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arXiv:nlin/0205048 · Journal of Physics A Mathematical and General

Julia set describes quantum tunnelling in the presence of chaos

Akira Shudo, Yutaka Ishii, Kensuke S. Ikeda

Published Apr 19, 2002• 50 citations
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Abstract

We find that characteristics of quantum tunnelling in the presence of chaos can be regarded as a manifestation of the Julia set of the complex dynamical system. Several numerical pieces of evidence for the standard map, together with a rigorous statement for the Hénon map, are presented, demonstrating that the complex classical paths which contribute to the semiclassical propagator are dense in the Julia set. Chaotic tunnelling can thus be characterized by the transitivity of the dynamics and high density of the trajectories on the Julia set.

DOI
10.1088/0305-4470/35/17/101

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We find that characteristics of quantum tunnelling in the presence of chaos can be regarded as a manifestation of the Julia set of the complex dynamical system. Several numerical pieces of evidence for the standard map, together with a rigorous statement for the Hénon map, are presented, demonstrating that the complex classical paths which contribute to the semiclassical propagator are dense in the Julia set. Chaotic tunnelling can thus be characterized by the transitivity of the dynamics and high density of the trajectories on the Julia set.
Chaotic tunneling occurring in the standard map, which is one of the most familiar paradigms in simple chaotic mappings, is analyzed in terms of complex semiclassical theory in the time domain. It is shown that the complex classical trajectories successfully describe the transition to the region where the real classical path cannot reach. In connection with our previous result (Physica D115 (1998), 234), the universality of the organizing rule to find out dominantly contributing complex paths is suggested. Several fundamental problems in the treatment of semiclassical theory in the complex domain are also discussed.
In generic Hamiltonian systems that are neither completely integrable nor fully chaotic, phase space consists of a mixture of regular and chaotic components. In classical dynamics, transitions between different invariant sets in phase space are strictly forbidden, and these sets act as dynamical barriers to one another. In quantum mechanics, in contrast, wave effects allow transitions through such dynamical barriers. This process, known as dynamical tunneling, refers to penetration through dynamical barriers in phase space and was first recognized in the early 1980s. Since then, various aspects of dynamical tunneling have been elucidated, significantly advancing our understanding of such a novel quantum phenomenon. In this article, we provide an overview of several phenomenological perspectives of dynamical tunneling, including chaos-assisted and resonance-assisted tunneling, and also introduce approaches based on classical mechanics extended into the complex domain. In particular, we seek to clarify what is meant by the common claim that "chaos leads to an enhancement of the tunneling probability", which is often made when dynamical tunneling is dressed. We discuss what regime this refers to and, if such an enhancement occurs, what its likely origin is.
We develop the semiclassical method of complex trajectories in application to chaotic dynamical tunneling. First, we suggest a systematic numerical technique for obtaining complex tunneling trajectories by the gradual deformation of the classical ones. This provides a natural classification of the tunneling solutions. Second, we present a heuristic procedure for sorting out the least suppressed trajectory. As an illustration, we apply our technique to the process of chaotic tunneling in a quantum mechanical model with two degrees of freedom. Our analysis reveals rich dynamics of the system. At the classical level, there exists an infinite set of unstable solutions forming a fractal structure. This structure is inherited by the complex tunneling paths and plays a central role in the semiclassical study. The process we consider exhibits the phenomenon of optimal tunneling: the suppression exponent of the tunneling probability has a local minimum at a certain energy which is thus (locally) the optimal energy for tunneling. We test the proposed method by comparison of the semiclassical results with the results of the exact quantum computations and find a good agreement.
The interplay between classical chaos and quantum tunneling is examined in driven nonlinear systems, with emphasis on how semi classical phase space structures influence purely quantum transport phenomena. We show that, in the presence of external driving and stochastic perturbations, tunneling rates acquire an activated form determined by effective classical barriers, providing a transparent link between chaotic dynamics and quantum tunneling. Within this framework, chaos assisted tunneling and coherent destruction of tunneling emerge as closely related manifestations of the same underlying phase space restructuring and interference effects induced by driving. The results offer a unified perspective on tunneling control in non integrable systems and remain relevant for modern studies of driven quantum dynamics and decoherence resistant transport.
We discuss and give evidence for the existence of a new mechanism for quantum dynamical tunneling. It may occur when a quantum system's underlying classical dynamics are far from integrability. In these circumstances, we show that the dominant tunneling contributions arise through chaos-assisted processes. This leads to behavior that is drastically different from that found in integrable and quasi-integrable systems. In particular, one can observe a marked crossing mechanism when a chaotic level passes nearby the tunneling ones, and the distributions of splitting (due to tunneling) can be modeled using properly designed ensembles of random matrices. Such tunneling should be amenable to experimental detection.