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Can Waves Be Chaotic?
Jen-Hao Yeh, Biniyam Taddese, James Hart, Elliott BradshawEdward Ott, Thomas Antonsen, Steven M. Anlage
Research funded by AFOSR and the ONR-MURI and DURIP programs
Karlsruhe Institute of Technology
16 April, 2010
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Outline
• Classical Chaos
• What is Wave Chaos?
• Universal Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• Statistical Properties of Closed Wave Chaotic Systems
• Properties of Open / Scattering Wave Chaotic Systems
• The Problem of Non-Universal and System-Specific Effects
• ‘De-Coherence’ in Quantum Transport
• Conclusions
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Simple Chaos1-Dimensional Iterated Maps
The Logistic Map: )1(41 nnn xxx Parameter: Initial condition: 0x
10 15 20 25 30
0.2
0.3
0.4
0.5
5.0
100.00 x
Iteration number
x
10 15 20 25 30
0.2
0.3
0.4
0.5
0.6
0.7
0.8
8.0
Iteration number
x
10 15 20 25 30
0.2
0.4
0.6
0.8
1
0.1
Iteration number
x
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Extreme Sensitivity to Initial Conditions1-Dimensional Iterated Maps
The Logistic Map: )1(41 nnn xxx 0.1
Change the initial condition (x0) slightly…
10 15 20 25 30
0.2
0.4
0.6
0.8
1
x
Iteration Number
101.00 x
100.00 x
Although this is a deterministic system, Difficulty in making long-term predictions Sensitivity to noise
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Classical Chaos in BilliardsBest characterized as “extreme sensitivity to initial conditions”
qi, pi qi+qi, pi +pi
Regular system
2-Dimensional “billiard” tables
Newtonianparticletrajectories
ii
ii
qHppHq
/
/
VTH Hamiltonian
qi+qi, pi +piqi, pi
Chaotic system
)0(
)(t
)0(1x
)0(2x )(1 tx
)(2 tx
tet )0()(
Lyapunov exponent
0
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Outline
• Classical Chaos
• What is Wave Chaos?
• Universal Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• Statistical Properties of Closed Wave Chaotic Systems
• Properties of Open / Scattering Wave Chaotic Systems
• The Problem of Non-Universal and System-Specific Effects
• ‘De-Coherence’ in Quantum Transport
• Conclusions
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It makes no sense to talk about“diverging trajectories” for waves
1) Waves do not have trajectoriesWave 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 linear wave equations which, in the semiclassical limit, can be described by chaotic ray trajectories
In the ray-limitit is possible to define chaos
“ray chaos”
Maxwell’s equations, Schrödinger’s equation are linear
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How Common is Wave Chaos?
Consider an infinite square-well potential (i.e. a billiard) that shows chaos in the classical limit:
Solve the wave equation in the same potential well
Examine the solutions in the semiclassical regime: 0 < << L
Hard Walls Bow-tie
L
Sinai billiard
Bunimovich stadium
YES But how?
Bunimovich Billiard
Some example physical systems:Nuclei, 2D electron gas billiards, acoustic waves in irregular blocks or rooms, electromagnetic waves in enclosures
Will the chaos present in the classical limit have an affect on the wave properties?
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Outline
• Classical Chaos
• What is Wave Chaos?
• Universal Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• Statistical Properties of Closed Wave Chaotic Systems
• Properties of Open / Scattering Wave Chaotic Systems
• The Problem of Non-Universal and System-Specific Effects
• ‘De-Coherence’ in Quantum Transport
• Conclusions
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H
Random Matrix Theory (RMT)Wigner; Dyson; Mehta; Bohigas …
The RMT Approach: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
This hypothesis has been tested in many systems:Nuclei, atoms, molecules, quantum dots, acoustics (room, solid body, seismic), optical resonators, random lasers,…
Some Questions:Is this hypothesis supported by data in other systems?
Can losses / decoherence be included?What causes deviations from RMT predictions?
Hypothesis: Complicated Quantum/Wave systems that have chaotic classical/raycounterparts possess universal statistical properties described byRandom Matrix Theory (RMT) “BGS Conjecture”
Cassati, 1980 Bohigas, 1984
EH
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Outline
• Classical Chaos
• What is Wave Chaos?
• Universal Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• Statistical Properties of Closed Wave Chaotic Systems
• Properties of Open / Scattering Wave Chaotic Systems
• The Problem of Non-Universal and System-Specific Effects
• ‘De-Coherence’ in Quantum Transport
• Conclusions
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Microwave Cavity Analog of a 2D Quantum Infinite Square Well
Table-top experiment!
Ez
Bx By
boundariesatwith
VEm
n
nnn
0
022
2
Schrödinger equation
boundariesatEwith
EkE
nz
nznnz
0
0
,
,2
,2
Helmholtz equation
Stöckmann + Stein, 1990Doron+Smilansky+Frenkel, 1990Sridhar, 1991Richter, 1992
d ≈ 8 mm
An empty “two-dimensional” electromagnetic resonator
A. Gokirmak, et al. Rev. Sci. Instrum. 69, 3410 (1998)
~ 50 cmThe only propagatingmode for f < c/d: Metal walls
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The Experiment:A simplified model of wave-chaotic scattering systems
A thin metal box
ports
Side view
21.6 cm
43.2 cm
0.8 cm
λ
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Microwave-Cavity Analog of a 2DInfinite Square Well
with Coupling to Scattering States
Network Analyzer (measures S-matrix vs. frequency)
Thin Microwave Cavity PortsElectromagnetWe measure from 500 MHz – 19 GHz, covering about 750 modes in the semi-classical limit
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Outline
• Classical Chaos
• What is Wave Chaos?
• Universal Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• Statistical Properties of Closed Wave Chaotic Systems
• Properties of Open / Scattering Wave Chaotic Systems
• The Problem of Non-Universal and System-Specific Effects
• ‘De-Coherence’ in Quantum Transport
• Conclusions
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Wave Chaotic Eigenfunctions (~ closed system)with and without Time Reversal Invariance (TRI)
D. H. Wu and S. M. Anlage, Phys. Rev. Lett. 81, 2890 (1998).
TRI Broken(GUE)
TRI Preserved(GOE)
r = 106.7 cm
r = 64.8 cm
0 10 20 30 40
20
10
0
20
10
00 10 20 30 40
x (cm)
y (c
m)
181512840
A2||
Ferrite
a)
b)
13.69 GHz
13.62 GHz
A magnetizedferrite in thecavity breaks
TRI
De-magnetizedferrite
Bext
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0.001
0.01
0.1
1
10
0 1 2 3 4 5 6 72
P (
)
2
GUE (TRSB)
GOE (TRS)GUE (TRSB)
GOE (TRS)
0
1
0 1 2
D. H. Wu, et al. Phys. Rev. Lett. 81, 2890 (1998).
Probability Amplitude Fluctuationswith and without Time Reversal Invariance (TRI)
P() = (2)-1/2 e-/2 TRI (GOE)e-TRI Broken (GUE)
“Hot Spots”
RMT
Prediction:
Broken TRI
(TRI)
(TRI)
(Broken TRI)
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Outline
• Classical Chaos
• What is Wave Chaos?
• Universal Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• Statistical Properties of Closed Wave Chaotic Systems
• Properties of Open / Scattering Wave Chaotic Systems
• The Problem of Non-Universal and System-Specific Effects
• ‘De-Coherence’ in Quantum Transport
• Conclusions
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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
(k
) m
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
dd
Proton energy
Compound nuclear reaction
12
Incoming Channel
Outgoing Channel
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Universal Scattering Statistics
Despite the very different physical circumstances, these measuredscattering fluctuations have a common underlying origin!
Universal Properties of the Scattering Matrix:
Re[S]
Im[S]
RMT prediction: Eigenphases of S uniformly distributed on the unit circle
mn
iiN
mn eeP ~),...,( 21
GSE 4GUE 2GOE 1
Eigenphaserepulsion
Nuclear ScatteringCross Section
dd
2D Electron GasQuantum Dot
Resistance)(BR
Microwave CavityScattering Matrix,
Impedance, Admittance, etc.
ieSS Unitary Case
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Outline
• Classical Chaos
• What is Wave Chaos?
• Universal Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• Statistical Properties of Closed Wave Chaotic Systems
• Properties of Open / Scattering Wave Chaotic Systems
• The Problem of Non-Universal and System-Specific Effects
• ‘De-Coherence’ in Quantum Transport
• Conclusions
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The Most Common Non-Universal Effects:1) Non-Ideal Coupling between external scattering states and internal modes (i.e. Port properties)
Universal Fluctuations are Usually Obscured by Non-Universal System-Specific Details
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 port and fixed walls of the billiard
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N-Port Description of an Arbitrary Scattering System
N – Port
System
N Ports
Voltages and Currents,
Incoming and Outgoing Waves
Z matrixS matrix
1V
1VV1 , I1
VN , IN
NV
NV
Complicated Functions of frequency Detail Specific (Non-Universal)
)(),( SZ)()( 0
10 ZZZZS
NN V
V
V
S
V
V
V
2
1
2
1
][
NN I
II
V
VV
2
1
2
1
][
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iRiXZ avgavg Universally Fluctuating ComplexQuantity with Mean 1 (0) for the
Real (Imaginary) Part.Predicted by RMT
SemiclassicalExpansion over
Short Orbits
Complex Radiation Impedance(characterizes the non-universal coupling)
Index of ‘Short Orbit’of length l Stability of orbit
Action of orbit
Theory of Non-Universal Wave Scattering Properties
James Hart, T. Antonsen, E. Ott, Phys. Rev. E 80, 041109 (2009)
Port
ZCavity
Port
ZR
The waves donot return to the port
Perfectly absorbingboundaryCavity
Orbit Stability Factor:►Segment length►Angle of incidence►Radius of curvature of wallAssumes foci and caustics are absent!
Orbit Action:►Segment length►Wavenumber►Number of Wall Bounces
1-Port, Loss-less case:
b(l)
iLliklblbRRavg
PorteDpRZZ 4/)()()(
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Prob
abili
ty D
ensi
ty
-2 -1 0 1 20.0
0.3
0.62a=0.635mm
2a=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 BASECAVITY BASE
CrossSection View
CAVITY LIDCAVITY 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
RXX
jR
Rz
RAW Impedance PDF NORMALIZED Impedance PDF
Metallic Perturbations
Port 1
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SSZZ
11
0
ieSS RMT prediction:
The distribution of the phase of S should be uniform from 0 ~ 2π
Independent of Loss!
Guhr, Müller-Groeling, Weidenmüller, Physics Reports 299, 189 (1998)
A universal property:uniform phase of the scattering parameter
From 100 realizations
10.0 ~ 10.5 GHz (about 14 modes)
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From 100 realizations
10.0 ~ 10.5 GHz (about 14 modes)
SSZZ
11
0
ieSS RMT prediction:
The distribution of the phase of S should be uniform from 0 ~ 2π
Independent of Loss!
Guhr, Müller-Groeling, Weidenmüller, Physics Reports 299, 189 (1998)
A universal property:uniform phase of the scattering parameter
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28Jen-Hao Yeh, J. Hart, E. Bradshaw, T. Antonsen, E. Ott, S. M. Anlage, Phys. Rev. E 81, 025201(R) (2010)
From 100 realizations
10.0 ~ 10.5 GHz (about 14 modes)
Short-orbit correction up to 200 cm
SSZZ
11
0
ieSS RMT prediction:
The distribution of the phase of S should be uniform from 0 ~ 2π
Independent of Loss!
A universal property:uniform phase of the scattering parameter
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Outline
• Classical Chaos
• What is Wave Chaos?
• Universal Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• Statistical Properties of Closed Wave Chaotic Systems
• Properties of Open / Scattering Wave Chaotic Systems
• The Problem of Non-Universal and System-Specific Effects
• ‘De-Coherence’ in Quantum Transport
• Conclusions
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Quantum TransportMesoscopic and Nanoscopic systems show quantum effects in transport:
Conductance ~ e2/h per channelWave interference effects“Universal” statistical properties
However these effects are partially hidden by finite-temperatures, electron de-phasing, and electron-electron interactions
Also theory calculates many quantities that are difficult to observe experimentally, e.g.scattering matrix elementscomplex wavefunctionscorrelation functions
Develop a simpler experiment that demonstrates the wave properties without all of the complications► Electromagnetic resonator analog of quantum transport
Was
hbur
n+W
ebb
(198
6)
B (T)
G (e
2 /h)
C. M
. Mar
cus,
et a
l. (1
992)
B (T)
R (k
)
Pono
mar
enko
Sci
ence
(200
8)
Graphene Quantum Dot
T = 4 K
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Quantum vs. Classical Transport in Quantum Dots
)/( 212 VVIG
Lead 1:Waveguide with
N1 modes
Lead 2:Waveguide with
N2 modes
Ray-Chaotic 2-DimensionalQuantum Dot
Incoherent Semi-Classical Transport
N1=N2=1 for ourexperiment
C. M. Marcus (1992)
2-D Electron Gas
electron mean free path >> system size
Ballistic Quantum Transport
Quantum interference Fluctuations in G ~ e2/h“Universal Conductance Fluctuations”
An ensemble of quantum dots has a distributionof conductance values:
(N1=N2=1)
Landauer-Büttiker
RMTPrediction
1 2
1 1
222 N
n
N
mnmS
heG
)/2/(2/1)(
2 heGGP
212 2
heG
212)(
2
heGGP
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De-Phasing in (Chaotic) Quantum TransportConductance measurements through 2-Dimensional quantum dotsshow behavior that is intermediate between:
Ballistic Quantum transportIncoherent Classical transport
Why? “De-Phasing” of the electrons
One class of models: Add a “de-phasing lead” with N modes with transparency Electrons that visit the lead are re-injected with random phase.
G/G0
P(G/G0)
G0 = 2e2/h
1/2 1
“semi-classical”
ballistic
G/G0
P(G/G0)
G0 = 2e2/h
1/2 1
“semi-classical”
ballistic
Actuallymeasured
incoherent
P(G)
Bro
uwer
+Bee
nakk
er (1
997)
We can test these predictions in detail:
)/( 212 VVIG
De-phasinglead
Büttiker (1986)
= 0 Pure quantum transport ∞ Incoherent classical limit
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The Microwave Cavity Mimics the Scattering Properties of a 2-Dimensional Quantum Dot
Uniformly-distributed microwave losses are equivalent to quantum “de-phasing”
Microwave Losses Quantum De-Phasing
3dB bandwidthof resonances
dB3
Loss Parameter:
Mean spacingbetween resonances
= 0 Pure quantum transport ∞ Classical limit
is varied by adding microwave absorber to the walls determined from fits to PDF(Z)
is determined from fits to PDF(eigenvalues of SS+)
By comparing the Poynting theorem for a cavity with uniform lossesto the continuity equation for probability density, one finds:
Brouwer+Beenakker (1997)
No absorbers
Many absorbers
f
|S21|
f
|S21|
1
1
4
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0.40 0.45 0.50 0.550
10
20
30
40
50 2.82T
1.35T
1.272T)7(x
P(G
)
G
Conductance Fluctuations of the Surrogate Quantum Dot
0.3 0.4 0.5 0.60
4
82.11T
G
P(G
)
RMT predictions (solid lines)(valid only for >> 1)
Data (symbols)
)/2/( 2 he
)/2/( 2 he
HighLoss / Dephasing
LowLoss / Dephasing
RMT prediction (valid only for >> 1)
Data (symbols)
RM Monte Carlo computation
222
221
212
211
221
222
212
2112
122
2)1)(1(
)/2/(ssssssss
sheG
Ordinary Transmission Correction for waves that visit the “parasitic channels”
SurrogateConductance
Beenakker RMP (97)
Ensemble Measurementsof the Microwave Cavity
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Conclusions
When the wave properties of ray-chaotic systems are studied, one finds certain universal properties:
Eigenvalue repulsion, statisticsStrong Eigenfunction fluctuationsScattering fluctuations
Chaos does play a role in Electromagnetism and Quantum Mechanics in the semi-classical limit
Our microwave analog experiment directly simulates quantum mechanical systems with “de-phasing”
Ongoing experiments:Tests of RMT in the loss-less limit: Superconducting cavity
Pulse-propagation and tests of RMT in the time-domainClassical analogs of quantum fidelity and the Loschmidt echo
Time-reversed electromagnetics and quantum mechanics
Many thanks to: P. Brouwer, M. Fink, S. Fishman, Y. Fyodorov, T. Guhr, U. Kuhl, P. Mello, R. Prange, A. Richter, D. Savin, F. Schafer, L. Sirko, H.-J. Stöckmann, J.-P. Parmantier
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The Maryland Wave Chaos Group
Tom Antonsen Steve AnlageEd Ott
Elliott Bradshaw Jen-Hao Yeh James Hart Biniyam Taddese