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Adiabatic Eigenfunction Based Approach to Coherent Transfer: Application to the Fenna−Matthews−Olson (FMO) Complex and the Role of Correlations in the Efficiency of Energy Transfer Pallavi Bhattacharyya* and K. L. Sebastian* Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India ABSTRACT: We have recently suggested a method (Pallavi Bhattacharyya and K. L. Sebastian, Physical Review E 2013, 87, 062712) for the analysis of coherence in finite-level systems that are coupled to the surroundings and used it to study the process of energy transfer in the Fenna−Matthews−Olson (FMO) complex. The method makes use of adiabatic eigenstates of the Hamiltonian, with a subsequent transformation of the Hamiltonian into a form where the terms responsible for decoherence and population relaxation could be separated out at the lowest order. Thus one can account for decoherence nonperturbatively, and a Markovian type of master equation could be used for evaluating the population relaxation. In this paper, we apply this method to a two-level system as well as to a seven-level system. Comparisons with exact numerical results show that the method works quite well and is in good agreement with numerical calculations. The technique can be applied with ease to systems with larger numbers of levels as well. We also investigate how the presence of correlations among the bath degrees of freedom of the different bacteriochlorophyll a molecules of the FMO Complex affect the rate of energy transfer. Surprisingly, in the cases that we studied, our calculations suggest that the presence of anticorrelations, in contrast to correlations, make the excitation transfer more facile.

■

INTRODUCTION In the recent past, very interesting experiments by Fleming and co-workers1−3 on the Fenna−Matthews−Olson (FMO) complex of the photosystem of green sulfur bacteria revealed the existence of long-lived coherences in excitonic energy transfer. In the photosystem, the FMO complex plays the role of a bridge between the antenna and the reaction center and aids in the transfer of energy. The complex is a trimer of identical subunits, each subunit again consisting of seven “bacteriochlorophyll a” molecules. Two-dimensional electronic spectroscopic studies conducted in the recent past on this complex found survival of coherence for time scales as long as 660 fs at 77 K2 and 300 fs at 277 K.4 In all likelihood, the system interacts strongly with its environment and hence one would expect rapid loss of coherence to occur.5−12 The loss of coherence,13 caused by the quantum nature of the surroundings is referred to as decoherence. One would expect that such a loss could be avoided only if the system could be kept sufficiently isolated, and/or the temperatures could be kept significantly low. However, these experimental results imply that even in wet and hot physiological systems, one could expect coherence to persist for significant time scales. This has kindled a lot of excitement in Quantum Biology.14,15 Theoretical calculations for coherence were done by the same group.16,17 The calculations predicted the survival of coherence even at room temperature, and this was confirmed experimentally.4,18,19 This obviously is of great interest in quantum computing too, where the challenge is the prevention of decoherence. One of the reasons suggested for such long-lived coherences20 was that the presence of correlations among the bath degrees of freedom © 2013 American Chemical Society

would reduce the entanglement of the system with the surroundings, reducing the loss of phase information, thus preventing rapid decoherence. The excitation would then behave in a wavelike manner for a longer time scale and one would expect it to sample the sites (bacteriochlorophyll a molecules) faster and more effectively. Consequently, the excitation transfer from the antenna to the reaction center is expected to be more efficient and less time-consuming if there are correlations existing among the bath degrees of freedom of the different bacteriochlorophyll molecules. Theoretical modeling can help us understand the factors responsible for the retention of coherence. There has been a recent study21 on the effect of correlated bath fluctuations on the energy transfer in a different system LH2, between two adjacent B850 bacteriochlorophyll molecules. We intend to investigate the effect of correlations in the FMO Complex which is a bigger (seven-level) system. The principal challenge is to account for decoherence as exactly as possible. Ideally one should use an exact numerical procedure like the one used by Nalbach et al.22 This, however, is a rather difficult calculation. We have recently developed an approach23 which makes use of the adiabatic eigenfunctions for the electronic part and used it for the calculation of excitation transfer. It was found that the method agrees very closely with the Special Issue: Structure and Dynamics: ESDMC, IACS-2013 Received: May 7, 2013 Revised: June 24, 2013 Published: June 27, 2013 8806

dx.doi.org/10.1021/jp4045463 | J. Phys. Chem. A 2013, 117, 8806−8813

The Journal of Physical Chemistry A

Article

We first express the Hamiltonian in terms of the adiabatic electronic states. Let the eigenfunctions of the electronic Hamiltonian Hel + Hel−ph, having the energy εm(Q) be denoted by |m(Q)⟩, where we use the notation Q = (Q1,Q2,...). As Hel + Hel−ph is diagonal in terms of them, we have

exact results of Nalbach et al.8 As this method makes the calculations easier, in this paper we report results for the sevenlevel system with and without correlations using our method. We also did calculations for cases with anticorrelations. We find that correlations actually reduce the rate of energy transfer to the reaction center. Further, in general, anticorrelations are found to enhance the energy transfer. This kind of result has been found in a recent paper by Tiwari et al.,24 who reported it for a two-level system. We find that in general, the result is true for the sevenlevel system too. We now give a description of the contents of the paper. In the next section, we give a brief outline of our method. This is followed by its application to the two-level system for which exact numerical results are available. This is followed by results for the FMO complex, which has seven levels. Results are presented for correlated and anticorrelated bath.

H=

c†m(Q)

̂ Q) −iℏ∇Q T (Q) = T (Q) A(

j Â (Q) =

(1)

∑ A nmj(Q)cn†cm (8)

n≠m

where = −iℏ⟨n(Q)|(∂/∂Qj)m(Q)⟩. can be evaluated using the Hellmann−Feynman theorem and the result is Ajnm(Q)

∑ εj|j⟩⟨j| + ∑ (Vij|i⟩⟨j| + Vji|j⟩⟨i|) j

i,j

(2)

j = 1, 2, ..., N, denote the different sites in the photosystem. |j⟩ is the state in which the excitation is on the site j and it has the energy εj. Vibrations of the surroundings are represented by ⎞ ⎛ p̂ 2 1 jk = ∑ ⎜⎜ + mjk ωjk 2qjk 2⎟⎟ 2 j , k ⎝ mjk ⎠

(3)

H̅ = H̅ 0 + H̅ na

∑ mjkνjkqjk k

(9)

(10)

where (4)

qjk denotes the position of the kth harmonic oscillator associated with the jth site, having mass mjk, angular frequency ωjk and momentum p̂jk. Qj may be thought of as the shift in the energy of |j⟩ due to its interaction with the vibrations. The shift Qj is dependent on positions of harmonic oscillators and is taken to be given by Qj =

⟨n(Q)|j⟩⟨j|m(Q)⟩ εn(Q) − εm(Q)

In the following, we use the symbols a, b, c, i, and j for orbitals on the sites and m, n, r, and s for adiabatic eigenstates evaluated at Q = 0. We now work with the transformed Hamiltonian H̅ = T†(Q)HT(Q), which may be calculated to be

∑ Q j|j⟩⟨j| j

Ajnm(Q)

j A nm (Q) = −iℏ

and the coupling between them and the excitons by Hel − ph =

(7)

Â (Q) is a vector of dimension N whose jth component is the operator Â j(Q) given by

with

Hph

c†m(Q)|0⟩

In the above is the creation operator defined by = |m(Q)⟩. c†m(Q) is dependent on Q, which is very inconvenient. We therefore wish to rewrite the Hamiltonian in terms of c†m(Q=0), which are the operators creating electronic states appropriate for the equilibrium position of the bath oscillators. We shall use the notation c†m = c†m(Q=0). c†m will create the state |m(0)⟩ (=|m⟩). To rewrite the Hamiltonian, we introduce a unitary transformation T(Q) which maps |m(Q)⟩ to |m⟩ by |m(Q)⟩ = T(Q)|m⟩. It obeys the equation26−28

THE METHOD The Hamiltonian that is popularly used16,17 has an electronic part representing the excitons, a collection of harmonic oscillators representing the surroundings and a coupling between the two. It is given by

Hel =

(6)

m

■

H = Hel + Hph + Hel − ph

∑ εm(Q) cm†(Q) cm(Q) + Hph

∑

H̅ 0 =

εm(Q)cm† cm

m

+

1 2

1 + 2

⎧ ⎫ ̂2 ⎪ pjk ⎪ 2 2 ⎬ ⎨ ∑ ⎪ + mjkωjk qjk ⎪ j , k ⎩ mjk ⎭

∑ mjkνjk 2(Â j (Q))2 (11)

j,k

and H̅ na =

(5)

This term has an alternative interpretation: the presence of the excitation on the jth site shifts the equilibrium positions of all the harmonic oscillators coupled to that site and the shift is determined by the value of νjk. In the usual approaches,17,25 Hel−ph is considered as the perturbation and a master equation is derived for the time evolution of the reduced density matrix. The master equation is then evaluated approximately using various techniques. In this type of approach, it is the same term (Hel−ph) that would result in both decoherence and population relaxation. This term is treated perturbatively and would be truncated at some order in Qj’s. Realizing the importance of this term, we formulate a different approach to the problem, as outlined below.

1 2

j

j

̂ ̂ ̂ ̂ ∑ {PA j (Q) + A (Q)Pj} j

(12)

with P̂ j = ∑kvjkp̂jk. Now, the retention of coherence in the photosystem would imply that the excitation transfer occurs mostly through the delocalized adiabatic states. Consequently, one would expect the terms quadratic in Â j(Q) to be small and we shall be neglecting them. It would be sufficient to include the nonadiabatic term linear in Aj(Q). We justify this later for the photosystem (see Figure 4 and the associated discussion in the subsequent section on the FMO complex). The most important term in H̅ is εm(Q). This term fluctuates, owing to variations in Q and this would lead to decoherence. On the other hand, the nonadiabatic term H̅ na causes transfer of excitation between the different adiabatic states |m⟩ and would result in population 8807



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changes. Hence the two important effects of Hel−ph, viz., decoherence and population relaxation are separated out at the lowest order and can be accounted for separately, in a natural fashion. We imagine that that the system is initially described by ρ(0) = |α⟩⟨α|ρph(0), where ρph(0) is the density operator for the bath of oscillators, assumed to be in equilibrium at a temperature T. Up to the initial time t = 0, the system is in the ground state and the bath is in equilibrium with it. At t = 0, the excitation is created and one expects this state to be reasonably well described by a density operator that is a product. As time evolves, the system and environment are no longer independent and the system properties could be extracted only by tracing out the bath degrees of freedom. We calculate the importance of the coherence |γ⟩⟨β| at the time t. We shall specify the states |α⟩, |β⟩, |γ⟩ later. Thus we find ρβγ(t) = Trph{|γ⟩⟨β|ρ(t)}, where ρ(t) = e−iHt/ℏ|α⟩⟨α|ρph(0)e−iHt/ℏ. ρβγ(t) may be written as ρβγ(t) = Tr{cβe−iHt/ℏcα† |0⟩⟨0|ρph(0)cαeiHt/ℏcγ†, where |0⟩ denotes the vacuum state. Using the identity H = T(Q)H̅ T†(Q), one can write this as

Trph{⟨m|UI(t )|n⟩(T̂ e−i ∫0 dt1εn(Q(t1))dt1/ ℏ)ρph (0) ≈ Trph{⟨m|UI(t )|n⟩ρph (0)⟨n′|UI†(t )|m′⟩} t

× Trph{(T̂ e−i ∫0 dt1εn(Q(t1))dt1/ ℏ)ρph (0)

Thus we get ρbc (t ) ≈

∑

⟨b|m⟩⟨n|a⟩⟨a|n⟩⟨m′|c⟩pmm ′ , nn (t ) ⟨b|m⟩⟨n|a⟩⟨a|n′⟩⟨m′|c⟩pmm ′ , nn ′ (t )Dnn ′(t )

m , m ′ , n ′≠ n

(18)

where we define pmm ′ , nn ′ (t ) = Trph{⟨m|UI(t )|n⟩ρph (0)⟨n′|UI†(t )|m′⟩}

(19)

and Dnn ′(t ) (13)

t

†

t

= Trph{(T̂ e i ∫0 εn(Q(t1))dt1/ ℏ)ρph (0)(T̂ e−i ∫0 εn′(Q(t2))dt2 / ℏ)} (20)

The time evolution operator UI(t) obeys ∂UI(t ) = UI(t )H̅ na(t ) ∂t

(21)

H̅ na(t ) = e iH̅ 0t / ℏH̅ nae−iH̅ 0t / ℏ

(22)

iℏ

with

As we know that the photosystem is close to being adiabatic, we take H̅ na as a perturbation, and calculate the population relaxation using it. Further, we approximate H̅ na by its value at Q = 0 and derive a Lindblad type master equation using the standard projection operator technique and the Markovian assumption. For this we work with ρ̃(t) defined by

(14)

⟨b|m⟩⟨n|a⟩⟨a|n′⟩⟨m′|c⟩Smn , n ′ m ′(t ) (15)

m,n,m′,n′

∑

+

and arrive at23 ρbc (t ) =

∑ m,n,m′

We denote T†(Q)c†α|0⟩ = |α̅ ⟩. The calculations would turn out to be simple if |α̅⟩ is taken to be a state in which the ath site is excited, viz., |a⟩. (The alternate choice would be to take c†α|0⟩ to be |a⟩, and this would result in slightly more involved expressions, without any significant addition to the physics.) Thus in the following we shall take |α̅⟩ = |a⟩, |β̅⟩ = |b⟩ and |γ⟩̅ = |c⟩, as the calculations are rendered simpler. We now make use of an interaction picture where we take the unperturbed Hamiltonian to be H̅ 0. For this we use UI(t ) = e−iHt̅ / ℏe iH̅ 0t / ℏ

t

†

× (T̂ e i ∫0 dt1εn′(Q(t1))dt1/ ℏ)}

ρβγ (t ) = Tr{cβT (Q)e−iHt̅ / ℏT†(Q)cα†|0⟩ × ⟨0|ρph (0)cαT (Q)e iHt̅ / ℏT†(Q)cγ†}

t

†

× (T̂ e i ∫0 dt1εn′(Q(t1))dt1/ ℏ)⟨n′|UI†(t )|m′⟩}

pmm ′ , nn ′ (t ) = ⟨m|ρñ , n ′ (t )|m′⟩

where

(23)

ρ̃(t) obeys the Liouville equation ∂ρnñ ′ (t )

t

Smn , n ′ m ′(t ) = Trph{⟨m|UI(t )|n⟩(T̂ e−i ∫0 dt1εn(Q(t1))dt1/ ℏ)ρph (0) †

∂t

t


i = − [H̅ na , 0 , ρnñ ′ (t )] = Lna(t ) ρnñ ′ (t ) ℏ

T̂ and T̂ † are the time ordering and antitime ordering operators. On the right-hand side in the above equation, we write contribution from the diagonal (initial populations) and offdiagonal terms (initial coherences) separately to get

∂ρ1̃ (t ) ∂t

= − ∑ Br , r ′ , s [X r ′ , s|r ⟩⟨s|ρ1̃ (t ) + X r ′ , r ρ1̃ (t )|r ⟩⟨s|] r ,r′,s

+

Smn , n ′ m ′(t ) = Trph{⟨m|UI(t )|n⟩ρph (0)⟨n|UI†(t )|m′⟩δnn ′}

∑

Br′, r ′ , s ′ , s (X r ′ , r + Xs ′ , s)[|r ⟩⟨r′|ρ1̃ (t )|s′⟩⟨s|]

r ,r′,s′,s

(25)

t

+ (1 − δnn ′)Trph{⟨m|UI(t )|n⟩(T̂ e−i ∫0 dt1εn(Q(t1))dt1/ ℏ)ρph (0) †

with the initial condition ρ̃1(0) = |n⟩⟨n′|. We do not explicitly write the initial coherence nn′ in ρ̃1(t) to prevent undue cluttering. Note that

t


(24)

Defining ρ̃1(t) = ρ̃ph(0)Trphρ̃(t) and following well-known methods, we see that ρ̃1(t) obeys the master equation

(16)

(17)

The last term above contains both decoherence and population relaxation. The calculations can be made easier if one decouples decoherence and population relaxation by making the approximation

ρ1,̃ mm ′ (t ) = ⟨m|ρ1̃ (t )|m′⟩

(26)

|r⟩, |r′⟩, |s′⟩, and |s⟩ are the adiabatic states, at Q = 0. Br,r′,s = ∑jAjr,r′,Ajr′s; Br,r′,s′,s = ∑jAjr,r′,Ajs′s; and ωrr′ = (εr−εr′)/ℏ. 8808


The Journal of Physical Chemistry A ⎛ π J (ω )ω 2 rr ′ rr ′ ⎜ if ωrr ′ > 0 ⎜ ℏ e βℏωrr′ − 1 Xr ,r′ = ⎜ 2 ⎜ π J(ωr ′ r )ωr ′ r if ω > 0 ⎜ r′r ⎝ ℏ 1 − e−βℏωr′r

Article

Re(ϕn , n ′(t )) =

(27)

⎛ β ℏω ⎞ 1 − cos(ωt ) ⎟ coth⎜ 2 ⎝ 2 ⎠ ω 2 ⎞ ⎛ ∂εn ∂εn ′ ⎟ ⎜ × ∑ ⎜ − ∂Q j ⎟⎠ j = 1,2,...,7 ⎝ ∂Q j Q =0 1 ℏ

∫0

∞

dω J(ω)

j

(31)

Here J(ω) is the spectral density defined by J(ω) =

∑ k

mjk νjk 2 2ωjk

and δ(ω − ωjk)

Im(ϕn , n ′(t )) =

(28)

■

APPLICATION TO THE TWO-LEVEL SYSTEM We now apply the above formalism to a two-level system with sites denoted as 1 and 2. We write the Hamiltonian for the twolevel system as

H̅ 0 =

∑ m

1 + 2

J(ω) =

(29)

(32)

2λ ωωc π ω 2 + ωc 2

(33)

density one gets 2 ⎞ ⎛ ∂ ε ∂ ε n n ′ ⎟ Re(ϕn , n ′(t )) = f (t ) ∑ ⎜⎜ − ∂Q j ∂Q j ⎟⎠ j ⎝ Q =0

⎧ ⎫ ̂2 ⎪ pjk ⎪ ∑ ⎨⎪ + mjkωjk 2qjk 2⎬⎪ m j , k ⎩ jk ⎭

j

(34)

where ⎛ λ ⎛ β ℏωc ⎞ −ω t ⎟(e c + ωct − 1) f (t ) = ⎜⎜ cot⎜ ⎝ 2 ⎠ ⎝ ℏωc +

t

t

4λωc 2

ℏβ

∞

∑ n=1

⎞ 1 (e−νnt + νnt − 1)⎟⎟ 2 νn(νn − ωc ) ⎠ 2

(35)

and

× (T̂ e i ∫0 dt1εn′(Q (t1))dt1/ ℏ)}

⎛ λ −ω t ⎞ Im(ϕn , n ′(t )) = ⎜ − (e c + ωct − 1)⎟ ⎝ ℏωc ⎠

= Trph{⟨n|e−iH̅ 0t / ℏ|n⟩ρph (0)⟨n′|e iH̅ 0t / ℏ|n′⟩} = Trph{⟨n|e−iH0t / ℏe iH0t / ℏe−iH̅ 0t / ℏ|n⟩ρph (0)

⎛ 2 2⎞ ⎛ ⎞ ⎟ ⎜⎛ ∂εn ⎞ ∂εn ′ ⎟ ⎜ ⎟ ⎜ × ∑ ⎜⎜ −⎜ ⎟ ⎟ ⎟ ∂Q j ⎠ ⎟ j ⎜⎝ ∂Q j ⎠ ⎝ Q j= 0 Q j= 0 ⎠ ⎝

× ⟨n′|e iH̅ 0t / ℏe−iH0t / ℏe iH0t / ℏ|n′⟩} = e−iεnn′t / ℏTrph{⟨n|e iH0t / ℏe−iH̅ 0t / ℏ|n⟩ρph (0) × ⟨n′|e iH̅ 0t / ℏe−iH0t / ℏ|n′⟩}

(36)

For the above Hamiltonian, ∑j[(∂ε+/∂Qj) − (∂ε−/∂Qj)Qj=02))

t

= e−iεnn′t / ℏTrph{(T̂ e−i ∫0 dt1VI(t1)/ ℏ)ρph (0)

= 2 cos2 θ, ∑j([(∂ε+/∂Qj)]Qj=02 − [(∂ε−/∂QjWi)]Qj=02) = 0 and

t

× (T̂ e i ∫0 dt1VI(t1)/ ℏ)}

hence ϕ+,−(t) is real and is given by

Using cumulant expansion and writing Qj(t) in terms of creation and annihilation operators, we can evaluate the above expression. Then we get23 Dnn ′(t ) = e−iεnn′t / ℏe−ϕn,n′(t )

sin(ωt ) − ωt ω2

where λ = ∫ 0∞dω J(ω)/ω is the reorganization energy. With this

Dnn ′(t ) = Trph{(T̂ e−i ∫0 dt1εn(Q (t1))dt1/ ℏ)ρph (0)

†

dω J(ω)

density (also referred to as Debye spectral density)

We expand εm(Q) up to first order in with respect to Q. Consequently, εm(Q) = ε0m + ∑j(∂εm/∂Qj)Qj=0Qj. Therefore, H̅ 0 = H0 + V where H0 = ∑mε0mc†mcm + 1/2∑j,k{(p̂jk2/mjk) + mjkωjk2qjk2} and V = ∑j,m(∂εm/∂Qj)Qj=0Qjc†mcm. Now we can write

†

∞

Following Nalbach et al.,8 we consider the Drude spectral

Then the energy eigenvalues are e− = −Δ with the eigenvector (−cos(θ/2), sin(θ/2)) and e+ = Δ with the eigenvector (sin(θ/ 2), cos(θ/2)). We can evaluate the quantity Dnn′(t) in eq 20 in the following manner: We have from eq 11 εm(Q)cm† cm

∫0

⎛ 2 2⎞ ⎞ ⎛ ⎜⎛ ∂εn ⎞ ⎟ ∂ ε ⎟ × ∑ ⎜⎜⎜ − ⎜⎜ n ′ ⎟⎟ ⎟ ⎟ ∂Q j ⎠ ⎟ j ⎜⎝ ∂Q j ⎠ ⎝ Q j= 0 Q j= 0 ⎠ ⎝

Note that this is taken to be the same for all the sites and that there are no correlations between sites having different j values.

⎛−cos θ sin θ ⎞ Hclosed = Δ⎜ ⎟ ⎝ sin θ cos θ ⎠

1 ℏ

ϕ(t ) = 2 cos2 θf (t )

(37)

where we have omitted the subscripts on ϕ. For population relaxation, ρ̃1,mm′(t) from eqs 25 and 26 can be

(30)

where εnn′ = ε0n − ε0n′,

evaluated using the matrix equation 8809



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⎛ ρ ̃ ̇ (t )⎞ ⎜ 1, −− ⎟ ⎜ ρ ̃ ̇ (t )⎟ ⎜ 1, ++ ⎟ ⎜ ̇ ⎟ ⎜ ρ1,̃ −+ (t )⎟ ⎜⎜ ⎟⎟ ̇ ⎝ ρ1,̃ +− (t )⎠ ⎛−Γ+− ⎜ ⎜ Γ+− =⎜ ⎜0 ⎜⎜ ⎝0

Γ −+ −Γ −+ 0 0

⎞⎛⎜ ρ1,̃ −− (t )⎞⎟ ⎟⎜ 0 0 ⎟⎜ ρ1,̃ ++ (t )⎟⎟ ⎟⎜ −(Γ+−+Γ −+)/2 0 ⎟⎜ ρ1,̃ −+ (t )⎟⎟ ⎟⎟ 0 −(Γ+−+Γ −+)/2 ⎠⎜⎜ ρ ̃ (t )⎟⎟ ⎝ 1, +− ⎠ 0

0

(38)

We have defined ω+− = ω+−ω− = 2Δ and ∑j(⟨−|j⟩⟨j|+⟩) = 2 × cos2(θ/2) sin2(θ/2). This gives Γ+− = exp(2βℏΔ)Γ−+ with 2

Γ+− = =

2π J(ω+−) ℏ e βℏω+− − 1

Figure 2. Initial excitation on site 2. Results are shown for evolution of population on site 2, for different values of the reorganization energy λ. A,C, E, G: decoherence without population relaxation. B, D, F, H: decoherence with population relaxation.

∑ (⟨−|j⟩⟨j|+⟩)2 j

16 λΔωc 1 (cos 2(θ /2) sin 2(θ /2)) ℏ 4Δ2 + ωc 2 e βℏ(2Δ) − 1 (39)

Initially, we provide a comparison of our results with the results obtained by exact path integral calculations of Nalbach et al.8 to provide a flavor of how well our method works (Figure 1).

Figure 3. Initial excitation on site 2. Results are shown for evolution of population on site 2, for different values of temperature. A, C, E: decoherence without population relaxation. B, D ,F: decoherence with population relaxation.

they are seen to persist only up to 400 fs. At λ = 300 cm−1 (curve H), the oscillations persist for only ∼150 fs. In Figure 3 we have taken a moderate value of λ and looked at the coherence as a function of the temperature. It shows that at 77 K the oscillations survive for about 1000 fs (curve B) whereas increasing the temperature to 150 K causes this time to reduce to about 600 fs (curve D). At 300 K, the damping increases significantly and oscillations are seen to persist only for ≈400 fs. The important points to note are (1) we have taken decoherence into account nonperturbatively, (2) the damping of the oscillations is essentially present in the plots which neglect population relaxation, and (3) even if we used a better approximation to calculate population relaxation, the oscillations are not going to survive longer.

Figure 1. Initial excitation on site 2. Results are presented for time evolution of population on site 2 and comparison of our results with those of Nalbach et al.

The values of the parameters used are phonon relaxation time τc = 100 fs, Δ cos θ = 50 cm−1, Δ sin θ = 100 cm−1, and λ = 20 cm−1, and we consider the second site to be at higher energy. The excitation is initially assumed to be on this site. The temperature is 300 K. Figure 1 shows the agreement to be very good. We also did the following: At 77 K, with Δ cos θ = 75 cm−1 and Δ sin θ = 150 cm−1, calculations were done for four different values of the reorganization energy viz., λ = 3, 30, 100, and 300 cm−1 and the results are shown in Figure 2. The effect of population relaxation can be switched off by neglecting H̅ na (curves A, C, E, and G), whereas curves B, D, F, and H account for population relaxation too. As one would expect, for λ = 3 cm−1, the oscillations continue beyond 1000 fs (curve B), whereas even a moderate value of λ = 30 cm−1 causes the oscillation to be damped significantly (curve D). A higher value of λ = 100 cm−1 (curve F) damps the oscillations still more and

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THE FMO COMPLEX: EFFECTS OF CORRELATIONS IN THE BATH We have demonstrated above the application of our method to a two-level system. However, the approach, being analytical, can be easily extended to systems with larger number of levels. Here, we 8810



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⎛ ⎞ ⎛ ⎜⎛ ∂ε ⎞ ⎜ ∂εn ⎟ − Wn , n ′ = ∑ Cij⎜⎜⎜ n ⎟⎟ ⎟ ⎜ ⎜⎝ ∂Q i ⎠Q = 0 ⎝ ∂Q j ⎠ i,j Q j= 0 i ⎝

will apply the technique to the seven-level FMO complex and use the Hamiltonian used by Nalbach and co-workers.22 However, the method would hold provided that the nonadiabatic effects are small so that they could be treated perturbatively. The fact that nonadiabatic effects are small may be seen by looking at the variation of εm(Q) with respect to Qi for typical values of m and i (Figure 4). For Qi up to 100 cm−1, the values of ∂εm(Q)/∂Qi are

⎛ ⎞ ⎛ ∂ε ⎞ ⎜ ∂εn ′ ⎟ 2⎜⎜ n ⎟⎟ ⎜ ⎟ ⎝ ∂Q i ⎠Q = 0 ⎝ ∂Q j ⎠ i

Q j= 0

⎞ ⎟ ⎟ ⎟ Q j= 0⎠

⎛ ⎞ ⎛ ∂ε ⎞ ⎜ ∂εn ′ ⎟ + ⎜⎜ n ′ ⎟⎟ ⎜ ⎟ ⎝ ∂Q i ⎠Q = 0 ⎝ ∂Q j ⎠ i

(41)

where f(t) is defined in eq 35 and ⎛ λ −ω t ⎞ Im(ϕn , n ′(t )) = ⎜− (e c + ωct − 1)⎟ × ℏ ω ⎝ ⎠ c ⎛ ⎞⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ∂ε ⎞ ⎜ ⎟⎟ ⎜⎛ ∂εn ⎞ ∂ ε ∂ ε ′ ′ n n n ⎟ ⎜ ⎟ ⎜ ⎟⎟ ⎟⎟ ⎜∑ Cij⎜⎜⎜ − ⎜⎜ ⎟⎟ ⎟ ⎜ ⎟ ⎜ ⎜ i,j ⎝ ∂Q i ⎠Q = 0 ⎝ ∂Q j ⎠ ⎟⎟ ⎜⎝ ∂Q i ⎠Q = 0 ⎝ ∂Q j ⎠ Q j= 0 Q j = 0 ⎠⎠ i i ⎝ ⎝ (42)

Also, instead of eq 39 for the case of the two-level system, the population relaxation rate for the seven-level system becomes Γnn ′ =

Figure 4. Derivative of εm(Q̅ ) with respect to Qi. For instance, the purple curve (m = 1, i = 1) implies the variation of eigenenergy ε1, with respect to Q1.

(43)

and Γn′n = exp(βεnn′)Γnn′. Note that in the above expressions, ωnn′ = εnn′/ℏ > 0. Because the final site where the excitation has to reach before getting transferred to the reaction center is site 3, we compare the evolution of population at site 3 in the presence or absence of correlations. Figures 5 and 6 compare the evolution of population for correlated, uncorrelated and anticorrelated bath for two specific cases: (a) population evolution at site 3 when the initial excitation is at site 6 at 77 K and (b) population evolution at site 3 when the initial excitation is at site 6 at 300 K. From both Figures 5 and 6, we see that the excitation transfer to site 3 is fastest when the bath degrees of freedom are anticorrelated. Recently, similar conclusions were also reported by Tiwari et al.24 We have also obtained results for the population at site 3 when the initial excitation was at site 1 at the temperatures of 77 and 300 K, respectively (Figure 7 and Figure 8). Interestingly, we

seen to be less than 0.005, showing that εm(Q) are only weakly dependent on Q, and hence for reasonable values of Q, one does not expect any two εm(Q) values to cross. As mentioned earlier, there have been arguments claiming that the long-lived coherence observed in FMO might be a consequence of correlations existing among the bath degrees of freedom of the bacteriochlorophyll a molecules of the FMO. We now investigate the effects of correlations using our method. The spectral density used here is again the Drude spectral density given as J(ω) = (2λ/ π)(ωωc/(ω2 + ωc2)). Here, λ = 35 cm−1 and ωc−1 = 50 fs. We investigate, using our technique, how the presence of correlations among the bath degrees of freedom affects the process of light-harvesting. The excitons in the FMO complex are found to be delocalized on two chromophores mostly. The following pairs of chromophores are found to be strongly coupled: 1−2 and 5−6 and also 4−5 and 4−7. However, the electronic coupling among 4−5 and 4−7 is not as strong as the other two. The stronger electronic coupling indicates closer proximity and hence there may be correlated bath fluctuations too. To model correlations, we assume that ⟨QiQj⟩ = CijJ(ω), where we refer to Cij as the correlation matrix. Our previous calculations had assumed Cij to be equal to the Kronecker delta function δij. We now remove that and take the following correlation coefficients:29 C12 = C21 = C56 = C65 = 0.9; C45 = C54 = C47 = C74 = 0.4. We also look at how the presence of anticorrelations among the bath degrees of freedom might affect the population evolution. We achieve this easily by using C12 = C21 = C56 = C65 = −0.9 and C45 = C54 = C47 = C74 = −0.4. The presence of correlations modifies the expressions for ϕn,n′(t) to Re(ϕn , n (t )) = f (t )Wn , n ′ ′

2π J(ωnn ′) (∑ Cij(⟨n|i⟩⟨i|n′⟩)(⟨n|j⟩⟨j|n′⟩)) ℏ e βℏωnn′ − 1 i , j

Figure 5. Evolution of population at site 3 at 77 K for initial excitation at site 6.

(40) 8811



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inspection, we see that at a lower temperature (77 K) the presence of anticorrelations among the bath degrees of freedom render the transfer less efficient compared to the case where we have correlations. For 300 K, we see that the population grows almost together for the three cases. We have tuned the values of the correlation coefficients Cij but, again, the population evolution at site 3 remains mostly unaffected by the presence or absence of correlations or anticorrelations. It is to be noted that the sites 3 and 4 enjoy a moderately strong electronic coupling, as seen from the Hamiltonian. Therefore, it is likely that there is correlation between the baths of the two, though ref 29 does not take it to be so. We performed calculations by giving C34 (=C43) a nonzero value. We found that the energy transfer to site 3 is enhanced considerably for the case of the initial excitation at site 6 when this is done. The efficiency is more if the value of the correlation coefficient is more. However, if they are anticorrelated, the energy transfer efficiency is roughly the same. If the initial excitation is at site 1, the energy transfer efficiency is again roughly the same for both correlations and anticorrelations even if we introduce a high correlation/ anticorrelation coefficient for sites 3 and 4.


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CONCLUSIONS We have developed an analytical approach for treating coherent wavelike excitation transfer. Unlike the usual approaches that employ perturbative techniques, our treatment employs mapping onto the adiabatic basis, to obtain a Hamiltonian that accounts for decoherence and population relaxation separately. The treatment of decoherence is nonperturbative whereas population relaxation is evaluated using a Markovian master equation. Calculations have been presented for a two-level system and found to be in excellent agreement with exact numerical results. This computationally facile approach has also been applied to larger systems. We have used it to investigate the effect of correlations among the bath degrees of freedom of the bacteriochlorophyll a molecules of the FMO Complex, the light harvesting antenna complex found in green sulfur bacteria. We observe that anticorrelations as well as correlations could suppress or enhance the energy transfer, depending on the initial site of excitation. In general, for the FMO complex, it is found that anticorrelations among the bath degrees of freedom have a stronger effect than correlations in enhancing the rate of excitation transfer.


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AUTHOR INFORMATION

Corresponding Author

*E-mail: P.B., [email protected]; K.L.S., [email protected]. in. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work of K.L.S. was supported by the Department of Science and Technology under the J. C. Bose fellowship program. P.B. thanks the Indian Institute of Science for scholarship.

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REFERENCES

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observe that bath correlations do not have so significant effects when site 1 contains the initial excitation. The population at site 3 in this case grows almost together for the correlated, uncorrelated, and anticorrelated bath. However, on closer 8812



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