probability distribution of conductance and transmission eigenvalues zhou shi and azriel z. genack...

Probability Distribution of Conductance and Transmission Eigenvalues

Zhou Shi and Azriel Z. GenackQueens College of CUNY

Measurement of transmission matrix t

a

b

tba

Frequency range:10-10.24 GHz: Wave localized14.7-14.94 GHz: Diffusive wave

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

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).

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.

Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012)

Spectrum of transmittance T and n

1

N

nnT

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

Probability distribution of conductance

Probability distribution of conductance

Probability distribution of conductance

Probability distribution of conductance

Probability distribution of conductance

Probability distribution of conductance

Probability distribution of conductance

<lnτn> for different value of <lnT> for g = 0.37

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

Probability distribution of optical transmittance T

V. Gopar, K. A. Muttalib, and P. Wölfle, Phys. Rev. B 66, 174204 (2002).

Single parameter scaling

P. W. Anderson et al. Phys. Rev. B 22, 3519 (1980).

2 var(ln )T

Leff = L+2zb, zb: extrapolation length

Correlation of transmittance in frequency domain

( ) ( ) ( ) ( ) ( )C T T T T 22( 0) var( )C T T T

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).

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>

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

Level rigidity

F. J. Dyson and M. L. Mehta, J. Math. Phys. 4, 701 (1963).

Single configurationRandom ensemble

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

Level rigidity

Conclusions:

1. Relate the distribution of conductance to underlying transmission eigenvalues

Conclusions:

1. Relate the distribution of conductance to underlying transmission eigenvalues

2. Observe universal conductance fluctuation for classical waves

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