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