Role of Coherence in Biological Energy Transfer

Tomas MancalCharles University in Prague

QuEBS 09 8.7.2009 Lisbon

Collaborators: Jan Olšina, Vytautas Balevičius and Leonas Valkunas

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

Pisliakov, Mančal & Fleming, J. Chem. Phys, 124 (2006) 234505Kjellberg, Brüggemann & Pullerits, Phys. Rev. B 74 (2006) 024303

Diagonal cut through 2D spectrumof 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.

Electronic Coherence

2D photon echo of FMO complex

G. S. Engel et al., Nature 446 (2007) 782

• Spectrum reveals the predicted oscillations• Oscillations live longer than predicted• Also the contribution corresponding to energy relaxation oscillates

Conclusion: coherence transfer

Time evolution of a 2D spectrum

Vibrational CoherenceTask: 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) modesω ≈ 140 cm-1

andω ≈ 570 cm-1

Full Numerical CalculationEx

perim

ent Theory

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

Spectroscopic signal : )(tS

)(ˆˆ)( tWDtrtS )()(ˆ ttDtr

..)(ˆˆ)(ˆˆˆ .1010001000

1 chEetUDtUDtriDtr rki

First order response

Equivalent to:

0th order - contributes by zero

rkirkibath EetDEetttrD

.

01

2

10.

10

2

1011 )(ˆˆ)()(ˆ

Element of reduceddensity matrix

First order signal can be calculated from a Master equation for coherence elementsof the reduced density matrix!

Whole world density matrix: )()()(ˆ tttW

Spatial phasefactor

Response Functions and Density Matrix Propagation)(ˆ01 tFor some models , element can be calculated exactly from the master equation.

R. Doll et. al, Chem. Phys. 347 (2008) 243Second order terms:

rkirkibath eEtUDUDtUDtr

..2

1010001011021)(ˆˆ)(ˆˆ)(ˆˆ1

rkirkibath eEtUDUDtUDtr

..2

1101010001021)(ˆˆ)(ˆˆ)(ˆˆ2

21101

2

10 );(ˆˆˆ EtDD Excited state element of RDM, with special initial condition

20010

2

10 );(ˆˆˆ EtDD Ground state element of RDM, with special initial condition

Second order master equation is exact (if define correctly).0H

Response Functions and Density Matrix Propagation

In the perturbation expansion we visit different “corners” of the total density matrix

• For resonantly coupled 2 levelsystems the density matrix splitsinto decoupled blocks.

• Optical transitions occurbetween these blocks

• Spectroscopists often use thelanguage of these blocks.

“System is in the ground state”“We excited a coherence.”“We excited a population.”We excited a coherence betweenone-exciton and two-exciton band.

We can use this “language” as long as we keep in mind that it relates to the “current order”of perturbation theory!

1 N N(N-1)/2

eqW )1(W )2(W )3(W

Feynman Diagrams and Liouville Pathways

gg

eg ee

ge gg

t

T

Each pathway or diagram corresponds to three successivepropagations of the density matrix block

: )0(ˆ)()(ˆ)(ˆ gegegegege WUWW

T : )(ˆ),()(ˆ)(ˆ gggggggggg WTUTWW

)(ˆ),()(ˆ)(ˆ eeeeeeeeee WTUTWW

t : )(ˆ),()(ˆ)(ˆ TWTtUtWTW gegegegege

g g

g e

ee

e gt

T

Putting all this together we get a response function

}ˆ)()()({),,( eqegegeeeegegebath WUTUtUtrTtR

)0(ˆ)()(ˆ egegegeg WUW

Let us consider the coherence term

)0()0(ˆ)0(ˆ eqeg WW

After the excitation

Mean Field Approach

Time evolution

)(ˆ)0()0()(ˆ)(ˆ tUtUtW geeg

Reduced density matrix

eqegbatheg Wtr ˆ)0(ˆ})0()0({)0(ˆ

eqegeg WttW ˆ)(ˆ)(ˆ ?

Master Equations

Nakajima-Zwanzig

)(ˆ)()()(ˆ)(ˆ0

0

tMdtItiLtt

tPast evolution of the system

Convolution-less approach

)(ˆ)()()()(ˆ)(ˆ0

0 tUMdtItiLtt

t

Total evolution operator of the system

In Nakajima-Zwanzing one can introduce so-called “Markov” approximation

)(ˆ)(ˆ )()( tt II which accidentally leads to the same result as convolution-less approach, when we stay in second order in system-bath coupling.

eqegeg WttW ˆ)(ˆ)(ˆ But we can’t do much better than . So lets use it anyway.

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.

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

One of the major results of Greg Engel’s experiment is that secular approximationdoes not work well for FMO.

}ˆ)()()({),,( eqegegeeeegegebands UTUtUtrTtR … and let’s assumewe can calculate spectroscopy from reduced quantities.

Quantum Dynamics and Photosynthetic Systems

Some non-trivial Coherence Effects

• What if we drop secular approximation.

Coherence Transfer Effect in Absorption Spectroscopy

T. Mancal, L. Valkunas, and G. R. Fleming, Chem. Phys. Lett. 432 (2006) 301

Long wavelength partof the bacterial reactioncenter absorption spectrum

)()(

)()(

0

0

22021

12110

0

0

tt

ii

tt

t c

e

c

e

c

e

200 ec

21122

221100 4)]([21Re iec

Eigenfrequencies

Coupled coherences

Can we simulate what we measure?

• Photosynthetic systems are not Markovian.• Coherence transfer leads to troubles.

Comparison of Relaxation Theories

Breakdownof positivity

Populations of a molecular dimer

Oscillations due tocoupling to coherences

• Non-secular Markov QMEis not satisfactory at long times

• Oscillations of the populationseem to be a “real” effect

J. Olsina and T. Mancal, in preparation

Comparison of Relaxation TheoriesSurvival of coherencesdue to memory

Coherence in a molecular dimer

Stationary coherencein non-secular dynamics

• In non-Markov dynamicscoherence lives longer; Population dynamics doesnot matter.

• Stationary coherenceleads to the break-downof the positivity in non-secular Markov theory

Relaxation Theories and 2D Spectrum

1d

2d

3d

3

1

223J

12J

13J

Simple Trimer 2321

11 10000 cm

1500 cm1

132312 100 cmJJJ

Overdamped Brownian oscillator modelfor energy gap correlation function

130 cmfsc 50

Absorption Spectrum

Relaxation Theories and 2D SpectrumPopulations of a molecular trimer

General results from dimer system remain valid

• When relaxation is slow2D spectrum depends mostlyone the evolution of coherences.

• Representative coherenceevolutions are given by the fullQME and secular Markov 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

Trimer

Secular Markov Full QME

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

}ˆ)()()({),,( eqegegeeeegege WUTUtUtrTtR

How good was our calculation?

We used a projection operators: eqbath WAtrAP ˆ}ˆ{ˆ Complementary operator: PQ 1

}ˆ)())(())((){( eqegegeeeegege WPUQPTUQPtUQPtr

}ˆ)()()({ eqegegeeeegege WPPUTPUtPUtr

}ˆ)()()({ eqegegeeeegege UTUtUtr

A rather crude approximation!

Response Functions as Multi-point Correlation Functions

T. Mancal, in preparation

To calculate response function from Master equations

0ˆ WQat all three occurrences.

Each interval has to be calculated with different projector, i.e. by a different Masterequation.

For first coherence interval the projector will do the job. eqbath WAtrAP ˆ}ˆ{ˆ

)()(ˆˆ)(ˆ}ˆ{ˆ g

eeqgbath eUWUAtrAP For population interval we need

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 ClassicalityCoherent states are the best quantum approximationsof classical states!• Relaxation of harmonic oscillator

Gaussian wavepacket vs. pointin the phase space

• Optical coherent states Coherent state vs. classicalelectromagnetic wave

Ultr

afas

t exc

itatio

n

Relaxationof a coherentwavepacket

Initial state = linear combination of some vibrational states

Final state = linear combination of different vibrational states

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