can waves be chaotic?

1

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

http://index.html/

http://www.fzk.de/fzk/idcplg?IdcService=KIT&lang=en

2

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

3

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

4

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

5

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

6

Outline

• Classical Chaos








• Conclusions

7

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

8

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?

9

Outline

• Classical Chaos








• Conclusions

10

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

11

Outline

• Classical Chaos








• Conclusions

12

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

13

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

14

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

15

Outline

• Classical Chaos








• Conclusions

16

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

17

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)

18

Outline

• Classical Chaos








• Conclusions

19

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

20

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

21

Outline

• Classical Chaos








• Conclusions

22

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

23

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

][

24

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/)()()(

25

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

26

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)

27



SSZZ

11

0




Guhr, Müller-Groeling, Weidenmüller, Physics Reports 299, 189 (1998)


28Jen-Hao Yeh, J. Hart, E. Bradshaw, T. Antonsen, E. Ott, S. M. Anlage, Phys. Rev. E 81, 025201(R) (2010)



Short-orbit correction up to 200 cm

SSZZ

11

0





29

Outline

• Classical Chaos








• Conclusions

30

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

31

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

32

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

33

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

34

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

35

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

http://index.html/

36

The Maryland Wave Chaos Group

Tom Antonsen Steve AnlageEd Ott

Elliott Bradshaw Jen-Hao Yeh James Hart Biniyam Taddese

can waves be chaotic?

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