Statistical Mechanics of Disordered Systems:Optics Applications

Candidate: Fabrizio Antenucci

Supervisor: Dr. Luca Leuzzi

Sapienza University - Graduate School “Vito Volterra”

27 February 2013

Overview

1 What is a random laser?DefinitionRL Experimental factsRL Theory

2 Statistical Mechanics Models for RLsHandle quenched disorderA RL model: Mean-field for Slow Amplitude

3 What I am going to doAnalyticallyNumericallyStatistical inference

What is a random laser?

1958: Schawlow, Townes1967: Letokhov (th.)1994: Lawandy (exp.)

credit by Science Magazine & R.Tandy

1 light is multiply scattered due to randomness and amplified

2 there exists a threshold above which total gain is larger than total loss

RL Experimental Facts - Isotropic emission

credit by D.S.Wiersma 2012

Speckle pattern of a disordered structure: the randomly sparse lines correspond tothe light emission directions, reflecting the mode-structure of the RL

RL Experimental Facts - Transition CW-RL regime

Emission Spectra ZnO powder [H.Cao et al. PRL ’99]

Above the threshold:

very narrow spikes emerge

the integrated emission intensity increasemuch more rapidly with the pump power

What is the physical origin of such spikes?

Anderson-localized modes

Extended modes

RL Experimental Facts - Chaotic Behavior Regime

Single-shot emission spectra

[D.S.Wiersma et al. PRA ’07]

modes competition + quenched disorder =good candidate to PHYSICAL REPLICAS

Spectral(#) and speckle(�) correlation coefficient

What is the physical origin for the C.B.?

H: Complex free energy landascape

RL Theory - Multimode Laser Master Equation

Maxwell equations in presence of nonlinear polarization in a CLOSED cavity

∇∧H = ∂tD = ε0 n2(r) ∂tE + ∂tPNL

∇∧ E = −µ0∂tH

admit solutions in the form of superposition of the normal modes

E = <

[∑k

√ωk ak(t) Ek(r) e−iωk t

], H = <

[∑k

√ωk ak(t) Hk(r) e−iωk t

]

with E =∑

k ωk |ak |2, if the time evolution of the amplitudes is

dakdt

= −√ωk

4i

∫E?k(r) · Pk(r)dV ,

where to the leading order we have (α = x , y , z ; j = 1, . . . , N)

Pαj (r) =∑klm|

ωj +ωl =ωk+ωm

χ(3)αβγδ(ωj ;ωk ,−ωl , ωm; r)Eβk (r)Eγl (r)E δm(r)

√ωkωlωm aka

?l am

RL Theory - Langevin and Hamiltonian Formulation

Defining

G(4)jklm = −

√ωjωkωlωm

8i

∫χ

(3)αβγδ(ωj ;ωk ,−ωl , ωm; r)E?j

α(r)Eβk (r)E?lγ(r)E δm(r) dV

the Langevin equations for the modes in the OPEN cavity regime are

dajdt

=∑k

G(2)jk ak +

∑klm|

ωj +ωl =ωk+ωm

2G(4)jklm ak a

?l am + ηj ,

with 〈η?j (t) ηk(t ′))〉 = 2kBT δjk δ(t − t ′).The Hamiltonian for the modes has the form (with real G - no Kerr lens effect)

H = −<

∑jk

G(2)jk a?j ak +

∑jklm|

ωj +ωl =ωk+ωm

G(4)jklm a?j ak a

?l am

, E =∑k

ωk |ak |2 .

Statistical Mechanics for RLs - Models

What is specific to Random Laser?Random Spatial Modes Distribution (unknow)�

Quenched Disordered Interactions

H = −<

∑jk

G(2)jk a?j ak +

∑ωj+ωl=ωk+ωm

G(4)jklm a?j ak a

?l am

, E =∑k

ωk |ak |2

Statistical Mechanics provides several kinds of RL models characterized by

degree of disorder ←→ coupling values distribution

extension of modes localization ←→ lattice/graph structure

geometry and dimension ←→ lattice/graph structure

pumping intensity ←→ temperature

contribute to clarify the basic physics of random laser

first direct measure of the overlap order parameter in a real world system

How to handle quenched disorder (when you can)

Free energy for a mean-field quenched disordered system

−βΦ = limN→∞

1

NlogZJ

replica trick−−−−−−→ limN→∞

limn→0

1

nNlogZ n

J →

average over disorder−−−−−−−−−−−−→ limN→∞

limn→0

1

nNlog

∫DQ DΛ e−NG(Q,Λ) →

limits exchange−−−−−−−−→ limn→0

1

nextrG [Q]

Ansatz for the overlap n × n matrix Q:

Replica Symmetric → unstable at low T

Replica Symmetry Broken → Parisi Scheme (iterative)

kRSB: Qab = Qa∩b = qr with r = 0, . . . k + 1

Physical Meaning of the overlap matrix (1-component spin σ)

limn→0

2

n(n − 1)

∑a<b

Qab =1

N

∑k

〈σk〉2 ,

The (main) RLs models class: 1RSB

Thermodynamic transition at Ts

Discontinuous jump in order parameter

Dynamic transition at Td > Ts

Ergodicity breaking

for Ts < T < Td there is a nonzero complexity Σ = 1N logN

cumputable as Legendre trasform of the replicated free energy Φ

Σ(m) = minm

[−βmΦ(m) + βmf ] = βm2 ∂Φ

∂m, f =

∂(m Φ)

∂m

very complex landscape

A RL model: Mean-field for Slow Amplitude

fully connected modes network

each modes is localized in the whole pumped regionstrongly peaked spectral distribution: ωj + ωl w 2ω0, ∀ j , l

strong cavity limit: G(2)kl = G

(2)kk δkl

quenched amplitudes: |ak | slowly varying compare to the phase φk = arg(ak)

gaussian disorder: Jjklm = J0/N3 ,(Jjklm − Jjklm

)2= σ2

J/N3

H = −∑

j<k<l<m

Jjklm cos (φj − φk + φl − φm) , P2 = βJ0

Need magnetization NMaa =∑

k 〈e iφk 〉 (for J0 6= 0) and two overlap matrices

limn→0

1

n(n − 1)

∑a 6=b

Qab =1

N

∑k

|〈e iφk 〉|2 , Qaa ≡ 1 ,

limn→0

1

n(n − 1)

∑a 6=b

Rab =1

N

∑k

〈e iφk 〉2 , Raa =1

N

∑k

〈e2iφk 〉 ,

Mean-field for Slow Amplitude: Phase-diagram

H = −∑

j<k<l<m

Jjklm cos (φj − φk + φl − φm) , P2 = βJ0

Conti, Leuzzi PRB 83, 134204 (2010)

What I am going to do

fully connected modes network

each modes is localized in the whole pumped regionstrongly peaked spectral distribution: ωj + ωl w 2ω0, ∀ j , l

strong cavity limit: G(2)kl = G

(2)kk δkl

quenched amplitudes: |ak | slowly varying compare to the phase φk = arg(ak)

gaussian disorder: Jjklm = J0/N3 ,(Jjklm − Jjklm

)2= σ2

J/N3

1 Analytical Analysis:Fast Varying Mode Amplitudes → Two Components Spherical Spin

Mean-field 2 + 4 Mode Amplitude Interacting ModelsMean-field M-p Mode Amplitude Model with 4-body Interaction

2 Numerical Simulation on CUDA GPUs:

Finite Size Analysis

Mean-field 2 + 4 Mode Amplitude Interacting ModelsMean-field M-p Mode Amplitude Model with 4-body Interaction

What I am going to do

fully connected modes network

each modes is localized in the whole pumped regionstrongly peaked spectral distribution: ωj + ωl w 2ω0, ∀ j , l

strong cavity limit: G(2)kl = G

(2)kk δkl

quenched amplitudes: |ak | slowly varying compare to the phase φk = arg(ak)

gaussian disorder: Jjklm = J0/N3 ,(Jjklm − Jjklm

)2= σ2

J/N3

1 Numerical Simulation on CUDA GPUs:

Finite Dimensional Analysis + Mode-Locking

M-p Mode Amplitude Model on Levy (diluite) graph

What I am going to do

fully connected modes network

each modes is localized in the whole pumped regionstrongly peaked spectral distribution: ωj + ωl w 2ω0, ∀ j , l

strong cavity limit: G(2)kl = G

(2)kk δkl

quenched amplitudes: |ak | slowly varying compare to the phase φk = arg(ak)

gaussian disorder: Jjklm = J0/N3 ,(Jjklm − Jjklm

)2= σ2

J/N3

1 Statistical Inference Theory:

Inverse Problem for the RL model