analog experiments on quantum chaotic scattering and transportanlage.umd.edu/anlage aps march...
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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
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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
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)
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Outline
• What is Wave Chaos?
• Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• One Experimental Result: Universal Conductance Fluctuations
• Ongoing Work
• Conclusions
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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”
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Outline
• What is Wave Chaos?
• Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• One Experimental Result: Universal Conductance Fluctuations
• Ongoing Work
• Conclusions
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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:
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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
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Outline
• What is Wave Chaos?
• Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• One Experimental Result: Universal Conductance Fluctuations
• Ongoing Work
• Conclusions
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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
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Outline
• What is Wave Chaos?
• Statistical Properties of Wave Chaotic Systems
• Our Microwave Analog Experiment
• One Experimental Result: Universal Conductance Fluctuations
• Ongoing Work
• Conclusions
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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
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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
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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