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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