Interplay between coherence and decoherence in LHCII photosynthetic complex
aa r X i v : . [ qu a n t - ph ] J un Interplay between coherence and decoherence in LHCII photosynthetic complex
Paolo Giorda, Silvano Garnerone, Paolo Zanardi, and Seth Lloyd ISI Foundation, I-10133 Torino, Italy Department of Physics and Astronomy and Center for Quantum Information Science & Technology,University of Southern California, Los Angeles, CA 90089 Massachusetts Institute of Technology Research Lab of Electronics and Dept. ofMechanical Engineering 77 Massachusetts Avenue, Cambridge, MA 02139, USA
PACS numbers:
ABSTRACT
This paper investigates the dynamics of excitonic transport in photocomplex LHCII, the primary component ofthe photosynthetic apparatus in green plants. The dynamics exhibits a strong interplay between coherent processesmediated by the excitonic Hamiltonian, and incoherent processes due to interactions with the environment. Thespreading of the exciton over a single monomer is well described by a proper measure of delocalization that allowsone to identify two relevant time scales. An exciton initially localized in one chromophore first spreads coherently toneighboring chromophores. During this initial coherent spreading, quantum effects such as entanglement play a role.As the effects of a decohering environment come into play, coherence and decoherence interact to give rise to efficientand robust excitonic transport, reaching a maximum efficiency at the levels of decoherence found in physiologicalconditions. We analyze the efficiency for different possible topologies (monomer, dimer, trimer, tetramer) and showhow the trimer has a particular role both in the antenna and the wire configuration.
INTRODUCTION
Recent experiments probing excitonic transport in green sulphur bacteria suggest that quantum coherence playsan important role in photosynthesis [2–4]. Detailed models of the interplay between coherent exciton dynamicsand decoherence and relaxation induced by the exciton’s environment show that the resulting transport is robustand efficient [5–7], an effect known as environmentally assisted quantum transport (ENAQT). This paper extendsthese analyses to the light-harvesting complex of green plants [18], specifically, transport through the light-harvestingcomplex LHCII . Excitonic transport through sets of coupled LHCII complexes [8–11] differs in significant ways fromthe transport through the Fenna-Matthews-Olson complex (FMO) of green bacteria . Notably, the LHCII can actboth as antennae and ’wires’ capable of transferring the excitons captured by other complexes through the structure.In order to move through a sequence of LHCII complexes, the exciton must move both up and down in energy, aprocess mediated by interactions with the environment. LHCII complexes can be found in different forms allowing forvarious regulation activities. In particular, some LHCII complexes can migrate, under specific light conditions, fromthe Photosystem II (PSII) to Photosystem I (PSI) in order to optimize the photosynthetic process (state transition),or disassamble into monomeric subunits in order to favor the regulation of light harvesting in excess light [14].This paper applies a purely-decohering Haken-Strobl model to analyze interactions between coherent and decoherentdynamics in excitonic transport in LHCII [12, 15]. The advantage of using Haken-Strobl model is that it is the simplestmodel that allows this transport to be investigated in the strongly-coupled, non-perturbative regime. The disadvantageof a purely decohering model is that it does not include relaxation, and so will over-estimate transport rates as theexciton moves up in energy, and under-estimate them as it moves down. Nonetheless, our analysis shows that puredecoherence is a surprisingly effective transport mechanism even when the exciton is moving down through an energyfunnel.To analyze the interplay of coherence and decoherence, we use familiar tools from quantum information theory.Quantum mutual information between sets of sites is used to track the pathways by which correlations spread throughthe LHCII monomeric complex due to exciton motion. Negativity and concurrence are used to demonstrate andquantify entanglement. The comparison among the different measures allows us to identify two timescales. Overthe first half picosecond, an initially localized exciton spreads coherently to neighboring chromophore. This coherentspreading is accompanied by rapid oscillations in quantum mutual information and negativity, indicating the presenceof significant coherence and entanglement. Next, over the course of several picoseconds, the coherent oscillationsdisappear, the negativity decreases, and quantum mutual information between sites grows as decoherence kicks in and
FIG. 1: The chains of Chls in the trimeric LHCII. Blue: CHls-b; green: CHls-a. the exciton diffuses partially incoherently to more distant sites. The interplay between coherence and decoherence givesrise to highly efficient transport through the complex. We analyze the efficiency of transport for various geometries ofLCHII (monomer, dimer, trimer, and tetramer) and for the two possible configurations: antenna and wire. We showthat efficiency increases as the number of subunits increases, saturating at the trimer. That is, the dimer exhibitsmore efficient transport than the monomer; the trimer is more efficient than the dimer; and the trimer and tetramergive the same efficiency.
BACKGROUND
Photoabsorption, the initial step of photosynthesis, takes place in photosynthetic complexes formed by groups ofpigments (chlorophylls, Chl) and proteins placed within the thylakoid membrane [1]. The light-harvesting pigmentsare arranged in protein matrices in such a way that the photo-excitation is funneled to the reaction center, where theenergy carried by the excitation is used for fueling chemical reactions.One of the main elements in the photosynthetic complexes of higher plants is the light-harvesting complex LHCII.In the following we will refer to the LHCII that can be found in the
Arabidopsis thaliana , which is a flowering plantthat has been extensively studied as a model in plant biology and genomics. In this plants, LHCII can be found asperipheral antenna in the supercomplex PSII [13]. Its role in the supercomplex is to collect photons from the incominglight or excitons from the neighbouring complexes (other LHCIIs or other minor antenna complexes) and transferthem to other LHCII complexes or to the reaction center.The LHCII crystal structure has been determined up to 2 . A in resolution [16, 17]: it is composed of three similarmonomeric subunits (Lhcb1-3) each containing 14 Chls molecules embedded in a protein matrix (see Fig.1). Twodifferent types of Chls are present: 8 Chl-a type, and 6 Chl-b type. The main difference between the two is thepresence in Chl-b of a carbonyl group that allows for higher excitation energies. The Chls are disposed in two layerswhich are called stromal and lumenal , the first being oriented towards the outer part of the thylakoid membrane, thesecond towards its inner part (see fig. 2).Within each layer, the Chls can be grouped on the basis of their relative distance and orientations which determinethe strength of the electronic interaction. The main groups of Chls are depicted in fig. 3; different groups havedifferent linear absorption spectra and two main bands can be identified: the Chl-a band centered at 14925 cm − , andthe Chl-b band centered at 15385 cm − .The exciton energies and the relative pigment participations in LHCII have been determined in [9]. The energy FIG. 2: LHCII monomer in the membrane, and its top and bottom view. Blue: CHls-b; green: CHls-a relaxation pathways of the system were described by van Grondelle (see [8]) and refined in [10] by means of theexperimental data obtained by applying 2D spectroscopic techniques to samples of LHCII at 77 K ; in the same paper,the Hamiltonian for the monomeric unit we will use in the following was also derived and optimized in order toaccount for the experimental results. The Hamiltonian is written in the site basis: the diagonal terms are the siteenergies, determined by fitting the linear absorptions spectrum and the off-diagonal terms account for the Coulombinteractions between pairs of Chls (see [10] and references therein). The experimental evidences allows to identify twomain downhill relaxation pathways [8, 10]. One takes entirely place at the stromal level (stromal-stromal) while thesecond starts in the lumenal level (lumenal-stromal) (see fig. 4).One of the fundamental mechanisms that is at the basis of the energy transfer through the LHCII is given bythe interplay between strong electronic couplings between nearby Chls. The coupling allows both for the energysplitting between excitonic levels and for the exciton delocalization. If we focus for example on the group of Chls a , a , a
612 (fig. 3) we see that the contiguity and the relative orientation of these molecules result in a stronginteraction which in turn it allows for the presence of three exciton levels (levels 5 , , < f s ) relaxation processes within the group-excitonic band(levels 5 , , b , b , b
609 which allows for a fast ( < f s )relaxation process within the stromal b-band (levels 13 , ,
10 in fig. 4).In analogy with what happens for the previous groups of Chls, the intra-group energy pathways can be describedby the same mechanism. In particular, in the lumenal-stromal pathways, one can identify a bottleneck of the pathwaythat is given by the Chls b
605 and a FIG. 3: Stromal (A) and lumenal (B) layers of the LHCII. The groups of strongly coupled Chls are enclosed in thick circles.FIG. 4: Energy relaxation pathways as experimentally determined in [10].
The dynamics of the bottleneck will be seen in action in the simulations of exciton transport through the monomerin the next section.The lowest energy states of the monomer are two of the excitonic states localized in the a , a , a
612 group ofChls. These sites are therefore indicated as the output sites of the monomer unit (donors), which can be coupledto other LHCII monomeric units, or other complexes in the PSII. The coupling can be between exitonic statescorresponding to similar energies but located on different neighbouring complexes. In [10] these two output states havebeen studied in terms of their directionality, which is determined by the interplay between the site basis contributionto the specific excitonic state (delocalization over the group of Chls a , a , a i ) a sites based coupling, where eachoutput site (Chls a , a , a ii ) anexciton based coupling, where the two lowest excitonic states are independently coupled with an outer sink.For the inter-monomeric couplings, the experimental evidences in [17] ([10]), show that the main coupling shouldbe localized in the stromal side between Chls of b type ( b , b , b cm − , (35 cm − ). In [10] this inter-monomeric coupling has been neglected in the determination of thesingle monomer Hamiltonian. While this has the effect of shifting the excitonic b-band, it should not significantlyaffect the other transitions which are localized within each monomer. In order to have a description of the wholetrimeric LHCII, we reintroduce the coupling between Chl-b pertaining to different adjacent monomers; in particularwe choose two different configurations: b ↔ b
609 and b ↔ b , b MODEL AND TOOLS
In the following we will focus on transport properties of the monomeric, dimeric, trimeric and tetrameric LHCIIsystems. We study the dynamics of the open quantum system in the presence of dephasing processes (Haken- Stroblformalism) in the presence of recombination and trapping mechanisms. In this model we resort to the description usedin [5, 6], where recombination and trapping processes are modeled by adding non-Hermitian terms to the Hamiltonian.The effects of static disorder will be taken into account below. The equation of motion for the density matrix of themonomer subunit can be written as dρdt = − i ~ [ H monomer , ρ ( t )] + L deph ρ ( t ) − { H recomb + H trapping , ρ } ≡≡ L [ ρ ( t )] , (1)The free Hamiltonian of the monomer is a tight binding Hamiltonian and it is expressed in terms of the site energiesand couplings given in [10] H monomer = X m ǫ m | m ih m | + X m 612 are supposed to be singularly linked with other surroundingcomplexes H trapping ≡ − ik trap X m =610 , , | m ih m | , (4)or with respect to the two lowest exciton eigenstates that are localized in the a , a , a 612 sites: H trapping ≡ − ik trap ( | E ih E | + | E ih E | ) . (5)The main difference between the two pictures should be that with the site-based trapping mechanism the sink actson the output sites, including the case where only the highest exciton localized on the output states (number 5 infigure 4) is populated. By contrast, with the excitonic-based mechanism only the two lowest energy excitonic statesare involved in the trapping dynamics.The functionals we use to evaluate the efficiency of the transport are the efficiency η , defined as η = 2 k trap X m Z ∞ dt h m | ρ ( t ) | m i (6) c m − X 671 nm633 nm FIG. 5: Energy levels for the Hamiltonian of the monomer and the trimer. and the average transfer time τ , defined as τ = 2 η k trap X m Z ∞ dt t h m | ρ ( t ) | m i (7)The system dynamics and the efficiency η can be evaluated numerically by vectorizing the density matrix andconstructing the proper linear super operator associated to Lρ ( t ) → | ρ ( t ) ii L → L| dρ ( t ) dt ii = L| ρ ( t ) ii = ⇒ | ρ ( t ) ii = e L t | ρ (0) ii . L has been constructed using the identity | ABC ii = A ⊗ C t | B ii , where A,B and C are matrices of size n and |·ii isa vector of size n . We can now compute η in terms of L as η = − k X m hh m |L − | ρ (0) ii . Before passing to analyze the results of our simulations a comment on the monomer Hamiltonian is in order. TheHamiltonian given in [10] has been obtained after many optimization processes with the goal of faithfully reproducingthe 2-D spectroscopy experimental results. This implies that the actual eigenstates that can be derived by diagonalizingthe Hamiltonian are delocalized over groups of Chls which are sometimes different by the ones showed in fig. 4, wherethe relative pigment participations are derived via other optimization processes. The differences manly involve thehighest energy eigenstates of the b lumenal and stromal band. In our analysis, we stick to the experimentally optimizedHamiltonian and use its actual eigenstates; the spectrum of the monomer and the trimer is given in figure 5. Themaximal, minimal and average difference in energy for the monomer are ∆ min ≈ cm − , ∆ max ≈ cm − , ∆ Av =59 cm − ( k B T ≈ cm − at room temperature). The differences in energies for the dimer and the trimer are verysimilar.In order to study the dynamics of the monomer we make use of well established quantum information measures ofcorrelations. In particular, we measure the total amount of correlations between two subsystems of chromophores A and B with the quantum mutual information [20] I AB = ( S ) A + ( S ) B − ( S ) AB (8)where ( S ) X = P i λ i log λ i is the von Neumann entropy of the reduced density matrix ρ X of subsystem X evaluatedin terms of its eigenvalues { λ i } . As for the quantum correlations between subsystems composed by arbitrary numberof chromophores we use the negativity N AB [21] that in the single exciton manifold can be written as [22] N AB = vuut a + 4 k X n =1 N X m = k +1 | h n | ρ AB | m i | − a (9)where a is the element corresponding to the zero exciton subspace and A = 1 , · · · , k , B = k + 1 , · · · , N are twogeneric subsystems of chromophores. The quantum correlations between pairs of sites m, n are also measured bymeans of the concurrence [23] which in the single site exciton has the simplified form [24] C m,n = 2 | ρ m,m | . (10)In order to study the relationship between the dynamics of the above described correlations measures and thedelocalization of the exciton over the monomeric structure we define a measure of delocalization D ( t ) that involvesthe single site populations. In keeping with this paper’s methods of using information-based measures to characterizethe excitonic transport, we use Shannon entropy as a measure of delocalization. If h n i i is the population of site i attime t, then by using the normalized populations λ i = h n i i / P i h n i i we can define D ( t ) = − X i λ i log λ i (11)as the Shannon entropy of the normalized populations. The higher D ( t ), the flatter the probability distribution { λ i } ,the higher the delocalization of the exciton over the complex. MONOMER DYNAMICS In the following we describe the dynamics of the monomer. The values of the recombination and trapping coefficientsused for the simulations are the ones used for the FMO complex in [5, 18]. The recombination coefficient Γ = 10 − ps − takes into account the estimated lifetime of the exciton, 1 ns . The trapping coefficient is k trap = 1 ps − and is assumedto be equal for each output exciton state. Our results do not depend sensitively on the exact value of the excitonlifetime and the trapping rate: what is important for the analysis is that the exciton has a relatively long lifetimecompared with coupling rates, and that the trapping rate is comparable to those rates.We first focus on the time simulation of the evolution of the monomer in order to identify the possible energy transferpathways. We therefore fix the value of the dephasing rate γ φ ≈ ps − that corresponds to 77 K (temperature atwhich the experiments were done in [10]). The dephasing rate can be written in terms of the bath correlation functionas [6]: γ φ ( T ) = 2 π k B T ~ ∂ ω J ( ω ) | ω =0 = 2 π k B T ~ E r ~ ω c (12)where we have supposed to have an Ohmic correlation function J ( ω ) = π k B T E r ~ ωω c exp ω/ω c (super- and sub-Ohmiccorrelation functions give a similar dependence on T , E r , and ω c ). The recombination energy E r = 35 cm − and thecut-off frequency ω c = 150 cm − are chosen to be the ones used for the FMO simulations. Again, the qualitativebehavior of the excitonic transport does not depend sensitively on the precise values of E r and ω c . Energy and correlation pathways We want to describe the time evolution of the state of the monomer coupled with the environment. We first try toidentify and characterize the existence of two possible pathways, stromal-stromal and lumenal-stromal, by which theexciton, starting from a high energy state belonging to the b band, can reach the output sites. We therefore focus our aout-Stromalaint-Stromala-Lumenalab-Lumenal b-Stromalb-Lumenal FIG. 6: Schematic picture of the bipartitions used for the analysis of correlations dynamics attention on the behaviour of the system without any trapping and for a value of the dephasing rate that correspondsto the temperature of 77 K used in [10]. As noted above, since the Haken-Strobl model is purely dephasing and doesnot include an explicit relaxation term, we expect our analysis to underestimate the rate of transfer from high energystates to low energy states. Nonetheless, as will now be seen, pure decoherence gives efficient excitonic transportdown the energy ladder.The distinct pathways can be studied by plotting the following populations: i ) the populations of the excitonicstates which are mostly localized on Chl-b molecules that belong to the stromal (lumenal) side P bStrom ( P bLum ) ii ) thepopulations of the excitons states which are mostly localized on Chl-a molecules that belong to the stromal (lumenal)side P aStrom ( P aLum ).The different excitonic behaviors along the two distinct pathways can also be highlighted by using measures ofquantum correlations. In particular, we study the quantum mutual information between the relevant subsystems thatare naturally suggested by the energy landscape in fig. 4; in this way we can identify the correlation pathways andtheir dynamics. On the lumenal side we select the subsystems bL = { , } , abL = { , } , aL = { , } ,while on the stromal side we select the subsystems bS = { , , } , aintS = { , } , aoutS = { , , } and aS = aintS ∪ aoutS ; the bipartitions are schematically depicted in fig. 6. The growth of quantum correlationsbetween subsystems is a signature of the spreading of the initially localized exciton between subsystems. The formthat these correlations take over time reveals the mechanism of this spreading – an almost purely coherent initialpropagation followed by semi-coherent diffusion.For the stromal-stromal pathway we choose as initial state of our simulations the highest energy eigenstate | E i of the Hamiltonian, which is mostly localized on the b 601 Chl. In Fig.7 we see how the exciton mostly flows fromthe stromal b-band to the stromal a-band on a very short time scale ( ≈ ps ). On a slower time scale the populationpartially delocalizes over the lumenal band. The flow of population between the two b-bands was highlighted in theenergy pathway given in [8] (but not highlighted in [10]), where the possibility of a flow from the b-lumenal to theb-stromal band is estimated to be of the order of 2 ps . Here we observe the inverse passage b-stromal to b- lumenal and this is due also to the partial delocalization of the | E i on the b-lumenal sites. The global population decreasesbecause of excitonic decay with a time scale of the order of 1 ns .The dynamics can be further analyzed by considering the correlation pathways. Figure 8 refers to the stromal-stromal pathway starting with | E i . The plots shows that the dynamics mostly takes place on the stromal side. Inparticular (left plot) the bS subsystem initially gets correlated with a-band stromal sites as a whole ( M IaSbS ); in thefirst few picoseconds most of the correlations are established between the subsystems bS - aintS , and the subsystems aintS - aoutS . From the first picosecond on the bS sites get directly correlated with the output sites aoutS . Theright plot in fig. 8 shows that while there are correlations between the stromal and the lumenal b-bands ( M IbLbS ),the correlations among the subsystems on the lumenal side and the lumenal-stromal correlations at the level of thea-bands are negligible ( ≈ one order of magnitude smaller). This picture is consistent with a dynamics mostly localizedon the stromal side of the monomer. popu l a t i on FIG. 7: Stromal-Stromal pathway: populations for initial state | E i Mutual informationIntial state |E >Stromal side ps MIaSbS MIaintSbS MIaintSaS MIaoutSbS Mutual informationIntial state |E >Lumenal side ps MIbLbS MIbLabL MIabLaL MIaLaoutS MIaoutSbL FIG. 8: Stromal-Stromal pathway: mutual information for initial state | E i ; left: stromal side; right: lumenal side We now pass to analyze the behaviour of the system when the exciton starts on a a high energy eigenstate | E i which is mostly localized on the lumenal side (in particular on the site b a 604 (which is far from the other a-lumenal sites), andon the other hand to the large energy separation with respect to the excitons localized on the neighbouring lumenalsites b , b , b 607 (∆ E ≈ cm − eigenvalue 8 with respect to eigenvalue 9 in fig. 5). Indeed, while in the firstfew picoseconds, the population of the b-lumenal sites decreases in favor of the population of a-lumenal band, part ofthe b-lumenal population start to flow toward the b-stromal band. The overall effect is that the a-stromal sites, andtherefore the output sites, become populated with a smaller rate than in the stromal-stromal case.The trapping effect on the lumenal side can also interpreted in terms of correlation pathways. As it is shown in theright plot of fig. 10 the b-lumenal band is initially highly correlated with the b-stromal band (black line, right plot).On one hand this is due to the fact that the eigenstate | E i is partially delocalized on the sites b , b − ps , and, despite the small level of interaction ( ≤ cm − , see Ham) it is consistent with the contiguity of theb-lumenal and b-stromal sites in the monomer.While the correlations between b-lumenal and b-stromal bands rapidly decay in the first few picoseconds, there isnot a corresponding growth of correlations on the lumenal side: the b-lumenal and a-lumenal sites remain very poorlycorrelated among each other and with the rest of the a-stromal sites. At the same time there is a clear enhancementof the correlations on the stromal side, which become rapidly greater than those on the lumenal side, suggesting the0 popu l a t i on FIG. 9: Lumenal-Stromal pathway: populations for initial state | E i Mutual informationIntial state |E >Stromal side ps MIaSbS MIaintSbS MIaintSaS MIaoutSbS Mutual informationIntial state |E >Lumenal side ps MIbLbS MIbLabL MIabLaL MIaLaoutS MIaoutSbL FIG. 10: Lumenal-Stromal pathway: mutual information for initial state | E i ; left: stromal side; right: lumenal side activation of the stromal-stromal pathway.The above analysis is robust with respect to the choice of the initial state; for example in fig. 11 the same simulationshave been carried out for an initial state | b i fully localized on the b-lumenal site b ps the b-lumenal subsystem gets correlated both with the sites b , a 604 ( M IabLbL )and with the b-stromal band ( M IbLbS right plot); then the correlations are mostly built onthe stromal side (left plot) while on the same time scales the correlations with the output sites on the lumenal sideare build with a slower rate.The net effect of the presence of the above mentioned bottleneck on the lumenal side is a substantial slowdownof the lumenal-stromal dynamics. There are a number of possible functional advantages for this slowdown. Onepossibility is to assist in photoprotection, the elimination of triplet exciton states that can create harmful singletoxygen. This elimination takes place primarily by the transfer of triplet excitons to triplet carotenoid states, and hasa relatively slow timescale (a fraction of a microsecond) compared with singlet exciton transfer [19]. Triplet transferis a Dexter process, mediated by wave function overlap, and requires the carotenoids to be physically close to thechromophore carrying the triplet. X-ray crystallography studies of LHCII suggest that the a 604 chromophore in thelumenal bottleneck is close enough to a lutein carotenoid to allow triplet transfer, a process confirmed by spectroscopy[17, 19].A second possible function for the lumenal bottleneck is that the bottleneck b 605 chromophore could mediateexcitonic transport from one trimer to another [17]. This chromophore ‘sticks out’ from the others in the LHCIIcrystallographic structure, giving both weaker couplings to the other chromophores within the LHCII monomer,and potentially stronger couplings to chromophores in neighboring triples. Crystallographic investigations of LHCII1 Mutual informationIntial state |b607>Stromal side ps MIaSbS MIaintSbS MIaintSaS MIaoutSbS Mutual informationIntial state |b607>Lumenal side ps MIbLbS MIbLabL MIabLaL MIaLaoutS MIaoutSbL FIG. 11: Lumenal-Stromal pathway: mutual information for initial state | b i ; left: stromal side; right: lumenal side suggest that the b 605 is positioned to transfer excitons to the b 606 and a 614 chromophores of neighboring trimersin LHCII aggregates [17]. This transfer pathway could also participate in photoprotection via non-photochemicalfluorescence quenching.A third possible reason for the slower lumenal pathway is that it might allow two excitons to propagate throughthe LHCII complex simultaneously without quenching. The weak coupling between the bottleneck chromophores ofthe lumenal pathways and the chromophores of the stromal pathway, together with their relative spatial separation,could allow an exciton localized in the lumenal pathway to wait for a stromal exciton in the same complex to passthrough the stromal trapping states, before passing through itself.To summarize, the dynamics of the slow lumenal side and the fast stromal side have many have a rich set ofpotential biological functions. These dynamics are in turn based on a rich quantum structure, which the next sectionelucidates. Delocalization and quantum correlations In the following we examine the monomer dynamics from the point of view of the time scales that characterize thedelocalization of the excitons over the whole monomeric structure and the quantum correlations between the varioussubsystems. In order to estimate the delocalization time we plot the populations of the various bands in fig. 12(left plot), and the previously defined measure of delocalization D ( t ) (11). Since we want to study the dynamics ofthe spreading of the correlations we first focus on the initial state | b i localized on the Chl b 601 only. We choosethis state because it is very close to the highest energy eigenstate | E i , which is mostly localized on the same sitebut has non-negligible quantum correlations with the rest of the system. We want to start with a state localized ona single chromophore, in order to study how the correlations spread through the structure. In fig. 12 we plot thedelocalization function D ( t ) for γ φ = 3 ps ( ≈ K ), Γ = 0 . 001 and no trapping. The delocalization has a very fastgrowth and it can be well represented by a function D ( t ) = y A − t/t + A − t/t where two time scales t ≈ f s and t ≈ . ps appear. The relevance of these time scales can be understood by studying the quantumcorrelations between the relevant subsystems.In the left panel of fig. 13 we plot the mutual information for the relevant subsystems over a time of 10 ps . As itcan be easily seen, the second time scale t ≈ . ps can be correlated with the growth of the mutual informationbetween the various subsystems which reach its maximum at a time t very close to t . In particular the correlationsbetween the b-band and the a-band on the stromal side has a maximum at t ≈ . ps . In the right panel of fig. 13 weshow the dynamics of correlations within the first picosecond. Here the dynamics displays an initial fast growth of allcorrelations ( ≈ . ps ) which are characterized by an oscillating behaviour; the time scale of these oscillations is ofthe order of 0 . ps . These oscillations are signatures of the high degree of quantum coherence in the initial spreading.The growth becomes regular within the first 400 f s ≈ . t ; within the same period of time the correlations spreadtoward the lumenal side ( M IbLbS )2 bStromal aStromal bLumenal aLumenal TotPop FIG. 12: Initial state | b i ( ≈ | E i ), state localized on the Chl b 601 on the stromal side. Populations (left) and delocalization(right) D ( t ) with fitting curve. t max MI bSaS = 2.63 ps t deloc1 = 0.25 ps t deloc2 = 2.68 ps ps MIaSbS MIaintSbS MIaintSaS MIaoutSbS MIbLbS , MIaSbS MIaintSbS MIaintSaS MIaoutSbS MIbLbS FIG. 13: Initial state | b i ( ≈ | E i ), state localized on the Chl b 601 on the stromal side. Mutual information betweensubsystems on the stromal side and between b-lumenal and b-stromal band ( MIbLbS ); the right plot displays the dynamicsduring the first 1 ps . The overall effect can therefore be described in terms of the two relevant timescales: the delocalization takes placeover the structure with an initial fast rate, and subsequently it grows toward its asymptotic behaviour with a slowerrate. The time scale of the initial rapid and oscillating growth is consistent with the dynamics of the quantumcorrelations present in the system. In fig. 14 (left plot) we show the behaviour of the negativities among the relevantsubsystems over the first 10 ps (NegbLbS is the negativity between the lumenal and the stromal b-bands). In the firstpicosecond (right plot), after a first rapid growth ( ≈ f s ) they show the same oscillatory behaviour of the mutualinformation and they then decay in a smooth way after the first ≈ ps (left plot). This behaviour is also shown bythe concurrences between sites: in fig. 15 we show the relevant (non-negligible) concurrences between the site b NegaSbS NegaintSbS NegaintSaS NegaoutSbS , NegaSbS NegaintSbS NegaintSaS NegaoutSbS NegbLbS FIG. 14: Initial state | b i ( ≈ | E i ), state localized on the Chl b 601 on the stromal side. (Un-normalized) Negativitiesbetween subsystems on the stromal side and between b-lumenal and b-stromal band ( NegbLbS ); the right plot displays thedynamics during the first 1 ps . ps C1_2 C1_3 C1_6 C1_11 FIG. 15: Initial state | b i ( ≈ | E i ), state localized on the Chl b 601 on the stromal side. Relevant, i.e. non-negligible,concurrences between site b 601 and other sites on the stromal ( b b 11) and lumenal ( b ps . Effect of the trapping on the monomer dynamics In order to see how the trapping modifies the monomer dynamics we focus on the stromal-stromal dynamics. Theinitial exciton state is | E i and we choose to fix the value of the dephasing rate to γ φ = 12 ps − (that approximatelycorresponds to the ambient temperature if one uses 12). In fig. 16, the left plot displays the populations of thevarious subsystems. The comparison of the populations dynamics with the site based trapping mechanism and theexciton based mechanism shows that the first one is obviously more efficient in reducing the total population and thepopulation of the various band (in particular the a-lumenal one). As noted above, this difference arises in our modelbecause site-based trapping operates on three sites, while the exciton based mechanism acts only when the two lowesteigenstates get populated.In the right plot of fig. 16 we show the delocalization functional for no trapping and for site/exciton based trapping.4 bStromal aStromal bLuminal aLuminal TotPop DExcTrap DSitesTrap DNoTrap FIG. 16: Initial state | E i . Dephasing γ φ = 12 ps − ; trapping k trap = 1 ps − ; recombination Γ = 0 . ps − . Left: populationsof the different bands with exciton/site base trapping; right: delocalization functional D ( t ) with exciton/site base trapping andno trapping The presence of the trapping mechanism starts to be relevant already after the first picosecond, i.e. before the fulldelocalization of the exciton has occurred. Indeed for γ φ = 12 ps − the typical time scales of the delocalization withouttrapping are t ≈ f s and t ≈ . ps . This means that, in presence of trapping, the delocalization due to the initialdynamics of the quantum correlations has a fundamental role in the energy transfer process. The monomer as wire An interesting problem is to determine the typical time scales that govern the delocalization of an exciton initiallylocalized on the output sites a , a , a 612 of the monomer. These quantities become relevant when one describesthe behaviour of the dimer and the trimer when they act as quantum wires. Indeed, the LHCII complex can inprinciple be activated by other neighbouring LHCII complexes and this should happen when the output sites of apair of LHCII are sufficiently close. In this process, the a a a 612 sites on an LHCII monomer acceptan exciton from the same sites on a neighboring trimer. The exciton then spreads first through the monomer, andthen throughout the three LHCII units of the trimer. When it reaches another set of output sites, the exciton can betransferred to another trimer, and the process repeats.Accordingly, we now analyze the interaction between donors a , a , a 612 on one monomer within the trimer,and “acceptor” a , a , a 612 sites on a second monomer within the trimer. The acceptor LHCII sites behaveas a sink with a trapping rate that depends on how fast the exciton localized on those “acceptor” sites diffuses toa neighboring LHCII trimer or to some other part of the overall photocomplex. In order to study this diffusionprocess, we first focus on a single monomer, initialized in the monomer Hamiltonian ground state | E i (localized on a , a , a D ( t ) which is plotted in the right part of fig. 17.The simulations employ a fixed value of the dephasing rate γ φ = 12 ps − that approximately corresponds to ambienttemperature. The population of the acceptor sites decreases and becomes of the same order of the other populationsin about 10 ps . If we take D ( t ) = y A t/t + A t/t to estimate the delocalization process over the wholemonomer ( D ( t ) is a site based measure of delocalization) we see that the relevant time scale t ≈ ps (while for theinitial fast growth we have t ≈ f s ).The value of t is of the same order of the inverse of the trapping rate which is characteristic of the FMO k − trap = 1 ps that we used for the monomer in the previous section, and that we will use for the dimer, the trimer and the tetramerin the following section.5 ps aoutS aintS bStromal aLumenal blumenal totPop FIG. 17: Initial state | E i , ground state of the monomer localized over the output sites. Populations of the different subsystemsand delocalization functional D ( t ) EFFICIENCY: A COMPARISON BETWEEN CLUSTERS OF MONOMERS In this section we examine the efficiency and typical transfer time of the single monomer and of groups of 2, 3,and 4 monomers. The clusters are build by connecting monomeric subunits via a site-site interaction: as pointed outin [10] there is evidence of a relatively strong coupling in the cluster { b , b , b } , where b 601 belongs to onemonomer and { b , b } to another one. In [17], where the LHCII of the spinach is analyzed, the link between the b 601 and the b 609 sites is estimated to be 42 cm − . We therefore write the overall Hamiltonian of the complex, saythe trimer, as: H trimer = X i =1 , , H imonomer + H int + H int + H int , (13)where the terms H intij = V b i ,b j ( | b i ih b j | + | b j ih b i | account for the Coulomb interaction between theChl b 601 on the i − th monomer and the Chl b 609 on the j − th monomer. V b i ,b j is chosen to be equal to 42 cm − [10]. We will also consider the case when the coupling b − b 608 is present and equal to 42 cm − .The recombination parameter is fixed at Γ = 0 . ps − , which again is the value used for the FMO in [6]. Trappingis supposed to be similar to the monomeric case; the output sites are now the group of sites a , a , 612 on eachmonomer, and the trapping Hamiltonian is P i k trap ( | a i ih a i | + | a i ih a i | + | a i ih a i | ), where i labelsthe monomers, with k trap = 1 ps − .The main result of these simulations is that there is a clear evidence for a dephasing-assisted mechanism thatenhances the transport efficiency of the systems. This mechanism is an example of environmentally assisted quantumtransport (ENAQT) [5–7]. The result is independent of the structure analyzed. Efficiencies and the average transfertimes have their optimal values in correspondence of a dephasing rate γ φ ≥ ps − , i.e. the rate corresponding toambient temperature. The antenna configuration We first describe the efficiency of the various complexes when they act as antennae. We start by focusing on therelevant figures of merit for the single monomeric unit. In fig. 18 we show the efficiency (left) and the average transfertime (right) for two different initial states: | E i , mostly localized on the stromal side and | E i , mostly localizedon the lumenal side. The plots show that the differences highlighted by our analysis of the dynamics in the previoussection have relevant effects also in terms of the transport efficiency of the monomer. The stromal-stromal pathwaythat starts with | E i results in general in a better efficiency and a smaller τ over the whole range of dephasing rates.In particular, at γ φ ≥ τ is ≈ . −3 −2 −1 −1 ] h |E >|E > 10 −3 −2 −1 −1 ] t |E >|E > FIG. 18: Efficiency η ( γ φ ) (left) and Average transfer time τ ( γ φ ) (right) for the monomer with different initial states: | E i (mostly localized on the lumenal side) and | E i (mostly localized on the stromal side). We now describe in detail the results of our simulations for more complex structures. We first examine the case inwhich there is an active sink attached to each of the monomers of a given structure. The initial state used for thevarious simulations is the highest excited state of the Hamiltonian, which is basically a copy over different monomersof the eigenstate | E i of one single monomer: it is thus mainly localized on the b-stromal Chls of each structure. InFig. 19 (left) we compare the efficiency of the energy transport for different geometries in presence of only one inter-monomeric coupling. While the dephasing assisted mechanism is always present, we see that for physically relevantvalues of the dephasing parameter the change of geometry does not provide significant modifications in the efficiency.Fig. 19 (right) shows the result of the simulations for τ , the characteristic transfer time. The value corresponding tothe highest efficiency in the transport is around 10-15 ps. This transport time is consistent with that observed forthe FMO complex, taking into account the fact that each monomer subunit of the LHCII has twice the number ofchromophores of FMO.The results we obtain in this paper are robust with respect to the introduction of static disorder. In Fig.20 we addthe effects of static disorder (on site-energy values) and compare the efficiency for different numbers of monomers.The strength of disorder is taken to be that reported in [26]. The qualitative behavior of the efficiency with disorderis the same as that without. The monomer is more affected by disorder than the dimer and trimer, indicating greaterrobustness for the more complicated geometries. This is consistent with the fact that fluctuations due to disorder arestronger for systems of smaller size.Differences between the various geometries can be observed when the inter-monomeric coupling is supposed to bepresent between the b601 Chl in one monomer and both b608 and b609 in the neighbouring monomer, see fig. 21.The efficiency of the clusters of monomers benefits from this kind of coupling: η is enhanced with respect the singlemonomer case over a wide range of dephasing values. Moreover, while the dimer is slightly less efficient than the trimerand the tetramer structures that have a higher number of traps, our simulations suggest that there is no advantage inadding more than three subunits: the trimer behaves just as well as the tetramer. This results could be an indicationfor a functional selection of the trimeric configuration with respect to the other ones: the trimer could be the resultof an optimization with respect to the “cost” of the structure.Interesting differences between the behaviour of the different structures appear when one considers a variable numberof active sinks. In Fig. 22 we compare the transport in monomeric and dimeric, trimeric and tetrameric antennas withonly one inter-monomeric coupling, and with different numbers of sinks attached to the available monomeric subunits.The initial state in these simulations is always the highest excited state of the structure. The main feature here isthat the efficiency at physiological temperatures is always greater for the structure with a number of sinks equal tothe number of monomers. For example, with a single sink attached, the monomer is more efficient than the dimer,trimer and tetramer. This behaviour is reasonable since the exciton is initially delocalized over the whole structureand in particular over those monomers that do not have any sink attached. A qualitative explanation of the relativebehavior for the different configurations is based on the competition between an enhancement of the efficiency due toa greater number of sinks, and the depletion of efficiency due to a greater number of chromophores, where the exciton7 −3 −2 −1 −1 ] h −3 −2 −1 −1 ] t FIG. 19: Efficiency η (left) and average transfer time τ ( ps ) (right)for the different topologies (monomer, dimer, trimer andcomplex with 4 monomers) with single intra-monomeric coupling for various values of the dephasing rate γ φ . Initial state:highest energy eigenstate of the given structure. −2 −1 ] h averaged monomeraveraged dimeraveraged trimermonomerdimertrimer FIG. 20: Efficiency η for different topologies (monomer, dimer, trimer) in the presence of static disorder, varying the dephasingrate γ φ . Solid lines represent the averaged value, while dashed lines represent the result for the non-disordered Hamiltonianthat we use in this work. can delocalize and dissipate in the bath, causing a slow down in the funneling process. As already noticed, Fig. 19,the trimer saturates the enhancement of efficiency due to the number of exits: the tetrameric complex with four sinksand the trimer complex with three sinks have the same efficiency. The wire configuration We now analyze the multi-monomer structures when they are used as “wires”. As noted above, the LHCII complexescan function as connecting structures between different units of the PSII complex. In the following we analyze thiscase by fixing as the initial state of the dynamics the ground state | E i of one single monomer in the cluster and byconnecting a sink to each of the other monomers in the complex.Fig. 23 shows the efficiency and the characteristic time of the dimer, the trimer and of a cluster of four monomers8 −3 −2 −1 −1 ] h −3 −2 −1 −1 ] t FIG. 21: Efficiency η (left) and average transfer time τ ( ps ) (right)for the different topologies (monomer, dimer, trimer andcomplex with 4 monomers) with two intra-monomeric coupling for various values of the dephasing rate γ φ . Initial state: highestenergy eigenstate of the given structure. −2 −1 −1 ] h D1D2T1T2T3Q1Q2Q3Q4monomer 10 −1 −1 ] h T12T13T23Q12Q13Q14Q23Q24Q34dimer 10 −2 −1 −1 ] h Q123Q124Q234Q314trimer FIG. 22: Efficiency of the transport for dimer, trimer and tetramer clusters, with one inter-monomeric coupling, acting asANTENNAS In the legend Q23 means that we consider a tetrameric complex where the exciton is captured in the firstmonomer, and can exit only from the II and III monomers. Analoguosly for the trimer (T) and the dimer (D). coupled together with only one inter-monomer connection. The simulation suggests again a special role played bythe trimer. It is significantly more efficient than the dimer, but almost indistinguishable from the cluster of fourmonomers. In the wire configuration, the exciton passing through the dimer has only one trapping site that it canreach, while in the trimer, tetramer, and higher order polymers, the exciton entering at the lowest energy site has twotrapping sites that it can reach – those on the two monomers adjacent to the entrance site. Consequently, the efficiencyof trapping is higher for the trimer and tetramer than for the dimer: two traps are better than one. Moreover, thedynamics for passing from the entrance site to those two adjacent traps are identical for the trimer, tetramer, andhigher order polymers. Consequently, the efficiency profiles for the trimer, tetramer, and higher order polymers areidentical.We finally describe how the number of attached sinks modifies the overall efficiency of the different quantum wires.In Fig. 24 the simulations refer to a situation where the initial state of the structure is the ground-state | E i localizedin the a , a , a 612 Chls of a single monomer of the structure. Here again we see a behaviour similar to theantenna confirguration case: at fixed number of sinks attached the efficiency is higher for those structures with asmaller number of chromophores . The trimer is again a limiting case, it is always more or as efficient as the tetramer,and, as already shown in fig. (23) the trimer with two sinks has the same performance of the four-monomer structurewith three sinks. An analogue comparison holds true for the dimeric vs trimeric cluster. When only one sink is active,at the relevant values of dephasing – i.e. where the efficiency is maximal – the dimer can perform slightly better than9 −3 −2 −1 −1 ] h −3 −2 −1 −1 ] t FIG. 23: Efficiency η (left) and average transfer time τ ( ps ) (right) for the different topologies (dimer, trimer and complex with4 monomers) with single intra-monomeric coupling for various values of the dephasing rate γ φ . Initial state | E i , i.e. groundstate of a single monomer, localized on the ”acceptor” sites a , a , a 612 . −2 −1 −1 ] h Q23Q24D1to2D2to1Q2Q3T2Q4T3T23 FIG. 24: Efficiency of the transport for dimer, trimer and tetramer clusters, with one inter-monomeric coupling, acting asWIRES. In the legend Q23 means that we consider a tetrameric complex where the exciton is captured in the first monomer,and can exit only from the II and III monomers. Analoguosly for the trimer (T) and the dimer (D). the trimer. From our simulations it is also evident that, within our assumptions for the inter-monomer links, thevarious structures show a directionality of transport. For example the efficiency of the dimer changes significantlyover a wide range of dephasing values depending on the direction of the energy flow. CONCLUSION We have analyzed excitonic transport in the primary component of the photosynthetic apparatus in green plants:LHCII. By means of a simple decoherence model (Haken-Strobl), the analysis shows that an exciton initially localizedon a single chromophore moves through the LCHII photocomplex in a two-step process which clearly signaled by a0quantum information motivated measure of delocalization. First, over the timescale of a few hundreds of picosec-ond, the exciton spreads coherently to neighboring chromophores. This coherent spreading exhibits rapid quantumoscillations and entanglement. Second, as the environment decoheres the exciton’s position, the exciton diffuses semi-coherently throughout the complex over a timescale of ten to twenty picoseconds. Although the Haken-Strobl modeldoes not include relaxation, we expect this two-step, coherent–semicoherent model to hold for more detailed modelsof the dynamics as well, for example, in the full hierarchy approach of [25].Detailed comparison of different measures of correlations also allows us to identify how the two main downhillrelaxation pathways unveiled by the experiments (stromal-stromal, lumenal-stromal) can be dynamically described interms of correlations pathways: the correlations among subsytems of pigments are mostly established on the stromalside of the monomeric complex even when the excitation is initially localized on the lumenal side. This behaviour hasimportant consequences when the efficiency of the transport is considered: the stromal pathway is in general moreefficient than the lumenal one.In general, the analysis shows that even in the absence of relaxation, pure dephasing induces effective transportdown the LHCII energy funnel. The transport is efficient and robust in the presence of static disorder, and exhibitsthe characteristic signature of characteristic environmentally assisted quantum transport (ENAQT) – low efficiency atlow temperature due to transient localization, followed by a robust maximum efficiency at physiological temperature,with a falling off of efficiency at very high temperature. In the second part of the paper, we compared the efficiencyof transport through LHCII structures with different topologies (monomers, dimers, trimers, and tetramers) anddifferent configurations (antenna and wire). We find that the efficiency of transport increases as the number ofsubunits increases, saturating at the level of the trimer. These results provide evidence for the functional selection ofthe trimeric configuration.PG would like to thank A. Ishizaki, L. Valkunas and T. Mancal for useful discussions and suggestions.PZ acknowledges support from NSF grants PHY-803304, PHY-0969969 and DMR-0804914. [1] R.E. Blankenship, Molecular Mechanisms of Photosynthesis, Blackwell Science Ltd., London, 2002.[2] G.S. Engel, T. R. Calhoun, E. L. Read, T. K. Ahn, T. Mancal, Y. C. Cheng, R. E. Blankenship and G. R. Fleming, Nature , 782 (2007).[3] E. Collini, C. Y.Wong, K. E.Wilk, P.M. Curmi, P. Brumer, and G. D. Scholes, Nature , 644 (2010).[4] G. Panitchayangkoon, D. Hayes, K. A. Fransted, J. R. Caram, E. Harel, J. Wen, R. E. Blankenship, G. S. Engel, Proc.Nat. Acad. Sci. , 12766 (2010).[5] M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, J. Chem. Phys. , 174106 (2008).[6] P. Rebentrost, M. 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