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Hanada Y, Shudo A. Quantum Tunneling and Complex Dynamics in the Suris's Integrable Map. ENTROPY (BASEL, SWITZERLAND) 2024; 26:414. [PMID: 38785663 PMCID: PMC11119662 DOI: 10.3390/e26050414] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/23/2024] [Revised: 05/01/2024] [Accepted: 05/08/2024] [Indexed: 05/25/2024]
Abstract
Quantum tunneling in a two-dimensional integrable map is studied. The orbits of the map are all confined to the curves specified by the one-dimensional Hamiltonian. It is found that the behavior of tunneling splitting for the integrable map and the associated Hamiltonian system is qualitatively the same, with only a slight difference in magnitude. However, the tunneling tails of the wave functions, obtained by superposing the eigenfunctions that form the doublet, exhibit significant differences. To explore the origin of the difference, we observe the classical dynamics in the complex plane and find that the existence of branch points appearing in the potential function of the integrable map could play the role of yielding non-trivial behavior in the tunneling tail. The result highlights the subtlety of quantum tunneling, which cannot be captured in nature only by the dynamics in the real plane.
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Affiliation(s)
- Yasutaka Hanada
- Department of Information Science, Faculty of Arts and Sciences, Showa University, Yamanashi 403-0005, Japan
- Department of Physics, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan;
| | - Akira Shudo
- Department of Physics, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan;
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2
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Takahashi K, Ikeda KS. Sawtooth structure in tunneling probability for a periodically perturbed rounded-rectangular potential. Phys Rev E 2024; 109:044203. [PMID: 38755861 DOI: 10.1103/physreve.109.044203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/02/2023] [Accepted: 03/06/2024] [Indexed: 05/18/2024]
Abstract
Sawtooth structures are observed in tunneling probabilities with changing Planck's constant for a periodically perturbed rounded-rectangular potential with a sufficiently wide width for which instanton tunneling is substantially prohibited. The sawtooth structure is a manifestation of the essential nature of multiquanta absorption tunneling. Namely, the periodic perturbation creates an energy ladder of harmonic channels at E_{n}=E_{I}+nℏω, where E_{I} is an incident energy and ω is an angular frequency of the perturbation. The harmonic channel that absorbs the minimum amount of quanta of n=n[over ¯], such that V_{0}
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Affiliation(s)
- Kin'ya Takahashi
- Department of Physics and Information Technology, Kyushu Institute of Technology, Kawazu 680-4, Iizuka 820-8502, Japan; Research Institute for Information Technology, Kyushu University, 744 Motooka Nishi-ku, Fukuoka 819-0395, Japan; and AcsiomA Ltd, 3-8-33 Momochihama Sawara-ku, Fukuoka 814-0001, Japan
| | - Kensuke S Ikeda
- Department of Physical Sciences, Ritsumeikan University, Noji-higashi 1-1-1, Kusatsu 525-8577, Japan
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3
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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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Firmbach M, Fritzsch F, Ketzmerick R, Bäcker A. Resonance-assisted tunneling in four-dimensional normal-form Hamiltonians. Phys Rev E 2019; 99:042213. [PMID: 31108719 DOI: 10.1103/physreve.99.042213] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2019] [Indexed: 11/07/2022]
Abstract
Nonlinear resonances in the classical phase space lead to a significant enhancement of tunneling. We demonstrate that the double resonance gives rise to a complicated tunneling peak structure. Such double resonances occur in Hamiltonian systems with an at least four-dimensional phase space. To explain the tunneling peak structure, we use the universal description of single and double resonances by the four-dimensional normal-form Hamiltonians. By applying perturbative methods, we reveal the underlying mechanism of enhancement and suppression of tunneling and obtain excellent quantitative agreement. Using a minimal matrix model, we obtain an intuitive understanding.
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Affiliation(s)
- Markus Firmbach
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany.,Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Felix Fritzsch
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - Roland Ketzmerick
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany.,Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Arnd Bäcker
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany.,Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
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Zamastil J, Šimsa D. Quantum effects and quantum chaos in multidimensional tunneling. Phys Rev E 2017; 96:062201. [PMID: 29347439 DOI: 10.1103/physreve.96.062201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/12/2017] [Indexed: 11/07/2022]
Abstract
The ground-state energy splitting due to tunneling in two-dimensional double wells of the form V(x,y)=(x^{2}-R^{2})^{2}/8R^{2}+x^{2}-R^{2}/R^{2}γy+ω^{2}/2y^{2} is calculated. Several results are reported. First, we give a systematic WKB expansion of the splitting in series in powers of R^{-2}, each term of the series being a finite polynomial in γ^{2}. We find an ascending sequence of the values of the parameter γ characterizing the curvature of the classical path, for which the successive corrections to the leading order vanish. This effect arises because curvature of the path and quantum nature of motion cancel each other; it does not appear for one-dimensional double wells. Second, we find that for large curvatures, such as for those describing the proton transfer in a malonaldehyde and hydroxalate anion, this expansion is of no practical use. Thus, the WKB expansion is reordered to a strong coupling form, each term of the series in powers of R^{-2} being an infinite series in powers of γ[over ¯]^{2}, γ[over ¯]=γ/R. Third, we find that the radius of convergence of the series is determined by the singularity at γ[over ¯]_{s}=ω/2. At the singularity the system changes its character from being a double well to become a single well. Close to this singularity the classical action and its first quantum correction are found to be nonanalytic functions of γ[over ¯], most likely of the form [1-(γ[over ¯]/γ[over ¯]_{s})^{2}]^{α}, where α=1/2 and α=-1/2 for the classical action and its first quantum correction, respectively. Since in the semiclassical regime of large R the splitting is exponentially dependent on the value of the classical action and its first quantum correction, close to the singularity we establish strong sensitivity of the splitting on slight variations of the parameter γ[over ¯] entering the Hamiltonian linearly.
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Affiliation(s)
- J Zamastil
- Department of Chemical Physics and Optics, Charles University, Faculty of Mathematics and Physics, Ke Karlovu 3, 121 16 Prague 2, Czech Republic
| | - D Šimsa
- Department of Chemical Physics and Optics, Charles University, Faculty of Mathematics and Physics, Ke Karlovu 3, 121 16 Prague 2, Czech Republic
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Fritzsch F, Bäcker A, Ketzmerick R, Mertig N. Complex-path prediction of resonance-assisted tunneling in mixed systems. Phys Rev E 2017; 95:020202. [PMID: 28297952 DOI: 10.1103/physreve.95.020202] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2016] [Indexed: 06/06/2023]
Abstract
We present a semiclassical prediction of regular-to-chaotic tunneling in systems with a mixed phase space, including the effect of a nonlinear resonance chain. We identify complex paths for direct and resonance-assisted tunneling in the phase space of an integrable approximation with one nonlinear resonance chain. We evaluate the resonance-assisted contribution analytically and give a prediction based on just a few properties of the classical phase space. For the standard map excellent agreement with numerically determined tunneling rates is observed. The results should similarly apply to ionization rates and quality factors.
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Affiliation(s)
- Felix Fritzsch
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Arnd Bäcker
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Roland Ketzmerick
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Normann Mertig
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
- Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji 192-0397, Japan
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Mertig N, Kullig J, Löbner C, Bäcker A, Ketzmerick R. Perturbation-free prediction of resonance-assisted tunneling in mixed regular-chaotic systems. Phys Rev E 2016; 94:062220. [PMID: 28085465 DOI: 10.1103/physreve.94.062220] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2016] [Indexed: 06/06/2023]
Abstract
For generic Hamiltonian systems we derive predictions for dynamical tunneling from regular to chaotic phase-space regions. In contrast to previous approaches, we account for the resonance-assisted enhancement of regular-to-chaotic tunneling in a nonperturbative way. This provides the foundation for future semiclassical complex-path evaluations of resonance-assisted regular-to-chaotic tunneling. Our approach is based on a new class of integrable approximations which mimic the regular phase-space region and its dominant nonlinear resonance chain in a mixed regular-chaotic system. We illustrate the method for the standard map.
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Affiliation(s)
- Normann Mertig
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
- Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji 192-0397, Japan
| | - Julius Kullig
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
- Institut für Theoretische Physik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, 39016 Magdeburg, Germany
| | - Clemens Löbner
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Arnd Bäcker
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Roland Ketzmerick
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
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Wisniacki DA, Schlagheck P. Quantum manifestations of classical nonlinear resonances. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062923. [PMID: 26764790 DOI: 10.1103/physreve.92.062923] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2015] [Indexed: 06/05/2023]
Abstract
When an integrable classical system is perturbed, nonlinear resonances are born, grow, and eventually disappear due to chaos. In this paper the quantum manifestations of such a transition are studied in the standard map. We show that nonlinear resonances act as a perturbation that break eigenphase degeneracies for unperturbed states with quantum numbers that differ in a multiple of the order of the resonance. We show that the eigenphase splittings are well described by a semiclassical expression based on an integrable approximation of the Hamiltonian in the vicinity of the resonance. The morphology in phase space of these states is also studied. We show that the nonlinear resonance imprints a systematic influence in their localization properties.
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Affiliation(s)
- Diego A Wisniacki
- Departamento de Física and IFIBA, FCEyN, UBA Ciudad Universitaria, Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina
| | - Peter Schlagheck
- Departement de Physique, University of Liege, 4000 Liège, Belgium
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