Quantum Redirection of Antenna Absorption to Photosynthetic

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Quantum Redirection of Antenna Absorption to Photosynthetic Reaction Centres Felipe Caycedo-Soler, Christopher A. Schroeder, Caroline Autenrieth, Arne Pick, Robin Ghosh, Susana F. Huelga, and Martin B. Plenio J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b02714 • Publication Date (Web): 29 Nov 2017 Downloaded from http://pubs.acs.org on December 1, 2017

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The Journal of Physical Chemistry Letters

Quantum Redirection of Antenna Absorption to Photosynthetic Reaction Centres Felipe Caycedo-Soler,∗,† Christopher A Schroeder,†,¶ Caroline Autenrieth ,‡ Arne Pick,† Robin Ghosh ,‡ Susana F. Huelga,† and Martin B. Plenio∗,† Institute of Theoretical Physics and Integrated Quantum Science and Technology IQST, University of Ulm, Albert-Einstein-Allee 11, D - 89069 Ulm, Germany, and Department of Bioenergetics, Institute of Biomaterials and Biomolecular Systems, University of Stuttgart, Pfaffenwaldring 57, D - 70569 Stuttgart, Germany E-mail: [email protected]; [email protected]

∗ To

whom correspondence should be addressed of Theoretical Physics and Integrated Quantum Science and Technology IQST, University of Ulm, Albert-Einstein-Allee 11, D - 89069 Ulm, Germany ‡ Department of Bioenergetics, Institute of Biomaterials and Biomolecular Systems, University of Stuttgart, Pfaffenwaldring 57, D - 70569 Stuttgart, Germany ¶ Joint Quantum Institute, Department of Physics, University of Maryland and National Institute of Standards and Technology, College Park, MD 20742, USA † Institute

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The Journal of Physical Chemistry Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract The early steps of photosynthesis involve the photo-excitation of reaction centres (RCs) and light-harvesting (LH) units. Here, we show that the –historically overlooked– excitonic delocalisation across RC and LH pigments results in a redistribution of absorption amplitudes that benefits the absorption cross section of the optical bands associated with the RC of several species. While we prove that this redistribution is robust to the microscopic details of the dephasing between these units in the purple bacterium Rhodospirillum rubrum, we are able to show that the redistribution witnesses a more fragile, but persistent, coherent population dynamics which directs excitations from the LH towards the RC units under incoherent illumination and physiological conditions. Even though the redirection does not seem to affect importantly the overall efficiency in photosynthesis, stochastic optimisation allows us to delineate clear guidelines and develop simple analytic expressions, in order to amplify the coherent redirection in artificial nano-structures.

Photosynthesis – the conversion of sunlight to chemical energy – is fundamental for supporting life on our planet. Despite its importance, the physical principles that underpin the primary steps of photosynthesis, from photon absorption, to excitonic dynamics and electronic charge separation, remain to be understood in full. Excitonic delocalisation between tightly-packed pigments, such as within the RC or within the LH units, has been recognised to be of considerable importance for characterising their individual optical responses and for determining the associated time-scales for excitation energy transfer steps. 1–6 More recently, the study of coherent effects in these biologically relevant systems has attracted increasing attention due to the observation of long-lasting 2


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oscillatory signals measured with optical time-resolved techniques. 7–13 These results have driven a wave of theoretical work aimed at understanding the microscopic mechanisms that may underpin persistent coherent signals 14–20 and encouraged the discussion of the significance of coherent dynamics for efficient energy transfer. 21–25 These studies, though, have discussed less on the impact of the coherent RC-LH dynamics on optical spectra, likely biased by the observation that energy migration between RC and LH units is mainly driven by incoherent excitonic transfer. 25 As we will show here, the theoretical examination of the rapid time-scale inherent of absorption spectra underlines its importance as a useful tool to encode in the amplitudes of these spectra -and not in their shifts-, the effects of the moderate coupling between RC and LH pigments, despite the aggressive dephasing environment intrinsic to these photosynthetic complexes. Across several species, we find a noticeable redistribution of absorption amplitudes, increasing the bands associated to RC transitions when the RC pigments interact with the LH units, with respect to these bands from the isolated RC pigments. We show that the absorption redistribution indicates a subtle phenomenon present under physiological conditions, which we refer to as population redirection, and that represents an increase in the RC population driven by coherent dynamics, albeit is triggered by incoherent illumination. Although this redirection in photosynthetic structures is small with respect to the population driven by the subsequent incoherent dynamics, we are able to show how it can be largely amplified in artificial devices. Our model system for exemplifying the role of RC-LH coherent dynamics, is the core complex of the purple bacterium R. rubrum, 26 composed of an LH (LH1) ring of 16 dimers NLH = 32 pigments) of bacteriochlorophyll pigments (BChl) and a “special pair” (P, with NRC = 2 pigments) of BChl molecules which mediate the primary process of light-induced charge separation, presented in Fig.1 A. The fundamental principle that motivates this work is illustrated in this Figure in which we stress the several time-scales affecting the dynamics of excitations in the core complex. In the very early stages of these dynamics, the coherent interaction between LH and RC pigments results in excitonic delocalisation extended over RC and LH1 pigments. Thereby, observables like the absorption spectra, which as it will be clearly stated, are able to probe such a short time-scale, will

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Absorption

2

1 0.5

0.5

0

50

-0.2 0 840

λ(nm) 900

840

47

0.2

-0.4

0

860 880 λ(nm)

900

Absorption

B

850 900 λ(nm)

48

0.4

∆ A( λ)

1

C

1.5

0

A(λ)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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D

46 45 44 43

875 880 885 λ(nm)

Figure 1: Temporal evolution of coherence and spectral signatures of full core complex delocalisation A Schematic representation of the R. rubrum RC-LH1 BChl pigments, with excitons that within a few tens to hundred femtoseconds delocalise over the entire core complex, before environment-induced dephasing forces excitations to reside either on the antenna or the RC (blue shading). B Spectra from isolated RC (blue theory, circles experiment). In red is presented the difference between the calculated RC-like and the isolated RC spectra (red). The inset in B shows the calculated (continuous) and experimental (circles) spectra of isolated LH1. In C RC (blue) and RC-like (red) spectra are compared. In D the LH1 (blue) and LH1 -like (red) spectra are shown. In C and D yellow areas and arrows highlight the magnitude and the direction of the absorption redistribution due to the RC-LH1 coherent coupling. All spectra are normalised to the maximum height of the P870 peak.

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provide specific signatures that allow to determine the impact of the short lived coherent RC-LH exchange. In all the simulations of the core complex that we present, we use the excitonic Hamiltonian

H =

N X

ωi |ii hi| +

i

=

N X η=α,β

N X

Ji j (|ii h j| + | ji hi|)

i, j

ωη |ηi hη| +

X

Vα,β (|αi hβ| + |βi hα|) =

X α0

αRC βLH

ω0α α0 hα0| ,

(1)

that considers all Qy transitions to exciton states |ii of the 32 LH1 and the two P pigments. Due to the fact that recently published spectra confirmed that purple bacteria ecosystems of peat lakes and costal waters are dominated by red and near-IR light, 27 the near-IR Qy transition is the most relevant in R. rubrum under physiological conditions. This Hamiltonian can also be expressed P P β in terms of, either, excitonic eigenstates |αi = iNRC cαi |ii, |βi = iNLH ci |ii in the absence of the RC-LH coherent interaction Vα,β and therefore delocalised over either the RC or the LH1, respecP tively, or, in terms of the full core complex excitonic eigenstates |α0i = i cα0 i |ii delocalised over the entire core complex, i.e., over the RC and the LH1 pigments. In this article we use primed variables to denote quantities associated to |α0i states, greek unprimed for those related to |ηi = |αi , |βi states, and latin letters to denote pigments. The states |α0i will be labelled as RC-like or LH1-like states, since they still present a delocalisation which extends mostly over the RC or LH1 pigments, respectively, given that their mutual coupling Vα,β is smaller than the energy gap ωα − ωβ . The coherent excitonic dynamics provided by the Hamiltonian eq.(1) are degraded by relaxation and dephasing, captured by a master equation ∂t A = −i[H, A] + LA = UA with a dissipative part in P the Lindblad form LA = k Ok AO+k − 21 O+k Ok A − 12 AO+k Ok , used for a Markovian (memory-less) relaxation. A realistic model must resort in experimental observations. We model with operators √ Oη α,β = γη |ηi hη| pure dephasing between excitons |αi or |βi, and intra-ring incoherent transfer √ with Oη β,ν β = γη,ν |ηi hν| for η , ν (rates γη,ν proportional to the spectral density at the transition frequency ωη − ων fulfilling detailed balance). Our estimates, based upon the rates and processes

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identified by experiments, 28 lead to γα = 254 cm−1 , γβ = 172 cm−1 and γη,ν tuned such that it reflects the experimental observation 29 of equilibration in the LH1 ring within 400 fs (see SI for further details). Results correspond to averages of stochastic realisations of ωi and couplings Ji,i±1 in equation (1), taking into consideration the full set of pigments of the LH1 and P. These inhomogeneities, termed static disorder, are complementary to the homogeneous broadening obtained by the rates γη and γη,ν , for the full width of optical bands. The redistribution of absorption amplitudes: robust signatures of RC-LH moderate coupling The absorption spectrum A(ω = 2πc/λ) of the Qy transition of the P pigments peaks around λ = 870 nm (circles in Figs. 1B and D), whereas the LH1 exhibits a single absorption band at λ = 880 nm (inset in Fig. 1B and Fig. 1D), denoted the P870 and B880 bands, respectively. The single LH1 band arises mainly from a doublet of states which are bright due to the pigments’ circular arrangement with the individual pigments’ Qy transition dipoles d~i arranged almost tangentially to the LH1 circumference. 30,31 A relevant observation of this article is presented with the red line in Fig. 1B: an important difference in the absorption spectra between the full core complex (Vα,β , 0) and the addition of individual RC and LH units (Vα,β = 0), is obtained. This difference is commensurate to the isolated RC spectra, also shown in this Figure, and may arise, both, from the energy shifts of the frequencies ω0α with respect to the uncoupled RC and LH1 systems frequencies ωα , ωβ , and/or, from the increase or decrease in the amplitude of absorption in the full core for specific wavelengths. The most straightforward tool to indicate the origin of this difference is a proper deconvolution of the full core complex spectrum into LH1-like and RC-like contributions, compared with the isolated RC and LH1 spectra. Calculations of absorption spectra were performed by direct Fourier transform of the evolution super-operator F [eUt ] = 1/(iω + U) 15 and a RC-like LH1-like deconvolution was performed by the partition of this evolution kernel based upon |α0 i and |β0 i states (see SI for further details). Figures 1C and D show spectra from isolated RC and LH1 with a considerably different am-

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plitude than those from the calculated RC-like and LH1-like spectra of the interacting core complex, with an increase of about 60 % in the area of the P870 band when the RC is within the core complex. On the other hand, peak maxima from isolated RC and LH1 spectra do not differ substantially from the RC-like and LH1-like spectra. Equivalent amplitude enhancements are obtained in Fig. S4 for different phenomenological dephasing models, which stand for a site dephas√ ing (full homogeneous width given by operators Oi = γi |ii hi|) or collective dephasing scenarios p (constructed by operators Oα0 = γα0 |α0 i hα0 | for the dephasing contribution). The absorption R∞ spectrum, A(ω) = 0 DDCF(t)eiωt dt is the Fourier-Laplace transform of the dipole-dipole correlation function DDCF(t) = Tr{DeUt Dρ ss } and depends on the action of the dipole moment operator P D = i d~i · Eˆ |ii h0| + h.c on the stationary state ρ ss . Under moderate laser intensity excitation (with P a field polarised along the Eˆ direction), Dρ ss = i d~i · Eˆ |ii h0|. The robustness of the increase in amplitude of the P870 band can be explained by the fact that DDCF(t) is the time evolution of the these optical coherences, thereby dependent on the magnitude of the dephasing between ground and excited states, which is equal in all cases (discussed in detail in the SI). Since the optical coherences display many oscillations with a period of 2-3 fs, they are able to imprint the details of the moderate coherent interactions, before the dephasing processes with a much longer 70-100 fs time-scale 7,8,28 take over the dynamics. The similarity of the result for the three dephasing models opens up the possibility to trace back the spectral changes with any of the models studied. In the collective dephasing model, DDCF(t) = P 0 2 (iω0α −γα0 )t, which results in an absorption spectrum A(ω) = P γ|D0 |2 /[γ2 + (ω − ω0 )]2 for α |Dα | e α0 α α a single realisation within the inhomogeneous ensemble. Hence, the increase of the absorption cross section of the P870 stems from the difference between |D0α |2 and |Dα |2 . In order to obtain this difference, for the moment, let us consider a single bright state of the P870 band and a single bright state of the B880 band, with an energy difference ∆E and which couple according to the Hamiltonian H = ∆E |P870i hP870| + V(|P870i |B880i + |B880i hP870|).

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(2)


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Typically the coupling V between bright P870 and B880 states is smaller than our calculation between the brightest LH1 and RC states in absence of static disorder, V ≈ 13 cm−1 . The energy difference of of P870 and B880 bright states will be commensurate to the gap between the peaks of RC and LH1 absorption spectra, ∆E ≈ 130 cm−1 . The above Hamiltonian leads to the RC-like dipole strength |D0P870 |2 , |D0P870 |2 = | cos θDP870 + sin θDB880 |2

(3)

where the mixing angle θ = 21 arctan(2V/∆E) will be, in general, small. Accordingly, expanding eq.(3) and using bright states from typical realisations for which cos2 θ ≈ 1 and sin2 θ ≈ 0, we obtain the difference in the dipole strength |D0P870 |2 − |DP870 |2 ≈ Re{D∗P870 DB880 sin(2θ)}. This accounts for a redistribution of dipole strength, which compensates the rather small mixing angle sin(2θ) ' 2V/∆E 1, with the large transition dipole of the LH unit DB880 . Such a magnification √ does not occur for the spectral shifts due to this moderate interaction ∆E 0 = ∆E 2 + 4V 2 ≈ ∆E(1 + 4V 2 /∆E 2 ) which only depends on the energy difference-moderate coupling ratio V/∆E. Thereby, any mechanism that amplifies the B880 dipole strength will cooperate for a greater absorption redistribution towards the RC, e.g. excitonic delocalisation over several harvesting pigments in the LH1, 25,32,33 or the intensity borrowing occurring from the Soret band of individual BChl pigments to the Qy B880 band when pigments associate to form the R. rubrum LH1 ring. 34 In order to understand why the P870 band increases its amplitude due to V, it is useful to appeal to the analytical expressions for the ring states |βi obtained in absence of static disorder, which permits to reduce the RC-LH1 interaction to the Hamiltonian Eq.(2). As a result we obtain a explicit dependence of V and DB880 on the ring geometry (see the SI), and a change of P870 dipole √ 3 2 (1 − 3 sin ∆γ)/( 2R ∆E). Here Φ(1) captures strength |D0P870 |2 − |DP870 |2 ≈ DB880 cos2 Φ(1)+γ 2 the difference between intra- and inter-dimer coupling in the ring, γ is the angular separation between pigments at any dimer, and R is the average distance between ring and P pigments. From this expression follows that, given the higher energy of the P870 transition with respect to the B880

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transition (∆E > 0), a positive change of dipole strength depends on (1 − 3 sin2 ∆γ) > 0, which is possible due to ∆γ 1, i.e., a small angle ∆γ of the LH1 BChl Qy transition dipoles with respect to the ring’s tangent (cf. Fig.1 A). 30,31 The description of the full ring with static disorder, presents many bright states which couple to the single bright P870 state. We show in the SI that for θ 1 the increase of the RC transition dipole is nearly the addition of the individual contributions from each LH bright state, hence, supporting this qualitative discussion of the figures of merit underlying the absorption redistribution for the complex under static disorder, based in the two state model of Eqs.(2) and (3).

0.08

A 0.03

Population in RC

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.05

B

P870: 850 nm B880: 878 nm

0.025

as from ρA

0.02

0.01 0.025 0

500

1000 t(fs)

1500

2000

B880: 872 nm P870: 901 nm

C

0

200 t(fs)

400

Figure 2: Population dynamics in the core complex under incoherent light illumination. A Ensemble population of the P pigments (103 realisations of equation (1)). The arrow shows the theoretical estimate for the maximum coherent population redirection to the RC according to ρA ∝ P 02 0 0 α0 |Dα | |α i hα |. B and C Individual realisations of the ensemble where the P870 lies with an energy above and below the B880 band, respectively. The identification of B880 or P870 is made based on the brightest state |αi associated to either RC or LH1, from diagonalisation of H with V = 0, and respective energies that correspond to the wavelength in the inset of each figure. Green, blue and red correspond to site dephasing, partial excitons dephasing and collective dephasing models, as explained in the main text.

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Population redirection towards the RC

The main dynamic consequence of the absorption re-

distribution is the coherent redirection from populations reflecting the transition dipoles of pigments, to populations reflecting the transition dipoles of the excitonically coupled system. As we will show, this population redirection occurs albeit the incoherent nature of illumination. The Hamiltonian for the illumination of the full core complex by thermal light H f ull = H + h¯ Ωa+ a + PNRC +NLH ~ ˆ + gdi · E(a |0i hi| + a |ii h0|) introduces a mode with creation (annihilation) operators a+ (a), i √ broadened by a Markovian reservoir modeled by Lindblad dynamics operators O↑ = Γ(n + 1)a+ √ and O↓ = Γn a where n is the Boltzmann mean occupation at the mode’s frequency Ω and sun temperature 5000 K. In order to guarantee the description of incoherent broadband illumination, we use g = 4 × 10−7 cm−1 /Debye, associated with an intensity of 0.1 W/m2 , proper of purple bacteria growth 27 and a value Γ = 104 cm−1 , much greater than ∆E and gauged such that further increase of its value, did not change the presented results (the SI presents further details). This model results in a Lorentzian spectral density for the electronic interaction with the light field, 35 which is different than the Lorentz-Drude spectral density used in previous models of incoherent illumination, 36 widely used to describe the interaction of electrons with overdamped phonon modes. 37 Both models are comparable, as it is expected that the large broadening Γ ∆E for the incoherent field, reduces the relevance of specific details of the functional form of the spectral densities considered. Figure 2A shows that the incoherent illumination excites the P pigments according to their indiP ˆ 2 /I ≈ 0.025 at very early times t ≈ 0 fs. Here I = PiRC,LH1 |d~i · E| ˆ2 vidual dipole strength iRC |d~i · E| is the transition dipole of the full complex. After initial excitation, follows a very fast population dynamics towards the RC within a time-scale shorter than the inverse of the effective dephasing rate which is, thereby, partially coherent. In the absence of any relaxation, this redirection would not be possible since populations would display coherent reversible oscillations between RC and √ LH pigments, with a period given by 2π/ ∆E 2 + 4V 2 in our reduced two level model. Thereby, an initial pull driven by coherent dynamics becomes directed, hence irreversible, due to the establishment of incoherent dynamics, manifest by the change of slope common to all curves of Fig.2 after t∗ ≈120-150 fs. Since ∆E V in the core complex, this population increase is fast thanks to

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the energy mismatch, ∆E, but nevertheless originated by V. This transient shows how excitonic properties prevail over pigment properties after the excitonic interaction had the time to influence the dynamics. In Fig. 2A, the populations at t ≈ 0 in the three models are equal and proportional to the pigment transition dipoles. They redistribute in the collective dephasing model (red) after some tens of femtoseconds into populations that reflect a P density operator ρA = α0 |D0α |2 |α0 i hα0 | /I, hence weighted by the dipole strengths |D0α |2 , reflecting the properties of the states |α0 i. This fact illustrates that the dephasing operators O0α together with H mold dynamically the state of the system, until observables start to reflect the properties of the |α0 i states. Since this population redirection depend on RC-LH1 delocalisation which is protected by the collective dephasing model, unsurprisingly, the site and realistic dephasing models present in Fig.2A a smaller amount of population redirection (15% increase for site and realistic versus 80%P population increase in the collective dephasing model) after about ' 100 fs. Notice that the population redirection occurs towards the RC when ∆E > 0 or away from the RC when ∆E < 0, as can be observed in Figs. 2B and C for specific realisations of the ensemble that present such energy landscape. Therefore, the rather robust absorption redistribution towards the RC band in the core complex shown in Figs. 1B and S4, represents a clear signature of a more delicate, but nonetheless persistent, coherent population redirection towards the RC pigments under natural illumination. As a general feature, the quantum mechanical description of energy transfer at longer timescales, has been shown to be very similar to a classical description based in rate equations for the core complex, 38 and for other structures as the FMO complex of green sulphur bacteria. 39 The rate 1/ f

1/c

equations regarding the excitonic transfer LH1 ↔ P, the charge separation P → P+ and dissipation at 1/b

rate 1/l, are used in order to check the impact of the population redirection for the overall efficiency of the core complex (see SI for details). In this model, the P population after the mentioned coherent redirection P(t∗ ), is taken as an initial condition for the rate equations describing the longer time-scale dynamics. The percent change in charge transfer efficiency by means of this redirection follows ∆ = f × (P(t∗ ) − P(0))/(l + f P(0)), where P(0) is the initial P population for the rate model, if the coherent population redistribution is disregarded. Using previous estimates

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of f =40 ps and b= 8.1 ps, 38,40–43 l = 250 ps, 43 and the experimentally determined value c=3 ps for charge separation, 44 the transient redirection enhances by ∆ = 0.3 − 0.05% the overall efficiency of the core complex, with limits corresponding to the enhancements due to the obtained redirection for collective and site dephasing models. Even though this increase is indeed vanishingly small, the principle can be exploited to optimise the redirection and provide guidelines for more relevant enhancements in specific scenarios where either f (l) are long (short).

Predicted absorption redistribution in general photosynthetic structures: uncovering principles for optimisation of artificial light harvesters

The possibility to quantify the absorption

redistribution based on the dipole strengths |D0α |2 as weights, allows us to predict an analogous absorption spectra redistribution in photosystem (PS) 1 and PS2 monomer of higher plants, with a 37% and 50% increase of the P band, respectively, due to excitonic delocalisation over antenna pigments independent of the microcopics of dephasing acting in these structures. The natural dimeric structure of PS2 presents, however, almost no enhancement, and hence no potential redirection to the RC pigments (see SI for additional details). The prediction of a significant absorption redistribution conserved across some species, and hence, the possibility to address a more fragile population redirection in these natural structures, draws our attention towards the theoretical limits that stand behind this coherent effect. We have developed an optimisation procedure of the population of a single target pigment according to ρA based on stochastic variations of positions and orientations of a set of N identical harvesters (further details in SI). Figure 3A shows that the dipole strengths |DLH |2 associated to the N harvesters of the optimal configurations, is concentrated almost entirely in a single transition |ηi P ˆ 2 ' |Dη |2 , with the remainder of states therefore dark, |Dν,η |2 ≈ 0. This such that DLH = iN |d~i · E| finding permits to describe the optimal configurations by a simple two level system as described by a Hamiltonian equation (2), and which under a collective dephasing, will develop a population

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in the target state |T i, PT =

1 4I

n o |DT |2 (3 + cos 4θ) + 2< D∗T DLH sin 4θ + |DLH |2 (1 − cos 4θ)

(4)

expressed in terms of the target and light-harvesting dipoles DT and DLH . This population, for tan(4θmax ) = 2

Quantum Redirection of Antenna Absorption to Photosynthetic

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