distribution of scattering matrix elements in quantum … · 2014. 1. 9. · santosh kumar shiv...

Santosh KumarShiv Nadar UniversityDadri, India

DISTRIBUTION OF SCATTERING MATRIX ELEMENTS IN QUANTUM CHAOTIC SCATTERING

Joint work with

A. Nock*, H.-J. Sommers, T. Guhr Universität Duisburg-Essen, Duisburg.*Queen Mary University of London, London

B. Dietz, M. Miski-Oglu, A. Richter, F. Schäfer Institut für Kernphysik, Technische Universität Darmstadt

IX Brunel-Bielefeld Workshop on Random Matrix TheoryBielefeld 2013

Distribution of S-matrix element in Fourier Space

OUTLINE

Introduction

Brief sketch of the derivation

Results and comparison with simulations and experiments

Conclusion

SCATTERING

Scattering of light waves by clouds Scattering of radio waves

Rutherford-Geiger-Marsden experiment

Scattering of laser beam

Electron scattering inside aquantum dot

A candidate scattering event at LHC leading to the discovery of Higgs Boson

Deviation of a wave or a particle from its trajectory because of some localized non-uniformity in the medium.

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SCATTERING: GENERIC SET-UP

States before the scattering event

Interaction region(Scattering event) States after the

scattering event

Channels of reaction (Characterized by total

energy E and other quantum numbers)

Scattering matrix (S-matrix) connects states existing asymptotically before and after the scattering event

(Relates incoming and outgoing waves)

B1(k)A2(k)

�=

S11 S12S21 S22

� A1(k)B2(k)

� 1(x) = A1(k)e

ikx + B1(k)e�ikx

2(x) = A2(k)eikx + B2(k)e

�ikx

SS† = S†S = 1

SCATTERING MATRIX

H

Relates incoming and outgoing waves

2 channel example:

S- matrix is unitary owing to the flux conservation,

N(� 1) bound states : hl|mi = �lmM channels : ha,E1|b, E2i = �ab �(E1 � E2)

HAMILTONIAN FORMULATION

H =X

lm

|liHlmhm|+X

c

ZdE|c, EiEhc, E|

+

X

l,c

✓|li

ZdEWlchc, E|+ herm. conj.

◆

Hab

d

e

f

c

Schematic view of the general scattering problem: Different channels of reaction (labeled a,b,...) are

connected via a compact interaction region described by a Hamiltonian H.

Schematic view of the general scattering problem: Different channels of reaction (labeled a,b,...) are

connected via a compact interaction region described by a Hamiltonian H.

GaAs based quantum dot(Source: http://www.newton.ac.uk/reports/9798/dqc1.gif)

• C. Mahaux and H. A. Weidenmüller, Shell Model Approach to Nuclear Reactions (North Holland, Amsterdam, 1969)

SCATTERING MATRIX

The resolvent G(E) is given by:

The N-component coupling vectors are assumed to satisfy the orthogonality

Sab(E) = �ab � i2⇡W †aG(E)Wb

G(E) =

E1N �H + i⇡

MX

c=1

WcW†c

!�1

W †cWd =�c⇡�cd; c, d = 1, · · · ,M

• C. Mahaux and H. A. Weidenmüller, Shell Model Approach to Nuclear Reactions (North Holland, Amsterdam, 1969)

STATISTICAL DESCRIPTION

Scattering process is quite often of chaotic nature.

Complicated dependence on

‣ Parameters of incoming waves (e.g., energy)

‣ The scattering region (e.g., the form or strength of the scattering potential.)

Scattering description of S-matrix is needed!

S-matrix, itself, is treated as a stochastic quantity and is described by the Poisson kernel.

(Based on the assumption of minimal information content)

M: Dimension of the S-matrix (Number of channels): Average S-matrixβ: Symmetry class

Dependence on the parameters not obvious!

P (S) / |det�1M � ShS†i

�|�(�M+2��)

MEXICO APPROACH

• P. Mello, P. Pereyra and T. H. Seligman, Ann. Phys. (NY) 161, 254 (1985)• H. U. Baranger and P. Mello , Phys. Rev. Lett. 73,142 (1994); Europh. Lett. 33, 465 (1996)• P. Mello and H. Baranger , Physica A 220, 15 (1995)

HEIDELBERG APPROACHIntroduces stochasticity on the level of the Hamiltonian describing the scattering center.

Random Matrix Universality: Universal and generic features can be extracted by modeling the Hamiltonian describing the scattering center using appropriate ensemble of Random matrices

N: Dimension of the matrices Hv: Energy scale

β: Symmetry class

β=1 Time reversal invariant “spinless” systems (Real-Symmetric H) β=2 Time reversal noninvariant systems (Hermitian H)

• D. Agassi, H. A. Weidenmüller and Z. Mantzouranis, Phys. Rep. 22, 145 (1975)

P(H) / exp✓��N

4v2trH2

◆

KNOWN RESULTS

• Two-point correlation function (β=1) J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer Phys. Rep. 129, 367 (1985)

• Up to fourth moment (β=1) E. D. Davis and D. Boose Phys. Lett. B 211, 379 (1988); Z. Phys. A 332, 427 (1989)

• Two-point correlation function (β=2) D. V. Savin, Y. V. Fyodorov, H.-J. Sommers Acta Phys. Pol. A 109, 53-64 (2006)

• Two-point correlation function (β=2, Arbitrary UN invariant distribution for H) S. Mandt and M. R. Zirnbauer, J. Phys. A: Math. Theor. 43, 025201 (2010)

KNOWN RESULTS





‣ Distribution of diagonal elements (β=1, 2): Y. V. Fyodorov, D. V. Savin, and H.-J. Sommers J. Phys. A: Math. Gen 38, 10731 (2005)

KNOWN RESULTS





‣ Distribution of diagonal elements (β=1, 2): Y. V. Fyodorov, D. V. Savin, and H.-J. Sommers J. Phys. A: Math. Gen 38, 10731 (2005)

The problem of finding distribution of off-diagonal S-matrix elements remained unsolved!

SOME OBSERVATIONS

Already in 1975 numerical simulations revealed that

The distributions, in general, exhibit Non - Gaussian behavior (expected because of Unitarity constraint).

The real and imaginary parts show different deviations from the Gaussian behavior for β=1.

Similar conclusions were arrived at from the data obtained from experiments on microwave resonators.

• J.W.Tepel, Z.Physik A 273, 59 (1975). • J.Richert, M. H. Simbel, and H. A. Weidenmüller,Z.Physik A 273, 195 (1975).• B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A. Richter, F. Schäfer, and H. A.

Weidenmüller, Phys. Rev. E 81, 036205 (2010).

Ps(xs) =

Zd[H]P(H)�(xs � }s(Sab)); s = 1, 2

GOAL

Distribution of off-diagonal S-matrix elements

Real part of Sab:

Imaginary part of Sab:

}1(Sab) =Sab + S⇤ab

2= �i⇡(W †aGWb �W

†bG

†Wa)

}2(Sab) =Sab � S⇤ab

2i= �⇡(W †aGWb +W

†bG

†Wa)

Rs(k) =

Zd[H]P(H) exp(�ik}s(Sab)); s = 1, 2

Ps(xs) =1

2⇡

Z 1

�1dkRs(k) exp(ikxs)

CHARACTERISTIC FUNCTIONS

The characteristic function also serves as the moment generating function

The distributions can be obtained as the Fourier transform of Rs(k):

Rs(k) =

Zd[H]P(H) exp(�ik}s(Sab)); s = 1, 2

W =

WaWb

�As =

0 (�i)sG

isG† 0

�

Rs(k) =

Zd[H]P(H) exp(�ik⇡W †AsW )

P(H) / exp⇣� �N

4v2trH2

⌘

CHARACTERISTIC FUNCTIONS

Introduce a 2N-dimensional vector W and a 2N×2N dimensional matrix As

H appears in the denominator of G: Ensemble averaging nontrivial !

G =

E1N �H + i⇡

MX

c=1

WcW†c

!�1

SUPERMATHEMATICSAnticommuting (Grassmann or Fermionic) variables:

Any function of the anticommuting variables is a finite polynomial,e.g., exp(α)=1+α

“Complex conjugate”

Conventions:

•F. A. Berezin, Introduction to Superanalysis (Reidel, Dordrecht, 1987)• K. B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, 1997)

↵1↵2 = �↵2↵1↵2 = 0

↵ ↵⇤

(↵⇤)⇤ = �↵ (↵�)⇤ = ↵⇤�⇤(↵⇤↵)⇤ = (↵⇤)⇤↵⇤ = �↵↵⇤ = ↵⇤↵

SUPERMATHEMATICSIntegrals (Berezin Integrals):

In contrast, for the ordinary complex variables

Superintegral:

•F. A. Berezin, Introduction to Superanalysis (Reidel, Dordrecht, 1987)• K. B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, 1997)

Zd↵ = 0,

Zd↵↵ =

1p2⇡Z Z

d↵⇤ d↵ exp(iq ↵⇤↵) =q

2⇡i

Z Z Z Zdz⇤dz d↵⇤d↵ exp(iq(z⇤z + ↵⇤↵)) = 1

Z Zdz⇤ dz exp(iq z⇤z) =

2⇡i

q

str� = tra� trb

sdet� =det(a� µb�1⌫)

det b=

det a

det(b� ⌫a�1µ)

SUPERMATHEMATICS

Supervectors:

Supermatrices:

DefinitionsSupertrace:

Superdeterminant:

=

z⇣

� † = [z† ⇣†]

� =

a µ⌫ b

��T =

aT ⌫T

�µT bT�

�† =

a† ⌫†

�µ† b†�

z =

zazb

�⇣ =

⇣a⇣b

�A�1s =

0 (�i)s(G�1)†

isG�1 0

�W =

WaWb

�

Zd[⇣] exp

✓i

4⇡k⇣†A�1s ⇣

◆= det

A�1si8⇡2k

!

SUPERMATHEMATICSMultivariate Gaussian Integrals:

Using vectors with commuting entries:

Using vectors with anticommuting entries:

Zd[z] exp

✓i

4⇡kz†A�1s z

◆exp

✓i

2

(W †z + z†W )

◆= det

�1

A�1si8⇡2k

!exp(�ik⇡W †AsW )

Zd[z]

Zd[⇣] exp

✓i

4⇡k(z†A�1s z + ⇣

†A�1s ⇣)

◆exp

✓i

2

(W †z + z†W )

◆= exp(�ik⇡W †AsW )

SUPERMATHEMATICS

Combining the above integral results we obtain

Zd[z]

Zd[⇣] exp

✓i

4⇡k(z†A�1s z + ⇣

†A�1s ⇣)

◆exp

✓i

2

(W †z + z†W )

◆= exp(�ik⇡W †AsW )

Rs(k) =

Zd[H]P(H) exp(�ik⇡W †AsW )

SUPERMATHEMATICS

Combining the above integral results we obtain

The exponential on RHS is exactly the factor in our expression for Rs(k)

Rs(k) =

Zd[ ] exp

⇣ i2

(W† + †W)⌘Z

d[H]P(H) exp⇣ i4⇡k

†A�1s ⌘

W =

2

664

WaWb00

3

775 =

2

664

zazb⇣a⇣b

3

775 A�1s =

2

666664

0 (�i)s(G�1)†

isG�1 00

00 (�i)s(G�1)†

isG�1 0

3

777775

G�1 = E1N �H + i⇡MX

c=1

WcW†c

Rs(k) =

Zd[ ] exp

⇣ i2

(W† + †W)⌘Z


†A�1s ⌘

W =

2

664

WaWb00

3

775 =

2

664

zazb⇣a⇣b

3

775 A�1s =

2

666664

0 (�i)s(G�1)†

isG�1 00

00 (�i)s(G�1)†

isG�1 0

3

777775

G�1 = E1N �H + i⇡MX

c=1

WcW†c

H is now linear in the exponent containing the supervectors

Rs(k) =

Zd[ ] exp

⇣ i2

(W† + †W)⌘Z


†A�1s ⌘

W =

2

664

WaWb00

3

775 =

2

664

zazb⇣a⇣b

3

775 A�1s =

2

666664

0 (�i)s(G�1)†

isG�1 00

00 (�i)s(G�1)†

isG�1 0

3

777775

G�1 = E1N �H + i⇡MX

c=1

WcW†c

H is now linear in the exponent containing the supervectors

As-1 is not block diagonal!

!⌅+ 00 2�⌅

�

�

† ! †⌅± =

0 ±(�i)s1N

�is1N 0

�

Ψ and Ψ† can be treated as independent complex quantities and therefore admit independent transformations.

Jacobian: (-1)N 2-2N for β= 1 and (-1)N for β= 2

The choice of Ξ± ensures proper convergence requirements for the supermatrix introduced later

Rs(k) = (�1)NZ

d[ ] exp⇣ i2

(U†s + †W)

⌘Zd[H]P(H) exp

⇣ i4⇡k

†A�1 ⌘

A�1 = diag[�(G�1)†, G�1,�(G�1)†,�G�1]

β = 2 (HERMITIAN H)

Rs(k) = (�1)NZ

d[ ] exp⇣ i2

(U†s + †W)

⌘Zd[H]P(H) exp

⇣ i4⇡k

†A�1 ⌘

=

2

664

zazb⇣a⇣b

3

775

A�1 = diag[�(G�1)†, G�1,�(G�1)†,�G�1]


Rs(k) = (�1)NZ

d[ ] exp⇣ i2

(U†s + †W)

⌘Zd[H]P(H) exp

⇣ i4⇡k

†A�1 ⌘

z†aHza � z†bHzb + ⇣

†aH⇣a + ⇣

†bH⇣b

= tr(HD)

D = zaz†a � zbz

†b � ⇣a⇣

†a � ⇣b⇣

†b

H-part in exponent involving the supervectors:

where

A�1 = diag[�(G�1)†, G�1,�(G�1)†,�G�1]


Rs(k) = (�1)NZ

d[ ] exp�i †Vs

� Zd[H]P(H) exp

⇣ i4⇡k

†A�1 ⌘

A�1 = diag(�(G�1)†, G�1,�(G�1)†,�G�1)⌦ 12

β= 1 (REAL SYMMETRIC H)

=

2

66666666664

xa

ya

xb

yb

⇣a

⇣

⇤a

⇣b

⇣

⇤b

3

77777777775

Rs(k) = (�1)NZ

d[ ] exp�i †Vs

� Zd[H]P(H) exp

⇣ i4⇡k

†A�1 ⌘

A�1 = diag(�(G�1)†, G�1,�(G�1)†,�G�1)⌦ 12


Rs(k) = (�1)NZ

d[ ] exp�i †Vs

� Zd[H]P(H) exp

⇣ i4⇡k

†A�1 ⌘

xTaHxa + yTa Hya � xTb Hxb � yTb Hyb + ⇣†aH⇣a � ⇣Ta H⇣⇤a + ⇣

†bH⇣b � ⇣

Tb H⇣

⇤b

= tr(HD)

D = xaxTa + yay

Ta � xbxTb � ybyTb � ⇣a⇣†a + ⇣⇤a⇣Ta � ⇣b⇣

†b + ⇣

⇤b ⇣

Tb

A�1 = diag(�(G�1)†, G�1,�(G�1)†,�G�1)⌦ 12H-part in exponent involving the supervectors:

where


Zd[H]P(H) exp

⇣ i4⇡k

trHD⌘= exp

⇣� 1

4rtrD2

⌘

= exp

⇣� 1

4rstr (K1/2BK1/2)2

⌘

r =4�⇡2k2N

v2

Bmn =NX

j=1

( m)j( †n)j ; m,n = 1, 2, .., 8/�

K = diag(1,�1, 1, 1)⌦ 12/�

ENSEMBLE AVERAGING

where

exp

⇣� 1

4rstr (K1/2BK1/2)2

⌘=

Zd[�] exp

�� r str�2 + i str�K1/2BK1/2

�

=

Zd[�] exp

�� r str�2 + i †K1/2(� ⌦ 1N )K1/2

�

HUBBARD-STRATONOVICH TRANSFORMATION

σ is an 8/β-dimensional supermatrix having same structure as B, and K = K ⌦ 12/�

Rs(k) = (�1)NZ

d[�] exp(�r str�2)Z

d[ ] exp⇣i †K1/2⌃K1/2 + i †Vs

⌘

Rs(k) = (�1)NZ


d[ ] exphi †K1/2⌃K1/2 +

i

2

(U†s + †W)

i

⌃ =⇣� � E

4⇡k18/�

⌘⌦ 1N +

i

4kL⌦

MX

c=1

WcW†c

L = diag(1,�1, 1,�1)⌦ 12/�

β=1

β=2

Rs(k) = (�1)NZ


d[ ] exp⇣i †K1/2⌃K1/2 + i †Vs

⌘

Rs(k) = (�1)NZ


d[ ] exphi †K1/2⌃K1/2 +

i

2

(U†s + †W)

i

⌃ =⇣� � E

4⇡k18/�

⌘⌦ 1N +

i

4kL⌦

MX

c=1

WcW†c

L = diag(1,�1, 1,�1)⌦ 12/�

Integral over the supervector can now be performed

β=1

β=2

Rs(k) =

Zd[�] exp

⇣� r str�2 � �

2

str ln⌃� i4

Fs⌘

⌃ =⇣� � E

4⇡k18/�

⌘⌦ 1N +

i

4kL⌦

MX

c=1

WcW†c

L = L⌦ 1N , L = diag(1,�1, 1,�1)⌦ 12/�

Fs =

(VTs L

�1/2⌃�1L�1/2Vs, � = 1

U†sL�1/2⌃�1L�1/2W, � = 2

REPRESENTATION IN SUPERMATRIX SPACE

β=1 32 independent integration variables β=2 16 independent integration variables

Rs(k) =

Zd[�] exp

⇣� r str�2 � �

2

str ln⌃� i4

Fs⌘

⌃ =⇣� � E

4⇡k18/�

⌘⌦ 1N +

i

4kL⌦

MX

c=1

WcW†c

L = L⌦ 1N , L = diag(1,�1, 1,�1)⌦ 12/�

Fs =

(VTs L

�1/2⌃�1L�1/2Vs, � = 1

U†sL�1/2⌃�1L�1/2W, � = 2



Drastic reduction in the number of integration variables!

Rs(k) =

Zd[�] exp

⇣� r str�2 � �

2

str ln⌃� i4

Fs⌘

⌃ =⇣� � E

4⇡k18/�

⌘⌦ 1N +

i

4kL⌦

MX

c=1

WcW†c

L = L⌦ 1N , L = diag(1,�1, 1,�1)⌦ 12/�

Fs =

(VTs L

�1/2⌃�1L�1/2Vs, � = 1

U†sL�1/2⌃�1L�1/2W, � = 2



Drastic reduction in the number of integration variables!Form similar to that of generating function for correlations

Rs(k) =

Zd[�] exp

⇣� r str�2 � �

2

str ln⌃� i4

Fs⌘

⌃ =⇣� � E

4⇡k18/�

⌘⌦ 1N +

i

4kL⌦

MX

c=1

WcW†c

L = L⌦ 1N , L = diag(1,�1, 1,�1)⌦ 12/�

Fs =

(VTs L

�1/2⌃�1L�1/2Vs, � = 1

U†sL�1/2⌃�1L�1/2W, � = 2



Drastic reduction in the number of integration variables!Form similar to that of generating function for correlations

(Verbaarschot, Weidenmüller, Zirnbauer) apart from the Fs part

Rs(k) =

Zd[�] exp(�L� �L)

L = N 4�⇡2k2

v2str�2 +N

�

2str ln

⇣� � E

4⇡k18/�

⌘

�L =MX

c=1

str ln⇣18/� +

i�c4⇡k

⇣� � E

4⇡k18/�

⌘�1L⌘+

i

4Fs

�0 =1

8⇡k

�E ± i

p4v2 � E2

�

SADDLE POINT ANALYSIS

We are interested in N >> M limit. We fix M and let N → ∞

Fs is a linear combination of matrix elements of multiplied with γc, where c=a, b.

Saddle point equation:

Scalar solution:

8�⇡2k2

v2� +

�

2

⇣� � E

4⇡k18/�

⌘�1= 0

�G =1

8⇡k

�E18/� �

p4v2 � E2 Q

�

Q = �i T�1LT ; strQ = 0; Q2 = �18/�

L = diag(1,�1, 1,�1)⌦ 12/�

MANIFOLD OF SOLUTIONS

The dominant part of the free energy is invariant under the application of T

β=1:T belongs to Lie supergroup UOSP(2,2/4) Q belongs to the coset superspace UOSP(2,2/4)/(UOSP(2/2)×UOSP(2/2))

β=2:T belongs to Lie supergroup U(1,1/2) Q belongs to the coset superspace U(1,1/2)/(U(1/1)×U(1/1))

•K. B. Efetov, Adv. Phys. 32, 53 (1983) • J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer, Phys. Rep. 129, 367 (1985)• Y. V. Fyodorov, and H.-J. Sommers, J. Math. Phys. 38,1918 (1997)

� = �G + ��

SEPARATING “GOLDSTONE” AND “MASSIVE” MODES

Expand up to the second power in δσ. The integrals involving Goldstone and Massive modes factorize. Symbolically:

The part involving Massive modes are Gaussian integrals and yields unity.

Z(�) =

Z(�G)

Z(��)

• L. Schäfer and F. Wegner, Z. Phys. B 38, 113 (1980)• J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer, Phys. Rep. 129, 367 (1985)

Rs(k) =

Zdµ(�G)e

�iFs/4MY

c=1

sdet��2

⇣18/� +

i�c4⇡k

��1E L⌘

�G =1

8⇡k

�E18/� �

p4v2 � E2 Q

� �E = �G �E

4⇡k18/�

NONLINEAR SIGMA MODEL

Parametrization of Q:β=1: Eight commuting variables Eight anticommuting variables

β=2: Four commuting variables, Four anticommuting variables

• K. B. Efetov, Adv. Phys. 32, 53 (1983) • J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer, Phys. Rep. 129, 367 (1985)• Y. V. Fyodorov, and H.-J. Sommers, J. Math. Phys. 38,1918 (1997)

Rs(k) = 1�Z 1

1d�1

Z 1

�1d�2

k2

4(�1 � �2)2FU(�1,�2)

�t1at

1b + t

2at

2b

�J0

⇣kq

t1at1b

⌘

Ps(xs) =@

2f(xs)

@x

2s

,

f(x) = x⇥(x) +

Z 1

1d�1

Z 1

�1d�2

FU(�1,�2)4⇡(�1 � �2)2

t

1at

1b + t

2at

2b�

t

1at

1b � x2

�1/2⇥(t1at

1b � x2)

FU =MY

c=1

gc + �2gc + �1

gc =v2 + �2c

�cp4v2 � E2

=2

Tc� 1

tjc =

q|�2j � 1|

(gc + �j), j = 1, 2

RESULTS (β=2)

Identical results for real (s=1) and imaginary (s=2) parts

Characteristic Function

Distribution

Ps(xs) = �(xs) +@f

(s)1

@xs+

@

2f

(s)2

@x

2s

+@

3f

(s)3

@x

3s

+@

4f

(s)4

@x

4s

FO =MY

c=1

gc + �0(gc + �1)1/2(gc + �2)1/2

J = (1� �20)|�1 � �2|

2(�21 � 1)1/2(�22 � 1)1/2(�1 � �0)2(�2 � �0)2

Rs(k) = 1 +1

8⇡

Z 1

�1d�0

Z 1

1d�1

Z 1

1d�2

Z 2⇡

0d J (�0,�1,�2)FO(�0,�1,�2)

4X

n=1

(s)n kn

RESULTS (β=1)Different results for real (s=1) and imaginary (s=2) parts

Characteristic Function

Distribution

EXPERIMENTS WITH MICROWAVE RESONATORS

Equivalence in mathematical structure of the time-independent Schrödinger and Hemholtz equations (two-dimensions)

The shape of microwave cavity is such that the dynamics of the corresponding classical billiard is chaotic

Not only moduli, but both real and imaginary parts of the S-matrix elements can be measured

(r2 + k2) = 0 (r2 + k2)Ez = 0

COMPARISON WITH EXPERIMENTAL DATA (β=1)

Characteristic functions for the real and imaginary parts of S12 for the frequency range 10-11 GHz

Characteristic functions for the real and imaginary parts of S12 for the frequency range 24-25 GHz

COMPARISON WITH EXPERIMENTAL DATA (β=1)

Distributions for the real and imaginary parts of S12 for the frequency range 18-19 GHz

Distributions for the real and imaginary parts of S12 for the frequency range 24-25 GHz

COMPARISON WITH NUMERICAL SIMULATIONS (β=1)

COMPARISON WITH NUMERICAL SIMULATIONS (β=2)

CONCLUSION

We solved a long-standing problem of finding the exact results (in the N→∞ limit) for distributions of off-diagonal S- matrix elements.

We accomplished this task using a novel route to the nonlinear sigma model based on the characteristic function.

We validated our results with experimental data obtained with chaotic microwave billiards, and thus presented a new confirmation of the random matrix universality conjecture.

• S. Kumar, A. Nock, H.-J. Sommers, T. Guhr, B. Dietz, M. Miski-Oglu, A. Richter, and F. Schäfer, Phys. Rev. Lett. 111, 030403 (2013)• A. Nock, S. Kumar, H.-J. Sommers, T. Guhr, Ann. Phys. (In press); Preprint: arXiv:1307.4739

Thank You!

distribution of scattering matrix elements in quantum … · 2014. 1. 9. · santosh kumar shiv...

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