1
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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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2
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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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3
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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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4
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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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5
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Arnal M, Chatelain G, Martinez M, Dupont N, Giraud O, Ullmo D, Georgeot B, Lemarié G, Billy J, Guéry-Odelin D. Chaos-assisted tunneling resonances in a synthetic Floquet superlattice. SCIENCE ADVANCES 2020; 6:6/38/eabc4886. [PMID: 32948592 PMCID: PMC7500923 DOI: 10.1126/sciadv.abc4886] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2020] [Accepted: 07/29/2020] [Indexed: 05/22/2023]
Abstract
The field of quantum simulation, which aims at using a tunable quantum system to simulate another, has been developing fast in the past years as an alternative to the all-purpose quantum computer. So far, most efforts in this domain have been directed to either fully regular or fully chaotic systems. Here, we focus on the intermediate regime, where regular orbits are surrounded by a large sea of chaotic trajectories. We observe a quantum chaos transport mechanism, called chaos-assisted tunneling, that translates in sharp resonances of the tunneling rate and provides previously unexplored possibilities for quantum simulation. More specifically, using Bose-Einstein condensates in a driven optical lattice, we experimentally demonstrate and characterize these resonances. Our work paves the way for quantum simulations with long-range transport and quantum control through complexity.
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Affiliation(s)
- M Arnal
- Laboratoire Collisions Agrégats Réactivité, IRSAMC, Université de Toulouse, CNRS, UPS, France
| | - G Chatelain
- Laboratoire Collisions Agrégats Réactivité, IRSAMC, Université de Toulouse, CNRS, UPS, France
| | - M Martinez
- Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France
| | - N Dupont
- Laboratoire Collisions Agrégats Réactivité, IRSAMC, Université de Toulouse, CNRS, UPS, France
| | - O Giraud
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
| | - D Ullmo
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
| | - B Georgeot
- Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France
| | - G Lemarié
- Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France
| | - J Billy
- Laboratoire Collisions Agrégats Réactivité, IRSAMC, Université de Toulouse, CNRS, UPS, France
| | - D Guéry-Odelin
- Laboratoire Collisions Agrégats Réactivité, IRSAMC, Université de Toulouse, CNRS, UPS, France.
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6
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Díaz GI, Palmero MS, Caldas IL, Leonel ED. Diffusion entropy analysis in billiard systems. Phys Rev E 2019; 100:042207. [PMID: 31770867 DOI: 10.1103/physreve.100.042207] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/26/2019] [Indexed: 11/07/2022]
Abstract
In this work we investigate how the behavior of the Shannon entropy can be used to measure the diffusion exponent of a set of initial conditions in two systems: (i) standard map and (ii) the oval billiard. We are interested in the diffusion near the main island in the phase space, where stickiness is observed. We calculate the diffusion exponent for many values of the nonlinear parameter of the standard map where the size and shape of the main island change as the parameter varies. We show that the changes of behavior in the diffusion exponent are related with the changes in the area of the main island and show that when the area of the main island is abruptly reduced, due to the destruction of invariant tori and, consequently, creation of hyperbolic and elliptic fixed points, the diffusion exponent grows.
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Affiliation(s)
- Gabriel I Díaz
- Instituto de Física, IFUSP-Universidade de São Paulo, Rua do Matão, Cidade Universitária, 05314-970 São Paulo, SP, Brazil
| | - Matheus S Palmero
- Instituto de Física, IFUSP-Universidade de São Paulo, Rua do Matão, Cidade Universitária, 05314-970 São Paulo, SP, Brazil
| | - Iberê Luiz Caldas
- Instituto de Física, IFUSP-Universidade de São Paulo, Rua do Matão, Cidade Universitária, 05314-970 São Paulo, SP, Brazil
| | - Edson D Leonel
- Departamento de Física, UNESP-Universidade Estadual Paulista, Bela Vista, 13506-900 Rio Claro, SP, Brazil
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7
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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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8
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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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9
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Schnell A, Ketzmerick R, Eckardt A. On the number of Bose-selected modes in driven-dissipative ideal Bose gases. Phys Rev E 2018; 97:032136. [PMID: 29776161 DOI: 10.1103/physreve.97.032136] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2018] [Indexed: 06/08/2023]
Abstract
In an ideal Bose gas that is driven into a steady state far from thermal equilibrium, a generalized form of Bose condensation can occur. Namely, the single-particle states unambiguously separate into two groups: the group of Bose-selected states, whose occupations increase linearly with the total particle number, and the group of all other states whose occupations saturate [Phys. Rev. Lett. 111, 240405 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.240405]. However, so far very little is known about how the number of Bose-selected states depends on the properties of the system and its coupling to the environment. The answer to this question is crucial since systems hosting a single, a few, or an extensive number of Bose-selected states will show rather different behavior. While in the former two scenarios each selected mode acquires a macroscopic occupation, corresponding to (fragmented) Bose condensation, the latter case rather bears resemblance to a high-temperature state of matter. In this paper, we systematically investigate the number of Bose-selected states, considering different classes of the rate matrices that characterize the driven-dissipative ideal Bose gases in the limit of weak system-bath coupling. These include rate matrices with continuum limit, rate matrices of chaotic driven systems, random rate matrices, and rate matrices resulting from thermal baths that couple to a few observables only.
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Affiliation(s)
- Alexander Schnell
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Roland Ketzmerick
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - André Eckardt
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
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10
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Doggen EVH, Georgeot B, Lemarié G. Chaos-assisted tunneling in the presence of Anderson localization. Phys Rev E 2017; 96:040201. [PMID: 29347601 DOI: 10.1103/physreve.96.040201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2016] [Indexed: 06/07/2023]
Abstract
Tunneling between two classically disconnected regular regions can be strongly affected by the presence of a chaotic sea in between. This phenomenon, known as chaos-assisted tunneling, gives rise to large fluctuations of the tunneling rate. Here we study chaos-assisted tunneling in the presence of Anderson localization effects in the chaotic sea. Our results show that the standard tunneling rate distribution is strongly modified by localization, going from the Cauchy distribution in the ergodic regime to a log-normal distribution in the strongly localized case, for both a deterministic and a disordered model. We develop a single-parameter scaling description which accurately describes the numerical data. Several possible experimental implementations using cold atoms, photonic lattices, or microwave billiards are discussed.
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Affiliation(s)
- Elmer V H Doggen
- Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France
| | - Bertrand Georgeot
- Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France
| | - Gabriel Lemarié
- Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France
- Department of Physics, Sapienza University of Rome, P.le A. Moro 2, 00185 Rome, Italy
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11
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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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12
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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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13
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Gehler S, Löck S, Shinohara S, Bäcker A, Ketzmerick R, Kuhl U, Stöckmann HJ. Experimental Observation of Resonance-Assisted Tunneling. PHYSICAL REVIEW LETTERS 2015; 115:104101. [PMID: 26382678 DOI: 10.1103/physrevlett.115.104101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2015] [Indexed: 06/05/2023]
Abstract
We present the first experimental observation of resonance-assisted tunneling, a wave phenomenon, where regular-to-chaotic tunneling is strongly enhanced by the presence of a classical nonlinear resonance chain. For this we use a microwave cavity made of oxygen free copper with the shape of a desymmetrized cosine billiard designed with a large nonlinear resonance chain in the regular region. It is opened in a region, where only chaotic dynamics takes place, such that the tunneling rate of a regular mode to the chaotic region increases the line width of the mode. Resonance-assisted tunneling is demonstrated by (i) a parametric variation and (ii) the characteristic plateau and peak structure towards the semiclassical limit.
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Affiliation(s)
- Stefan Gehler
- Fachbereich Physik, Philipps-Universität Marburg, Renthof 5, D-35032 Marburg, Germany
- Department of Energy Management and Power System Operation, University of Kassel, D-34121 Kassel, Germany
| | - Steffen Löck
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- OncoRay, National Center for Radiation Research in Oncology, Faculty of Medicine and University Hospital Carl Gustav Carus, Technische Universität Dresden, Helmholtz-Zentrum Dresden-Rossendorf, Fetscherstraß e 74, PF 41, 01307 Dresden, Germany
| | - Susumu Shinohara
- 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
| | - Ulrich Kuhl
- Fachbereich Physik, Philipps-Universität Marburg, Renthof 5, D-35032 Marburg, Germany
- Université Nice Sophia Antipolis, CNRS, Laboratoire de Physique de la Matière Condensée, UMR 7336 Parc Valrose, 06100 Nice, France
| | - Hans-Jürgen Stöckmann
- Fachbereich Physik, Philipps-Universität Marburg, Renthof 5, D-35032 Marburg, Germany
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14
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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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15
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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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Dietz B, Guhr T, Gutkin B, Miski-Oglu M, Richter A. Spectral properties and dynamical tunneling in constant-width billiards. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022903. [PMID: 25215795 DOI: 10.1103/physreve.90.022903] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2014] [Indexed: 06/03/2023]
Abstract
We determine with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics of the constant-width billiards is unidirectional, a change of the direction of motion is possible in the corresponding quantum system via dynamical tunneling. This becomes manifest in a splitting of the vast majority of resonances into doublets of nearly degenerate ones. The fluctuation properties of the two respective spectra are demonstrated to coincide with those of a random-matrix model for systems with violated time-reversal invariance and a mixed dynamics. Furthermore, we investigate tunneling in terms of the splittings of the doublet partners. On the basis of the random-matrix model we derive an analytical expression for the splitting distribution which is generally applicable to systems exhibiting dynamical tunneling between two regions with (predominantly) chaotic dynamics.
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Affiliation(s)
- B Dietz
- Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany
| | - T Guhr
- Fakultät für Physik, Universität Duisburg-Essen, Lotharstraße 1, D-47048 Duisburg, Germany
| | - B Gutkin
- Fakultät für Physik, Universität Duisburg-Essen, Lotharstraße 1, D-47048 Duisburg, Germany
| | - M Miski-Oglu
- Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany
| | - A Richter
- Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany
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Bittrich L, Bäcker A, Ketzmerick R. Temporal flooding of regular islands by chaotic wave packets. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:032922. [PMID: 24730928 DOI: 10.1103/physreve.89.032922] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2014] [Indexed: 06/03/2023]
Abstract
We investigate the time evolution of wave packets in systems with a mixed phase space where regular islands and chaotic motion coexist. For wave packets started in the chaotic sea on average the weight on a quantized torus of the regular island increases due to dynamical tunneling. This flooding weight initially increases linearly and saturates to a value which varies from torus to torus. We demonstrate for the asymptotic flooding weight universal scaling with an effective tunneling coupling for quantum maps and the mushroom billiard. This universality is reproduced by a suitable random matrix model.
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Affiliation(s)
- Lars Bittrich
- 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
| | - 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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Löbner C, Löck S, Bäcker A, Ketzmerick R. Integrable approximation of regular islands: the iterative canonical transformation method. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062901. [PMID: 24483525 DOI: 10.1103/physreve.88.062901] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2013] [Indexed: 06/03/2023]
Abstract
Generic Hamiltonian systems have a mixed phase space, where classically disjoint regions of regular and chaotic motion coexist. We present an iterative method to construct an integrable approximation H(reg), which resembles the regular dynamics of a given mixed system H and extends it into the chaotic region. The method is based on the construction of an integrable approximation in action representation which is then improved in phase space by iterative applications of canonical transformations. This method works for strongly perturbed systems and arbitrary degrees of freedom. We apply it to the standard map and the cosine billiard.
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Affiliation(s)
- 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
| | - Steffen Löck
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany and OncoRay, National Center for Radiation Research in Oncology, TU Dresden, Fetscherstraße 74, 01307 Dresden, Germany
| | - 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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Le Deunff J, Mouchet A, Schlagheck P. Semiclassical description of resonance-assisted tunneling in one-dimensional integrable models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042927. [PMID: 24229269 DOI: 10.1103/physreve.88.042927] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2013] [Indexed: 06/02/2023]
Abstract
Resonance-assisted tunneling is investigated within the framework of one-dimensional integrable systems. We present a systematic recipe, based on Hamiltonian normal forms, to construct one-dimensional integrable models that exhibit resonance island chain structures with accurately controlled sizes and positions of the islands. Using complex classical trajectories that evolve along suitably defined paths in the complex time domain, we construct a semiclassical theory of the resonance-assisted tunneling process. This semiclassical approach yields a compact analytical expression for tunnelling-induced level splittings which is found to be in very good agreement with the exact splittings obtained through numerical diagonalization.
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Affiliation(s)
- Jérémy Le Deunff
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
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Shudo A, Ikeda KS. Tunneling effect and the natural boundary of invariant tori. PHYSICAL REVIEW LETTERS 2012; 109:154102. [PMID: 23102311 DOI: 10.1103/physrevlett.109.154102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2012] [Indexed: 06/01/2023]
Abstract
The invariant torus of nonintegrable systems breaks up in complexified phase space. The breaking border is expected to form a natural boundary (NB) along which singularities are densely condensed. The NB cuts off the instanton orbit controlling the tunneling transport from a quantized invariant torus, which might result in a serious effect on the tunneling process. In the present Letter, we provide clear evidence showing that the presence of the NB is observable as an anomalous enhancement of the tunneling wave amplitude in the immediate outer side of the NB.
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Affiliation(s)
- Akira Shudo
- Department of Physics, Tokyo Metropolitan University, Tokyo, 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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Kruscha A, Ketzmerick R, Kantz H. Biased diffusion inside regular islands under random symplectic perturbations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:066210. [PMID: 23005199 DOI: 10.1103/physreve.85.066210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2012] [Indexed: 06/01/2023]
Abstract
We study the random concatenation of slightly different two-dimensional Hamiltonian maps with a mixed phase space. We consider a regular island whose fixed point is identical for all maps. Trajectories of the concatenated maps near this fixed point are no longer confined to invariant tori. We derive a stochastic model for the distance from the fixed point, which turns out to be a biased random walk with multiplicative noise. We give an analytical prediction of the survival probability of trajectories inside the regular island, which asymptotically is the product of a power law and an exponential. We confirm these results numerically for the parametrically perturbed standard map.
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Affiliation(s)
- Alexandra Kruscha
- Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzerstrasse 38, 01187 Dresden, Germany
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Rudolf T, Mertig N, Löck S, Bäcker A. Consequences of flooding on spectral statistics. Phys Rev E 2012; 85:036213. [PMID: 22587167 DOI: 10.1103/physreve.85.036213] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/20/2012] [Indexed: 11/07/2022]
Abstract
We study spectral statistics in systems with a mixed phase space, in which regions of regular and chaotic motion coexist. Increasing their density of states, we observe a transition of the level-spacing distribution P(s) from Berry-Robnik to Wigner statistics, although the underlying classical phase-space structure and the effective Planck constant h(eff) remain unchanged. This transition is induced by flooding, i.e., the disappearance of regular states due to increasing regular-to-chaotic couplings. We account for this effect by a flooding-improved Berry-Robnik distribution, in which an effectively reduced size of the regular island enters. To additionally describe power-law level repulsion at small spacings, we extend this prediction by explicitly considering the tunneling couplings between regular and chaotic states. This results in a flooding- and tunneling-improved Berry-Robnik distribution which is in excellent agreement with numerical data.
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Affiliation(s)
- Torsten Rudolf
- Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
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Creagh SC, White MM. Differences between emission patterns and internal modes of optical resonators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:015201. [PMID: 22400610 DOI: 10.1103/physreve.85.015201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2011] [Revised: 12/09/2011] [Indexed: 05/31/2023]
Abstract
The evanescent wave field outside an optical resonator is typically strongly directional when the shape deviates even very slightly from being perfectly circular or spherical. In this Rapid Communication we show that the tunneling mechanism underlying escape from such weakly deformed resonators can lead to emission patterns that look quite different to the corresponding internal mode. A direct short-wavelength analysis is not possible because the required complex ray data cannot be found due to the presence of natural boundaries. We show, however, that an approach based on perturbative approximation of the ray families successfully describes this phenomenon. In appropriate limits, the emission pattern allows one effectively to perform a direct observation of solutions of the one-dimensional Schrödinger equation in the complex plane.
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Affiliation(s)
- Stephen C Creagh
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
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Löck S, Bäcker A, Ketzmerick R. Coupling of bouncing-ball modes to the chaotic sea and their counting function. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:016210. [PMID: 22400646 DOI: 10.1103/physreve.85.016210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/28/2011] [Indexed: 05/31/2023]
Abstract
We study the coupling of bouncing-ball modes to chaotic modes in two-dimensional billiards with two parallel boundary segments. Analytically, we predict the corresponding decay rates using the fictitious integrable system approach. Agreement with numerically determined rates is found for the stadium and the cosine billiard. We use this result to predict the asymptotic behavior of the counting function N_{bb}(E)∼E^{δ}. For the stadium billiard we find agreement with the previous result δ=3/4. For the cosine billiard we derive δ=5/8, which is confirmed numerically and is well below the previously predicted upper bound δ=9/10.
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Affiliation(s)
- Steffen Löck
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
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Bäcker A, Ketzmerick R, Löck S, Mertig N. Fractional-power-law level statistics due to dynamical tunneling. PHYSICAL REVIEW LETTERS 2011; 106:024101. [PMID: 21405229 DOI: 10.1103/physrevlett.106.024101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2010] [Indexed: 05/30/2023]
Abstract
For systems with a mixed phase space we demonstrate that dynamical tunneling universally leads to a fractional power law of the level-spacing distribution P(s) over a wide range of small spacings s. Going beyond Berry-Robnik statistics, we take into account that dynamical tunneling rates between the regular and the chaotic region vary over many orders of magnitude. This results in a prediction of P(s) which excellently describes the spectral data of the standard map. Moreover, we show that the power-law exponent is proportional to the effective Planck constant h(eff).
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Affiliation(s)
- Arnd Bäcker
- Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
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