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
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 ...
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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)
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
√
2Φ
invariance in the n1 = n2 direction
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Introduction Theory α-Helix: 1D and 3D model Energy redistribution Conclusion
PP Spectra of an α-helix: Amide-I (C=O)
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
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
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
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
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
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
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
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 ))
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 ))
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
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,. . .
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
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
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
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
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
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
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
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
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
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
××××
++++
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
×××
+++
×+
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
××++
××++
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
×+ ×××
+++
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
××××
++++
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
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
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 ?
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
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
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
Appendix
Remerciements
Vincent PouthierChris Eilbeck de l’université Heriot-Watt, EdimbourgPeter Hamm et Julian Edler de l’université de ZurichCNRS et Région Franche-Comté
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