Environment-Assisted Quantum Coherence in Photosynthetic

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Environment Assisted Quantum Coherence in Photosynthetic Complex Rajesh Dutta, and Biman Bagchi J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b02480 • Publication Date (Web): 30 Oct 2017 Downloaded from http://pubs.acs.org on November 1, 2017

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The Journal of Physical Chemistry Letters

Environment Assisted Quantum Coherence in Photosynthetic Complex Rajesh Dutta1 and Biman Bagchi1,* 1

SSCU, Indian Institute of Science, Bangalore 560012, India. *Email: [email protected]

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Abstract Recent experiments [Nature, 2007, 446, 782-786] revealed the existence of surprisingly long lived quantum coherence in noisy biological environment of photosynthetic Fenna-MatthewsOlson (FMO) complex. Such coherence can clearly play important role in facilitating efficient energy transfer. The occurrence of quantum coherence in quantum transport is also implicated in excitation transport processes in conjugated polymers [Science, 2009, 323, 369-373]. Even though these systems are strongly correlated, most theoretical studies invoke Markovian approximation where the temporal correlation of bath fluctuations is neglected. We use an elegant non-perturbative method based on Kubo’s quantum stochastic Liouville equation (QSLE) to study the effects of correlated non-Markovian bath fluctuations in several different limits and find the interesting result that fluctuations not only destroy coherence but under appropriate conditions can also facilitate it. We show that temperature has the most pronounced effect in the intermediate coupling limit where it can promote transition from coherent to incoherent transfer.

TOC Graphic

KEYWORDS: FMO complex, non-Markovian bath, Excitation energy transfer, Fluctuation assisted coherence, Coherent and Incoherent transfer


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Quantum transport of charge and energy can be quite distinct from their classical analogues. In the classical limit, we can imagine a relation between diffusion and fluctuating forces in terms of Einstein’s picture. The situation is quite different in the quantum case, as exemplified by profound effects of disorder in Anderson localization.1 Existence of coherence between different sites and/or energy levels plays an important role in quantum transport. Presence of disorder can destroy this coherence. A large number of studies have been devoted to include the effects of static disorder on quantum transport or diffusion (as in Anderson localization problem or in phonon localization).2 In contrast, relatively fewer studies have considered effects of dynamical disorder in transport properties.3 The situation has undergone a see-saw change in recent years, triggered by recent results in excitation energy transfer (EET) in photosynthetic complex4-9 and conjugate polymers.10-13 The main reason behind the attention is the discovery of coherent transport in the noisy biological environment. The important application of EET is that the efficiency of energy transfer from chromophores to reaction center in photosynthetic system is near about unity.14 If it can be mimicked in artificial solar cells, one can design more efficient solar cells. Historically, most of the studies of EET considered either the Markovian limit or quantum master equation approach where perturbative truncation is necessary. However, these approaches could be problematic in the cases of both photosynthetic complex and conjugate polymers where the EET timescales and the environment bath correlation time scales are comparable.5-7 In a series of pioneering studies using 2D Fourier transform electronic spectroscopy, Fleming and co-workers4 observed long lived coherent EET in photosynthetic complexes. Collini and Scholes10,11 have observed coherent intra-chain EET in conjugated polymer at room temperature. Later Ishizaki and Fleming studied theoretically explore the role of environment in energy



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transport through the consideration of reorganization for FMO15 complex. In this approach infinite set of equations are truncated after finite numbers. Another well-known approach is based on polaron transfer technique developed by Jang and co-workers16 to investigate the energy transfer dynamics. However, the second-order perturbative truncation with respect to the renormalized electron-phonon coupling is very crude approximation in case photosynthetic systems. Few years ago, Aspuru-Guzik and co-workers17,18 have described environment assisted quantum transport in real photosynthetic complex and model systems. Silbey and co-workers19-23 have computed the efficiency of exciton transfer in case of dimer model system and population relaxation for FMO complex. However, use of Bourret’s integrals is limited to fast fluctuations or short bath correlation time. Despite all the studies, however, there is yet no study on the effects of temporal correlation of bath in a non-Markovian environment on transport in an extended system. In this study we explore coherent and incoherent transport of excitation energy as a function of bath correlation time and the fluctuation strength. In our earlier studies we observed quantum diffusion, line shape and occupation probability in presence of dynamical disorder for several model systems.24,25 Recently Moix, Khasin and Cao26 have concluded that although the static disorder leads to lack of diffusion and localization, a small amount of dynamic disorder is enough to continue the exciton transport. In the present study we consider real photosynthetic FMO complex such that we can investigate the localization and delocalization excitation energy in presence of both heterogeneity of site energies and dynamic disorders. Inclusion of memory effects in the combined system-bath dynamics is highly non-trivial and is rarely attempted. Usual treatments invoke a Markovian approximation where bath dynamics is delta correlated in time. There is however, an alternative powerful approach to


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include memory effects introduced by Kubo which is known as quantum stochastic Liouville equation (QSLE).27 In this approach, one needs to specify the form of bath evolution operator as described below. In the present study, we consider the total Hamiltonian in the interaction representation of the fluctuating bath Hamiltonian, so that the interaction V is time dependent and can be modeled by a stochastic function with known statistical properties. In this interaction representation, the Hamiltonian can be represented as follows,

H (t) = HS + V (t)

(1)

We use the Hamiltonian based on the theories of Haken-Strobl-Reineker28,29 and Silbey30 where the system (exciton) Hamiltonian (Hs) is defined as

HS =

∑E

k

k

k

k +

∑J

kl

k

l

.

(2)

k ,l k ≠l

Here, Ek is the energy of an electronic exciton localized at site k in absence of any coupling and

Jkl is the time independent off-diagonal interaction or inter-site coupling between excitations at site k and l. We assume J kl = J kl* = Jlk = J . Thus in the interaction representation, the time dependent coupling potential can be written as,

V (t ) = Vd (t )∑ k k + Vod (t )∑ k l k

k ,l k ≠l

.

(3)

Vint is the interaction Hamiltonian between the system (exciton) and the environment. Here, Vd ( t ) and Vod ( t ) denote diagonal (local) and off-diagonal (non-local) parts of the fluctuating Hamiltonian Vint . We assume V(t) to be described either by a Poisson stochastic process or a 5 ACS Paragon Plus Environment


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Gaussian Markov process.31 We use Kubo’s QSLE (see Supporting Information (SI)) and the equation is provided as follows ∂σ i = − [ H (t ), σ ] + Γ vσ . ∂t h

(4)

where σ is the reduced density matrix (averaged over system variables) defined using joint probability distribution, Γ v is the stochastic diffusion operator (see SI) and V is the same random energy variable. The QSLE was used to explain exciton transport in one dimensional lattice considering correlated and uncorrelated bath for continuum model.32 The non-perturbative approach can be used to study both strong coupling non-Markovian limit and weak coupling Markovian limit. In weak coupling and Markovian limit EET ensues with the equilibrium phonon states and classical rate expression is appropriate to explain the EET dynamics. The behavior can be used to explain FRET.33 We consider two systems cyclic hetero trimer and heptamer; and use system parameters from FMO Hamiltonian (SI). All the calculations are performed considering uncorrelated diagonal fluctuations because of negligibly small off-diagonal fluctuations in photosynthetic systems. Here uncorrelated diagonal fluctuations refer to the situation where all the fluctuating diagonal elements in the Hamiltonian are independent of each other. The coupled equations of motion (EOM) for cyclic trimer is given as follows x

∂σ jmn ∂t

x

i  = −  ∑ Ek k k  σ jmn h  k =1  3

 3  1 x i i −  ∑ J kl k l  σ jmnp − Vd ∑ (δ j +1, j ′ + δ j −1, j′ ) ( 1 1 ) σ j ′mn   h  k =1,l =1 h j′= 0   k ≠l 

1 i − Vd ∑ (δ m +1,m′ + δ m −1,m′ ) ( 2 2 h m′ = 0

)

x

i h

1

σ jm′n − Vd ∑ (δ n +1,n′ + δ n −1,n′ ) ( 3 3 ) σ jmn′ x

n′ = 0

− jbdσ jmnp − mbdσ jmnp − nbdσ jmnp .

(5) 6

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where, Vd is the strength of the fluctuation and bd corresponds to the rate of fluctuation. We assume uncorrelated nearest neighbor diagonal fluctuations have same strength and rate. We have numerically solved the coupled EOM using Runge-Kutta fourth order method. Population of each site can be represented as follows, Pn ( t ) = n σ

N

∏ ai

n

(6)

i =1

where, n is the site number, N is the total number of site and a1 , a2 , a3 ,....aN all the elements are zero that denotes the equilibrium states of Poisson bath. 1

0.6

0.8

0.4 0.2 0 0

0.6 0.4 0.2

0.5

1 1.5 Time (ps)

2

0 0

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2

2.5

0.3

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(c)

||

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||

(d) 0.25

||

Off-diagonal

||

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||

0.2 0.1 0 0

Site1 Site2 Site3 Site1 Site2 Site3

(b) Population

0.8 Population

1

Site1 Site2 Site3 Site1 Site2 Site3

(a)

Off-diagonal

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|| ||

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||

0.15 0.1 0.05

0.5

1 1.5 Time (ps)

2

2.5

0 0

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1 1.5 Time (ps)

2

2.5 −1

Figure 1.(a)-(b) depict population dynamics in trimer system. (a) Solid line :Vd= 50 cm-1 and bd = −1

−1

10 fs., dashed line :Vd= 50 cm-1 and bd = 250 fs. (b) Solid line :Vd= 350 cm-1 and bd = 10 fs., dashed 7 ACS Paragon Plus Environment


−1

line : Vd= 350 cm-1 and bd = 250 fs. Figure (c)-(d) indicate absolute value of coherence(c) Solid line −1

−1

:Vd= 50 cm-1 and bd = 10 fs., dashed line Vd= 50 cm-1 and bd = 250 fs. (d) Solid line :Vd= 350 cm-1 −1

−1

and bd = 10 fs, dashed line : Vd= 350 cm-1 and bd = 250 fs. 1

1 Site1 Site2 Site3 Site4 Site5 Site6 Site7

Population

0.8 0.6

0.8

0.4

0.6 0.4 0.2

0.2 0 0

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1 1.5 Time (ps)

0.5

2

0 0

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Off-diagonal

|| ||

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(d)

||

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(c) Off-diagonal

Site1 Site2 Site3 Site4 Site5 Site6 Site7

(b) Population

(a)

|| ||

0.3

||

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||

0.1

0 0

0.5

1 1.5 Time (ps)

2

0 0

2.5

0.5

1 1.5 Time (ps)

2

2.5

Figure 2. .(a)-(b) depict population dynamics of FMO complex (cyclic seven sites system). (a) Vd= 50 −1 −1 cm-1 and bd = 10 fs. (b) Vd= 50 cm-1 and bd = 250 fs. Figure (c)-(d) signify the absolute value of −1

coherences of FMO complex (cyclic seven sites system), with (c): Vd= 50 cm-1 and bd = 10 fs. −1

(d):Vd= 50 cm-1 and bd = 250 fs..

1

1 Site1 Site2 Site3 Site4 Site5 Site6 Site7

0.8 0.6 0.4

0.8

0.2 0 0

Site1 Site2 Site3 Site4 Site5 Site6 Site7

(b) Population

(a) Population

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0.6 0.4 0.2

0.5

1 1.5 Time (ps)

2

2.5

0 0


0.5

1 1.5 Time (ps)

2

2.5 8

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0.25

0.3

||

(c)

Off-diagonal

|| ||

0.15

||

0.1

||

0.05 0 0

||

(d)

0.25

||

0.2 Off-diagonal

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||

0.2

||

0.15

||

|| ||

0.1 0.05

0.2

0.4 0.6 Time (ps)

0.8

1

0 0

0.2

0.4 0.6 Time (ps)

0.8

1

Figure 3.(a)-(b) depict population dynamics of FMO complex (cyclic seven sites system). (a)Vd= 350 −1 −1 cm-1 and bd = 10 fs. (b) Vd= 350 cm-1 and bd = 250 fs. Figure (c)-(d) indicates absolute value of −1

coherences of FMO complex (cyclic seven sites system), with (c): Vd= 350 cm-1 and bd = 10 fs. −1

(d):Vd= 350 cm-1 and bd = 250 fs..

In the absence of bath fluctuations, coherence is trivial because in this case it occurs due to the presence of the constant inter-site coupling J. However, in the presence of bath fluctuations several possibilities arise, one must consider both the role of bath fluctuation strength (Vd) and −1

bath correlation time ( bd ). In the absence of any off-diagonal coupling J, transition from nonMarkovian to Markovian limit can be obtained by increasing bd alone, at fixed Vd which is the real picture in case of vibrational energy relaxation.31 However, the situation is complex in the presence of a significant J, as shown below. According to recent studies mentioned earlier, EET in photosynthetic complex as well as in conjugated polymers occurs in the intermediate regime where EET and bath correlation time scales are almost similar. These studies indicate that EET in photosynthetic or conjugated polymer occurs in non-Markovian limit. In this study strong coupling denotes dominance of time independent off-diagonal coupling J. For the fixed value of J in case of FMO complex, value of −1 Vd= 50 cm-1 and bd = 250 fs. falls in the strong coupling limit. However, in this case weak −1 coupling refers Vd= 350 cm-1 and bd = 10 fs. In this study we consider intermediate limits as



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−1 −1 Vd= 50 cm-1 and bd = 10 fs. as well as Vd= 350 cm-1 and bd = 250 fs. Both strong coupling and

intermediate regime correspond to non-Markovian limit. Markovian or weak coupling limit can −1 be obtained when Vd is large and bd is small. In this limit

Vd2 act as an effective bath relaxation bd

Vd2 rate. If is much larger than the energy transfer time or time the scale associated with J, bd energy transfer can be incoherent. The Figures 1-3 show population and absolute value of coherences for trimer as well as FMO complex. Initially site 2 gains most of the population by taking the advantage of quantum coherence. In case of strong coupling limit, population shows oscillatory dynamics i.e. EET is completely coherent up to 3 ps. However, in intermediate coupling regime duration of oscillation decreases and we observe transition from coherent to incoherent transport of energy at 1 ps. In Fig. 1a and 1b we plot population for small as well as large bath correlation time, for two fixed values of fluctuation strength. Even when the bath correlation time (inverse of bd) is small, increase in the fluctuation strength (Vd) may not be suffice to reduce or remove the effect of the electronic coupling J. Consequently, coherence and oscillations may survive for a long time. One of the main parameters that determines the transport mechanism is

Vd2 and not only bd bd

alone, but also J. A major outcome of the present work is the understanding of the precise role played by the static (time averaged) off-diagonal coupling J in a non-Markovian description. This aspect is often not realized in a Markovian description that does not consider Vd and bd separately.


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Vd2 If the ratio ( ) is much greater than J, and at the same time both bd and Vd are larger bd than J, then the motion can become incoherent. However, if bd alone becomes large at a fixed Vd, the ratio

Vd2 may become smaller than J. In this limit, oscillation could last for a long time. bd

Similarly one can explain the results for FMO complex as well. In the intermediate coupling regime when both fluctuation strength and bath correlation time are large, coherence could be destroyed by high fluctuation strength. In this case, one can induce a crossover at intermediate times from coherence to incoherence by changing the bath correlation time. One can also explain similar behavior where fluctuation strength and bath correlation time scale are small or fluctuation is very fast. The asymmetry between the behavior of both Vd and

bd−1 as well as the

Vd2 ratio helps in coherent transport and consequently one can term this as an bd

environment or fluctuation assisted coherent EET. We also plot the absolute value of coherences in equilibrium bath states in all the limits. All the non-local coherences actually denote the non-local quantum correlations and create new pathways of EET and interference between pathways as well. The non-local terms have less contribution when environment effect is strong. Coherences in excited bath states have negligibly small contributions in the FMO complex that vary from 10-2 to 10-6 with increasing bath correlation time scales. In case of FMO complex all the site energies and coupling elements are different. Consequently there is a possibility of competition between localization due to site energy heterogeneities and delocalization due to dynamic disorders (bath fluctuations). In this study we observe slow non11 ACS Paragon Plus Environment


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oscillatory EET dynamics which essentially denote the dominance of the effect of heterogeneity in site energies. Thus the only possibility of energy transfer is the incoherent or hopping mechanism in the long time limit. We consider the temperature corrected quantum stochastic Liouville equation derived by Tanimura and Kubo.34,35 Temperature independent quantum stochastic Liouville equation ignores the reaction of the system to the bath. The high temperature classical bath approximation leads to equal population of all sites in long time and the system does not attain Boltzmann distribution. The main difference between temperature corrected and the temperature independent stochastic Liouville equation is that, in addition to the stochastic interaction term it contains dissipation term which is missing in the stochastic model and helps to capture the effect of bath on system dynamics. These two terms are related through the fluctuation–dissipation theorem leads the system to attain Boltzmann distribution at finite temperature and at long time limit. Temperature corrected stochastic Liouville equation for Gaussian modulation case can be written as follows ∂σ ( V,t ) ∂t

 i ∂  ∂  iβ b  ∂  o =  − H (t ) x − b V + − V +  V σ ( V,t ) ∂V  ∂V  2  ∂V    h

(7)

where, as before, H (t ) = H ex + V (t ) and V is the random variable for the Gaussian modulation. First two terms on the right hand side of Eq. (7) is essentially same as the QSLE equation. Here,

β=

1 o , O f = Of + fO . The term associated with β is the reaction of the system to the bath k BT

and is the temperature correction term that helps the system to reach thermal equilibrium. The set of coupled EOM in the hierarchy is now given as follows ∂σ m β bh o  i i i = − H exx σ m − V xσ m +1 −  V x − V  σ m −1 − mbσ m ∂t h h h 2 

(8) 12


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It is well known that the truncation after second bath state (i.e. considering only 0 and 1 bath states) in QSLE equation provides the exact expression for Poisson bath or the two state jump model. In the present case, where we consider fluctuation as a Gaussian stochastic process, we include up to fourth bath state from the infinite number of bath states. (see SI, Eq.S13 for coupled EOM). 1

0.6

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0.4

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2

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1 1.5 Time (ps)

1 Site1 (Temp. Dep.) Site2 (Temp. Dep.) Site3 (Temp. Dep.) Site1 (Temp. Indep.) Site2 (Temp. Indep.) Site3 (Temp. Indep.)

(c)

0 0

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2

2.5 Site1 (Temp. Dep.) Site2 (Temp. Dep.) Site3 (Temp. Dep.) Site1 (Temp. Indep.) Site2 (Temp. Indep.) Site3 (Temp. Indep.)

(d) 0.8 Population

0.5

1

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Site1 (Temp. Dep.) Site2 (Temp. Dep.) Site3 (Temp. Dep.) Site1 (Temp. Indep.) Site2 (Temp. Indep.) Site3 (Temp. Indep.)

(b) Population

0.8 Population

1

Site1 (Temp. Dep.) Site2 (Temp. Dep.) Site3 (Temp. Dep.) Site1 (Temp. Indep.) Site2 (Temp. Indep.) Site3 (Temp. Indep.)

(a)

Population

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0.5

1 1.5 Time (ps)

2

2.5

0 0

0.5

1 1.5 Time (ps)

2

2.5 −1

Figure 4.(a)-(d) corresponds to population dynamics of trimer. (a): Vd= 50 cm-1 and bd = 10 fs. (b): −1

−1

−1

Vd= 50 cm-1 and bd = 250 fs. (c): Vd= 350 cm-1 and bd = 10 fs. (d): Vd= 350 cm-1 and bd = 250 fs. Solid line corresponds to temperature independent study whereas dotted line corresponds to temperature independent study. Effect of temperature is investigated at T = 300 K.

Figure 4 shows the effects of temperature in excitation transfer dynamics. Temperature promotes incoherent transport by destruction of coherence. In strong coupling limit when environment effect is negligible (Figure 4a), EET occurs via coherent mechanism and there is very small difference between temperature dependent and independent theories. However, in



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long time limit after the transition from coherent to incoherent transport, the difference increases. Similarly when environment effect is very strong there is negligibly small effect of temperature on dynamics (Figure 4c) as environment contribution is quite larger than temperature. In the intermediate coupling regime when there is transition from coherent to incoherent transport the effect of temperature is prominent (Figures 4b and 4d). In this study, we use both temperature dependent and independent QSLE to investigate the EET dynamics of trimer and FMO complex. We consider fluctuation as a Poisson stochastic process for temperature independent studies. However, for temperature dependent studies we consider Gaussian bath. The main results of the present study are summarized as follows (1) The condition of coherence is that, the constant off-diagonal coupling J has to be larger than Vd. That is, to get incoherence, J must be overwhelmed. Thus, in extended systems with multiple chromophores, J may be significant, and coherence is natural. In nonMarkovian or strong coupling limit energy transport is fully coherent up to long time (~3 ps.). However, in the intermediate coupling regime we observe transition from coherent to incoherent excitation transport at 1 ps.. The asymmetry between the behavior Vd2 of both Vd and b as well as the ratio makes the study of EET dynamics non-trivial. bd −1 d

−1

In the Markovian and weak coupling limit, Vd and bd are combined together in the form Vd2 , the ratio dictates the nature EET dynamics and energy transfer occurs through bd incoherent or hopping mechanism. Excitation transfer in the intermediate regime can be termed as fluctuation or environment assisted coherent transport. In the absence of J (or,


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very small J) where one would expect incoherence, one can still get coherence in the −1 non-Markovian limit where Vd is large, bd is also large and baths are temporally

correlated. This is an important result of this work. (2) Absolute values of coherence in different limits indicate the non-local quantum correlation between different sites that lead to create new pathways for EET and consequently the EET dynamics is facilitated. (3) Presence of site energy heterogeneities and dynamic disorders creates competition between localization and delocalization of EET.

At intermediate time regime

heterogeneities in the site energies localizes the excitation energy whereas dynamic disorders help in delocalization of the same. At long time limit slow dynamics indicate the localization of excitation energy on each site. (4) Effects of temperature are investigated by the comparison between results obtained from temperature dependent and independent theories. In the strong coupling and in the nonMarkovian limit, the difference between both theories can be observed in the long time limit. As the temperature independent theory provides equal population in long time limit, difference between two theories increases in this limit. In the weak coupling or Markovian limit when environment effect is strong, both theories predict more or less same dynamics. However, in the intermediate regime when transition from coherent to incoherent transport occurs, temperature effect is more prominent. Experimentally it was observed that for FMO complex coherence can survive upto 1 ps. In our study Figures 2(b) and 3(b) show that coherence in the intermediate limit can survive upto 1 ps. which is similar to the experimental time scale of coherent EET in photosynthetic complex.



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Sibey and co-workers studied efficiency of conversion and dependence on the trapping rate using extended Haken-Strobl model and they observed that efficiency is less than that obtained from full quantum calculation. They also concluded that EET efficiency in FMO complex is dominated by Förster hopping rate. To obtain greater efficiency, just coherent energy transfer is not enough. Presence of incoherent mechanism in long time limit after initial coherent dynamics helps in increasing the efficiency. To study the scenario in detail, one needs to consider both the effects of temperature and vibration in the dynamics in the non-Markovian limit.

ACKNOWLEDGEMENTS BB thanks Department of Science and Technology (DST, India) and Sir J. C. Bose fellowship for providing partial financial support. RD thanks Dr. Sarmistha Sarkar, Mr. Sayantan Mondal and Mr. Saumyak Mukherjee for carefully reading the manuscript.

Supporting Information Available Detailed description about temperature independent quantum stochastic Liouville equation and temperature dependent coupled equations of motion is provided.

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Environment-Assisted Quantum Coherence in Photosynthetic

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