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Reichl LE. Chaos-Assisted Tunneling. ENTROPY (BASEL, SWITZERLAND) 2024; 26:144. [PMID: 38392399 PMCID: PMC10887951 DOI: 10.3390/e26020144] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2024] [Revised: 02/01/2024] [Accepted: 02/03/2024] [Indexed: 02/24/2024]
Abstract
The ability of particles to "tunnel" through potential energy barriers is a purely quantum phenomenon. A classical particle in a symmetric double-well potential, with energy below the potential barrier, will be trapped on one side of the potential well. A quantum particle, however, can sit on both sides, in either a symmetric state or an antisymmetric state. An analogous phenomenon occurs in conservative classical systems with two degrees of freedom and no potential barriers. If only the energy is conserved, the phase space will be a mixture of regular "islands" embedded in a sea of chaos. Classically, a particle sitting in one regular island cannot reach another symmetrically located regular island when the islands are separated by chaos. However, a quantum particle can sit on both regular islands, in symmetric and antisymmetric states, due to chaos-assisted tunneling. Here, we give an overview of the theory and recent experimental observations of this phenomenon.
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Affiliation(s)
- Linda E Reichl
- Center for Complex Quantum Systems, Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA
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2
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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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3
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Martinez M, Giraud O, Ullmo D, Billy J, Guéry-Odelin D, Georgeot B, Lemarié G. Chaos-Assisted Long-Range Tunneling for Quantum Simulation. PHYSICAL REVIEW LETTERS 2021; 126:174102. [PMID: 33988390 DOI: 10.1103/physrevlett.126.174102] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2020] [Revised: 03/02/2021] [Accepted: 03/29/2021] [Indexed: 06/12/2023]
Abstract
We present an extension of the chaos-assisted tunneling mechanism to spatially periodic lattice systems. We demonstrate that driving such lattice systems in an intermediate regime of modulation maps them onto tight-binding Hamiltonians with chaos-induced long-range hoppings t_{n}∝1/n between sites at a distance n. We provide a numerical demonstration of the robustness of the results and derive an analytical prediction for the hopping term law. Such systems can thus be used to enlarge the scope of quantum simulations to experimentally realize long-range models of condensed matter.
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Affiliation(s)
- Maxime Martinez
- Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, 31062 Toulouse France
| | - Olivier Giraud
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
| | - Denis Ullmo
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
| | - Juliette Billy
- Laboratoire Collisions Agrégats Réactivité, Université de Toulouse, CNRS, UPS, 31062 Toulouse France
| | - David Guéry-Odelin
- Laboratoire Collisions Agrégats Réactivité, Université de Toulouse, CNRS, UPS, 31062 Toulouse France
| | - Bertrand Georgeot
- Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, 31062 Toulouse France
| | - Gabriel Lemarié
- Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, 31062 Toulouse France
- MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Unit, 117543 Singapore
- Centre for Quantum Technologies, National University of Singapore, 117543 Singapore
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4
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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: 8] [Impact Index Per Article: 2.0] [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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5
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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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6
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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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7
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Wüster S, Dąbrowska-Wüster BJ, Davis MJ. Macroscopic quantum self-trapping in dynamical tunneling. PHYSICAL REVIEW LETTERS 2012; 109:080401. [PMID: 23002725 DOI: 10.1103/physrevlett.109.080401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/09/2011] [Indexed: 06/01/2023]
Abstract
It is well known that increasing the nonlinearity due to repulsive atomic interactions in a double-well Bose-Einstein condensate suppresses quantum tunneling between the two sites. Here we find analogous behavior in the dynamical tunneling of a Bose-Einstein condensate between period-one resonances in a single driven potential well. For small nonlinearities we find unhindered tunneling between the resonances, but with an increasing period as compared to the noninteracting system. For nonlinearities above a critical value we generally observe that the tunneling shuts down. However, for certain regimes of modulation parameters we find that dynamical tunneling reemerges for large enough nonlinearities, an effect not present in spatial double-well tunneling. We develop a two-mode model in good agreement with full numerical simulations over a wide range of parameters, which allows the suppression of tunneling to be attributed to macroscopic quantum self-trapping.
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Affiliation(s)
- Sebastian Wüster
- The University of Queensland, School of Mathematics and Physics, Brisbane, Queensland 4072, Australia
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8
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Bäcker A, Ketzmerick R, Löck S. Direct regular-to-chaotic tunneling rates using the fictitious-integrable-system approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:056208. [PMID: 21230564 DOI: 10.1103/physreve.82.056208] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2010] [Indexed: 05/30/2023]
Abstract
We review the fictitious integrable system approach which predicts dynamical tunneling rates from regular states to the chaotic region in systems with a mixed phase space. It is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the regular island. We focus on the direct regular-to-chaotic tunneling process which dominates if nonlinear resonances within the regular island are not relevant. For quantum maps, billiard systems, and optical microcavities, we find excellent agreement with numerical rates for all regular states.
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Affiliation(s)
- Arnd Bäcker
- Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
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10
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Mouchet A, Eltschka C, Schlagheck P. Influence of classical resonances on chaotic tunneling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:026211. [PMID: 17025529 DOI: 10.1103/physreve.74.026211] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2006] [Indexed: 05/12/2023]
Abstract
Dynamical tunneling between symmetry-related stable modes is studied in the periodically driven pendulum. We present strong evidence that the tunneling process is governed by nonlinear resonances that manifest within the regular phase-space islands on which the stable modes are localized. By means of a quantitative numerical study of the corresponding Floquet problem, we identify the trace of such resonances not only in the level splittings between near-degenerate quantum states, where they lead to prominent plateau structures, but also in overlap matrix elements of the Floquet eigenstates, which reveal characteristic sequences of avoided crossings in the Floquet spectrum. The semiclassical theory of resonance-assisted tunneling yields good overall agreement with the quantum-tunneling rates, and indicates that partial barriers within the chaos might play a prominent role.
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Affiliation(s)
- Amaury Mouchet
- Laboratoire de Mathématiques et de Physique Théorique, CNRS UMR 6083, Université François Rabelais Avenue Monge, Parc de Grandmont, 37200 Tours, France
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11
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Hofferbert R, Alt H, Dembowski C, Gräf HD, Harney HL, Heine A, Rehfeld H, Richter A. Experimental investigations of chaos-assisted tunneling in a microwave annular billiard. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:046201. [PMID: 15903764 DOI: 10.1103/physreve.71.046201] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/21/2004] [Indexed: 05/02/2023]
Abstract
We present detailed investigations of the experimental signatures of chaos-assisted tunneling in the two-dimensional annular billiard, as already summarized in Phys. Rev. Lett. 84, 867 (2000). We have performed analog experiments with two-dimensional, electromagnetic resonators allowing for a direct simulation of the corresponding quantum system. Spectra from a superconducting cavity with a high-frequency resolution are combined with electromagnetic intensity distributions of high spatial resolution experimentally determined using a normal conducting twin cavity. Thereby all eigenmodes were obtained with properly identified quantum numbers. Besides distributions of quasi-doublet splittings, which serve as fundamental observables for the tunneling between whispering gallery types of modes, we also focus on the distributions of resonance widths of the doublets. These directly reflect the role of lifetime of certain modes in the tunneling process. Here, as theoretically expected, the class of so-called beach modes is found to play a particular role in mediating between regular and chaotic states to enhance the tunneling strength. This behavior is found in the spectrum and also in the structure of the wave functions.
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Affiliation(s)
- R Hofferbert
- Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany
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Abstract
This work establishes a firm relationship between classical nonlinear resonances and the phenomenon of dynamical tunneling. It is shown that the classical phase space with its hierarchy of resonance islands completely characterizes dynamical tunneling and explicit forms of the dynamical barriers can be obtained only by identifying the key resonances. Relationship between the phase space viewpoint and the quantum mechanical superexchange approach is discussed in near-integrable and mixed regular-chaotic situations. For near-integrable systems with sufficient anharmonicity the effect of multiple resonances, i.e., resonance-assisted tunneling, can be incorporated approximately. It is also argued that the presumed relation of avoided crossings to nonlinear resonances does not have to be invoked in order to understand dynamical tunneling. For molecules with low density of states the resonance-assisted mechanism is expected to be dominant.
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13
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Egydio de Carvalho R, Mijolaro AP. Fluctuations of doublet splittings using the annular billiard. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:056212. [PMID: 15600733 DOI: 10.1103/physreve.70.056212] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2004] [Indexed: 05/24/2023]
Abstract
We study the statistical distribution of quantum energy splittings due to a dynamical tunneling. The system, the annular billiard, has whispering quasimodes due to a discrete symmetry that exists even when chaos is present in the underlying classical dynamics. Symmetric and antisymmetric combinations of these quasimodes correspond to quantum doublet states whose degeneracies decrease as the circles become more eccentric. We construct numerical ensembles composed of splittings for two distinct regimes, one which we call semiclassical for high quantum numbers and high energies where the whispering regions are connected by chaos, and other which we call quantal for low quantum numbers, low energies, and near integrable where dynamical tunneling is not a dominant mechanism. In both cases we observe a variation on the fluctuation amplitudes, but their mean behaviors follow the formula of Leyvraz and Ullmo [J. Phys. A 29, 2529 (1996)]. A description of a three-level collision involving a doublet and a singlet is also provided through a numerical example.
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Affiliation(s)
- R Egydio de Carvalho
- Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista-UNESP 13500-230 Rio Claro, SP, Brazil.
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Podolskiy VA, Narimanov EE. Semiclassical description of chaos-assisted tunneling. PHYSICAL REVIEW LETTERS 2003; 91:263601. [PMID: 14754050 DOI: 10.1103/physrevlett.91.263601] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2003] [Indexed: 05/24/2023]
Abstract
We study tunneling between regular and chaotic regions in the phase space of Hamiltonian systems. We analytically calculate the transition rate and show that its variation depends only on corresponding phase space area and in this sense is universal. We derive the distribution of level splittings associated with the pairs of quasidegenerate regular eigenstates which in the general case is different from a Cauchy distribution. We show that chaos-assisted tunneling leads to level repulsion between regular eigenstates, solving the longstanding problem of level-spacing distribution in mixed systems.
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Affiliation(s)
- Viktor A Podolskiy
- Electrical Engineering Department, Princeton University, Princeton, New Jersey 08544, USA
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Onishi T, Shudo A, Ikeda KS, Takahashi K. Semiclassical study on tunneling processes via complex-domain chaos. ACTA ACUST UNITED AC 2003; 68:056211. [PMID: 14682875 DOI: 10.1103/physreve.68.056211] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2002] [Revised: 05/20/2003] [Indexed: 11/07/2022]
Abstract
We investigate the semiclassical mechanism of tunneling processes in nonintegrable systems. The significant role of complex-phase-space chaos in the description of the tunneling processes is elucidated by studying a kicked scattering model. Behaviors of tunneling orbits are encoded into symbolic sequences based on the structure of a complex homoclinic tangle. By means of the symbolic coding, the phase space itineraries of tunneling orbits are related with the amounts of imaginary parts of actions gained by the orbits, so that the systematic search of dominant tunneling orbits becomes possible.
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Affiliation(s)
- T Onishi
- Department of Physics, Tokyo Metropolitan University, Minami-Ohsawa, Hachioji 192-0397, Japan.
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Artuso R, Rebuzzini L. Effects of a nonlinear perturbation on dynamical tunneling in cold atoms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:036221. [PMID: 14524882 DOI: 10.1103/physreve.68.036221] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2003] [Indexed: 11/07/2022]
Abstract
We perform a numerical analysis of the effects of a nonlinear perturbation on the quantum dynamics of two models describing noninteracting cold atoms in a standing wave of light with a periodical modulated amplitude A(t). One model is the driven pendulum, recently considered by D.A. Steck, W.H. Oskay, and M.G. Raizen [Science 293, 274 (2001)], and the other is a variant of the well-known kicked rotator model. In absence of the nonlinear perturbation, the system is invariant under some discrete symmetries and quantum dynamical tunneling between symmetric classical islands is found. The presence of nonlinearity destroys tunneling, breaking the symmetries of the system. Finally, further consequences of nonlinearity in the kicked rotator case are considered.
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Affiliation(s)
- Roberto Artuso
- Center for Nonlinear and Complex Systems, Università dell'Insubria a Como, Via Valleggio 11, 22100 Como, Italy
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