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Role of Coherence in Biological Energy Transfer Tomas Mancal Charles University in Prague QuEBS Lisbon Collaborators: Jan Olšina, Vytautas Balevičius and Leonas Valkunas

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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.

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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

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Coherence in 2D Photon Echo Spectroscopy

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

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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

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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

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

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Quantum Dynamics and Spectroscopy

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

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

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Response Functions and Density Matrix Propagation For some models, element can be calculated exactly from the master equation. 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 Second order master equation is exact (if define correctly).

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Response Functions and Density Matrix Propagation In the perturbation expansion we visit different “corners” of the total density matrix 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. “System is in the ground state”“We excited a coherence.”“We excited a population.” We excited a coherence between one-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! 1NN(N-1)/2

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

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

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Master Equations Nakajima-Zwanzig Past evolution of the system Convolution-less approach Total evolution operator of the system 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. But we can’t do much better than. So lets use it anyway.

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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 approximation does not work well for FMO. … and let’s assume we can calculate spectroscopy from reduced quantities.

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Quantum Dynamics and Photosynthetic Systems

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

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

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

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Comparison of Relaxation Theories Breakdown of positivity Populations of a molecular dimer 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

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Comparison of Relaxation Theories Survival of coherences due to memory Coherence in a molecular dimer 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

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

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Relaxation Theories and 2D Spectrum Populations of a molecular trimer General results from dimer system remain valid When relaxation is slow 2D spectrum depends mostly one the evolution of coherences. Representative coherence evolutions are given by the full QME and secular Markov QME.

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T = 0 fs Trimer Secular MarkovFull QME

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T = 25 fs Trimer Secular MarkovFull QME

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T = 60 fs Trimer Secular MarkovFull QME

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T = 90 fs Trimer Secular MarkovFull QME

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T = 125 fs Trimer Secular MarkovFull QME

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T = 200 fs Trimer Secular MarkovFull QME

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Quantum Dynamics and Multi-point Correlation Functions

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

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

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Response Functions as Multi-point Correlation Functions T. Mancal, in preparation 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

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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

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What is Quantum on Quantum Coherences?

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Coherent States and Classicality

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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 Ultrafast excitation Relaxation of a coherent wavepacket Initial state = linear combination of some vibrational states Final state = linear combination of different vibrational states In between there is coherence transfer!

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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.

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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

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