Hidden Symmetries of Quantum Transport in Photosynthesis

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Hidden Symmetries of Quantum Transport in Photosynthesis Diploma Thesis by Tobias Zech Juni 2012 Supervisors: Prof. Dr. Roberto Mulet PD. Dr. Thomas Wellens Professor: Prof. Dr. Andreas Buchleiter Quantum Optics and Statistics Physikalisches Institut Fakultät für Mathematik und Physik Albert-Ludwigs-Universität Freiburg

Transcript of Hidden Symmetries of Quantum Transport in Photosynthesis

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Hidden Symmetries of QuantumTransport in Photosynthesis

Diploma Thesis by

Tobias Zech

Juni 2012

Supervisors: Prof. Dr. Roberto Mulet

PD. Dr. Thomas Wellens

Professor: Prof. Dr. Andreas Buchleiter

4Corporate Design Manual – Albert-Ludwigs-Universität Freiburg

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Albert-Ludwigs-Universität Freiburg

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Zusammenfassung

In der vorliegende Arbeit wird der Zusammenhang zwischen Transporteffizienzenund versteckten Symmetrien in molekularen Netzwerken, welche photosynthetischeKomplexe beschreiben, untersucht.Es ist bekannt, dass der Transport von Anregungen in photosynthetischen Kom-

plexen hocheffizient ist und die Struktur dieser Komplexe zu den bestaufgelöstenin der Biochemie gehören. Experimente aus der nichtlinearen optischen Spektrosko-pie beobachten des Weiteren kohärente Oszillationen, welche langlebiger sind als vonquantenchemischen Berechnungen vorhergesagt. Diese drei Teile des Puzzles – Effizi-enz, Struktur und Kohärenz – sind bisher noch nicht einheitlich beschrieben worden.Wir modellieren die photosynthetischen Pigmentproteinkomplexe durch ein Zu-

fallsensemble von molekularen Netzwerken und konzentrieren uns dabei auf den ko-härenten Transport auf kurzen Zeitskalen. Auf Grundlage numerischer Simulationenkleiner Netzwerke richten wir unser Augenmerk auf die Charakterisierung von Netz-werken hoher Effizienz. Wir zeigen, dass solche hocheffizienten Netzwerke in der Re-gel eine versteckte Symmetrie, genannt Spiegel- oder Zentrumssymmetrie aufweisen.Unser Modell legt nahe, dass diese Symmetrie mit der Transporteffizienz statistischkorreliert ist.Wir zeigen weiter, dass die beobachtete Korrelation unter dem Einfluss von lokaler

Dephasierung und in Gegenwart zusätzlicher Freiheitsgraden fortbesteht. Schließlichüberprüfen wir die Konsistenz der beobachteten Korrelation mit experimentellen Da-ten zu dem photosynthetischen Fenna Matthews Olson (FMO)-Komplex von Schwe-felbakterien und dem Kryptophytkomplex1 PC645 von Meeresalgen.Da unsere Simulationen die prinzipelle Existenz der Struktur-Effizienz-Korrelation

als Folge kohärenter Quantendynamik beweisen, sind wir überzeugt, dass weitereund speziell experimentelle Untersuchungen unter realer Umgebungskopplung dieBedeutung unserer Idee als wichtiges Prinzip für kohärenzinduzierte, hocheffizienteTransportprozesse in der Natur bestätigen werden.

1Kryptophyte können sich während der ungünstigen Jahreszeit in den Boden oder unter die Was-seroberfläche zurückziehen.

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Abstract

This diploma thesis deals with the relation between transport efficiencies and hiddensymmetries in a molecular network which describes photosynthetic complexes.We know that the excitation transport in photosynthetic complexes is highly ef-

ficient, and the structure of these complexes belongs to the best-resolved ones inbiochemistry. In addition, non-linear optical spectroscopy experiments reported co-herent oscillations which last longer than expected from quantum chemistry calcu-lations. These three pieces, efficiency, structure, and coherence have not yet beensatisfactorily unified.We model the pigment-protein complexes of photosynthesis by a random ensemble

of molecular networks, and concentrate on the coherent transport on short timescales. On the basis of numerical simulations of small size networks, we focus on thecharacterisation of networks with high efficiencies. We find that these highly efficientnetworks exhibit a hidden symmetry, called mirror or centro symmetry. Our modelsuggests that this symmetry is statistically correlated with the transport efficiency.We verify that this correlation persists while introducing local dephasing and extra

degrees of freedom. Finally, we check for consistency of the observed correlation withexperimental data on photosynthetic complexes, namely the Fenna-Matthews-Olson(FMO) complex of sulfur bacteria and the cryptophyte2 PC645 complex of marinealgae.Since our simulations are a proof of principle, we are confident that further and in

particular experimental investigations with more realistic environments will confirmour idea as an important principle for transport processes in biology.

2Cryptophytes can seasonally live beneath the soil or water surface.

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Thanks

I thank Andreas Buchleitner for his support over several years, I own him greatopportunities of attending great conferences and meeting the big and small guys inthe business. But most of all, that I could freely talk to him about every scientificand private problem!Next I thank Roberto Mulet, for supervising my thesis and being patient with my

resistance to work on it. I know, you give a shit and I promise to relax in the future.And I thank Thomas Wellens, especially for final reading.I thank Torsten Scholak for any kind of assistance even as an oversee service.I am very glad of fruitful discussions with, Akihito Ishizaki, Thomas Renger, Rienk

van Grondelle, Yoshitka Tanimura, Shaul Mukamel, and Elizabeth van Hauff.I am proud to work in an awesome group of people, who never escaped fun, espe-

cially my room mates Hannah Venzl and Tycho Stange.I thank Klaus Zimmermann, for being my boss at the FMF, and for teaching me

everything about electronic data processing units, and how to work in a structuredway.Special thanks goes to the IT-guys, Stefan Weber and Gerald Endres, and the

CBOOs3 Gislinde Bühler and Susanne Bergmann.I thank all me friends in Freiburg, ranging from physics, and climbing to the fire

brigade. You successfully ensure that I stay grounded. Especially, I thank Julia, whokicked my ass concerning writing and job application, and Jerry for scheduling ourtraining according to me needs.I thank my flat mates, Honeybunny and Ralfibumbalfi, Rebbi and Esquihasi for

the patience to bear my mess in the moments of intensive work and in the remainingtime.I am very sad, that I cannot show my thesis to you, Karin. I miss you. I thank

Beate and Rudi for reminding me what it means to be a Zech.Matthias, Sabine, and Leonidas, thank you for being a great family, which accepts

that I emigrated to the South.Finally, I thank Caro, for waking me up in the morning and listing to me in the

evening. You gave me the motivation to finalise this work.

3Chief Buchleitner Office Officiers

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Contents

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 The Problem 11.1 Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Fenna Matthew Olson Protein (FMO) . . . . . . . . . . . . . 51.1.2 Cryptophyte Phycocyanin 645 (PC 645) . . . . . . . . . . . . 6

1.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Quantum Efficiency . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Energy Conversion Efficiency . . . . . . . . . . . . . . . . . . 91.2.3 Quantum Efficiency of a Solar Cell . . . . . . . . . . . . . . . 10

1.3 2d Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Localisation / Environment . . . . . . . . . . . . . . . . . . . . . . . 14

2 The Engineered Solution 172.1 Coherent Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Perfect State Transfer (PST) . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Linear Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Composite Chains . . . . . . . . . . . . . . . . . . . . . . . . 202.2.3 Conditions for PST in One Dimension . . . . . . . . . . . . . 222.2.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.5 Perturbations and Disorder in PST Systems . . . . . . . . . . 25

3 Statistical Approach 273.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Statistics of Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Matrix Elements and Dynamics: Low Compared to High Efficiency . 313.4 Distance Between Hamiltonians . . . . . . . . . . . . . . . . . . . . . 363.5 Mirror symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6 Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.7 Angular Part of the Dipolar Couplings . . . . . . . . . . . . . . . . . 48

4 Photosynthetic Systems 514.1 Uncorrelated Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 The Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Dynamics of the FMO8 and PC645 . . . . . . . . . . . . . . . . . . . 57

5 Conclusion & Outlook 61

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6 Appendix 636.1 Transformation of Units . . . . . . . . . . . . . . . . . . . . . . . . . 636.2 Average Hamiltonian of Genetic Algorithm . . . . . . . . . . . . . . . 646.3 Eigensystems of Small Networks . . . . . . . . . . . . . . . . . . . . . 656.4 Conjugacy Class of Reflection for Small Sizes . . . . . . . . . . . . . 67

Bibliography 69

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DefinitionsADP Adenosine diphosphateATP Adenosine triphosphateC4 Square graph, circular graph of size fourdHS Hilbert-Schmidt(HS) distancedminHS HS distance minimised while relabeling the intermediate sitesε = dmirror,minHS Measure of the mirror symmetryFMO Fenna-Matthew-Olson proteinFMO8 FMO monomer with eight pigments∆G Standard Gibbs free energy of formationK2 Graph of two sitesNADPH Nicotinamide adenine dinucleotide phosphateP (t) Output probability at time tP3 Linear graph of three sitesPST Perfect state transferP Transport efficiency/Maximal output efficiencyPC645 Cryptophyte Phycocyanin 645rin,out Distance between input and output sitermin Exclusion radiusSN Symmetric group, permutation of N elementsτ Transfer timeT Rabi timeT Reference timeω Reflection, element of SN , generates mirror symmetry

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1 The Problem

Life on earth is fed by organic material. Organic matter is produced by plantsusing photosynthesis. Not only plants, also other species like algae and bacteriaperform photosynthesis [9, 104]. Some of them must operate under extremely lowlight conditions [75]. These conditions require efficient handling of the captured light.Experiments with photosynthetic complexes determine the maximal efficiency of

excitation transport within these complexes to a value larger than 95% [9, 20, 25,111]. How does the molecular structure of these complexes lead to these high ef-ficiencies? Even though these complexes belong to the best resolved structures inbiochemistry [9, 100, 101, 104], the relationship between molecular structure andbiological function is still not fully understood. Historically, the excitation transportin this type of systems was described by diffusive rate equations [56, 59, 87, 104, 112].But from a diffusive approach, it is hard to justify the high value of the efficiency.The reason is that the structural data show a high degree of disorder in most of thesystems, and a diffusive process on such a random structure leads to equipartitionof the excitation over the entire system instead of directed motion towards a spe-cific location, where it is needed [80]. A solution are energy funnels,1 which preventequipartition. The excitation energy, therein, descends an energy cascade. This,however, leads to complete excitation transfer only in the long time limit. Long timescales for efficient transport are bounded from above by the observed fast transferrates in the transport process [9, 25, 26, 104]. Hence, fast and efficient transport hasnot yet been fully explained by diffusive models based on the observed structures.Oscillations, found in recent experiments of the excitation transfer mechanism in

light-harvesting complexes [15, 33, 61, 108], offer the possibility to consider transportmodels of coherent instead of diffusive dynamics [21]. These experiments infer thecoupling of excitonic states and associated level splittings from laser spectroscopydata, which are often obtained by two-dimensional spectroscopy, as a special appli-cation of four-wave mixing techniques. The oscillations were found in some spectralfeatures, and persist much beyond the expected decoherence time [24, 61, 76, 90, 91].Decoherence times on the order of the time scales of the transport process suggestthat the observed oscillations reflect a functional role of coherence for the excitationtransfer.The origin of these oscillations is still a mystery [103]. One idea is that they are

traces of quantum mechanical dynamics, which survive the semiclassical limit as sin-gle atoms are assembled to macromolecular structures. In biological terms, this ideawould imply that nature uses quantum mechanics to overcome the limitations of dif-

1Energy is collected by energetically high lying states, and relaxed to a low lying target state.

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fusive motion [33]. So far, there is no unambiguous method, neither on the theory noron the experimental side, to clearly identify quantum signatures in macromolecularcomplexes that show oscillatory dynamics.In order to understand the different concepts for transport models, it helps to look

at the scientific communities involved in unraveling the mystery of the oscillationsin photosynthetic complexes. Besides biologists, there are physical chemistists, whoapply advanced spectroscopic tools. With short-pulse technology, they monitor dy-namics on the femto-second scale. The dynamical response is then translated into thespectral domain. They are presently capable to do experiments which, in principle,return spectra and couplings directly [62]. This community has detailed knowledgeon excitonic states, and how to address these individually [14, 44, 94, 103], but theirspectral data, in particular for measurements of photosynthetic complexes, are highlyconvoluted.On the other side, there are the findings of the quantum optics and quantum infor-

mation community. This is the bottom-up approach. The Holy Grail is the perfectlycontrolled quantum computer [73]. Usually, the model system under considerationhas been tested and verified for long times, and the important system parameters, likesite energies and couplings, are well controlled. Noise, produced by an environment,or by unwanted degrees of freedom, is successfully suppressed [97, 105]. Regardingtransport, the question here is which are the appropriate parameter values neededto faithfully transport a quantum state form input to output [12, 23, 53, 78]. Butsuch high level of control is so far only achieved for rather small systems. In orderto harvest real advantages of this technology, it needs to be scaled up.Clearly, there is still a gap between both communities. What a quantum chemist

means by control is hardly sufficient for quantum information, whereas the complex-ity of a pigment-protein2 complex, found in a light-harvesting system, is far beyondthe issue of adding an extra two-level atom to a quantum gate3 [36, 97, 105].When the quantum optics community noticed the observation of oscillations in

light-harvesting complexes, they came up with new models to explain the efficiencyissue and the long-lasting oscillations (see section 1.4), distinct from classical mod-els of diffusive rate equations. Soon it was clear that there might be a problemwith a strictly coherent quantum excitation transport. Firstly, the internal cou-plings of the system are of the same order of magnitude as the coupling to theenvironment, what also invalidates the purely diffusive transport picture [112]. Sec-ondly, infinitely extended disordered systems suppress coherent transport, due toAnderson-like localisation. In a nutshell, Anderson localisation is caused by destruc-tive multipath-interference [4]. And thirdly, strictly coherent dynamics, as generatedby the Hamiltonians derived from experimental data, gave bad efficiencies [1, 84].Mohseni et al. [67] assume a direct connection between the obtained bad efficienciesand Anderson localisation, and destroy the suppression of transport by adding en-

2Pigments are macromolecules with the ability to capture visible light. Proteins, here, usuallybuild the backbone of the structures [9, 104].

3Elementary logical device of a quantum computer [73].

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vironmental noise to the quantum model. The noise disrupts the interference, andin particular so destructive interference. Thus localisation is broken, and efficienttransport is recovered.4

There is a drawback of this idea: The environment is also responsible for thedecay of the oscillations. Therefore, the environment must assist transport whilepreserving coherence. This is not easy to achieve, since the information flow to ageneric Markovian environment is not reversible, and phase information, the crucialingredient of coherence, is irrevocably destroyed in simple environmental models[13, 49].But there is still a different option. The quite variable assessment of the exper-

imental data by the diverse communities as described above, might give a wrongimpression of the actual accuracy of the Hamiltonians inferred from the spectro-scopic data. The errors of these Hamiltonians are of the same order of magnitudeas their entries’ absolute values [1]. Hence, instead of tuning the environment toallow for high efficiencies, the matrix elements of the Hamiltonian might be tuned,as well. Scholak et al. [82–86] found that it is possible to generate Hamiltonians fora network model which mimics the network of pigments in a photosynthetic com-plex, and exhibits highly efficient transport for strictly coherent dynamics. Moreover,uncertainties in the Hamiltonian are modelled by a random ensemble. The modelshows that the likelihood for high efficient samples in the random ensemble is notexponentially suppressed. This idea of a random ensemble of molecular networks isthe starting point of our present work.We focus on the characterisation of samples with high efficiencies of this random

ensemble. Our main result, which we infer from Monte Carlo simulations of thisensemble, is a correlation between a symmetry property of the Hamiltonian, calledmirror symmetry, and its transport efficiency.

The remaining chapter 1 introduces those photosynthetic structures, the Fenna-Matthews-Olson (FMO) complex of sulfur bacteria, and the cryptophyte PC645 com-plex of marine algae, which are used in section 4 to compare with the abstract modelof molecular networks. Thereafter, we consider the issue of the efficiency, wherewe distinguish between quantum efficiency of excitation transport and energy con-version efficiency. After this discussion, we briefly compare the results to efficiencymeasurements of solar cells. Next, we explain the background of two-dimensionalspectroscopy, and try to create some intuition for the complications in interpretingthe 2d-spectra. The last section of chapter 1 deals with the relation between local-isation and disorder. Furthermore, we will briefly present models for environmentalcoupling, which allow to describe the decoherence-induced destruction of localisationeffects.Chapter 2 explains the model of coherent quantum transport and takes the per-

spective of a quantum engineer who wants to build a quantum wire. The latterperspective results in the theory of perfect state transfer (PST), and we recollect

4 This is not a completely new idea, see for instance, [31, 42, 65]

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conditions for a Hamiltonian to mediate perfect state transfer.5 In particular, a spa-tial symmetry property of one-dimensional finite chains will be defined, called mirrorsymmetry, on which we will build in our statistical approach.Chapter 3 describes the statistical model of random networks and our original

results. We show that efficiency can be related to mirror symmetry through statisticalanalysis, while here – in contrast to the result on one-dimensional chains collectedin the second chapter – transport across 3D random networks is considered. Localdephasing is added to the model, in order to check whether the correlation betweenmirror symmetry and efficiency persists while coupling to an environment. Also therole of dipolar orientations is addressed.Finally chapter 4 is concerned with the consistency of the correlations found in

our random network model with the experimentally obtained Hamiltonians of FMOand PC645, before chapter 5 concludes the present thesis.

1.1 Functionality

Photosynthesis is a complex biochemical process which consists of many steps. First,the sunlight is captured by an antenna system, then transferred to a reaction centre,where the charge separation takes place, and finally converted into chemically storedenergy [110]. We are here interested in the second step, the excitation transfer. Thereason for our interest is the above-mentioned high efficiency of this transfer. Wewill elaborate on this topic in section 3.2. In the following, however, we first brieflydiscuss the biological complexes considered in this thesis.There exist many different species in nature which do photosynthesis. They are

subdivided into bacteria and eukarya, the latter including algae and plants. Eachgroup developed slightly different ways to fulfill the task [9, 104], but all share thestructural feature that the excitation transfer is mediated by macromolecular com-plexes, which mainly consist of chlorophylls, bacteriochlorophylls and carotenoids.These functional components, called pigments, are embedded in a protein backbone.The most distinctive feature of different photosynthetic architectures lies in the

symmetry properties of the pigment arrangement. There are the Light-harvestingsystems one (LH1) and two (LH2) of bacteria, which exhibit a ring shape. In contrast,the Fenna Matthews Olson (FMO) protein of green sulfur bacteria [21], CryptophytePhycocyanin 645 (PC645) from Chroomonas CCMP270 [28], the Photosystems one(PS1) and two (PS2), and the Light-harvesting-complex (LHCII) of higher plantshave pigment configurations which look much disordered by inspection [8]. We focuson this second class of configurations. From almost all branches of physics it is knownthat symmetries, in general, induce specific and often robust system features. In theabsence of symmetries, it is difficult to identify robust system properties. Still, somehidden symmetries might persist and govern the system dynamics. We anticipatesuch symmetries in the above, apparently disordered, photosynthetic complexes.

5The transport efficiency is 100%; for a precise definition, see section 2.1.

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The Fenna Matthews Olson protein is a protein-pigment complex which is part ofthe light-harvesting system of green sulfur bacteria. It connects the antenna, wherethe light is converted to electronic excitation, to the reaction centre, where the chargeseparation takes place. Hence it acts like a wire [107]. Besides the enormous amountof experimental interest, this fact makes it a perfect testing ground for our theoreticalapproach, because light-capturing is here strictly separated from excitation transfer.We will collect the current state of knowledge on the FMO protein in section 1.1.1.Furthermore, we will introduce a second light-harvesting complex PC645, which

is similar both in function and size (see Section 1.1.2). In both pigment-protein-complexes oscillations in the two-dimensional pump-probe spectra were found [21,28].

1.1.1 Fenna Matthew Olson Protein (FMO)

The green sulfur bacteria live at the bottom of the Pacific Ocean, and make a livingfrom the rare sunlight which percolates the water. Their photosynthetic systemis built of an antenna (chlorosome), the reaction center and the FMO protein inbetween. The antenna collects each photon while converting it to an electronicexcitation. The FMO protein transfers the excitation, with almost no loss, to thereaction centre. There, the charge separation takes place, and is transformed intochemical energy.The FMO protein is a trimer. Each monomer consists of eight bacteriochlorophyll

a (BChl a) pigments. They take the form of a disordered network. One pigmentis assumed to be the input for the excitation, nearest to the antenna, and anotherone mediates the final transfer to the reaction centre. The structural informationwas obtained by crystallographic X-ray spectroscopy and mass spectrometry-basedfootprinting [34, 100, 101, 107]. The pigment arrangement is verified, in addition,by electronic structure calculations for the site energies. Assuming an energy funnel,the pigment closest to the antenna shall have the highest, and the one close to thereaction centre the lowest site energy. The results are not inconsistent with thishypothesis, however the values of the site energies have large uncertainties, and varyfor different methods. The error bars of the site energies, for instance, overlap forcertain pairs, thus even the ordering of the site energies depends on the chosenmethod [81]. While the collection of evidences that we have a precise picture of theFMO’s structure is strong, each single dataset seems to have a weak point. This isnot at all meant to be a blame, but notwithstanding very important to be kept inmind, when performing simulations that rely on this basis of data.The pigment network is embedded in a pocket of proteins. It has axial dimensions

45× 35× 15 Å3 [34, 100]. It is also interesting to note that green sulfur bacteria useH2S to fix CO2 [9]. This is possible since the generalised equation of photosynthesis

CO2 + 2H2Ahν−→ (CH2O) + 2A + H2O (1.1)

permits different electron donors H2A. Photosynthetic bacteria, in particular green

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sulfur bacteria, can even use distinct electron donors simultaneously. This is inter-esting, because it helps to understand the energetics of equation (1.1) by testing itwith electron donors of different chemical potential [60]. This fact can be used todistinguish quantum from energy conversion efficiency, that we will discuss in thesection 1.2.

1.1.2 Cryptophyte Phycocyanin 645 (PC 645)

Alike the FMO, PC645 is a pigment complex imbedded in a protein scaffold. It is alight-harvesting antenna complex of marine algae and has also the ability to operateunder low-light conditions. In contrast to FMO, it does not only transfer excitations,but also captures light. It consists of eight phycobilinproteins, differently from higherplants, which use chlorophylls for light-capturing [66, 106].Light is captured at a specific protein and converted to electronic excitation before

being transferred towards a reaction centre. As in the FMO, the mechanism is usuallyassumed to be an energy funnel, described by a diffusive rate equation [66]. In thesame work, a deviation of the experimentally observed time scale of certain transfersteps was found with respect to theoretical results from standard quantum chemicalcalculations. These new time scales did not necessarily contradict a rate model, butnonetheless open the opportunity to suggest models of coherent transport. This waslater supported by the experimental observation of coherent oscillations in non-linearspectra [28].

1.2 Efficiency

Central to our subject is the issue of the unexplained high efficiency. In order to avoidconfusion, we will address various definitions of efficiency related to photosyntheticprocesses here. We will start with experiments determining the quantum efficiency.When the high efficiency of photosynthesis is mentioned, this is what is meant. Thenwe will connect this to the poor efficiency of natural light-harvesting in energeticterms. Last we discuss the relation to human-made light-collecting machines, namelysolar cells. The expectations of using biology-inspired technology to solve the energyissue will be put into perspective. When we introduce our theoretical model in thesubsequent chapters, we will refer to the quantum efficiency as defined here.

1.2.1 Quantum Efficiency

It is claimed that the efficiency of the excitation transport in photosynthesis showstransport efficiencies of about 95% [9, 20, 25, 40, 79, 111]. A historical note [39], anda review of the variation of reported quantum efficiency [92], as well as its theoreticallimitations [64] can be found as cited. Older results, specifically for the green sulfurbacteria, which employ the FMO, appear in [60].Let us start with clarifying which kind of efficiency is meant here. It is not the

conversion efficiency of solar energy into chemical energy by the complete process

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of photosynthesis. As a matter of fact, this latter value is rather small, as we willdiscuss in the next section. Rather, the 95% refer to the efficiency of how manyphotons are needed to generate one separated charge pair,

µQE :=number of charge pairs

number of absorbed photons. (1.2)

Equivalently, the quantum requirement is defined as the number of absorbed photonsneeded to generate one charge pair. Quantification of the quantum efficiency is astandard measurement in photovoltaics, and we will provide a comparison in section1.2.3.The experimental idea is pretty simple: Set up a light source and determine the

amount of photon flux hitting the sample. Then measure the number of chargesseparated by the flux and determine the quotient of both. The detection of thecharge separations is the subtle part. How can we determine the number of chargeseparations from a macroscopic quantity? The black box approach measures theamount of oxygen produced or carbon dioxide fixed [32, 40, 64]. Hence, its startingpoint is the equation of photosynthesis [9, Ch. 3, Eq. 3.2],

CO2 + H2Ohν−→ (CH2O) + O2 , (1.3)

with (CH2O) being a placeholder for a carbonhydrate.The problem is that it is not clear which is the minimal number of charge sep-

arations required to actually realise (1.3). From energetic arguments three to fourphotons would suffice (see Section 1.2.2). But equation (1.3) is realised in varioussteps, where each step might require a single excitation, which corresponds to a singleelectron transferred from one to another reactant. This latter idea is borrowed fromquantum chemistry and quite appealing for the discussion of quantum coherence inphotosynthesis. Unfortunately, it is not so straightforward. The reason is that notevery step can be associated with electron transfer between reactants.The constitutive steps of photosynethsis’ chemistry help to understand. Equa-

tion (1.3) is not directly implemented. The steps discussed in the previous sectionsjust describe the gathering of light followed by the transfer to the reaction centre.The charge separation happens therein. For long time storage of the chemical energy,recombination must be avoided. Therefore, a number of fast reactions separate thecharges spatially. After this, there are two possible forms of intermediate chemicalstorage – as ATP or as NADPH. Both fuel more complicated chemical reactions and,in particular, drive the Calvin cycle which, finally, produces the carbohydrates, andthus implements the last step of equation (1.3).The ferredoxin-NADP+E reductase produces the NADPH [110]. This process is

part of the electron transfer chain connecting the different photosynthetic compo-nents, and, thus, there might exist a sequence of electron transfer steps, which con-nect the NADPH formation one-to-one to the charge-separated state of the reactioncentre. Hence the assumption of quantised electron transfer seems to be valid here.

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ATP, on the contrary, is created by a proton gradient through a membrane. Thehydrogen ions, then, drive ATP synthesis via ATP synthase. ATP synthase is amolecular machine in a membrane, which is driven by the proton gradient [109, 110].It converts the energy stored in the gradient to mechanical deformation of its ownstructure. This deformation is responsible for the reaction of ADP to ATP. Sincethe reaction is related to the original charge separation in the reaction centre viathe proton gradient, it is not apparent why the relation between charges and ATPshould be quantised.Whereas the assumption of an integer value for the ratio of the number of separated

charges and of generated NADPH is reasonable, it is not for the generation of ATP.Hence, the theoretical basis for a minimal quantum requirement for the equation ofphotosynthesis is not on solid grounds, as far the author can see.Disregarding this fact, we present the result of the determination of the quantum

efficiency of photosynthesis. From our perspective, the best results were obtainedby Chain et al. [20], who measured the ATP formation, instead of the oxygen orcarbon dioxide concentration. They measured a quantum requirement of 2.1 photonsper formed ATP, and stated that theoretically two charge pairs are needed for ATPsynthesis. How this theoretical value of two charge pairs to synthesise ATP is derivedlacks an explanation. Following their reasoning results in a quantum efficiency of

µQE =2

2.1≈ 0.95% . (1.4)

This value is obtained for light at the red edge of the absorption spectrum, namely(715 ± 10) nm. For shorter wave lengths, the quantum efficiency decreases. At(545 ± 20) nm Chain et al. found a minimal quantum requirement of 3.8 photons,what halves the quantum efficiency. One of the reasons is that, at large wavelengths,only one out of two photosystems is active.In the same experiment Chain et al. measured also the NADPH production. This

is done by the photosystem which is inactive at 715 nm. Referring to the complica-tions for a theoretically justified minimal quantum requirement above, this choice ofNADPH measurement has the advantage of an association with a well-known elec-tron transfer pathway. However, the number of photons which contribute to NADPHproduction cannot be separated from those producing ATP. Therefore, the efficiencyvalue derived from the ATP generation above is currently probably the best valuefrom black box experiments, since it measures the most elementary chemical productswhich can be experimentally separated. Further improvements in the determinationof the actual value of µQE can only be made by including microscopic knowledge, anddirectly populate and read-out the electronic states. Non-linear optical spectroscopyexperiments, discussed in Section 1.3, attempt this.Since we are mainly interested in the smallest pigment networks like the FMO,

we must also relate our studies to experiments specific to them. In black box ex-periments, Larsen et al. [60] found a quantum requirement of ten photons per fixedcarbon dioxide molecule for green sulfur bacteria. This value coincides with quantumrequirements measured for green plants [32, 64] and for algae [40].

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As we mentioned when discussing equation (1.1), green sulfur bacteria are capa-ble to process electron donors different than water. This changes the energetics ofthe equation. Notwithstanding, the experimentally inferred quantum requirementsremain unchanged. This is an indicator that the energy balance is not crucial for thequantum efficiency, and hence cannot serve as an indicator for a theoretical valueof the minimal quantum requirement. We will discuss the relation of the energetics,namely the energy conversion efficiency of photosynthesis, and the quantum efficiencyin the next section.

1.2.2 Energy Conversion Efficiency

It seems that the quantum efficiency of the excitation transport can achieve ratherhigh values, as we have seen in the previous section. Accepting this, does it implythat photosynthesis is energetically efficient?It is actually known that the overall energy conversion efficiency of land-based

photosynthesis is pretty bad. Of the incident sunlight’s energy not more than 0.2%is transformed into biomass [9, Ch. 3]. However, this includes all side effects, fromlight absorption to plant respiration. Thus, the isolated transport process of theexcitation transport might still be energetically efficient, and the hypothesis of abiological wire without energy losses would still be reasonable. Instead of citing theliterature, we here only want to show that a high quantum efficiency does not requirethat energy be used efficiently.The energy conversion efficiency is defined by the ratio of the chemically stored

energy to the energy of the incoming photons. The chemical energy, for instance ofATP synthesis, is:6

ATP +H2O → ADP + Pi ⇒ ∆GATP→ADP ≈ −57kJ/mol (1.5)

= −36 · 1022eV/mol ,

where Pi stands for an inorganic phosphate. The energy in one mol of photons withλ = 715 nm, as used in the section above, is

Ephotonmol = NAhc

λ= 1.7eV ·NA (1.6)

= 102 · 1022 eV .

Given the theoretical quantum requirement of two photons (see previous section),this results in an energy conversion efficiency of

ηE :=∆GATP→ADP

2Ephotonmol

≈ 17% . (1.7)

Hence, more than 80% are lost. The weak point of such argument is, however,that we assumed that no other processes than ATP synthesis are supplied by the

6Transformations between different energy units can be found in the Appendix 6.1 (Equa-tion (6.11)).

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collected photon energy. However, estimations of the energy conversion efficiencyunder consideration of oxygen generation obtain similar values [9, Ch. 3].

1.2.3 Quantum Efficiency of a Solar Cell

As we improve our understanding of photosynthesis, there is some hope to harvestknowledge from biology to resolve world’s energy problem. In special focus is thearea of photovoltaics (PV). There is a recent comparison between various aspectsof photosynthesis and PV [10], which, in particular, analyses the system energyconversion efficiencies on both sides. Since high quantum efficiencies spurred theinterest of quantum physicists in photosynthesis, we will now relate our discussionof the previous sections and the issue of the quantum efficiency of a solar cell.

As we have already mentioned earlier, the measurement of the quantum efficiencyis by far simpler for a solar cell, since the separated charges can be detected as theelectric output current of the cell. Usually one distinguishes internal and externalquantum efficiency, where the former is measured, and the latter is computed byaccounting for the reflection of photons, measured in a second experiment. The set-up is straightforward. A lamp behind a spectral filter illuminates both the sampleand a reference cell. To determine the number of generated charges, the short circuitcurrents7 of both solar cells are measured. Since the external quantum efficiency ofthe reference cell is known, the quotient of the circuits determines the efficiency.

Quantum efficiencies of a high efficiency solar cell can reach up to 90%, see forinstance Fig. 8 in [95]. Hence, as single purpose devices, they are already competitiveif not superior to nature. Honestly, this is just the highly efficient edge of solar cells.They reach energy conversion efficiencies above 30%, but are produced by using veryexpensive mono-crystalline GaAs, and implementing triple junction design. But eventhe more disordered, organic solar cells reach rather high quantum efficiencies. Thereare examples of polymer-fullerene cells with internal quantum efficiencies of close to100% [77]. However, their energy conversion efficiency is around 5%. Again, thisshows that high quantum efficiency is a necessary, but not a sufficient conditionfor good energy conversion. Deibel et al. reviewed the topic of organic solar cellsincluding discussions of efficiency issues in [30].

In conclusion, today’s solar cell easily outperforms natural photosynthesis in termsof both, quantum and energy conversion efficiency. One might argue that this com-parison is not fair in the end, since photosynthesis produces organic material insteadof electricity. But even if one includes the efficiency of a fuel cell into the calculations,the artificial device is superior to the natural archetype [10].

7This is the current without a resistance in the circuit. It is at the same time the maximal currentobtainable.

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1.3 2d Spectroscopy

During the past decade many groups found temporal oscillations in the spectraldata of systems related to photosynthesis. They mainly used two-dimensional spec-troscopy, a technique well-known in the physics of nuclear magnetic resonance (NMR),but difficult to apply and interpret in the case of laser light interacting with molec-ular samples. We now introduce this technique, and what we consider to be thoseresults of relevance for our subsequent discussion.Two-dimensional spectroscopy is the most important experimental technique to

study coherence in biological systems. While other spectroscopic methods [9, 104]are capable to deduce the same information, 2d spectroscopy, in general, provides amore direct access. For a good review see, for instance, [43, 70, 104].The technique, including theoretical calculations, is nicely introduced by Hamm

and Zanni [44]. The only drawback of their text, for our current purpose, is thatthey consider infrared, instead of visible light which is used in most experiments withphotosynthetic systems. The difference manifests, for instance, in the explanation ofasymmetries of the 2d spectra. In the infrared regime, asymmetry is mainly causedby anharmonicities of the vibrational potential. This is different in the visible range,where the laser frequency which drives the electronic transition is much larger thanthe band width of the vibrational level splitting. For a detailed discussion, see Turneret al. [103].We start out with an explanation of absorption spectroscopy, since there the con-

nection between the experimental protocol, the notation in terms of diagrams, andthe interpretation connecting both are still easily comprehensible. Some commonmisinterpretations can be avoided when starting with this simplest case. Subse-quently, we will move on to two-dimensional pump-probe spectroscopy, and willfinally conclude with time-resolved two-dimensional spectroscopy, the variant usedin the cited experiments.In the case of linear absorption spectroscopy, just one laser pulse interacts with

the sample, and the frequency-resolved intensity of the scattered light is detected.With the help of a local oscillator, simply provided by the incoming laser light, thesignal is homodyne detected – meaning that phase information is retrieved. Fromthe point of view of theory, the observable here is the linear approximation to thepolarisation inscribed by the pump field into the sample. This signal can be derivedfrom a time-dependent perturbation theory, details of which can be found in Hamm’sMukamel for Dummies, with Zanni, and for technically strong people, in Mukamel:Nonlinear Spectroscopy [43, 44, 69]. Importantly for us, the perturbative expansionresults in a diagrammatic representation. These diagrams provide one of the mainsources for the interpretation of the spectra obtained for photosynthetic complexes.Unfortunately, they also include one of the sources for misinterpretations, and theyshow the infinitesimal changes of the bra and of the ket states of the density matrix.Let us have a look at such a diagram. For simplicity, we assume that we have a two-

level system. The density matrix before laser interaction represents the ground state,

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because the electronic energy splitting in these types of systems is much larger thanthe thermal energy. The laser pulse creates a coherent superposition between theground and the excited state. Since this is not a stationary state, it will exhibit Rabioscillations, hence population is constantly transferred between the ground and theexcited state. The diagram, however, only shows the generation of the coherence,since, as a linear approximation, it only captures the dynamics for short times.When experiments with many pulses are considered, we often find the statementthat the first pulse creates a coherence, and the second a population. This statement,however, can be misleading, since we just indicated that the laser pulse in a linearabsorption spectrum induces a Rabi oscillation, which comes along with a transferof population from the ground to the excited state. In order to get the absorptionspectrum from the diagram, one here must calculate the Fourier transformation ofthe Rabi oscillation.In linear spectroscopy, only the time interval between the pulse and the signal is a

tunable parameter, and, therefore, only the first order response function is accessible.With two time delays, in principle, the second order response function is accessible.However, due to symmetry arguments, it vanishes for most situations. Only atsurfaces, for instance, can it be observed. The next non-trivial response function isof third order. Therefore, we need three time intervals in a pulsed experiment.Two-dimensional spectroscopy uses four pulses. This allows to monitor not only

the characteristic frequencies of the system eigenstates, but to also learn about theelectronic couplings. Since there are three time intervals, we have different possibil-ities to represent our data. A commonly used version is to Fourier-transform withrespect to the first and the last time interval. The first and the last Fourier variableare called pump, and probe, respectively. The underlying interpretation is as follows:If a signal is measured for a certain pump frequency, then there exists a state withthis frequency in the system. By the first two laser pulses we excite two wave-packetscentred around this state. The following time interval is not transformed and referredto as population time, for reasons briefly discussed for the linear absorption. If, af-ter a certain population time, there is also a signal at a distinct probe frequency,then population transfer between two states must have occurred. One was excitedby the pump pulse and the other was de-excited by the probe pulse. In this way,2d-spectroscopy opens the possibility to directly examine couplings.But why is this such a cutting edge technology? In general it is not. Two-

dimensional spectroscopy with nuclear magnetic resonance (NMR) is known sincethe 1970’s. But there are significant differences. First, NMR techniques require onlyradio-frequency pulses. In order to extract the response function from the signal, it isnecessary that the pulse width be much shorter than the characteristic system timescales. If this condition is not fulfilled, one only detects the envelope of the pulse,since the measured signal is the convolution of the pulse shape with the system re-sponse. In NMR, short pulses are easily realised in the radio frequency regime. Forlaser light, instead, only the advent of femtosecond pulses made it possible to probethe electronic dynamics in biological tissue. Another difference is the interpretation

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of the experimental procedure. NMR experiments induce strong coupling betweenthe electromagnetic field and the sample, while the interaction with the electromag-netic field is only weak in our present case, consequently the induced dynamics isperturbative, and it is only a small fraction of the wave function which contributesto the signal.An additional advantage of NMR over optical spectroscopy is that there are well-

established models for nuclear spins. For molecular 2d spectroscopy this is not thecase, which makes the interpretation of the spectroscopic data hard. Nowadays,there is a whole bunch of models, taking into account the environment, which claimto describe the biological systems correctly. We will come back to them in the nextsection. Here we want to focus on the most-cited method to extract an effectiveHamiltonian for the FMO.Adolphs and Renger presented the first Hamiltonian for the FMO protein [1]. They

used the structural data [100] of the pigment positions and orientations from whichthey built up a charge density calculation. After they determined the electroniccouplings via this method, they fit the site energies to the spectral data. For theinteraction with the environment they assumed a certain spectral density. Overall,this is a fairly complicated method, which required many steps from the experimentaldata to the obtained site energies and couplings. Hence error bars are large. Whenan eighth pigment was discovered, the calculations were repeated. In [81, Tab. 1] theauthors compare the site energies obtained by different methods. Even the energeticordering of the site energies changes for different methods! We learn that the obtainedvalues must be related to the method that was used to infer them.Here we encounter one of the points were the two communities of quantum chem-

istry and quantum optics clash. For the physical chemists, this is a reasonablyaccurate result obtained from somewhat intrinsically dirty experimental data. Aquantum optician, in contrast, associates at this point a microscopic Hamiltonianwhich generates the dynamics of an exciton in a molecular network. This discrep-ancy is the rational for us to choose a statistical approach to the problem. Doingso, we avoid the pitfall of underestimating the error bars, that garnish any effectiveHamiltonian, and draw conclusions which are statistically robust.To conclude this section, we recollect the history of recent key experiments in

the field of 2d-spectroscopy on photosynthesis. Brixner et al. applied the techniqueto the FMO complex [15]. They concluded from their spectra that the picture ofa stepwise cascade of energy might be wrong. Engel et al. [33] confirmed this byobserving signatures of coherent oscillations in the 2d-spectra of the FMO complex.Both these results came as a surprise. Not only that the established models of inter-molecular energy transfer were questioned. Also the observed coherence times of660 fs exceeded the expected values of a few femtoseconds [76].Both experiments were performed at 77 K, cooled by liquid nitrogen, what might

possibly explain the long-lasting coherences. Similar findings by Lee et al. [61] onthe reaction centre of purple bacteria were performed at 77 K and 180 K. Theseauthors also suggested an environmental protection of the oscillations. Collini et al.

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[27, 28] extended the studies to ambient temperatures and still found coherences. Intheir first work, they analysed the dynamics in a conjugated polymer. The secondwork dealt with PC645. Coherence times of about 130 fs were observed.One of the questions which arises from the experiments above is that on the origin

of the oscillations. According to the general interpretation of the Hamiltonian ofAdolphs and Renger, they are of electronic nature. Still, it might be possible thatvibrational degrees of freedom are excited. Those can not be resolved in the spectra.Turner et al. [103] analysed the problem for PC645 and added, for the first time,error bars to the observed 2D signal.So far, all experiments are performed on bulk samples, which imply ensemble

averages. The isolated pigment complexes usually come in liquid solution, and arehomogeneously illuminated by the laser. There are single molecule experiments byHildner et al. [47], which avoid the ensemble average. Still it will take some time toreach the spectral resolution of bulk measurements for single molecule experiments.

1.4 Localisation / Environment

When propagating the Hamlitonian of Adolphs and Renger [1, 81] and monitoringthe population of the site nearest to the reaction center, the result does not pointtowards highly efficient excitation transfer. On biologically significant times it neverreaches efficiencies above 4% [84]. We will discuss the details of such a propagationlater. At this point, it is enough to convince ourselves that destructive quantuminterference prohibits good transport for this Hamiltonian.What is the reason that a biological systems, which we believe to have the function

of a wire, shows features of an insulator rather than of a conductor? One possibilityis that the elements in the Hamiltonian are wrong. Another one that the model isincomplete. We will discuss variations of the elements of the Hamiltonian in depthlater. First we want to consider a different line of thought [19, 67, 68, 71, 89].The argument goes as follows: The Hamiltonian is obtained from experiments and

does not exhibit the efficient transport we want. But there is also a large amount ofuncertainty on the matrix elements. And, overall, biological systems are just prettyunstructured, better said disordered. But there is a theorem about suppression ofwave transport in disordered media. It was found by Anderson [4] and nowadaysis even put on mathematically strict grounds. Hence it is not at all surprising thattransport is suppressed, and we now know that this phenomenon is called Andersonlocalisation. But how to avoid this effect? What about destroying coherence a littlebit? This will break destructive interference as well, and, additionally, the interac-tion with the environment must be strong in biological systems anyway, especiallyat room temperature. And what about experimentally observed coherences? Letus just construct an environment in the right way. Then we can keep the Hamil-tonian obtained from experiments, see our oscillations, which are protected by ourenvironment, and recover the biologically relevant efficiency.Even if a well-tuned environment solves the problem, there is a fundamental flaw

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in the argument. When Anderson derived his result on localisation, he assumedan infinite linear chain. Modern findings show that his results holds also for othergeometries, but in order to see a signature of localisation, the sample length mustexceed the localisation length. For systems of size seven or eight, this is unlikely tobe the case.Disregarding the Anderson argument, is it still true that the transport efficiency

can be changed form bad to good by the assistance of an environment?First this is trivially true, since the dynamics induced by the environment can theo-

retically be made dominant. The transport is, then, of classical, diffusive type insteadof coherent. In this limit, the coupling with the environment is much larger than thecouplings within the system. The latter are usually inscribed in the Hamiltonian.This is an old approach, well-tested in many molecular structures for transport, andknown as Förster theory [59, 87, 104, 112].Second, the argument still holds for weak perturbations due to an environment. As

mentioned above, coherence is very fragile with respect to environmental disturbance.Hence, if transport is suppressed by destructive interference, perturbing the relativephases will destroy the interference, and thus the resulting suppression. In the longtime limit, weak perturbations destroy the disorder-induced destructive interference.Therefore, the transport is no longer suppressed, and the finite transfer rate alwaysyields large values of the transport efficiency on asymptotic time scales.In summary: First, a functional role of coherence in photosynthesis is incompatible

with an environment that dominates the dynamics. Second, interference conditionsare fragile, but not only with respect to incoherent perturbations due to the envi-ronment, but also under changes of the different matrix entries of the Hamiltonian.We will take this latter fact for our advantage, and tune the Hamiltonian in orderto optimise transport, in the following chapters. There still might be some coopera-tive interplay between coherent and incoherent dynamics, but we will mainly focuson the former. The reason is that we expect that coherent dynamics only plays arole on short time scales. On such scales non-dominant coupling to an environmentinfluences the dynamics only weakly [11, 63, 96]. Third, short time scales requirequantifiers of the transport efficiency which are non-asymptotic on the time-axis. Ifcoherent transport is important for the function of photosynthesis, then it must doits job on these short time scales.Before we present our results, we introduce an idea from quantum information.

As mentioned, if efficient excitation transfer is due to quantum coherence, then theelements of the Hamiltonian must be precisely tuned. The theory of perfect statetransfer deals exactly with the construction of such Hamiltonians.

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2 The Engineered Solution

When the first evidence of wavelike dynamics in photosynthetic complexes appeared,this raised two questions:

1. How does nature manage to protect the oscillations from decoherence?

2. Why did nature choose coherent dynamics?

We already touched upon the discussion in the literature concerning the first ques-tion, in the previous chapter. The second question was quickly connected to theefficiency issue. Being convinced that quantum physics was at the root of the obser-vation, Engel et al. [33] borrowed an idea from another topic of high interest in quan-tum physics – quantum computations. They suggested that nature uses quantumcoherence in the same way a quantum computer does to speed up search algorithms[41]. This suggestion was later disproven [67], but the idea lasts that quantum in-formation offers tools to unravel the puzzle. Work closest to the suggested quantumsearch has been done by Childs et al. [22]. They mapped the transport problemon a network to the graph traversal problem – a well-defined problem in computerscience. They provided bounds on the transfer time and compared them with aclassical model.While mapping a transfer process in a biological system to a quantum search

algorithm is a rather bold step, comparison to the theory of perfect state transfer(PST) is, maybe, more direct. The purpose of PST is to find conditions under whicha quantum state is losslessly transferred from one place to another. This is importantfor quantum computation, since the information stored in qubits1 must be routedthrough the device [73]. In other words, the aim of PST is to build a quantum wire.In this chapter we describe how coherent quantum transport in finite systems is

modelled. Afterwards, we introduce the notions of efficiency and transfer time, inthe context of PST in simple systems. The definition of efficiencies and time scaleswill also be of great importance for our statistical model, in the following chapter.Next we discuss analytic results for perfect state transfer on one-dimensional chains,which allows us to relate efficiency with symmetry properties of the system. Thisrelation will be the key point for our results in chapter 3. We close the chapter byreferencing to relevant experiments and publications which achieve PST under theinfluence of disorder.

1Natural unit to store information in a quantum computer. Analogous to a bit in a classicalcomputer

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2.1 Coherent Transport

We start with coherent quantum transport of excitations. For this purpose, weintroduce states, the equation of motion and an efficiency measure.A quantum mechanical state in the most general form is described by a density

operator, but since we consider purely coherent dynamics, it suffices to take unitvectors in a Hilbert space [13, 29]. According to Dirac’s 〈Bra|C |ket〉 notation,the states are denoted by |Ψ(t)〉, where Ψ labels the state, whereas t denotes itstime dependence. For a time-independent state, the latter will be omitted. Whenrepresented in a specific basis, the state will also be called wave function.Coherent quantum dynamics is fully determined by the Hamiltonian H. It gener-

ates the dynamics through the Schrödinger equation:

i~∂t |Ψ(t)〉 = H |Ψ(t)〉 , (2.1)

with the solution given by the time evolution operator U acting on the system’sinitial state |Ψ(0)〉. For a time-independent Hamiltonian we have

|Ψ(t)〉 = U(t) |Ψ(0)〉 , U(t) = exp

(− i~Ht

). (2.2)

Through the entire thesis, we focus on the situtation where only a single excitationis injected into the system. We, furthermore, deal with finite networks, composedof N sites. The set of states |j〉 where the excitation is localised at site j forms abasis of state space. The connectivity of the network and the different connections’weights are described by the Hamiltonian. A single excitation’s propagation on thenodes is governed by the Schrödinger equation. Two sites of the network are special:The site where the excitation is initially localised, which we call the input, and thesite where the excitation is to be delivered, the output. We usually define the inputat site 1 and the output at site N . The central question of perfect state transfer is:

How must the Hamiltonian look like to fully transfer the excitation from input tooutput in a finite time?

With the transfer time denoted by τ ,

PST ⇔ ∃τ > 0 : U(τ) |1〉 = eiθ |N〉 . (2.3)

The phase θ is global and has no physical relevance, since the probability to find theexcitation at site j is defined as:

Pj(t) := | 〈j|U(t) |1〉 |2 . (2.4)

Thus, the transfer is perfect if the output probability PN (t) is one at some instanceof time.

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In the case of imperfect state transfer, where PN (t) 6= 1 ∀t, one can define anefficiency measure P, which we call the transport efficiency :2

P := maxt∈[0,T ]

PN (t) = maxt∈[0,T ]

| 〈N |U(t) |1〉 |2 . (2.5)

From the definition of P, the transfer time τ can also be defined for imperfect transfer.τ is the smallest t ∈ [0, T ] for which PN (t) takes its maximum value,

τ := min {t ∈ [0, T ] | PN (t) = P} . (2.6)

This includes the above definition of τ as the PST transfer time with T = ∞,implying that

PST ⇔ P = 1 . (2.7)

Later on in this thesis, we will consider finite time intervals [0, T ]. We alreadymotivated this restriction by our interest in the short time scales on which quantumphysics can play a functional role (see section 1.4). However, for a meaningful defi-nition of T , we need further knowledge on the elements of the Hamiltonian matrix,since such restriction must be related to the characteristic system time scales. Wewill get to this point once we introduce our model, in the next chapter.

2.2 Perfect State Transfer (PST)

The aim of PST is the creation of a quantum wire. This wire is usually modelledas a network of two-level systems, where the excitation enters at the input site ofthe network, and is hopefully found, after the transfer time τ , with certainity at theoutput port.

2.2.1 Linear Chains

To gain some intuition, let us start with the smallest possible system to study statetransfer – a network of two coupled sites, called K2 in graph theory. This system’sstate space is two-dimensional, and we solve the equations of motion by diagonalisa-tion of the Hamiltonian

HK2 =

(ε1 JJ ε2

). (2.8)

The transfer is perfect if and only if both diagonal entries, ε1 and ε2, called the siteenergies, are equal. The probability of finding the excitation at the output port isthen

PK2(t) = sin2(Jt) ⇒ PK2

(τK2 =

π

2J

)= 1 . (2.9)

2The definition of the efficiency measure is in accordance with the work of Scholak et al. [82, 84, 85],where a more detailed discussion of the properties of this particular measure, and relations toother definitions of transfer efficiency can be found.

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Next, we enlarge the system by one site, and consider a linear chain with equalnearest-neighbour couplings, named P3. The Hamiltonian and the output probability

Figure 2.1: The graph P3

read:

HP3 =

0 J 0J 0 J0 J 0

, (2.10)

PP3(t) =

∣∣∣∣12 (cos(√

2Jt)− 1)∣∣∣∣2 , (2.11)

τP3 =π√2J

. (2.12)

We see that, for equal coupling constants, J , the transfer time for three sites islonger, but we still have perfect transfer. One, hence, might guess that linear chainsof arbitrary length exhibit perfect transfer, only with increasingly longer transfertimes.This is, however, not the case. In fact, only the two previous examples, N = 2

and N = 3, exhibit perfect state transfer. It can be shown that for a linear chain ofN sites and uniform nearest-neighour coupling, J = 1, the output probability reads[23]:

Plc(t) =

∣∣∣∣∣ 2

N + 1

N∑k=1

sin

(πk

N + 1

)sin

(πkN

N + 1

)e−iEkt

∣∣∣∣∣2

. (2.13)

Thus there are no linear chains with nearest-neighbour coupling which exhibit PST,expect those of length two or three.3

2.2.2 Composite Chains

We have seen that finite linear chains with uniform nearest-neighbour couplings arenot a candidate for scalable quantum wires. But one can use their short instancesto construct new networks. The trick, presented in [23], is to build hypercubes, outof length-two and -three chains. It is then shown that these again exhibit PST. Infact, the principle is much more general, and is based on the Cartesian product ofnetworks. Each network, which is a Cartesian product of PST networks, exhibits

3This might appear counterintuitive as to what is known on the Bloch states of electrons onperiodic lattices, as well-known from text books on solid state theory (see, for instance, [6]).The crucial difference between our present problem and paradigmatic solid state models is thatwe are dealing with finite systems, which lack translational invariance, in contrast to a typicallattice in the solid state.

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itself PST. We will not cover the topic here, but refer the interested reader to thereferences [7, 23, 55].4

The simplest hypercube is the square C4 – the product of two length-two networksK2.

Figure 2.2: The graph C4, which is the product of two K2 networks.

The Hamiltonian and the transport probability for the square C4 read (calculationsin appendix 6.3):

HC4 =

0 J J 0J 0 0 JJ 0 0 J0 J J 0

, (2.14)

PC4(t) =

∣∣∣∣12 (cos (2Jt)− 1)

∣∣∣∣2 , (2.15)

τC4 =π

2J. (2.16)

The transfer time is as large as for the two site network K2, and therefore faster thanfor the linear chain of three sites. This is generic for hypercubic networks. Theirtransfer time is independent of their size and only determined by the transfer timeof their elementary building blocks.Let us leave the abstract world of networks for a second and take a more physical

point of view. We ask the question whether it is better to put two instead of onesite in between the in- and the output. We assume that their is only one couplingconstant in each network which, for instance, scales with the inverse distance to thepower α, J ∝ 1

rα . The latter is the situation which we will encounter in our statisticalmodel. With these assumptions, we can extract a geometric constraint. We assumethat the distance between the in- and the output for both networks is fixed to thesame value, then the distances between the sites can be extracted trigonometrically.In order to ensure faster transport for the square, τC4 > τP3 , the angle θ between theline connecting in- and output and the edge to one of the intermediate sites in C4

(see Figure 2.3) has to obey θ < arccos(

2−12α

). For a dipole-dipole type interaction,

with α = 3, we obtain an angle θ < 27◦. Note that we neither assumed directcoupling between the in- and the output nor between the intermediate sites. Whilethe former might be negligible, the latter assumption is rather unrealistic for such

4 Note that these contributions are, in substance, very close to the theory of quantum graphs[38, 58, 93], while little cross-talk of both communities can be found in the literature.

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θ

Figure 2.3: C4 with angle θ

angles. Further down, we will provide examples where PST exists for fully connectednetworks.5

One way to obtain a PST network with weighted edges is a consequence of theCartesian product approach, and already included in [23]. The idea is to project thehypercube onto a one dimensional chain with varying couplings. So, the Hamiltonianwe project onto, looks like follows:

H1d,nn =

0 J1 0 . . . 0J1 0 J2 . . . 00 J2 0 . . . 0...

......

. . . JN−10 0 0 JN−1 0

. (2.17)

If we take the hypercube based on K2, and use the column method [22], whichdecomposes the dynamics of the network’s Hamiltonian into invariant subspaces, weget

Jn =√n(N − n) (2.18)

for the couplings.

2.2.3 Conditions for PST in One Dimension

There exist necessary and sufficient conditions for PST. The necessary conditionconcerns a spatial symmetry of the system. In order to understand it, we will first givea more general condition which every PST system must fulfill. Form the definition(2.3) of PST we have that application of the time evolution operator at transfer timeτ onto the input state |1〉 renders the output state |N〉, up to a global phase,

U(τ) |1〉 = eiθ |N〉 .

Expanding the equation over the orthonormal eigenbasis |Ej〉, we obtain a set ofequations for the overlaps of the in- and of the output state, respectively, with theeigenstates:

U(τ) |1〉 =∑j

e−iEjt 〈Ej |1〉 |Ej〉Equation (2.3)

= eiθ∑j

〈Ej |N〉 |Ej〉 (2.19)

⇒ e−iEjt 〈Ej |1〉 = eiθ 〈Ej |N〉 . (2.20)5A network is fully connected if there is an edge between each pair of sites.

22

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This can be decomposed into a set of equations for the moduli of the overlaps andfor the phases [54],

|〈Ej |1〉| = |〈Ej |N〉| (2.21)

(Ejt− ϕj + θ ≡ 0) mod 2π, with ϕj := arg

( 〈Ej |1〉〈Ej |N〉

). (2.22)

The latter can be written explicitly:

Ejt− ϕj + θ = 2πmj , with mj ∈ Z . (2.23)

The freedom in the choice of the mj shows that there are infinitely many solutionsto the problem of PST, and [57] showed that these solutions are not isomorphic, butrepresent distinct dynamics.At a first glance, finding a solution of (2.21) and of (2.22) does not seem too

involved. They may appear rather uncoupled. But they are not. What is notexplicitly stated is that the states belonging to in- and output are orthonormal,since they represent spatially localised and separated excitations:

‖|1〉‖ =∑j

|〈Ej |1〉|2 = 1 =∑j

|〈Ej |N〉|2 = ‖|N〉‖ , (2.24)

〈N |1〉 =∑j

〈N |Ej〉 〈Ej |1〉 = 0 . (2.25)

This correlates the eigenvectors’ moduli and phases and prevents us to solve equations(2.21) and (2.22) separately.However, if there is a symmetry which already fixes the phases ϕj of the overlaps,

then both equations do separate. The probably simplest solution would be givenby pairing the overlaps. Taking, for instance, 〈1 |E1〉 = −〈1 |EN 〉 and so on wouldguarantee the orthogonality between |1〉 and |N〉.6 For one-dimensional chains withnearest-neighbour coupling, this requirement on the eigenvectors is so strong that itinduces a full symmetry of the system, and is a necessary condition for PST [51, 53].The symmetry is known as mirror symmetry, in the mathematical literature as

centro symmetry. It is described by a symmetry operator,

1 2 3 4 5 6 7

Figure 2.4: Permutation which keeps a mirror symmetric linear chain invariant.

6 ∑j 〈N |Ej〉 〈Ej |1〉

(2.21)∝

∑j eiφj |〈Ej |1〉|2 =

∑φj=0 |〈Ej |1〉|

2 −∑φj=π

|〈Ej |1〉|2 = 0.

23

Page 36: Hidden Symmetries of Quantum Transport in Photosynthesis

SC =∑n

|n〉 〈N + 1− n| (2.26)

which commutes with a mirrorsymmetric Hamiltonian, [H,SC ] = 0 (see also Fig-ure 2.4). For the overlaps of the site basis |n〉 with the eigenstates |Ej〉 we have

〈n |Ej〉 = (−1)j 〈N + 1− n |Ej〉 , ∀n,Ej > Ej−1 . (2.27)

The ordering of the energies is only necessary to give a closed expression. Impor-tantly, the eigenstates come in pairs of symmetric and anti-symmetric vectors in thesite basis [16, 99], according to their parity with respect to mirror reflections at thecentre of the chain.Equation (2.27) immediately satisfies (2.21), and reduces the condition (2.22) on

the phases to

Ejt− πj + θ = 2πmj with mj ∈ Z , (2.28)

hence the phase matching equation for the spectrum and phases separates from thecondition on the eigenstates. Analytic solutions for the coupling elements of theHamiltonian, in some cases of mirror symmetric systems can be found in [72].This result is not only useful in one-dimensional nearest-neighbour chains, but

works as a starting point for many constructions of PST Hamiltonians [52, 115].While it is neither necessary nor sufficient for systems beyond nearest-neighbourcoupling and/or of higher dimensionality, it is neat for constructions. The number offree parameters to describe the Hamiltonian decreases significantly and an efficientalgorithm to solve the inverse eigenvalue problem can be found [52]. We will turnthe problem around, and show that, in certain systems, it is statistically favourableto use mirror symmetry in order to find PST. But before let us look at experimentalrealisations, and scrutinise the effect of perturbations on systems that exhibit PST.

2.2.4 Experiments

Before we discuss the influence of perturbations and disorder on PST, we brieflypresent some experimental realisations of PST protocols, and point out the problemsthey are facing. A link to theory and further experiments can be found in [12, 17].The highest level of control over finite size quantum systems is currently held by

nuclear magnetic resonance (NMR). There, the nuclear spins of small molecules areused to encode quantum states, and manipulate them by sequences of radio frequencypulses. In some well-known and well-controlled molecules, even individual spins canbe addressed. For instance, the only demonstration of Shor’s factorisation algorithmhas been implemented by NMR in a seven spin system [105]. Zhang et al. [114]implemented the method derived by Christandl et al. [23], but only in a three-spinsystem. Therefore, effects induced by mirror symmetry or by couplings as definedby the semi-circle equation (2.18) were not observed. For scaling up the system size,perturbations due to the environment and imperfections of the coupling constants

24

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leading to disorder are a major obstacle. Therefore, we now consult the literature onthe impact of disorder in PST systems, and then present our approach of modellingthese systems as statistical ensembles.

2.2.5 Perturbations and Disorder in PST Systems

Environmental interactions and deviations in the engineered system parameters areexperimental challenges for quantum information processing systems, and for statetransfer. To find out which protocol fits better the experimental constraints is ahard task. On the one hand, this requires more accurate models including faithfuldescriptions of the specific environments. On the other hand, a result that does notdepend on the details of the external perturbations is preferable, since environmentsare in some sense badly characterised by their very definition. With the addition ofnoise, the problem becomes very similar to the transport problem in photosyntheticsystems. While here, in the specific perspective of the present section, the focus lieson a recipe for construction, photosynthesis is mainly reverse-engineering. Still, bothsolutions for optimal transfer must be robust with respect to external perturbations.There are many publications discussing noise in PST [3, 52, 78, 113, 115]. There

are usually two different questions. One concerns the robustness with respect toperturbations of a given set of parameters in the Hamiltonian, which we are interestedin. Another deals with external control, for instance by pulse sequences in NMRexperiments. In [78], the authors provide a comparison of three different controlschemes. All three exhibit PST, without disorder. The first is a sequence of SWAPoperations.7 It requires a high accuracy in timing to achieve high transfer efficiencies.The second is the strictly static chain, without external control. It is more robustthan the first scheme, but less than the third one, which uses a smooth, adiabatictime-dependence of the control pulse. We do not want to extend this discussionsince, motivated by the biological networks, we focus on transport without externalcontrol.What are the essential ingredients to make PST robust with respect to noise?

When studying PST in the presence of disorder, one usually makes a distinctionbetween diagonal and off-diagonal disorder [113]. We want to briefly review a workwhich takes a different path [115].Their model is a linear, nearest-neighbour coupled chain with mirror symmetry.

For such a model, the eigenvalues uniquely define the Hamiltonian, from which thecoupling elements of the Hamiltonian can be determined by solving an inverse eigen-value problem. Remember that in this situation the condition for PST reduces tothe condition (2.28) on the phases. The sequence of the eigenvalues within oneHamiltonian is characterised by an exponent α, and by a reference index jβ ,

Ej(jβ, α) ∝ sgn(j − j0)(jβ − |j − j0|α − jαβ ) , (2.29)

7Shape a pulse such that the state on one site is exchanged with that on its next neighbour to theleft, for instance.

25

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where j0 marks the centre of the spectrum. For α = 1, the eigenvalues dependlinearly on their index j. If further jβ is set to zero, then we will end up withan equidistant spectrum, which is equivalent to couplings given by the semicircleequation (2.18). A quadratic dependence is obtained for α = 2, which leads tostrong couplings concentrating even more in the middle of the chain, as comparedto the linear case. For smaller values of α, the distribution of the couplings getsbroader.The PST Hamiltonian is then perturbed by variation of the couplings – the transfer

is no longer perfect. The authors of [115] analyse how the choice of the spectrum,i.e. of different values of α, influences the robustness of the transport efficiency. It isfound that the quadratic spectrum is most robust, followed by the linear spectrum.For smaller α with broader coupling distributions, the performance decreases.The reasons they found out, were, first, that the dynamics of the unperturbed

system already indicates criteria for robustness. For quadratic and linear spectra,the transfer time coincides with the first maximum in the transfer probability. Thisis important, since disorder strongly affects every peak. In addition, if the numberof peaks is small, then no high frequencies modulate the dynamics, and, hence, thetransfer probability is smoother, and the maximum of the output population flatter.This leads to a larger time interval in which almost perfect state transfer can occur,what implies a high robustness with respect to timing errors. We will also find in ourmodel that highly efficient samples usually attain their maximal output probabilityat the first or second peak of the output population signal (see figure 3.2).Second, the eigenstates for quadratic and linear spectra are more localised on the

network, with strongest localisation found for the quadratic spectrum. Hence, whenexpanding the initial state in the eigenstates, only few frequencies enter the game.Therefore, it is at least plausible that the impact of a variation of the eigenfrequencies,due to the disorder in the couplings, is weaker than for a broader expansion. It isintuitively clear that the phase matching condition (2.28) is more easily fulfilled foronly a few frequencies. This observation was also made in [57].Third, since a quadratic spectrum leads to strong couplings in the centre and

weak ones at the end of the chain, perturbations of the couplings strength mainlyinfluence the eigenvalues belonging to eigenstates with large overlaps at the centreof the chain. Since the initial state is localised on that part of the spectrum which isleast influenced by the perturbation, also the dynamics persists, in the quadratic case.

We now move on to introduce our own approach to this problem, through a sta-tistical model. In contrast to the above, our model includes long-range interactionson a three dimensional graph. Instead of starting from a mirror symmetric system,we will identify this property as an essential ingredient to the statistical likelihoodof efficient transport.

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3 Statistical Approach

A variety of models are introduced to describe the dynamics of pigment-proteincomplexes, especially for the FMO. Many of them rely on the electronic Hamiltonianextracted from experimental data [1, 66, 81], and on different approaches for treatingits coupling to the environment, such that the dynamics and long coherence timesare reproduced in accordance with experimental observations [18, 24, 46, 50, 67, 68].But the experimental data exhibit a large spread, and the experiments are highlydemanding (see Chapter 1, Section 1.1.1). Very detailed microscopic models need tobe supplemented by parameter values not directly accessible in experiments, as forinstance the spectral density [74].There are methods in statistical physics to approach problems with a large de-

gree of uncertainty. Systems dynamics are sampled, for instance, by Monte Carlosimulations. We follow a somewhat similar approach on the basis of a simple mi-croscopic model for quantum transport on a random network of pigments. Then weconstruct a random ensemble on this models’ basis (section 3.1). When calculatingthe transport efficiency for each realisation within this ensemble (section 3.2), we findthat we cannot identify highly efficient networks by inspection of their Hamiltonians,neither on a single sample level (section 3.3) nor when considering the statistics ofdistances between Hamiltonians (section 3.4). However, the transport efficiency isstrongly correlated to a symmetry property of the system, namely its mirror symme-try (section 3.5). We check the persistence of this correlation against local dephasing(section 3.6) or the introduction of new degrees of freedom, such as variable dipolarorientations (section 3.7). We return to the macromolecular Hamiltonians inferredfrom the experimental data, in chapter 4, and examine the correlation between effi-ciency and mirror symmetry, as unveiled for the random network model of chapter3.

3.1 Model

Our model is based on a random molecular network, introduced by Scholak et al. [82,84, 85]. This is an abstraction of the biological systems of chapter 1. Each pigmentwhich participates in the transport is associated with one single site/node in thenetwork. The links/edges connecting different sites represent the couplings betweenthe pigments. Futhermore, we assume the pigments to be coupled via isotropicdipole-dipole interaction. Consequently, the inter-pigment distances determine thestrength of the links. We assume the site energies to be equal, and can, therefore,

27

Page 40: Hidden Symmetries of Quantum Transport in Photosynthesis

set them to zero. Accordingly, the Hamiltonian reads

Hij =

{0, for i = j ,

Vij , otherwise ,(3.1)

Vij =c

|ri − rj |3, (3.2)

where the ri are the positions of the sites in configuration space. In our ensemble,they are distributed randomly and uniformly within the unit sphere. Site 1, whichis the input site, is located on the sphere’s north pole, and site N, the output, onthe south pole. The proportionality factor c has no relevance in our model, sinceit drops out when we define our time scale for the transfer based on the couplingmatrix element between site 1 and site N (see equations (3.3), (3.4)).We now paraphrase the transport problem as follows:

Which spatial configurations of sites within the sphere allow for efficient transportfrom the north to the south pole,. . .

What is still unspecified here is the term ‘efficient’. We already defined an efficiencymeasure in (2.5), but left open the specific choice of the time scale T . The referencetime T was set to infinity for the definition of perfect state transfer in section 2.1above. In contrast, it will be finite now. Due to the full connectivity of the network– dipole-dipole interactions are long range – we have a well-defined intrinsic timescale in the system. It is determined by the coupling of the in- to the output site,and given by the Rabi period for the oscillation between input and output, in theabsence of all intermediate sites:

T :=π

2 |H1N |. (3.3)

Since the distance between the input and the output site is the largest in the system,the coupling between both sites is the weakest in the system, and, hence, the Rabiperiod T is here the largest time scale of a single oscillatory mode. For the transportproblem, the Rabi period has the following meaning: All transport which happenson a time scale equal or larger than T cannot be considered more efficient than if nointermediate sites were in the network at all. Hence, we complete our above guidingquestion by one further important characterisation of efficiency :

. . . while increasing the speed of transfer?

Thus, the reference time T on which the dynamics is monitored must be much smallerthan the intrinsic system time scale T ,

T � T . (3.4)

Typically we choose the reference time as one tenth of the intrinsic time scale,T = T/10. If we choose longer reference times, then the likelihood to find of highefficiencies of random networks increases.1

1 A detailed discussion of system time scales versus efficiency can be found in [82, chapter 5//2].

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Besides the reference time T , there is a second parameter in the model. While,by construction, the sites within the sphere are sampled from a uniform distribu-tion, they can, in principle, come arbitrarily close to each other. In this case, thedipole-dipole interaction diverges (see equation (3.2)). Since this is unrealistic in thebiological systems we want to mimic here, we include an exclusion volume with ra-dius rmin around each site. Our numerical sampling algorithm rejects configurationsof inter-site distances smaller than the exclusion radius. It is measured in units ofthe distance between input and output site, rin,out. The effect of a non-vanishingexclusion volume is that the coupling matrix elements of the Hamiltonian (3.2) arebounded from above by Vij ≤ c/r3min. Hence, also the fastest system time scale islimited, which has an impact on the transport efficiency, since we demand the trans-fer to happen much faster than the Rabi time (see equation (3.4)). We touch theissue of different exclusion radii in section 3.7.2

In the following, we will now first restrict ourselves to purely coherent time evolu-tion (see equation (2.2)), and examine the connection between the transport efficiencyon the reference time scale (section 3.2) and characteristic properties of the systemHamiltonian – the generator of the coherent dynamics (section 3.3, 3.4). Later, wewill check whether our results are robust with respect to incoherent perturbations,due to incoherent coupling to an environment (section 3.6), and extend our model(3.2) by adding variable dipolar orientations (section 3.7).

3.2 Statistics of Efficiencies

The model of a random network, as introduced in the previous section, leads to adistribution of transport efficiencies induced by the distribution of underlying Hamil-tonians. Since our model is motivated by the pigment network of the FMO, we setthe size of the network to N = 7.For an ensemble of random networks with N = 7 sites, and an exclusion radius

rmin = 0.05 rin,out, the histogram of the transport efficiencies P is shown in figure3.1, for a reference time T = T/10. The histogram is plotted once with a logarith-mic ordinate, to highlight small probabilities, whereas the second graph shows thedistribution of the efficiencies’ logarithm, logP, which we will use later (see sections3.5 ff.). The most probable value of the network’s transport efficiency is at about5% ≈ e−3. Transport efficiencies above 20% are exponentially suppressed. At thesame time, high efficiencies P > 0.8 still occur with a probability larger than 10−5.The shape of the distribution and its dependence on the sampling parameters T andrmin are exhaustively discussed in [82].We here focus on the characterisation of the subset of networks with high efficien-

cies, which carry finite weight in the ensemble, even though they are exponentiallysuppressed. Our observation that there is a sufficiently large volume of Hamiltonianparameters that lead to high transport efficiencies is supported by an optimisation

2 More details about the dependence of the transport efficiency on the exclusion radius can befound in [82, chapter 5//3].

29

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0 0.2 0.4 0.6 0.8 110−4

10−3

10−2

10−1

100

101

P

ρ(P

)

−10 −8 −6 −4 −2 00

0.1

0.2

loge P

ρ(P

)

Figure 3.1: Histogram of (left) the transport efficiencies P (2.5), and (right) its nat-ural logarithm loge P, after sampling over 106 random arrangements ofN = 7 molecular sites within a sphere. In- and output site are fixed tothe north and the south pole, respectively; also see (3.1), (3.2). Referencetime T = T

10 ; exclusion radius rmin = 0.05 rin,out.

analysis. In [82] a genetic algorithm was employed to find optimal network config-urations. This analysis indicates that the volume in parameter space even of thoseconfigurations which exhibit almost unit efficiency is non-vanishing.

We will use a similar algorithm to extend our study from random networks tobiological structure data (see section 4). Before, however, we will concentrate on thecharacterisation of highly efficient networks in our random ensemble.

30

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3.3 Matrix Elements and Dynamics: Low Compared toHigh Efficiency

Our random network model leads to a broad distribution of transport efficiencies, aswe have already seen in Section 3.2. Now we want to investigate how to distinguishnetworks which exhibit high and low efficiencies.For this purpose, let us first compare two single realisations both drawn from the

same random ensemble (3.1), (3.2), but one with high and one with low efficiency(P = 92% vs. P = 7%). Figure 3.2 shows the site occupations (also called popu-lations), of the network under coherent time evolution (2.4). The site occupationsdo not contain the full information of the state of the system. There are also thecoherences, which are the off-diagonal elements of the density operator. However,we only show the occupations here, since transport properties must finally manifestin a transfer of the excitation to a specific output site.In our setup, the excitation is initially located at site 1, thus P1(0) = 1, and

Pj(0) = 0 for all j ≥ 2. As time increases, the excitation is transferred to theremaining sites of the network. Hence, the population of site 1 decreases, while theother site populations grow. Since we consider strictly coherent quantum transport,which obeys the Schrödinger equation (2.1), the transfer of populations does not leadto diffusive spreading over the whole network, which would typically reveal itself ina monotonous increase or decrease of the populations, but instead to oscillations ofthe populations over time. The oscillations result from constructive or destructiveinterference between different transfer pathways through the network,3 dependingon the phase relation between the different paths.The transport efficiency P (2.5) is determined by the maximal site occupation of

the output site, which is P7 in figure 3.2. As explained in section 3.1, we restrictthe time window within which the transfer must occur to one tenth of the Rabiperiod T . The population of site 7 grows until it reaches a value of P7(t) = 0.92at time t = 0.08T . The value 0.92 is then assigned as the transport efficiency P ofthis network, under the given choice of the reference time T = T/10. Since we plotthe population dynamics for the entire Rabi period T , we see that the populationof the output site decreases for values larger than the transfer time τ = 0.08T ,and shows oscillations with period 2τ between zero and values larger than 75%.Similar oscillations, shifted by the transfer time τ , are displayed by the input site.Remarkably, the sum of both populations exceeds 29% within the entire time intervalfrom zero to one Rabi period, i.e. P1(t) + P7(t) ≥ 0.29, for all t ∈ [0, T ]. Notethat, if the sum of the in- and of the output site occupation carried 100% of thetotal population, then there would be only direct coupling between both sites, nointermediate sites would contribute to the transfer, and the transfer would occur asin a two-level system, on the time scale of the full Rabi period. Since we require the

3A pathway is defined by a sequence of sites which the excitation visits during its transfer throughthe network under the time evolution. A single classical particle takes only one pathway, whereasthe amplitude describing a quantum particle can take many paths simultaneously [35].

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Page 44: Hidden Symmetries of Quantum Transport in Photosynthesis

transfer to be ten times faster than T , the intermediate sites are needed to speed upthe transfer, and, thus, must also carry populations during the transfer, as can beseen in figure 3.2, especially at the intermediate times t ≈ 0.04T .Let us now consider the network with transport efficiency P = 7% (see figure 3.3).

The excitation is again initially placed at site 1. We observe almost perfect revivalsof the population at site 1, as in figure 3.2. In contrast to the network with highefficiency in figure 3.2, the excitations mainly oscillates between site 1 and site 6. Site4 and site 5 also exhibit population up to slightly above 40%, whereas the populationof site 7, the output site, hardly exceeds 25% during the entire Rabi period. Thepopulations of site 2 and 3 effectively vanish over the entire time interval.The vanishing population of site 2 and 3 is due to an effect which is called pair

localisation [82, Section 5//3], as confirmed by inspection of the coupling elementsof the Hamiltonian which is associated with the network. The Hamiltonian is thegenerator of the dynamics, and thus contains the full information concerning thecoherent quantum time evolution. Indeed, the Hamiltonian of the network with lowefficiency (see figure 3.3 (top left)) shows a coupling between site 2 and site 3 whichis one order of magnitude larger than the average. This strong coupling leads to ashift of the levels of the eigenstates which are localised at site 2 and site 3. Thisenergetic shift effectively decouples this pair of sites from the dynamics. How doesthis pair localisation effect influence the transport efficiency of the network? If oneof the strongly coupled sites was the in- or the output site, then the excitation wouldbe trapped at the input, or would never reach the output, respectively. On theother hand, if the pair of sites consists of two intermediate sites, the dynamics of theexcitation is essentially ‘blind’ to this pair. The latter is the case in the exemplarydynamics of figure 3.3. We are left with an effective network of five sites.To unmask the couplings between these remaining five sites, we monitor the Hamil-

tonian entries in a finite interval between zero and rather low values in modulus, onthe top right of figure 3.3. The matrix plot structure appears rather random, muchlike that of the network with high efficiency (see figure 3.2). There are no apparentstructural elements that allow to predict the various Hamiltonians’ transport effi-ciencies. The same applies for a second network with low efficiency P = 10%, infigure 3.4.

To unveil the distinctive structural properties of highly efficient as compared tolow efficient networks, we ,therefore, set out for a more extensive statistical analysison a large number of networks drawn from the random ensemble, in the next section.

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1 2 3 4 5 6 7

1234567

j

i

Good sample

0

20

40

Hij

|H1N |

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.250.5

0.75

t/T

P1(t)

P2(t)

P3(t)

P4(t)

P5(t)

P6(t)

P7(t)

Figure 3.2: Top: Matrix elements of a random Hamiltonian with efficiency P =92%. The colour code is in units of the in/out coupling’s modulus |H1N |.Reference time T = T

10 . Bottom: Dynamics of the site occupations Pi(t)generated by this Hamiltonian.

33

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1 2 3 4 5 6 7

1234567

j

iBad sample

0

500

1,000

Hij

|H1N |

1 2 3 4 5 6 7

1234567

j

i

Cutoff

0

10

20

30

40

Hij

|H1N |

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.250.5

0.75

t/T

P1(t)

P2(t)

P3(t)

P4(t)

P5(t)

P6(t)

P7(t)

Figure 3.3: Top left: Matrix elements of a random Hamiltonian with efficiency P =7%. The colour code is in units of the in/out coupling’s modulus |H1N |.Reference time T = T

10 . Top right: Same as top left, but the colour codelinearly interpolating the interval from vanishing entries to 40 |H1N |,such as to resolve the matrix structure for subdominant entries. Bottom:Dynamics of the site occupations Pi(t) generated by this Hamiltonian.

34

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1 2 3 4 5 6 7

1234567

j

i

Another bad sample

0

20

40

Hij

|H1N |

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

00.250.5

0.75

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.250.5

0.75

t/T

P1(t)

P2(t)

P3(t)

P4(t)

P5(t)

P6(t)

P7(t)

Figure 3.4: Top: Matrix elements of a random Hamiltonian with efficiency P =10%. The colour code is in units of the in/out coupling’s modulus |H1N |.Reference time T = T

10 . Bottom: Dynamics of the site occupations Pi(t)generated by this Hamiltonian.

35

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3.4 Distance Between Hamiltonians

We have seen that the differences in transport efficiencies can not be extracted bysimple inspection of the Hamiltonian’s entries for two individual networks. In a sec-ond attempt, we now perform a statistical analysis. We introduce a distance measureon the space of Hamiltonians, and plot its distribution over a random ensemble ofmolecular networks. Our hypothesis is that Hamiltonians with high transport effi-ciencies are ‘closer’ to each other than to Hamiltonians with low efficiencies.Our first choice of a distance measure on the matrix space which represents the

Hamiltonians is the Hilbert-Schmidt distance defined by [48, Section 5.6],

dHS(A,B) :=

√√√√ n∑i,j=1

(Aij −Bij)2 , (3.5)

where A and B are N × N matrices. Unfortunately, this measure gives a non-vanishing value even if the two matrices differ only by exchanging basis vectors,consider for instance

dHS

((1 00 2

),

(2 00 1

))= 2 .

This is problematic, since the efficiency measure P is invariant under permutationsof the labels of the intermediate sites. This invariance of the transport efficiencyresults from the fact that any unitary transformation S acting only on the statespace spanned by the intermediate sites commutes with the projection onto theinput and the output site,

[S, |1〉〈1|] and [S, |N〉〈N |] = 0 . (3.6)

From this it follows that

P (t) = |〈N |U(t) |1〉|2

= 〈N |U(t) |1〉 〈1|U †(t) |N〉= Tr

{|N〉〈N |U(t)|1〉〈1|U †(t)

}= Tr

{|N〉〈N |U(t) S†S︸︷︷︸

1

|1〉〈1|U †(t) S†S︸︷︷︸1

}Eq.(3.6)

= Tr{|N〉〈N |SU(t)S†|1〉〈1|SU †(t)S†

}=∣∣∣〈N |SU(t)S† |1〉

∣∣∣2= PS(t) , (3.7)

i.e. the output populations P (t) and PS(t) for the time evolution operator U andthe transformed time evolution operator SU(t)S† are identical. This agrees with

36

Page 49: Hidden Symmetries of Quantum Transport in Photosynthesis

the intuition that a relabelling of the intermediate sites has no physical meaning.Hence, our distance measure in the space of the Hamiltonians should also obey thisinvariance. Consequently, we amend our definition of distance in conformation spaceby taking the minimum over all permutations of the intermediate sites’ labels,

dminHS := minσ∈Sn−2

dHS(Aσ, B) , (3.8)

Aσ = ST (σ)AS(σ) , (3.9)

where S(σ) represents the permutation group SN−2 which permutes the N − 2 innersites, while leaving in- and output site unaffected.We now inspect the distance of each element of a large sample of random molecular

networks to the most efficient one of this sample. We would like to know whetherthe networks with high efficiencies cluster around the most efficient one, or if thedistribution of distances to the most efficient network is independent of the othernetwork’s efficiency. We, therefore, generate a sample of 105 such random molecularnetworks and identify their most efficient element in terms of the transport efficiencyP (2.5). Next, the distance of each element of the sample to this most efficient net-work is calculated using the modified Hilbert-Schmidt distance (3.8). We normalisethe distance with respect to the constant matrix element X1N = c/r3in,out (3.2) inorder to restore scale invariance with respect to rinout. Then we subdivide this sam-ple of random molecular networks into the set with transport efficiencies P > 0.2(blue), and those with P < 0.2 (green), and compare, for both cases, the normalisedprobability distribution of distances with each other in figure 3.5. Small distances(dminHS ≤ 0.2X1N ) are more probable for the efficient subset (blue) than the inefficientone (green), and vice versa for intermediate distances (0.2 < dminHS /X1N < 0.5). Forlarge distances (dminHS > 0.5X1N ), no significant difference between both distributionsis visible in figure 3.5 – apart from the stronger fluctuations of the blue sample dueto its smaller sampling size.Note that, due to the pair localisation effect, the decoupling of pairs from the

system dynamics (see section 3.3) is also a source of noise in the distribution ofdistances. Indeed, two networks can exhibit the same efficiency, while one networkcontains a pair of sites, and the other does not. The strong coupling which resultsfrom the pair of sites will dominate the distance to any other network. The distanceof both such networks to a third one will, hence, be very different, even though bothnetworks have the same transport efficiency.

Our initial expectation – that efficient networks are close to each other – is thereforefulfilled only in a quite weak statistical sense, as discussed above. In the next sectionwe examine another property of the Hamiltonian related to spatial symmetry, whichwill turn out to be strongly correlated with the transport efficiency.

37

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

dminHS (X,Hbest)/X1N

ρ(d

min

HS(X,H

best)/X

1N)

Figure 3.5: Normalised probability distribution of distances to the most efficient net-work. Hilbert-Schmidt distance dminHS (3.8) of 105 random molecular net-works X (3.1), (3.2) to the most efficient network represented by theHamiltonian Hbest with P(Hbest) = 73%, for different efficiency thresh-olds P(X) > 0.2 (blue), P(X) < 0.2 (green). Reference time T = T

10 ;exclusion radius rmin = 0.05 rin,out.

38

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3.5 Mirror symmetry

In the last section, we have seen that a statistical analysis of the distances betweenHamiltonians does not suffice to reliably distinguish networks with high from thosewith low efficiencies. From the theory of perfect state transfer, as presented in section2.2.3, it is known that in one-dimensional, nearest-neighbour coupled chains mirrorsymmetry is a necessary condition for perfect transfer [53]. In the general case, wehad found the condition |〈Ej |1〉| = |〈Ej |N〉| for the eigenstates |Ej〉 as necessarycondition for perfect state transfer (see equation (2.21)), which also indicates a sym-metry under exchange of the sites 1 and N . In this chapter, we therefore generalisethe concept of mirror symmetry to the case of arbitrary networks, and then examinehow this mirror symmetry is related to transport efficiency.

1 2 3 4 5 6 7

Figure 3.6: Definition of the permutation ω that characterises the mirror symmetryof a linear chain.

In a linear chain, mirror symmetry is defined with respect to the chain’s cen-tre. Figure 3.6 shows the permutation which, by definition, leaves mirror symmetricchains invariant. This permutation is called a reflection ω which, in cycle represen-tation,4 reads (1 N)(2 N − 1)(3 N − 2) . . . . We measure a deviation from mirrorsymmetry by first permuting the sites according to the reflection ω, and then com-puting the distance of the permuted HamiltonianHω to the originalH. As a distancemeasure we use again the Hilbert-Schmidt distance [48, Section 5.6]:

dmirrorHS (H) := dHS (Hω, H) SN 3 ω := (1 N)(2 N − 1)(3 N − 2) . . .

=

√∑ij

(Hij −HN+1−j,N+1−i)2 . (3.10)

Consequently, we have dmirrorHS (H) = 0 if and only if H is mirror-symmetric.For the three dimensional molecular networks which we are interested in, it is

not as straightforward to define mirror symmetry. As mentioned in section 3.4 onthe inter-Hamiltonian distances, the intermediate sites do not have an unambiguous

4The elements of the permutation group SN can be represented by its cycles. (a b c) means thata is mapped to b, b is mapped to c, and c is mapped to a. All permutations can be uniquelydecomposed into such cycles.

39

Page 52: Hidden Symmetries of Quantum Transport in Photosynthesis

ordering as in a linear chain. Therefore, we again take the minimum of (3.10) overall permutations of the intermediate sites,

dmirror,minHS (H) := minσ∈SN−2

dmirrorHS (Hσ) = minσ∈SN−2

dHS (Hσω, Hσ) . (3.11)

The number of possible permutations grows exponentially with the system size N .Fortunately, we do not need to evaluate the minimum over the entire group, but,due to the following identity:

ε :=dmirror,minHS (H)

N=

1

Nmin

σ∈SN−2

dHS (Hσω, Hσ)

=1

Nmin

σ∈SN−2

dHS

(Hσωσ−1

, H)

=1

Nmin

α∈CN−2(ω)dHS (Hα, H) , (3.12)

only over the conjugacy class

CN−2(ω) :={σωσ−1 | σ ∈ SN−2

}(3.13)

of the reflection ω. Each element of CN−2(ω) only contains elements which have thesame number of cycles with the same length as ω, since a conjugation with an elementof SN−2 just amounts to a relabelling of the sites (see example in appendix 6.4). Thereflection ω contains bN2 c5 two-cycles, and hence the conjugacy class consists of alldifferent sets of two-cycles of the inner sites. The evaluation of the minimum, thus,scales only quadratically in N . Finally, we normalise the measure with respect tothe number of sites N , such that ε is of the order of a single matrix element of theHamiltonian H.We now calculate the mirror symmetry measure ε, as defined in (3.12), for our

ensemble of random networks (3.1), (3.2). The mirror symmetry is measured inunits of the modulus of the coupling between the input and the output site H1N .We then compute the two-dimensional histogram of the network’s mirror symmetryε and their transport efficiency P. Figure 3.7 shows the 2D histogram on a doublelogarithmic scale for ε and P, while the probability density of a pair (loge ε, loge P)is linearly colour coded. We see that the bulk of the distribution is aligned along theanti-diagonal, and has a comet-like shape, with the comet’s tail containing networkswith extremely low efficiencies of P < 0.0025 ≈ e−6 and large deviations from amirror symmetric arrangement, ε > 3 ≈ e1. The bulk of the comet is centred aroundthe point ε ≈ 1 = e0 and P ≈ 5% ≈ e−3. We immediately note that the probabilitiesof finding an almost mirror symmetric (ε < 0.3 ≈ e−1) network with efficiencybelow 2% ≈ e−4, or, vice versa, a completely asymmetric (ε > 5 ≈ e1.5) networkwith efficiencies above 0.3 ≈ e−1 is strongly suppressed. These observations indicatethat there is a strong correlation between the mirror symmetry and the transport

40

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−2 −1 0 1 2

−8

−6

−4

−2

loge ε

logeP

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Figure 3.7: Correlations of mirror symmetry ε (3.12) vs. efficiency P (2.5), on adouble logarithmic scale. The color code shows the probability density ofa pair (log(ε), log(P)) obtained from a sample of 108 random molecularnetworks (3.1, 3.2). Reference time T = T

10 ; exclusion radius rmin =0.05 rin,out. White dot: Configuration taken from [84]; found by a geneticalgorithm (courtesy of T. Scholak).

efficiency in our random ensemble of molecular networks, which spans over severalorders of magnitude, from log ε = −2 to 2, and from logP = −8 to −1.Despite this observation of the correlation, we see that this comet-like distribution

exhibits a non-vanishing width perpendicularly to the anti-diagonal. Hence, we de-duce that the correlation between mirror symmetry and transport efficiency is onlyof statistical nature. To investigate the probability of finding networks which do notfollow this correlation, we redo the same plot, but now with logarithmic colour codefor the probability density (see figure 3.8). The logarithmic scale resolves the rareevents of those networks which are far off the anti-diagonal defined by the comet.With the help of figure 3.8, we can, therefore, quantify how rare an event is whichdoes not follow the correlation. The blue boundaries in the logarithmic colour code,for instance, indicate regions, which are 104 times less probable than the centre ofthe comet. We see that the correlation that we observed in figure 3.7 reaches out toefficiencies close to 100%, in figure 3.8.

5The floor value, which is obtained by the largest integer smaller of equal to N/2.

41

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−2 −1 0 1 2

−8

−6

−4

−2

loge ε

logeP

−10

−8

−6

−4

−2

0

Figure 3.8: Correlation between mirror symmetry ε (3.12) and transport efficiency P(2.5), on a triple logarithmic scale. The color code shows the natural log-arithm of the probability density of a pair (log(ε), log(P)) obtained froma sample of 108 random molecular networks (3.1, 3.2). Reference timeT = T

10 ; exclusion radius rmin = 0.05 rin,out. White dot: Configurationtaken from [84]; found by a genetic algorithm (courtesy of T. Scholak).

Since our main focus lies on the characterisation of networks with high transportefficiencies, we examine the range of high efficiencies by zooming into this region infigure 3.9. The x axis ranges from mirror symmetry values ε = 0 to the beginning ofthe comet’s tail at ε = 2, but now in linear scale, whereas the y axis is also linearlyscaled, but displays only transport efficiencies P above 20%. The colour encodesthe logarithm of the probability density for a pair (ε,P). The plot shows that thehighest probability for transport efficiencies P above 80% can be found at a valuefor the mirror symmetry of about ε ≈ 0.25. On the other hand, for the same mirrorsymmetry, it is 104 times more likely to find a network with transport efficiency of20%. There are, in addition, networks with efficiencies above 80% which are lesssymmetric than the average, i.e. ε > 1.5.One might think that the pair localisation effect is again responsible for blurring

the correlation between ε and P, as already discussed in section 3.4. This is, however,not the case: Large coupling matrix elements in the Hamiltonian, for pairs whichneither involve the input nor the output site, and therefore do not affect the transport

42

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ε

P

0

2

4

6

8

Figure 3.9: Histogram of mirror symmetry ε (3.12) and transport efficiency P (2.5)in the range of high efficiencies. Note the linear scale of ε and P. Theprobability density of a pair (ε,P) is colour-coded on a logarithmic scale.The histogram is obtained from the subset with efficiencies P > 0.2 ofthe entire sample of 108 random molecular networks (3.1, 3.2). Referencetime T = T

10 ; exclusion radius rmin = 0.05 rin,out. White dot: Config-uration taken from [84]; found by a genetic algorithm (courtesy of T.Scholak).

efficiency, do neither contribute to the mirror symmetry, since they drop out whensubtracting the appropriately permuted Hamiltonian in (3.12). On the other hand,pairs involving the input or the output site lead to an extremely small efficiency Pand, correspondingly, to an extremely large deviation ε from mirror symmetry. This,in fact, leads to the strongly correlated comet’s tail at small efficiencies, as alreadymentioned above.In summary, the finite width of the comet-like distribution shows that the cor-

relation between mirror symmetry and efficiency is not perfect. Therefore, the keystatement we can distill from the this observation is:

A mirror symmetric Hamiltonian is statistically favourable for efficient transport,within the model of molecular random networks.

Since our zoom into the high efficiency region of the correlation (figure 3.9) suggeststhat the probability to find a network with unit efficiency increases for decreasing

43

Page 56: Hidden Symmetries of Quantum Transport in Photosynthesis

mirror symmetry, up to ε ≈ 0.2, we wonder whether a network which is optimisedto exhibit unit efficiency utilises the observed correlation. We, therefore, take theoptimised network of [84], which uses the same network model as ours, and mark itstransport efficiency and mirror symmetry in the previous plots by a white dot (seefigures 3.7, 3.8, 3.9). The optimised network lies above the tip of the distribution, aswe expected (best seen in figure 3.9). This agrees with the intuition that an optimi-sation algorithm will end up with higher probability in a state which is statisticallyfavourable.

Given the above, strong correlation between mirror symmetry and transport ef-ficiency in our model of coherent quantum transport, we whether it persists in thepresence of noise, in the next section. Then, we add variable dipolar orientationsas extra degrees of freedom to our model, in order to approach the type of thephotosynthetic complexes we would like to imitate.

44

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3.6 Dephasing

For the biological systems of interest, interactions with the environment are of vi-tal importance. The coupling to external degrees of freedom is assumed to be ofcomparable strength as the electronic inter-site couplings of typical photosyntheticcomplexes (see section 1.4). Therefore, we must test the correlations detected inthe last section for robustness with respect to external perturbations. As mentionedabove, a large variety of models which describe the interaction with the environmentcompete for the most accurate description of the dynamics in these biological sys-tems (section 1.4). We do not want to get involved in this competition, but rathersubject our result, which was attained under conditions of purely coherent dynamics,to the simplest scenario of open system dynamics, to determine the influences of anenvironment.We are particularly interested in the short time scales on which quantum dynamics

matters (see section 1.4). Therefore, we need to identify that mechanism of environ-mental interaction which interferes on the shortest time scale, since this will destroyour hypothesis of coherent transport first. The theory of decoherence predicts that, inopen quantum systems with many degrees of freedom, the effect of the environmentfirst destroys the coherences, before the dissipation of energy and/or the transfer ofpopulation sets in [49]. This means that the phase relation between two states in acoherent superposition is most fragile with respect to perturbations. We, therefore,restrict ourselves to a dephasing model of the environment.We compute the open system dynamics via the master equation in Lindblad form.

The Hamiltonian of the coherent part is the same as before, and the Lindblad oper-ator models a dephasing in the site basis |k〉 [13, 49]:

ρ̇(t) =− i [H, ρ(t)]− 4γ∑k 6=l〈k| ρ(t) |l〉 |k〉 〈l| . (3.14)

The density operator ρ describes the state, with populations 〈k| ρ |k〉 on the diagonal,and coherences 〈k| ρ |l〉 on the off-diagonal. Solving the master equation without thecoherent part results in an exponential decay, with decay constant 4γ of all off-diagonal matrix elements. The effect of the dephasing strongly depends on the basison which it acts [37]. We choose the site basis, first for simplicity, and second sincemirror symmetry is defined with respect to this basis.We generate the same ensemble of molecular random networks as in the previous

section 3.5, and compute their mirror symmetry. In contrast to the previous results,we do not propagate the system coherently, but instead use the master equation(3.14). The initial state is again defined by the excitation localised on the firstsite. The transport probability PN (t) is now given by the Nth diagonal entry ofthe density operator ρ. Accordingly, we define the transport efficiency for the opensystem dynamics, as

Popen := max[0,T ]〈N | ρ(t) |N〉 . (3.15)

45

Page 58: Hidden Symmetries of Quantum Transport in Photosynthesis

Figure 3.10 shows histograms of mirror symmetry ε versus transport efficiency Popen,for different dephasing strengths γ. When the dephasing rate is on the order ofthe inverse Rabi period T−1 = 2 |H1N | /π which defines the lower bound of thesystem’s coupling matrix elements, there is no qualitative change noticeable (see topright of figure 3.10). Only ten times larger dephasing rates6 widen the comet-likedistribution (see bottom left of figure 3.10). In addition, the gradient of the cometgets flatter. This means that it becomes harder to distinguish less efficient frommore efficient networks by their mirror symmetry. A dephasing rate of ten times theinverse Rabi period is also the typical scale for system’s coupling matrix elements,since for uniformly distributed points within a sphere the average inter-site distanceis about one half of the distance between the input and the output [102]. Given thedipolar coupling in our model (3.2), the average matrix element is then on the orderof 23 ≈ 10 [82, section 5//3]. When the dephasing rate dominates over the couplingelements of the Hamiltonian, the correlation between ε and P is lost (see bottomright of figure 3.10). This is very plausible, since the ambient noise then rapidlywashes out the Hamiltonian’s specific structure.We thus conclude that the correlation between mirror symmetry and transport

efficiency is robust enough to persist even when environment-induced dephasing ratesare comparable to the Hamiltonian coupling matrix elements.Having convinced ourselves that the result first observed in figure 3.7 is stable in

the theoretical world of quantum open system dynamics, we now move closer to realbiological systems, by first introducing dipolar orientations, as a more faithful modelof the inter-site coupling.

6 A dephasing rate which is on the order of the reference time T corresponds to one incoherentevent during the reference time T = T/10

46

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−2 0 2

−8

−6

−4

−2

loge ε

logeP o

pen

−2 0 2

−8

−6

−4

−2

loge ε

logeP o

pen

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

−2 0 2

−8

−6

−4

−2

loge ε

logeP o

pen

−2 0 2

−8

−6

−4

−2

loge ε

logeP o

pen

Figure 3.10: Histogram of the mirror symmetry ε (3.12) and of the transport effi-ciency Popen, under open system dynamics (3.15), for different valuesof the dephasing rate in the site basis, γ = 0 (top left), 1 (top right),10 (bottom left), 100 (bottom right), in units of the characteristic timescale T−1 = 2 |H1N | /π. The histogram is generated from 108 randomnetworks for γ = 0, and 105 for γ > 0; for details see (3.1, 3.2). Thecolour code shows the probability density of the histogram on a linearscale, as in figure 3.7. Reference time T = T

10 , and exclusion radiusrmin = 0.05 rin,out.

47

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3.7 Angular Part of the Dipolar Couplings

While we have so far used only isotropic dipole-dipole interactions, see equation (3.2),we now include the angular part of the dipole-dipole interaction for a more realisticmodelling of the electronic inter-site couplings [1]. We replace the couplings Vij inthe Hamiltonian (3.1) by

Vij ∝di · dj − 3(di · rij)(dj · rij)

|rij |3, rij := ri − rj . (3.16)

Thus, the dipoles di are new additional random variables. They are sampled froma uniform distribution on the surface of a sphere. Hence, the ensemble of couplingsmatrix elements Vij is obtained from the distribution of distances within a sphere,taking into account a finite exclusion radius, and from sampling the dipolar ori-entations of all sites. The reference time T with respect to which the transportefficiency is evaluated, see equation 2.5, is chosen again as T = π/(20 r−3in,out), seeequation (3.4), corresponding to one tenth of the Rabi oscillation of the two-levelsystem formed by the input and output site with identical dipoles perpendicular torin − rout.Besides the addition of the variable relative orientations of the dipoles, we also

investigate the effect of changing the exclusion radius rmin. The motivation comesfrom the fact that the structure of the FMO complex is known with high spatial res-olution [100, 101]. Hence, there is not much freedom in the choice of the pigments’positions. If we increase the exclusion radius, we simulate this effect by enhanc-ing the spatial constraints. Therefore, we first enlarge the exclusion radius rmin inour model, without variable dipolar orientations, from a weak spatial constraint onthe sites’ positions (rmin = 0.05 rin,out) to strong constraint (rmin = 0.45 rin,out).The left column of figure 3.11 shows that the probability distribution is much moreconcentrated for the larger exclusion radius compared to the smaller (bottom left).This is mainly caused by the fact that the comet’s tail, which is a consequence ofpair localisation including the input or output site, is progressively cut off when in-creasing rmin. At the same time, also the width of comet increases. Thus, we inferthat a large exclusion radius reduces the correlation between mirror symmetry andtransport efficiency. However, the width of the comet along the line from top left tobottom right is still larger than its width perpendicular to this line. This shows thatthe correlation between P and ε still prevails, but is now less pronounced than forsmaller rmin.As compared to the exclusion radius, we find the influence of variable dipolar ori-

entations to be of minor importance. The latter is illustrated on the right hand sideof figure 3.11, both for small (top right) and large exclusion radius (bottom right).We see that, in the dipolar case the upper left corner of the distribution around(log ε, logP) ≈ (−1,−2) is suppressed as compared to the isotropic case. This leadsessentially to a reduction of the comet’s width along the line from top left to bot-tom right, which, however, in both cases, still remains larger than its perpendicular

48

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width. The correlation between transfer efficiency and mirror symmetry is thereforestill present in the case of dipolar orientations.

We saw that the correlation of mirror symmetry and transport efficiency, whichwe found for coherent quantum transport, persists under the influence of a simplemodel for the environment. Furthermore, the correlation is also found to be robustwhen employing a more realistic model with dipolar orientations. We could followthis direction, and increase the level of details in our model. However, this is notthe path we want to proceed. Instead, we test our hypothesis directly with theexperimentally inferred Hamiltonians from the biochemical literature in the nextchapter.

49

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−1 0 1 2

−8

−6

−4

−2

log ε

logP

−1 0 1 2

−8

−6

−4

−2

log ε

logP

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

−1 0 1 2

−8

−6

−4

−2

log ε

logP

−1 0 1 2

−8

−6

−4

−2

log ε

logP

Figure 3.11: Histogram of the mirror symmetries ε and transport efficiencies P forthe random network model with variable dipole orientations, equation(3.16). Top left: Same plot as in figure 3.7, with no angular part of thedipolar couplings, and exclusion radius rmin = 0.05 in units of rin,out.Bottom left: Same as top left, but with exclusion radius rmin = 0.45,and 106 realisations. Histograms with dipolar orientations are generatedby sampling over 105 random networks with isotropically distributeddipoles and variable site positions. The probability density of pairs(ε,P) is indicated by the linear colour code, and is plotted for differentexclusion radii rmin = 0.05 (top right), 0.45 (bottom right), in units ofrin,out. Reference time T = T

10 .

50

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4 Photosynthetic Systems

The dynamics of the Hamiltonian extracted from experimental data for pigment-protein complexes (see [1, 50, 81]) show that it performs badly in terms of ourefficiency measure (2.5). This was already noted in [67, 84], and can be read offthe dynamics of the site occupations in figures 4.4 and 4.5 (see below). This mightbe due to the fact that environmental effects are neglected for the dynamics (seesection 1.4). Another possibility is that there is a highly efficient Hamiltonian ‘close’to the experimentally inferred structure. Since we have seen for random networksthat efficient realisations are not all close to each other in terms of the Hamiltoniandistance measure in section 3.4, it is reasonable to expect higher transport efficiencieswithin the error margin of the experimentally determined Hamiltonians describingthe photosynthetic complexes.[84] showed that the efficiency of coherent transport can be dramatically enhanced

by sampling within the experimental error margin. Speaking in numbers, the effi-ciency for the FMO Hamiltonian [1] can be changed from about 4% to above 40%.As we will show below, we can go even further and obtain perfect state transferby optimising the original Hamiltonian, while at the same time staying close to theoriginal parameter values. We already mentioned in Section 1.4 that Sener et al. [88]follow a similar line of thought, and optimised photosystem I for high quantum yield.Since in contrast to our approach, however, these authors use a diffusive transportmodel, their results are not directly comparable to ours.Furthermore, apart from the possibility of optimising the experimentally inferred

Hamiltonians, we will focus our attention in this chapter on the question whethersuch an optimisation procedure exploits the correlation between mirror symmetryand transport efficiency which we found in the last chapter. Therefore, while run-ning a genetic algorithm in order to improve the transport efficiency, we will si-multaneously monitor the evolution of the mirror symmetry induced by the geneticalgorithm. We expect the algorithm to converge towards highly efficient mirror sym-metric Hamiltonians. But before doing so, we will first analyse the degree of mirrorsymmetries of the experimental Hamiltonians.

4.1 Uncorrelated Ensemble

If we take one of the experimentally obtained Hamiltonians, either FMO8 [81] orPC645 [28], and compute its mirror symmetry, we face a problem: We do not knowthe proper scale to compare the obtained mirror symmetry value ε (3.12) with. Sincethe off-diagonal elements of the Hamiltonians are calculated from structural data via

51

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charge density calculations (see [1, 81]), they are correlated by their mutual spatialdistances and dipolar orientations. Therefore, if the two species of Hamiltoniansturn out to exhibit special properties concerning their degree of mirror symmetry,then such correlations might be the reason. In order to check this, we remove thecorrelations by generating an ensemble of random matrices which consist of uncorre-lated, normally distributed matrix elements, and gauge the mirror symmetry of theoriginal experimental Hamiltonians against this ensemble. The parameters of therandom ensemble are fixed by the mean and the variance of the normal distribution,which we compute from the matrix elements of the experimentally obtained Hamilto-nians. Since we want to eliminate all correlations between basis states in the randomensemble, we assume that all the matrix elements of the ensemble obey the samenormal distribution, except for the fact that we discriminate between the diagonaland off-diagonal part. For the off-diagonal part, the mean and the variance of thenormal distribution for each matrix element of the random ensemble are determinedby

µ =2

N(N − 1)

N∑i<j

Hij , (4.1)

σ2 =2

N(N − 1)

N∑i<j

(Hij − µ)2 , (4.2)

where Hij are the coupling matrix elements of the Hamiltonians of FMO8 [81, SITable 5] or of PC645 [28, Table S.9.], respectively (see also appendix 6.2).

The diagonal part is handled differently. If we allow for non-vanishing matrixelements on the diagonal, we observed that, when changing the elements of theHamiltonians to optimise for high efficiency values, some pairs of diagonal matrixelements of the Hamiltonian are adjusted to be approximately equal to each other.For efficient coherent quantum transport, this tuning is reasonable, since a resonanceof two site energies favours transfer between both sites. The change in the value ofthe mirror symmetry due to the off-diagonal part, however, is then masked by thiseffect which only concerns the diagonal part. In order to resolve deviations in themirror symmetry of the site couplings, we therefore decided to exclude the diagonalpart of the Hamiltonian for the entire analysis, and set the diagonal to zero.

There is an additional argument in favour of the disregard of the diagonal: Through-out most of this thesis, we concentrate on the case of coherent quantum dynamics,whereas the differences in the diagonal elements (also called site energies) originateonly from the different couplings of the pigments, which are associated to each site,to the environment.

Consequently, after having determined the parameters of the normal distribution(4.2), the matrix elements of the random ensemble Mij are drawn from the distribu-

52

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tion

ρ(Mij) =

1√2πσ2

e−(Mij−µ)

2

2σ2 for i 6= j ,

δ(Mij) else ,(4.3)

where δ(x) denotes the delta distribution [5]. We calculate the distribution of mirrorsymmetries ε for this random ensemble (see figure 4.1). Then we compare the valueof the mirror symmetry of the experimental Hamiltonians with the distribution ofthe uncorrelated ensemble (see grey lines in figure 4.1), where also only the off-diagonal elements of the experimental Hamiltonians were accounted for to calculatetheir mirror symmetries.

15 35 550

0.02

0.04

0.06

FM

O8

ε

ρ(ε)

25 45 65 850

0.01

0.02

0.03

PC

645

ε

ρ(ε)

Figure 4.1: Distribution of the mirror symmetry ε of the uncorrelated ensemble basedon (left) FMO8 [81, SI Table 5], (right) PC645 [28, Table S.9.]. Ensembleswere generated by sampling over 103 Hamiltonians of 8× (8− 1)/2 = 28uncorrelated, normally distributed coupling elements (4.3). The meanand the variance of these normal distributions were derived from theoff-diagonal part of the FMO8, and of the PC645 Hamiltonian, respec-tively, according to (4.2). The mirror symmetries as calculated from theexperimental data are indicated by the vertical grey lines.

The two histograms of FMO and of PC645 in figure 4.1 show that the experimentalHamiltonian lies in the bulk of the distribution, in both cases. This means that theyare not particularly mirror symmetric, when compared to the random ensemble ofuncorrelated coupling matrix elements.

While the experimental Hamiltonians do not exhibit a significant degree of mirrorsymmetry, they also perform badly in terms of coherent transport (see figure 4.4, 4.5,and [1]). This fact together with the non-significant mirror symmetries of the experi-mental Hamiltonians is in accordance with the correlation between mirror symmetry

53

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and transport efficiency of the last chapter. However, what happens if we optimisethe experimentally obtained data for high transport efficiency [84]? We follow thisidea and check whether the mirror symmetry increases after optimisation.

4.2 The Genetic Algorithm

Given the experimental uncertainty of the Hamiltonian, there is still space for vari-ations within its error bars [1, 81].We use a genetic algorithm to optimise the couplings in such a way that high

efficiencies are reached. Therefore, we start with the experimental Hamiltonianswithout the diagonal part as explained in section 4.1 above. Then, we perturb eachcoupling Hij within ten percent of its value according to a Gaussian distribution withmean Hij and standard deviation |Hij/10|, and compute the transport efficiency P(2.5) of the new Hamiltonian with respect to the reference time T = π/(20 |H1N |).After repeating this step m times, we obtain m different perturbed Hamiltonianswith, correspondingly, m different values for the efficiency. We choose the one withthe highest efficiency and reiterate the process with this Hamiltonian. In order tokeep the iterated Hamiltonians close to the original one, we divide the standarddeviation for the matrix element Hij , initially set to |Hij/10|, by the number ofprevious iterations. After n iterations, we stop the algorithm, and get a collection ofnew, optimised couplings. We used m = 100 Hamiltonians in each step, and n = 104

steps. Given these parameters for the optimisation routine, we always end up witha transport efficiency larger than 90%.Running the algorithm 100 times with different seeds for the random number gen-

erator, we therefore obtain 100 optimised Hamiltonians with transport efficienciesabove 90%. The distribution of the mirror symmetry ε (3.12) for this set of opti-mised Hamiltonians is shown in blue in figure 4.2. It has about the same width as thedistribution of mirror symmetries of the random ensemble of uncorrelated matrices(4.3) (green), however, the mean value of the mirror symmetry distribution for theoptimised Hamiltonians is shifted to smaller values of ε compared to the random en-semble, for both initial Hamiltonians of FMO8 and PC645. We infer the significanceof the shift from a comparison with the scale induced by the random ensemble. Theshift is as large as half of the width of the distribution of mirror symmetries obtainedfrom the random ensemble of uncorrelated matrices.We are left with the question whether the shift is influenced by the underlying

experimental Hamiltonian, or entirely caused by the nature of the genetic algorithm.Therefore, we run the algorithm again, but this time seed the algorithm by a realisa-tion of the random ensemble of normally distributed, uncorrelated matrix elementsfrom section 4.1, instead of the Hamiltonians of FMO8 or PC645. If the shift of themirror symmetries which is observed in figure 4.2 is not caused by properties spe-cific to the experimental Hamiltonians, such as correlations between different matrixelement or non-Gaussian, higher order statistical moments for the distribution ofindividual matrix elements, then we must see the same shift for the random ensem-

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15 35 550

0.02

0.04

0.06

0.08

FM

O8

ε

ρ(ε)

25 45 65 850

0.02

0.04

PC

645

ε

ρ(ε)

Figure 4.2: Distribution of the mirror symmetry ε after optimisation. The genetic al-gorithm generates a distribution of optimised Hamiltonians after n = 104

steps. It starts from (left) the off-diagonal part of the FMO8 [81, SI Ta-ble 5], and (right) of the PC645 [28, Table S.9.] with a population ofm = 100. The histogram of ε of the thus generated Hamiltonians is plot-ted (blue), and compared to the distribution of the (green) uncorrelatedensemble (4.3), as in figure 4.1. Mirror symmetries of both original, ex-perimental Hamiltonians are indicated by the vertical grey lines, as infigure 4.1.

ble after optimisation. We pick ten realisations of the random ensemble, and runthe algorithm 100 times for each realisation, such that we end up with 1000 finalHamiltonians. We retain the parameters m = 100 and n = 104 of the algorithm.The distribution of the mirror symmetries for the random ensemble after optimi-

sation is shown in figure 4.3 (orange). We do not observe a shift between the mirrorsymmetries of the optimised Hamiltonians seeded by the random ensemble, and themirror symmetries of the random ensemble itself.

Since we based our observation on the shift of the distributions’ mean values, wewould like to assign a single Hamiltonian to this value. For the optimised cases shownin figure 4.2 (blue), we expect to find both an increase in mirror symmetry and ahigh transport efficiency. For this purpose, we will study in the following sectionthe average of the 100 optimised Hamiltonians, each of which individually displaysa transport efficiency above 90%, and compare its dynamics and its matrix elementsto the original, experimental Hamiltonian.

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15 35 550

0.02

0.04

0.06

0.08

FM

O8

ε

ρ(ε)

25 45 65 850

0.02

0.04

PC

645

ε

ρ(ε)

Figure 4.3: Distribution of the mirror symmetry ε after optimising the uncorrelatedensemble. The genetic algorithm generates a distribution of optimisedHamiltonians after n = 104 steps, for a population of n = 100, start-ing from 10 different Hamiltonians of the uncorrelated ensemble of theFMO8 (left), and PC645 (right). The histogram of mirror symmetries ofthese optimised Hamiltonians (orange) is compared to the distribution ofthe uncorrelated ensemble before optimisation (green), as in figure 4.1.Mirror symmetries of experimental Hamiltonians, FMO8 and PC645, areindicated by the vertical grey lines.

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4.3 Dynamics of the FMO8 and PC645

We want to find out whether the average Hamiltonian generated by the geneticalgorithm has again a high transport efficiency, and how its dynamics in the siteoccupations compares to the original, experimental Hamiltonian, which seeds theoptimisation routine.The dynamics of the experimental Hamiltonians is computed by coherent quantum

time evolution (2.1) as in the previous chapter. We order the sites according to ourconvention of the input being labelled as site 1, and the output site as site 8.1

Figures 4.4 and 4.5 shows the dynamics of the Hamiltonians of FMO8 and PC645,respectively, before (left column) and after (right column) optimisation. Concerningthe latter case, we averaged over all 100 optimised Hamiltonians which were seededby the experimental Hamiltonians2 (see the blue distribution in figure 4.2). We seethat, for both experimental Hamiltonians, the population of the output site does notexceed values of 20%. While both experimentally inferred Hamiltonian do not exhibithigh transport efficiencies for purely coherent quantum transport, they show distinctbehaviours in their dynamics of the site occupations. For FMO8, besides sites 4and 6, all others participate in the transfer. Site 1 and site 7 exhibit alternating,large oscillations. However, the excitation is not trapped within site 1 and site 7, incontrast to PC645, where the excitation never escapes the pair of sites 1 and 2.For FMO8, the dynamics of the average Hamiltonian after optimisation (figure 4.4

(right)) shows that only four out of eight sites participate. The population of site8, which determines the transport efficiency, reaches values up to P = 94%. This isremarkable, since we averaged over many different Hamiltonians, and the ensembleaverage and the coherent quantum dynamics, in general, do not commute. In otherwords, if two Hamiltonians H1 and H2 exhibit perfect state transfer, it is not at allobvious why we should expect the same for their average (H1+H2)/2. At present, wecannot give a convincing explanation why this average Hamiltonian nevertheless alsoexhibits almost perfect state transfer. One might think that the genetic algorithmconverges to a single fix point, and that the set of optimised Hamiltonians which weaveraged is homogenous. However, this contradicts with the distribution of mirrorsymmetries of this set, which proves that the Hamiltonians of this set are not simplyidentical. Moreover, we have also checked that they do not commute with eachother. On the other hand, it must be noted that perfect state transfer between well-defined states is a significantly weaker condition than the generation of the sametime evolution (by H1, H2, (H1 +H2)/2) of arbitrary initial statesFor PC645, the average Hamiltonian exhibits fast oscillations in the site popula-

tions (figure 4.5 (right)). In addition, all sites are populated during the transfer.Hence, in this respect the effect of the genetic algorithm on PC645 is opposite to itsimpact on the FMO8, where the number of effectively participating sites is reduced.

1See appendix 6.2 for details about the ordering.2See appendix 6.2 for the matrix elements of both the original and the average of the optimisedHamiltonians.

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However, the same observation about the transport efficiency which we made forFMO8 also holds true for PC645: The average over all 100 optimised Hamiltoniansagain yields a high transport efficiency, P = 93%.Last, we convince ourselves that the new couplings of the average Hamiltonians

do not differ much from the original ones (see section 6.2). Half of the couplings areessentially unchanged. The other half is changed on a scale comparable to the meancoupling, except for a few entries which show large deviations. We believe that wecould achieve the same results with deviations in the matrix elements below errorbounds of one tenth of the values in the Hamiltonian, by using a more sophisticatedgenetic algorithm.

In summary, the results of this chapter show that the idea of enhancing efficiencyby means of mirror symmetry not only applies to the random networks studiedin chapter 3, but is also consistent with the experimental data available on smallphotosynthetic complexes.

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00.5

00.5

00.5

00.5

00.5

00.5

00.5

0 0.2 0.4 0.6 0.8 10

0.5

t/T

0 0.2 0.4 0.6 0.8 1

t/T

P1(t)

P2(t)

P3(t)

P4(t)

P5(t)

P6(t)

P7(t)

P8(t)

Figure 4.4: Comparison of the dynamics of the site occupations Pj(t) (2.4) of theFMO8 Hamiltonian [81, SI Table 5] before (left) and after (right) optimi-sation (see figure 6.1). The initial excitation on site 1 is propagated by(2.1), where the diagonal of the Hamiltonian is always set to zero. TheFMO8 Hamiltonian is optimised by the genetic algorithm for maximalvalues of P8, with n = 104 steps and a population of m = 100. Theresults of 100 runs are averaged and then propagated by (2.1), to obtainthe dynamics, as displayed in the right figure. Time t is given in units ofthe Rabi period T .

59

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00.5

00.5

00.5

00.5

00.5

00.5

00.5

0 0.2 0.4 0.6 0.8 10

0.5

t/T

0 0.2 0.4 0.6 0.8 1

t/T

P1(t)

P2(t)

P3(t)

P4(t)

P5(t)

P6(t)

P7(t)

P8(t)

Figure 4.5: Same as figure 4.4, for the PC645 Hamiltonian [28, Table S.9.] before(left) and after (right) optimisation (see figure 6.2).

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5 Conclusion & Outlook

In this work, we showed that hidden symmetries may play a role in quantum trans-port in biological systems – specifically related to excitation transfer in photosyn-thesis.After we argued why we expect that coherence can affect transport in these systems

only on short time scales, we restricted ourselves to a model of purely coherentquantum transport, and found that a spatial symmetry – mirror symmetry – playsan important role.In a statistical model of random networks, we found their transport efficiency to

be strongly correlated with mirror symmetry: networks with higher degree of mirrorsymmetries tend to exhibit, on average, a higher transport efficiency, and vice versa.We found that this correlation between mirror symmetry and transport efficiency

ranges over several orders of magnitude, and can, in particular, also be observed in theregime of high efficiencies (section 3.5). It is robust with respect to local dephasing,on short time scales, and when adding extra degrees of freedom by variable dipolarorientations (section 3.6).In our last chapter 4, we verified that the correlation of mirror symmetry and

transport efficiency is consistent with available experimental data for the Hamil-tonian of photosynthetic complexes (section 4). Whereas the original, experimentalHamiltonians for two different photosynthetic complexes (FMO8 and PC645) exhibitneither a particularly high degree of mirror symmetry, nor a high coherent transportefficiency, we found that, when optimising both Hamiltonians towards high transportefficiencies by means of a genetic algorithm, also their degree of mirror symmetrysignificantly increases. Moreover, most matrix elements of the optimised Hamilto-nians – which both exhibit a transport efficiency larger than 90% – do not stronglydiffer from those of the original, experimental Hamiltonians. If we average the op-timised Hamiltonians which are obtained by many independent runs of the geneticalgorithm, we find, as discussed in chapter 4, that the resulting Hamiltonian is againefficient. This is a remarkable result in itself, the implications of which remain to beclarified in future work.Another important step will be the experimental observation of the correlation

between transport efficiency and mirror symmetry. This poses challenges not just onthe experimental side. While it is theoretically convenient to define the Hamiltonianand its symmetry in the site basis, it is not so for spectroscopists. Their laser pulsesusually address eigenstates which, in general, do not commute with the site basis.There are recent experiments with, both, high temporal and spatial resolution [2],but they cannot resolve single pigments yet. A promising approach might be offeredby the use of chirality. In [98], it was shown how to extract an experimental signal

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with chirality selectivity. Since chirality is sensitive to reflections, it might be well-suited to see the correlation between mirror symmetry, which is characterised byreflection (see section 3.5), and transport efficiency. Another, bottom-up approachwould be to consider different types of systems with a higher degree of control. Forinstance, arrays of nano-crystals exhibit similar features, in terms of transport andcoherence, and their spectroscopic features are much better understood [62]. Thiswould open the way to engineer samples with variable mirror symmetry, and to probethe correlation between transport efficiency and mirror symmetry here predicted.

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6 Appendix

6.1 Transformation of Units

E

[kJmol

]= NA E [kJ] (6.1)

= NA · 10−3 E[J] (6.2)

= NA · 10−3 · e E[eV] (6.3)

= 6.022 · 1023mol−1 · 10−3 · 1.6 · 10−19C E[eV] (6.4)

≈ 102mol−1C E[eV] (6.5)

= 10−3NA E

[eVmol

](6.6)

≈ 3

2· 10−22C E

[eVmol

](6.7)

E[J ] = hcE[cm−1] (6.8)

E[~s−1] = 2π c E[~ m−1] (6.9)

= 200π c E[~ cm−1] (6.10)

≈ 1.88 · 1011E[~ cm−1] (6.11)

[45]

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6.2 Average Hamiltonian of Genetic Algorithm

. . . −12.7/− 11.1 35.9/39.5 3.6/7.9 −1.6/− 1.6 4.8/4.4 −7.0/− 9.1 1.4/1.4

. . . . . . −12.8/− 12.2 6.4/5.7 −96.4/− 62.3 −4.9/− 4.6 21.9/35.1 3.8/3.4

. . . . . . . . . −96.2/− 98.0 −6.0/− 5.9 7.9/7.1 −12.4/− 15.1 2.8/5.5

. . . . . . . . . . . . 7.9/7.6 1.5/1.6 12.7/13.1 35.8/29.8

. . . . . . . . . . . . . . . −40.9/− 64.0 −16.2/− 17.4 −9.3/− 58.9

. . . . . . . . . . . . . . . . . . 134.9/89.5 −1.2/− 1.2

. . . . . . . . . . . . . . . . . . . . . −7.9/− 9.3

. . . . . . . . . . . . . . . . . . . . . . . .

Figure 6.1: Off-diagonal elements of the average Hamiltonian after optimising theFM08 with a genetic algorithm (see section 4.2) with m = 100 andn = 104 (left), and of the original, experimental FMO8 Hamiltonian[81, SI Table 5] (right). The ordering of the sites is according to ourconvention of the input site located at site 1 and the output site at siteN = 8. Compared to the standard notation from the literature, the sitescorresponds to the labels 8, 1, 2, 4, 5, 6, 7, and 3.

. . . −10.6/− 46.8 106.4/319.4 −5.7/− 9.6 −23.8/− 43.9 18.4/20.3 3.2/25.3 −0.7/− 20.0

. . . . . . 14.4/21.5 −17.1/− 15.8 56.1/53.8 10.9/11.0 29.2/29.0 5.1/48.0

. . . . . . . . . 30.2/43.9 7.1/7.7 16.3/30.5 23.3/29.0 105.4/48.0

. . . . . . . . . . . . 4.5/4.3 −154.4/− 86.7 −3.0/− 2.9 14.5/49.3

. . . . . . . . . . . . . . . 3.5/3.4 108.4/86.2 −20.2/− 14.7

. . . . . . . . . . . . . . . . . . 7.9/7.8 11.4/10.0

. . . . . . . . . . . . . . . . . . . . . −7.6/− 10.7

. . . . . . . . . . . . . . . . . . . . . . . .

Figure 6.2: Off-diagonal elements of the average Hamiltonian after optimising thePC645 with a genetic algorithm (see section 4.2) with m = 100 andn = 104 (left), and of the original, experimental PC645 Hamiltonian[28, Table S.9.] (right). The ordering of the sites is according to ourconvention of the input site located at site 1 and the output site at siteN = 8. Compared to the standard notation from the literature, thesites corresponds to the labels DBVc, PCBc82, DBVd, MBVa, MBVb,PCBc158, PCBd158, and PCBd82.

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6.3 Eigensystems of Small Networks

Taken equal weights on all edges, the Hamiltonians for P3 and C4 = K2 ×K2 are,

HP3 =

0 J 0J 0 J0 J 0

, HC4 =

0 J J 0J 0 0 JJ 0 0 J0 J J 0

, (6.12)

where we get

E1 = 0 , v1 =1√2

10−1

, (6.13)

E2 =√

2J , v2 =1√4

1√2

1

, (6.14)

E3 = −√

2J , v3 =1√4

1

−√

21

, (6.15)

for HP3 , and

E1 = 0 , v1 =1√2

100−1

, (6.16)

E2 = 0 , v2 =1√2

01−10

, (6.17)

E3 = 2J , v3 =1√4

1111

, (6.18)

E4 = −2J , v4 =1√4

1−1−11

, (6.19)

65

Page 78: Hidden Symmetries of Quantum Transport in Photosynthesis

for HC4 . This results in the dynamics

〈3| exp (−iHP3t) |1〉 =1

2

(cos(√

2Jt)− 1)

, (6.20)

〈4| exp (−iHC4t) |1〉 =1

2(cos (2Jt)− 1) , (6.21)

P (τ) = 1 ⇒ τP3 =π√2J

, τC4 =π

2J. (6.22)

.For a Hamiltonian of the type

H =

0 J . . . J 0J 0 . . . 0 J...

.... . .

......

J 0 . . . 0 J0 J . . . J 0

, (6.23)

the transport probability gives:

P (t) = (6.24)

N = 4 ⇒ | sin(Jt)|4 , (6.25)

N = 5 ⇒ 1

6

∣∣∣sin(√6Jt)∣∣∣2 , (6.26)

N = 6 ⇒ 1

8

∣∣∣sin(2√

2Jt)∣∣∣2 , (6.27)

N = 7 ⇒ 1

10

∣∣∣sin(√10Jt)∣∣∣2 , (6.28)

N = 8 ⇒ 1

12

∣∣∣sin(2√

3Jt)∣∣∣2 , (6.29)

N = 9 ⇒ 1

14

∣∣∣sin(√14Jt)∣∣∣2 , (6.30)

N = 10 ⇒ 1

16| sin(4Jt)|2 , (6.31)

...

∀N ≥ 5 ⇒ 1

2N − 4

∣∣∣sin(√2N − 4 J t)∣∣∣2 . (6.32)

This was found numerically, and the general formula deduced from the results. PSToccurs only for N = 4.

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6.4 Conjugacy Class of Reflection for Small Sizes

The reflection ω for a chain of length N = 4 can be expressed in cycle presentationas

ω ∼= (14)(23) . (6.33)

When we conjugate ω by the whole symmetric group S4 for N = 4, we obtain theconjugacy class

C4(ω) :={σωσ−1 | σ ∈ S4

}= {(14)(23), (13)(24), (12)(34)} . (6.34)

In chapter 3, we deal with networks of size N = 7. In addition, we allow only forpermutations of the intermediate, i.e., of all sites but sites 1 and 7. Hence, thereflection writes

ω′ ∼= (17)(26)(35)(4) . (6.35)

The conjugacy class with respect to the group acting only on the intermediate sites2 to 6 is given by

C5(ω′) = {(17)(26)(35)(4),

(17)(36)(25)(4),

(17)(23)(56)(4),

(17)(34)(56)(2),

(17)(35)(46)(2),

(17)(36)(45)(2),

(17)(26)(45)(3),

(17)(24)(56)(3),

(17)(25)(46)(3),

(17)(23)(46)(5),

(17)(24)(36)(5),

(17)(26)(34)(5),

(17)(23)(45)(6),

(17)(24)(35)(6),

(17)(25)(34)(6)} . (6.36)

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Erklärung der Urheberschaft

Ich, Tobias Zech, erkläre hiermit, dass ich die vorliegende Arbeit selbstständig undohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe. Fremdesgedankliches Eigentum ist als solches gekennzeichnet und in den Quellen belegt. Vonmir wurde keine ähnliche Arbeit bei einer anderen Prüfungsbehörde vorgelegt oderveröffentlicht.

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