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Dynamics of Anderson localization

Laboratoire de Physique et Modélisation des Milieux Condensés. Dynamics of Anderson localization. S.E. Skipetrov CNRS/Grenoble http://lpm2c.grenoble.cnrs.fr/People/Skipetrov/ (Work done in collaboration with Bart A. van Tiggelen). Multiple scattering of light. Incident wave. Detector.

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Dynamics of Anderson localization

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  1. Laboratoire de Physique et Modélisation des Milieux Condensés Dynamics ofAnderson localization S.E. Skipetrov CNRS/Grenoble http://lpm2c.grenoble.cnrs.fr/People/Skipetrov/ (Work done in collaboration with Bart A. van Tiggelen)

  2. Multiple scattering of light Incident wave Detector Random medium

  3. Multiple scattering of light Incident wave Detector l l Random medium L

  4. From single scattering toAnderson localization Wavelength l Mean free paths l, l* Localization length x Size L of the medium Ballistic propagation (no scattering) Strong (Anderson) localization Multiple scattering (diffusion) Single scattering 0 ‘Strength’ of disorder

  5. Anderson localization of light: Experimental signatures Exponential scaling of average transmission with L Diffuse regime: Localized regime: L Measured by D.S. Wiersma et al.,Nature390, 671 (1997)

  6. Anderson localization of light: Experimental signatures Rounding of the coherent backscattering cone Measured by J.P. Schuurmans et al.,PRL83, 2183 (1999)

  7. Anderson localization of light: Experimental signatures Enhanced fluctuations of transmission Diffuse regime: Localized regime: Measured by A.A. Chabanov et al.,Nature404, 850 (2000)

  8. And what if we look in dynamics ? L

  9. Time-dependent transmission:diffuse regime (L) Diffusion equation + Boundary conditions

  10. Time-dependent transmission:diffuse regime (L) How will be modified when localization is approached ?

  11. Theoretical description ofAnderson localization • Supersymmetric nonlinear s –model • Random matrix theory • Self-consistent theory of Anderson localization • Lattice models • Random walk models

  12. Theoretical description ofAnderson localization • Supersymmetric nonlinear s –model • Random matrix theory • Self-consistent theory of Anderson localization • Lattice models • Random walk models

  13. Self-consistent theory ofAnderson localization

  14. Self-consistent theory ofAnderson localization The presence of loops increases return probability as compared to ‘normal’ diffusion Diffusion slows down Diffusion constant should be renormalized

  15. Generalization to open media Loops are less probable near the boundaries Slowing down of diffusion is spatially heterogeneous Diffusion constant becomes position-dependent

  16. Quasi-1D disordered waveguide Number of transverse modes: Dimensionless conductance: Localization length:

  17. Mathematical formulation Diffusion equation + Self-consistency condition + Boundary conditions

  18. Stationary transmission: W = 0

  19. ‘Normal’ diffusion: g Path of integration Diffusion poles

  20. ‘Normal’ diffusion: g

  21. ‘Normal’ diffusion: g

  22. From poles to branch cuts: g Branch cuts

  23. Leakage function PT()

  24. Time-dependent diffusion constant Diffuse regime: Closeness of localized regime is manifested by

  25. Time-dependent diffusion constant

  26. Time-dependent diffusion constant Diffusion constant Time Data by A.A. Chabanov et al. PRL90, 203903 (2003)

  27. Time-dependent diffusion constant Width Center of mass

  28. Time-dependent diffusion constant Consistent with supersymmetric nonlinear s-model [A.D. Mirlin, Phys. Rep.326, 259 (2000)] for

  29. Breakdown of the theory for t > tHMode picture Diffuse regime:

  30. Breakdown of the theory for t > tHMode picture Diffuse regime: ‘Prelocalized’ mode The spectrum is continuous

  31. Breakdown of the theory for t > tHMode picture Diffuse regime: ‘Prelocalized’ mode Only the narrowest mode survives

  32. Breakdown of the theory for t > tHMode picture Diffuse regime: Localized regime: The spectrum is continuous There are many modes

  33. Breakdown of the theory for t > tHMode picture Diffuse regime: Localized regime: Only the narrowest mode survives in both cases Long-time dynamics identical ?

  34. Breakdown of the theory for t > tHPath picture

  35. Breakdown of the theory for t > tHPath picture

  36. Beyond the Heisenberg time Randomly placed screens with random transmission coefficients

  37. Time-dependent reflection ‘Normal’ diffusion Localization Reflection coefficient Time Consistent with RMT result: M. Titov and C.W.J. Beenakker, PRL85, 3388 (2000) and 1D result (N = 1): B. White et al. PRL59, 1918 (1987)

  38. Generalization to higher dimensions • Our approach remains valid in 2D and 3D • For and we get Consistent with numerical simulations in 2D: M. Haney and R. Snieder, PRL91, 093902 (2003)

  39. Conclusions • Dynamics of multiple-scattered waves in quasi-1D disordered media can be described by a self-consistent diffusion model up to • For and we find a linear decrease of the time-dependent diffusion constant with in any dimension • Our results are consistent with recent microwave experiments, supersymmetric nonlinear s-model, random matrix theory, and numerical simulations

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