Overview of the Random Coupling Model

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. “ray chaos” In the ray-limit it is possible to define chaos qi, pi qi+Dqi, pi +Dpi Wave Chaos? 1) Classical chaotic systems have diverging trajectories 2-Dimensional “billiard” tables with hard wall boundaries Chaotic system Regular system Newtonian particle trajectories qi, pi qi+Dqi, pi +Dpi 2) Linear wave systems can’t be chaotic It makes no sense to talk about “diverging trajectories” for waves 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

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

4. 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 Replace with a Hamiltonian with matrix elements chosen randomly from a Gaussian distribution Examine the statistical properties of the resulting Hamiltonians • RMT predicts universal statistical properties: Open Systems Closed Systems • Scattering matrix statistics: |S|, fS • Impedance matrix (Z) statistics (K matrix) • Transmission matrix (T = SS†), conductance statistics • etc. • Eigenvalue nearest neighbor spacing • Eigenvalue long-range correlations • Eigenfunction 1-pt, 2-pt correlations • etc.

5. Chaos and Scattering Compound nuclear reaction Nuclear scattering: Ericson fluctuations Billiard Incoming Channel Outgoing Channel Proton energy 2 mm Transport in 2D quantum dots: Universal Conductance Fluctuations 1 Outgoing Voltage waves B (T) Incoming Voltage waves Resistance (kW) Hypothesis: Random Matrix Theory quantitatively describes the statistical properties of all wave chaotic systems (closed and open) Incoming Channel Outgoing Channel Electromagnetic Cavities: Complicated S11, S22, S21 versus frequency

6. Z-mismatch at interface of port and cavity. Short Orbits “Prompt” Reflection due to Z-Mismatch between antenna and cavity Transmitted wave Universal Fluctuations are Usually Obscured by Non-Universal System-Specific Details • The Most Common Non-Universal Effects: • 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 Ray-Chaotic Cavity Port Incoming wave 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

7. Z matrix S matrix • Complicated Functions of frequency • Detail Specific (Non-Universal) N-Port Description of an Arbitrary Scattering System V1 , I1 • N Ports • Voltages and Currents, • Incoming and Outgoing Waves N – Port System VN , IN

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

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

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/l - 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).

11. Applications of Wave Chaos Ideas to Practical Problems • 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. 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 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

13. The Maryland Wave Chaos Group Elliott Bradshaw Jen-Hao Yeh James Hart Biniyam Taddese Tom Antonsen Steve Anlage Ed Ott