Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Progress of Theoretical Physics Supplement

Complex Trajectory Description for Chaotic Tunneling

Akira Shudo, Kensuke S. Ikeda

Published Jan 1, 2000• 5 citations
Publisher

Abstract

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.

DOI
10.1143/ptps.139.246

Related papers

Julia set describes quantum tunnelling in the presence of chaos

Akira Shudo, Yutaka Ishii, Kensuke S. Ikeda

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.

arXiv:nlin/0205048•Journal of Physics A Mathematical and General•Apr 19, 2002•50 citations

Complex trajectories in chaotic dynamical tunneling

D. G. Levkov, A. G. Panin, Sergey M. Sibiryakov

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.

arXiv:nlin/0701063•Physical Review E•Oct 9, 2007•21 citations

Complex-Domain Semiclassical Theory and Periodically Perturbed Barrier Tunneling Problems

Kin’ya Takahashi, Kensuke S. Ikeda

In recent papers we demonstrated a new class of barrier tunneling phenomena which can be observed in periodically driven one dimensional barrier systems (i.e., 1.5D barrier tunneling systems). In such systems the tunneling component is in general accompanied by remarkable interference fringe patterns as the perturbation strength exceeds a certain characteristic value. Such a phenomenon is quite general and is not restricted to periodically driven 1D systems, and it may be also observed in intrinsic two dimensional autonomous systems. The complex-domain semiclassical theory developed by ourselves was applied to the problem, and we succeeded in reconstructing every detail of the fringed tunneling wavefunction only by using the complexified classical tunneling trajectories. This fact greatly motivates us to investigate the mechanism of the tunneling in terms of “classical mechanics”. The later part of the present paper is devoted to asserting the generality of the “classical mechanism” of the fringed tunneling. We develop detailed semiclassical analyses in the complex domain and demonstrate that the key trajectories contributing to the fringed tunneling are quite different from the instanton-like trajectories, which works only in unperturbed or weakly-perturbed regimes. Indeed, such trajectories are traveling in the complex phase space being guided by the complexified stable and unstable manifolds which exhibit heteroclinic-like entanglement with the incoming wave manifolds.

Journal of the Physical Society of Japan•Jan 1, 2003•0 citations

On quantum tunneling in real time

Neil Turok

A detailed real time description of quantum tunneling in the semiclassical limit is given, using complex classical trajectories. This picture connects naturally with the ideas of post-selection and weak measurement introduced by Aharonov and collaborators. I show that one can precisely identify the complex classical trajectory which a post-selected tunneling particle has followed, and which dominates the path integral in the limit as Planck's constant tends to zero. Detailed analytical calculations are presented for tunneling in cubic and quartic potentials. For a long post-selected tunneling time, the imaginary part of the tunneling coordinate is found to achieve very large values just before the particle tunnels. I discuss how the real and imaginary parts of the particle's coordinate may, in principle, be independently measured using weak measurements. It would be very interesting to observe this effect, which would demonstrate the essential role of complex numbers in our closest possible classical description of reality. Extensions to quantum field theory and general relativity are briefly discussed.

arXiv:1312.1772•New Journal of Physics•Jun 5, 2014•41 citations

Coherent Classical-Path Description of Deep Tunneling

Dong Hui Zhang, Eli Pollak

A central challenge to the semiclassical description of quantum mechanics is the quantum phenomenon of "deep" tunneling. Here we show that real time classical trajectories suffice to account correctly even for deep quantum tunneling, using a recently formulated semiclassical initial value representation series of the quantum propagator and a prefactor free semiclassical propagator. Deep quantum tunneling is effected through what we term as coherent classical paths which are composed of one or more classical trajectories that lead from reactant to product but are discontinuous along the way. The end and initial phase space points of consecutive classical trajectories contributing to the coherent path are close to each other in the sense that the distance between them is weighted by a coherent state overlap matrix element. Results are presented for thermal and energy dependent tunneling through a symmetric Eckart barrier.

Physical Review Letters•Sep 27, 2004•31 citations

The Role of Interference in the Semiclassical Approximation to Chaotic Motion

Eric J. Heller

Interference of distinct contributions to a given amplitude is a familiar aspect of quantum mechanics. Semiclassically, such interference arises from distinct classical paths connecting the appropriate initial and final points. In chaotic systems the number of distinct contributions grows very rapidly with the length or the duration of the trajectory. After a short time hundreds or millions of distinct classical paths are contributing. We discuss the accumulation of errors and the sometimes surprising result of many interfering contributions.

Progress of Theoretical Physics Supplement•Jan 1, 1994•3 citations
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.
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.
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.
In recent papers we demonstrated a new class of barrier tunneling phenomena which can be observed in periodically driven one dimensional barrier systems (i.e., 1.5D barrier tunneling systems). In such systems the tunneling component is in general accompanied by remarkable interference fringe patterns as the perturbation strength exceeds a certain characteristic value. Such a phenomenon is quite general and is not restricted to periodically driven 1D systems, and it may be also observed in intrinsic two dimensional autonomous systems. The complex-domain semiclassical theory developed by ourselves was applied to the problem, and we succeeded in reconstructing every detail of the fringed tunneling wavefunction only by using the complexified classical tunneling trajectories. This fact greatly motivates us to investigate the mechanism of the tunneling in terms of “classical mechanics”. The later part of the present paper is devoted to asserting the generality of the “classical mechanism” of the fringed tunneling. We develop detailed semiclassical analyses in the complex domain and demonstrate that the key trajectories contributing to the fringed tunneling are quite different from the instanton-like trajectories, which works only in unperturbed or weakly-perturbed regimes. Indeed, such trajectories are traveling in the complex phase space being guided by the complexified stable and unstable manifolds which exhibit heteroclinic-like entanglement with the incoming wave manifolds.
A detailed real time description of quantum tunneling in the semiclassical limit is given, using complex classical trajectories. This picture connects naturally with the ideas of post-selection and weak measurement introduced by Aharonov and collaborators. I show that one can precisely identify the complex classical trajectory which a post-selected tunneling particle has followed, and which dominates the path integral in the limit as Planck's constant tends to zero. Detailed analytical calculations are presented for tunneling in cubic and quartic potentials. For a long post-selected tunneling time, the imaginary part of the tunneling coordinate is found to achieve very large values just before the particle tunnels. I discuss how the real and imaginary parts of the particle's coordinate may, in principle, be independently measured using weak measurements. It would be very interesting to observe this effect, which would demonstrate the essential role of complex numbers in our closest possible classical description of reality. Extensions to quantum field theory and general relativity are briefly discussed.
A central challenge to the semiclassical description of quantum mechanics is the quantum phenomenon of "deep" tunneling. Here we show that real time classical trajectories suffice to account correctly even for deep quantum tunneling, using a recently formulated semiclassical initial value representation series of the quantum propagator and a prefactor free semiclassical propagator. Deep quantum tunneling is effected through what we term as coherent classical paths which are composed of one or more classical trajectories that lead from reactant to product but are discontinuous along the way. The end and initial phase space points of consecutive classical trajectories contributing to the coherent path are close to each other in the sense that the distance between them is weighted by a coherent state overlap matrix element. Results are presented for thermal and energy dependent tunneling through a symmetric Eckart barrier.
Interference of distinct contributions to a given amplitude is a familiar aspect of quantum mechanics. Semiclassically, such interference arises from distinct classical paths connecting the appropriate initial and final points. In chaotic systems the number of distinct contributions grows very rapidly with the length or the duration of the trajectory. After a short time hundreds or millions of distinct classical paths are contributing. We discuss the accumulation of errors and the sometimes surprising result of many interfering contributions.