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

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

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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.

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

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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

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Probability distribution of conductance

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for different value of for g = 0.37

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

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

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Correlation of transmittance in frequency domain

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

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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).

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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 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

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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 approaching Anderson Localization

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