Multi-quanta vibrational dynamics in nonlinear quantum ... › tel-00128559 › file ›...

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion Multi-quanta vibrational dynamics in nonlinear quantum lattices Polarons and bi-polarons in bio-polymers and molecular nanostructures C. Falvo Laboratoire de Physique Moléculaire Université de Franche-Comté December 12th 2006

Transcript of Multi-quanta vibrational dynamics in nonlinear quantum ... › tel-00128559 › file ›...

Page 1: Multi-quanta vibrational dynamics in nonlinear quantum ... › tel-00128559 › file › soutenance.pdf · Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Multi-quanta vibrational dynamics in nonlinearquantum lattices

Polarons and bi-polarons in bio-polymers and molecularnanostructures

C. Falvo

Laboratoire de Physique MoléculaireUniversité de Franche-Comté

December 12th 2006

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Linear wave theory

A large number of physical systems are explained by alinear wave theoryExamples:

Electromagnetic wavesHydrodynamic waves...

Linearity induces dispersion and limits the transportproperties

However, Nature is nonlinear ...

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Discovery of nonlinearity: Soliton

Nonlinearity induces interactions between plane wavesand counterbalances the dispersion. It yields surprisingobjects not predicable by a perturbative theoryThe "Solitary wave" discovered by J.S. Russel in 1834signs the begining of a new theoretical area

Linear wavepacket

Nonlinear wave: thesoliton

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Solitons in microscopic systems: Davydov Theory

First theoretical description of the vibrational dynamics inmolecular lattices: The Davydov theory on the energytransfer in α-helices

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Solitons in microscopic systems: Davydov Theory

α-helix ≡ 1D modelAmide group O=C−N−HInternal vibration: Amide-I (C=O), Amide-A (N−H)

O=C−N−H O=C−N−H O=C−N−H O=C−N−H? ? ? ?J J J

ATP

C=Ovibration

ω0

hydrolysis vibronψ

n

J dipole-dipole

couplingphonons

γ nonlinear

coupling

Nonlinear Schrodinger Equation (NLS)

iψn = ω0ψn − J(ψn+1 + ψn−1) − γ|ψn|2ψn

solitondiscrete breather

No experimental evidenceSemi-classical approach → full quantum model

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Objectives

To give a theoretical formalism to describe the vibrationaldynamics in molecular latticesIn molecular lattices, vibrations are ruled by nonlineareffects which permit the energy localization or the energycoherent transfer in a translationally invariant systemIn molecular lattices, vibration are ruled by quantummechanics which states that in a translationally invariantsystem all the quantum states are delocalized (Blochtheorem)

Theoretical Challenge:Combine quantum and nonlinear physics in a lattice

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Summary

1 TheoryEffective HamiltonianA simple example: the 1D Hubbard Hamiltonian

2 α-Helix: 1D and 3D modelHelix geometry1D modelPump-probe spectroscopy3D model

3 Energy redistributionHamiltonian and quantum analysisNumerical analysisSimplified equivalent lattice

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Vibrational Hamiltonian

Qn−1 Qn Qn+1

ωn

ωn-2An hn =P2

n2µ

+12

V (2)n Q2

n +13!

V (3)n Q3

n +14!

V (4)n Q4

n + . . .

≈ ωnb†nbn − Anb†2

n b2n

ενn = ωnνn − Anνn(νn − 1)

H =∑

n

[ωnb†

nbn − Anb†2n b2

n

]︸ ︷︷ ︸

anharmonic oscilators

+∑n 6=m

Φnmb†nbm︸ ︷︷ ︸

lateral coupling

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Vibrational Hamiltonian

Qn−1 Qn Qn+1

ωn

ωn-2An hn =P2

n2µ

+12

V (2)n Q2

n +13!

V (3)n Q3

n +14!

V (4)n Q4

n + . . .

≈ ωnb†nbn − Anb†2

n b2n

ενn = ωnνn − Anνn(νn − 1)

H =∑

n

ωnb†nbn +

∑n 6=m

Φnmb†nbm︸ ︷︷ ︸

free vibrons

−∑

n

Anb†2n b2

n︸ ︷︷ ︸vibron-vibron coupling

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Vibrational Hamiltonian

Qn−1 Qn Qn+1

un−1 unun+1

ωn

ωn-2An hn =P2

n2µ

+12

V (2)n Q2

n +13!

V (3)n Q3

n +14!

V (4)n Q4

n + . . .

≈ ωnb†nbn − Anb†2

n b2n

ενn = ωnνn − Anνn(νn − 1)

H =∑

n

ωn(−→u )b†

nbn +∑n 6=m

Φnm(−→u )b†

nbm︸ ︷︷ ︸free vibrons

−∑

n

An(−→u )b†2

n b2n︸ ︷︷ ︸

vibron-vibron coupling

+

phonon dynamics︷︸︸︷Hp

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Vibron-Phonon coupling

linear dependence of the vibron parameters

ωn(−→u ) ≈ ω0 +

∑m

ξnmum

An(−→u ) ≈ A +

∑m

ξ′nmum

Φnm(−→u ) ≈ Φnm

harmonic approximation for the phonon Hamiltonian

Hp =∑

n

p2n

2M+

12

∑nm

Wnmunum =∑

q

Ωq(a†qaq + 1/2)

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Vibrational Hamiltonian

The vibrational dynamics is divided into three parts

H =∑

n

[ω0b†

nbn − Ab†2n b2

n

]+

∑n 6=m

Φnmb†nbm

+∑nq

[∆nqa†q + ∆∗

nqaq]b†nbn + [∆′

nqa†q + ∆′∗

nqaq]b†2n b2

n

+∑

q

Ωq(a†qaq + 1/2)

H = Hv + ∆Hvp + Hp

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Vibrational Hamiltonian

The vibrational dynamics is divided into three parts

H =∑

n

[ω0b†

nbn − Ab†2n b2

n

]+

∑n 6=m

Φnmb†nbm

+∑nq

[∆nqa†q + ∆∗

nqaq]b†nbn + [∆′

nqa†q + ∆′∗

nqaq]b†2n b2

n

+∑

q

Ωq(a†qaq + 1/2)

H = Hv + ∆Hvp + Hp

Strong vibron-phonon coupling |∆nq| ∼ A ∼ |Φnm|perturbation theory is not suitablenonperturbative theory: the polaron point of view

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Polaron point of view

Polaron theory is based on the Lang-Firsov transformationwhich renormalizes the vibron-phonon coupling. H = THT †

The creation of a vibron induces a local lattice distortionThe lattice distortion follows the motion of the vibration,modifies its dynamics and yields the formation of a polaron

×+

polaron = vibron + lattice distortion

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Polaron point of view

Polaron theory is based on the Lang-Firsov transformationwhich renormalizes the vibron-phonon coupling. H = THT †

The creation of a vibron induces a local lattice distortionThe lattice distortion follows the motion of the vibration,modifies its dynamics and yields the formation of a polaron

×+

polaron = vibron + lattice distortion

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Polaron point of view

Polaron theory is based on the Lang-Firsov transformationwhich renormalizes the vibron-phonon coupling. H = THT †

The creation of a vibron induces a local lattice distortionThe lattice distortion follows the motion of the vibration,modifies its dynamics and yields the formation of a polaron

×+ ×+

polaron = vibron + lattice distortion

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Polaron point of view

Polaron theory is based on the Lang-Firsov transformationwhich renormalizes the vibron-phonon coupling. H = THT †

The creation of a vibron induces a local lattice distortionThe lattice distortion follows the motion of the vibration,modifies its dynamics and yields the formation of a polaron

×+

polaron = vibron + lattice distortion

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Effective Hamiltonian

How the vibron-phonon coupling modifies the vibrationaldynamics ?

Since the phonons are in thermal equilibrium attemperature T , the dynamics is governed by an effectiveHamiltonian

Heff = 〈H − Hp〉

Renormalization of the vibron parameters

Heff =∑

n

[(ω0−ε

(1)nn

)b†

nbn −(

A+ε(1)nn + 2ε(2)

nn

)b†2

n b2n

]−

∑n 6=m

ε(1)nmb†

nb†mbnbm +

∑n 6=m

Φnme−Snm(T )b†nbm

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

1D Hubbard Model for bosons

H =∑

n

ω0b†nbn − Ab†2

n b2n + Φb†

n(bn−1 + bn+1)

H conserves the vibron population: [H, v ] = 0 wherev =

∑n b†

nbn

H is translationally invariant: [H, T ] = 0 where T is thetranslation operator defined by T b†

n = b†n+1T with

eigenvalues τ = exp(ik)

H is bloc diagonal according to the good quantumnumbers v and k

v ≡ total number of vibron in the latticek ≡ wavevector corresponding to the delocalization of thevibron barycenter

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

One vibron states

one quantum = free particleeigenstates = plane wavesquantum state in the local basis

|ψ1(k)〉 =1√N

N∑n=1

eikn|n〉

The dispersion curve describes a delocalized particle

ω1(k) = ω0 + 2Φ cos(k)

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Two vibron states

quantum state in the local basis

|ψ2〉 =∑

n16n2

ψ(n1,n2)|n1,n2〉

n1 and n2 correspond to the position of each quantumtwo vibrons on a 1D lattice ≡ one particle on a 2D lattice

n1

n2

2ω0 − 2A

2ω0 Φ

invariance in the n1 = n2 direction

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Two vibron states

quantum state is delocalized according to the vibronbarycenter

ψ(n1,n2) ≈ ψk (m = n2 − n1) exp(

ikn1 + n2

2

)k ≡ wavevector corresponding to the delocalization of thevibron barycenterm ≡ interdistance between the two vibronsHψk = ω2(k)ψk

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Two vibrons states

a continuum = free states: two independent vibronsω2(k = k1 + k2) = ω1(k1) + ω1(k2)an isolated band = bound states: two trapped vibrons witha delocalized barycenter

ω2(k) = 2ω0 − 2√

A2 + 4Φ2 cos2(k/2) ; ∆ω ≈ 4Φ2

A

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Helix geometry

1D model ≡ 1 spine3D model ≡ 3interacting spines

~R(n) = R0 cos nθ0~ex + R0 sin nθ0~ey + nh~ez

R0 = 2.8Åθ0 = 100

h = 1.5Å

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Helix geometry

n

1D model ≡ 1 spine3D model ≡ 3interacting spines

~R(n) = R0 cos nθ0~ex + R0 sin nθ0~ey + nh~ez

R0 = 2.8Åθ0 = 100

h = 1.5Å

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Helix geometry

n

n+3

1D model ≡ 1 spine3D model ≡ 3interacting spines

~R(n) = R0 cos nθ0~ex + R0 sin nθ0~ey + nh~ez

R0 = 2.8Åθ0 = 100

h = 1.5Å

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Helix geometry

n

n+3

n+6

1D model ≡ 1 spine3D model ≡ 3interacting spines

~R(n) = R0 cos nθ0~ex + R0 sin nθ0~ey + nh~ez

R0 = 2.8Åθ0 = 100

h = 1.5Å

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Helix geometry

n

n+3

n+6

1D model ≡ 1 spine3D model ≡ 3interacting spines

~R(n) = R0 cos nθ0~ex + R0 sin nθ0~ey + nh~ez

R0 = 2.8Åθ0 = 100

h = 1.5Å

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Helix geometry

n

n+1n+3

n+6

1D model ≡ 1 spine3D model ≡ 3interacting spines

~R(n) = R0 cos nθ0~ex + R0 sin nθ0~ey + nh~ez

R0 = 2.8Åθ0 = 100

h = 1.5Å

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Helix geometry

n

n+1

n+2

n+3

n+6

1D model ≡ 1 spine3D model ≡ 3interacting spines

~R(n) = R0 cos nθ0~ex + R0 sin nθ0~ey + nh~ez

R0 = 2.8Åθ0 = 100

h = 1.5Å

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Helix geometry

n

n+1

n+2

n+3

n+6

1D model ≡ 1 spine3D model ≡ 3interacting spines

~R(n) = R0 cos nθ0~ex + R0 sin nθ0~ey + nh~ez

R0 = 2.8Åθ0 = 100

h = 1.5Å

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Vibron-phonon Hamiltonian

nearest neighbor interaction

Hv =∑

n

ω0b†nbn − Ab†2

n b2n − Jb†

n (bn+1 + bn−1)

acoustical phonons

Hp =∑

n

p2n

2M+

12

W (un+1 − un)2

vibron-phonon coupling

∆Hvp =∑

n

ξ(un+1 − un−1)b†nbn − ξ′(un+1 − un−1)b

†2n b2

n

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Effective Hamiltonian

Heff =∑

n

[ω0b†

nbn − Ab†2n b2

n − Bb†nb†

n+1bnbn+1

]−

∑n

Je−S(T )(b†nbn+1 + b†

n+1bn)

ω0 = ω0 − ε

A = A + ε− 4yε Local nonlinearity

B = ε Nonlocal nonlinearity

The vibron-phonon coupling is characterized by theparameters ε = ξ2/W and y = ξ′/ξ

two kinds of nonlinearities → two kinds of bound states

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Two polarons states: CO vibration

ω0 = 1664 cm−1, A = 8 cm−1, J = 7.8 cm−1, T = 310 K

ε = 5 cm−1

y = 0

ε = 15 cm−1

y = 0

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

What is pump probe spectroscopy ?

Pump-probe spectroscopy of an isolated anharmonicvibration

0

1

2

ω0

ω0-2A

ω0-2A ω0

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

PP Spectra of an α-helix: Amide-I (C=O)

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

PP Spectra of an α-helix: Amide-I (C=O)

PP Spectroscopy of CO ofPBLG

From J. Edler, Ph.D. thesis,University of Zurich, with the

courtesy of J. Edler

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

PP Spectra of an α-helix: Amide-I (C=O)

PP Spectroscopy of CO ofPBLG

From J. Edler, Ph.D. thesis,University of Zurich, with the

courtesy of J. Edler

Parameter ε to recover the experiment:ε = 4.2 cm−1

y = 0

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

PP Spectra of an α-helix: N−H

PP Spectroscopy of NH mode ofPBLG1

J. Edler, R. Pfister, V. Pouthier, C. Falvo andP. Hamm PRL 93,106405 (2004)

at T = 260K, PBLG is in random coil → one positive peakat T = 293K, PBLG is in α-helix → two positive peaks

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

PP Spectra of an α-helix: N−H

PP Spectroscopy of NH mode ofPBLG1

J. Edler, R. Pfister, V. Pouthier, C. Falvo andP. Hamm PRL 93,106405 (2004)

at T = 260K, isolated vibration modelA = 60 cm−1

at T = 293K, α-helix 1D modelε = 119.4 cm−1, y = 0.0877

at T = 260K, PBLG is in random coil → one positive peakat T = 293K, PBLG is in α-helix → two positive peaks

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

PP Spectra of an α-helix: N−H

PP Spectroscopy of NH mode ofPBLG1

J. Edler, R. Pfister, V. Pouthier, C. Falvo andP. Hamm PRL 93,106405 (2004)

ω0 = 3400 cm−1, A = 60 cm−1,ε = 119.4 cm−1, y = 0.0877

at T = 260K, PBLG is in random coil → one positive peakat T = 293K, PBLG is in α-helix → two positive peaks

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Vibrons Hamiltonian

Long range interactionsJ(3) intra-spine couplingJ(1), J(2) inter-spine couplings

Hv =∑

n

ω0b†nbn − Ab†2

n b2n

− 12

∑n 6=n′

J(|n − n′|)b†nbn′

n

n+1n+2

n+3

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Phonons Hamiltonian

Hp =∑nα

p2α(n)

2M+

12

∑nα,n′β

Φαβ(nn′)uα(n)uβ(n′)

coordinates are expanded asBloch waves in the local framesDispersion curves

Hp =∑qs

Ωqs(a†qsaqs +

12)

0

20

40

60

80

100

120

140

0 0.5 1 1.5 2 2.5 3

Ωkσ

(cm

-1)

k

Acoustical branch: longitudinal phonons polarized alongthe hydrogen bondsOptical branch: breathing mode of the helix radius

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Effective Hamiltonian

0.000

1.000

2.000

3.000

4.000

5.000

6.000

2 4 6 8 10 12 14 16 18 20

S a(n

,T)

n

T=310 KT=150 K

T=5 K

0.000

0.005

0.010

0.015

0.020

0.025

0.030

2 4 6 8 10 12 14 16 18 20

S o(n

,T)

n

Heff =∑

n

ω0b†nbn − Ab†2

n b2n −

∑n 6=n′

12

B(n − n′)b†nb†

n′bnbn′

−∑n 6=n′

Jeff (|n − n′|)b†nbn′

with Jeff (n) = J(n) exp(−S(n,T ))

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Effective Hamiltonian

0 1 2 3 4 5 6 7 8 9

10

0 50 100 150 200 250 300 350

|Jef

f(n)|

(cm

-1)

T (K)

n=1n=2n=3

Heff =∑

n

ω0b†nbn − Ab†2

n b2n −

∑n 6=n′

12

B(n − n′)b†nb†

n′bnbn′

−∑n 6=n′

Jeff (|n − n′|)b†nbn′

with Jeff (n) = J(n) exp(−S(n,T ))

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Two polarons states: CO vibration

T = 5 K

The 3D nature of the quantumstates is important

T = 310 K

The helix is equivalent to 3independant 1D chains

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Nonlinear quantum lattice

In molecular vibrations: two kinds of nonlinearity

− Local nonlinearity (intramolecular anharmonicityand vibron-phonon coupling)

− Nonlocal nonlinearity (vibron-phonon coupling only)

In molecular vibrations: multi-quanta states

− Nonlinear spectroscopy− Fermi resonance 2ωN−H ≈ 4ωC=O− Local excitation (STM)

Nonlinear quantum lattice: a general issue

In molecular vibrations but also in Bose Einstein conden-sates, quantum spin lattices, electronic excitons in molecu-lar lattices,. . .

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Generalized Hubbard model

Dynamics of v quanta interacting on a 1D lattice

H =∑

n

ω0b†nbn−Ab†

nb†nbnbn−Bb†

n+1b†nbn+1bn+Φ

[b†

n+1bn + b†nbn+1

]Quantum dynamics i

ddt|Ψ(t)〉 = H|Ψ(t)〉

Initial condition: localized state |Ψ(0)〉 =b†v

n0√v !|∅〉

An analytical and computational challenge

Dv =(N + v − 1)!

v !(N − 1)!, e.g. N = 91,

v 2 3 4 5Dv 4 186 129 766 3 049 501 57 940 519

Dv/N 46 1 426 33 511 636 709

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Local population evolution

Pn(t) = 〈Ψ(t)|b†nbn|Ψ(t)〉

v = 3, N = 91, A = 3Φ

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Local population evolution

Pn(t) = 〈Ψ(t)|b†nbn|Ψ(t)〉

v = 3, N = 91, A = 3Φ

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Local population evolution

Pn(t) = 〈Ψ(t)|b†nbn|Ψ(t)〉

v = 3, N = 91, A = 3Φ

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Local population evolution

Pn(t) = 〈Ψ(t)|b†nbn|Ψ(t)〉

v = 3, N = 91, A = 3Φ

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Local population evolution

Pn(t) = 〈Ψ(t)|b†nbn|Ψ(t)〉

v = 3, N = 91, A = 3Φ

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Local population evolution

Pn(t) = 〈Ψ(t)|b†nbn|Ψ(t)〉

v = 3, N = 91, A = 3Φ

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Local population evolution

Pn(t) = 〈Ψ(t)|b†nbn|Ψ(t)〉

v = 3, N = 91, A = 3Φ

Dynamical transition for B = 2A

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Survival probability

S0(t) = |〈Ψ(0)|Ψ(t)〉|2

v=3 v=4 v=5

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

|s0(

t)|2

Φt

B=0

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Φt

B=2A

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Simplified equivalent lattice

relevant states

|n,p〉 = |0, . . . ,0, v − p︸ ︷︷ ︸n

, p︸︷︷︸n+1

,0, . . . ,0〉

example v = 4

|n,0〉 = |0, . . . ,0, 4︸︷︷︸n

, 0︸︷︷︸n+1

,0, . . . ,0〉

n− 2 n− 1 n n + 1 n + 1

××××

++++

Page 58: Multi-quanta vibrational dynamics in nonlinear quantum ... › tel-00128559 › file › soutenance.pdf · Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Simplified equivalent lattice

relevant states

|n,p〉 = |0, . . . ,0, v − p︸ ︷︷ ︸n

, p︸︷︷︸n+1

,0, . . . ,0〉

example v = 4

|n,1〉 = |0, . . . ,0, 3︸︷︷︸n

, 1︸︷︷︸n+1

,0, . . . ,0〉

n− 2 n− 1 n n + 1 n + 1

×××

+++

×+

Page 59: Multi-quanta vibrational dynamics in nonlinear quantum ... › tel-00128559 › file › soutenance.pdf · Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Simplified equivalent lattice

relevant states

|n,p〉 = |0, . . . ,0, v − p︸ ︷︷ ︸n

, p︸︷︷︸n+1

,0, . . . ,0〉

example v = 4

|n,2〉 = |0, . . . ,0, 2︸︷︷︸n

, 2︸︷︷︸n+1

,0, . . . ,0〉

n− 2 n− 1 n n + 1 n + 1

××++

××++

Page 60: Multi-quanta vibrational dynamics in nonlinear quantum ... › tel-00128559 › file › soutenance.pdf · Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Simplified equivalent lattice

relevant states

|n,p〉 = |0, . . . ,0, v − p︸ ︷︷ ︸n

, p︸︷︷︸n+1

,0, . . . ,0〉

example v = 4

|n,3〉 = |0, . . . ,0, 1︸︷︷︸n

, 3︸︷︷︸n+1

,0, . . . ,0〉

n− 2 n− 1 n n + 1 n + 1

×+ ×××

+++

Page 61: Multi-quanta vibrational dynamics in nonlinear quantum ... › tel-00128559 › file › soutenance.pdf · Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Simplified equivalent lattice

relevant states

|n,p〉 = |0, . . . ,0, v − p︸ ︷︷ ︸n

, p︸︷︷︸n+1

,0, . . . ,0〉

example v = 4

|n + 1,0〉 = |0, . . . ,0, 0︸︷︷︸n

, 4︸︷︷︸n+1

,0, . . . ,0〉

n− 2 n− 1 n n + 1 n + 1

××××

++++

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Resonance at B = 2A

Equivalent lattice ≡ 1 particle in a 1D lattice with a periodicpotential

+ + + + + + + +

+

|n − 1, 0〉 |n, 0〉 |n, 1〉 |n, 2〉 |n, 3〉 |n + 1, 0〉

Φ0

Φ1 Φ2

Φ3

ε0

ε1

ε2

E

Φp =√

(p + 1)(v − p)Φ ε0 = vω0 − v(v − 1)Aεp = ε0 + p(v − p)(2A− B)

|2A− B| Φ → strong energy barrier

The particle delocalizes through tunneling effect

Delocalization time: τ ∼ |2A− B|v−1

Φv

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Resonance at B = 2A

Equivalent lattice ≡ 1 particle in a 1D lattice with a periodicpotential

|n − 1, 0〉 |n, 0〉 |n, 1〉 |n, 2〉 |n, 3〉 |n + 1, 0〉

Φ0 Φ1 Φ2 Φ3

ε0

E

Φp =√

(p + 1)(v − p)Φ ε0 = vω0 − v(v − 1)Aεp = ε0 + p(v − p)(2A− B)

B = 2A → no energy barrier

The particle can freely propagates

Delocalization time: τ ∼ Φ−1

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Vibrational lattice

In molecular lattice: A0 ≡ intramolecular anharmonicityε ≡ vibron phonon coupling

local nonlinear coupling: A ≈ A0 + εnonlocal nonlinear coupling: B ≈ ε

2A− B = 2A0 + ε

vibration A0 ε Φ 2A− BN−H 60 cm−1 80 cm−1 5 cm−1 200 cm−1

C=O 8 cm−1 4 cm−1 7.8 cm−1 20 cm−1

Amide-II ? ? ? ?

Amide-II ≡ N−H bendingsmall global anharmonicity A ≈ A0 + ε . 8 cm−1

symetric potential → A0 < 0 ?2A− B ∼ 0 ?

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Conclusion

TheoryDipole-dipole coupling and translation symetry induce adelocalisation of high frequency vibrationsTwo kinds of nonlinear sources: intramolecularanharmonicity and vibron-phonon couplingNonlinearity favours the occurrence of bound states

α-helices: model 1D and model 3DThe two polarons spectrum in 1D model shows theexistence of two kinds of bound statesEach bound state has been observed in a pump-probeexperimentIn 3D model, a low temperature the 3D nature of quantumstates is important. At biological temperature, helixgeometry induce a separation of the three spines

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Conclusion

Energy redistribution

In a generalized Hubbard model we show the occurence ofa dynamical transition. Bound states can either localize ortransfer the energy depending on the nonlinearities

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Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion

Perspectives

Molecular vibrationAmide-II vibrationStudy of the polaron relaxationMore realistic protein modelMulti-dimensional spectroscopy with relaxation

Quantum nonlinear latticeWith a low number of quanta → number state methodWith a large number of quanta → quasi-classicalapproximation (NLS equation)Intermediate situation → Simplified equivalent latticeGeneration of quantum breather through coherent states

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Appendix

Remerciements

Vincent PouthierChris Eilbeck de l’université Heriot-Watt, EdimbourgPeter Hamm et Julian Edler de l’université de ZurichCNRS et Région Franche-Comté