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Probability Distribution of Conductance and Transmission Eigenvalues
Zhou Shi and Azriel Z. GenackQueens College of CUNY
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Measurement of transmission matrix t
a
b
tba
Frequency range:10-10.24 GHz: Wave localized14.7-14.94 GHz: Diffusive wave
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Number of waveguide modes : N~ 30 localized frequency range N~ 66 diffusive frequency range
Measurement of transmission matrix t
N/2 points from each polarization
t : N×N
L = 23, 40, 61 and 102 cm
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Transmission eigenvalues n
τn : eigenvalue of the matrix product tt†
2 †
, 1 1( )
N N
ba na b nT t Tr tt
g T
Landauer, Fisher-Lee relation
R. Landauer, Philos. Mag. 21, 863 (1970).
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Transmission eigenvalues n
1
N
nng
O. N. Dorokhov, Solid State Commun. 51, 381 (1984).
Y. Imry, Euro. Phys. Lett. 1, 249 (1986).
Most of channels are “closed” with τn 1/e.Neff ~ g channels are “open” with τn ≥ 1/e.
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Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012)
Spectrum of transmittance T and n
1
N
nnT
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Scaling and fluctuation of conductance
1
N
nng
~ 1g P(lng) is predicted to be highly asymmetric
K. A. Muttalib and P. Wölfle, Phys. Rev. Lett. 83, 3013 (1999).
1 2 1 21( ) ( ) ( , )
N
n n nnP g g P d d d
P(lng) is Gaussian with variance of lng, σ2 = -<lng>
1g
P(g) is a Gaussian distribution1g
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Probability distribution of conductance
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Probability distribution of conductance
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Probability distribution of conductance
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Probability distribution of conductance
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Probability distribution of conductance
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Probability distribution of conductance
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Probability distribution of conductance
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<lnτn> for different value of <lnT> for g = 0.37
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Probability distribution of the spacing of lnτn, s
1
1
ln ln
ln lnn n
n n
s
242
2
32( )
sP s s e
Wigner-Surmise for GUE
t is a complex matrix
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Probability distribution of optical transmittance T
V. Gopar, K. A. Muttalib, and P. Wölfle, Phys. Rev. B 66, 174204 (2002).
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Single parameter scaling
P. W. Anderson et al. Phys. Rev. B 22, 3519 (1980).
2 var(ln )T
Leff = L+2zb, zb: extrapolation length
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Correlation of transmittance in frequency domain
( ) ( ) ( ) ( ) ( )C T T T T 22( 0) var( )C T T T
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Universal conductance fluctuation
R. A. Webb et. al., Phys. Rev. Lett. 54, 2696 (1985). P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).B. L. Altshuler, JETP Lett. 41, 648 (1985).
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Y. Imry, Euro. Phys. Lett. 1, 249 (1986).
Level repulsion
Neff ~ g with τn ≥ 1/e.
1
N
nng
Poisson process: var(Neff)~ <Neff>
var(g)~ <g>
Observation: var(g) independent of <g>
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Level repulsion and Wigner distribution
Y. Imry, Euro. Phys. Lett. 1, 249 (1986).K. A. Muttalib, J. L. Pichard and A. D. Stone, Phys. Rev. Lett. 59, 2475 (1987).
1
1
ln ln
ln lnn n
n n
s
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Level rigidity
F. J. Dyson and M. L. Mehta, J. Math. Phys. 4, 701 (1963).
Single configurationRandom ensemble
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Level rigidity
2
,
1( ( ) ) dmin
i
i
L
A B
N x Ax BL
In an interval of length L, it is defined as the least-squares deviation of the stair case function N(L) from the best fit to a straight line
Poisson Distribution Δ(L)=L/15
Wigner for GUE1
2
1 5( ) ln(2π ) ( )
2π 4L L L
Ο
F. J. Dyson and M. L. Mehta, J. Math. Phys. 4, 701 (1963).
L
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Level rigidity
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Conclusions:
1. Relate the distribution of conductance to underlying transmission eigenvalues
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Conclusions:
1. Relate the distribution of conductance to underlying transmission eigenvalues
2. Observe universal conductance fluctuation for classical waves
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Conclusions:
1. Relate the distribution of conductance to underlying transmission eigenvalues
2. Observe universal conductance fluctuation for classical waves
3. Observe weakening of level rigidity when approachingAnderson Localization