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Probability Distribution of Conductance and Transmission Eigenvalues

Probability Distribution of Conductance and Transmission Eigenvalues. Zhou Shi and Azriel Z. Genack Queens College of CUNY. Measurement of transmission matrix t. Frequency range: 10-10.24 GHz: Wave localized 14.7-14.94 GHz: Diffusive wave. b. a. t ba.

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Probability Distribution of Conductance and Transmission Eigenvalues

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

  2. Measurement of transmission matrix t Frequency range: 10-10.24 GHz: Wave localized 14.7-14.94 GHz: Diffusive wave b a tba

  3. Measurement of transmission matrix t Number of waveguide modes : N~ 30 localized frequency range N~ 66 diffusive frequency range N/2 points from each polarization t : N×N L = 23, 40, 61 and 102 cm

  4. Transmission eigenvalues tn τn: eigenvalue of the matrix product tt† Landauer, Fisher-Lee relation R. Landauer, Philos. Mag. 21, 863 (1970).

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

  6. Spectrum of transmittance T and tn Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012)

  7. Scaling and fluctuation of conductance P(g) is a Gaussian distribution P(lng) is predicted to be highly asymmetric P(lng) is Gaussian with variance of lng, σ2 = -<lng> K. A. Muttalib and P. Wölfle, Phys. Rev. Lett. 83, 3013 (1999).

  8. Probability distribution of conductance

  9. Probability distribution of conductance

  10. Probability distribution of conductance

  11. Probability distribution of conductance

  12. Probability distribution of conductance

  13. Probability distribution of conductance

  14. Probability distribution of conductance

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

  16. Probability distribution of the spacing of lnτn, s t is a complex matrix Wigner-Surmise for GUE

  17. Probability distribution of optical transmittance T V. Gopar, K. A. Muttalib, and P. Wölfle, Phys. Rev. B 66, 174204 (2002).

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

  19. Correlation of transmittance in frequency domain

  20. 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).

  21. Level repulsion Neff ~ g with τn ≥ 1/e. Poisson process: var(Neff)~ <Neff> var(g)~ <g> Observation: var(g) independent of <g> Y. Imry, Euro. Phys. Lett. 1, 249 (1986).

  22. 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).

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

  24. Level rigidity N(L) L 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 Wigner for GUE Poisson Distribution Δ(L)=L/15 F. J. Dyson and M. L. Mehta, J. Math. Phys. 4, 701 (1963).

  25. Level rigidity

  26. Conclusions: 1. Relate the distribution of conductance to underlying transmission eigenvalues

  27. Conclusions: 1. Relate the distribution of conductance to underlying transmission eigenvalues 2. Observe universal conductance fluctuation for classical waves

  28. 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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