Report

Overview of the Random Coupling Model Jen-Hao Yeh, Sameer Hemmady, Xing Zheng, James Hart, Edward Ott, Thomas Antonsen, Steven M. Anlage Research funded by AFOSR and the ONR/UMD AppEl, ONR-MURI and DURIP programs 1 Wave Chaos? 1) Classical chaotic systems have diverging trajectories 2-Dimensional “billiard” tables with hard wall boundaries Chaotic system Regular system Newtonian particle trajectories q i, p i qi+Dqi, pi +Dpi qi, pi qi+Dqi, pi +Dpi 2) Linear wave systems can’t be chaotic It makes no sense to talk about In the ray-limit “ray chaos” “diverging trajectories” for waves it is possible to define chaos 3) However in the semiclassical limit, you can think about rays 2 Wave Chaos concerns solutions of wave equations which, in the semiclassical limit, can be described by chaotic ray trajectories Ray Chaos Many enclosed three-dimensional spaces display ray chaos 3 UNIVERSALITY IN WAVE CHAOTIC SYSTEMS • Wave Chaotic Systems are expected to show universal statistical properties, as predicted by Random Matrix Theory (RMT) Bohigas, Giannoni, Schmidt, PRL (1984) The RMT Approach: Wigner; Dyson; Mehta; Bohigas … Complicated Hamiltonian: e.g. Nucleus: Solve H E Replace with a Hamiltonian with matrix elements chosen randomly from a Gaussian distribution Examine the statistical properties of the resulting Hamiltonians H • RMT predicts universal statistical properties: Closed Systems • Eigenvalue nearest neighbor spacing • Eigenvalue long-range correlations Open Systems • Scattering matrix statistics: |S|, fS • Impedance matrix (Z) statistics (K matrix) • Eigenfunction 1-pt, 2-pt correlations • Transmission matrix (T = SS†), conductance 1000 1.0 1.000 statistics • etc. • etc. T2 500 0.5000 MLE 12 k 2 /(Dkn2Q) 0.72 00 0.5 4 0.5 T 1.0 Chaos and Scattering Hypothesis: Random Matrix Theory quantitatively describes the statistical properties of all wave chaotic systems (closed and open) Compound nuclear reaction Nuclear scattering: Ericson fluctuations d d Incoming Channel Outgoing Channel Proton energy Billiard Outgoing Voltage waves S matrix V 1 V 1 V 2 V 2 [S ] V N 5 V N Incoming Channel Outgoing Channel mm Resistance (kW) Transport in 2D quantum dots: Universal Conductance Fluctuations B (T) | S xx | Incoming Voltage waves Electromagnetic Cavities: Complicated S11, S22, S21 versus frequency 2 1 |S11| |S22| |S21| Frequency (GHz) Universal Fluctuations are Usually Obscured by Non-Universal System-Specific Details The Most Common Non-Universal Effects: 1) Non-Ideal Coupling between external scattering states and internal modes (i.e. Antenna properties) 2) Short-Orbits between the antenna and fixed walls of the billiards Z-mismatch at interface of port and cavity. Ray-Chaotic Cavity Port Incoming wave “Prompt” Reflection due to Z-Mismatch Transmitted between antenna and wave cavity We have developed a new way to remove these non-universal effects using the Impedance Z We measure the non-universal details in separate experiments and use them to normalize the raw impedance to get an impedance z that displays universal fluctuating properties described by Random Matrix Theory 6 N-Port Description of an Arbitrary Scattering System N Ports V1 V1 V1 , I1 Voltages and Currents, N – Port Incoming and Outgoing Waves System VN VN S matrix V 1 V 1 V 2 V 2 [S ] V N V N 7 VN , IN Z matrix V1 I1 V I 2 2 [ ] VN I N S (Z Z0 )1 (Z Z0 ) Z ( ), S ( ) Complicated Functions of frequency Detail Specific (Non-Universal) Step 1: Remove the Non-Universal Coupling Form the Normalized Impedance (z) Coupling is normalized away at all energies Port ZCavity ZCavity RCavity j X Cavity Combine Cavity z RCavity RRad j Port ZRad Perfectly absorbing boundary 8 X Cavity X Rad RRad X. Zheng, et al. Electromagnetics (2006) Z Rad RRad j X Rad Radiation Losses Reactive Impedance of The waves do not return to the port Antenna ZRad: A separate, deterministic, measurement of port properties Port 1Details Testing Insensitivity to System Coaxial Cable Freq. Range : 9 to 9.75 GHz CAVITY LID Cross Section View Cavity Height : h= 7.87mm Statistics drawn from 100,125 pts. Metallic Perturbations Radius (a) z CAVITY BASE Probability Density 0.015 RAW Impedance PDF 2a=0.635mm RRad j X Cavity X Rad RRad NORMALIZED Impedance PDF 2a=0.635mm 2a=1.27mm 0.010 0.3 2a=1.27mm 0.005 0.000 -500 -250 9 0.6 RCavity 0 250 Im(ZCav )(W) 0.0 -2 500 -1 0 Im(z ) 1 2 Step 2: Short-Orbit Theory Loss-Less case (J. Hart et al., Phys. Rev. E 80, 041109 (2009)) Original Random Coupling Model: where 0 Z iX Rad i RRad 0 RRad is Lorentzian distributed (loss-less case) Now, including short-orbits, this becomes: Z iX pavg i R pavg R pavg where is a Lorentzian distributed random matrix projected into the 2L/l - dimensional ‘semi-classical’ subspace Z pavg 1 T iX Rad v v 1 T Z Rad 2 v T v 1 with vv RRad T is the ensemble average of the semiclassical Bogomolny transfer operator … and T can be calculated semiclassically… Experiments: J. H. Yeh, et al., Phys. Rev. E 81, 025201(R) (2010); Phys. Rev. E 82, 041114 (2010). 10 Applications of Wave Chaos Ideas to Practical Problems 1) Understanding and mitigating HPM Effects in electronics Random Coupling Model “Terp RCM Solver” predicts PDF of induced voltages for electronics inside complicated enclosures 2) Using universal statistics + short orbits to understand time-domain data Extended Random Coupling Model Fading statistics predictions Identification of short-orbit communication paths 3) Quantum graphs applied to Electromagnetic Topology 11 Conclusions The microwave analog experiments can provide clean, definitive tests of many theories of quantum chaotic scattering Demonstrated the advantage of impedance (reaction matrix) in removing non-universal features Some Relevant Publications: S. Hemmady, et al., Phys. Rev. Lett. 94, 014102 (2005) S. Hemmady, et al., Phys. Rev. E 71, 056215 (2005) Xing Zheng, T. M. Antonsen Jr., E. Ott, Electromagnetics 26, 3 (2006) Xing Zheng, T. M. Antonsen Jr., E. Ott, Electromagnetics 26, 37 (2006) Xing Zheng, et al., Phys. Rev. E 73 , 046208 (2006) S. Hemmady, et al., Phys. Rev. B 74, 195326 (2006) S. M. Anlage, et al., Acta Physica Polonica A 112, 569 (2007) http://www.cnam.umd.edu/anlage/AnlageQChaos.htm 12 Many thanks to: R. Prange, S. Fishman, Y. Fyodorov, D. Savin, P. Brouwer, P. Mello, F. Schafer, J. Rodgers, A. Richter, M. Fink, L. Sirko, J.-P. Parmantier The Maryland Wave Chaos Group Elliott Bradshaw Ed Ott 13 Jen-Hao Yeh Tom Antonsen James Hart Biniyam Taddese Steve Anlage