Ishikawa Akiyuki, Tanaka Atushi, Ikeda Kensuke S, Shudo Akira
Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Sep;86(3 Pt 2):036208. doi: 10.1103/PhysRevE.86.036208. Epub 2012 Sep 13.
The role of diffraction is investigated for two-dimensional area-preserving maps with sharply or almost sharply divided phase space, in relation to the issue of dynamical tunneling. The diffraction effect is known to appear in general when the system contains indifferentiable or discontinuous points. We find that it controls the quantum transition between regular and chaotic regions in mixed phase space in the case where the border between these regions is set to be sharp. However, its manifestation is rather subtle: it would be possible to identify the diffraction effect under suitable coordinates if the support of the wave function contains indifferentiable or discontinuous points, whereas it is mixed with the tunneling effect and the whole process becomes hybrid if the support does not contain the sources of diffraction. We make detailed analyses, including the semiclassical treatment of edge contributions of the one-step propagator, to clarify the nature of diffraction in mixed phase space. Our result implies that chaos does not play any roles in the regular-to-chaotic transition process when the phase space is sharply divided.
针对具有急剧或几乎急剧划分相空间的二维面积守恒映射，研究了衍射在动力学隧穿问题中的作用。已知当系统包含不可微或不连续点时，一般会出现衍射效应。我们发现，在这些区域之间的边界设定为清晰的情况下，它控制着混合相空间中规则区域和混沌区域之间的量子跃迁。然而，它的表现相当微妙：如果波函数的支集包含不可微或不连续点，那么在合适的坐标下有可能识别出衍射效应，而如果支集不包含衍射源，它就会与隧穿效应混合，整个过程就会变成混合的。我们进行了详细分析，包括对单步传播子边缘贡献的半经典处理，以阐明混合相空间中衍射的本质。我们的结果表明，当相空间急剧划分时，混沌在规则到混沌的跃迁过程中不起任何作用。
