1. Dynamics of Anderson localization
Laboratoire de Physique et Modélisation
des Milieux Condensés
Dynamics of Anderson localization
S.E. Skipetrov
CNRS/Grenoble
(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 to Anderson localization
Wavelength l Mean free paths l, l* Localization length x
Size L of the medium
Ballistic propagation
(no scattering)
Multiple scattering
(diffusion)
Strong (Anderson)
localization
Single scattering
‘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., Nature 390, 671 (1997)

6. Anderson localization of light: Experimental signatures
Rounding of the coherent backscattering cone
Measured by J.P. Schuurmans et al., PRL 83, 2183 (1999)

7. Anderson localization of light: Experimental signatures
Enhanced fluctuations of transmission
Diffuse regime:
Localized regime:
Measured by A.A. Chabanov et al., Nature 404, 850 (2000)

8. And what if we look in dynamics ?

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 of Anderson localization
Supersymmetric nonlinear s –model
Random matrix theory
Self-consistent theory of Anderson localization
Lattice models
Random walk models

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

13. Self-consistent theory of Anderson localization

14. Self-consistent theory of Anderson 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  

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
Data by A.A. Chabanov et al. PRL 90, (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 > tH Mode picture
Diffuse regime:

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

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

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

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

34. Breakdown of the theory for t > tH Path picture

35. Breakdown of the theory for t > tH Path 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, PRL 85, 3388 (2000)
and 1D result (N = 1): B. White et al. PRL 59, 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, PRL 91, (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