of Laputa chain is nothing more than Julia set. The wavefunction tunnelling through the dynamical barrier is constructed along the real component of J^-. These facts means that the major trajectories tunnells being guided by the complexified stable manifolds of saddles dense in the chaotic sea, and are scatterd along their unstable manifold.(2) Confining ourselves to a class of barrier tunnelling process, we elucidate how multi-dimensionality of the system results in a new universal mechanism causing complicated tunnelling phenomena peculiar to multi-dimensional barrier systems. First we showed that the complex semiclassical theory surely reproduces the purely quantum wave matrix even in the strong coupling regime, where the tunnelling component is accompanied by complicated fringes. Next, it is shown that the complexified trajectories guided by complexified stable and unstable manifolds are responsible for the fringed tunneling effect. Such a mechanism provides a new picture of tunnelling quite different from the classical instanton mechanism. Seen from a mathematical side, the mechanism is explained in terms of a divergent shift of movable singularities, which are the origins of the multivaluedness of complexified trajectories in one-dimensional tunnelling problem. Less

系统的多维性从根本上影响隧道现象。特别是，如果该系统是经典的不可汇总，复杂的隧道现象，则观察到这是由于存在混乱的套件而被称为混乱的隧道。通过使用扩展到完全复杂的领域的经典动力学，即复杂的半经典方法。（1）研究在混乱的存在下进行隧穿来研究混乱隧穿的基本机制，以进行广泛的数值研究表明，隧道轨迹揭示了动态隧道的构成非常有限的链接的构成构成构成的特征，从而构成了构成最初的表现。研究了这种集合的数学意义，将恒星动力学理论的结果应用于数值澄清的本质。主要结果是，Laputa链的闭合是由两组Namel…More Y，Jula Set J^+和填充的Julia Set K^+分别从下和更高的界定。进一步猜想k^+= j^+。如果是这种情况，Laputa链的关闭无非是朱莉娅的设定。沿J^ - 的实际组成部分构建了通过动态屏障的波功能隧道。 These facts mean that the major trajectories tunnelling are guided by the complexed stable manifolds of Saddles dense in the chaotic sea, and are scattered along their unstable manifold.(2) Confining ourselves to a class of barrier tunnelling process, we elucidate how multi-dimensionality of the system results in a new universal mechanism causing complicated tunnelling phenomena peculiar to multi-dimensional barrier systems.首先，我们表明复杂的半古老理论肯定会重现纯量子波矩阵，即使在强耦合方向上，隧道成分伴随着复杂的条纹。接下来，表明，由复合稳定和不稳定的歧管引导的复合轨迹是边缘隧道效应的原因。这样的机制提供了与经典的intanton机制完全不同的隧道的新图片。从数学方面可以看出，该机制是用可移动奇点的不同转移来解释的，这是一维隧道问题中综合轨迹多相关性的起源。较少的