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Multidimensional Tunnelling Effect and Complexified Classical Dynamics

Research Project

Project/Area Number 13640410
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 物理学一般
Research InstitutionRitsumeikan University

Principal Investigator

IKEDA Kensuke  Ritsumeikan Univ, Department of Physics, Professor, 理工学部, 教授 (40151287)

Co-Investigator(Kenkyū-buntansha) TAKAHASHI Kin'ya  Kyushu Inst. of Technology, Physics Laboratories, Associate Professor, 情報工学部, 助教授 (70188001)
SHUDO Akira  Tokyo Metropolitan Univ, Department of Physics, Associate Professor, 理学研究科, 助教授 (60206258)
ISHII Yutaka  Kyushu Univ., Dept of Mathematics, Research Associate, 大学院・数理学研究院, 助手 (20304727)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥4,000,000 (Direct Cost: ¥4,000,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2002: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2001: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordstunnelling effect / chaos / complex dynamical systems / semiclassical mechanics / multi-dimensional tunnelling / chaotic tunnelling / 多自由度トンネル効果 / 半古典理論 / 複素領域半古典理論 / ストークス問題 / ジュリア集合
Research Abstract

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

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (22 results)

All 2005 2003 2002 2001 Other

All Journal Article (12 results) Publications (10 results)

  • [Journal Article] Classical mechanism of multidimensional barrier tunneling2005

    • Author(s)
      K.Takahashi, K.S.Ikeda
    • Journal Title

      Adv.Chem.Phys Vol.130

      Pages: 401-434

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Classical mechanism of multidimensional barrier tunneling2005

    • Author(s)
      K.Takahashi, K.S.Ikeda
    • Journal Title

      Adv.Chem.Phys. Vol.130

      Pages: 401-434

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Complex-classical mechanism of tunneling process in strongly coupled 1.5 dimensional barrier systems2003

    • Author(s)
      K.Takahashi, K.S.Ikeda
    • Journal Title

      J.Phys. A36

      Pages: 7953-7987

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Complex-domain semiclassical theory and periodically perturbed barrier tunneling problems2003

    • Author(s)
      K.Takahashi, K.S.Ikeda
    • Journal Title

      Supplement C to J.Phys.Soc.Japan 72

      Pages: 87-96

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Semiclassical study on tunnelling processes via complex domain chaos2003

    • Author(s)
      T.Onishi, A.Shudo, K.S.Ikeda, K.Takahashi
    • Journal Title

      Phys.Rev.E 68

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Complex-classical mechanism of tunneling process in strongly coupled 1.5 dimensional barrier systems2003

    • Author(s)
      K.Takahashi, K.S.Ikeda
    • Journal Title

      J.Phys.A 36

      Pages: 7953-7987

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Complex-domain semiclassical theory and periodically perturbed barrier tunneling problems2003

    • Author(s)
      K.Takahashi, K.S.Ikeda
    • Journal Title

      Supplement C to J. Phys. Soc. Japan 72

      Pages: 87-96

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Julia set describes quantum tunnelling in the presence of chaos2002

    • Author(s)
      A.Shudo, Y.Ishii, K.S.Ikeda
    • Journal Title

      J.Phys. A35

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Julia set describes quantum tunnelling in the presence of chaos2002

    • Author(s)
      A.Shudo, Y.Ishii, K.S.Ikeda
    • Journal Title

      J.Phys.A 35

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Complex domain semiclassical theory : application to time-dependent barrier tunneling problem2001

    • Author(s)
      K.Takahashi, K.S.Ikeda
    • Journal Title

      Foundations of Physics Vol.31

      Pages: 177-201

    • NAID

      120002440694

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Classical mechanism of multidimensional barrier tunneling

    • Author(s)
      K.Takahashi, K.S.Ikeda
    • Journal Title

      Adv.Chem.Phys. (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] An intrinsic multi-dimensional mechanism of barrier tunnelling

    • Author(s)
      K.Takahashi, K.S.Ikeda
    • Journal Title

      Euro.Phys.Lett. (出版予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] A.Shudo, T.Onishi, K.S.Ikeda, K.Takahashi: "Dynamical and Energetic Barrier Tunneling in the Presence of Chaos in Complex Phase Space"Proc.Japan-Italy Joint Workshop on Quantum Open Systems (World Scientific, Singapore). 157-176 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Takahashi, K.S.Ikeda: "Complex-classical mechanism of tunneling process in strongly coupled 1.5 dimensional barrier systems"J.Phys.A. 36. 7953-7987 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Takahashi, K.S.Ikeda: "Complex-Domain Semiclassical Theory and Periodically Perturbed Barrier Tunneling Problems"Supplement C to J.Phys.Soc.Jpn.. 72. 87-96 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Onishi, A.Shudo, K.S.Ikeda, K.Takahashi: "Tunneling mechanism due to chaos in a complex phase space"Phys.Rev.E. 63. 025202-1-025202-4 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Shudo, K.S.Ikeda: "Quantum chaos in mixed phase space and the Julia set"Prog.Theor.Phys.Suppl.. 150. 267-280 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Shudo, Y.Ishii, K.S.Ikeda: "Julia set describes tunnelling in the presence of chaos"Journal of Physics A. 35. L225-L231 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Takahashi, K.S.Ikeda: "Movable singularities, complex-domain heteroclinicity, and fringed tunneling in multi-dimensional systems"Physics Letters A. 297. 370-375 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Takahashi, K.S.Ikeda: "Complex Domain Semiclassical Theory: Application to Time-dependent Barrier Tunneling Problem"Foundation of Physics. 31. 177-201 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Onishi, A.Shudo, K.S.Ikeda, K.Takahashi: "Tunneling mechanism due to chaos in complex phase space"Phys. Rev. E. 64. 025201-1-025201-4 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Takahashi, A.Yoshimoto, K.S.Ikeda: "Movable singularities, complex-domain heteroclinicity and fringed tunneling in multi-dimensional systems"Phys. Lett. A. (印刷中).

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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