1. Gabriel Cwilich Yeshiva University NeMeSyS Symposium 10/26/2008 Luis S. Froufe Perez Ecole Centrale, Paris Juan Jose Saenz Univ. Autonoma, Madrid Antonio Garcia Martin CSIC, Madrid INTENSITY CORRELATIONS IN COHERENT TRANSPORT IN DISORDERED MEDIA

2. OUTLINE Coherent transport Correlations Experimental motivation Transport Random Matrix Theory Numerical Results Conclusions

3. COHERENT PROPAGATION Elastic scattering Phase preserved Direction randomized over l l < L < l in Weak Localization Universal Conductance Fluctuations Berkovits & Feng, Phys Rep 238 (1994) Mesoscopic Physics of electrons and Photons E. Akkermans and G. Montambaux l No need for low T !! No need for miniaturization !!

4. CORRELATIONS Feng, Kane, Lee & Stone PRL 61, 834 (1988) Diagrammatic theory for I - I correlations g = (e 2 /h)  ab T ab (Landauer) T ab = | t ab | 2 Transmission: T a =  b T ab ; g =  ab T ab = N = N 2 follows from isotropy

5. C 1 is essentially (F E ) 2 (square of field correlation) In transmission: C aba’b’ = { C 1  aa’  bb’ + C 2 (  aa’ +  bb’ ) + C 3 } This leads to interesting experimental effects: C aba’b’ = = C 1 + C 2 + C 3 No crossing, one crossing, two crossings prop to 1, 1/g, 1/g 2

6. Fluctuations of the intensity at one speckle pattern (Shapiro, PRL 86) SHORT RANGE Fluctuations in total transmission (Stephen & GC, PRL 87). LONG RANGE Fluctuations in g UCF (Feng et al, PRL88) INFINITE RANGE C aba’b’ = { C 1 d aa’ d bb’ + C 2 ( d aa’ + d bb’ ) + C 3 }

7. The experiment Spatial-Field Correlation: The Building Block of Mesoscopic Fluctuations P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro Similar results in 2003-2004 for polarization Van Tiggelen

8. Microwaves: Tumbles system and averages ~ 700 configurations Places a source at 50 positions on the input face, and detector at the output face. 100 cm 7.5 cm r R Measures E(r,R) I(r,R) = |E(r,R)| 2

10. C 1 (  r,d) = (0.002) C 1 (  r,0) C 1 is a product of two identical functions of the source and detector [C- C 1 ](  r,  R) falls to ½ when one of the variables increases beyond a certain value. C 2 is a sum of the same two identical functions of the source and detector C(  r,0) = C( ,  R) C 1 (  r,0) = C 1 ( ,  R) There is complete symmetry source-detector

11. THE RMT APPROACH The coefficients of the scattering matrix have to verify the physical constraints Flux conservation S S + = 1 Also: S = S t (if there is no time reversal breaking mechanism)

12. Expand in eigenfunctions of the clean cavity Quasi – 1D - geometry Transverse Longitudinal Statistical averages

13. Transport RMT theory to evaluate the averages ( Mello, Beenakker ) polar decomposition The average over transmission coefficients factorizes in a geometric part which is independent of the transport regime ISOTROPY Unitary matrices All information about transport is contained in the t ( transport eigenvalues) Key idea: maximum entropy

14. Indeed the symmetry source-detector; and the product (C 1 ) and the sum (C 2 ) have been reproduced. Also C 3 = C 1 -1 One can prove results for the averages of the u i = (1/N) d jj’ d nn’ e.c.a

15. FIELD CORRELATION Torres and Saenz, J.Phys. Soc. Japan 73, 2182 Comparison between the large N limit and a solution on a cavity of cross section W with W/( l /2) = 41/2 ( N= 20, g  1.2) In the limit of large N

16. Intensity correlations The correlations for sources at  R 12 =0 and  R 12 >>, comparing our expressions with the numerical simulation Not perturbative results

17. Dependence of correlations with the length of the system Dorokhov and Mello-Pereyra-Kumar introduced an equation for the joint distribution of the transport eigenvalues  as a function of the length s = L/ l (Fokker-Planck) Exact solution only for  =2, but there are approximations in the localized case s >> N, and the diffusive limit 1 << s << N. Monte Carlo method Froufe-Perez et al, PRL (2003) defined as s

18. Dependence of correlations with the length of the system Using our numerical values for C(  r,  R) we adjust with the RMT expression obtained and find C 1, C 2, C 3 at different scales The continuous line is the DMPK solution C 2 goes negative at small scales s = L/ l

20. Intuitive explanation in terms of flux conservation

21. Conclusions RMT is a useful tool to understand transport properties, in particular space correlations, for specific geometries. The structure of the correlations is independent of the transport regime. We get good agreement with numerical simulations. We can predict dependence of the correlations with length of the cavity, which can be tested. More details in: Phys Rev E74, 045603(R), 2006 Physica A386, 625, 2007