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Fan L, Yan X, Wang H, Wang LV. Real-time observation and control of optical chaos. SCIENCE ADVANCES 2021; 7:7/3/eabc8448. [PMID: 33523874 PMCID: PMC7806228 DOI: 10.1126/sciadv.abc8448] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/17/2020] [Accepted: 11/18/2020] [Indexed: 06/01/2023]
Abstract
Optical chaotic system is a central research topic due to its scientific importance and practical relevance in key photonic applications such as laser optics and optical communication. Because of the ultrafast propagation of light, all previous studies on optical chaos are based on either static imaging or spectral measurement, which shows only time-averaged phenomena. The ability to reveal real-time optical chaotic dynamics and, hence, control its behavior is critical to the further understanding and engineering of these systems. Here, we report a real-time spatial-temporal imaging of an optical chaotic system, using compressed ultrafast photography. The time evolution of the system's phase map is imaged without repeating measurement. We also demonstrate the ability to simultaneously control and monitor optical chaotic systems in real time. Our work introduces a new angle to the study of nonrepeatable optical chaos, paving the way for fully understanding and using chaotic systems in various disciplines.
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Affiliation(s)
- Linran Fan
- Caltech Optical Imaging Laboratory, Andrew and Peggy Cherng Department of Medical Engineering and Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
| | - Xiaodong Yan
- Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA
| | - Han Wang
- Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA.
- Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, CA 90089, USA
| | - Lihong V Wang
- Caltech Optical Imaging Laboratory, Andrew and Peggy Cherng Department of Medical Engineering and Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA.
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Prakash R, Pandey A. Saturation of number variance in embedded random-matrix ensembles. Phys Rev E 2016; 93:052225. [PMID: 27300898 DOI: 10.1103/physreve.93.052225] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2015] [Indexed: 11/07/2022]
Abstract
We study fluctuation properties of embedded random matrix ensembles of noninteracting particles. For ensemble of two noninteracting particle systems, we find that unlike the spectra of classical random matrices, correlation functions are nonstationary. In the locally stationary region of spectra, we study the number variance and the spacing distributions. The spacing distributions follow the Poisson statistics, which is a key behavior of uncorrelated spectra. The number variance varies linearly as in the Poisson case for short correlation lengths but a kind of regularization occurs for large correlation lengths, and the number variance approaches saturation values. These results are known in the study of integrable systems but are being demonstrated for the first time in random matrix theory. We conjecture that the interacting particle cases, which exhibit the characteristics of classical random matrices for short correlation lengths, will also show saturation effects for large correlation lengths.
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Affiliation(s)
- Ravi Prakash
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
| | - Akhilesh Pandey
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
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Nagler J, Krieger M, Linke M, Schönke J, Wiersig J. Leaking billiards. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:046204. [PMID: 17500975 DOI: 10.1103/physreve.75.046204] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2006] [Indexed: 05/15/2023]
Abstract
Billiards are idealizations for systems where particles or waves are confined to cavities, or to other homogeneous regions. In billiard systems a point particle moves freely except for specular reflections from rigid walls. However, billiard walls are not always completely reflective and measurements inside can also open the billiard. Since boundary openings have been studied extensively in the literature, we rather model leakages inside the billiard. In particular, we investigate the classical dynamics of a leakage for a continuous family of billiard systems, that is, the stadium-lemon-billiard family. With a single parameter the geometry of the billiard can be tuned from stadium (being fully hyperbolic) over circle (integrable) to the lemon-shaped billiard (mixed chaotic). For the stadium billiard we found an algebraically decaying mean escape time with the linear size of the leakage n(esc) approximately epsilon-1 together with an exponential decay of the survival probability distribution. The finding is nearly independent of the position and size of the leakage, as long as the leakage is much smaller than the system size, and it is in good agreement with a stochastic map approximation of the dynamics. Due to the mixed phase space for lemon billiards, the mean escape time depends both on the position and geometry of the leakage. For systems where quasiregular motion dominates, we found a linear dependence of the mean escape time, n(esc) approximately 1-epsilon, which we refer to as flooding law. Our findings are helpful in understanding dynamics of leaking Hamiltonian systems.
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Affiliation(s)
- Jan Nagler
- Institut für Theoretische Physik, Universität Bremen, Otto-Hahn-Allee, D-28334 Bremen, Germany
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Timberlake T, Petruzielo F, Reichl LE. Localization of Floquet states along a continuous line of periodic orbits. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:016208. [PMID: 16090067 DOI: 10.1103/physreve.72.016208] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2005] [Indexed: 05/03/2023]
Abstract
A periodically driven particle in an infinite square well is shown to exhibit quantum localization due to a continuous line of periodic orbits in the classical system. Individual Floquet eigenstates localized along this line of periodic orbits are identified. The enhanced localization persists for field strengths beyond that at which the continuous line of orbits is destroyed in the classical dynamics. These results may be relevant to experiments involving trapping potentials with flat regions.
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Affiliation(s)
- T Timberlake
- Department of Physics, Astronomy, and Geology, Berry College, Mount Berry, Georgia 30149-5004, USA
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Dietz B, Heine A, Heuveline V, Richter A. Test of a numerical approach to the quantization of billiards. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:026703. [PMID: 15783452 DOI: 10.1103/physreve.71.026703] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/31/2004] [Indexed: 05/24/2023]
Abstract
A method for computing large numbers of eigenvalues of self-adjoint elliptic operators [J. Comput. Phys. 184, 321 (2003)] is tested in numerical studies of the spectral properties of quantum billiards. To this extent, we study a time-reversal invariant quantum billiard of threefold symmetry, that undergoes a transformation in its symmetry properties from C(3v) to C3 . Thereby a transition from Gaussian orthogonal to Gaussian unitary ensemble statistics is observed, verifying earlier experimental indications and theoretical predictions. At the same time our numerical ansatz is shown to be applicable to arbitrary billiard shapes.
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Affiliation(s)
- B Dietz
- Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany
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Li W, Reichl LE, Wu B. Quantum chaos in a ripple billiard. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:056220. [PMID: 12059693 DOI: 10.1103/physreve.65.056220] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2001] [Indexed: 05/23/2023]
Abstract
We study the quantum chaos of a ripple billiard that has sinusoidal walls. We show that this type of ripple billiard has a Hamiltonian matrix that can be found exactly in terms of elementary functions. This feature greatly improves computation efficiency; a complete set of eigenstates from the ground state up to the 10,000th level can be calculated simultaneously. Nearest neighbor spacing of energy levels of a chaotic ripple billiard shows a Brody distribution (with a confidence level of 99% by chi(2) test) instead of the Gaussian orthogonal ensemble prediction. For high energy levels we observe scars and interesting patterns that have no resemblance to classical periodic orbits. Momentum localization of scarred eigenstates is also observed. We compare the scar associated localization with quantum dynamical Anderson localization by drawing the wave function distribution on basis state coefficients.
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Affiliation(s)
- Wenjun Li
- Center for Studies in Statistical Mechanics and Complex Systems and Department of Physics, The University of Texas at Austin, Texas 78712, USA
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Wilkinson PB, Fromhold TM, Taylor RP, Micolich AP. Electromagnetic wave chaos in gradient refractive index optical cavities. PHYSICAL REVIEW LETTERS 2001; 86:5466-5469. [PMID: 11415277 DOI: 10.1103/physrevlett.86.5466] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/28/2000] [Indexed: 05/23/2023]
Abstract
Electromagnetic wave chaos is investigated using two-dimensional optical cavities formed in a cylindrical gradient refractive index lens with reflective surfaces. When the planar ends of the lens are cut at an angle to its axis, the geometrical ray paths are chaotic. In this regime, the electromagnetic mode spectrum of the cavity is modulated by both real and ghost periodic ray paths, which also "scar" the electric field intensity distributions of many modes. When the cavity is coupled to waveguides, the eigenmodes generate complex series of resonant peaks in the electromagnetic transmission spectrum.
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Affiliation(s)
- P B Wilkinson
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
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Hlushchuk Y, Sirko L, Kuhl U, Barth M, Stöckmann HJ. Experimental investigation of a regime of Wigner ergodicity in microwave rough billiards. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:046208. [PMID: 11308931 DOI: 10.1103/physreve.63.046208] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2000] [Revised: 01/04/2001] [Indexed: 05/23/2023]
Abstract
We study experimentally a new regime of Wigner ergodicity [K.M. Frahm and D.L. Shepelyansky, Phys. Rev. Lett. 79, 1833 (1997)] in a microwave rough billiard. We show that in the Wigner regime, eigenstates are extended over the whole energy surface but have a strongly peaked nonergodic structure. The Shannon width of the eigenstate distributions is calculated to estimate their spreads and to find their departure from the ergodic distributions.
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Affiliation(s)
- Y Hlushchuk
- Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
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Dembowski C, Graf HD, Heine A, Rehfeld H, Richter A, Schmit C. Gaussian unitary ensemble statistics in a time-reversal invariant microwave triangular billiard. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:R4516-R4519. [PMID: 11089072 DOI: 10.1103/physreve.62.r4516] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2000] [Indexed: 05/23/2023]
Abstract
The spectrum of a chaotic two-dimensional quantum billiard with threefold symmetry has been studied in an experiment with a superconducting microwave cavity. In total 622 eigenvalues were identified experimentally and compared with numerical calculations. The statistical analysis of the data shows that Gaussian unitary ensemble statistics can be observed for a spectrum of a time-reversal invariant system.
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Affiliation(s)
- C Dembowski
- Institut fur Kernphysik, Technische Universitat Darmstadt, D-64289 Darmstadt, Germany
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Lu W, Viola L, Pance K, Rose M, Sridhar S. Microwave study of quantum n-disk scattering. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:3652-3663. [PMID: 11088143 DOI: 10.1103/physreve.61.3652] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/1999] [Indexed: 05/23/2023]
Abstract
We describe a wave-mechanical implementation of classically chaotic n-disk scattering based on thin two-dimensional microwave cavities. Two-, three-, and four-disk scatterings are investigated in detail. The experiments, which are able to probe the stationary Green's function of the system, yield both frequencies and widths of the low-lying quantum resonances. The observed spectra are found to be in good agreement with calculations based on semiclassical periodic orbit theory. Wave-vector autocorrelation functions are analyzed for various scattering geometries, the small wave-vector behavior allowing one to extract the escape rate from the quantum repeller. Quantitative agreement is found with the value predicted from classical scattering theory. For intermediate energies, nonuniversal oscillations are detected in the autocorrelation function, reflecting the presence of periodic orbits.
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Affiliation(s)
- W Lu
- Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA
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