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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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2
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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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Stránský P, Šindelka M, Kloc M, Cejnar P. Complex Density of Continuum States in Resonant Quantum Tunneling. PHYSICAL REVIEW LETTERS 2020; 125:020401. [PMID: 32701313 DOI: 10.1103/physrevlett.125.020401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2020] [Accepted: 06/16/2020] [Indexed: 06/11/2023]
Abstract
We introduce a complex-extended continuum level density and apply it to one-dimensional scattering problems involving tunneling through finite-range potentials. We show that the real part of the density is proportional to a real "time shift" of the transmitted particle, while the imaginary part reflects the imaginary time of an instantonlike tunneling trajectory. We confirm these assumptions for several potentials using the complex scaling method. In particular, we show that stationary points of the potentials give rise to specific singularities of both real and imaginary densities which represent close analogues of excited-state quantum phase transitions in bound systems.
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Affiliation(s)
- Pavel Stránský
- Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 18000 Prague, Czech Republic
| | - Milan Šindelka
- Institute of Plasma Physics, Academy of Sciences of the Czech Republic, Za Slovankou 3, 18200 Prague, Czech Republic
| | - Michal Kloc
- Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 18000 Prague, Czech Republic
- Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - Pavel Cejnar
- Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 18000 Prague, Czech Republic
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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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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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Hanada Y, Shudo A, Ikeda KS. Origin of the enhancement of tunneling probability in the nearly integrable system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042913. [PMID: 25974568 DOI: 10.1103/physreve.91.042913] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2014] [Indexed: 06/04/2023]
Abstract
The enhancement of tunneling probability in the nearly integrable system is closely examined, focusing on tunneling splittings plotted as a function of the inverse of the Planck's constant. On the basis of the analysis using the absorber which efficiently suppresses the coupling, creating spikes in the plot, we found that the splitting curve should be viewed as the staircase-shaped skeleton accompanied by spikes. We further introduce renormalized integrable Hamiltonians and explore the origin of such a staircase structure by investigating the nature of eigenfunctions closely. It is found that the origin of the staircase structure could trace back to the anomalous structure in tunneling tail which manifests itself in the representation using renormalized action bases. This also explains the reason why the staircase does not appear in the completely integrable system.
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Affiliation(s)
- Yasutaka Hanada
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
| | - Akira Shudo
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
| | - Kensuke S Ikeda
- Department of Physics, Ritsumeikan University, Kusatsu, Shiga 525-0577, Japan
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Kullig J, Löbner C, Mertig N, Bäcker A, Ketzmerick R. Integrable approximation of regular regions with a nonlinear resonance chain. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:052906. [PMID: 25493857 DOI: 10.1103/physreve.90.052906] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2014] [Indexed: 06/04/2023]
Abstract
Generic Hamiltonian systems have a mixed phase space where regions of regular and chaotic motion coexist. We present a method for constructing an integrable approximation to such regular phase-space regions including a nonlinear resonance chain. This approach generalizes the recently introduced iterative canonical transformation method. In the first step of the method a normal-form Hamiltonian with a resonance chain is adapted such that actions and frequencies match with those of the nonintegrable system. In the second step a sequence of canonical transformations is applied to the integrable approximation to match the shape of regular tori. We demonstrate the method for the generic standard map at various parameters.
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Affiliation(s)
- Julius Kullig
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany and Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany and Institut für Theoretische Physik, Universität Magdeburg, Postfach 4120, 39016 Magdeburg, Germany
| | - Clemens Löbner
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany and 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 and Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany and Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji 192-0397, Japan
| | - Arnd Bäcker
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany and 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 and Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
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