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Hanada Y, Ikeda KS, Shudo A. Dynamical tunneling across the separatrix. Phys Rev E 2023; 108:064210. [PMID: 38243542 DOI: 10.1103/physreve.108.064210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2023] [Accepted: 11/20/2023] [Indexed: 01/21/2024]
Abstract
The strong enhancement of tunneling couplings typically observed in tunneling splittings in the quantum map is investigated. We show that the transition from instanton to noninstanton tunneling, which is known to occur in tunneling splittings in the space of the inverse Planck constant, takes place in a parameter space as well. By applying the absorbing perturbation technique, we find that the enhancement invoked as a result of local avoided crossings and that originating from globally spread interactions over many states should be distinguished and that the latter is responsible for the strong and persistent enhancement. We also provide evidence showing that the coupling across the separatrix in phase space is crucial in explaining the behavior of tunneling splittings by performing the wave-function-based observation. In the light of these findings, we examine the validity of the resonance-assisted tunneling theory.
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Affiliation(s)
- Yasutaka Hanada
- Department of Information Science, Showa University, Yamanashi 403-0005, Japan and Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
| | - Kensuke S Ikeda
- Department of Physics, Ritsumeikan University, Kusatsu, Shiga 525-0577, Japan
| | - Akira Shudo
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
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Koda R, Hanada Y, Shudo A. Ergodicity of complex dynamics and quantum tunneling in nonintegrable systems. Phys Rev E 2023; 108:054219. [PMID: 38115491 DOI: 10.1103/physreve.108.054219] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2023] [Accepted: 10/11/2023] [Indexed: 12/21/2023]
Abstract
The instanton approximation is a widely used approach to construct the semiclassical theory of tunneling. The instanton path bridges the regions that are not connected by classical dynamics, but the connection can be achieved only if the two regions have the same energy. This is a major obstacle when applying the instanton method to nonintegrable systems. Here we show that the ergodicity of complex orbits in the Julia set ensures the connection between arbitrary regions and thus provides an alternative to the instanton path in the nonintegrable system. This fact is verified using the ultra-near integrable system in which none of the visible structures inherent in nonintegrability exist in the classical phase space, yet nonmonotonic tunneling tails emerge in the corresponding wave functions. The simplicity of the complex phase space allows us to explore the origin of the nontrivial tunneling tails in terms of semiclassical analysis in the time domain. In particular, it is shown that not only the imaginary part but also the real part of the classical action plays a role in creating the characteristic step structure of the tunneling tail that appears as a result of the quantum resonance.
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Affiliation(s)
- Ryonosuke Koda
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
| | - Yasutaka Hanada
- Department of Information Science, Faculty of Arts and Sciences, Showa University, Yamanashi 403-0005, Japan
| | - Akira Shudo
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
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Iijima R, Koda R, Hanada Y, Shudo A. Quantum tunneling in ultra-near-integrable systems. Phys Rev E 2022; 106:064205. [PMID: 36671098 DOI: 10.1103/physreve.106.064205] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2022] [Accepted: 11/16/2022] [Indexed: 06/17/2023]
Abstract
We study the tunneling tail of eigenfunctions of the quantum map using arbitrary precision arithmetic and find that nonmonotonic decaying tails accompanied by step structures appear even when the corresponding classical system is extremely close to the integrable limit. Using the integrable basis constructed with the Baker-Campbell-Hausdorff (BCH) formula, we clarify that the observed structure emerges due to the coupling with excited states via the quantum resonance mechanism. Further calculations reveal that the step structure gives stretched exponential decay as a function of the inverse Planck constant, which is not expected to appear in normal tunneling processes.
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Affiliation(s)
- Riku Iijima
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
| | - Ryonosuke Koda
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
| | - Yasutaka Hanada
- Department of Information Science, Faculty of Arts and Sciences, Showa University, Yamanashi 403-0005, Japan
| | - Akira Shudo
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
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Hanada Y, Shudo A, Okushima T, Ikeda KS. Renormalized perturbation approach to instanton-noninstanton transition in nearly integrable tunneling processes. Phys Rev E 2019; 99:052201. [PMID: 31212559 DOI: 10.1103/physreve.99.052201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2018] [Indexed: 11/07/2022]
Abstract
A renormalized perturbation method is developed for quantum maps of periodically kicked rotor models to study the tunneling effect in the nearly integrable regime. Integrable Hamiltonians closely approximating the nonintegrable quantum map are systematically generated by the Baker-Hausdorff-Campbell (BHC) expansion for symmetrized quantum maps. The procedure results in an effective integrable renormalization, and the unrenormalized residual part is treated as the perturbation. If a sufficiently high-order BHC expansion is used as the base of perturbation theory, the lowest order perturbation well reproduces tunneling characteristics of the quasibound eigenstates, including the transition from the instanton tunneling to a noninstanton one. This approach enables a comprehensive understanding of the purely quantum mechanisms of tunneling in the nearly integrable regime. In particular, the staircase structure of tunneling probability dependence on quantum number can be clearly explained as the successive transition among multiquanta excitation processes. The transition matrix elements of the residual interaction have resonantly enhanced invariant components that are not removed by the renormalization. Eigenmodes coupled via these invariant components form noninstanton (NI) tunneling channels of two types contributing to the two regions of each step of the staircase structure: one type of channel is inside the separatrix, and the other goes across the separatrix. The amplitude of NI tunneling across the separatrix is insensitive to the Planck constant but shows an essentially singular dependence upon the nonintegrablity parameter. Its relation to the Melnikov integral, which characterizes the scale of classical chaos emerging close to the saddle on the potential top, is discussed.
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Affiliation(s)
- Yasutaka Hanada
- Department of Electrical Engineering, Faculty of Science and Engineering, Kyushu Sangyo University, 3-1-2 Matsukadai, Higashi-ku, Fukuoka 813-8503, Japan
| | - Akira Shudo
- Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
| | - Teruaki Okushima
- Science and Technology Section, General Education Division, College of Engineering, Chubu University, Matsumoto-cho, Kasugai, Aichi 487-8501, Japan
| | - Kensuke S Ikeda
- College of Science and Engineering, Ritsumeikan University Noji-higashi 1-1-1, Kusatsu 525, Japan
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Takahashi K, Ikeda KS. Instanton and noninstanton tunneling in periodically perturbed barriers: semiclassical and quantum interpretations. Phys Rev E 2012; 86:056206. [PMID: 23214856 DOI: 10.1103/physreve.86.056206] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2012] [Indexed: 11/07/2022]
Abstract
In multidimensional barrier tunneling, there exist two different types of tunneling mechanisms, instanton-type tunneling and noninstanton tunneling. In this paper we investigate transitions between the two tunneling mechanisms from the semiclassical and quantum viewpoints taking two simple models: a periodically perturbed Eckart barrier for the semiclassical analysis and a periodically perturbed rectangular barrier for the quantum analysis. As a result, similar transitions are observed with change of the perturbation frequency ω for both systems, and we obtain a comprehensive scenario from both semiclassical and quantum viewpoints for them. In the middle range of ω, in which the plateau spectrum is observed, noninstanton tunneling dominates the tunneling process, and the tunneling amplitude takes the maximum value. Noninstanton tunneling explained by stable-unstable manifold guided tunneling (SUMGT) from the semiclassical viewpoint is interpreted as multiphoton-assisted tunneling from the quantum viewpoint. However, in the limit ω→0, instanton-type tunneling takes the place of noninstanton tunneling, and the tunneling amplitude converges on a constant value depending on the perturbation strength. The spectrum localized around the input energy is observed, and there is a scaling law with respect to the width of the spectrum envelope, i.e., the width ∝ℏω. In the limit ω→∞, the tunneling amplitude converges on that of the unperturbed system, i.e., the instanton of the unperturbed system.
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Affiliation(s)
- Kin'ya Takahashi
- The Physics Laboratories, Kyushu Institute of Technology, Kawazu 680-4, Iizuka 820-8502, Japan
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Ishikawa A, Tanaka A, Ikeda KS, Shudo A. Diffraction and tunneling in systems with mixed phase space. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:036208. [PMID: 23030998 DOI: 10.1103/physreve.86.036208] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2012] [Indexed: 06/01/2023]
Abstract
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.
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Affiliation(s)
- Akiyuki Ishikawa
- Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
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Takahashi K, Ikeda KS. Non-instanton tunneling: semiclassical and quantum interpretations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:026203. [PMID: 21929077 DOI: 10.1103/physreve.84.026203] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2011] [Indexed: 05/31/2023]
Abstract
Tunneling essentially different from instanton-type tunneling, say noninstanton tunneling, is studied from both semiclassical and quantum viewpoints. Taking a periodically perturbed rounded-off step potential for which the instanton-type tunneling is substantially prohibited, we analyze change of the tunneling probability with change of the perturbation frequency based on the stable-unstable manifold-guided tunneling (SUMGT) theory, which we have recently introduced. In the large and small limits of the frequency, the tunneling rate rapidly decays, but it is markedly enhanced in an intermediate range. We will also make a quantum interpretation of the noninstanton tunneling by using an exactly solvable model--a periodically perturbed right-angled step potential. Analysis with this model shows that SUMGT is considered as a sort of photoassisted tunneling through a large energy gap induced with absorbing a huge number quanta, which is completely different from the instanton-type tunneling. Both approaches from the semiclassical and quantum viewpoints complement each other to cause a better understanding of noninstanton tunneling.
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Affiliation(s)
- Kin'ya Takahashi
- The Physics Laboratories, Kyushu Institute of Technology, Kawazu 680-4, Iizuka 820-8502, Japan
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Ishikawa A, Tanaka A, Shudo A. Dynamical tunneling in many-dimensional chaotic systems. PHYSICAL REVIEW LETTERS 2010; 104:224102. [PMID: 20867174 DOI: 10.1103/physrevlett.104.224102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/15/2009] [Indexed: 05/29/2023]
Abstract
We investigate dynamical tunneling in many-dimensional systems using a quasiperiodically modulated kicked rotor, and find that the tunneling rate from the torus to the chaotic region is drastically enhanced when the chaotic states become delocalized as a result of the Anderson transition. This result strongly suggests that amphibious states, which were discovered for a one-dimensional kicked rotor with transporting islands [L. Hufnagel, Phys. Rev. Lett. 89, 154101 (2002)], quite commonly appear in many-dimensional systems.
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Affiliation(s)
- Akiyuki Ishikawa
- Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
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Ishikawa A, Tanaka A, Shudo A. Recovery of chaotic tunneling due to destruction of dynamical localization by external noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:046204. [PMID: 19905412 DOI: 10.1103/physreve.80.046204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2009] [Indexed: 05/28/2023]
Abstract
Quantum tunneling in the presence of chaos is analyzed, focusing especially on the interplay between quantum tunneling and dynamical localization. We observed flooding of potentially existing tunneling amplitude by adding noise to the chaotic sea to attenuate the destructive interference generating dynamical localization. This phenomenon is related to the nature of complex orbits describing tunneling between torus and chaotic regions. The tunneling rate is found to obey a perturbative scaling with noise intensity when the noise intensity is sufficiently small and then saturate in a large noise intensity regime. A relation between the tunneling rate and the localization length of the chaotic states is also demonstrated. It is shown that due to the competition between dynamical tunneling and dynamical localization, the tunneling rate is not a monotonically increasing function of Planck's constant. The above results are obtained for a system with a sharp border between torus and chaotic regions. The validity of the results for a system with a smoothed border is also explained.
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Affiliation(s)
- Akiyuki Ishikawa
- Department of Physics, Tokyo Metropolitan University, Tokyo, Japan
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Levkov DG, Panin AG, Sibiryakov SM. Complex trajectories in chaotic dynamical tunneling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:046209. [PMID: 17995084 DOI: 10.1103/physreve.76.046209] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2007] [Revised: 07/27/2007] [Indexed: 05/25/2023]
Abstract
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.
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Affiliation(s)
- D G Levkov
- Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary prospect 7a, Moscow 117312, Russia.
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Takahashi K, Ikeda KS. Anomalously long passage through a rounded-off-step potential due to a new mechanism of multidimensional tunneling. PHYSICAL REVIEW LETTERS 2006; 97:240403. [PMID: 17280258 DOI: 10.1103/physrevlett.97.240403] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/03/2006] [Indexed: 05/13/2023]
Abstract
The fully complex domain semiclassical theory based upon the complexified stable-unstable manifold theory, which we have developed in our recent studies, is successfully applied to explain anomalous tunneling phenomena numerically observed in a periodically modulated round-off-step potential. Numerical experiments show that tunneling through the oscillating step potential is characterized by a spatially nondecaying tunneling tail and an anomalously slow relaxation. The key is the existence of a critical trajectory exhibiting singular behavior, and the analysis of neighboring trajectories around it reproduces the essence of such anomalous phenomena.
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Affiliation(s)
- Kin'ya Takahashi
- The Physics Laboratories, Kyushu Institute of Technology, Kawazu 680-4, Iizuka 820-8502, Japan
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Mouchet A, Eltschka C, Schlagheck P. Influence of classical resonances on chaotic tunneling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:026211. [PMID: 17025529 DOI: 10.1103/physreve.74.026211] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2006] [Indexed: 05/12/2023]
Abstract
Dynamical tunneling between symmetry-related stable modes is studied in the periodically driven pendulum. We present strong evidence that the tunneling process is governed by nonlinear resonances that manifest within the regular phase-space islands on which the stable modes are localized. By means of a quantitative numerical study of the corresponding Floquet problem, we identify the trace of such resonances not only in the level splittings between near-degenerate quantum states, where they lead to prominent plateau structures, but also in overlap matrix elements of the Floquet eigenstates, which reveal characteristic sequences of avoided crossings in the Floquet spectrum. The semiclassical theory of resonance-assisted tunneling yields good overall agreement with the quantum-tunneling rates, and indicates that partial barriers within the chaos might play a prominent role.
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Affiliation(s)
- Amaury Mouchet
- Laboratoire de Mathématiques et de Physique Théorique, CNRS UMR 6083, Université François Rabelais Avenue Monge, Parc de Grandmont, 37200 Tours, France
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Onishi T, Shudo A, Ikeda KS, Takahashi K. Semiclassical study on tunneling processes via complex-domain chaos. ACTA ACUST UNITED AC 2003; 68:056211. [PMID: 14682875 DOI: 10.1103/physreve.68.056211] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2002] [Revised: 05/20/2003] [Indexed: 11/07/2022]
Abstract
We investigate the semiclassical mechanism of tunneling processes in nonintegrable systems. The significant role of complex-phase-space chaos in the description of the tunneling processes is elucidated by studying a kicked scattering model. Behaviors of tunneling orbits are encoded into symbolic sequences based on the structure of a complex homoclinic tangle. By means of the symbolic coding, the phase space itineraries of tunneling orbits are related with the amounts of imaginary parts of actions gained by the orbits, so that the systematic search of dominant tunneling orbits becomes possible.
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Affiliation(s)
- T Onishi
- Department of Physics, Tokyo Metropolitan University, Minami-Ohsawa, Hachioji 192-0397, Japan.
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Takahashi KY, Ikeda KS. Complex-classical mechanism of the tunnelling process in strongly coupled 1.5-dimensional barrier systems. ACTA ACUST UNITED AC 2003. [DOI: 10.1088/0305-4470/36/29/305] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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