Quantum dynamics in complex environments { towards ... · Bittner & co-workers, in: Energy Transfer...
Transcript of Quantum dynamics in complex environments { towards ... · Bittner & co-workers, in: Energy Transfer...
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Quantum dynamics in complex environments –towards biological and nanostructured systems
Chris Engelbrecht Summer School on Quantum Biology
Lecture 4
Irene Burghardt
Department of Physical and Theoretical Chemistry, Goethe University Frankfurt
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Agenda
Lecture 1 Introduction – some general perspectivesLecture 2 Photophysics of extended molecular systemsLecture 3 Electronic, excitonic, and vibrational coherenceLecture 4 Excitation energy transfer (EET) – coherent or not coherent?Lecture 5 Non-Markovian system-environment interactions
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Excitonic coherence in multi-chromophoric systems
• Two basic (single-excited) configurations:
|φ1〉 = |e(1)〉 ⊗ |g(2)〉 |φ2〉 = |g(1)〉 ⊗ |e(2)〉
• For two identical, non-interacting monomers, these configurations aredegenerate
• once an excitonic coupling is included, the degeneracy is removed
• Frenkel exciton state = superposition of these configurations:
|Ψexciton(t)〉 = c1(t)|φ1〉+ c2(t)|φ2〉3
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Exciton model Hamiltonian
• matrix representation in the basis of single (locally) excitedconfigurations (site basis)
Hexciton =
ε(1) JJ ε(2) J
J ε(3) J. . . . . . . . .
J ε(n−1) JJ ε(n)
where J appears as a diabatic coupling
• alternative way of writing the same Hamiltonian employingcreating/annihilation operators:
H =∑n
ε(n)a†nan + J(a†nan+1 + a†n+1an)4
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What is the excitonic coupling?
• Coulomb coupling matrix element between donor (D) and acceptor (A)permitting transitions between single-excited configurations:
J = 〈eDgA|VCoulomb|gDeA〉
• this can be re-written in terms of coupled transition densities ρ(ge)A and
ρ(eg)D (noting that D/A exchange interactions have been disregarded!):
J =∫dr3Ddr3
A
ρ(eg)D (rD)ρ(ge)
A (rA)
|rD − rA|
• at large D-A distances, one can approximate transition densities by
transition dipoles: ρ(eg)D µ
(eg)D
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Alternatively: diabatization approach
0.8 eV
Bright Dark ∆E/2 = 0.86 eV
Bright-Dark exciton gap
Coupling between localized excitons
• start from a “super-molecular” calculation forthe (DA) system
• employ a diabatizationprocedure which trans-forms the adiabatic (Born-Oppenheimer) electronicstates to a diabatic basisof site-local states
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Exciton model including vibrations
Hexciton =N∑i
ωi
2
(p2i + x2
i
)
+
ε(1) +
∑i κ
(1)i xi J
J ε(2) +∑i κ
(2)i xi J
J ε(3) +∑i κ
(3)i xi J
. . . . . . . . .
J ε(n)+∑i κ
(n)i xi
• here, we assumed that the vibrational modes couple diagonally
• the coupling parameters can be obtained from the monomerHamiltonian; for example, for a mode that couples exclusivelyto monomer 1: h1(x1) ' c(1)
g x1|g(1)〉〈g(1)|+ c(1)e x1|e(1)〉〈e(1)|,
contributions appear for all exciton configurations 7
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Vibronic coupling picture of excitation energytransfer (EET)
Donor Acceptor
FRET
De Ag
Donor Acceptor
Dg Ae
Vibrational modes0
Initial state Final state
absorption
D A
Vibrational modes of
DA supermolecule
FRET
• FRET = fluorescence resonance energy transfer (equivalent to EET)8
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Exciton transfer can be coherent and ultrafast
quantum dynamics simulation:
0
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1
0 500 1000 1500
site
pop
ulat
ion
time [fs]
site 1: initial excitation
23 4 5 6 7 8 9
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0 200 400 600 800
site
pop
ulat
ion/
cohe
renc
etime [fs]
site 1: initial excitation
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12
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23 coherence
• site-site excitation energy transfer |φn〉 |φn+1〉• transfer is mediated by coherences |φn〉〈φn+1|• but the environment could rapidly induce “decoherence”
• coherent transfer can be observed experimentally! 9
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Correlations and coherenceǫ1
ǫ2
ǫ3
ǫ4• • •
ǫn• Do correlated fluctuations favor
conservation of coherence?
0
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0 500 1000 1500
po
pu
latio
ns
time [fs]
J=0.01 eV
pop 1pop 2pop 3pop 4pop 5
−0.6
−0.4
−0.2
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0 500 1000 1500
co
he
ren
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s
time [fs]
Im coh 12Im coh 13
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po
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time [fs]
J=0.01 eV
pop 1pop 2pop 3pop 4pop 5
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co
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time [fs]
Im coh 12Im coh 13
(a) site-local couplings by high-frequencyand low-frequency modes
(b) nearest-neighbor correlatedcouplings, with identical overallcoupling strengths as in (a) 10
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Excitonic eigenstate picture
electronic densities of stacked
adenine pentamer corresponding to two
ππ∗ states with different degrees of
localisation
Bittner & co-workers, in: Energy TransferDynamics in Biomaterial Systems, Burghardt etal. (eds), Springer (2009).
• the excitonic eigenstates are obtained by diagonalizing the excitonHamiltonian, as a function of the nuclear coordinates:
|Ψexciton(R; t)〉 =∑n
cn(R, t)|φn〉11
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Exciton transfer in the FMO complex
Brixner & collaborators, Nature 434, 625 (2005)
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5, 6
4
3
3, 7
2
2, 4
1
1
• FMO = Fenna-Matthews-Olson bacteriochlorophyll a (BChl) protein ofgreen sulphur bacteria
• antenna system that collects light and channels excitation to a reactioncenter where charge transfer takes place 12
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“Coherence dynamics in photosynthesis: proteinprotection of excitonic coherence”
Lee, Cheng, Fleming, Science 316, 1462 (2007)
k1
k3 k2
ks=-k1+k2+k3
t1t2B
Fig. 1. The 2CECPE experiment. (A) The 77 K
• 2CECPE = two-color electronic coherence photon echo experiment13
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2D photon echoes
pulse 1 −−− t1 −−− pulse 2 −−− t2 −−− pulse 3 −−− t3 −−−
ρeg ρee′ ρfe
• e single-exciton manifold ; f double-exciton manifold
• 2D Fourier transform Secho(ω1, t2, ω3), for various t2’s
• cross peaks appear, indicating a transfer process via ρee′ in the t2 interval
cf. Abramavicius & Mukamel, JCP 133, 064510 (2010)
Cheng & Fleming, Annu. Rev. Phys. Chem. 60, 241 (2009)
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Simulated 2D PE spectrum (RC of photosystem II)
064510-4 D. Abramavicius and S. Mukamel J. Chem. Phys. 133, 064510 ~2010!
• main diagonal peaks: D1 (e1) and D2 (e2)
• main cross peak: C1 (e1 e2)
• left: Lindblad model (QT)
• right: Classical transfer model (CT)
from: Abramavicius & Mukamel, JCP 133, 064510 (2010)
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Can we use perturbative limits?
consider site-site coupling (J) vs. electron-phonon coupling (κ)
• case 1: J � κ
Forster theory – non-coherent hopping between sites
e.g., Scholes & collaborators, J. Phys. Chem. B, 113, 656 (2009): EET in semiconducting polymers
• case 2: κ� J
excitonic eigenstates in a vibrational bath
e.g., Abramavicius & Mukamel, J. Chem. Phys. 133, 064510 (2010): EET in Photosystem II
• but in most systems of interest: κ ∼ J
in principle, we need the full dynamics on the Born-Oppenheimer surfacesof the oligomer/aggregate species 16
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Two examples
• Case 1: Fluorescent markers in biology: Forster theory
• Case 2: EET in conjugated polymers
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Fluorescent markers in biological applications
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Derivation of Forster theory, in a nutshell:Fermi’s Golden Rule + dipole-dipole interaction
thermally averaged rate (2nd order perturbation theory):
kDA =2π
h
∑D/A vib. states
f(E∗D)f(E0A) |〈ψ∗Dψ
0A|V
dipDA|ψ
0Dψ∗A〉|
2δ(E∗D+E0A−E
∗A−E
0D)
〈ψ∗Dψ0A|V
dipDA|ψ
0Dψ∗A〉 = 〈ψ∗Dψ
0A|µD · µA|rDA|3
−3(rDA · µD)(rDA · µA)
|rDA|5|ψ0Dψ∗A〉
= κDA〈ψ∗D|µD|ψ0
D〉〈ψ0A|µA|ψ∗A〉
|rDA|3
Now identify the ingredients of this formula with the expressions for thedonor emission / acceptor absorption spectra spectral overlap 19
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Forster theory = rate theory for resonant energytransfer between two dipoles
kDA(rDA) = kradD
( rFrDA
)6
r6F =
9c4 κ2DA
8πη4
∫ ∞0
dω1
ω4ID(ω)αA(ω)
“Forster radius” rF ∝ orientational factor × spectral overlap
ID(ω) = donor emission spectrum
αA(ω) = acceptor absorption spectrum
κDA = orientational factor:
κDA = sinθD sinθA cosΦARD − 2 cosθD cosθA
Standard Forster expression: isotropic average κ2DA = 2/3 20
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Forster theory, cont’d
kDA(rDA) = kradD
( rFrDA
)6; r6
F =9c4 κ2
DA
8πη4
∫ ∞0
dω1
ω4ID(ω)αA(ω)
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Exciton migration and long-lived coherences
Collini, Scholes, Science 323, 369 (2009), Bredas & Silbey, Science 323, 348 (2009)
V lin = ∑i
κ(1)
i xi λ (12)i xi
λ (12)i xi κ(2)
i xi λ (23)i xi
λ (23)i xi κ(3)
i xi λ (34)i xi
λ (43)i xi κ(3)
i xi λ (45)i xi
... ... ...
• coherent “surfing” rather than Forster-type hopping?• role of site-local vs. shared (correlated) modes• do shared modes contribute to the conservation of electronic coherence?
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How do Excitons Migrate across a Torsional Defect (Kink)?
1 2 3 4 5−0.2
0
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1
0 250 500 750 1000
time [fs]
cos(theta)
p 4+5
Im coh(34)
• exciton initially localized on left fragment (1 + 2 + 3)• the transfer to the (4 + 5) fragment sets in as a function of the torsional
displacement at the (34) junction• this transfer is mediated by the excitonic coherence ρ34
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Electronic Structure of Oligomers (OPV’s)
(Collaboration Lischka group (Vienna, Texas Tech))
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• high-level electronicstructure methods(ADC, CC2, MRCI)
• exciton trapping, due toBLA modes, describedcorrectly
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Panda, Plasser et al., in preparation, see also: Sterpone, Rossky, JPCB 112, 4983 (2008), Nayyar et al., JPCL 2, 566 (2011)
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Ring Torsions in OPV’s
0 30
monom
er
pote
ntial
torsion [deg]
S1
S0
●FC
• calculate torsion profiles ofoligomer excited states
• effective, correlated torsionalmodes, e.g., (T1-T1a) ≡ θ
nn′
• likewise, correlated BLA modes
0
0 0.5
1 1.5
2 2.5
3 3.5
4 4.5
5
0 30 60 90 120 150 180
Angle(deg)
T3
S0S1S2
Panda,Pla
sser
etal.,to
be
subm
itte
d(2
011)
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Fit Data to Exciton-Phonon Model
H = H0 +∑n Hmonomern (x1n
, . . . ,xNn)+∑nn′ δn′,n+1H
junction
nn′(θ torsion
nn′,xBLA
nn′)
• express H in the basis of single-excited configurations:|n〉= |G(1)
G(2) . . .E(n) . . .G(N)〉
• initial state = lowest excited (adiabatic) exciton state @ FC geometry; thenevolve in time: |ψFrenkel(t)〉= ∑n cn({xi,θi}, t) |n〉
• diagonal potentials (torsion + BLA):∝ V0(1− cos(σ(θnn′ −θ0)))+ k/2(xnn′ − x
0nn′
)2
• nearest-neighbor couplings modulated by torsions:
〈n|Hjunction
nn′|n′ = n+1〉= Jnm cosθnn′ (1−α(xnn′ − x0))
Sterpone, Martinazzo, Panda, Burghardt, Z. Phys. Chem. 225, 541 (2011)26
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Quantum Dynamics based on Model Hamiltonian
0
0.2
0 250 500 750 1000
time [fs]
p 4+5
p 4
Im coh(34)
0 300
600 900
1 2
3 4
5
1
po
pu
latio
n
time [fs]site index
• T = 0 simulations
• fluctuations due to high and low frequency modes reduce excitonic coherencebeyond 200 fs (but, marked dependence on model parameters)
• Ohmic damping (HO bath) acting upon torsional mode @junction
• extend to include (i) non-zero T and (ii) static disorder27
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EET Time Scales
tens of fs partial localization/trapping of excitons due to e-ph(BLA) coupling; formation of “spectroscopic units”
∼200 fs initial polarization anisotropy decay due to exciton mi-gration, possibly involving coherent intra-chain EET
100 fs- few ps torsional geometry relaxation interfering with EET
1-10 ps inter-chain EETps-ns thermally assisted hopping, mainly intra-chain
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