Yevgeny Krivolapov, Michael Wilkinson, SF Liad Levy, Moti Segev

1. Yevgeny Krivolapov, Michael Wilkinson, SF Liad Levy, Moti Segev Transport in potentials random in space and time: From Anderson localization to super-ballistic motion

2. Physics of Noise • Is noise uninteresting? • Rules of noise? • Universality classes of noisy behavior? • Random can be systematic!! • Central limit theorem • Statistical Mechanics!!

6. Optics : Transverse Localization Scheme Suggested by De Raedt, Lagengijk & de Vries (PRL 1987) Wave Equation • Scalar and Time harmonic • slow variations of index of refraction • slowly varying amplitude solution + Solid State Physics Optics

8. Disorder and Anderson Localization Periodic Potential: Bloch waves (extended states) What happens if the disorder is also random in time? A wave packet propagating freely through the medium exhibits Ballistic Transport/Diffraction Localization is destroyed – but what else ? - Localization Length Disordered Potential: Localized States Typical scale ξ The wave remains confined in some region of the potential Philip W. Anderson, 1958 (Nobel Prize 1977)

9. Experimental Method – optical induction in a photorefractive We utilize the photorefractive screening effect inside a 10mm-long SBN:60 crystal, in which separation of optically-excited charges (electrons) creates a varying electric space-charge field, which in turn induces a local change in the refractive index via the electro-optic effect. “Writing” the z (time) dependent refractive index (potential) . Ensemble average on many disorder realizations Ky Fourier plane DIFFUSER at Fourier plane Random phase Different kz yields z- variations. Correlations distance in z estimated by the beating length (z) of the 2 most separated plane waves Kx z-variations in the refractive index Same kz for all plane waves yields an Interference pattern constant in z. Ky Fourier plane micron Kx

10. Hyper Transport of Light (experimental results)

11. High momentum Semi-classical

12. Theory for Particles in 1D Independent random variables Interest in spreading to high momentum Wave nature is probably not important. mass=1 Natural to use Fourier expansion in optics and in atom optics

13. Typically for large Scaling solution Class 1

14. 3. Class 3 Singular correlations 2. Class 2 truncated as is the case in many experiments in optics and atom optics

15. No overlap

16. (color online) A phase-space, x'=x-p^{\text{res}}t,p , for two overlapping resonances. The double arrows indicate the region of the resonances. Resonances (1) and (2) overlap. Different colors (shades) describe different trajectories. The horizontal black solid and dashed lines, designate the edges of the resonance chains. The initial conditions were uniformly distributed on the p axis (x(t=0)=0). The system was integrated up-to t=10^4 (dimensionless variables).

17. 1D-wide distribution of the initial conditions Distribution of Fourier components of the potential uniform (color online) as a function of t . Dashed gray line are numerical solution of the equation of motion for H for 1000 initial conditions taken from a Gaussian distribution and averaged over 10 realizations of the potential, V . Different shades and styles of lines designate different N (see legend). The solid thick line is the numerical solution of the Focker-Planck equation. The parameters are, kR=0.1 and A=1e-4.

18. 2D –Gaussian distribution of the Fourier components Of the potential (color online) Same as previous figure, but for a two-dimensional system, for N=800 and parameters commonly used in the experiment, kR=0.0294 and A=2.06e-4.

19. A log-log plot of average squared velocity as a function of time for one dimensional system with an optical potential and a uniform distribution of wave-numbers over a segment. The blue dots represent the result of the Monte-Carlo solution averaged over 20 realizations and the black solid line is the numerical solution of the Fokker-Planck equation for the velocity. The dashed black and red lines are guides for the eye with the corresponding slopes of 2 , 1 and 0. The initial condition was a narrow distribution of velocities for the Fokker-Planck and x=v=0 for the Monte-Carlo calculation. The parameters used for this simulation are, V0=1e-4 , kr=0.1 .

20. A log-log plot of average squared velocity as a function of time for a two dimensional system with an optical potential and a uniform distribution of wave-numbers over a disk. The blue dots represent the result of the Monte-Carlo solution averaged over 20 realizations and the black solid line is the numerical solution of the Fokker-Planck equation for the velocity. The dashed black and red lines are guides for the eye with the corresponding slopes of 2 ,1. The initial condition was a narrow distribution of velocities for the Fokker-Planck and x=v=0 for the Monte-Carlo calculation. The parameters used for this simulation are, V0=1e-2 , kr=0.1 .

21. For large momentum makes sense to use classical limit Is there clash of limits problem??

22. summary • A formula for the diffusion coefficient in momentum in terms of the distribution function of the Fourier components of the potential was developed. Natural for potentials used in optics and atom optics • Classification into Universality classes and Identification of new classes. • Identification of the uniform acceleration regime for short time.

23. Open problems • Identification of regimes that are not uniform acceleration where Fokker—Planck fails • An analytic theory for spreading in coordinate space. • Is there a regime of high momentum where waves spread in a way fundamentally different from particles?