Transcript Role of Coherence in Biological Energy Transfer
Role of Coherence in Biological Energy Transfer
Tomas Mancal Charles University in Prague Collaborators: Jan Olšina, Vytautas Balevičius and Leonas Valkunas QuEBS 09 8.7.2009 Lisbon
System of Interest: Photosynthetic Aggregates of Chlorophylls • • • System of “two-level” molecules.
Resonance coupling results in delocalization.
Coupling to “vibrational” bath leads to energy relaxation and “decoherence”.
• • Well studied biological systems governed by quantum mechanics.
Some surprising new results appeared.
Spectroscopy of Molecular Aggregates • • • • Non-linear spectroscopy maps dynamics of the system to a spectroscopic signal There is a well-developed formalism which describes this mapping Mapping is provided by response functions = correlation functions of dynamics in different time intervals Signal is a mixture of response functions corresponding to different types of dynamics
Coherence in 2D Photon Echo Spectroscopy
Electronic Coherence
Diagonal cut through 2D spectrum of molecular dimer • All peaks change shape • with frequencies corresponding • to transitions between • excitonic states • • 2D spectrum reveals • the motion • of the electronic wavepacket • Oscillation were predicted • for photosynthetic protein • FMO.
Pisliakov, Mančal & Fleming, J. Chem. Phys, 124 (2006) 234505 Kjellberg, Brüggemann & Pullerits, Phys. Rev. B 74 (2006) 024303
2D photon echo of FMO complex Time evolution of a 2D spectrum • Spectrum reveals the predicted oscillations • Oscillations live longer than predicted • Also the contribution corresponding to energy relaxation oscillates Conclusion: coherence transfer G. S. Engel et al., Nature 446 (2007) 782
Vibrational Coherence
Task: to clarify the role of vibrational contributions to the beating.
We need a system that cannot exhibit electronic wavepackets.
Fast mode ω ≈ 1500 cm -1 Slow(er) modes ω ≈ 140 cm -1 and ω ≈ 570 cm -1
Full Numerical Calculation A. Nemeth et al., Chem. Phys. Lett. 459 (2008) 94
Quantum Dynamics and Spectroscopy
Can we see what we want to see?
• (Non-linear) spectroscopy gives us a partial view of the system’s density matrix
Response Functions and Density Matrix Propagation Whole world density matrix:
S
(
t tr
) 0
tr
0 (
t
)
i
tr
(
t
)
tr
10
U
ˆ 0 (
t
) (
t
) (
t
) 0 (
t
) (
t
) 0 Spectroscopic signal :
S
(
t
) Spatial phase 01
U
ˆ 1 (
t
)
Ee
i
k
1 .
r
factor
h
.
c
.
0 th order - contributes by zero First order response Equivalent to: ˆ 10 2
tr bath
0 (
t
) 1 (
t
)
Ee
i k
1 .
r
ˆ 10 2 ˆ 01 (
t
)
Ee
i k
1 .
r
Element of reduced density matrix First order signal can be calculated from a Master equation for coherence elements of the reduced density matrix!
Response Functions and Density Matrix Propagation For some models , element ˆ 01 (
t
) can be calculated exactly from the master equation.
H
R. Doll et. al, Chem. Phys. 347 (2008) 243 Second order terms: 1
tr bath
ˆ 10
U
ˆ 1 (
t
) ˆ 10
U
ˆ 0 ( ) 0 0 ˆ 01
U
ˆ 1 (
t
)
E
2
e
i k
1 .
r
i k
2 .
r
2 ˆ 10 2 ˆ 01 ˆ 11 (
t
; )
E
2
tr bath
ˆ 10
U
ˆ 0 (
t
) 0 0 Excited state element of RDM, with special initial condition ˆ 01
U
ˆ 1 ( ) ˆ 10
U
ˆ 1 (
t
)
E
2
e
i k
1 .
r
i k
2 .
r
ˆ 10 2 ˆ 10 ˆ 00 (
t
; )
E
2 Ground state element of RDM, with special initial condition
Response Functions and Density Matrix Propagation In the perturbation expansion we visit different “corners” of the total density matrix 1 N N(N-1)/2 • For resonantly coupled 2 level systems the density matrix splits into decoupled blocks.
• Optical transitions occur between these blocks ˆ ( ( ( 1 2 ) ) )
eq
• Spectroscopists often use the language of these blocks. We excited a coherence between “System is in the ground state” We can use this “language” as long as we keep in mind that it relates to the “current order” of perturbation theory!
gg e e g g eg Feynman Diagrams and Liouville Pathways
t
g Each pathway or diagram corresponds to three successive propagations of the density matrix block e
T
e g :
T
:
t
:
ge
( )
ge
( )
U gege
( )
gg
( )
ee
( )
gg
(
T
)
W ee
(
T
)
U gggg
(
T
, )
U eeee
(
T
, )
ge
( 0 )
gg
( )
ee
( )
W ge
(
T
)
ge
(
t
)
U gege
(
t
,
T
)
ge
(
T
) ee
T
Putting all this together we get a response function
R
(
t
,
T
, )
tr bath
{
U gege
(
t
)
U eeee
(
T
)
U egeg
( ) ˆ
eq
} ge
t
gg
Mean Field Approach
Let us consider the coherence term
eg
( )
U egeg
( )
eg
( 0 ) After the excitation ˆ
eg W eg
( 0 )
W eq
Reduced density matrix ( 0 ) Time evolution
W eg
(
t
)
U
ˆ
e
(
t
) ( 0 ) ( 0 ) ( 0 )
U
ˆ
g
(
t
) ( 0 )
tr bath
{ ( 0 ) ( 0 ) } ˆ
eg
( 0 ) ˆ
eq
ˆ
eg
(
t
) ˆ
eg
(
t
) ˆ
eq
?
Master Equations
ˆ
eg
(
t
) ˆ
eg
(
t
)
W
Nakajima-Zwanzig
t
ˆ (
t
)
iL
ˆ (
t
)
I
0 (
t
)
t
0
d
M
( ) ˆ (
t
) Past evolution of the system Total evolution operator of the system Convolution-less approach
t
ˆ (
t
)
iL
ˆ (
t
)
I
0 (
t
)
t
0
d
M
( )
U
( ) ˆ (
t
) In Nakajima-Zwanzing one can introduce so-called “Markov” approximation ˆ (
I
) (
t
) ˆ (
I
) (
t
) which accidentally leads to the same result as convolution-less approach, when we stay in second order in system-bath coupling.
Master Equations
A common approximation in the relaxation tensor is so-called Secular approximation = decoupling of populations and coherences, and even decoupling of different coherences from each other.
One of the major results of Greg Engel’s experiment is that secular approximation does not work well for FMO.
Further in this talk we will assume four types of relaxation equations: • • • • Full second order Nakajima-Zwanzig (QME) Full second order convolution-less relaxation equation (Markov) Secular QME Secular Markov … and let’s assume we can calculate spectroscopy from reduced quantities.
R
(
t
,
T
, )
tr bands
{
U gege
(
t
)
U eeee
(
T
)
U egeg
( ) ˆ
eq
}
Quantum Dynamics and Photosynthetic Systems
Some non-trivial Coherence Effects
• What if we drop secular approximation.
Coherence Transfer Effect in Absorption Spectroscopy Coupled coherences
t
e
0
c
0 (
t
(
t
) )
i
e
0 21 11
i
c
0 12 22
e
0
c
0 (
t
(
t
) ) Long wavelength part of the bacterial reaction center absorption spectrum Eigenfrequencies
c
0
e
0 2 Re 1 2 [
c
0
e
0
i
( 11 22 )] 2 4 12 21 T. Mancal, L. Valkunas, and G. R. Fleming, Chem. Phys. Lett. 432 (2006) 301
Can we simulate what we measure?
• • Photosynthetic systems are not Markovian.
Coherence transfer leads to troubles.
Comparison of Relaxation Theories
Populations of a molecular dimer Breakdown of positivity Oscillations due to coupling to coherences • Non-secular Markov QME is not satisfactory at long times • Oscillations of the population seem to be a “real” effect J. Olsina and T. Mancal, in preparation
Comparison of Relaxation Theories
Coherence in a molecular dimer Survival of coherences due to memory Stationary coherence in non-secular dynamics • In non-Markov dynamics coherence lives longer; Population dynamics does not matter.
• Stationary coherence leads to the break-down of the positivity in non secular Markov theory
Relaxation Theories and 2D Spectrum 1 1 Simple Trimer 2 3 10000
cm
1 2 500
cm
1
d
1 1
J
13
J
12
d
3
J
23 3
d
2 2
J
12
J
23
J
13 100
cm
1 Absorption Spectrum Overdamped Brownian oscillator model for energy gap correlation function
c
30
cm
1 50
fs
Relaxation Theories and 2D Spectrum Populations of a molecular trimer General results from dimer system remain valid • Representative coherence evolutions are given by the full QME and secular Markov QME. • When relaxation is slow 2D spectrum depends mostly one the evolution of coherences.
Trimer
Secular Markov Full QME T = 0 fs
Trimer
Secular Markov Full QME T = 25 fs
Trimer
Secular Markov Full QME T = 60 fs
Trimer
Secular Markov Full QME T = 90 fs
Trimer
Secular Markov Full QME T = 125 fs
Trimer
Secular Markov Full QME T = 200 fs
Quantum Dynamics and Multi-point Correlation Functions
Can we simulate what we measure?
• Response functions are multi-point time correlation functions – very difficult to evaluate by Master equations.
Response Functions as Multi-point Correlation Functions How good was our calculation?
We used a projection operators: Complementary operator:
P
ˆ
tr bath
{ ˆ }
Q
1
P
ˆ
eq R
(
t
,
T
, )
tr
{
U gege
(
t
)
U eeee
(
T
)
U egeg
( ) ˆ
eq
}
tr
{(
P
Q
)
U gege
(
t
)(
P
Q
)
U eeee
(
T
)(
P
Q
)
U egeg
( )
P W eq
}
tr
{
PU gege
(
t
)
PU eeee
(
T
)
PU egeg
( )
P
ˆ
eq
}
tr
{
U gege
(
t
)
U eeee
(
T
)
U egeg
( ) ˆ
eq
} A rather crude approximation!
Response Functions as Multi-point Correlation Functions To calculate response function from Master equations
Q W
0 at all three occurrences. Each interval has to be calculated with different projector, i.e. by a different Master equation.
P
ˆ
tr
{ ˆ }
W P
For population interval we need
tr bath
{ }
U
ˆ
g
( )
W eq U
ˆ
e
( )
e g
( ) T. Mancal, in preparation
What was not discussed here.
• • • • • Correlated fluctuations Finite laser pulse length effects – Wavepacket preparation – Influence on relaxation Non-adiabatic effects Polarization of laser pulses And probably many other issues
What is Quantum on Quantum Coherences?
Coherent States and Classicality
Coherent States and Classicality
Coherent states are the best quantum approximations of classical states!
• Relaxation of harmonic oscillator Gaussian wavepacket vs. point in the phase space • Optical coherent states Coherent state vs. classical electromagnetic wave Initial state = linear combination of some vibrational states Final state = linear combination of different vibrational states Relaxation of a coherent wavepacket In between there is coherence transfer!
Conclusions
• • • • “Realistic” description of ultrafast energy relaxation and transfer in biological systems has to account for electronic and vibrational coherence.
Often memory effects are of importance.
We view the dynamics through a very distorted “magnifying glass” the effects of which are not immediately obvious.
Coherent effects are perhaps more classical and more ubiquitous than we think.
Acknowledgements
• • • • • 2D electronic spectra: Graham R. Fleming Group, Berkeley Greg S. Engel, Tessa R. Calhoun, Elizabeth L. Read and others Electronic 2D on vibrations: Harald Kauffmann Group, University of Vienna – Alexandra Nemeth, Jaroslaw Sperling and Franz Milota QME calculations – Jan Olšina, Charles University in Prague Leonas Valkunas, Vytautas Balevičius, Vilnius University, Lithuania Money: – Czech Science Foundation (GACR) grant nr. 202/07/P278 – Ministry of Education, Youth and Sports of the Czech Republic, grant KONTAKT me899