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
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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 
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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
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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
vv  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).
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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
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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
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Jen-Hao Yeh
Tom Antonsen
James Hart
Biniyam Taddese
Steve Anlage