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Coherent Transport and Energy Flow Patterns in Photosynthesis under Incoherent Excitation Kenley M. Pelzer,† Tankut Can,‡ Stephen K. Gray,§ Dirk K. Morr,‡,∥,* and Gregory S. Engel†,* †

Department of Chemistry, The James Franck Institute, and the Institute for Biophysical Dynamics, The University of Chicago, 929 East 57th Street, Chicago, Illinois 60637, United States ‡ The James Franck Institute and the Department of Physics, The University of Chicago, 929 East 57th Street, Chicago, Illinois 60637, United States § Center for Nanoscale Materials, Argonne National Laboratory, 9700 South Cass Avenue, Building 440, Argonne, Illinois 60439, United States ∥ Department of Physics, University of Illinois at Chicago, 845 West Taylor Street M/C 273, Chicago, Illinois 60607, United States ABSTRACT: Long-lived coherences have been observed in photosynthetic complexes after laser excitation, inspiring new theories regarding the extreme quantum efficiency of photosynthetic energy transfer. Whether coherent (ballistic) transport occurs in nature and whether it improves photosynthetic efficiency remain topics of debate. Here, we use a nonequilibrium Green’s function analysis to model exciton transport after excitation from an incoherent source (as opposed to coherent laser excitation). We find that even with an incoherent source, the rate of environmental dephasing strongly affects exciton transport efficiency, suggesting that the relationship between dephasing and efficiency is not an artifact of coherent excitation. The Green’s function analysis provides a clear view of both the pattern of excitonic fluxes among chromophores and the multidirectionality of energy transfer that is a feature of coherent transport. We see that even in the presence of an incoherent source, transport occurs by qualitatively different mechanisms as dephasing increases. Our approach can be generalized to complex synthetic systems and may provide a new tool for optimizing synthetic light harvesting materials.

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INTRODUCTION Over the past decade, much effort has been devoted to understanding exciton transport processes1−6 in photosynthetic antenna complexes. The Fenna-Matthews-Olson complex (FMO),7,8 a photosynthetic light-harvesting complex in green sulfur bacteria, serves as a simple canonical model system for excitonic energy transfer. FMO shuttles excitations from the chlorosome, where photons are absorbed, to the reaction center, where charge separation takes place.9 Exciton transport in this photosynthetic system occurs with a quantum efficiency approaching 100%;10−13 however, the mechanisms behind this high efficiency remain a topic of debate. Ultrafast spectroscopic observations of dephasing demonstrate that, after excitation by short laser pulses, quantum coherences persist for time scales relevant to exciton transport.14,15 These coherences, or “quantum beats,” arise from coherent evolution of superpositions of Hamiltonian eigenstates. Although vibrational coherences may contribute to the oscillating features observed in these spectra and may play an important role in shaping energy transfer,16,17 studies employing isotopic substitution,18 temperature dependence of dephasing, differences in nonrephasing vs rephasing signals,20,21 pump− probe anisotropy data,22 crosspeak location,23,24 and analysis of bacteriochlorophyll controls25 to distinguish between vibrational and electronic coherences all indicate that observed coherences are largely or fully electronic in nature. Theoretical models of © 2014 American Chemical Society

FMO and other photosynthetic protein−pigment complexes predict that efficiency of transport is sensitive to the dephasing rate.10,26−31 It is therefore important to develop a theoretical framework that (a) establishes the microscopic conditions for observation of quantum coherence and coherent transport, thus clarifying the relation between these two phenomena, (b) allows exploration and understanding of the relationship between dephasing and efficiency in FMO, and (c) can be generalized to large networks of chromophores. Such a framework will provide insights into the nature of exciton transport in biology and guide new approaches for controlling excitonic motion in synthetic systems. In this study, we use the Keldysh Green’s function32−35 formalism to model exciton transport in FMO in the nonequilibrium steady-state limit using an incoherent energy source. FMO was recently modeled by Manzano in a nonequilibrium steady-state using a time-domain formalism; Manzano argues that because photon absorption occurs continuously in photosynthesis and no specific excitation time is measured, a continuous, steady-state picture is appropriate.36 Here we model the steady-state in the frequency domain rather than the Received: January 22, 2014 Revised: February 3, 2014 Published: February 5, 2014 2693

dx.doi.org/10.1021/jp500746a | J. Phys. Chem. B 2014, 118, 2693−2702

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time-domain, allowing us to decompose excitonic fluxes according to frequency and explore the unidirectionality or multidirectionality of transport. We use this formalism to explore the relationships between dephasing, efficiency, and the spatial form of exciton flux through FMO. While time-domain theory and spectroscopic work has suggested that energy transfer in FMO occurs via coherent evolution of superpositions of energy eigenstates, our frequency domain model provides different, complementary information. Because our results represent a steady-state limit, our system does not possess “coherence” among eigenstates as it is usually defined. (Formally, our results depend only on population elements of the density matrix represented in the system Hamiltonian eigenbasis in the infinite time limit.) The fact that oscillatory coherences cannot be present in our steady-state model does not necessarily imply that transport does not involve coherent mechanisms. Our insights into the nature of the transport process suggest that “coherent transport”, the transfer of energy via a delocalized, wavelike, multidirectional process (which we distinguish from “coherence” as we define it above) is possible even in the presence of an incoherent source of energy. The definition of the energy source as “incoherent”which is a result of thermalization and uncertainty in energyis explained in detail in the Theoretical Methods section. We demonstrate that the coupling of FMO to its environment yields a nonmonotonic relationship between dephasing and efficiency where efficiency is optimized at intermediate levels of dephasing.37 This finding demonstrates that the key results of past time-domain models of environmentally assisted quantum transport (EnAQT)27,28 hold with an explicitly incoherent energy source. Moreover, we show that the same microscopic property, excitonic dephasing, controls both the long-lived coherence in FMO (as measured in spectroscopic experiments of quantum beating) and the coherent transport through FMO in the steady-state. As a result, we demonstrate that coherent transport mechanisms are operative even in the presence of an incoherent source. Coherence and coherent transport both depend on coupling to the surrounding environment. Weak coupling to the environment, low temperature, or correlated spectral motions give rise to slow excitonic dephasing rates. While we show that observations of “coherence” and existence of “coherent transport” both require a sufficiently small excitonic dephasing rate, the absence of coherences does not imply the absence of coherent transport. Indeed, the electronic quantum coherences observed via spectroscopy14,15 are phenomena unique to spectroscopic experiments. Coherences detected as quantum beats in spectroscopic experiments exist only when the system possesses a sufficiently small dephasing rate and is excited by a coherent source (i.e., a laser). For the same reason, theoretical studies that utilize coherent excitation typically through a defined time and/or phase of excitation also observe coherences. In contrast, the presence or absence of coherent transport is only determined by the excitonic dephasing rate, and not by the nature (coherent versus incoherent) of the source. A similar conclusion regarding the origin of coherent charge currents was first reached by Landauer.38 Recent work by Kassal et al. argues that coherent transport (which they refer to as “process coherence”) occurs whenever the strength of coupling to the environment (dephasing) is small relative to the interchromophore coupling, and points out that coherent transport is possible even if the initial excited state is an incoherent mixture of eigenstates.39 Indeed, our results for excitonic transport

discussed below are obtained in the presence of an incoherent source, but show all the hallmarks of coherent transport. Thus, the analysis of environmental effects measured with coherent spectroscopy is not simply an artifact of laser excitation. Beyond identifying the origin of coherence and coherent transport, the formalism employed also naturally provides a clear picture of exciton flux between chromophores that complements previous theoretical models for FMO that have worked within density matrix master equation formalisms. Our model therefore provides a view of how environmental effects determine energy flow in photosynthetic complexes over the entire range from coherent to incoherent transport.

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THEORETICAL METHODS

After absorption of a photon, an exciton is created in the chlorosome, which enters the FMO complex’s bacteriochlorophyll chromophores (shown in Figure 1). The FMO complex itself is a trimeric complex, with each monomer containing seven strongly coupled bacteriochlorophylls (BChls) alongside a recently discovered eighth BChl near the chlorosome.40 In this work, we therefore consider only the seven strongly coupled BChls to provide a direct comparison to earlier models. The excitation is likely to enter the complex through BChls 1 and 6, the chromophores closest to the chlorosome.41 Due to the coupling of the chromophores, the exciton can move through the FMO complex and exit to the reaction center through BChl 3. At each chromophore, the exciton has a nonzero probability either to recombine (emitting a photon) or to be scattered by a phonon. The former process leads to loss of excitons from FMO, while the latter models the interactions with the environment of the FMO complex and leads to dephasing of the excitons. As we show below, this dephasing process crucially determines the form of the excitonic transport through the FMO complex, and provides information on the role of quantum mechanics in the transport process. To examine the flow of excitons through the FMO complex, we employ a nonequilibrium Keldysh Green’s function formalism. This formalism maps seamlessly from the coherent transport regime to the classical hopping regime.42 It also provides detailed information on the flow of excitons between chromophores. For calculation of the exciton flux, we treat the chlorosome as an explicitly incoherent source for excitons entering FMO. Our model consists of seven sites representing the seven chromophores of FMO, a chlorosome, a reaction center, and seven independent radiation baths (one coupled to each chromophore site). Flux from a chromophore to a radiation bath site represents the process of exciton recombination. In photosynthesis, the supply of excitons from the chlorosome varies based on the environmental light levels, but is consistently low enough to assume a one-exciton manifold for the seven chromophores. Thus, we set the coupling between the chlorosome and FMO to be low enough to enforce a very low exciton density in FMO. Our model does not include any intrinsic restrictions regarding the number of excitons in the system or on a given site; therefore, this low coupling is necessary to enforce the realistic condition that no more than one exciton is present in FMO in the steady-state. The system is described by the Hamiltonian: H = HFMO + Hrec + Hphonon + Hc + Hr + HT 2694

(1)



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Figure 1. Exciton transport and flux patterns through FMO with incoherent excitation. Dephasing affects exciton transport through the FMO complex in the presence of an incoherent source. Dependence of total flux through FMO to the reaction center (1a) represents the product of the fractional efficiency of transport (1b, solid) and the flux into FMO from the chlorosome (1b, dashed). The spatial distribution (1c) of excitonic flux between chromophore sites in FMO for several values of dephasing are shown. Thickness of arrows is linearly proportional to the magnitude of the flux.

with c and f destroying excitons in the chlorosome and reaction center, respectively. The hopping amplitude Vr is determined by the requirement that the lifetime of the exciton at site 3 is 1 ps, yielding Vr = 16.69 cm−1. We take Vc = 0.42 cm−1 ensuring a low flux into the FMO complex. Finally, while Hc/r describes the excitonic structure of the chlorosome and the reaction center, it is beyond the scope of this study to explicitly compute them. We therefore choose a phenomenological form of their respective Green’s functions, as shown below. The flow of excitons through FMO is driven by a difference in the exciton occupation number between the chlorosome and reaction center, as demonstrated in detail below. The derivation of the exciton flux between two sites i and j connected by a nonzero Vij is similar to that of a charge current32 and given by (setting ℏ = 1 in the following):

Here 7

HFMO =

∑

Eidi†di +

i=1

∑

Vijdi†dj (2)

i≠j

describes the site energies, Ei, of the seven chromophores in the FMO complex, and the coupling, Vij, between them, with d†i (di) being bosonic operators which create (annihilate) an exciton at site i in FMO. The values for Ei and Vij are taken from Cho et al.3 Moreover, 7

Hrec =

∑

Vb(ei†di + h. c . )

(3)

i=1

describes the exciton recombination in FMO, which is accompanied by the emission of a photon with e†i being the photon creation operator at site i. The reverse process is allowed, but strongly suppressed due to a vanishing photon density (see below). The recombination amplitude, Vb, is determined by the requirement that the resulting recombination time of the exciton is 1 ns, yielding Vb = 0.84 cm−1. The scattering of excitons by local phonons is described by the following: 7

Hphonon = g ∑ di†di(pi† + pi ) + i=1

Jij (t ) = Vij

ω0pi† pi

i=1

where is the full excitonic (bosonic) Keldysh Green’s function between sites i and j. In order to compute GKij , we define a Green’s function matrix, Ĝ K whose ij-element is GKij (ω) (and similarly for the retarded r and advanced a Green’s functions). The number of excitons at a given BChl i (i.e., the exciton density) is given by the following:

(4)

1⎡ Ni = − ⎢1 + 2⎣

p†i

where creates a phonon at site i, ω0 is the energy of the local phonon modes, and g is the exciton−phonon interaction strength.43 The tunneling of excitons between the chlorosome/reaction center and the FMO complex is described by the following: HT = Vc(d1†c + d6†c + h. c . ) + Vr(f † d3 + h. c . )

(6)

GKij (ω)

7

∑

dω Re[GijK (ω)] 2π

∫

∞

∫−∞

⎤ dω Im[GiiK (ω)]⎥ ⎦ 2π

(7)

with the total exciton number given by the following: 7

Ntot =

(5)

∑ i=1

2695

Ni

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while one has for the exciton density of states, 1 Ni(ω) = − Im[Giir (ω)] π

(9)

The Dyson equations for the Keldysh and retarded Green’s function matrices are then given by the following: K K r a Ĝ = Ĝ [(g ̂r )−1g ̂ K (g ̂a )−1 + Σ̂ ph]Ĝ

(10)

r r r Ĝ = g ̂r + g ̂r [V̂ + Σ̂ ph]Ĝ

(11)

(12)

is the noninteracting and decoupled Green’s function matrix. Here, ĝFMO is a (7 × 7) diagonal matrix whose elements, [ĝFMO]jj describe the local Green’s functions44 of the excitonic site j, given by (with δ → 0+), [gFMO ̂ r ]jj =

1 r = gFMO (ω , j ) ω − Ej + iδ

[gFMO ̂ K ]jj = 2i[1 + 2nBj (ω)]Im[gFMO ̂ r ]jj

(13) (14)

where is the Bose distribution function for excitons at site j. Note that the form of grFMO(ω, j) results after Fourier transformation and analytic continuation from the definition of the excitonic Green’s function in imaginary time, τ, gFMO(τ , j) =

gc/r ̂α

(16)

1 ω − Eβ + iηβ

gβK (ω) = 2i[1 + 2nBβ (ω)]Im[gβr (ω)]

(17) (18)

Here, Ec = 480.1 cm−1 and Er = −24.2 cm−1 are chosen such that they coincide with the largest and smallest exciton energy, respectively, in FMO, and ηβ = 105 cm−1. Moreover, to describe the photonic bath, we use the following: gb̂ r = −iπ1b̂

(19)

gb̂ K = 2i[1 + 2nBb(ω)]Im[gb̂ r ]

(20)

= −2πi[1 + 2nBb(ω)]1b̂

(21)

=1

(25)

(26)

Vĉ ,FMO = (Vc , 0, 0, 0, 0, Vc , 0)

(27)

Vr̂ ,FMO = (0, 0, Vr , 0, 0, 0, 0)

(28)

Σijα(ω) =

where 1̂ is the (7 × 7) identity matrix. This definition implies that the photonic density of states is unity. Moreover, for simplicity, we set, nBc (ω)

nBFMO(ω) = 0

To account for the effects of the exciton−phonon interaction for general g, ω0 and T is computationally demanding and beyond the scope of this work. However, since we are mainly interested in the effects of exciton dephasing, we can consider the high-temperature approximation introduced in Bihary and Ratner.45 In this approximation, the phonons do not exchange energy with the chromophores during the transport process. This approximation allows an analytic solution of the Keldysh Green’s function and significantly decreases the computational cost of the model. Work is in progress to consider phonons of a finite energy that exchange energy with the chromophores. Here, we take kBT ≫ ω0, implying a large thermal population of the phonon modes. As a result, one retains in the Dyson equation only those terms that contain a factor of nB(ω0), and obtains that the excitonic self-energy at a site in FMO in the self-consistent Born approximation is given by the following:

where (with ηβ ≠ 0 and β = r, c), gβr (ω) =

(24)

where V̂ FMO is a (7 × 7) matrix with the [V̂ FMO]ij = Vij, VFMO,b = Vb1̂b, and,

(15)

where Tτ is the τ-ordering operator. Moreover, ⎛ g α (ω) 0 ⎞ c ⎟ =⎜ α ⎜ 0 gr (ω)⎟⎠ ⎝

nBb(ω) = 0

⎛ V̂ ⎞ V̂ V̂ V̂ ⎜ FMO FMO, c FMO, r FMO, b ⎟ ⎜ V̂ 0 0 0 ⎟⎟ ⎜ c ,FMO ̂ V =⎜ ⎟ ̂ 0 0 0 ⎟ ⎜ Vr ,FMO ⎜⎜ ⎟ 0 0 0 ⎟⎠ ⎝ Vb̂ ,FMO

njB(ω)

−⟨Tτbj(τ )b†j (0)⟩

(23)

which guarantees that (a) the number of excitons in the chlorosome is larger than the number of excitons in the reaction center, thus inhibiting the process in which an exciton moves from the reaction center to the FMO complex [more generally, this is achieved by choosing ncB(ω) > nrB(ω)], (b) that the density of photons is zero, thus suppressing the creation of excitons through photon absorption in the FMO complex, but allowing exciton recombination via emission of a photon, (c) that for an FMO in equilibrium, the number of excitons in FMO is zero. In the nonequilibrium steady-state, the number of excitations in FMO will generally be nonzero. We note that, for the purpose of exciton transport, it is not necessary to incorporate the extended spatial structure of the chlorosome or the reaction center; the only relevant information is the excitonic structure of the parts that are directly coupled to the FMO complex. Therefore, it is sufficient to describe the chlorosome and the reaction center as individual sites. Finally, V̂ describes the exciton coupling within FMO, as well as the coupling to the chlorosome, reaction center, and the photonic bath, given by the following:

where Σ̂ph is the excitonic self-energy matrix arising from the electron−phonon interaction. Moreover (with α = K, r, a), ⎛ ĝα 0 0 ⎞ ⎜ FMO ⎟ gc/r 0 ⎟ ̂α g ̂α = ⎜ 0 ⎜ ⎟ ⎜ 0 α ⎟ g 0 ̂ ⎝ bath ⎠

nBr (ω) = 0

ig 2 2

∫

dυ K D (υ)Gijα(ω − υ) 2π

(29)

where D K = 2iπ (1 + 2nBph(ω))[δ(ω + ω0) − δ(ω − ω0)]

(22)

(30) 2696



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is the Keldysh phonon Green’s function, which we assume to remain unchanged by the exciton−phonon interactions, and nph B (ω) is the phonon Bose distribution function (we assume that the phonons remain in thermal equilibrium). A further simplification is achieved by considering the limit ω0 → 0 in which the self-energy, to leading order in kBT/ω0, is given by the following: Σijα(ω) = 2g 2

kBT α Gij (ω) ≡ ζGijα(ω) ω0

Re[G3,Kr (ω)] = 4π 2VrVc2Nc(ω)Nr(ω)[nBc (ω) − nBr (ω)] 7 r r 2 × {|G31 + G36 | + ζ ∑ |G3rk |2 |Gkr1 + Gkr6|2 k=1 7

+ ζ2

(31)

⎧G αδ if i is a site in the FMO complex ̃ ̂ α]ij = ⎨ ij ij [DG ⎩0 otherwise

where Nc/r(ω) is the density of states of the chlorosome/reaction center. Because the exciton flux between site 3 and the reaction center is obtained from eq 6 using eq 41, we immediately find that Jout is driven by the difference in the exciton distribution functions between the chlorosome and the reaction center. It also reflects the nonlocal full Green’s function inside FMO, which includes the effects of exciton−phonon interactions. Finally, to obtain the relation between ζ and the dephasing rate, we consider the retarded Green’s function of a single excitonic site with energy level E = 0 coupled to a phonon, which according to eq 36 is given by the following:45

⎪

⎪

(32)

and thus, α

(33)

Gr = [ω + iδ − ζGr ]−1

We next define the operator Û that acts on a matrix X̂ via, ̂ ̂ = Ĝ r XG ̂ ̂a UX

K ̃ ̂ ]−1 Λ̂ Ĝ = Û [1 − ζDU

Gr =

(36)

where we defined the diagonal matrix Λ̂ = (ĝ ) ĝ (ĝ ) . The only nonzero elements of Λ̂ij are those where i is either the chlorosome, the reaction center, or the photonic bath. Therefore, as already pointed out by Keldysh,35 the exciton flux is independent of the value set for nFMO (ω) at equilibrium. B After self-consistently solving eq 36 for Ĝ r, Ĝ K can be obtained in a closed expression. To demonstrate this, one expands the right-hand side of eq 35 to obtain the following: r −1 K

l

+ζ

a −1

Γ=

m a Q̂ lmQ̂ mp Λ̂ pp + . . .]Ĝ lj

∑

(37)

m,p

where, ⎧|G r |2 if l is a site in FMO Q̂ lm = ⎨ lm ⎩0 otherwise ⎪

⎪

(38)

Defining the next vector λ with λm = Λ̂mm, we finally obtain the following: K Giĵ =

∑

r a Gil̂ [(1 − ζQ̂ )−1λ]l Ĝ lj

l

̂K

̂r

(39)

̂a

or G = G Σ̃G , where the diagonal matrix Σ̃ is defined via the following:

Σ̃ll = [(1 − ζQ̂ )−1λ]l

(43)

3ζ /2

(44)

This level of dephasing, Γ, reflects the coupling of the chromophores to the phonon bath. This model is designed to represent the source of energy as incoherent. It is important to recognize that FMO does not receive photons directly from incoherent sunlight; rather, the chlorosome receives photons from the sun, excitons are generated within the chlorosome, and this excitonic energy is transferred to FMO. Much controversy exists about both the appropriate initial state of excitation within FMO, the dependence of this initial state on the source of light, and how this initial state determines the nature of the transport process.46−50 Here we do not attempt to address the problem of defining FMO’s initial state of excitation, as a steady-state model involves no concept of an “initial state.” We take advantage of the fact that the steady-state is by definition unaffected by initial states of excitation to demonstrate which phenomena are present regardless of the nature of the initial excited state. To enforce the incoherence involved in solar illumination, we define an incoherent chlorosome, which is the only part of the system that typically absorbs sunlight. Transport through the system is derived in the steady-state (long-time limit under constant illumination).35 By assigning the value of ηc in eq 17 to 105 cm−1, our source is given a continuous density of states of noninfinitesimal width. ηc therefore dephases the chlorosome by adding uncertainty in energy. The steadystate represents a t = ∞ limit and as such is longer than any and all time scales for dephasing. In this way, the source of all energy transferred to FMO can be defined as incoherent regardless of environmental dephasing. Another way to view the incoherence

∑ Gil̂ r [Λ̂ll + ζ ∑ Q̂ lmΛ̂mm 2

ω + iδ 1 ± (ω + iδ)2 − 2ζ 2ζ 2ζ

The sign is fixed at any given ω by requiring the density of states N(ω) = −Im[Gr(ω)]/π to be positive. The above form of Gr implies that the density of states possesses a peak centered at E = 0 and vanishes for |ω| > (2ζ)1/2. The half-width at half-maximum of the peak, which is equal to the dephasing rate, is given by the following:

(35)

r ̃ ̂ r )]−1 g ̂r Ĝ = [1 − g ̂r (V̂ + ζDG

(42)

whose solution is as follows: (34)

The solutions of the above Dyson equations are then given by the following:

K Giĵ =

|G3rk |2 |Gklr|2 |Glr1 + Glr6|2 + . . . } (41)

We next introduce the superoperator D̃ which, when operating on a Green’s function matrix, returns the same matrix with all elements set to zero except for the diagonal elements corresponding to FMO sites in the matrix, e.g.,

̃ ̂ Σα(ω) = ζDG

∑ k ,l=1

(40)

To gain further insight into the microscopic form of the exciton flux between site 3 and the reaction center, i.e., Jout, we expand eq 37 and obtain the following: 2697



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of the source is through thermalization at equilibrium: The form of our Green’s function comes from solving the Dyson equation with an equilibrium assumption, meaning that although our simulation is out of equilibrium, the source is always fully thermalized. Thus, the source is incoherent and has lost all knowledge of any initial coherences.

■

RESULTS AND DISCUSSION

Transport of excitons through FMO to the reaction center, Jout, shown in Figure 1a reveals a strong, nonmonotonic dependence as a function of dephasing rate (Γ), arising from coupling to a noisy environment.37 In particular, the flux to the reaction center exhibits a maximum around Γ = 350 cm−1, in agreement with the EnAQT results presented by Aspuru, Lloyd, and co-workers,26,28 and Plenio and Huelga.27 This exciton flux to the reaction center is likely relevant to evolutionary fitness in light-starved green sulfur bacteria.51 To understand this nonmonotonic variation of flux through the FMO complex, we first examine the dynamics in the absence of dephasing (Γ = 0 cm−1), which represents the coherent limit for excitonic transport. In this case, disorder among the site energies of the seven chromophores inhibits transport: because the various sites of FMO have different energies, Ei, without dephasing adding uncertainty to the site energies, there is little overlap between the density of states of various sites (this lack of overlap of the density of states is discussed in more detail below). Thus, transport is inhibited, and Figure 1b demonstrates that only a small fraction of excitons entering FMO ultimately reach the reaction center. As the dephasing rates increases, destructive interference, and hence localization, is suppressed.52 However, at dephasing levels greater than Γ = 100 cm−1, increasing dephasing reduces the flow of excitons into the FMO complex as shown in Figure 1b. Aspuru, Lloyd, and co-workers attribute this behavior at high dephasing rates to the quantum Zeno effect.28,53,54 The same effect is well-known in the context of charge transport (i.e., for electrical currents) where resistivity increases as electrons are scattered by phonons.55 This quantum Zeno effect via exciton− phonon interactions inhibits transport irrespective of whether the transport is permitted by considerations of energetics and site occupations. As a result of these competing effects arising from scattering by phononsdestruction of localization versus inhibiting the flow of excitonsexcitonic flux through FMO exhibits a maximum near Γ = 350 cm−1. This physical explanation is also reflected in the spatial pattern of exciton flow among chromophores in FMO (Figure 1c) and dependence of the recombination rates on dephasing shown in Figure 2. For Γ = 0, localization blocks exciton flux at BChls 2, 5, and 6 leading to large recombination rates at these sites. However, for large dephasing rates, corresponding to incoherent (i.e., classical) transport, the excitons flow along the paths where the interchromophoric couplings, Vij, are largest. The resulting flow patterns are qualitatively similar to those predicted by Brixner et al. from a Förster treatment of the FMO Hamiltonian,1 and akin to those predicted in a master equation treatment by Rebentrost et al.56 Yet, there is more to this process than spectral overlap alone can explain. The recombination rate decreases not because of faster transport through FMO, but because the Quantum Zeno effect blocks flux into FMO forcing the recombination rate to decrease proportionally. Thus, the reduction in flux to the reaction center at very large dephasing rates does not arise from changing efficiency in FMO, but rather because entry into the complex is blocked. In a biological system, blocking

Figure 2. Exciton losses from FMO as dephasing increases. Exciton recombination decreases with increasing dephasing rate. Losses from the system in the form of recombination (smaller arrows) and trapping by the reaction center (large arrows from site 3) are shown for various dephasing rates. The thickness of the arrows is proportional to the recombination rate at each site.

entry to FMO would result in radiative losses from the chlorosome and decreased photosynthetic production. The subtle evolution of the spatial flow patterns of the excitons for intermediate dephasing rates reflects the transition from coherent to incoherent transport. A direct signature of this transition is a reversal in the direction of the excitonic fluxes between some pairs of chromophores, as occurs, for example, in the flux between sites 1 and 4 (Figure 1c). For small dephasing, this flux exemplifies backflow: excitons flow from a site further from the source and with a lower on-site energy, to a site closer to the source with a higher on-site energy.42 Even more interesting is the flux between sites 2 and 6, which changes direction twice with increasing dephasing. At Γ = 0 cm−1, excitons flow from site 2 to site 6. As dephasing increases to Γ = 100 cm−1, the direction of the exciton flux reverses, now flowing from sites 6 to 2. As dephasing increases further to Γ = 350 cm−1, the flux changes direction once again. To understand these changes in the exciton flux pattern, we examine in Figure 3 the energy resolved flux between sites 2 and 6, J26(ω), over which we integrate to obtain the total excitonic flux. A positive (negative) value of J26(ω) implies that the exciton at energy ℏω flows from chromophore 2 to 6 (6 to 2). We observe both positive and negative regions of J26(ω) simultaneously at Γ = 100 cm−1 and Γ = 350 cm−1 demonstrating energy-resolved multidirectional flow between these two sites. Interestingly, this phenomenon occurs precisely at the intermediate levels of dephasing where previous literature suggests that “environment-assisted quantum transport (EnAQT)” (a highly efficient transport process that is neither fully coherent nor fully incoherent) thrives.27,28 It is not intuitive that a backflow occurs when transport is optimal, but in a disordered system reversible sampling of sites may help to avoid trapping and localization. The total number of excitons, Ntot, in the FMO complex varies with dephasing (Figure 4). As expected, we find that the number of excitons is greatest for Γ = 0 because localization leads to 2698



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Figure 3. Bidirectional transport through FMO. The exciton flux between sites 2 and 6, represented by J26(ω), demonstrates bidirectional flow for intermediate dephasing levels. A positive (negative) value of the flux corresponds to exciton flow from site 2 to site 6 (6 to 2). When both positive and negative regions are present, exciton density flows in both directions simultaneously.

Figure 4. Exciton density and patterns in FMO as dephasing increases. Exciton density in FMO changes in magnitude and pattern with dephasing. Colored clouds represent the density at each site (top) for various magnitudes of dephasing. The size and opacity of the cloud represents site density, with larger and darker circles reflecting a larger density. The data for each dephasing rate were normalized separately. The total density of excitons (lower) decreases with increasing dephasing. 2699



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Figure 5. Excitonic band structure of FMO. Energetic structure of the density of states in the FMO complex changes with dephasing rate. The density of states (DOS) for each site in FMO is shown for dephasing values Γ = 10 cm−1 and Γ = 350 cm−1. Highlighted in violet are regions of the excitonic spectrum that contribute to transport to the reaction center. The far right panel (green) shows the energetic structure of the flux to the reaction center.

“trapping” of the excitons at sites 2, 5 and 6, resulting in a longer “dwell-time” of the excitons at these sites, and consequently, a larger Ni, as shown in Figure 4 (lower) (the largest exciton density is found at site 6 because it is directly coupled to the chlorosome and its excitonic energy is close to that of the chlorosome). With increasing dephasing, localization is suppressed leading to increased transport and consequently a decrease in Ni. At the same time, dephasing counteracts the effects of disorder and leads to an exciton population that is more evenly distributed throughout the FMO complex for Γ = 700 cm−1, as shown in Figure 4 (upper). For all dephasing rates, the total excitonic population in the complex is much less than one, consistent with the biologically relevant situation in a light starved organism. To further illuminate the relation between the excitonic structure of FMO and the changes in flux to the reaction center (Jout) and excitonic population (Ntot) with increasing dephasing discussed above, we consider the frequency-resolved excitonic density of states (DOS). The densities of states at the seven chromophore sites for dephasing values of Γ = 10 cm−1 and Γ = 350 cm−1 are shown in Figure 5 alongside the energy resolved flux between site 3 and the reaction center. Peaks in the flux to the reaction center correspond to the excitonic states involved in transport to the reaction center, which are highlighted in violet. For low dephasing values, the localization of the excitonic wave functions is reflected by the lack of overlap between the density of states at different sites and forces the excitons to remain predominantly trapped at sites 2, 5, and 6. The observed transport to the reaction center arises from the small (but nonzero) overlap in the density of states with site 3 as well as tunneling through site 3 directly into the reaction center.

The density of states provides an alternative view of the dephasing-induced changes in the spatial flux pattern. With increasing dephasing, the exciton states broaden in energy, and the localization is reduced, until around Γ = 350 cm−1, a continuous “band” of states extends throughout the FMO complex, as shown in Figure 5. In contrast to the narrow states that carry the excitons for Γ = 10 cm−1, a broad band of states highlighted in violet now contributes to excitonic transport for Γ = 350 cm−1. As a result, the spatial path of the excitons through FMO is now determined by the magnitude of the couplings between the FMO sites. Thus, two dominant paths through FMO emerge with increasing dephasing: one leading from site 1 via site 2 to site 3, where the excitons exit FMO to the reaction center, and another one from site 6 via site 4 (either directly, or via site 5 or 7) to site 3. For dephasing rates above Γ = 350 cm−1, the overlap between the density of states at all seven chromophore sites changes only weakly, thus explaining the minor dependence of the flow pattern on dephasing.

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CONCLUSIONS Transport through FMO depends strongly on the dephasing rate even for incoherent excitation of the system. This same dephasing rate determines the lifetime of laser-induced coherence in FMO, as observed in spectroscopic experiments.14,15,57,58 Thus, the dephasing rate represents the fundamental link between coherent transport and long-lived coherence in FMO. We show that this dependence on the dephasing rate exists in a steady-state, frequency domain model. Coherent excitation is only a tool to measure this dephasing rate in spectroscopic experiments by observing decay of the time-dependent oscillations 2700



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Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DEFG02-05ER46225.

of electronic coherences. Thus, our results suggest that even though the dynamic oscillations of quantum electronic coherences that are observed in spectroscopy are absent in nature, the dependence of transport on dephasing should still be present in nature. The useful link between spectroscopic observations and natural processes is the information relating to dephasing, not the dynamic oscillating coherences themselves. We demonstrate that exciton transport efficiency through complex multichromophoric systems is sensitive to dephasing even in the absence of coherent initial excitation. Thus, even with incoherent excitation, transport does not necessarily occur only via the localized hopping mechanism that is often referred to as “incoherent transport.” By including dephasing, we represent the same environmental effects that have been explored in the timedomain and used as probes for the presence of long-lived quantum mechanical effects. Our approach allows us to understand the patterns of excitonic energy flow through the FMO complex as the dephasing rate changes, and the efficiencies that we observe are consistent with the key findings of past time-domain models of environmentally assisted quantum transport.27,28 Our results support the recent argument by Kassal et al. that the nonmonotonic relationship between dephasing and efficiency proposed by the EnAQT model exists under incoherent excitation.39 As the system moves from manifestly coherent to completely classical, we see dramatic changes in the transport physics. At low dephasing rates, excitons enter FMO, but cannot move through the complex. At high dephasing rates, their entry into the complex is blocked. Just as in EnAQT models, the optimal transport is achieved by a balance of incoherent and coherent effects. By examining fluxes between chromophores, we also demonstrate that dephasing affects the paths by which excitons travel through FMO. Interestingly, we observe backflow among sites in precisely the optimal regime as predicted by EnAQT models. Prior studies involving quantum coherences have proposed that the presence of long-lived coherence after laser excitation suggests persistence of quantum mechanical effects over time scales relevant to biological processes. By exploring dynamics after incoherent excitation, we show that quantum mechanical effects exist regardless of the nature of the initial excitation. This approach may be generalized to model and optimize large, device-scale synthetic systems, as well as larger photosynthetic complexes.

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REFERENCES

(1) Brixner, T.; Stenger, J.; Vaswani, H. M.; Cho, M.; Blankenship, R. E.; Fleming, G. R. Two-Dimensional Spectroscopy of Electronic Couplings in Photosynthesis. Nature 2005, 434, 625−628. (2) Vulto, S. I. E.; Neerken, S.; Louwe, R. J. W.; de Baat, M. A.; Amesz, J.; Aartsma, T. J. Excited-State Structure and Dynamics in FMO Antenna Complexes from Photosynthetic Green Sulfur Bacteria. J. Phys. Chem. B 1998, 102, 10630−10635. (3) Cho, M. H.; Vaswani, H. M.; Brixner, T.; Stenger, J.; Fleming, G. R. Exciton Analysis in 2d Electronic Spectroscopy. J. Phys. Chem. B 2005, 109, 10542−10556. (4) Cao, J. S.; Silbey, R. J. Optimization of Exciton Trapping in Energy Transfer Processes. J. Phys. Chem. A 2009, 113, 13825−13838. (5) Zhu, J.; Kais, S.; Rebentrost, P.; Aspuru-Guzik, A. Modified Scaled Hierarchical Equation of Motion Approach for the Study of Quantum Coherence in Photosynthetic Complexes. J. Phys. Chem. B 2011, 115, 1531−1537. (6) Sarovar, M.; Ishizaki, A.; Fleming, G. R.; Whaley, K. B. Quantum Entanglement in Photosynthetic Light-Harvesting Complexes. Nat. Phys. 2010, 6, 462−467. (7) Wen, J. Z.; Harada, J.; Buyle, K.; Yuan, K.; Tamiaki, H.; Oh-oka, H.; Loomis, R. A.; Blankenship, R. E. Characterization of an Fmo Variant of Chlorobaculum Tepidum Carrying Bacteriochlorophyll a Esterified by Geranylgeraniol. Biochemistry 2010, 49, 5455−5463. (8) Fenna, R. E.; Matthews, B. W.; Olson, J. M.; Shaw, E. K. Structure of a Bacteriochlorophyll Protein from Green Photosynthetic Bacterium Chlorobium limicolaCrystallographic Evidence for a Trimer. J. Mol. Biol. 1974, 84, 231−234. (9) Sauer, K. PhotosynthesisLight Reactions. Annu. Rev. Phys. Chem. 1979, 30, 155−178. (10) Ishizaki, A.; Calhoun, T. R.; Schlau-Cohen, G. S.; Fleming, G. R. Quantum Coherence and Its Interplay with Protein Environments in Photosynthetic Electronic Energy Transfer. Phys. Chem. Chem. Phys. 2010, 12, 7319−7337. (11) Mimuro, M.; Nozawa, T.; Tamai, N.; Shimada, K.; Yamazaki, I.; Lin, S.; Knox, R. S.; Wittmershaus, B. P.; Brune, D. C.; Blankenship, R. E. Excitation-Energy Flow in Chlorosome Antennas of Green Photosynthetic Bacteria. J. Phys. Chem. 1989, 93, 7503−7509. (12) Vasmel, H.; Vandorssen, R. J.; Devos, G. J.; Amesz, J. Pigment Organization and Energy-Transfer in the Green Photosynthetic Bacterium Chlorof lexus aurantiacus 1. The Cytoplasmic Membrane. Photosynth. Res. 1986, 7, 281−294. (13) Vos, M.; Nuijs, A. M.; Vangrondelle, R.; Vandorssen, R. J.; Gerola, P. D.; Amesz, J. Excitation Transfer in Chlorosomes of Green Photosynthetic Bacteria. Biochim. Biophys. Acta 1987, 891, 275−285. (14) Engel, G. S.; Calhoun, T. R.; Read, E. L.; Ahn, T. K.; Mancal, T.; Cheng, Y. C.; Blankenship, R. E.; Fleming, G. R. Evidence for Wavelike Energy Transfer through Quantum Coherence in Photosynthetic Systems. Nature 2007, 446, 782−786. (15) Panitchayangkoon, G.; Hayes, D.; Fransted, K. A.; Caram, J. R.; Harel, E.; Wen, J. Z.; Blankenship, R. E.; Engel, G. S. Long-Lived Quantum Coherence in Photosynthetic Complexes at Physiological Temperature. Proc. Natl. Acad. Sci., U.S.A. 2010, 107, 12766−12770. (16) Pullerits, T.; Zigmantas, D.; Sundström, V. Beatings in Electronic 2d Spectroscopy Suggest Another Role of Vibrations in Photosynthetic Light Harvesting. Proc. Natl. Acad. Sci., U.S.A. 2013, 110, 1148−1149. (17) Tiwari, V.; Peters, W. K.; Jonas, D. M. Electronic Resonance with Anticorrelated Pigment Vibrations Drives Photosynthetic Energy Transfer Outside the Adiabatic Framework. Proc. Natl. Acad. Sci., U.S.A. 2013, 110, 1203−1208. (18) Hayes, D.; Wen, J.; Panitchayangkoon, G.; Blankenship, R. E.; Engel, G. S. Robustness of Electronic Coherence in the FennaMatthews-Olson Complex to Vibronic and Structural Modifications. Faraday Discuss. 2011, 150, 459−469.

AUTHOR INFORMATION

Corresponding Author

*Phone: 773-834-0818; e-mail: [email protected]; gsengel@ uchicago.edu. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge support from the NSF MRSEC (DMR 08-02054), AFOSR (FA9550-09-1-0117), DTRA (HDTRA1-10-1-0091), and the DARPA QuBE program (N66001-10-1-4060) for supporting portions of this work. K.M.P. acknowledges the support of the DOE Computational Science Graduate Fellowship under Grant No. DE-FG0297ER25308. Use of the Center for Nanoscale Materials was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC0206CH11357. D.K.M. acknowledges support by the U.S. 2701



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(19) Fidler, A. F.; Caram, J. R.; Hayes, D.; Engel, G. S. Towards a Coherent Picture of Excitonic Coherence in the Fenna-Matthews-Olson Complex. J. Phys. B: Atmos., Mol. Opt. Phys. 2012, 45, 154013. (20) Turner, D. B.; Wilk, K. E.; Curmi, P. M. G.; Scholes, G. D. Comparison of Electronic and Vibrational Coherence Measured by Two-Dimensional Electronic Spectroscopy. J. Phys. Chem. Lett. 2011, 2, 1904−1911. (21) Fransted, K. A.; Engel, G. S. Probing Vibrational Dynamics of Pm650 with Two-Dimensional Electronic Spectroscopy. Chem. Phys. 2012, 403, 59−67. (22) Savikhin, S.; Buck, D. R.; Struve, W. S. Oscillating Anisotropies in a Bacteriochlorophyll Protein: Evidence for Quantum Beating between Exciton Levels. Chem. Phys. 1997, 303−312. (23) Caram, J. R.; Fidler, A. F.; Engel, G. S. Excited and Ground State Vibrational Dynamics Revealed by Two-Dimensional Electronic Spectroscopy. J. Chem. Phys. 2012, 137, 024507. (24) Caram, J. R.; Lewis, N. H. C.; Fidler, A. F.; Engel, G. S. Signatures of Correlated Excitonic Dynamics in Two-Dimensional Spectroscopy of the Fenna-Matthew-Olson Photosynthetic Complex. J. Chem. Phys. 2012, 136, 104505. (25) Fransted, K. A.; Caram, J. R.; Hayes, D.; Engel, G. S. TwoDimensional Electronic Spectroscopy of Bacteriochlorophyll a in Solution: Elucidating the Coherence Dynamics of the FennaMatthews-Olson Complex Using Its Chromophore as a Control. J. Chem. Phys. 2012, 137, 125101. (26) Mohseni, M.; Rebentrost, P.; Lloyd, S.; Aspuru-Guzik, A. Environment-Assisted Quantum Walks in Photosynthetic Energy Transfer. J. Chem. Phys. 2008, 129, 174106. (27) Plenio, M. B.; Huelga, S. F. Dephasing-Assisted Transport: Quantum Networks and Biomolecules. New J. Phys. 2008, 10, 113019. (28) Rebentrost, P.; Mohseni, M.; Kassal, I.; Lloyd, S.; Aspuru-Guzik, A. Environment-Assisted Quantum Transport. New J. Phys. 2009, 11, 033003. (29) Pollock, F. A.; McCutcheon, D. P. S.; Lovett, B. W.; Gauger, E. M.; Nazir, A. A Multi-Site Variational Master Equation Approach to Dissipative Energy Transfer. New J. Phys. 2013, 15, 075018. (30) Nalbach, P.; Ishizaki, A.; Fleming, G. R.; Thorwart, M. Iterative Path-Integral Algorithm Versus Cumulant Time-Nonlocal Master Equation Approach for Dissipative Biomolecular Exciton Transport. New J. Phys. 2011, 13, 063040. (31) Chen, X.; Silbey, R. J. Excitation Energy Transfer in a NonMarkovian Dynamical Disordered Environment: Localization, Narrowing, and Transfer Efficiency. J. Phys. Chem. B 2011, 115, 5499−5509. (32) Caroli, C.; Combesco, R.; Nozieres, P.; Saintjam, D. Direct Calculation of Tunneling Current. J. Phys. Part C Solid 1971, 4, 916− 929. (33) Cresti, A.; Farchioni, R.; Grosso, G.; Parravicini, G. P. KeldyshGreen Function Formalism for Current Profiles in Mesoscopic Systems. Phys. Rev. B 2003, 68, 075306. (34) Rammer, J.; Smith, H. Quantum Field-Theoretical Methods in Transport-Theory of Metals. Rev. Mod. Phys. 1986, 58, 323−359. (35) Keldysh, L. V. Diagram Technique for Nonequilibrium Processes. Sov. Phys. Jetp-USSR 1965, 20, 1018−1026. (36) Manzano, D. Quantum Transport in Networks and Photosynthetic Complexes at the Steady State. PLOS One 2013, 8, e57041. (37) Lloyd, S.; Mohseni, M.; Shabani, A.; Rabitz, H. The Quantum Goldilocks Effect: On the Convergence of Timescales in Quantum Transport. arXiv:1111.4982 [quant-ph], 2012. (38) Landauer, R. Spatial Variation of Currents and Fields Due to Localized Scatterers in Metallic Conduction. IBM J. Res. Dev. 1957, 1, 223−231. (39) Kassal, I.; Yuen-Zhou, J.; Rahimi-Keshari, S. Does Coherence Enhance Transport in Photosynthesis? J. Phys. Chem. Lett. 2013, 4, 362− 367. (40) Tronrud, D. E.; Wen, J. Z.; Gay, L.; Blankenship, R. E. The Structural Basis for the Difference in Absorbance Spectra for the Fmo Antenna Protein from Various Green Sulfur Bacteria. Photosynth. Res. 2009, 100, 79−87.

(41) Wen, J. Z.; Zhang, H.; Gross, M. L.; Blankenship, R. E. Membrane Orientation of the FMO Antenna Protein from Chlorobaculum tepidum as Determined by Mass Spectrometry-Based Footprinting. Proc. Natl. Acad. Sci., U.S.A. 2009, 106, 6134−6139. (42) Can, T.; Dai, H.; Morr, D. K. Current Eigenmodes and Dephasing in Nanoscopic Quantum Networks. Phys. Rev. B 2012, 85, 195459. (43) Suna, A. Greens Function Approach to Exciton-Phonon Interactions. Phys. Rev. 1964, 135, A111−A123. (44) Segall, B.; Mahan, G. D. Phonon-Assisted Recombination of Free Excitons in Compound Semiconductors. Phys. Rev. 1968, 171, 935−948. (45) Bihary, Z.; Ratner, M. A. Dephasing Effects in Molecular Junction Conduction: An Analytical Treatment. Phys. Rev. B 2005, 72, 115439. (46) Fassioli, F.; Olaya-Castro, A.; Scholes, G. D. Coherent Energy Transfer under Incoherent Light Conditions. J. Phys. Chem. Lett. 2012, 3, 3136−3142. (47) Hoki, K.; Brumer, P. Excitation of Biomolecules by Coherent vs Incoherent Light: Model Rhodopsin Photoisomerization. Chem. Procedia 2011, 3, 122−131. (48) Brumer, P.; Shapiro, M. Molecular Response in One-Photon Absorption Via Natural Thermal Light vs Pulsed Laser Excitation. Proc. Natl. Acad. Sci., U.S.A. 2012, 109, 19575−19578. (49) Chenu, A.; Maly, P.; Mancal, T.Dynamic Coherence in Excitonic Molecular Complexes under Various Excitation Conditions. arXiv: 1306:1693v2, 2013. (50) Mancal, T.; Valkunas, L. Exciton Dynamics in Photosynthetic Complexes: Excitation by Coherent and Incoherent Light. New J. Phys. 2010, 12, 065044. (51) Scholes, G. D.; Fleming, G. R.; Olaya-Castro, A.; van Grondelle, R. Lessons from Nature About Solar Light Harvesting. Nat. Chem. 2011, 3, 763−774. (52) Caruso, F.; Huelga, S. F.; Plenio, M. B. Noise-Enhanced Classical and Quantum Capacities in Communication Networks. Phys. Rev. Lett. 2010, 105, 190501. (53) Peres, A. Zeno Paradox in Quantum-Theory. Am. J. Phys. 1980, 48, 931−932. (54) Singh, I.; Whitaker, M. A. B. Role of the Observer in QuantumMechanics and the Zeno Paradox. Am. J. Phys. 1982, 50, 882−887. (55) Ashcroft, N. W.; Mermin, N. D. Solid State Physics; Holt: New York, 1976. (56) Rebentrost, P.; Mohseni, M.; Aspuru-Guzik, A. Role of Quantum Coherence and Environmental Fluctuations in Chromophoric Energy Transport. J. Phys. Chem. B 2009, 113, 9942−9947. (57) Collini, E.; Scholes, G. D. Coherent Intrachain Energy Migration in a Conjugated Polymer at Room Temperature. Science 2009, 323, 369−373. (58) Lee, H.; Cheng, Y. C.; Fleming, G. R. Coherence Dynamics in Photosynthesis: Protein Protection of Excitonic Coherence. Science 2007, 316, 1462−1465.

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