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Hsieh YH, Yu YT, Tuan PH, Tung JC, Huang KF, Chen YF. Extracting trajectory equations of classical periodic orbits from the quantum eigenmodes in two-dimensional integrable billiards. Phys Rev E 2017; 95:022214. [PMID: 28297938 DOI: 10.1103/physreve.95.022214] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2016] [Indexed: 11/07/2022]
Abstract
The trajectory equations for classical periodic orbits in the equilateral-triangular and circular billiards are systematically extracted from quantum stationary coherent states. The relationship between the phase factors of quantum stationary coherent states and the initial positions of classical periodic orbits is analytically derived. In addition, the stationary coherent states with noncoprime parametric numbers are shown to correspond to the multiple periodic orbits, which cannot be explicable in the one-particle picture. The stationary coherent states are further verified to be linked to the resonant modes that are generally observed in the experimental wave system excited by a localized and unidirectional source. The excellent agreement between the resonant modes and the stationary coherent states not only manifests the importance of classical features in experimental systems but also paves the way to manipulate the mesoscopic wave functions localized on the periodic orbits for applications.
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Affiliation(s)
- Y H Hsieh
- Department of Electrophysics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan
| | - Y T Yu
- Department of Electrophysics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan
| | - P H Tuan
- Department of Electrophysics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan
| | - J C Tung
- Department of Electrophysics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan
| | - K F Huang
- Department of Electrophysics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan
| | - Y F Chen
- Department of Electrophysics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan
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Yu YT, Tuan PH, Chang KC, Hsieh YH, Huang KF, Chen YF. Exploiting broad-area surface emitting lasers to manifest the path-length distributions of finite-potential quantum billiards. OPTICS EXPRESS 2016; 24:82-91. [PMID: 26832239 DOI: 10.1364/oe.24.000082] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Broad-area vertical-cavity surface-emitting lasers (VCSELs) with different cavity sizes are experimentally exploited to manifest the influence of the finite confinement strength on the path-length distribution of quantum billiards. The subthreshold emission spectra of VCSELs are measured to obtain the path-length distributions by using the Fourier transform. It is verified that the number of the resonant peaks in the path-length distribution decreases with decreasing the confinement strength. Theoretical analyses for finite-potential quantum billiards are numerically performed to confirm that the mesoscopic phenomena of quantum billiards with finite confinement strength can be analogously revealed by using broad-area VCSELs.
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Lafargue C, Lebental M, Grigis A, Ulysse C, Gozhyk I, Djellali N, Zyss J, Bittner S. Localized lasing modes of triangular organic microlasers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:052922. [PMID: 25493873 DOI: 10.1103/physreve.90.052922] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2014] [Indexed: 06/04/2023]
Abstract
We investigated experimentally the ray-wave correspondence in organic microlasers of various triangular shapes. Triangular billiards are of interest since they are the simplest cases of polygonal billiards and the existence and properties of periodic orbits in triangles are not yet fully understood. The microlasers with symmetric shapes that were investigated exhibited states localized on simple periodic orbits, and their lasing characteristics like spectra and far-field distributions could be well explained by the properties of the periodic orbits. Furthermore, asymmetric triangles that do not feature simple periodic orbits were studied. Their lasing properties were found to be more complicated and could not be explained by periodic orbits.
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Affiliation(s)
- C Lafargue
- Laboratoire de Photonique Quantique et Moléculaire, CNRS UMR 8537, Institut d'Alembert FR 3242, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, F-94235 Cachan, France
| | - M Lebental
- Laboratoire de Photonique Quantique et Moléculaire, CNRS UMR 8537, Institut d'Alembert FR 3242, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, F-94235 Cachan, France
| | - A Grigis
- Laboratoire Analyse, Géométrie et Applications, CNRS UMR 7539, Université Paris 13, Sorbonne-Paris-Cité, 99 avenue J. B. Clément, F-93430 Villetaneuse, France
| | - C Ulysse
- Laboratoire de Photonique et Nanostructures, CNRS UPR20, Route de Nozay, F-91460 Marcoussis, France
| | - I Gozhyk
- Laboratoire de Photonique Quantique et Moléculaire, CNRS UMR 8537, Institut d'Alembert FR 3242, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, F-94235 Cachan, France
| | - N Djellali
- Laboratoire de Photonique Quantique et Moléculaire, CNRS UMR 8537, Institut d'Alembert FR 3242, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, F-94235 Cachan, France
| | - J Zyss
- Laboratoire de Photonique Quantique et Moléculaire, CNRS UMR 8537, Institut d'Alembert FR 3242, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, F-94235 Cachan, France
| | - S Bittner
- Laboratoire de Photonique Quantique et Moléculaire, CNRS UMR 8537, Institut d'Alembert FR 3242, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, F-94235 Cachan, France
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Tuan PH, Yu YT, Chiang PY, Liang HC, Huang KF, Chen YF. Level statistics and eigenfunctions of square torus billiards: manifesting the transition from regular to chaotic behaviors. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:026202. [PMID: 22463295 DOI: 10.1103/physreve.85.026202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/27/2011] [Revised: 01/16/2012] [Indexed: 05/31/2023]
Abstract
We thoroughly analyze the level statistics and eigenfunctions in concentric as well as nonconcentric square torus billiards. We confirm the characteristics of quantum and classical correspondence and the existence of scarred and superscarred modes in concentric square torus billiards. Furthermore, we not only verify that the transition from regular to chaotic behaviors can be manifested in nonconcentric square torus billiards, but also develop an analytical distribution to excellently fit the numerical level statistics. Finally, we intriguingly observe that numerous eigenstates commonly exhibit the wave patterns to be an ensemble of classical diamond trajectories, as the effective wavelengths are considerably shorter than the size of internal hole.
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Affiliation(s)
- P H Tuan
- Department of Electrophysics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan
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