Probability Distribution of Conductance and Transmission Eigenvalues Zhou Shi and Azriel Z. Genack Queens College of CUNY

Measurement of transmission matrix t a b t ba Frequency range: GHz: Wave localized 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 τ n : eigenvalue of the matrix product tt † Landauer, Fisher-Lee relation R. Landauer, Philos. Mag. 21, 863 (1970).

Transmission eigenvalues  n O. N. Dorokhov, Solid State Commun. 51, 381 (1984). Y. Imry, Euro. Phys. Lett. 1, 249 (1986). Most of channels are “closed” with τ n 1/e. N eff ~ g channels are “open” with τ n ≥ 1/e.

Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, (2012) Spectrum of transmittance T and  n

Scaling and fluctuation of conductance P(lng) is predicted to be highly asymmetric K. A. Muttalib and P. Wölfle, Phys. Rev. Lett. 83, 3013 (1999). P(lng) is Gaussian with variance of lng, σ 2 = - P(g) is a Gaussian distribution

Probability distribution of conductance

for different value of for g = 0.37

Probability distribution of the spacing of lnτ n, s 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, (2002).

Single parameter scaling P. W. Anderson et al. Phys. Rev. B 22, 3519 (1980). L eff = L+2z b, z b : extrapolation length

Correlation of transmittance in frequency domain

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 N eff ~ g with τ n ≥ 1/e. Poisson process: var(N eff )~ var(g)~ Observation: var(g) independent of

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).

Level rigidity F. J. Dyson and M. L. Mehta, J. Math. Phys. 4, 701 (1963). Single configurationRandom ensemble

Level rigidity 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 GUE 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 approaching Anderson Localization