Multidimensional Tunnelling Effect and Complexified Classical Dynamics

系统的多维性从根本上影响着隧道现象。特别地，如果系统是经典不可积的，则观察到由于混沌集的存在而产生的复杂隧穿现象，称为混沌隧穿。将经典动力学扩展到全复域，即复半经典方法，研究了混沌隧穿的基本机理。(1)研究了一类量子映射系统在混沌存在下的隧穿现象。大量的数值研究表明，对动力隧穿过程起主导作用的隧穿轨迹在初始流形中形成了非常有限的一类集合，根据其特征形状称为“Laputa链”。将全态动力学理论的结果应用于数值性质研究了这类集合的数学意义。主要结果是，Laputa链的闭包由两个集合组成，分别从下往上是Jula集合J^+，从上往上是填充Julia集合K^+。进一步推测K^+=J^+。如果是这样的话，Laputa链的闭包就是Julia集。沿着J^-的实分量构造了穿过动力势垒的波函数。这些事实意味着主要的轨迹隧道被混沌海中密集的鞍形复杂稳定流形所引导，并沿着它们的不稳定流形分散。(2)以一类势垒隧穿过程为例，阐明了系统的多维性如何导致一种新的普遍机制，从而导致了多维势垒系统特有的复杂隧穿现象。首先，我们证明了即使在强耦合状态下，复杂半经典理论也肯定地再现了纯量子波矩阵，其中隧穿分量伴随着复杂条纹。其次，研究了由稳定流形和不稳定流形引导的复杂轨迹是造成条纹隧穿效应的主要原因。这种机制提供了一种完全不同于经典瞬子机制的隧穿新图景。从数学的角度来看，这一机制是用可移动奇点的发散位移来解释的，这是一维隧道问题中复杂轨迹多值性的根源。少

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