Statistical Mechanics of Disordered Systems:Applications in Optics
Candidate: Fabrizio Antenucci
Supervisor: Dr. Luca Leuzzi
In collaboration with: Claudio Conti, Andrea Crisanti and Miguel Ibañez Berganza
Sapienza University - Graduate School “Vito Volterra”
30 October 2014
What? Multimode Systems
• (Many) Well-defined Modes ak
Space Ek (r) Frequency ωk (� ∆ω)
• Space - Time Separation of the Electromagnetic Field
E(r, t) = <[∑
k
ak (t) Ek (r)
]with ak (t) ∼ exp(−iωk t)
• Dynamics for the Complex Mode Amplitudes• Near Lasing Regime�
�dajdt =
∑k
Gjk ak +∑klm
Gjklm ak a∗l am + ηj
What are the values for G s? Hardly known in Most CasesGeneral Properties of G s → Statistical Mechanics
What? Multimode Systems
• (Many) Well-defined Modes ak
Space Ek (r) Frequency ωk (� ∆ω)
• Space - Time Separation of the Electromagnetic Field
E(r, t) = <[∑
k
ak (t) Ek (r)
]with ak (t) ∼ exp(−iωk t)
• Dynamics for the Complex Mode Amplitudes• Near Lasing Regime�
�dajdt =
∑k
Gjk ak +∑klm
Gjklm ak a∗l am + ηj
What are the values for G s? Hardly known in Most CasesGeneral Properties of G s → Statistical Mechanics
What? Multimode Systems
• (Many) Well-defined Modes ak
Space Ek (r) Frequency ωk (� ∆ω)
• Space - Time Separation of the Electromagnetic Field
E(r, t) = <[∑
k
ak (t) Ek (r)
]with ak (t) ∼ exp(−iωk t)
• Dynamics for the Complex Mode Amplitudes• Near Lasing Regime�
�dajdt =
∑k
Gjk ak +∑klm
Gjklm ak a∗l am + ηj
What are the values for G s? Hardly known in Most CasesGeneral Properties of G s → Statistical Mechanics
What? Hamiltonian Multimode Systems
Langevin Dynamics for Complex Amplitudes
daj
dt= −
dHda∗j
+ ηj with H '∑jk
Gjk a∗j ak +
∑jklm
Gjklm a∗j ak a
∗l am
Main Working Hypothesis on G s:
H is real → Hamiltonian System
• Stability of the Steady-State• Spherical Constraint:
∑j |aj |2 ≡ �N
Machinery Well-Tested for Mean Field SML.Gordon, A. and Fisher, B. PRL 89, 103901 (2002)
What’s Next?
What? Hamiltonian Multimode Systems
Langevin Dynamics for Complex Amplitudes
daj
dt= −
dHda∗j
+ ηj with H '∑jk
Gjk a∗j ak +
∑jklm
Gjklm a∗j ak a
∗l am
Main Working Hypothesis on G s:
H is real → Hamiltonian System
• Stability of the Steady-State• Spherical Constraint:
∑j |aj |2 ≡ �N
Machinery Well-Tested for Mean Field SML.Gordon, A. and Fisher, B. PRL 89, 103901 (2002)
What’s Next?
What? Hamiltonian Multimode Systems
Langevin Dynamics for Complex Amplitudes
daj
dt= −
dHda∗j
+ ηj with H '∑jk
Gjk a∗j ak +
∑jklm
Gjklm a∗j ak a
∗l am
Main Working Hypothesis on G s:
H is real → Hamiltonian System
• Stability of the Steady-State• Spherical Constraint:
∑j |aj |2 ≡ �N
Machinery Well-Tested for Mean Field SML.Gordon, A. and Fisher, B. PRL 89, 103901 (2002)
What’s Next?
How? RL Mean Field TheoryAntenucci, F., Conti, C., Crisanti, A. and Leuzzi, L., arXiv:1406.7826 (2014)
• All modes are equal → MFT
Space: Extended Modes Frequency: Narrow Bandwidth
H = −1
2N
1,N∑sp
Jjkasa∗p −
1
4!N3
1,N∑spqr
Jspqrasa∗paqa
∗r ,
∑k
|ak |2 = �N .
with i.i.d. Jsp and Jspqr
Jsp =(1− α0)J0 Jspqr =α0J0
J2sp − Jsp2
=(1− α)2J2 J2spqr − Jspqr2
=α2J2
Control Parameters
• Degree of disorder�
�RJ = JJ0
• Pumping rate�� ��P = �√βJ0• Degree of nonlinearity�� ��α = α0
How? RL Mean Field TheoryAntenucci, F., Conti, C., Crisanti, A. and Leuzzi, L., arXiv:1406.7826 (2014)
• All modes are equal → MFT
Space: Extended Modes Frequency: Narrow Bandwidth
H = −1
2N
1,N∑sp
Jjkasa∗p −
1
4!N3
1,N∑spqr
Jspqrasa∗paqa
∗r ,
∑k
|ak |2 = �N .
with i.i.d. Jsp and Jspqr
Jsp =(1− α0)J0 Jspqr =α0J0
J2sp − Jsp2
=(1− α)2J2 J2spqr − Jspqr2
=α2J2
Control Parameters
• Degree of disorder�
�RJ = JJ0
• Pumping rate�� ��P = �√βJ0• Degree of nonlinearity�� ��α = α0
How? RL Mean Field Theory
“New Kind” Of Spin: a Complex Amplitude (XY + Spherical)
Order Parameters (Qaa ≡ 1 ↔ SC)
Qab =∑j
(aaj
)∗abj
�NRab =
∑j
<[aaj a
bj
]�N
Tab =∑j
=[aaj a
bj
]�N
mσ =
√2
�N
∑j
How? RL Mean Field Theory
“New Kind” Of Spin: a Complex Amplitude (XY + Spherical)
Order Parameters (Qaa ≡ 1 ↔ SC)
Qab =∑j
(aaj
)∗abj
�NRab =
∑j
<[aaj a
bj
]�N
Tab =∑j
=[aaj a
bj
]�N
mσ =
√2
�N
∑j
How? RL Mean Field Theory
“New Kind” Of Spin: a Complex Amplitude (XY + Spherical)
Order Parameters (Qaa ≡ 1 ↔ SC)
Qab =∑j
(aaj
)∗abj
�NRab =
∑j
<[aaj a
bj
]�N
Tab =∑j
=[aaj a
bj
]�N
mσ =
√2
�N
∑j
MFT: Phase Diagram (I)
10
20
30
40
50
60
70
80
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
P2
RJ
CW
PLW
RL
SMLα = α0 = 1
MFT: Phase Diagram (I)
10
20
30
40
50
60
70
80
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
P2
RJ
CW
PLW
RL
SMLα = α0 = 1
MFT: Phase Diagram (II)
2
4
6
8
10
12
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
P2
RJ
SML
CW
PWL
RL
α = α0 = 0.4
MFT: Phase Diagram (III)
0
0.2
0.4
0.6
0.8
1 0.2
0.4
0.6
0.8
1
10
20
30
40
50
60
70
80
P2
RJ α=α0
P2
MFT: Intensity Overlap
Hard to detect the mode phase correlations in Experiments:What about the Overlap between Intensity Fluctuations?
Iab ≡∑j
〈|aaj |2|abj |
2〉 − 〈|aaj |2〉〈|abj |
2〉�2N
In the MFT it holds (at m = 0)
Iab ≡ 2(Q2ab + R
2ab
)2+ 2
(Q2ab − R
2ab
)2Replica Symmetry is spontaneously Broken in the
Intensity Fluctuations Overlap in RL regime
MFT: Intensity Overlap
Hard to detect the mode phase correlations in Experiments:What about the Overlap between Intensity Fluctuations?
Iab ≡∑j
〈|aaj |2|abj |
2〉 − 〈|aaj |2〉〈|abj |
2〉�2N
In the MFT it holds (at m = 0)
Iab ≡ 2(Q2ab + R
2ab
)2+ 2
(Q2ab − R
2ab
)2Replica Symmetry is spontaneously Broken in the
Intensity Fluctuations Overlap in RL regime
MFT: Intensity Overlap - Experiments?
N. Ghofraniha et al., arXiv:1407.5428 (2014)
Replica Symmetry is spontaneously Broken in the
Intensity Fluctuations Overlap in RL regime
What? SML Beyond Mean FieldAntenucci, F., Ibañez Berganza, M. and Leuzzi, L., arXiv:1409.6345 (2014)
Consider the case of Standard Mode Locking Lasers
Space: Extended Modes Frequency: Comb δω � ∆ω = 2πc/2L
ak1 (t) · a∗k2
(t) · ak3 (t) · a∗k4
(t) ∼ exp[i(ωk1 − ωk2 + ωk3 − ωk4
)t]
The Hamiltonian is indeed (purely dissipative case)
H =−∑k
Gk |ak |2 −Γ
2
∑FMC(k)
ak1 · a∗k2· ak3 · a
∗k4,
∑k
|ak |2 = �N
with the Frequency Matching Condition
FMC(k) : |ωk1 − ωk2 + ωk3 − ωk4 | . δω
δω � ∆ω → k1 − k2 + k3 − k4 = 0
What? SML Beyond Mean FieldAntenucci, F., Ibañez Berganza, M. and Leuzzi, L., arXiv:1409.6345 (2014)
Consider the case of Standard Mode Locking Lasers
Space: Extended Modes Frequency: Comb δω � ∆ω = 2πc/2L
ak1 (t) · a∗k2
(t) · ak3 (t) · a∗k4
(t) ∼ exp[i(ωk1 − ωk2 + ωk3 − ωk4
)t]
The Hamiltonian is indeed (purely dissipative case)
H =−∑k
Gk |ak |2 −Γ
2
∑FMC(k)
ak1 · a∗k2· ak3 · a
∗k4,
∑k
|ak |2 = �N
with the Frequency Matching Condition
FMC(k) : |ωk1 − ωk2 + ωk3 − ωk4 | . δω
δω � ∆ω → k1 − k2 + k3 − k4 = 0
What? SML Beyond Mean Field
Homogeneous Dilution FMC Dilution
MC simulations: Results (I)
Energy
1.5 1.6 1.7 1.8 1.9-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
P
mean fie
ld
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
1.5 1.6 1.7 1.8 1.9
Pm
ean fie
ld
100200300400500-0.6
-0.4
-0.2
0
1.6 1.7
Pc(1
00)
Psp(1
00)
MC simulations: Results (II)
r =1
N
∑j
|aj |
0.76
0.8
0.84
0.88
0.92
0.96
1
1.5 1.55 1.6 1.65 1.7 1.75 1.8
P
(∞)
1.5 1.55 1.6 1.65 1.7 1.75 1.8 0.76
0.8
0.84
0.88
0.92
0.96
1
P
(∞)
100200300400500
√2/π
MC simulations: Results (III)
mx =1
N
∑j
MC simulations: Results (IV)
What is the origin of the lacking of the O(2) Symmetry Breaking?
Phase Waves
-1-0.5
0 0.5
1-1
-0.5
0
0.5
1
0
100
200
300
400
500
ω
Re[a]/|a|
Im[a]/|a|
ω
0
100
200
300
400
500
-π/2 -π/4 0 π/4 π/2
ω
atan(Im[a]/Re[a])
MC simulations: Results (IV)
What is the origin of the lacking of the O(2) Symmetry Breaking?
Phase Waves
-1-0.5
0 0.5
1-1
-0.5
0
0.5
1
0
100
200
300
400
500
ω
Re[a]/|a|
Im[a]/|a|
ω
0
100
200
300
400
500
-π/2 -π/4 0 π/4 π/2
ω
atan(Im[a]/Re[a])
MC simulations: Results (IV)
What is the origin of the lacking of the O(2) Symmetry Breaking?
Phase Waves
-1-0.5
0 0.5
1-1
-0.5
0
0.5
1
0
100
200
300
400
500
ω
Re[a]/|a|
Im[a]/|a|
ω
0
100
200
300
400
500
-π/2 -π/4 0 π/4 π/2
ω
atan(Im[a]/Re[a])
MC simulations: Results (IV)
What is the origin of the lacking of the O(2) Symmetry Breaking?
Phase Waves
-1-0.5
0 0.5
1-1
-0.5
0
0.5
1
0
100
200
300
400
500
ω
Re[a]/|a|
Im[a]/|a|
ω
0
100
200
300
400
500
-π/2 -π/4 0 π/4 π/2
ω
atan(Im[a]/Re[a])
MC simulations: Results (V)
aj = |aj | exp(iΦj)
: E(t|T ) =N∑j=1
|aj (T )| exp[i(2πωj t + Φj (T )
)], T � t
Phase Delay: Φj ' Φ0 + Φ′ωj
-π/2
-π/4
0
π/4
π/2
0 100 200 300 400 500
Φ
ω
-1
-0.5
0
0.5
1
199 200 201
E(t
)
t
MC simulations: Results (V)
aj = |aj | exp(iΦj)
: E(t|T ) =N∑j=1
|aj (T )| exp[i(2πωj t + Φj (T )
)], T � t
Phase Delay: Φj ' Φ0 + Φ′ωj
-π/2
-π/4
0
π/4
π/2
0 100 200 300 400 500
Φ
ω
-1
-0.5
0
0.5
1
199 200 201
E(t
)
t
MC simulations: Results (V)
aj = |aj | exp(iΦj)
: E(t|T ) =N∑j=1
|aj (T )| exp[i(2πωj t + Φj (T )
)], T � t
Phase Delay: Φj ' Φ0 + Φ′ωj
-π/2
-π/4
0
π/4
π/2
0 100 200 300 400 500
Φ
ω
-1
-0.5
0
0.5
1
200 201 202
E(t
)
t
MC simulations: Results (V)
aj = |aj | exp(iΦj)
: E(t|T ) =N∑j=1
|aj (T )| exp[i(2πωj t + Φj (T )
)], T � t
Phase Delay: Φj ' Φ0 + Φ′ωj
-π/2
-π/4
0
π/4
π/2
0 100 200 300 400 500
Φ
ω
-1
-0.5
0
0.5
1
199 200 201
E(t
)
t
MC simulations: Results (VI)
Evidence in the Spectra
0
1
2
3
4
5
6
760 800 840
I(λ
) (a
.u.)
λ (a.u.)
Pc(N)=1.60(2); N=150σg = 3885
4.472
1.630
1.593
0
1
2
3
4
5
6
760 800 840
I(λ
) (a
.u.)
λ (a.u.)
Pc(N)=1.60(2); N=150σg = 3885
760 800 840
λ (a.u.)
σg = 243
4.472
1.630
1.310
760 800 840
λ (a.u.)
σg = 243
g(λ)
Outline
Further “Surprises” in SML Beyond MFT:
• Metastability Vanishes in the Thermodynamic Limit
• Vanishing Two Point Correlation Functions
• Slow Dynamics of Phase Waves (different slopes as basins of the FEL)
• Power Condensation (O(1) modes take O(N) intensity in hyper-diluted systems)
• Synchronous MC Algorithm
Outlook:
• Random Laser Models Beyond MFT
Stvdivm Vrbis (photos by MIB)
Thanks forthe attention
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