Multidimensional Tunnelling Effect and Complexified Classical Dynamics

系统的多维性从根本上影响着隧穿现象。特别地，如果系统是经典的不可积的，则可以观察到由于混沌集的存在而产生的复杂的隧穿现象，称为混沌隧穿。利用推广到全复域的经典动力学方法，即复半经典方法，研究了混沌隧穿的基本机制。(1)研究了一类存在混沌的量子映射系统的隧穿。大量的数值研究表明，对动态隧穿过程起主要作用的隧穿轨迹在初始流形中形成了一类非常有限的集合，根据其特征形状被称为“拉普塔链”。应用钟形动力学理论的结果，研究了这种集合的数学意义，并用数值方法阐明了它的性质。主要结果是：Laputa链的闭包是由两个集合Namel…有界的更多y，Jula集J^+和填充Julia集K^+，分别从下到上。进一步猜想K^+=J^+。如果是这样的话，Laputa链的关闭无非是Julia Set。通过动力学势垒的波函数是沿着J^-的实部构造的。这些事实意味着，主要的轨道隧穿是由密集在混沌海洋中的复杂的稳定的鞍形流形引导的，并沿着它们的不稳定流形散布。(2)将我们限制在一类势垒隧穿过程中，我们阐明了系统的多维性如何导致了一种新的普遍机制，导致了多维势垒系统特有的复杂隧穿现象。首先，我们证明了即使在隧道分量伴随着复杂条纹的强耦合区域，复半经典理论也一定能重现纯量子波矩阵。其次，证明了由复杂的稳定流形和不稳定流形引导的复杂轨迹是产生边缘隧穿效应的原因。这种机制提供了一幅与经典瞬子机制截然不同的隧穿新图景。从数学的角度，用可移动奇点的发散位移来解释这一机制，这是一维隧道问题中复杂轨迹多值的根源。较少

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