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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 Cases General 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 Cases General 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 Cases General Properties of G s → Statistical Mechanics
What? Hamiltonian Multimode Systems
Langevin Dynamics for Complex Amplitudes
daj
dt = −
dH da∗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 = −
dH da∗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 = −
dH da∗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 Theory Antenucci, 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 − Jsp 2
=(1− α)2J2 J2spqr − Jspqr 2
=α2J2
Control Parameters
• Degree of disorder�
� RJ = JJ0
• Pumping rate�� ��P = �√βJ0 • Degree of nonlinearity�� ��α = α0
How? RL Mean Field Theory Antenucci, 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 − Jsp 2
=(1− α)2J2 J2spqr − Jspqr 2
=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
�N Rab =
∑ j
< [ aaj a
b j
] �N
Tab = ∑ j
= [ aaj a
b j
] �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
�N Rab =
∑ j
< [ aaj a
b j
] �N
Tab = ∑ j
= [ aaj a
b j
] �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
�N Rab =
∑ j
< [ aaj a
b j
] �N
Tab = ∑ j
= [ aaj a
b j
] �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
P 2
RJ
CW
PLW
RL
SML α = 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
P 2
RJ
CW
PLW
RL
SML α = 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
P 2
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
P 2
RJ α=α0
P 2
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
2 ab
)2 + 2
( Q2ab − R
2 ab
)2 Replica 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
2 ab
)2 + 2
( Q2ab − R
2 ab
)2 Replica 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 Field Antenucci, 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 Antenucci, F., Ibañez Berganza, M. and Leuzzi, L., arXiv:1409.6345 (2014)
Consider the case of Standard Mode Locking Lasers