1

Analog Experiments on Quantum Chaotic Scattering and Transport

Steven M. Anlage, Sameer Hemmady, Xing Zheng, James Hart,Edward Ott, Thomas Antonsen

Research funded by the AFOSR-MURI and DURIP programs

APS March Meeting8 March, 2007

2

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

Was

hbur

n+W

ebb

(198

6)

B (T)

ΔG

(e2 /h

)

C. M

. Mar

cus,

et a

l. (1

992)

B (T)

R (k

Ω)

3

Outline

• What is Wave Chaos?

• Statistical Properties of Wave Chaotic Systems

• Our Microwave Analog Experiment

• One Experimental Result: Universal Conductance Fluctuations

• Ongoing Work

• Conclusions

4

It makes no sense to talk about“diverging trajectories” for waves

1) Classical chaotic systems have diverging trajectories

qi, pi qi+Δqi, pi +Δpi

Regular system

2-Dimensional “billiard” tables with hard wall boundaries

Newtonianparticletrajectories

qi, pi qi+Δqi, pi +Δpi

Chaotic system

Wave Chaos?

2) Linear wave systems can’t be chaotic

3) However in the semiclassical limit, you can think about raysWave Chaos concerns solutions of wave equations which, in the semiclassical

limit, can be described by ray trajectories

In the ray-limitit is possible to define chaos

“ray chaos”

5

Outline

• What is Wave Chaos?

• Statistical Properties of Wave Chaotic Systems

• Our Microwave Analog Experiment

• One Experimental Result: Universal Conductance Fluctuations

• Ongoing Work

• Conclusions

6

Wave Chaos in Bounded RegionsWhich Billiards Show Ray Chaos?

Consider a two-dimensional infinite square-well potential (i.e. a billiard) which shows chaos in the classical limit:

Hard Walls

Bow-tie

L

Sinai billiard

Bunimovich stadium

r1

r2

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

The statistical properties of eigenvalues and eigenfunctions of closed billiard systems are in excellent agreement with the predictions of Random Matrix Theory (RMT)

Can RMT work for open systems?

Can RMT include the effects of losses or “de-coherence” in real systems?

Good general intro toRMT in quantum physics:

7

The Difficulty in Making Predictions in Wave Chaotic Systems…

8.7 8.8 8.9 9.0 9.1 9.2 9.3

200

400

600

800

1000

Abs

[Zca

v]

Frequency [GHz]

Perturbation Position 1

8.7 8.8 8.9 9.0 9.1 9.2 9.3

200

400

600

800

1000

Abs

[Zca

v]

Frequency [GHz]

Perturbation Position 2

Antenna Port

(Ω)

05.0~Lλ

Ele

ctro

mag

netic

W

ave

Impe

danc

e

Extreme sensitivityto small perturbations

We must resort to astatistical description

In our experiments we systematically move the perturber to generate many “realizations” of the system

8

Outline

• What is Wave Chaos?

• Statistical Properties of Wave Chaotic Systems

• Our Microwave Analog Experiment

• One Experimental Result: Universal Conductance Fluctuations

• Ongoing Work

• Conclusions

9

Microwave Cavity Analog of a 2DEGOur Experiment: A clean, zero temperature, quantum dot withno Coulomb or correlation effects! Table-top experiment!

Ez

Bx By

( )

boundariesatwith

VEm

n

nnn

0

022

2

=Ψ

=Ψ−+Ψ∇h

boundariesatEwithEkE

nz

nznnz

00

,

,2

,2

=

=+∇

Schrödinger equation

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 cm

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Microwave Scattering Experiment

Antenna Entry Points

Circular Arc

R=107 cmR=107 cm

Port 1

Port 2

|| 11Z

6 9 12 15 181

10

100

5

Log 10

(|Z|)

Frequency (GHz)

|| RadZ

|| z

The statistics of z are “universal” and are describedby Random Matrix Theory (RMT)

S. Hemmady, et al. PRL 94, 014102 (2005).

Cavity Base

Cavity Lid

Diameter (2a)

CoaxialCable

Height (h=7.8mm)Antenna CrossAntenna Cross--SectionSection

RadRadRad iXRZ +=Measured with outgoing boundary condition

iKiXRZ ⇔+=⎟⎟⎠

⎞⎜⎜⎝

⎛=

2221

1211

SSSS

S SSZZ

−+

=11

0

Wigner+Eisenbud (1947)Reaction matrix

6 9 12 15 181

10

100

5

Log 10

(|Z|)

Frequency (GHz)

|| 11Z

Rad

Rad

Rad RXX

jRR

z−

+=

Works for any number of ports/channels

11

Outline

• What is Wave Chaos?

• Statistical Properties of Wave Chaotic Systems

• Our Microwave Analog Experiment

• One Experimental Result: Universal Conductance Fluctuations

• Ongoing Work

• Conclusions

12

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

21

2122

NNNN

heG

+=

212 2

heG = N1=N2=1 for our

experiment

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

212)(

2

heGGP δ

C. M. Marcus (1992)

2-D Electron Gas

electron mean free path >> system size

Ballistic Quantum Transport

∑∑= =

=1 2

1 1

222 N

n

N

mnmS

heG

Quantum interference → Fluctuations in G ~ e2/h“Universal Conductance Fluctuations”

)/2/(2/1)(

2 heGGP =

An ensemble of quantum dots has a distributionof conductance values:

(N1=N2=1)

Landauer-Büttiker

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De-Phasing in Quantum TransportConductance measurements through 2-Dimensional quantum dotsshow behavior that is intermediate between:

Ballistic Quantum transportIncoherent Classical transport

Why? “De-Phasing” of the electrons

G/G0

P(G/G0)

G0 = 2e2/h

1/2 1

“semi-classical”

ballistic

One class of models: Add a “de-phasing lead” with Nφ modes with transparency Γφ.. 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

Actuallymeasured

incoherent

P(G)

Bro

uwer

+Bee

nakk

er(1

997)

We can test these predictions in detail:

)/( 212 VVIG −=

De-phasinglead

Büttiker (1986)

γ = 0 Pure quantum transportγ → ∞ Incoherent classical limit

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The Microwave Cavity Mimics a 2-Dimensional Quantum Dot

Uniformly-distributed microwave losses are equivalent to quantum “de-phasing”

Microwave Losses Quantum De-Phasing

3dB bandwidthof resonances

ωδωαΔ

= dB3

Loss Parameter:

Mean spacingbetween resonances

γ = 0 Pure quantum transportγ → ∞ Classical limit

By comparing the Poynting theorem for a cavity with uniform lossesto the continuity equation for probability density, one finds:

γπα ↔4

Brouwer+Beenakker (1997)

Metallic Perturbations

microwave Absorberx 0 : Loss Case 0x 16 : Loss Case 1x 32 : Loss Case 2

Port 22a=0.635 mm

Port 12a=1.27 mm

Metallic PerturbationsMetallic Perturbations

microwave Absorberx 0 : Loss Case 0x 16 : Loss Case 1x 32 : Loss Case 2

microwave Absorberx 0 : Loss Case 0x 16 : Loss Case 1x 32 : Loss Case 2

microwave Absorberx 0 : Loss Case 0x 16 : Loss Case 1x 32 : Loss Case 2

Port 22a=0.635 mm

Port 12a=1.27 mm

Variationof αNo absorbers

Many absorbers

8.0/~2 ≈Qk 8.0/~2 ≈Qk

DataData

14≈MLEγTheoryTheory

14≈MLEγ 14≈MLEγTheoryTheory

Determinationof γ: Fit to

PDF(eigenvalues of SS+)Pγ(T1, T2)

Brouwer+Beenakker PRB (1997)

~105 datapoints

10≅γ

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Potential with uniform imaginary part

Efetov (1995)McCann+Lerner (1996)

Why does the Microwave Cavity ↔ Quantum Dot Analogy Work?

It is known that for electromagnetic systems: Uniformly distributed losses are equivalent to a large number of “ports”, each with small transmission

PhysicalPorts

Uniformly Lossy Cavity

Lewenkopf, Müller, Doron (1992)Schanze (2005)

Potential with zero imaginary partBrouwer+Beenakker (1997)

φN channels

PhysicalPorts

Lossless cavity

Parasitic Ports (Equivalent Loss Channels)

“Locally weakabsorbing limit”

Zirnbauer (93)

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0.40 0.45 0.50 0.550

10

20

30

40

50 2.82≅⟩⟨Tγ

1.35≅⟩⟨Tγ

1.272≅⟩⟨Tγ)7(x

P(G

)

G

0.3 0.4 0.5 0.60

4

8

Conductance Fluctuations of the Surrogate Quantum Dot

2.11≅⟩⟨Tγ

G

P(G

)

RMT predictions (solid lines)(valid only for γ >> 1)

Data (symbols)

)/2/( 2 he

)/2/( 2 he

LowLoss / Dephasing

RMT prediction (valid only for γ >> 1)

Data (symbols)

HighLoss / Dephasing

RM Monte Carlo computation

-4 -3 -2 -1 0 1 2 3-2

-1

0

)2/1(2 −= Gx γ

])([10 γGPLog

]2

)||1([||

10

xexxLog−−+ 8.56≅⟩⟨Tγ

6.91≅⟩⟨Tγ

5.220≅⟩⟨Tγ

Scaling Prediction for P(G)Brouwer+Beenakker (1997)

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)

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Conclusions

Demonstrated the advantage of impedance (reaction matrix) inquantitative understanding of wave/quantum chaotic phenomena

Ongoing work on time-dependent scattering

The microwave analog experiment can provide clean, definitivetests of some theories of quantum chaotic scattering

Some Relevant Publications:Sameer Hemmady, et al., Phys. Rev. Lett. 94, 014102 (2005)

Sameer Hemmady, et al., Phys. Rev. E 71, 056215 (2005)Xing Zheng, Thomas M. Antonsen Jr., Edward Ott, Electromagnetics 26, 3 (2006)

Xing Zheng, Thomas M. Antonsen Jr., Edward Ott, Electromagnetics 26, 37 (2006)Xing Zheng, et al., Phys. Rev. E 73 , 046208 (2006)S. Hemmady, et al., Phys. Rev. E 74 , 036213 (2006)

Sameer Hemmady, et al., Phys. Rev. B 74, 195326 (2006)

Many thanks to: R. Prange, S. Fishman, Y. Fyodorov, D. Savin, P. Brouwer, P. Mello,

A. Richter, L. Sirko, J.-P. Parmantier

http://www.csr.umd.edu/anlage/AnlageQChaos.htm