Role of Coherence in Biological Energy Transfer

1. Role of Coherence in Biological Energy Transfer Tomas Mancal Charles University in Prague Collaborators: Jan Olšina, VytautasBalevičiusandLeonasValkunas QuEBS 09 8.7.2009 Lisbon

2. 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.

3. 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

4. Coherence in 2D Photon Echo Spectroscopy

5. 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

6. 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

7. 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

8. Full Numerical Calculation Theory Experiment A. Nemeth et al., Chem. Phys. Lett.459 (2008) 94

9. Quantum Dynamics and Spectroscopy

10. Can we see what we want to see? • (Non-linear) spectroscopy gives us a partial view of the system’s density matrix

11. Response Functions and Density Matrix Propagation Whole world density matrix: Spectroscopic signal : Spatial phase factor 0th order - contributes by zero First order response Equivalent to: Element of reduced density matrix First order signal can be calculated from a Master equation for coherence elements of the reduced density matrix!

12. Response Functions and Density Matrix Propagation can be calculated exactly from the master equation. For some models , element Second order master equation is exact (if define correctly). R. Doll et. al, Chem. Phys. 347 (2008) 243 Second order terms: Excited state element of RDM, with special initial condition Ground state element of RDM, with special initial condition

13. 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 • Spectroscopists often use the • language of these blocks. We excited a coherence between one-exciton and two-exciton band. “System is in the ground state” “We excited a coherence.” “We excited a population.” We can use this “language” as long as we keep in mind that it relates to the “current order” of perturbation theory!

14. Feynman Diagrams and Liouville Pathways e g Each pathway or diagram corresponds to three successive propagations of the density matrix block e e : g e : g g gg : eg ee Putting all this together we get a response function ge gg

15. Mean Field Approach Let us consider the coherence term After the excitation Time evolution Reduced density matrix ?

16. Master Equations But we can’t do much better than . So lets use it anyway. Nakajima-Zwanzig Past evolution of the system Total evolution operator of the system Convolution-less approach In Nakajima-Zwanzing one can introduce so-called “Markov” approximation which accidentally leads to the same result as convolution-less approach, when we stay in second order in system-bath coupling.

17. 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.

18. Quantum Dynamics and Photosynthetic Systems

19. Some non-trivial Coherence Effects • What if we drop secular approximation.

20. Coherence Transfer Effect in Absorption Spectroscopy Coupled coherences Long wavelength part of the bacterial reaction center absorption spectrum Eigenfrequencies T. Mancal, L. Valkunas, and G. R. Fleming, Chem. Phys. Lett.432 (2006) 301

21. Can we simulate what we measure? • Photosynthetic systems are not Markovian. • Coherence transfer leads to troubles.

22. Comparison of Relaxation Theories Populations of a molecular dimer Breakdown of positivity • Non-secular Markov QME • is not satisfactory at long times Oscillations due to coupling to coherences • Oscillations of the population • seem to be a “real” effect J. Olsina and T. Mancal, in preparation

23. Comparison of Relaxation Theories Coherence in a molecular dimer Survival of coherences due to memory • In non-Markov dynamics • coherence lives longer; • Population dynamics does • not matter. Stationary coherence in non-secular dynamics • Stationary coherence • leads to the break-down • of the positivity in non- • secular Markov theory

24. Relaxation Theories and 2D Spectrum Simple Trimer Absorption Spectrum Overdamped Brownian oscillator model for energy gap correlation function

25. 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.

26. Trimer Secular Markov Full QME T = 0 fs

27. Trimer Secular Markov Full QME T = 25 fs

28. Trimer Secular Markov Full QME T = 60 fs

29. Trimer Secular Markov Full QME T = 90 fs

30. Trimer Secular Markov Full QME T = 125 fs

31. Trimer Secular Markov Full QME T = 200 fs

32. Quantum Dynamics and Multi-point Correlation Functions

33. Can we simulate what we measure? • Response functions are multi-point time correlation functions – very difficult to evaluate by Master equations.

34. Response Functions as Multi-point Correlation Functions How good was our calculation? We used a projection operators: Complementary operator: A rather crude approximation!

35. Response Functions as Multi-point Correlation Functions To calculate response function from Master equations at all three occurrences. Each interval has to be calculated with different projector, i.e. by a different Master equation. For first coherence interval the projector will do the job. For population interval we need T. Mancal, in preparation

36. What was not discussed here. • Correlated fluctuations • Finite laser pulse lengtheffects • Wavepacket preparation • Influence on relaxation • Non-adiabatic effects • Polarizationof laser pulses • And probably many other issues

37. What is Quantum on Quantum Coherences?

38. Coherent States and Classicality

39. Coherent States and Classicality Coherentstates are thebestquantumapproximations ofclassicalstates! • Relaxationofharmonicoscillator • Gaussianwavepacket vs. point • in thephasespace • Opticalcoherentstates • Coherentstate vs. classical • electromagneticwave Relaxation of a coherent wavepacket Initialstate= linear combination of some vibrational states Ultrafastexcitation Finalstate = linear combination of different vibrational states In between there is coherence transfer!

40. 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.

41. 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, JaroslawSperling and Franz Milota • QME calculations – Jan Olšina, Charles University in Prague • LeonasValkunas, VytautasBalevičius, Vilnius University, Lithuania • Money: • Czech Science Foundation (GACR) grant nr. 202/07/P278 • Ministry ofEducation, YouthandSportsoftheCzechRepublic, grant KONTAKT me899