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

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

Newtonianparticletrajectories

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 semiclassicallimit, can be described by chaotic ray trajectories

qi+Dqi, pi +Dpiqi, pi

Chaotic system

qi, pi qi+Dqi, pi +Dpi

In the ray-limitit is possible to define chaos

“ray chaos”

3

Ray Chaos

Many enclosed three-dimensional spaces display ray chaos

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|, fS• Impedance matrix (Z) statistics

(K matrix)

• Transmission matrix (T = SS†), conductance statistics

• etc.

H

The RMT Approach: Wigner; Dyson; Mehta; Bohigas …

Complicated Hamiltonian: e.g. Nucleus: Solve

Replace with a Hamiltonian with matrix elements chosen randomlyfrom a Gaussian distribution

Examine the statistical properties of the resulting Hamiltonians

EH

0.5 1.0

0.5

1.0

0

0.5000

1.0001000

500

0

12MLE

72.0)/( 22 Qkk n

0.5 1.0

0.5

1.0

0

0.5000

1.0001000

500

0

12MLE

72.0)/( 22 Qkk n

12MLE

72.0)/( 22 Qkk n

2T

1T

5

Billiard

IncomingChannel

OutgoingChannel

Chaos and ScatteringHypothesis: Random Matrix Theory quantitatively describes the statistical

properties of all wave chaotic systems (closed and open)

|S|S1111||

|S|S2222||

|S|S2121||

Frequency (GHz)

|| xxS

|S|S1111||

|S|S2222||

|S|S2121||

Frequency (GHz)

|| xxS

Electromagnetic Cavities: Complicated S11, S22, S21 versus frequency

B (T)

Transport in 2D quantum dots: Universal Conductance Fluctuations

Res

ista

nce

(kW

) mm

S matrix

NN V

V

V

S

V

V

V

2

1

2

1

][

S matrix

NN V

V

V

S

V

V

V

2

1

2

1

][

Incoming Voltage waves

Outgoing Voltage waves

Nuclear scattering: Ericson fluctuations

d

d

Proton energy

Compound nuclear reaction

1

2

Incoming Channel

Outgoing Channel

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 impedanceto 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

ShortOrbits

2) Short-Orbits between the antenna and fixed walls of the billiards

7

N-Port Description of an Arbitrary Scattering System

N – Port

System

N Ports

Voltages and Currents,

Incoming and Outgoing Waves

Z matrix

NN I

I

I

V

V

V

2

1

2

1

][

S matrix

NN V

V

V

S

V

V

V

2

1

2

1

][

1V

1VV1 , I1

VN , INNVNV

)()( 01

0 ZZZZS

)(),( SZ Complicated

Functions of frequency

Detail Specific (Non-Universal)

8

Step 1: Remove the Non-Universal CouplingForm the Normalized Impedance (z)Coupling is normalized away at all energies

Port

ZCavity

Port

ZRad

CavityCavityCavity XjRZ

RadRadRad XjRZ

The waves donot return to the port

RadiationLosses

ReactiveImpedance ofAntenna

Perfectly absorbingboundary

Cavity

Rad

RadCavity

Rad

Cavity

R

XXj

R

Rz

Combine

X. Zheng, et al. Electromagnetics (2006)

ZRad: A separate, deterministic, measurement of port properties

9

Prob

abil

ity

Den

sity

-2 -1 0 1 20.0

0.3

0.6

2a=0.635mm2a=1.27mm

)Im(z-500 -250 0 250 500

0.000

0.005

0.010

0.015

2a=1.27mm

2a=0.635mm

))(Im( CavZ

Testing Insensitivity to System Details

CAVITY BASE

CrossSection View

CAVITY LID

Radius (a)

CoaxialCable Freq. Range : 9 to 9.75 GHz

Cavity Height : h= 7.87mm

Statistics drawn from 100,125 pts.

Rad

RadCavity

Rad

Cavity

R

XXj

R

Rz

RAW Impedance PDF NORMALIZED Impedance PDF

Metallic Perturbations

Port 1

10

Step 2: Short-Orbit TheoryLoss-Less case (J. Hart et al., Phys. Rev. E 80, 041109 (2009))

Original Random Coupling Model:

where is Lorentzian distributed (loss-less case)

vT

TviXZ Radpavg

1

1

Now, including short-orbits, this becomes:

RadRvv

RadRadRad RRiiXZ 0

0

pavgpavgpavg RRiiXZ

where is a Lorentzian distributed random matrix projected into the2L/ l - dimensional ‘semi-classical’ subspace

with

1

2

vTvZRad is the ensemble average of the semiclassical Bogomolny transfer operatorT

… and can be calculated semiclassically…T

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

1) Understanding and mitigating HPM Effects in electronicsRandom Coupling Model

“Terp RCM Solver” predicts PDF of induced voltagesfor electronics inside complicated enclosures

2) Using universal statistics + short orbits to understand time-domain dataExtended Random Coupling Model

Fading statistics predictionsIdentification of short-orbit communication paths

3) Quantum graphs applied to Electromagnetic Topology

12

Conclusions

Demonstrated the advantage of impedance (reaction matrix)in removing non-universal features

The microwave analog experiments can provide clean, definitivetests 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

The Maryland Wave Chaos Group

Tom Antonsen Steve AnlageEd Ott

Elliott Bradshaw Jen-Hao Yeh James Hart Biniyam Taddese