1. 1 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

2. 2 It makes no sense to talk about “diverging trajectories” for waves 1) Classical chaotic systems have diverging trajectories Regular system 2-Dimensional “billiard” tables with hard wall boundaries Newtonian particle trajectories Wave Chaos? 2) Linear wave systems can’t be chaotic 3) However in the semiclassical limit, you can think about rays Wave Chaos concerns solutions of wave equations which, in the semiclassical limit, can be described by chaotic ray trajectories q i +  q i, p i +  p i q i, p i Chaotic system q i, p i q i +  q i, p i +  p i In the ray-limit it is possible to define chaos “ray chaos”

3. 3 Ray Chaos Many enclosed three-dimensional spaces display ray chaos

4. 4 Wave Chaotic Systems are expected to show universal statistical properties, as predicted by Random Matrix Theory (RMT) Bohigas, Giannoni, Schmidt, PRL (1984) UNIVERSALITY IN WAVE CHAOTIC SYSTEMS RMT predicts universal statistical properties: Closed Systems Open Systems Eigenvalue nearest neighbor spacing Eigenvalue long-range correlations Eigenfunction 1-pt, 2-pt correlations etc. Scattering matrix statistics: |S|,  S Impedance matrix (Z) statistics (K matrix) Transmission matrix (T = SS † ), conductance statistics etc. The RMT Approach: Wigner; Dyson; Mehta; Bohigas … Complicated Hamiltonian: e.g. Nucleus: Solve Replace with a Hamiltonian with matrix elements chosen randomly from a Gaussian distribution Examine the statistical properties of the resulting Hamiltonians

5. 5 Billiard Incoming Channel Outgoing Channel Chaos and Scattering Hypothesis: Random Matrix Theory quantitatively describes the statistical properties of all wave chaotic systems (closed and open) Electromagnetic Cavities: Complicated S 11, S 22, S 21 versus frequency B (T) Transport in 2D quantum dots: Universal Conductance Fluctuations Resistance (k  ) mm Incoming Voltage waves Outgoing Voltage waves Nuclear scattering: Ericson fluctuations Proton energy Compound nuclear reaction 1 2 Incoming Channel Outgoing Channel

6. 6 The Most Common Non-Universal Effects: 1)Non-Ideal Coupling between external scattering states and internal modes (i.e. Antenna properties) Universal Fluctuations are Usually Obscured by Non-Universal System-Specific Details 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 Port Ray-Chaotic Cavity Incoming wave “Prompt” Reflection due to Z-Mismatch between antenna and cavity Z-mismatch at interface of port and cavity. Transmitted wave Short Orbits 2) Short-Orbits between the antenna and fixed walls of the billiards

7. 7 N-Port Description of an Arbitrary Scattering System N – Port System N Ports  Voltages and Currents,  Incoming and Outgoing Waves Z matrixS matrix V 1, I 1 V N, I N  Complicated Functions of frequency  Detail Specific (Non-Universal)

8. 8 Step 1: Remove the Non-Universal Coupling Form the Normalized Impedance (z) Coupling is normalized away at all energies Port Z Cavity Port Z Rad The waves do not return to the port Radiation Losses Reactive Impedance of Antenna Perfectly absorbing boundary Cavity Combine X. Zheng, et al. Electromagnetics (2006) Z Rad : A separate, deterministic, measurement of port properties

9. 9 Probability Density 2a=0.635mm 2a=1.27mm 2a=0.635mm Testing Insensitivity to System Details CAVITY BASE Cross Section View CAVITY LID Radius (a) Coaxial Cable  Freq. Range : 9 to 9.75 GHz  Cavity Height : h= 7.87mm  Statistics drawn from 100,125 pts. RAW Impedance PDF NORMALIZED Impedance PDF

10. 10 Step 2: Short-Orbit Theory Loss-Less case (J. Hart et al., Phys. Rev. E 80, 041109 (2009)) Original Random Coupling Model: where is Lorentzian distributed (loss-less case) Now, including short-orbits, this becomes: where is a Lorentzian distributed random matrix projected into the 2L/  - dimensional ‘semi-classical’ subspace with is the ensemble average of the semiclassical Bogomolny transfer operator … and can be calculated semiclassically… Experiments: J. H. Yeh, et al., Phys. Rev. E 81, 025201(R) (2010); Phys. Rev. E 82, 041114 (2010).Phys. Rev. E 81, 025201(R) (2010)Phys. Rev. E 82, 041114 (2010)

11. 11 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

12. 12 Conclusions Demonstrated the advantage of impedance (reaction matrix) in removing non-universal features The microwave analog experiments can provide clean, definitive tests of many theories of quantum chaotic scattering 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) 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 http://www.cnam.umd.edu/anlage/AnlageQChaos.htm

13. 13 The Maryland Wave Chaos Group Tom AntonsenSteve Anlage Ed Ott Elliott BradshawJen-Hao Yeh James HartBiniyam Taddese