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Gabriel Cwilich
Yeshiva University
NeMeSyS Symposium 10/26/2008
Luis S. Froufe Perez Ecole Centrale, Paris Juan Jose Saenz Univ. Autonoma, MadridAntonio Garcia Martin CSIC, Madrid
INTENSITY CORRELATIONS IN COHERENT TRANSPORT IN
DISORDERED MEDIA
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OUTLINE
• Coherent transport
• Correlations
• Experimental motivation
• Transport Random Matrix Theory
• Numerical Results
• Conclusions
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COHERENT PROPAGATION
Elastic scattering
Phase preserved
Direction randomized over l l < L < lin
Weak LocalizationUniversal Conductance Fluctuations
Berkovits & Feng, Phys Rep 238 (1994)Mesoscopic Physics of electrons and PhotonsE. Akkermans and G. Montambaux
l
No need for low T !!
No need for miniaturization !!
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CORRELATIONS
Feng, Kane, Lee & Stone PRL 61, 834 (1988)
Diagrammatic theory for I - I correlations
g = (e2/h) ab Tab (Landauer) Tab = | tab |2
Transmission: Ta = b Tab ; g = ab Tab
<g> = N<Ta> = N2 < Tab > follows from isotropy
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C1 is essentially (FE )2 (square of field correlation)
In transmission:
Caba’b’ = <Tab> < Ta’b’>{ C1 aa’ bb’ + C2 (aa’ +bb’) + C3 }
This leads to interesting
experimental effects:
Caba’b’ = <Tab Ta’b’> = C1 + C2 + C3
No crossing, one crossing, two crossings
prop to 1, 1/g , 1/g2
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Fluctuations of the intensity at one
speckle pattern (Shapiro, PRL 86)
SHORT RANGE
1 2 32
var{ }2ab
ababab
TC C C C
T
' 1 2 32, '
var{ } 1 1(1 )a
ababb ba
TC C C C
T N N
Fluctuations in total transmission
(Stephen & GC, PRL 87).
LONG RANGE
' ' 1 2 32 2, , ', '
var{ } 1 2aba b
a b a b
gC C C C
g N N Fluctuations in g UCF
(Feng et al, PRL88)
INFINITE RANGE
Caba’b’ = <Tab> < Ta’b’>{ C1 aa’ bb’ + C2 (aa’ +bb’) + C3}
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The experiment
Spatial-Field Correlation: The Building Block of Mesoscopic FluctuationsP. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro
Similar results in 2003-2004 for polarization
Van Tiggelen
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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 cmr
R
Measures E(r,R) I(r,R) = |E(r,R)|2
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C1(r,d) = (0.002) C1(r,0)
C1 is a product of two identical functions of the source and detector
[C- C1](r, R) falls to ½ when one of the variables increases beyond a certain value.
C2 is a sum of the same two identical functions of the source and detector
C(r,0) = C(,R)
C1(r,0) = C1(,R) There is complete symmetry source-detector
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THE RMT APPROACH
The coefficients of the scattering matrix have to verify the physical constraints
Flux conservation S S+ = 1
Also: S = St (if there is no time reversal breaking mechanism)
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Expand in eigenfunctions of the clean cavity
Quasi – 1D - geometryTransverse Longitudinal
Statistical averages
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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 transport eigenvalues)
Key idea: maximum entropy
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Indeed the symmetry source-detector; and the product (C1) and the sum (C2) have been reproduced. Also C3 = C1-1
One can prove results for the averages of the ui
<( uijn )(ui
j’n’)*> = (1/N) jj’nn’ e.c.a
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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/(/2) = 41/2 ( N= 20, g 1.2)
In the limit of large N
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Intensity correlations
The correlations for sources at R12=0 and R12 >> , comparing our expressions with the numerical simulation
Not perturbative results
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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 ass
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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 C1, C2, C3 at different scales
The continuous line is the DMPK solution
C2 goes negative at small scales s = L/l
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Intuitive explanation in terms of flux conservation
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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
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