statistical mechanics of disordered systems: optics applications · 2017. 3. 14. · statistical...
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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