Assistance of Molecular Vibrations on Coherent Energy Transfer inPhotosynthesis from the View of a Quantum Heat EngineZhedong Zhang† and Jin Wang*,†,‡,¶

†Department of Physics and Astronomy and ‡Department of Chemistry, SUNY Stony Brook, Stony Brook, New York 11794, UnitedStates¶State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences,Changchun, Jilin 130022, People’s Republic of China

*S Supporting Information

ABSTRACT: Recently, the quantum nature in the energytransport in solar cells and light-harvesting complexes hasattracted much attention as being triggered by theexperimental observations. We model the light-harvestingcomplex (i.e., PEB50 dimer) as a quantum heat engine (QHE)and study the effect of the undamped intramolecule vibrationalmodes on the coherent energy-transfer process and quantumtransport. We find that the exciton−vibration interaction hasnontrivial contribution to the promotion of quantum yield as well as transport properties of the QHE at steady state by enhancingthe quantum coherence quantified by entanglement entropy. The perfect quantum yield over 90% has been obtained, with theexciton−vibration coupling. We attribute these improvements to the renormalization of the electronic couplings effectivelyinduced by exciton−vibration interaction and the subsequent delocalization of excitons. Finally, we demonstrate that the thermalrelaxation and dephasing can help the excitation energy transfer in the PEB50 dimer.

■ INTRODUCTION

Recently, widespread interest in uncovering the quantumphenomena in solars cell and the photosynthetic process hasbeen triggered by experimental investigations of the electronicdynamics in light-harvesting and Fenna−Matthews−Olson(FMO) complexes.1−4 The transport of excitation energyabsorbed by antenna toward the reaction center (RC) occurswith a perfect efficiency over 90%,5,6 which was shown to bestrongly correlated with the long-lived quantum coherencebetween different molecules.7,8

Numerous research studies made it clear that the excitonenergy transport in photosynthetic organisms critically dependson the exciton−phonon interaction,9−12 besides the electroniccoupling between molecules. The exciton−phonon interactionshave two types: high-frequency modes from the nucleivibrations in molecules and low-frequency modes induced byenvironmental fluctuations. Actually, these two kinds ofexciton−phonon interactions in these complexes are associatedwith low-energy fluctuations of protein immersed in the solventand intramolecular vibrations,13,14 respectively. Because theintramolecular vibrations are undamped, it is evidently shownto have significant influence on the coherent energy transferwhen energy quanta of vibrational modes is in resonance withthe energy splitting of excitons.15−17

In this work, we study the effect of intramolecule vibrationson energy-transfer processes in the pairs of chromophores,which describes several light-harvesting antennae in the nature.Two important examples are the central PEB50c−PEB50d dimerin the cryptophyte antennae PE545 (phycoerythrin 545; see

Figure 1) and the Chlb601−Chla602 pair in light-harvestingcomplex II (LHCII). The natural photosynthetic organismfunctions in the presence of both solar radiation and low-energy

Received: February 16, 2015Revised: March 5, 2015Published: March 16, 2015

Figure 1. (top) Illustration of cryptophyte antennae phycoerythrin545 (PE545). (bottom) Schematic of our model for the PEB50c−PEB50d dimer in PE545. Standard parameters are taken from refs 29and 30, ω = 800 cm−1 being quasi-resonant with the intramolecularmode of frequency, ε1 − ε2 = 1042 cm−1, Δ = 92 cm−1, and T2 = 300K.

Article

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© 2015 American Chemical Society 4662 DOI: 10.1021/acs.jpcb.5b01569J. Phys. Chem. B 2015, 119, 4662−4667

fluctuations of the protein. In our model, there are threedifferent types of energies involved, namely, solar radiation,high-frequency modes from the nuclei vibrations, and low-frequency modes from the environmental fluctuations.According to the quantum thermodynamics, the quantumheat engine (QHE) converts hot thermal radiation into low-entropy useful work.18−20 We will include the dynamics ofdiscrete vibrational modes between intramolecules, togetherwith the dynamics of system, due to the comparable relaxationtime.8,21,22 We will find that the interaction with vibrationalmodes has a nontrivial contribution to the enhancement of thenonequilibrium flux23,24 trapped by the RC and energy transferefficiency (ETE) by improving the coherence.

■ POLARON TRANSFORM AND QUANTUM MASTEREQUATION (QME)

We consider a prototype dimer where each chromophore hasan excited state described by exciton with energy εi stronglycoupled to a quantized vibrational mode of frequency ω, withthe identical coupling strength λ for i = 1,2. The Hamiltonian ofthe dimer and vibrational modes reads

ε ε= | ⟩⟨ | + + + Δ +† † † †H E c c c c c c c c0 0 ( )ex 0 1 1 1 2 2 2 1 2 2 1 (1)

and Hvib = ℏω(b1†b1 + b2

†b2). The interaction term is

λ ω λ ω= ℏ + + ℏ +−† † † †H c c b b c c b b2 ( ) 2 ( )ex vib 1 1 1 1 2 2 2 2

(2)

In what follows, we will reduce the degrees of the exciton−vibration dynamics by introducing the correlated (+) andanticorrelated (−) vibrational coordinates: b± = (b1 ± b2)/√2,which rewrites the Hamiltonian as

λ ω

λ ω

= ℏ − +

+ ℏ + +−

† †− −

†

† †+ +

†

H c c c c b b

c c c c b b

( )( )

( )( )ex vib 1 1 2 2

1 1 2 2 (3)

where the first term in Hex−vib describes the vibrational modewith a phase difference of π while the second term describes thecenter-of-mass motion of vibrational modes. Thus, theanticorrelated vibrational mode is excited by optical phonons.25

Next, we will employ the polaron transformation26 to eliminatethe center-of-mass motion of intramolecule vibrations anduncover the effect of the exciton−vibration interaction. Thegenerating function can be written as S = λ[(c1

†c1 − c2†c2)(b−

† −b−) + (c1

†c1 + c2†c2)(b+

† − b+)], which is obviously anti-Hermitian,namely, S† = −S, to ensure the unitarity of the transform. Thenew Hamiltonian in the polaron frame is obtained by evaluatingeSHe−S, and it takes the form of

ε λ ω ε λ ω = − ℏ + − ℏ

+ Δ + +λ λ

† †

− † − − †−†

− −†

−

H c c c c

c c c c H

( 2 ) ( 2 )

[e e ]b b b b

12

1 1 22

2 2

2 ( )1 2

2 ( )2 1 vib (4)

where Hvib = ℏω(b+†b+ + b−

†b−). From eq 4, it is clear that thecenter-of-mass motion is independent of the dynamics of theexciton so that we can ignore its effect. The exciton−vibrationinteraction causes a renormalization of the electronic coupling,which, as will be shown later, will critically affect the quantumtransport. Notice that the one-particle approximation for theexciton is not used here and after.The Hamiltonian in the polaron frame provides a picture that

the transition of the exciton in the dimer is assisted byexchanging energy with the vibrational modes. Physically, wecan only consider the single-quanta process of vibrations, owing

to (i) the low probability of multiexcitation and (ii) the energyscale of the exciton−vibration interaction being in quasi-resonance with the gap between the adjacent levels ofvibrational modes. Hence, the dynamics of intramoleculevibrations can be restricted into the space spanned by {|m⟩, |m+1⟩}, as shown in details in the Supporting Information (SI).Moreover the occupation m here is the average of the particlenumber in its eigenmodes rather than the thermal occupationo f b o s o n s . T h e t o t a l H i l b e r t s p a c e i s

= ⊗ ⊗elec(1)

elec(2)

vib.In light-harvesting complexes, that is, the PEB50 dimer and

LHCII pair (or the FMO complex), the excitons need tointeract with the radiation environment from solar as well aslow-energy fluctuations of the protein immersed in the solventin order to realize the energy transfer to the RC. In addition, weneed a connection of one site in the dimer (labeled by 2 in ournotation) to the RC, described by a trapping rate Γ, in order togenerate the output work. The interactions to radiation andprotein reservoirs read (notice that this is not influenced by thepolaron transform introduced before because the polarontransform only operates in the subspace of the systemHamiltonian)

∑ ∑

∑ ∑ γ

= + +

+ +

=

†−†

=

†−†

H g c c a a

f c c b b

( )( )

( )

ii i

ii i i

kk k k

qq q q

int1

2

,pp p ,p

1

2

,s,s s ,s

(5)

where akp and bqs are the bosonic operators for radiation andlow-energy fluctuation reservoirs, respectively. p and s denotethe polarizations of the modes of radiation and low-energyfluctuation reservoirs, respectively.Thus, the total Hamiltonianof the system and environments reads H = H + Hbath + Hint,where Hbath = ∑k,p akp

† akp + ∑q,s bqs† bqs. On the basis of the

perturbation theory, the whole solution to the density operatorcan be written as ρSR = ρS(t) ⊗ ρR(0) + ρδ(t) with the tracelessterm in higher orders of coupling. Because the time scaleassociated with the environmental correlations is much smallerthan the time scale of the system over which the state variesappreciably, the QME for the reduced density matrix of thesystems can be derived under the so-called Markoffapproximation

∑ ∑

∑ ∑ ∑

ρ ρ ρ δ ρ

ρ δ ρ

ρ

=ℏ

+ℏ

| ⟩⟨ | | ⟩⟨ | − | ⟩⟨ |

+ℏ

| ⟩⟨ | | ⟩⟨ | − | ⟩⟨ | +

+ Γ

= =

= = =

tH T a b c d c b

R a b c c c b

dd

i[ , ]

1( )

1( ) h.c.

( )

a b c dad bc ad

a b cabc ac

2, 0

7

, 0

7

,

20

7

0

7

0

7

tr (6)

w h e r e t h e d e c a y r a t e i s T a d , b c / ℏ2 =

vdc∑k,l∑m,n=07 Γkl

T1vmnUkm† UakUnlUlb

† and the dephasing rate isRabc/ℏ

2 = ∑i=07 ∑k,l γkl

T2PiPcUikUli†Uka

† Ubl. The superoperatordescribing the trapping by the RC is ρ( )tr = ∑n=m

m+1 (|0,0,n⟩⟨0,1,n|ρ|0,1,n⟩⟨0,0,n| − |0,1,n⟩⟨0,1,n|ρ|0,1,n⟩⟨0,1,n|). Γkl

T1 =∫ dq/(2π)2 gq2(nq + 1)δ(ωkl′ − νq) for k > l, or ∫ dq/(2π)2 gq

2nqδ(ωkl′ + νq) for k < l. For the environment of theprotein in the solvent, we use the Debye spectral density: J(ω)= (2ER/πℏ)(ωωd/(ω

2 + ωd2)), where ER is the so-called

reorganization energy, and γklT2 = J(|ωkl′ |)n(|ωkl′ |) for k > l,

J(|ωkl′ |)[n(|ωkl′ |) + 1]for k < l, or (2ER/πℏ)(kBT2/ℏωd) for k = l.U is the unitary matrix that diagonalizes the Hamiltonian matrix

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H. In the Liouville space, the QME can be formulated as a two-component form

ρ

ρ

ρ

ρ∂∂

=⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟⎛⎝⎜⎜

⎞⎠⎟⎟t

p

c

p pc

cp c

p

c (7)

Here, ρp and ρc represent the population and coherencecomponents of the density matrix, respectively. To obtain thepopulation dynamics, we need to project the QME intopopulation space by eliminating the coherence componentsusing the Laplace transform.24 On the basis of the viewpoint ofQHE, the whole system should work at steady state, which iswhat we are interested in in this Letter. Thus, in the long-timelimit, the QME at steady state reads

ρ− =−( ) 0p pc c1

cp pss

(8)

where we introduce ≡ − −p pc c

1cp and can define

the transfer matrix as ρ=Tmn nm mp. Thus, the net non-

equilibrium flux is ρ ρ ρ= −c min( , )mn nm m nm m mn np p p ,

which can be decomposed into the superposition of severalclosed loops.24,27 Actually cmn quantifies the detailed balancebreaking and time irreversibility. In reality, people areprincipally able to observe the flux trapped by the RC, whichis defined as ρ ρ= Γ +( )trap 4

p5p . This is consistent with the

definition of cmn because the transition induced by the RC isunidirectional. As we will see later, trap will play an significantrole in the discussion of transport properties of this QHE.

■ COHERENT ENERGY TRANSFER AND QUANTUMTRANSPORT

ETE and Flux Trapped by the RC. Now, we are able todiscuss the coherent energy transfer in the dimer, after theabsorption of photons from the Sun. In the natural light-harvesting complexes, the reorganization energy is ER = 34cm−1, the cutoff frequency is ωd = 50 fs−1, the temperature of

the radiation reservoir is T1 = 5780 K,28 and the trapping rateby the RC is set to be Γ = 1 ps−1. In PEB50 dimer, the dipolemoments in eq 5 are f1 = 1.0 and f 2 = −0.9.29 First, weintroduce the ETE10,11

η =+ Ktrap

trap de (9)

w h e r e t h e d e c a y r a t e i sρ ρ ρ= ∑ ∑ − Γ +μ ν μν ν= =K ( )de 0

12

5 p4p

5p , which quantifies

the probability going back to the ground state per unit time.Figure 2a and b illustrates the effect of coupling strength of theexciton to intramolecule vibrations on the trapping flux by theRC as well as ETE. As we can see, both the flux and ETE showa sharp increase as the coupling strength becomes large, besidesa small decrease at the beginning. The reason for such adecrease is the corresponding small increase of population onsite 1, as shown in Figure 2c, which indicates that the excitonbecomes more localized at very weak coupling to molecularvibrations. Significantly, the large coupling to vibrations leads tothe optimization of ETE of the system, namely, over 90%.Physically, we can understand it as follows: the exchange of theenergy between excitons and vibrational modes leads to therenormalization of the electronic coupling, which effectivelyamplifies the magnitude of the electronic couplings by a factorof (m+1)1/2, as shown in the Hamiltonian. Consequently, theexciton transport is accelerated. On the other hand, Figure 2aalso indicates that the nonequilibriumness of the system can beenhanced by the exciton−vibration dynamics, as quantified bythe trapping flux. In the regime of strong exciton−vibrationcoupling, namely, λ ≫ 1, the states become mainly vibrational,and the excitonic transport is suppressed, as relfected by thedecay of the flux illustrated in Figure 2a, though the ETE stillsaturates to a perfect value of 100%. In this sense, we know thatthe ETE is necessary but not sufficient for the description of theexcitation energy transport in the photosynthesis antenna. Inaddition to this, other quantities, such as output work (heat

Figure 2. (a) Flux trapped by the RC, (b) ETE, (c) the population on site 1 (small figure) and the square root of ratio of the populations of twochromophores in the PE545 dimer (large figure), (d) the entanglement entropy (which also quantifies the coherence effect), (e) work generated byQHE to the RC, and (f) EPR vary as a function of coupling strength of the exciton to intramolecule vibrational modes. Red, purple, blue, and orangelines correspond to m = 0, 1, 3, and 5, respectively, where m represents the bosonic occupation of the vibrational modes. Standard paramenters are ω= 800 cm−1, ϵ1 − ϵ2 = 1042 cm−1, Δ = 92 cm−1, and T2 = 300 K.29,30

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current into the RC) and entropy production rate (EPR), areessential as well to completely describe the quality of this QHEon excitation energy transport.Moreover, by comparison of fluxes and ETE for different

occupations of vibrations, it is clear that the higher excitedmode of intramolecule vibrations is more efficient for theenhancement of ETE and flux trapped by the RC. However, theenvironment around the chromophores always satisfies thecondition ℏω ≈ kBT2; thus, only those lowest vibrational modescan be populated. Hence, we show the results for m = 0, 1, 3,and 5 here. Particularly, the ETE and flux for m = 3 are of 72and 75% improvements with respect to m = 0 at λ = 0.6.The energy transfer between the PEB50 dimer and RC

acquires the delocalization of the excitons. We now investigatethe trends of delocalization under the influence of exciton−vibration interaction. As is shown in Figure 2c, the populationof high-lying exciton state has a rapid, nonexponential decay asthe coupling to vibrational modes increases, which can betraced back to the coherent transition from |1,0,m⟩ to |0,1,m +1⟩ and from |1,0,m + 1⟩ to |0,1,m⟩. On the other hand, thedelocalization of the wave packet is demonstrated in the largefigure in Figure 2c by increasing the coupling strength tovibrations. In fact, the coupling between excitons andvibrational modes leads to an effective amplification ofelectronic coupling by a factor of (m + 1)1/2, as mentionedbefore, between the two molecules in a dimer. This isparticularly reflected in the matrix elements ⟨1,0,m|H|0,1,m +1⟩, ⟨1,0,m|H|0,1,m⟩, and ⟨1,0,m + 1|H|0,1,m⟩.Coherence effect. Because the dynamics of intramolecule

vibrations is considered together with excitons, we need tostudy the influence of the exciton−vibration interaction on thecoherence effect. Notice that the coherence effect mentionedhere refers to the entanglement entropy and the sum of the off-diagonal elements of the density matrix with different electronicstates (quantum coherence). Due to the strong correltaionbetween entanglement and coherence, we use the quantumentanglement to quantify the coherence effect hereafter, andthe coherence is included in SI. To calculate the entanglemententropy, we first need to diagolize the density matrix at steadystate ρ = ∑f Pf |ψf⟩⟨ψf |, where |ψf⟩ = ∑f fn|n1( f),n2( f),m( f)⟩and is the unitary transform matrix in the diagonalization ofthe density matrix. For each component of the pure state, thedensity matrix reads, by partial tracing over the freedoms ofother sites except the first one (n3 ≡ m)

∑

∑ ∑ ∏

ρ ν ν ψ ψ ν ν

δ δ

= ⟨ | ⟩⟨ | ⟩

= | ⟩⟨ |

ν=

=

†

= =

n m n m

n j n l

( ), ( ) ( ), ( )

( ) ( ) ( )

f f f

j ljf fl

k sn k n j n k n l

(1)

0

7

2 2

, 0

7

0

7

2

3

( ), ( ) ( ), ( ) 1 1s s s s

(10)

whose eigenvalues are λ±f . Then, the entanglement entropy of

each pure component is Sf(1) = −kB(λ+f ln λ+f + λ−

f ln λ−f ), which

subsequently gives the total entanglement entropy at steadystate

∑ λ λ λ λ= − +=

+ + − −S k P ( ln ln )f

ff f f f

B0

7

(11)

Actually, the formal definition of the entanglement entropy in amixture ensemble is still an open question. The rational for our

choice of the expression in eq 11 for our system is the excitonicbasis describing the delocalized excitons that gives the uniquediagonal form of the steady-state density matrix, and thesedelocalized excitons contain the coherence between the localsites and play an important role in experiments. Moreover, theoff-diagonal elements of the Hamiltonian also lead to theentanglement in the excitonic basis. As illustrated by Figure 2d,the exciton−vibration interaction improves the coherence atfirst, owing to the energy exchange between excitons andvibrational modes. However, on the basis of the discussion onthe delocalization above, we know that the interaction withvibrational modes leads to the enhancement of the delocaliza-tion of the exciton wave packet. However, we can further seefrom Figure 2c that the population on site 1 of the excitonicstate at large exciton−vibration coupling is closed to 0. Thisindicates the suppression of the back-and-forth oscillationbetween the two localized excitonic states, which consequentlycauses the downhill trends of the coherence. On the otherhand, the exciton−vibration leads to the Rabi oscillation withthe frequency ∼(δω2 + λ2ω2)1/2, which indicates a rapidoscillation as well as the mismatching between the energysplitting of excitons and quanta of vibrational modes at largecoupling strength. Thus, the quantum entanglement andcoherence will eventually vanish in the long-time limit. Thisis partially in contrast to the previous predictions.17,31

Furthermore, by the comparison between Figure 2a, b, and d,it is found that in the range of coupling to vibrations before theextremum of entanglement, the improvement of quantumcoherence gives rise to a rapid growth of flux trapped by the RCand ETE, namely, λ ≃ eq 0.39, 0.46, 0.62, and 0.8 for m = 5, 3,1, and 0, respectively. In other words, the coherent energytransfer can lead to a significant improvement of quantum yield.The incoherent energy transfer becomes important on thefurther improvement of ETE after the coherent process.

■ MACROSCOPIC TRANSPORT

According to quantum thermodynamics, the observables on themacroscopic level serve as an important role in the function ofphotosynthetic organisms as a QHE. We now investigate tworepresentative quantities: output work and EPR. The formerprovides a critical measure of the macroscopic quality of theQHE, and the latter quantifies the nonequilibriumness of thesystem on the macroscopic level. The first and second laws inthermodynamics give Q1 − Q2 − Qtr = E and σ + S = St, wherethe output work to the RC can be calculated by Qtr=Tr[H ρ( )trap ]. S and S t are the rates of system entropy andEPR, respectively. Due to the assumption of large environmentswith the negligible back influence from the system toenvironments, the entropy flux flowing from the system toenvironments reads σ = −(Q1/T1) + (Q2/T2) + (Qtr/TRC),where TRC is the temperature of the RC. At steady state, noenergy consumption occurs inside of the system, so that Qtr =Q1 − Q2 and St = σ. In several natural light-harvesting antennae,the temperature in the RC is always the same as that for low-energy solvated protein fluctuations, namely, around roomtemperature. In Figure 2e and f, the output work and EPR havesimilar behaviors; the coupling of excitons to vibrational modespromotes the coherent energy transport, which leads to a rapidgrowth of the work and EPR. The incoherent transport causedby higher coupling strength to vibrations further saturates thework and EPR to particular values (distinguished by differentlevels of intramolecule vibrations). For the strong exciton−

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vibration coupling for λ ≫ 1, the incoherent transport leads tothe suppression of output work as well as EPR because thevibration dominates the feature of the states.

■ EFFECT OF THERMAL RELAXATION ONCOHERENT ENERGY TRANSFER

Now, we study the influence of the low-energy thermal bathdescribed by a continuous distribution of harmonic oscillatorson the coherent energy transfer in the exciton−vibration dimer.The interaction between the exciton and bath is governed bythe Debye spectral density with the cutoff frequency of ωd = 50fs−1 < kBT2. Here, we consider two regimes of coupling strengthbetween the PEB50 dimer and bath: very weak coupling ER = 4cm−1 and intermediate coupling ER = 34 cm−1. As is shown inFigure 3, the dephasing and fluctuation induced by the low-energy protein motion gives rise to the promotion of quantumyield illustrated in Figure 3a because the motion of excitonsglobally becomes more delocalized when the couplingincreases, as reflected in Figure 3b.

■ SUMMARY AND REMARKS

In conclusion, we investigated the effect of intramoleculevibrational modes on the coherent energy transfer in the light-harvesting complexes. It was demonstrated that the exciton−vibration interaction led to a nontrivial improvement ofcoherent energy transfer by the enhancement of coherence.Furthermore, we also show that the exciton−vibration couplingcan give rise to a perfect quantum yield (over 90%) for energytransport at steady state.Our study provides the insights for the exploration of the

inter- and intramolecular vibrations on multimolecule systems,that is, the natural light-harvesting complexes LHCII with 32chromophores and FMO complex with 7 molecules, to see howthe exciton−vibration interaction affects the energy-transferpathways.

■ ASSOCIATED CONTENT

*S Supporting InformationDetails of the derivations of equations are supplied. Thismaterial is available free of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: jin.wang.1@stonybrook.edu.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

We acknowledge the support from Grant NSF-MCB-0947767.

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Figure 3. Comparison between the regimes of weak and intermediate couplings of the exciton−bath as a function of coupling to vibrational modes.Blue and purple curves correspond to the reorganization energies ER = 4 and 34 cm−1, respectively.

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