excitonic energy transfer in light-harvesting complexes in

Excitonic energy transfer in light-harvesting complexes inpurple bacteria

Citation Ye Jun Kewei Sun Yang Zhao Yunjin Yu Chee Kong Lee andJianshu Cao ldquoExcitonic energy transfer in light-harvestingcomplexes in purple bacteriardquo The Journal of Chemical Physics136 no 24 (2012) 245104 copy 2012 American Institute ofPhysics

As Published httpdxdoiorg10106314729786

Publisher American Institute of Physics (AIP)

Version Final published version

Accessed Wed Apr 03 163220 EDT 2019

Citable Link httphdlhandlenet1721181978

Terms of Use Article is made available in accordance with the publishers policyand may be subject to US copyright law Please refer to thepublishers site for terms of use

Detailed Terms

The MIT Faculty has made this article openly available Please sharehow this access benefits you Your story matters

Excitonic energy transfer in light-harvesting complexes in purple bacteriaJun Ye Kewei Sun Yang Zhao Yunjin Yu Chee Kong Lee et al Citation J Chem Phys 136 245104 (2012) doi 10106314729786 View online httpdxdoiorg10106314729786 View Table of Contents httpjcpaiporgresource1JCPSA6v136i24 Published by the American Institute of Physics Additional information on J Chem PhysJournal Homepage httpjcpaiporg Journal Information httpjcpaiporgaboutabout_the_journal Top downloads httpjcpaiporgfeaturesmost_downloaded Information for Authors httpjcpaiporgauthors

Downloaded 24 Oct 2012 to 18746138 Redistribution subject to AIP license or copyright see httpjcpaiporgaboutrights_and_permissions

THE JOURNAL OF CHEMICAL PHYSICS 136 245104 (2012)

Excitonic energy transfer in light-harvesting complexes in purple bacteriaJun Ye1 Kewei Sun1 Yang Zhao1a) Yunjin Yu12 Chee Kong Lee1 and Jianshu Cao3

1School of Materials Science and Engineering Nanyang Technological University Singapore 639798Singapore2College of Physics Science and Technology Shenzhen University Guangdong 518060 China3Department of Chemistry Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA

(Received 18 March 2012 accepted 4 June 2012 published online 28 June 2012)

Two distinct approaches the Frenkel-Dirac time-dependent variation and the Haken-Strobl modelare adopted to study energy transfer dynamics in single-ring and double-ring light-harvesting (LH)systems in purple bacteria It is found that the inclusion of long-range dipolar interactions in thetwo methods results in significant increase in intra- or inter-ring exciton transfer efficiency Thedependence of exciton transfer efficiency on trapping positions on single rings of LH2 (B850)and LH1 is similar to that in toy models with nearest-neighbor coupling only However owingto the symmetry breaking caused by the dimerization of BChls and dipolar couplings such de-pendence has been largely suppressed In the studies of coupled-ring systems both methods re-veal an interesting role of dipolar interactions in increasing energy transfer efficiency by introduc-ing multiple intrainter-ring transfer paths Importantly the time scale (4 ps) of inter-ring excitontransfer obtained from polaron dynamics is in good agreement with previous studies In a double-ring LH2 system non-nearest neighbor interactions can induce symmetry breaking which leadsto global and local minima of the average trapping time in the presence of a non-zero dephas-ing rate suggesting that environment dephasing helps preserve quantum coherent energy transferwhen the perfect circular symmetry in the hypothetic system is broken This study reveals thatdipolar coupling between chromophores may play an important role in the high energy transferefficiency in the LH systems of purple bacteria and many other natural photosynthetic systemscopy 2012 American Institute of Physics [httpdxdoiorg10106314729786]

I INTRODUCTION

An essential photochemical process in plants and pho-tosynthetic bacteria is light-harvesting (LH) in which solarphotons are first absorbed by antenna complexes (also knownas the antenna protein) and the resulting excitonic energyis transferred with a high efficiency to specialized pigment-protein complexes or the reaction centers In a photosyntheticsystem of bacteriochlorophylls aggregates embedded in pro-tein scaffolds it has been suggested that the protein envi-ronment plays a critical role in protecting excitonic coher-ence and consequently facilitating coherent energy transferThe entire process of coherent energy transfer takes place ina very short period of time from a few hundred femtosec-onds to a few picoseconds Recent studies have shown thatthe efficiency of energy transfer from antenna pigments to thereaction centers is remarkably high (eg gt95)1ndash3 Such ahigh efficiency of energy transfer has inspired the designs ofefficient artificial systems to convert solar energy into otherforms However an acute lack of reliable simulation tech-niques and theoretical tools for understanding the exact mech-anisms of long time quantum coherence in such systems hin-ders the realistic application of the highly efficient energytransfer in artificial systems Currently the Foumlrster resonanceenergy transfer (FRET) theory based on weak electronic cou-pling approximation and the Redfield equation based on

a)Electronic mail YZhaontuedusg

the Markovian approximation are the two commonly appliedmethods to deal with the energy transfer processes in thesecomplexes The classical Foumlrster theory treats electronic cou-pling perturbatively and also assumes the incoherent hop-ping of exciton between single donor and acceptor inducedby point dipole-dipole interaction of transition dipoles4 of thechromophores Assumption of point dipole is inadequate indescribing pigmentsrsquo aggregation which results in the break-down of the approximation5 To overcome the drawback ofthe point-dipole approximation Jang et al6ndash10 developed amuti-chromophore version of FRET theory which can be ap-plied to cope with the transfer pathway interference that isnot accounted for in earlier FRET treatments However whenthe electronic coupling is strong but the system-bath couplingis weak it is necessary to consider relaxation of delocalizedexciton states In this limit the dynamics of exciton energytransfer (EET) can be described by the coupled Redfield equa-tions in the exciton basis11ndash14 The Redfield equations on theother hand are applicable in the weak exciton-phonon cou-pling limit where the problem can be treated perturbativelyMore importantly in the Redfield equation15 the bath de-gree of freedom has been projected out to obtain a reducedmaster equation for the density matrix evolution under theMarkovian approximation However for a realistic photosyn-thetic system the exciton-phonon coupling is not negligibleand the Markovian approximation is a poor one in describ-ing the bath behavior since the typical phonon relaxation timescale is quite long Recently much work has been devoted to

0021-96062012136(24)24510417$3000 copy 2012 American Institute of Physics136 245104-1


245104-2 Ye et al J Chem Phys 136 245104 (2012)

circumvent the difficulties in the traditional methods andthe efforts include the method of hierarchy equations16ndash19

the linearized density matrix approach based on pathintegral15 20ndash22 and quantum master equations using polarontransformations23ndash28

In purple bacteria the antenna complexes have highlysymmetric multi-ring structures29ndash31 which are the subjectof numerous studies regarding the mechanism underlying thehighly efficient energy transfer processes3 32ndash37 Photons areabsorbed mainly by the peripheral antennae complex LH2 ofpurple bacteria where the photo-generated excitons are sub-sequently transferred to the reaction center within the harvest-ing complex LH1 in a few picoseconds38 The LH2 complexis an eight-unit circular aggregate built from αβ-heterodimersforming C8 symmetry29 30 Each unit contains a pair ofα and β apoproteins three bacteriochlorophylls-a(BChls-a)molecules and a carotenoid and the BChl-a molecules formtwo rings named according to their corresponding absorp-tion maxima at 800 nm and 850 nm as the B800 and B850rings respectively The B850 ring consists of 16 tightly po-sitioned BChls-a with the Mg-Mg distance about 936 Aring forthe 1αndash1β dimer and about 878 Aring for the 1αndash1β dimer33

There are many studies on the mechanism behind these highlyefficient photosynthetic processes3 32ndash37 In Ref 39 optimalenergy transfer conditions in linear and nonlinear systemswere analyzed and a similar approach was recently appliedto self-assembling light-harvesting arrays40 to study optimalefficiency For the antenna complexes of purple bacteria en-ergy transfer dynamics in LH1 and LH2 rings is inadequatelyunderstood and questions regarding how dipolar interactionssymmetry of the ring and inter-ring coupling etc affect theenergy transfer efficiency remain unanswered

Inspired by previous studies in this work we study in de-tail the effect of intra-ring dipolar interactions ring geometryand symmetry and more importantly inter-ring distances indetermining EET As a starting point of understanding EETevents two approaches have been adopted to investigate var-ious light-harvesting complexes in purple bacteria For thelow-temperature exciton dynamics we use the Frenkel-Diractime-dependent variational method to simulate the quan-tum dynamics of the Holstein polaron via the Davydov D1

Ansatz while for the high-temperature dynamics and trans-fer efficiency calculations the Haken-Strobl model41 equiv-alent to the Redfield equation at high temperatures has beenemployed

The rest of the paper is organized as follows In Sec IIthe Hamiltonian is introduced and the methodology to estab-lish possible optimal conditions and understand energy trans-fer dynamics in ring systems is elaborated The effect of var-ious parameters including dynamic disorder due to couplingto phonons and dephasing due to inclusion of environment onthe optimal energy transfer efficiency in photosynthetic sys-tems is discussed in detail in Sec III Finally the conclusionsare drawn in Sec IV

II METHODOLOGY

Two approaches are adopted to study the energy transferprocess in the light-harvesting apparatus of the purple bac-

teria the Frenkel-Dirac time-dependent variational methodmaking use of the Holstein model which can be generally re-garded as a low-temperature treatment and the Haken-Stroblmethod applicable to high-temperature scenarios

A Hamiltonian

The Holstein Hamiltonian for the exciton-phonon systemreads42 43

H = Hex + Hph + Hexminusph (1)

with

Hex = minusJsum

n

adaggern(an+1 + anminus1) (2)

Hph =sum

q

ωqbdaggerq bq (3)

Hexminusph = minus 1radicN

sumn

sumq

gqωqadaggernan(bqe

iqn + Hc) (4)

where adaggern (an) is the creation (annihilation) operator for an

exciton at the nth site J is the exciton transfer integralbdaggerq (bq) is the creation (annihilation) operator of a phonon with

momentum q and frequency ωq and the Planckrsquos constant isset as macr = 1 Hexminusph is the linear diagonal exciton-phononcoupling Hamiltonian with gq as the coupling constant withsum

q g2qωq = Sω0 Here S is the well-known Huang-Rhys fac-

tor and ω0 is the characteristic phonon frequency which isset to unity in this paper for simplicity It is noted that inthis method phonon dispersions in the Hamiltonian can havevarious forms to describe many phonon branches of differ-ent origins For example a linear phonon dispersion with ωq

= ω0[1 + W (2|q|π minus 1)] may be adopted where W is aconstant between 0 and 1 and the bandwidth of the phononfrequency is 2Wω0 Furthermore other forms of exciton-phonon coupling can also be included demonstrating thecapability of the current approach in imitating the phononspectral density obtained with methods such as moleculardynamics44

For a multiple-ring system by assuming only excitontransfer integral and dipolar interaction between rings theoriginal Holstein Hamiltonian can be modified as

H primeex =

sumr1r2

sumnm

J r1r2nm ar1dagger

n ar2m (5)

where r1 = 1 Nring and r2 = 1Nring label a pair of rings(with total number of Nring rings) if r1 = r2 and the summa-tion is over all rings However if r1 = r2 Eq (5) becomes

H primeprimeex =

sumr

sumnm

J rnmardagger

n arm (6)

where r again runs from 1 to Nring For each ring the ex-citon Hamiltonian takes the form of the Frenkel exciton



Hamiltonian with Jnm given by45

Jnm=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

ε1 J1 W13 middot middot middot J2

J1 ε2 J2 middot middot middot W22N

W31 J2 ε1 middot middot middot middotmiddot middot middot middot middot middot middotmiddot middot middot middot ε2 J2 W2Nminus22N

middot middot middot middot J2 ε1 J1

J2 middot middot middot W2N2Nminus2 J1 ε2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(7)where ε1 and ε2 are the on-site excitation energies of an in-dividual BChl-a J1 and J2 are the transfer integral betweennearest neighbors N equals 8 as the system is of C8 sym-metry and matrix elements Wij are the dipolar coupling forthe non-nearest neighbors The dipolar coupling between sitei and site j takes the form

Wij = C

[di middot dj

|rij |3 minus 3(di middot rij )(dj middot rij )

|rij |5] (8)

where C is the proportionality constant rij is the vector con-necting the ith and jth monomers and di are the unit vectors ofthe ith BChl-a In our calculations in this work the followingparameters33 are adopted J1 = 594 cmminus1 J2 = 491 cmminus1

and C = 640725 Aring3 cmminus1 When the Jnm matrix is appliedto the polaron dynamics calculations it must be scaled bythe characteristic phonon frequency ω0 The Mg-Mg distancebetween neighboring B850 BChls is 92 Aring within the αβ-heterodimer and 925 Aring between neighboring heterodimersFor simplicity the α and β proteins are assumed to be evenlydistributed with a distance of 925 Aring

For multiple-ring systems the phonon and exciton-phonon interaction Hamiltonian can be described as

H primeph =

sumr

sumq

ωrq b

rdaggerq br

q (9)

H primeexminusph = minus 1radic

N

sumr

sumn

sumq

grqω

rq a

rdaggern ar

n(eiqnbrq + Hmiddotcmiddot)

(10)where r is summed over 1 to Nring b

rdaggerq (br

q) is the creation(annihilation) operator of a phonon with momentum q andfrequency ωr

q in the rth ring Similar to the single-ring casewe can obtain the Huang-Rhys factor S from

sumq(gr

q)2ωrq

= Sω0 It is clearly shown in the formalism that the phononsin different LH2 rings are completely independent of eachother in the current treatment

B Polaron dynamics in LH2 systems

The family of the time-dependent Davydov Ansaumltze iethe D1 D and D2 trial states can be written in a general formas

|D(t)〉 =sum

n

αn(t)adaggernU

daggern(t)|0〉ex|0〉ph (11)

where αn(t) are the variational parameters representing exci-ton amplitudes and U

daggern(t) is the Glauber coherent operator

U daggern(t) equiv exp

sumq

[λnq(t)bdaggerq minus Hc]

(12)

For the D1 Ansatz λn q(t) are N times N variational parametersrepresenting phonon displacements The D2 and D Ansaumltzeare two simplified cases of the D1 Ansatz In the D2 Ansatzλn q(t) are replaced by N independent variational parametersβq(t) ie λn q(t) = βq(t) While in the D Ansatz λn q(t) arereplaced by 2N minus 1 variational parameters λ0(t) βq = 0(t)and γ q = 0(t) where λn q = 0(t) = λ0(t) and λn q(t) = βq(t)+ eminusiqnγ q(t) q = 0

The time evolution of the photo-excited state in a one-dimensional molecular aggregate follows the time-dependentSchroumldinger equation There are several approaches to solvethe time-dependent Schroumldinger equation In the Hilbertspace for example the time-dependent wave function |(t)〉for the Hamiltonian H is parameterized by a set of time-dependent variables αm(t) (m = 1 M)

|(t)〉 equiv |αm(t)〉 (13)

Assuming that |(t)〉 satisfies the time-dependentSchroumldinger equation one has

ipart

partt|(t)〉 = H |(t)〉 (14)

Explicitly putting in the Hamiltonian H of Eq (1) and writingpart|(t)〉partt in terms of αm(t) and their time-derivatives αm(t)(m = 1 M) one obtains(

ipart

parttminus H

)|(t)〉 = |αm(t) αm(t)〉 = 0 (15)

Projecting Eq (15) onto M different states |m〉 (m = 1 M) one obtains M equations of motion for the parametersαm(t)

〈m|αm(t) αm(t)〉 = 0 (16)

The approach we adopt in this work is the Lagrangianformalism of the Dirac-Frenkel time-dependent variationalmethod46 a powerful technique to obtain approximate dy-namics of many-body quantum systems for which exact so-lutions often elude researchers We formulate the LagrangianL as follows

L = 〈(t)| imacr2

larrrarrpart

parttminus H |(t)〉 (17)

From this Lagrangian equations of motion for the M func-tions of time parameters αm(t) and their time-derivativesαm(t) (m = 1 M) can be obtained by

d

dt

(partL

partαlowastm

)minus partL

partαlowastm

= 0 (18)

Furthermore to better elucidate the correlation between theexciton and the phonons the phonon displacement ξ n(t) is in-troduced and defined as

ξn(t) equiv 〈D(t)| bn + bdaggern

2|D(t)〉 (19)

where bdaggern = Nminus12 sum

q eminusiqnbdaggerq The reader is referred to Ap-

pendix A for discussions on the precision of the Davydovtrial states (defined as 13(t)) and Appendix B for detailedderivations of equations of motion for polaron dynamics ina multiple-ring system



Applicability of the time-dependent variation approachto an exciton-phonon system has been tested systematicallyfor a wide range of control parameters The readers are re-ferred to Refs 48 and 49 for details where it is shown thatthe D1 trial state has the best performance judging from therelative magnitude of the deviation vector (with the exact so-lution of the Schroumldinger equation as the reference) a quan-tity characterizing the trial-state accuracy It is noted that ourpolaron dynamics simulation is a Hamiltonian approach andas such it is performed in a microcanonical ensemble whichis at variance with other methods On the other hand exci-tonic energy transfer in realistic photosynthetic complexes of-ten takes place in a rather short period of time (sim hundreds offs)38 Therefore there is insufficient time to reach a full ther-mal equilibrium in those systems as pointed out by a numberof authors52ndash54 It is therefore interesting to compare a Hamil-tonian approach to various other methods which are oftenconstructed using canonical ensembles

To have a better comparison with the other approach inthis work the Haken-Strobl method it is beneficial to intro-duce an additional term describing simple excitonic trappingAs trapping is intrinsically dissipative such a term will haveto be non-Hermitian A trapping matrix γ is therefore addedto the Holstein Hamiltonian to characterize the excitonic trap-ping process and estimate the efficiency of exciton transfer Ifwe know the population on site n then the average trappingtime τ n which characterizes the transfer efficiency can beobtained as

τn =int infin

0ρnn(t)dt (20)

where ρnn(t) are the diagonal elements of the reduced excitondensity matrix ρ(t) defined as

ρ(t) = Trph[|D(t)〉〈D(t)|] (21)

The reader is referred to Appendix C for further derivationdetails Similar expressions for the trapping time will bederived when the Haken-Strobl model is introduced inSubsection II C

C The Haken-Strobl model

The second approach applied to the LH1LH2 complexesin this work is the Haken-Strobl model which has beenpreviously used to describe the exciton dynamics in Fenna-Matthews-Olson (FMO) (see eg Ref 39) We start with theLiouville equation for the exciton density matrix ρ

ρ(t) = minusLρ(t)

= minus[Lsys + Ldissip + Ldecay + Ltrap]ρ(t) (22)

Here Lsys is the free evolution super-operator of the pure ex-citon system and can be written as Lsys = i[H ρ]macr where[H]nm = (1 minus δnm)Jnm + δnmεn with Jnm the excitonic cou-pling between site n and site m and ε the site energy Ldissip

represents the dephasing and population effects within the ex-citon manifold For the convenience of analysis the couplingto the environment is given by the standard Bloch-Redfieldequation which can be written as [Ldissip]nm = (1 minus δnmnm)

where is the pure dephasing rate Ldecay represents the de-cay of the exciton to the ground state and can be expressedas [Ldecay]nm = (kdn + kdm)2 with kd n the decay rate onsite n Ltrap represents the trapping rate and can be describedas [Ltrap]nm = (ktn + ktm)2 with kt n the trapping rate onsite n

In fact among Ldecay and Ltrap which represent two pos-sible channels for irreversible exciton energy loss the formeris ineffective in the energy transfer Efficiency of energy trans-fer can be gauged by the quantum yield q namely the trap-ping probability37

q =sum

n ktnτnsumn ktnτn + sum

n kdnτn

(23)

where the mean residence time τ n can be expressed as τn

= int infin0 ρn(t)dt and the population ρn = ρnn is the diagonal

element of the density matrix In photosynthetic systems kminus1t

is on the order of ps and kminus1d is on the order of ns so the

trapping rate is much larger than decay rate and the quantumyield is close to unity The quantum yield can then be approx-imated as q asymp (1 + kd〈t〉)minus1 where 〈t〉 = sum

nτ n is the meanfirst passage time to the trap state without the presence of theconstant decay ie the average trapping time Quantum yieldand trapping time have been extensively studied in the contextof molecule photon statistics39 and experimental investigationwas also carried out in a photosynthetic system47 A shortertrapping time corresponds to a higher efficiency of excitontransfer

III RESULTS AND DISCUSSION

A Polaron dynamics in LH2 systems

With the aid of the Frenkel-Dirac time-dependent varia-tional method a hierarchy of variational wave functions hasbeen previously formulated to faithfully describe the dynam-ics of the Holstein polaron48 49 The accuracy of the DavydovAnsaumltze has been discussed in length in Refs 48 and 49 andit has been shown that for diagonal exciton-phonon couplingthe D1 Ansatz has the highest precision in the hierarchy ofthe Davydov trial states when applied to the solution of theHolstein Hamiltonian The increase in the precision of theD1 Ansatz is achieved mainly by the improved sophisticationof the phonon wave function48 49 Such increased sophistica-tion plays a dramatic role especially in the weak and moder-ate coupling regime where the plane-wave-like phonon com-ponents render it impractical to use a simple linear superpo-sition of phonon coherent states The vastly increased num-ber of variational parameters for the phonon wave functionultimately results in the higher precision of the D1 Ansatzwhich can be applied reliably to the detailed excitonic energytransfer dynamics in the LH2 system (with a known moderatevalue of Huang-Rhys factor as 05)

We first apply the approach to an excitonic polaron evolv-ing in an isolated LH2 ring To assure the accuracy of ourapproach the D1 Ansatz has been chosen to deal with the dy-namics in this paper For a case of a wide phonon bandwidth(W = 08) and a moderate exciton-phonon coupling strength(S = 05) the time evolution of exciton probability |αn(t)|2



and phonon displacement ξ n(t) in the presence of only near-est neighbor transfer integral have been displayed in Figs 1(a)and 1(b) The polaron state is initialized as follows αn(0)= δn 8 and λn q(0) = 0 The corresponding dynamics with re-alistic LH2 transfer integral matrix [cf Eq (7)] has been givenin Figs 1(c) and 1(d) Time-dependent exciton and phononenergies as well as the magnitude of deviation vector 13(t) forboth cases are plotted in Fig 1(e) For all calculations a char-acteristic phonon frequency of ω0 = 1670 cmminus1 and a Huang-Rhys factor of S = 05 are chosen51 The phonon frequencyobtained from molecular dynamics simulation is related to thestretching mode in either a C=O bond or a methine bridge

It is important to compare calculated absorption spectrawith measured ones in order to further validate our approachand lend support to our simulation of excitation transfer dy-namics in a photosynthetic complex Linear absorption spec-tra are calculated and compared to measured ones in Fig 2Detailed derivations for absorption spectral calculations canbe found in Refs 48 and 49 It is noted that results from ourapproach can also be compared to those from Ref 50 Mea-sured data for B850 complexes are obtained from Ref 55Reasonable agreement is found between theory and experi-ment pointing to the applicability of our model to the B850ring for the parameter set chosen We note that although thelong tail shown in the measured spectrum is roughly repro-duced by our approach the agreement is only qualitative andbetter understanding of the phonon sidebands is needed

Owing to the aforementioned higher accuracy of the D1

Ansatz as recorded in Refs 48 and 49 in comparison withother trial states as shown in Fig 1(e) it is therefore an idealcandidate for the study of excitation dynamics of the LH2system The real LH2 system contains pigments that inter-act via long-range dipolar couplings as given by the pure ex-citon Hamiltonian Eq (7) By including dipole-dipole inter-actions into the Hamiltonian the accuracy for the D1 Ansatzhas to be rechecked and compared to that for a model systemwith nearest neighbor coupling only A first step toward theaccuracy checkup is the comparison of polaron dynamics inFigs 1(a)ndash1(d) where dramatic changes can be found as aresult of dipolar interactions

In contrast to the case with only nearest neighbor J shownin Figs 1(a) and 1(b) many new features have emerged inFigs 1(c) and 1(d) One major finding is that the inclusionof realistic dipolar interactions in the LH2 system breaks thesymmetry of the ring structure resulting in asymmetric andfaster exciton dynamics compared to the model system withonly nearest neighbor J Such an effect can be related to anemergence of additional energy transfer paths when dipolarinteractions are included Although the dynamics here doesnot contain the coupling to the external heat bath the essentialphysics is captured to a great extent It is worth noting that inFigs 1(b) and 1(d) during the time evolution of phonon dis-placements ξ n(t) the dominant V-shaped feature survives thechange of the exciton Hamiltonian from nearest neighbor J toa realistic matrix of Jnm This can be explained as follows thephonon dynamics of the system is not only determined by thecoupling between the exciton and the phonons but also has itsown propagation pattern determined by the linear dispersionrelationship given earlier As a result once the excitation on

site 8 starts to propagate as shown in the exciton dynamicsthe strong lattice distortion in the vicinity of the excitationwill also propagate Moreover the effect of the exciton onphonon dynamics becomes apparent by comparing Figs 1(a)and 1(b) where an exact correspondence between the excitonand phonon dynamics can be found However the moderatevalue of exciton-phonon coupling strength S implies that theexciton generated on site 8 can easily propagate through alarger distance similar to a bare exciton

More interestingly the revival and localization phe-nomenon shown in Fig 1(a) is nearly absent in Fig 1(c) asa result of broken symmetry as well as multiple transfer chan-nels due to the presence of dipolar interaction within the ringHowever there are certain possible positions such as on site8 where it appears to exhibit possible revival of the excitonamplitude at around 6 phonon periods similar to those shownin Fig 1(a) on site 8 at around 10 phonon periods Theseobservations lend further support to the speed up of excitondynamics (in terms of delocalization) due to the inclusionof dipolar interactions From above discussions by includinglong-range excitonic coupling other than that between near-est neighbors (which is true for many realistic cases such asthe LH2 system investigated in this paper) the exciton canmore efficiently and evenly spread over the entire ring in thepresence of exciton-phonon coupling In other words naturehas designed a unique way for the light harvesting complexessuch as LH2 to maximize their excitonic energy transfer effi-ciency in spite of exciton-phonon coupling that is responsiblefor the localization and dephasing of the excitons It appearsthe symmetry breaking of the energy transfer pathways playsa positive role in LH2 complexes

Furthermore it is also important to note in Fig 1(e) thatfor the chosen parameter sets the D1 Ansatz is capable to de-scribe the realistic LH2 system with great precision Com-paring to the maximum absolute value of phonon or exciton-phonon interaction energy shown in the figure the magnitudeof deviation vector 13(t) is less then 30 of the average sys-tem energy (here we use phonon energy as the reference) In-clusion of complex dipolar coupling does not further increasethe error of the D1 Ansatz in describing the system dynamicsAlthough the deviation magnitude does not change much theexciton-phonon dynamics deviates substantially upon inclu-sion of dipolar coupling if one compares Figs 1(a)ndash1(d)

Using the D1 Ansatz we also examined the exciton-phonon dynamics of a realistic two-ring system with long-range dipolar interactions Results from the dynamics calcu-lations are shown in Figs 3 and 4 for two inter-ring distances(15 and 20 Aring respectively) The choice of the distances forthe two-ring system is in accordance with the one shownin Ref 51 which suggests the Mg-Mg distance of around24 Aring for the closest chromophores between two adjacent LH2rings Considering the size of the chromophores it is bene-ficial to slightly reduce the distance to consider the excitontransfer from the tip of one molecule to the other thus twodifferent distances ie 15 and 20 Aring are used in this studyTo have better understanding of the roles of intra- and inter-ring excitonic transfer the two-ring system is initialized asfollows αr

n(0) = δr1δn8 and λrnq(0) = 0 ie a single exci-

tation at site 8 of ring-1 after a subsequent spreading within



FIG 1 Real space dynamics of (a) exciton probability |αn(t)|2 and (b) phonon displacement ξn(t) for a single-ring with only nearest neighbor transfer integralJ = 03557 a moderate exciton-phonon coupling strength S = 05 and a wide bandwidth W = 08 of phonon dispersion Corresponding real space dynamicsare displayed for (c) exciton probability |αn(t)|2 and (d) phonon displacement ξn(t) for a single realistic LH2 ring transfer integral matrix The D1 Ansatz isused for the simulation The time evolution of energy components for exciton phonon exciton-phonon interaction and deviation magnitude 13(t) for both caseshas also been given in (e) Note that the value of the nearest neighbor transfer integral is taken from the largest element (scaled by the characteristic phononfrequency ω0) of the realistic LH2 transfer integral matrix



FIG 2 Comparison of calculated linear absorption spectra and measuredones in a single B850 ring A realistic matrix for the LH2 transfer integralis adopted Parameters used are S = 05 ω0= 1670 cmminus1 W = 08 and adamping factor of 012

ring-1 the exciton population is transferred to site 1 which isthe closet point to site 10 (and nearby sites) of ring-2 at about3 phonon periods from t = 0 This exciton transfer process isclearly revealed in Figs 3(c) and 4(c)

It is always helpful to compare the value of 13(t) rel-ative with system energies as the first check As shownin Figs 3(e) and 4(e) the accuracy of the D1 Ansatzapplied to the two-ring system with different inter-ringdistances does not alter much as compared to the one-ring system since the magnitude 13(t) remains close tothat for the one-ring case The value of 13(t) also be-comes fairly stable after initial 5 phonon periods therebyconfirming the validity of this method applied to studythe multiple-ring exciton dynamics using time-dependentvariation

At a first glimpse one may spot few differences betweenFigs 3 and 4 during the first 10 phonon periods One can alsofind a resemblance of the exciton probabilities and phonondisplacement between the two-ring system and the one-ringsystem in Fig 1 if comparison is made during the same pe-riod of time This is due to the fact that a very small amountof exciton population is transferred to the second ring ofthe two-ring system during this period as clearly revealed inFigs 3(c) and 4(c) Thus the dynamics of both the exciton andphonons are essentially those of the one-ring case Howeverafter 10 phonon periods as the transfer of exciton populationincreases |αn(t)|2 and ξ n(t) become quite different for the twocases of different inter-ring distances

Interestingly as one may observe from Figs 3(c) and4(c) many bright spots appear in the exciton probability con-tour of ring-2 indicating possible pathway interference withinthe ring The origins of the bright spots are explained as fol-lows Among the nearest neighbors of ring-1 [cf Fig 3(f)]sites 8 to 10 of the second ring show pronounced changes inexciton probability if one compares Figs 3(c) and 4(c) It isalso true that interactions between sites 1 2 and 16 of ring-1and sites 8 9 and 10 of ring-2 dominate inter-ring transferaccording to Eq (7) resulting in multiple inter-ring excitontransfer pathways

Due to strong interactions between site 1 of ring-1 andsite 9 of ring-2 owing to their proximity it is somewhat coun-terintuitive to observe the relatively smaller exciton probabil-ity on site 9 of ring-2 The explanation lies in the dimeriza-tion of LH2 rings which results in opposite signs of dipolemoments of site 1 of ring-1 and site 9 of ring-2 as shown inFig 3(f) Besides dimerization it is also helpful to considerthe intra-ring interference in ring-2 between exciton wavepackets that travel through multiple paths from ring-1 to ring-2 Figures 3(c) and 4(c) suggest that sites 8 and 10 of ring-2(especially site 10) can be treated as input sites strongly con-nected to ring-1 Appearance of bright spots on sites 8 and10 can be simply understood as a combined effect of excitontransfer from ring-1 to ring-2 and a constructive interferenceof exciton wave packets in ring-2 However on locations farfrom site 10 such as sites 2 5 13 and 15 of the second ringthe effect of constructive interference is more prominent Assites 2 and 16 of ring-2 are opposite to the aforementionedinput sites it is not surprising to observe bright spots on sites2 and 16 as a result of constructive interference as shown inFigs 3(c) and 4(c) Effects of multiple pathways for inter-ringexciton transfer may thus play a role in the appearance of ad-ditional bright spots on ring-2

Even more interestingly one can also calculate the exci-ton population of a given ring ie

sumn |αr

n(t)|2 in ring-r (withr the ring index) as a function of time to shed light on theoverall exciton dynamics of the two-ring system with inter-ring distance of 15 and 20 Aring Calculation results are displayedin Fig 5 where exciton populations corresponding to Figs 3and 4 are plotted as a function of time A supplementary cal-culation for a much longer time is also given in the inset ofFig 5 for an inter-ring distance of 15 Aring Short-time resultssuggest a nearly stepwise population transfer from ring-1 toring-2 The stepwise features of population profiles in Fig 5coincide with the increase in the exciton probabilities on sites8 to 10 in the second ring as shown in Figs 3(c) and 4(c)lending further support to our previous discussion on inter-ring exciton transfer Long-time calculation for the case withan inter-ring distance of 15 Aring reveals that it takes about 200phonon periods to have the complete transfer of the excitonpopulation as shown in the inset of Fig 5 As one phononperiod is equal to about 20 fs the estimated total transfer timeof 4 ps is in good agreement with the findings in Ref 45

Effects of static disorder on polaron dynamics have alsobeen examined in order to test the applicability of the currentapproach to photosynthetic systems Simulation results withGaussian-distributed static disorder are included in Fig 5The on-site energy average is set to zero and a standard devi-ation of σ = 01ω0 is adopted in accordance with Ref 56For simplicity only on-site energy disorder is included al-though it will not be difficult to examine off-diagonal dis-order Over 100 realizations have been simulated in the cal-culation Figure 5 provides a comparison on the excitonpopulation transfer with and without the energetic disorderWhile some population fluctuations have been smoothenedout the overall energy transfer efficiency is not signifi-cantly affected by the addition of static disorder Further-more in the absence of any exact or more reliable dy-namics studies our approach here provides a useful guide



FIG 3 Real space dynamics of (a) exciton probability |αn(t)|2 and (b) phonon displacement ξn(t) in ring-1 for a two-LH2-ring system with realistic LH2 ringtransfer integral matrix a moderate exciton-phonon coupling strength S = 05 and a wide bandwidth W = 08 of phonon dispersion The D1 Ansatz is usedfor the simulation Corresponding real space dynamics in ring-2 are displayed for (c) exciton probability |αn(t)|2 and (d) phonon displacement ξn(t) The timeevolution of energy components from exciton phonon exciton-phonon interaction deviation magnitude 13(t) has also been given in (e) The distance betweentwo rings is set to 15 Aring with interaction between two rings given by Eq (8) (f) The schematics of two connected LH2 complexes r is the distance between thetips of the rings



FIG 4 Real space dynamics of (a) exciton probability |αn(t)|2 and (b) phonon displacement ξn(t) in ring-1 for a two-LH2-ring system with realistic LH2 ringtransfer integral matrix a moderate exciton-phonon coupling strength S = 05 and a wide bandwidth W = 08 of phonon dispersion The D1 Ansatz is usedfor the simulation Corresponding real space dynamics in ring-2 are displayed for (c) exciton probability |αn(t)|2 and (d) phonon displacement ξn(t) The timeevolution of energy components from exciton phonon exciton-phonon interaction deviation magnitude 13(t) has also been given in (e) The distance betweentwo rings is set to 20 Aring with interaction between two rings given by Eq (8)



FIG 5 Exciton population dynamics (sum

n |αrn(t)|2) for the two different sep-

arations shown in Figs 2 and 3 where a long time dynamics of the excitonpopulation has also been given in the inset for the ring system with a distanceof 15 Aring Exciton population dynamics for the inter-ring distance of 15 Aring withon-site energy static disorder has also been given in the figure

to the dynamics of excitonic energy transfer in LH1-LH2systems

B Exciton population dynamics of N-site symmetricsimple ring systems and LH2 complexes

Dynamics study in Sec III A reveals the effect of dipo-lar interactions on exciton transfer efficiency at low tempera-tures but it is also interesting to look at the opposite limit ofhigh temperatures As a starting point it is preferable to usesimple ring systems to analyze what may affect the energytransfer efficiency at high temperatures Here we first studythe energy transfer in symmetric ring systems of three foursix seven eight and sixteen pigments Efficiency of the en-ergy transfer with respect to various trapping positions willbe examined and for an N-site ring system analytical solu-tions will be obtained using the stationary condition on theoff-diagonal density matrix element ρnm

For a three-site ring system we assume that all the siteenergies are the same ie the energy difference is zero Thecoupling between neighboring sites is assumed to be constantwith J12 = J21 = J13 = J31 = J23 = J32 = J Furthermorethe coupling with the environment (dephasing) is 12 = 21

= 13 = 31 = 23 = 32 = If the system is excited atsite 1 and the excitation is trapped at the same location theaverage trapping time is given by 〈t〉 = 3kminus1

t independent of and J If a photon is absorbed at site 1 and the excitation istrapped at site 2 the average trapping time takes the form

〈t〉 = 3

kt

+ 1

+ 1

2 + kt

+ 2 + kt

4J 2 (24)

which depends on and J It is greater than that of the previ-ous situation

For a four-site ring system if we only consider near-est neighbor couplings the analytical expression for the res-idence time and the average trapping time can be easily ob-tained If a photon is absorbed at site 1 and the excitation is

also trapped at site 1 the average trapping time is given by〈t〉 = 4kminus1

t If a photon is absorbed at site 1 and the excita-tion is trapped at site 2 (or equivalently site 4) the averagetrapping time is

〈t〉 = 4

kt

+ 1

+ 1

+ kt2+ 3(2 + kt )

8J 2 (25)

If a photon is absorbed at site 1 and the excitation is trappedat site 3 the average trapping time is

〈t〉 = 4

kt

+

J 2+ 3kt

8J 2 (26)

By comparing the three equations for the four-site ring sys-tem it is found that the average trapping time of the first sys-tem is the smallest among the three scenarios and that of thesecond is larger than that of the third

As indicated by the definition of 〈t〉 exciton population isproportional to the value of average trapping time Thus wecan also probe the dynamics of the system using the excitonpopulation It is interesting to note that the exciton populationof site 3 grows larger than that of site 2 and site 4 even thoughit is farther away from site 1 where the exciton is initially lo-cated The phenomenon also happens for the second and thirdscenarios To explain this one can consider the two distinctpaths for the exciton produced at site 1 in clockwise and andcounterclockwise directions They have the same phase whenthey meet at site 3 where the exciton population is greatly en-hanced due to quantum interference As a result the excitonpopulation of site 3 is larger than that of site 2 and site 4 eventhough they are closer to site 1

Furthermore we note a few differences among these sce-narios ρ1(t) is larger in the second scenario than others there-fore it has the largest average trapping time which leads tothe lowest energy transfer efficiency and ρ2(t) and ρ4(t) areexactly the same in the first scenario due to positional symme-try ρ4(t) is slightly larger in the third scenario than the firstbecause the trap at site 2 destroys the symmetry

C Energy transfer in single ring systems

It is more interesting to look into the energy transfer ef-ficiency of the ring system from the point of view of averagetrapping time and as a starting point simple single-ring sys-tems with six seven eight and sixteen sites have been stud-ied The average trapping times at different trapping positionsfor the ring systems (with other parameters kt = 2 t = 20J = 100) are given in Fig 6(a) Site 1 is located at the ori-gin (0) and other sites are distributed in the ring uniformlyTaking a six-site system as an example the zero-degree trap-ping corresponds to the case in which a photon is initiallyabsorbed at site one and the excitation is trapped at the samesite Similarly (180) trapping corresponds to the situationin which a photon is absorbed at site 1 and the excitation istrapped at site 4 It is interesting to find that the systems withan even number of sites have a comparatively smaller aver-age trapping time when the exciton is trapped at the exactlyopposite position (eg 180) of photon absorption Since thisphenomenon does not occur in systems with odd number ofsites it can be inferred from the simple ring systems that the



FIG 6 (a) The average trapping time versus varying trapping positions for simple single-ring systems From bottom upwards the square circle triangle downtriangle left triangle and right triangle correspond to three four six seven eight and sixteen-sites ring systems respectively For clarity the schematics of thering systems as well as trapping positions relative to the x-axis (with the angle measured in degrees) are shown in (b) For all systems the parameters chosen forthe Haken-Strobl model are kt = 2 t = 20 and J = 100 The same quantity obtained from polaron dynamics for a sixteen-site ring system is also displayedin (a) as red line and triangles The parameters chosen for polaron dynamics are ω0 = 1 γ n = 0007 J = 03 S = 05 and W = 08 Be noted that the energyunit for the polaron dynamics calculation is the characteristic phonon frequency ω0



constructive and destructive interference closely related tothe symmetry of a system plays a significant role in deter-mining the energy transfer efficiency

For a ring system with even number of sites it is possibleto have constructive interference at the site exactly oppositeto the photon absorption site in the ring where the excitonamplitudes are now in phase due to their symmetrical transferpath In this scenario the constructive interference providesalmost exactly the same average trapping time compared tothe case where exciton is created and trapped at the samesite However it should be noted that for systems with largersizes dephasing becomes increasingly important resulting ina slight reduction in the energy transfer efficiency

It is also interesting to compare our polaron dynamicssimulation with the Haken-Strobl approach Here we onlyconsider the case of a toy model with 16 sites and nearestneighbor transfer integrals The trapping rate on site n γ nis assumed to be 0007ω0 which translates into kt asymp 2psminus1To compare with the output of the Haken-Strobl approachthe polaron dynamics result is also included in Fig 6(a) (redline and triangles) As indicated in Fig 6(a) similar behaviorin the dependence of the average trapping time on trappingpositions is found in the polaron dynamics results and thosecalculated by the Haken-Strobl method despite many differ-ences between the two approaches This is especially true onsite 1 (0) and the site opposite to site 1 (180) where thetrapping time reaches minima On other sites off diagonal el-ements of the density matrix neglected in the Haken-Strobleapproach seem to play a more important role in determiningthe trapping time

Now we turn to more realistic systems of photosynthe-sis where capture of solar photons is often facilitated bylight-harvesting antennae and excitons first created in theantennae are transferred to reaction centers Motivatedby the results of simple ring systems we further examine howthe trapping position would affect excitonic energy transfer ina realistic ring system that includes dipole-dipole interactionsbetween all transition-dipole pairs that are not nearest neigh-bors Figure 7(a) displays the dependence of average trappingtime on the trapping position and the inset of Fig 7(a) aschematic of B850 ring of LH2 in which solid dots denotethe α-apropotein and the hollow dots the β-apropotein In-terestingly a dip corresponding to a minimal average trappingtime is found to appear at site 9 in addition to the one at site1 In comparison to the uniform ring systems shown in Fig6 there emerge zig-zag features in the trapping-time versussite-number plot in Fig 7(a) Although the steepest dip canbe simply explained with constructive interference the ap-pearance of multiple dips is directly related to the inclusionof long-range dipolar interactions and therefore multiple ex-ction transfer pathways It is also noted that the dependenceof average trapping time on the site index no longer has thesymmetries that exist for uniform rings Furthermore the non-symmetric dependence of 〈t〉 on the trapping position makes itpossible to have different clockwise and anti-clockwise pathsan effect that can be related to the dimerization of the LH2ring

Dependence of average trapping time on trapping posi-tion has also been examined for a single LH1 ring and the

FIG 7 Average trapping time versus trapping site for the inset schematicsof B850 ring of (a) LH2 and (b) LH1 Solid dots represent the α-apropoteinsand hollow dots the β-apropoteins The directions of the dipoles are tan-gential to the circle with α-apropoteins pointing in clockwise direction β-apropoteins anti-clockwise A photon is absorbed at 1α BChl trapped atdifferent BChls For all calculations the trapping rate kt is set to 1 and de-phasing rate is equal to 5 The rest of parameters ie Jnm are given inEqs (7) and (8)

results are shown in Fig 7(b) The exciton Hamiltonian ofLH1 has a form similar to Eq (7) with the total number ofsites increased from 16 to 32 Chromophores are assumed tobe evenly distributed with a distance of about 10 Aring and otherparameters for the LH1 system are taken from Ref 33 Re-sults similar to the LH2 case can be obtained from Fig 7(b)and a dip appears around the middle point of the ring at site 17with an obvious loss of symmetry Due to the increase in thetotal site number which mitigates interference effects a muchsmoother curve is obtained in Fig 7(b) For both the LH2 andLH1 rings we have found that the long-range dipolar cou-pling reduces substantially the effect of quantum inference inagreement with similar earlier findings in polaron dynamicssimulations



D Energy transfer in multiple connected ring systems

To expand our study of energy transfer efficiency tomultiple-ring systems two-ring toy models as depicted inFig 8 are examined first Intra-ring nearest-neighbor interac-tions are assumed with coupling strength J and site energiesare set to be the same The two rings are coupling throughsite 16 and site 17 with coupling strength V Our findingsare shown in Fig 8 and an optimal coupling strength for ef-ficient energy transfer is discovered at zero pure dephasingrate = 0 This optimal strength found is close to the valueof intra-ring coupling J a result of system symmetry loss Ifthe trapping site is moved from site 32 to site 31 the opti-mal trapping time is found at a nonzero pure dephrasing rate = 02 as shown in Fig 8(c) It can be speculated thatto optimize the energy transfer efficiency system symmetryimperfections can be compensated by the introduction of thepure dephasing or the dissipation of system coherence by theenvironment Such a finding is applicable for the design ofartificial photosynthetic systems in general

Next we will look at a realistic two-ring LH2 systemwith long-range dipolar interactions as illustrated in Fig 9(a)Two rings interact via dipole-dipole coupling according toEq (8) An initial state in which all sites in the first ringare evenly excited is selected ie at t = 0 ρnn = 116 forn = 1 2 16 in the first ring We also assume that the exci-tation is evenly trapped in the second ring It is found that thetrapping time increases with the edge-to-edge inter-ring dis-tance r as expected and for r gt 20 Aring (or with higher dephas-ing rates) it increases rapidly It is known that quantum coher-ence may help enhance the efficiency of energy transfer34 Athigher dephasing rates with the destruction of coherence theexcitation is transported classically resulting in longer trap-ping times and lower efficiency

The findings on average trapping time in Fig 9 suggestan extremely distance-dependent nature of energy transfer Asindicated in Ref 57 the sites too close to each other in certainorientations will lead to non-fluorescent dimers which canserve as excitation sinks that deplete excitations

One can also observe an optimal value of inter-ring dis-tance of around 5 Aring corresponding to the highest strengthof dipolar interaction between the nearest neighbor dipolesThis value is not close to the realistic inter-ring distance ofLH2 complexes in a close packed membrane Although it hasbeen pointed out in Ref 57 that typical inter-chromophorecenter-to-center distances of neighboring molecules are con-sequently close (6 to 25 Aring) Thus the global minimum ob-tained with the method introduced in this paper is only anideal case such that two molecular aggregates are in con-tact with each other without any support of protein scaffoldAlthough the value only corresponds to the ideal case withhighest inter-ring coupling strength the existence of the localminimum is non-negligible Such local minimum in aver-age trapping time with non-zero dephasing rate appears ina continuous fashion throughout a large range of distancesbetween the rings as shown in Figs 9(b) and 9(c) Its exis-tence can be explained by the role environment related de-phasing plays in increasing energy transfer efficiency for asystem with asymmetric properties The symmetry-breaking

FIG 8 (a) Schematic of a two-ring toy model (b) Contour map of 〈t〉 as afunction of and V with trapping site locating at site 32 of the second ringas shown in (a) (c) Contour map of 〈t〉 as a function of and V with trap-ping site locating at site 31 of the second ring The value of nearest neighborcoupling strength is set to J = 1 and the trapping rate kt = 1

dipolar interaction in the LH2 system leads to multiple non-nearest neighbor energy transfer pathways which can en-hance the energy transfer within the ring greatly as shown inthe polaron dynamics calculations however it is more prefer-able to have more exciton populations that can be trappedat certain site Thus it is preferable to have finite dephasingto increase the inter-ring transfer efficiency by maximizingthe amount of exciton populations that can be trapped in the



FIG 9 (a) Schematic of a two-ring LH2 system r is the distance betweenthe edges of the rings (b) Short distance 〈t〉 contour map as a function of

and r (c) Long distance 〈t〉 contour map as a function of and r In the calcu-lations of 〈t〉s the first ring on the left is evenly excited and exciton is evenlytrapped in the ring-2 Inter-ring interaction takes same form of the intra-ringdipole-dipole coupling The trapping rate in the calculations is assumed to bea constant with kt = 1

trapping site of the second ring through multiple pathway in-terference This small amount of dephasing is not sufficient todestroy the quantum coherence in the system which gives usa hint on how important the unique arrangements and inter-actions between chromophores in the LH2 complexes as wellas the way with which each complex interacts with others arein optimizing the energy transfer in the presence of an envi-ronment that is coupled to the system More interestingly in arecent work on FMO complex58 even for this finite-size dis-ordered molecular network can effectively preserve coherent

excitation energy transfer against ambient dephasing whichlends a further support to what we have revealed in this sec-tion Furthermore the role of dipolar interaction in increasingthe exciton transfer efficiency through introducing multiplepathways for inter-ring exciton transfer has been made amplyevident

IV CONCLUSIONS

In this paper two distinct approaches have been appliedto study the excitation energy transfer in the ring systemsFrom both methods non-nearest neighbor dipolar interactionis found to be helpful in increasing intra- or inter-ring excitontransfer efficiency as a result of multiple pathways

Using the Haken-Strobl model the energy transfer in thering systems with coupling to environment is studied in de-tails The energy trapping efficiency is found to be dependenton the trapping position in both the hypothetic and the real-istic ring systems For the hypothetic system with the trappositioned diametrically opposite to the site of the initial ex-citation (which is possible only in systems with even numberof sites) we observe a sudden drop of the trapping time due tothe constructive interference between the clockwise and anti-clockwise paths leading to the trap The situations in a LH2(B850) ring and LH1 ring are similar to the hypothetic sys-tems but owing to the broken symmetry caused by the dimer-ization of BChls and dipolar couplings the drop of the trap-ping time is not significant In other words the ring becomesmore homogeneous in terms of energy transfer which furthersupports the findings of polaron dynamics studies of one-ringsystem

The Haken-Strobl method is a high-temperature treat-ment centering on the population dynamics while neglect-ing coherence dynamics among neighboring pigments Incomparison the Holstein Hamiltonian dynamics adds an al-ternative low-temperature perspective to the problem Dueto the femtosecond nature of the energy transfer in photo-synthetic complexes full thermal equilibrium may be hardto attain before the exciton is transferred from an an-tenna complex to a reaction center and it makes sensethat the system is also modeled as a microcanonical en-semble The simulation on polaron dynamics adds far morequantum details to the femtosecond energy transfer using anon-equilibrium framework in a supposedly low-temperaturesetting

The polaron dynamics of the coupled rings revealed aninteresting role of dipolar interaction in increasing energytransfer efficiency by introducing multiple transfer paths be-tween the rings and the time scale of inter-ring exciton trans-fer is in good agreement with previous studies Moreoverwhen the Haken-Strobl model is applied for the coupled ringsan optimal coupling strength is obtained between the hypo-thetic two-ring systems with zero dephasing rate while in atwo-LH2 system owing to the intrinsic symmetry breakingthe global minimum and local minimum of average trappingtime is always found with non-zero depahsing rate In boththe cases the efficiency drops with a further increase of de-phasing rate due to the loss of quantum coherence The find-ings in this paper regarding exciton transfer processes in ring



systems might assist us in designing more efficient artificiallight harvesting systems

ACKNOWLEDGMENTS

Support from the Singapore National Research Founda-tion through the Competitive Research Programme (ProjectNo NRF-CRP5-2009-04) and the Singapore Ministry of Edu-cation through the Academic Research Fund (Tier 2) (ProjectNo T207B1214) is gratefully acknowledged

APPENDIX A PRECISION OF THE TIME-DEPENDENTDAVYDOV ANSAumlTZE

For a trial wave function |(t)〉 that does not strictly obeythe time-dependent Schroumldinger equation the deviation vec-tor |δ(t)〉 can be defined as

|δ(t)〉 equiv ipart

partt|(t)〉 minus H |(t)〉 (A1)

and the deviation amplitude 13(t) is defined as

13(t) equivradic

〈δ(t)|δ(t)〉 (A2)

For the Holstein Hamiltonian H defined in Eqs (1)ndash(4)one can derive the explicit expression of 〈δ(t)|δ(t)〉 for the D1

Ansatz

〈δD1 (t)|δD1 (t)〉=

sumn

|iαn(t) + Tn(t) + αn(t)Rn(t)|2

+sum

αn(t)=0

sumq

|αn(t)

[iλnq(t) + gqradic

Nωqe

minusiqn minus ωqλnq(t)

]

+nq (t)|2 + 132D(t) minus

sumαn(t)=0

sumq

|nq(t)|2 (A3)

with

Rn(t) equiv Resum

q

[iλnq(t)+ 2gqradic

Nωqe

minusiqnminusωqλnq(t)

]λlowast

nq(t)

(A4)where Tn(t) n q(t) and 132

D(t) are three expressions whichhave no item of αn(t) or λnq(t)

For the D2 Ansatz Eq (A3) can be further derived to

〈δD2 (t)|δD2 (t)〉=

sumn


+sum

q

|iβq(t) + gqradicN

ωq

sumn

|αn(t)|2eminusiqn minus ωqβq(t)|2

+ 1

N

sumq

g2qω

2q

[1 minus

∣∣∣sumn

|αn(t)|2eminusiqn∣∣∣2

] (A5)

And for the D Ansatz Eq (A3) can be further derived to

〈δD(t)|δD(t)〉=

sumn


+ |iλ0(t) + gqradicN

ω0 minus ω0λ0(t)|2

+sumq =0

∣∣∣iγq(t) + ξq(t)iβq(t) + yq(t)∣∣∣2

+sumq =0

[1 minus |ξq(t)|2]

∣∣∣∣iβq(t) + dq(t)

1 minus |ξq(t)|2∣∣∣∣2

+sumq =0

sumn

|nq(t)|2 minus[|yq(t)|2 + |dq(t)|2

1 minus |ξq(t)|2]

+132D(t) minus

sumn

sumq

|nq(t)|2 (A6)

where ξ q(t) n q(t) yq(t) and dq(t) are four expressionswhich have no item of αn(t) βq(t) γq(t) or λ0(t)

Substituting Eq (11) into Eqs (2)minus(4) one obtains theexpressions for the system energies by the Davydov Ansaumltzein the Holstein model

Eex(t) equiv 〈D(t)|Hex|D(t)〉= minus 2JRe

sumn

αlowastn(t)Snn+1(t)αn+1(t) (A7)

Eph(t) equiv 〈D(t)|Hph|D(t)〉=

sumn

[|αn(t)|2

sumq

ωq |λnq(t)|2] (A8)

and

Eexminusph(t) equiv 〈D(t)|Hexminusph|D(t)〉

= minus 2radicN

sumn

[|αn(t)|2Re

sumq

gqωqλnq(t)eiqn]

(A9)

where Sn m(t) is the Debye-Waller factorNote that since the unit of 13(t) is that of the energy by

comparing 13(t) with the main component of the system ener-gies such as Eph(t) and Eexminusph(t) one can observe whether thedeviation of an Ansatz from obeying the Schroumldinger equationis negligible or not in the concerned case From this perspec-tive the comparison between 13(t) and energy components ofthe system provides a good reference for the validity of anAnsatz

APPENDIX B DETAIL DERIVATION OF POLARONDYNAMICS IN MULTIPLE-RING SYSTEM

The system energies of the multiple-ring system canbe obtained as follows by apply Davydov Ansaumltze to the



modified Holstein Hamiltonian given in Eqs (5) (9)and (10)

Eex(t) =sumr1r2

sumnm

J r1r2nm αr1lowast

n (t)αr2m (t)Sr1r2

nm (t) (B1)

Eph(t) =sum

r

sumn

∣∣αrn(t)

∣∣2 sumq

ωrq

∣∣λrnq(t)

∣∣2 (B2)

Eexminusph(t) = minus 2radicN

sumr

sumn

sumq

grqω

rq

∣∣αrn(t)

∣∣2Re

[eiqnλr

nq(t)]

(B3)with Debye-Waller factor Sr1r2

nm (t) given by

Sr1r2nm (t) = exp

sumq

[λr1lowast

nq (t)λr2mq(t)δr1r2

minus1

2

∣∣λr1nq(t)

∣∣2 minus 1

2

∣∣λr2mq(t)

∣∣2] (B4)

and the index r1 and r2 all runs over 1 to NringSince D1 is the used for all calculations in this paper here

we only give the expression of 〈δ(t)|δ(t)〉 for D1 which can bederived as follows

〈δD1 (t)|δD1 (t)〉=

sumr

sumn

∣∣iαrn(t) + αr

n(t)Rrn(t) + T r

n (t)∣∣2

+sum

r

sumαr

n(t)=0

sumq

∣∣αrn(t)r

nq(t) + rnq(t)

∣∣2 + 132D(t)

minussum

r

sumαr

n(t)=0

sumq

∣∣rnq(t)

∣∣2 minussum

r

sumn

∣∣T rn (t)

∣∣2 (B5)

Minimization of the first two terms in Eq (B5) leads tothe equations of motions for the time-dependent variationalparameters αr

n(t) and λrnq(t) as

αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] (B6)

and

λrnq(t) = i

[r

nq(t)

αrn(t)

+ grqradicN

ωrqe

minusiqn minus ωrqλ

rnq(t)

] (B7)

Each collected term in Eqs (B6) and (B7) is given as

rnq(t) = iλr

nq(t) + grqradicN

ωrqe


rnq(t) (B8)

Rrn(t) = Re

sumq

[r

nq(t) + grqradicN

ωrqe

minusiqn

]λrlowast

nq(t) (B9)

T r1n (t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t) (B10)

r1nq(t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t)

[λr2

mq(t)δr1r2 minus λr1nq(t)

]

(B11)

and

132D(t) =

sumr1

sumn

[ sumr prime

2mprime

sumr2m

Jr1r

prime2

nmprime Jr1r2nm α

r prime2lowast

mprime (t)αr2m (t)S

r prime2r2

mprimem(t)

]

(B12)

APPENDIX C TRAPPING IN POLARONDYNAMICS SIMULATION

Based on the Trotter decomposition we have

|D(t)〉 = eminusiH tminus 12 γ t |D(0)〉

= lim13trarr0

[eminusiH13tminus 12 γ 13t ]N |D(0)〉

= lim13trarr0

[eminus 12 γ 13teminusiH13t ]N |D(0)〉 (C1)

where t = N13t with N an integer γ is the trapping matrix andH is the Holstein Hamiltonian The fourth-order Runge-Kuttamethod will have enough precision if 13t is sufficiently smallFrom Eq (C1) and assuming γ is a diagonal matrix for eachtime step 13t only a trapping factor γnprime is added at site nprime inEq (B6) which can be explicitly expressed as

αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] minus δnnprimeγnprimeαr

nprime(t) (C2)

The reduced density matrix is again

ρ(t) = Trph[|D(t)〉〈D(t)|] (C3)

We can obtain the reduced density matrix in the site represen-tation

ρmn(t) = 〈m|ρ(t)|n〉 (C4)

where the population is denoted by ρnn(t)

1K Sauer Annu Rev Phys Chem 30 155 (1979)2R E Blankenship Molecular Mechanisms of Photosynthesis (BlackwellScience OxfordMalden 2002)

3R van Grondelle and V I Novoderezhkin Phys Chem Chem Phys 8793 (2006)

4G D Scholes Annu Rev Phys Chem 54 57 (2003)5Y C Cheng and G R Fleming Annu Rev Phys Chem 60 241 (2009)6S Jang M D Newton and R J Silbey Phys Rev Lett 92 218301 (2004)7S Jang M D Netwton and R J Silbey J Phys Chem B 111 6807(2007)

8S Jang J Chem Phys 127 174710 (2007)9S Jang M D Newton and R J Silbey J Chem Phys 118 9312 (2003)

10S Jang and R J Silbey J Chem Phys 118 9324 (2003)11H P Breuer and F Petruccione The Theory of Open Quantum Systems

(Oxford University Press Oxford 2002)12A G Redfield IBM J Res Dev 1 19 (1957)13D J Heijs V A Malyshev and J Knoester Phys Rev Lett 95 177402

(2005)14M Schroumlder U Kleinekathofer and M Schreiber J Chem Phys 124

084903 (2006)15P Huo and D F Coker J Chem Phys 133 184108 (2010)16A Ishizaki and Y Tanimura J Phys Soc Jpn 74 3131 (2005)17M Tanaka and Y Tanimura J Phys Soc Jpn 78 073802 (2009)18M Tanaka and Y Tanimura J Chem Phys 132 214502 (2010)19A Sakurai and Y Tanimura J Phys Chem A 115 4009minus4022 (2011)20P Huo and D F Coker J Chem Phys 135 201101 (2011)21P Huo and D F Coker J Phys Chem Lett 2 825 (2011)22P Huo and D F Coker J Chem Phys 136 115102 (2012)23D P S McCutcheon and A Nazir New J Phys 12 113042 (2010)24D P S McCutcheon and A Nazir Phys Rev B 83 165101 (2011)



25D P S McCutcheon and A Nazir J Chem Phys 135 114501 (2011)26S Jang Y C Cheng D R Reichman and J D Eaves J Chem Phys 129

101104 (2008)27S Jang J Chem Phys 131 164101 (2009)28S Jang J Chem Phys 135 034105 (2011)29G McDermontt S M Prince A A Freer A M Hawthornthwaite M

Z Papiz R J Cogdell and N W Issacs Nature (London) 374 517(1995)

30J Koepke X Hu C Muenke K Schulten and H Michel Structure 4 581(1996)

31E Harel J Chem Phys 136 174104 (2012)32T Brixner J Stenger H M Vaswani M Cho R E Blankenship and G

R Fleming Nature (London) 434 625 (2005)33Y Zhao M F Ng and G H Chen Phys Rev E 69 032902 (2004)34A Olaya-Castro C F Lee F F Olsen and N F Johnson Phys Rev B

78 085115 (2008)35F Fassioli A Olaya-castro S Scheuring J N Sturgis and N F Johnson

Biophys J 97 2464 (2009)36F Muumlh M EI-A Madjet J Adolphs A Abdurahman B Rabenstein H

Ishikita E W Knapp and T Renger Proc Natl Acad Sci USA 10416862 (2007)

37A Ishizaki and G R Fleming Proc Natl Acad Sci USA 106 17255(2009)

38X Hu T Ritz A Damjanovic F Autenrieth and K Schulten Q RevBiophys 35 1 (2002)

39J S Cao and R J Silbey J Phys Chem A 113 13825 (2009)40J H Kim and J S Cao J Phys Chem B 114 16189 (2010)

41H Haken and G Strobl Z Phys 262 135 (1973)42T Holstein Ann Phys 8 325 (1959) 8 343 (1959)43G D Mahan Many Particle Physics 3rd ed (Kluwer 2000)44C Olbrich and U Kleinekathoumlfer J Phys Chem B 114 12427 (2010)45X Hu T Ritz A Damjanovic and K Schulten J Phys Chem B 101

3854 (1997)46C Itzykson and J Zuber Many-Particle Physics (McGraw-Hill New York

1980)47C Hofmann T J Aartsma H Michel and J Koumlhler Proc Natl Acad

Sci USA 100 15534 (2003)48B Luo J Ye C B Guan and Y Zhao Phys Chem Chem Phys 12

15073 (2010)49B Luo J Ye and Y Zhao Phys Status Solidi C 8 70 (2011)50S Jang R J Silbey R Kunz and C Hofmann and J Koumlhler J Phys

Chem B 115 12947 (2011)51A Damjanovic I Kosztin U Kleinekathoumlfer and K Schulten Phys Rev

E 65 031919 (2002)52T Ritz A Damjanovic and K Schulten Chem Phys Chem 3 243

(2002)53H D Kim Y Tanimura and M H Cho J Chem Phys 127 075101

(2007)54E O Potma and D A Wiersma J Chem Phys 108 4894 (1998)55J L Herek et al Biophys J 78 2590 (2000)56S Jang S E Dempster and R J Silbey J Phys Chem B 105 6655

(2001)57G D Scholes et al Nat Chem 3 763 (2011)58T Scholak et al Phys Rev E 83 021912 (2011)


Excitonic energy transfer in light-harvesting complexes in purple bacteriaJun Ye Kewei Sun Yang Zhao Yunjin Yu Chee Kong Lee et al Citation J Chem Phys 136 245104 (2012) doi 10106314729786 View online httpdxdoiorg10106314729786 View Table of Contents httpjcpaiporgresource1JCPSA6v136i24 Published by the American Institute of Physics Additional information on J Chem PhysJournal Homepage httpjcpaiporg Journal Information httpjcpaiporgaboutabout_the_journal Top downloads httpjcpaiporgfeaturesmost_downloaded Information for Authors httpjcpaiporgauthors







I INTRODUCTION














II METHODOLOGY



A Hamiltonian



with

Hex = minusJsum

n


Hph =sum

q

ωqbdaggerq bq (3)


sumn

sumq

gqωqadaggernan(bqe

iqn + Hc) (4)








H primeex =

sumr1r2

sumnm

J r1r2nm ar1dagger

n ar2m (5)


H primeprimeex =

sumr

sumnm

J rnmardagger

n arm (6)





Jnm=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝






⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠


Wij = C

[di middot dj


|rij |5] (8)




H primeph =

sumr

sumq

ωrq b

rdaggerq br

q (9)


N

sumr

sumn

sumq

grqω

rq a

rdaggern ar



rdaggerq (br



sumq(gr

q)2ωrq




|D(t)〉 =sum

n

αn(t)adaggernU





sumq


(12)



|(t)〉 equiv |αm(t)〉 (13)


ipart

partt|(t)〉 = H |(t)〉 (14)


ipart

parttminus H

)|(t)〉 = |αm(t) αm(t)〉 = 0 (15)


〈m|αm(t) αm(t)〉 = 0 (16)


L = 〈(t)| imacr2

larrrarrpart



d

dt

(partL

partαlowastm

)minus partL

partαlowastm

= 0 (18)



2|D(t)〉 (19)








τn =int infin

0ρnn(t)dt (20)


ρ(t) = Trph[|D(t)〉〈D(t)|] (21)




ρ(t) = minusLρ(t)






q =sum


n kdnτn

(23)




































sumn |αr




















〈t〉 = 3

kt

+ 1

+ 1

2 + kt

+ 2 + kt

4J 2 (24)





〈t〉 = 4

kt

+ 1

+ 1

+ kt2+ 3(2 + kt )

8J 2 (25)


〈t〉 = 4

kt

+

J 2+ 3kt

8J 2 (26)

































IV CONCLUSIONS








ACKNOWLEDGMENTS







13(t) equivradic

〈δ(t)|δ(t)〉 (A2)


Ansatz

〈δD1 (t)|δD1 (t)〉=

sumn


+sum

αn(t)=0

sumq

|αn(t)

[iλnq(t) + gqradic

Nωqe


]


sumαn(t)=0

sumq

|nq(t)|2 (A3)

with

Rn(t) equiv Resum

q

[iλnq(t)+ 2gqradic

Nωqe


]λlowast

nq(t)




〈δD2 (t)|δD2 (t)〉=

sumn


+sum

q

|iβq(t) + gqradicN

ωq

sumn


+ 1

N

sumq

g2qω

2q

[1 minus

∣∣∣sumn


] (A5)



sumn




+sumq =0


+sumq =0

[1 minus |ξq(t)|2]

∣∣∣∣iβq(t) + dq(t)

1 minus |ξq(t)|2∣∣∣∣2

+sumq =0

sumn


1 minus |ξq(t)|2]

+132D(t) minus

sumn

sumq

|nq(t)|2 (A6)




sumn



sumn

[|αn(t)|2

sumq

ωq |λnq(t)|2] (A8)

and


= minus 2radicN

sumn

[|αn(t)|2Re

sumq

gqωqλnq(t)eiqn]

(A9)








Eex(t) =sumr1r2

sumnm

J r1r2nm αr1lowast

n (t)αr2m (t)Sr1r2

nm (t) (B1)

Eph(t) =sum

r

sumn

∣∣αrn(t)

∣∣2 sumq

ωrq

∣∣λrnq(t)

∣∣2 (B2)


sumr

sumn

sumq

grqω

rq

∣∣αrn(t)

∣∣2Re

[eiqnλr

nq(t)]


nm (t) given by

Sr1r2nm (t) = exp

sumq

[λr1lowast


minus1

2

∣∣λr1nq(t)

∣∣2 minus 1

2

∣∣λr2mq(t)

∣∣2] (B4)



〈δD1 (t)|δD1 (t)〉=

sumr

sumn


n(t)Rrn(t) + T r

n (t)∣∣2

+sum

r

sumαr

n(t)=0

sumq

∣∣αrn(t)r

nq(t) + rnq(t)

∣∣2 + 132D(t)

minussum

r

sumαr

n(t)=0

sumq

∣∣rnq(t)

∣∣2 minussum

r

sumn

∣∣T rn (t)

∣∣2 (B5)



αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] (B6)

and

λrnq(t) = i

[r

nq(t)

αrn(t)

+ grqradicN

ωrqe


rnq(t)

] (B7)


rnq(t) = iλr

nq(t) + grqradicN

ωrqe


rnq(t) (B8)

Rrn(t) = Re

sumq

[r

nq(t) + grqradicN

ωrqe

minusiqn

]λrlowast

nq(t) (B9)

T r1n (t) = minus

sumr2

summ

J r1r2nm αr2


r1nq(t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t)

[λr2


]

(B11)

and

132D(t) =

sumr1

sumn

[ sumr prime

2mprime

sumr2m

Jr1r

prime2

nmprime Jr1r2nm α

r prime2lowast


r prime2r2

mprimem(t)

]

(B12)




= lim13trarr0


= lim13trarr0



αrn(t) = i

[T r

n (t) + αrn(t)Rr


nprime(t) (C2)


ρ(t) = Trph[|D(t)〉〈D(t)|] (C3)


ρmn(t) = 〈m|ρ(t)|n〉 (C4)








































I INTRODUCTION














II METHODOLOGY



A Hamiltonian



with

Hex = minusJsum

n


Hph =sum

q

ωqbdaggerq bq (3)


sumn

sumq

gqωqadaggernan(bqe

iqn + Hc) (4)








H primeex =

sumr1r2

sumnm

J r1r2nm ar1dagger

n ar2m (5)


H primeprimeex =

sumr

sumnm

J rnmardagger

n arm (6)





Jnm=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝






⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠


Wij = C

[di middot dj


|rij |5] (8)




H primeph =

sumr

sumq

ωrq b

rdaggerq br

q (9)


N

sumr

sumn

sumq

grqω

rq a

rdaggern ar



rdaggerq (br



sumq(gr

q)2ωrq




|D(t)〉 =sum

n

αn(t)adaggernU





sumq


(12)



|(t)〉 equiv |αm(t)〉 (13)


ipart

partt|(t)〉 = H |(t)〉 (14)


ipart

parttminus H

)|(t)〉 = |αm(t) αm(t)〉 = 0 (15)


〈m|αm(t) αm(t)〉 = 0 (16)


L = 〈(t)| imacr2

larrrarrpart



d

dt

(partL

partαlowastm

)minus partL

partαlowastm

= 0 (18)



2|D(t)〉 (19)








τn =int infin

0ρnn(t)dt (20)


ρ(t) = Trph[|D(t)〉〈D(t)|] (21)




ρ(t) = minusLρ(t)






q =sum


n kdnτn

(23)




































sumn |αr




















〈t〉 = 3

kt

+ 1

+ 1

2 + kt

+ 2 + kt

4J 2 (24)





〈t〉 = 4

kt

+ 1

+ 1

+ kt2+ 3(2 + kt )

8J 2 (25)


〈t〉 = 4

kt

+

J 2+ 3kt

8J 2 (26)

































IV CONCLUSIONS








ACKNOWLEDGMENTS







13(t) equivradic

〈δ(t)|δ(t)〉 (A2)


Ansatz

〈δD1 (t)|δD1 (t)〉=

sumn


+sum

αn(t)=0

sumq

|αn(t)

[iλnq(t) + gqradic

Nωqe


]


sumαn(t)=0

sumq

|nq(t)|2 (A3)

with

Rn(t) equiv Resum

q

[iλnq(t)+ 2gqradic

Nωqe


]λlowast

nq(t)




〈δD2 (t)|δD2 (t)〉=

sumn


+sum

q

|iβq(t) + gqradicN

ωq

sumn


+ 1

N

sumq

g2qω

2q

[1 minus

∣∣∣sumn


] (A5)



sumn




+sumq =0


+sumq =0

[1 minus |ξq(t)|2]

∣∣∣∣iβq(t) + dq(t)

1 minus |ξq(t)|2∣∣∣∣2

+sumq =0

sumn


1 minus |ξq(t)|2]

+132D(t) minus

sumn

sumq

|nq(t)|2 (A6)




sumn



sumn

[|αn(t)|2

sumq

ωq |λnq(t)|2] (A8)

and


= minus 2radicN

sumn

[|αn(t)|2Re

sumq

gqωqλnq(t)eiqn]

(A9)








Eex(t) =sumr1r2

sumnm

J r1r2nm αr1lowast

n (t)αr2m (t)Sr1r2

nm (t) (B1)

Eph(t) =sum

r

sumn

∣∣αrn(t)

∣∣2 sumq

ωrq

∣∣λrnq(t)

∣∣2 (B2)


sumr

sumn

sumq

grqω

rq

∣∣αrn(t)

∣∣2Re

[eiqnλr

nq(t)]


nm (t) given by

Sr1r2nm (t) = exp

sumq

[λr1lowast


minus1

2

∣∣λr1nq(t)

∣∣2 minus 1

2

∣∣λr2mq(t)

∣∣2] (B4)



〈δD1 (t)|δD1 (t)〉=

sumr

sumn


n(t)Rrn(t) + T r

n (t)∣∣2

+sum

r

sumαr

n(t)=0

sumq

∣∣αrn(t)r

nq(t) + rnq(t)

∣∣2 + 132D(t)

minussum

r

sumαr

n(t)=0

sumq

∣∣rnq(t)

∣∣2 minussum

r

sumn

∣∣T rn (t)

∣∣2 (B5)



αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] (B6)

and

λrnq(t) = i

[r

nq(t)

αrn(t)

+ grqradicN

ωrqe


rnq(t)

] (B7)


rnq(t) = iλr

nq(t) + grqradicN

ωrqe


rnq(t) (B8)

Rrn(t) = Re

sumq

[r

nq(t) + grqradicN

ωrqe

minusiqn

]λrlowast

nq(t) (B9)

T r1n (t) = minus

sumr2

summ

J r1r2nm αr2


r1nq(t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t)

[λr2


]

(B11)

and

132D(t) =

sumr1

sumn

[ sumr prime

2mprime

sumr2m

Jr1r

prime2

nmprime Jr1r2nm α

r prime2lowast


r prime2r2

mprimem(t)

]

(B12)




= lim13trarr0


= lim13trarr0



αrn(t) = i

[T r

n (t) + αrn(t)Rr


nprime(t) (C2)


ρ(t) = Trph[|D(t)〉〈D(t)|] (C3)


ρmn(t) = 〈m|ρ(t)|n〉 (C4)











































II METHODOLOGY



A Hamiltonian



with

Hex = minusJsum

n


Hph =sum

q

ωqbdaggerq bq (3)


sumn

sumq

gqωqadaggernan(bqe

iqn + Hc) (4)








H primeex =

sumr1r2

sumnm

J r1r2nm ar1dagger

n ar2m (5)


H primeprimeex =

sumr

sumnm

J rnmardagger

n arm (6)





Jnm=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝






⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠


Wij = C

[di middot dj


|rij |5] (8)




H primeph =

sumr

sumq

ωrq b

rdaggerq br

q (9)


N

sumr

sumn

sumq

grqω

rq a

rdaggern ar



rdaggerq (br



sumq(gr

q)2ωrq




|D(t)〉 =sum

n

αn(t)adaggernU





sumq


(12)



|(t)〉 equiv |αm(t)〉 (13)


ipart

partt|(t)〉 = H |(t)〉 (14)


ipart

parttminus H

)|(t)〉 = |αm(t) αm(t)〉 = 0 (15)


〈m|αm(t) αm(t)〉 = 0 (16)


L = 〈(t)| imacr2

larrrarrpart



d

dt

(partL

partαlowastm

)minus partL

partαlowastm

= 0 (18)



2|D(t)〉 (19)








τn =int infin

0ρnn(t)dt (20)


ρ(t) = Trph[|D(t)〉〈D(t)|] (21)




ρ(t) = minusLρ(t)






q =sum


n kdnτn

(23)




































sumn |αr




















〈t〉 = 3

kt

+ 1

+ 1

2 + kt

+ 2 + kt

4J 2 (24)





〈t〉 = 4

kt

+ 1

+ 1

+ kt2+ 3(2 + kt )

8J 2 (25)


〈t〉 = 4

kt

+

J 2+ 3kt

8J 2 (26)

































IV CONCLUSIONS








ACKNOWLEDGMENTS







13(t) equivradic

〈δ(t)|δ(t)〉 (A2)


Ansatz

〈δD1 (t)|δD1 (t)〉=

sumn


+sum

αn(t)=0

sumq

|αn(t)

[iλnq(t) + gqradic

Nωqe


]


sumαn(t)=0

sumq

|nq(t)|2 (A3)

with

Rn(t) equiv Resum

q

[iλnq(t)+ 2gqradic

Nωqe


]λlowast

nq(t)




〈δD2 (t)|δD2 (t)〉=

sumn


+sum

q

|iβq(t) + gqradicN

ωq

sumn


+ 1

N

sumq

g2qω

2q

[1 minus

∣∣∣sumn


] (A5)



sumn




+sumq =0


+sumq =0

[1 minus |ξq(t)|2]

∣∣∣∣iβq(t) + dq(t)

1 minus |ξq(t)|2∣∣∣∣2

+sumq =0

sumn


1 minus |ξq(t)|2]

+132D(t) minus

sumn

sumq

|nq(t)|2 (A6)




sumn



sumn

[|αn(t)|2

sumq

ωq |λnq(t)|2] (A8)

and


= minus 2radicN

sumn

[|αn(t)|2Re

sumq

gqωqλnq(t)eiqn]

(A9)








Eex(t) =sumr1r2

sumnm

J r1r2nm αr1lowast

n (t)αr2m (t)Sr1r2

nm (t) (B1)

Eph(t) =sum

r

sumn

∣∣αrn(t)

∣∣2 sumq

ωrq

∣∣λrnq(t)

∣∣2 (B2)


sumr

sumn

sumq

grqω

rq

∣∣αrn(t)

∣∣2Re

[eiqnλr

nq(t)]


nm (t) given by

Sr1r2nm (t) = exp

sumq

[λr1lowast


minus1

2

∣∣λr1nq(t)

∣∣2 minus 1

2

∣∣λr2mq(t)

∣∣2] (B4)



〈δD1 (t)|δD1 (t)〉=

sumr

sumn


n(t)Rrn(t) + T r

n (t)∣∣2

+sum

r

sumαr

n(t)=0

sumq

∣∣αrn(t)r

nq(t) + rnq(t)

∣∣2 + 132D(t)

minussum

r

sumαr

n(t)=0

sumq

∣∣rnq(t)

∣∣2 minussum

r

sumn

∣∣T rn (t)

∣∣2 (B5)



αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] (B6)

and

λrnq(t) = i

[r

nq(t)

αrn(t)

+ grqradicN

ωrqe


rnq(t)

] (B7)


rnq(t) = iλr

nq(t) + grqradicN

ωrqe


rnq(t) (B8)

Rrn(t) = Re

sumq

[r

nq(t) + grqradicN

ωrqe

minusiqn

]λrlowast

nq(t) (B9)

T r1n (t) = minus

sumr2

summ

J r1r2nm αr2


r1nq(t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t)

[λr2


]

(B11)

and

132D(t) =

sumr1

sumn

[ sumr prime

2mprime

sumr2m

Jr1r

prime2

nmprime Jr1r2nm α

r prime2lowast


r prime2r2

mprimem(t)

]

(B12)




= lim13trarr0


= lim13trarr0



αrn(t) = i

[T r

n (t) + αrn(t)Rr


nprime(t) (C2)


ρ(t) = Trph[|D(t)〉〈D(t)|] (C3)


ρmn(t) = 〈m|ρ(t)|n〉 (C4)





































Jnm=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝






⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠


Wij = C

[di middot dj


|rij |5] (8)




H primeph =

sumr

sumq

ωrq b

rdaggerq br

q (9)


N

sumr

sumn

sumq

grqω

rq a

rdaggern ar



rdaggerq (br



sumq(gr

q)2ωrq




|D(t)〉 =sum

n

αn(t)adaggernU





sumq


(12)



|(t)〉 equiv |αm(t)〉 (13)


ipart

partt|(t)〉 = H |(t)〉 (14)


ipart

parttminus H

)|(t)〉 = |αm(t) αm(t)〉 = 0 (15)


〈m|αm(t) αm(t)〉 = 0 (16)


L = 〈(t)| imacr2

larrrarrpart



d

dt

(partL

partαlowastm

)minus partL

partαlowastm

= 0 (18)



2|D(t)〉 (19)








τn =int infin

0ρnn(t)dt (20)


ρ(t) = Trph[|D(t)〉〈D(t)|] (21)




ρ(t) = minusLρ(t)






q =sum


n kdnτn

(23)




































sumn |αr




















〈t〉 = 3

kt

+ 1

+ 1

2 + kt

+ 2 + kt

4J 2 (24)





〈t〉 = 4

kt

+ 1

+ 1

+ kt2+ 3(2 + kt )

8J 2 (25)


〈t〉 = 4

kt

+

J 2+ 3kt

8J 2 (26)

































IV CONCLUSIONS








ACKNOWLEDGMENTS







13(t) equivradic

〈δ(t)|δ(t)〉 (A2)


Ansatz

〈δD1 (t)|δD1 (t)〉=

sumn


+sum

αn(t)=0

sumq

|αn(t)

[iλnq(t) + gqradic

Nωqe


]


sumαn(t)=0

sumq

|nq(t)|2 (A3)

with

Rn(t) equiv Resum

q

[iλnq(t)+ 2gqradic

Nωqe


]λlowast

nq(t)




〈δD2 (t)|δD2 (t)〉=

sumn


+sum

q

|iβq(t) + gqradicN

ωq

sumn


+ 1

N

sumq

g2qω

2q

[1 minus

∣∣∣sumn


] (A5)



sumn




+sumq =0


+sumq =0

[1 minus |ξq(t)|2]

∣∣∣∣iβq(t) + dq(t)

1 minus |ξq(t)|2∣∣∣∣2

+sumq =0

sumn


1 minus |ξq(t)|2]

+132D(t) minus

sumn

sumq

|nq(t)|2 (A6)




sumn



sumn

[|αn(t)|2

sumq

ωq |λnq(t)|2] (A8)

and


= minus 2radicN

sumn

[|αn(t)|2Re

sumq

gqωqλnq(t)eiqn]

(A9)








Eex(t) =sumr1r2

sumnm

J r1r2nm αr1lowast

n (t)αr2m (t)Sr1r2

nm (t) (B1)

Eph(t) =sum

r

sumn

∣∣αrn(t)

∣∣2 sumq

ωrq

∣∣λrnq(t)

∣∣2 (B2)


sumr

sumn

sumq

grqω

rq

∣∣αrn(t)

∣∣2Re

[eiqnλr

nq(t)]


nm (t) given by

Sr1r2nm (t) = exp

sumq

[λr1lowast


minus1

2

∣∣λr1nq(t)

∣∣2 minus 1

2

∣∣λr2mq(t)

∣∣2] (B4)



〈δD1 (t)|δD1 (t)〉=

sumr

sumn


n(t)Rrn(t) + T r

n (t)∣∣2

+sum

r

sumαr

n(t)=0

sumq

∣∣αrn(t)r

nq(t) + rnq(t)

∣∣2 + 132D(t)

minussum

r

sumαr

n(t)=0

sumq

∣∣rnq(t)

∣∣2 minussum

r

sumn

∣∣T rn (t)

∣∣2 (B5)



αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] (B6)

and

λrnq(t) = i

[r

nq(t)

αrn(t)

+ grqradicN

ωrqe


rnq(t)

] (B7)


rnq(t) = iλr

nq(t) + grqradicN

ωrqe


rnq(t) (B8)

Rrn(t) = Re

sumq

[r

nq(t) + grqradicN

ωrqe

minusiqn

]λrlowast

nq(t) (B9)

T r1n (t) = minus

sumr2

summ

J r1r2nm αr2


r1nq(t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t)

[λr2


]

(B11)

and

132D(t) =

sumr1

sumn

[ sumr prime

2mprime

sumr2m

Jr1r

prime2

nmprime Jr1r2nm α

r prime2lowast


r prime2r2

mprimem(t)

]

(B12)




= lim13trarr0


= lim13trarr0



αrn(t) = i

[T r

n (t) + αrn(t)Rr


nprime(t) (C2)


ρ(t) = Trph[|D(t)〉〈D(t)|] (C3)


ρmn(t) = 〈m|ρ(t)|n〉 (C4)






































τn =int infin

0ρnn(t)dt (20)


ρ(t) = Trph[|D(t)〉〈D(t)|] (21)




ρ(t) = minusLρ(t)






q =sum


n kdnτn

(23)




































sumn |αr




















〈t〉 = 3

kt

+ 1

+ 1

2 + kt

+ 2 + kt

4J 2 (24)





〈t〉 = 4

kt

+ 1

+ 1

+ kt2+ 3(2 + kt )

8J 2 (25)


〈t〉 = 4

kt

+

J 2+ 3kt

8J 2 (26)

































IV CONCLUSIONS








ACKNOWLEDGMENTS







13(t) equivradic

〈δ(t)|δ(t)〉 (A2)


Ansatz

〈δD1 (t)|δD1 (t)〉=

sumn


+sum

αn(t)=0

sumq

|αn(t)

[iλnq(t) + gqradic

Nωqe


]


sumαn(t)=0

sumq

|nq(t)|2 (A3)

with

Rn(t) equiv Resum

q

[iλnq(t)+ 2gqradic

Nωqe


]λlowast

nq(t)




〈δD2 (t)|δD2 (t)〉=

sumn


+sum

q

|iβq(t) + gqradicN

ωq

sumn


+ 1

N

sumq

g2qω

2q

[1 minus

∣∣∣sumn


] (A5)



sumn




+sumq =0


+sumq =0

[1 minus |ξq(t)|2]

∣∣∣∣iβq(t) + dq(t)

1 minus |ξq(t)|2∣∣∣∣2

+sumq =0

sumn


1 minus |ξq(t)|2]

+132D(t) minus

sumn

sumq

|nq(t)|2 (A6)




sumn



sumn

[|αn(t)|2

sumq

ωq |λnq(t)|2] (A8)

and


= minus 2radicN

sumn

[|αn(t)|2Re

sumq

gqωqλnq(t)eiqn]

(A9)








Eex(t) =sumr1r2

sumnm

J r1r2nm αr1lowast

n (t)αr2m (t)Sr1r2

nm (t) (B1)

Eph(t) =sum

r

sumn

∣∣αrn(t)

∣∣2 sumq

ωrq

∣∣λrnq(t)

∣∣2 (B2)


sumr

sumn

sumq

grqω

rq

∣∣αrn(t)

∣∣2Re

[eiqnλr

nq(t)]


nm (t) given by

Sr1r2nm (t) = exp

sumq

[λr1lowast


minus1

2

∣∣λr1nq(t)

∣∣2 minus 1

2

∣∣λr2mq(t)

∣∣2] (B4)



〈δD1 (t)|δD1 (t)〉=

sumr

sumn


n(t)Rrn(t) + T r

n (t)∣∣2

+sum

r

sumαr

n(t)=0

sumq

∣∣αrn(t)r

nq(t) + rnq(t)

∣∣2 + 132D(t)

minussum

r

sumαr

n(t)=0

sumq

∣∣rnq(t)

∣∣2 minussum

r

sumn

∣∣T rn (t)

∣∣2 (B5)



αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] (B6)

and

λrnq(t) = i

[r

nq(t)

αrn(t)

+ grqradicN

ωrqe


rnq(t)

] (B7)


rnq(t) = iλr

nq(t) + grqradicN

ωrqe


rnq(t) (B8)

Rrn(t) = Re

sumq

[r

nq(t) + grqradicN

ωrqe

minusiqn

]λrlowast

nq(t) (B9)

T r1n (t) = minus

sumr2

summ

J r1r2nm αr2


r1nq(t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t)

[λr2


]

(B11)

and

132D(t) =

sumr1

sumn

[ sumr prime

2mprime

sumr2m

Jr1r

prime2

nmprime Jr1r2nm α

r prime2lowast


r prime2r2

mprimem(t)

]

(B12)




= lim13trarr0


= lim13trarr0



αrn(t) = i

[T r

n (t) + αrn(t)Rr


nprime(t) (C2)


ρ(t) = Trph[|D(t)〉〈D(t)|] (C3)


ρmn(t) = 〈m|ρ(t)|n〉 (C4)



























































sumn |αr




















〈t〉 = 3

kt

+ 1

+ 1

2 + kt

+ 2 + kt

4J 2 (24)





〈t〉 = 4

kt

+ 1

+ 1

+ kt2+ 3(2 + kt )

8J 2 (25)


〈t〉 = 4

kt

+

J 2+ 3kt

8J 2 (26)

































IV CONCLUSIONS








ACKNOWLEDGMENTS







13(t) equivradic

〈δ(t)|δ(t)〉 (A2)


Ansatz

〈δD1 (t)|δD1 (t)〉=

sumn


+sum

αn(t)=0

sumq

|αn(t)

[iλnq(t) + gqradic

Nωqe


]


sumαn(t)=0

sumq

|nq(t)|2 (A3)

with

Rn(t) equiv Resum

q

[iλnq(t)+ 2gqradic

Nωqe


]λlowast

nq(t)




〈δD2 (t)|δD2 (t)〉=

sumn


+sum

q

|iβq(t) + gqradicN

ωq

sumn


+ 1

N

sumq

g2qω

2q

[1 minus

∣∣∣sumn


] (A5)



sumn




+sumq =0


+sumq =0

[1 minus |ξq(t)|2]

∣∣∣∣iβq(t) + dq(t)

1 minus |ξq(t)|2∣∣∣∣2

+sumq =0

sumn


1 minus |ξq(t)|2]

+132D(t) minus

sumn

sumq

|nq(t)|2 (A6)




sumn



sumn

[|αn(t)|2

sumq

ωq |λnq(t)|2] (A8)

and


= minus 2radicN

sumn

[|αn(t)|2Re

sumq

gqωqλnq(t)eiqn]

(A9)








Eex(t) =sumr1r2

sumnm

J r1r2nm αr1lowast

n (t)αr2m (t)Sr1r2

nm (t) (B1)

Eph(t) =sum

r

sumn

∣∣αrn(t)

∣∣2 sumq

ωrq

∣∣λrnq(t)

∣∣2 (B2)


sumr

sumn

sumq

grqω

rq

∣∣αrn(t)

∣∣2Re

[eiqnλr

nq(t)]


nm (t) given by

Sr1r2nm (t) = exp

sumq

[λr1lowast


minus1

2

∣∣λr1nq(t)

∣∣2 minus 1

2

∣∣λr2mq(t)

∣∣2] (B4)



〈δD1 (t)|δD1 (t)〉=

sumr

sumn


n(t)Rrn(t) + T r

n (t)∣∣2

+sum

r

sumαr

n(t)=0

sumq

∣∣αrn(t)r

nq(t) + rnq(t)

∣∣2 + 132D(t)

minussum

r

sumαr

n(t)=0

sumq

∣∣rnq(t)

∣∣2 minussum

r

sumn

∣∣T rn (t)

∣∣2 (B5)



αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] (B6)

and

λrnq(t) = i

[r

nq(t)

αrn(t)

+ grqradicN

ωrqe


rnq(t)

] (B7)


rnq(t) = iλr

nq(t) + grqradicN

ωrqe


rnq(t) (B8)

Rrn(t) = Re

sumq

[r

nq(t) + grqradicN

ωrqe

minusiqn

]λrlowast

nq(t) (B9)

T r1n (t) = minus

sumr2

summ

J r1r2nm αr2


r1nq(t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t)

[λr2


]

(B11)

and

132D(t) =

sumr1

sumn

[ sumr prime

2mprime

sumr2m

Jr1r

prime2

nmprime Jr1r2nm α

r prime2lowast


r prime2r2

mprimem(t)

]

(B12)




= lim13trarr0


= lim13trarr0



αrn(t) = i

[T r

n (t) + αrn(t)Rr


nprime(t) (C2)


ρ(t) = Trph[|D(t)〉〈D(t)|] (C3)


ρmn(t) = 〈m|ρ(t)|n〉 (C4)



















































〈t〉 = 3

kt

+ 1

+ 1

2 + kt

+ 2 + kt

4J 2 (24)





〈t〉 = 4

kt

+ 1

+ 1

+ kt2+ 3(2 + kt )

8J 2 (25)


〈t〉 = 4

kt

+

J 2+ 3kt

8J 2 (26)

































IV CONCLUSIONS








ACKNOWLEDGMENTS







13(t) equivradic

〈δ(t)|δ(t)〉 (A2)


Ansatz

〈δD1 (t)|δD1 (t)〉=

sumn


+sum

αn(t)=0

sumq

|αn(t)

[iλnq(t) + gqradic

Nωqe


]


sumαn(t)=0

sumq

|nq(t)|2 (A3)

with

Rn(t) equiv Resum

q

[iλnq(t)+ 2gqradic

Nωqe


]λlowast

nq(t)




〈δD2 (t)|δD2 (t)〉=

sumn


+sum

q

|iβq(t) + gqradicN

ωq

sumn


+ 1

N

sumq

g2qω

2q

[1 minus

∣∣∣sumn


] (A5)



sumn




+sumq =0


+sumq =0

[1 minus |ξq(t)|2]

∣∣∣∣iβq(t) + dq(t)

1 minus |ξq(t)|2∣∣∣∣2

+sumq =0

sumn


1 minus |ξq(t)|2]

+132D(t) minus

sumn

sumq

|nq(t)|2 (A6)




sumn



sumn

[|αn(t)|2

sumq

ωq |λnq(t)|2] (A8)

and


= minus 2radicN

sumn

[|αn(t)|2Re

sumq

gqωqλnq(t)eiqn]

(A9)








Eex(t) =sumr1r2

sumnm

J r1r2nm αr1lowast

n (t)αr2m (t)Sr1r2

nm (t) (B1)

Eph(t) =sum

r

sumn

∣∣αrn(t)

∣∣2 sumq

ωrq

∣∣λrnq(t)

∣∣2 (B2)


sumr

sumn

sumq

grqω

rq

∣∣αrn(t)

∣∣2Re

[eiqnλr

nq(t)]


nm (t) given by

Sr1r2nm (t) = exp

sumq

[λr1lowast


minus1

2

∣∣λr1nq(t)

∣∣2 minus 1

2

∣∣λr2mq(t)

∣∣2] (B4)



〈δD1 (t)|δD1 (t)〉=

sumr

sumn


n(t)Rrn(t) + T r

n (t)∣∣2

+sum

r

sumαr

n(t)=0

sumq

∣∣αrn(t)r

nq(t) + rnq(t)

∣∣2 + 132D(t)

minussum

r

sumαr

n(t)=0

sumq

∣∣rnq(t)

∣∣2 minussum

r

sumn

∣∣T rn (t)

∣∣2 (B5)



αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] (B6)

and

λrnq(t) = i

[r

nq(t)

αrn(t)

+ grqradicN

ωrqe


rnq(t)

] (B7)


rnq(t) = iλr

nq(t) + grqradicN

ωrqe


rnq(t) (B8)

Rrn(t) = Re

sumq

[r

nq(t) + grqradicN

ωrqe

minusiqn

]λrlowast

nq(t) (B9)

T r1n (t) = minus

sumr2

summ

J r1r2nm αr2


r1nq(t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t)

[λr2


]

(B11)

and

132D(t) =

sumr1

sumn

[ sumr prime

2mprime

sumr2m

Jr1r

prime2

nmprime Jr1r2nm α

r prime2lowast


r prime2r2

mprimem(t)

]

(B12)




= lim13trarr0


= lim13trarr0



αrn(t) = i

[T r

n (t) + αrn(t)Rr


nprime(t) (C2)


ρ(t) = Trph[|D(t)〉〈D(t)|] (C3)


ρmn(t) = 〈m|ρ(t)|n〉 (C4)





























































IV CONCLUSIONS








ACKNOWLEDGMENTS







13(t) equivradic

〈δ(t)|δ(t)〉 (A2)


Ansatz

〈δD1 (t)|δD1 (t)〉=

sumn


+sum

αn(t)=0

sumq

|αn(t)

[iλnq(t) + gqradic

Nωqe


]


sumαn(t)=0

sumq

|nq(t)|2 (A3)

with

Rn(t) equiv Resum

q

[iλnq(t)+ 2gqradic

Nωqe


]λlowast

nq(t)




〈δD2 (t)|δD2 (t)〉=

sumn


+sum

q

|iβq(t) + gqradicN

ωq

sumn


+ 1

N

sumq

g2qω

2q

[1 minus

∣∣∣sumn


] (A5)



sumn




+sumq =0


+sumq =0

[1 minus |ξq(t)|2]

∣∣∣∣iβq(t) + dq(t)

1 minus |ξq(t)|2∣∣∣∣2

+sumq =0

sumn


1 minus |ξq(t)|2]

+132D(t) minus

sumn

sumq

|nq(t)|2 (A6)




sumn



sumn

[|αn(t)|2

sumq

ωq |λnq(t)|2] (A8)

and


= minus 2radicN

sumn

[|αn(t)|2Re

sumq

gqωqλnq(t)eiqn]

(A9)








Eex(t) =sumr1r2

sumnm

J r1r2nm αr1lowast

n (t)αr2m (t)Sr1r2

nm (t) (B1)

Eph(t) =sum

r

sumn

∣∣αrn(t)

∣∣2 sumq

ωrq

∣∣λrnq(t)

∣∣2 (B2)


sumr

sumn

sumq

grqω

rq

∣∣αrn(t)

∣∣2Re

[eiqnλr

nq(t)]


nm (t) given by

Sr1r2nm (t) = exp

sumq

[λr1lowast


minus1

2

∣∣λr1nq(t)

∣∣2 minus 1

2

∣∣λr2mq(t)

∣∣2] (B4)



〈δD1 (t)|δD1 (t)〉=

sumr

sumn


n(t)Rrn(t) + T r

n (t)∣∣2

+sum

r

sumαr

n(t)=0

sumq

∣∣αrn(t)r

nq(t) + rnq(t)

∣∣2 + 132D(t)

minussum

r

sumαr

n(t)=0

sumq

∣∣rnq(t)

∣∣2 minussum

r

sumn

∣∣T rn (t)

∣∣2 (B5)



αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] (B6)

and

λrnq(t) = i

[r

nq(t)

αrn(t)

+ grqradicN

ωrqe


rnq(t)

] (B7)


rnq(t) = iλr

nq(t) + grqradicN

ωrqe


rnq(t) (B8)

Rrn(t) = Re

sumq

[r

nq(t) + grqradicN

ωrqe

minusiqn

]λrlowast

nq(t) (B9)

T r1n (t) = minus

sumr2

summ

J r1r2nm αr2


r1nq(t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t)

[λr2


]

(B11)

and

132D(t) =

sumr1

sumn

[ sumr prime

2mprime

sumr2m

Jr1r

prime2

nmprime Jr1r2nm α

r prime2lowast


r prime2r2

mprimem(t)

]

(B12)




= lim13trarr0


= lim13trarr0



αrn(t) = i

[T r

n (t) + αrn(t)Rr


nprime(t) (C2)


ρ(t) = Trph[|D(t)〉〈D(t)|] (C3)


ρmn(t) = 〈m|ρ(t)|n〉 (C4)





































ACKNOWLEDGMENTS







13(t) equivradic

〈δ(t)|δ(t)〉 (A2)


Ansatz

〈δD1 (t)|δD1 (t)〉=

sumn


+sum

αn(t)=0

sumq

|αn(t)

[iλnq(t) + gqradic

Nωqe


]


sumαn(t)=0

sumq

|nq(t)|2 (A3)

with

Rn(t) equiv Resum

q

[iλnq(t)+ 2gqradic

Nωqe


]λlowast

nq(t)




〈δD2 (t)|δD2 (t)〉=

sumn


+sum

q

|iβq(t) + gqradicN

ωq

sumn


+ 1

N

sumq

g2qω

2q

[1 minus

∣∣∣sumn


] (A5)



sumn




+sumq =0


+sumq =0

[1 minus |ξq(t)|2]

∣∣∣∣iβq(t) + dq(t)

1 minus |ξq(t)|2∣∣∣∣2

+sumq =0

sumn


1 minus |ξq(t)|2]

+132D(t) minus

sumn

sumq

|nq(t)|2 (A6)




sumn



sumn

[|αn(t)|2

sumq

ωq |λnq(t)|2] (A8)

and


= minus 2radicN

sumn

[|αn(t)|2Re

sumq

gqωqλnq(t)eiqn]

(A9)








Eex(t) =sumr1r2

sumnm

J r1r2nm αr1lowast

n (t)αr2m (t)Sr1r2

nm (t) (B1)

Eph(t) =sum

r

sumn

∣∣αrn(t)

∣∣2 sumq

ωrq

∣∣λrnq(t)

∣∣2 (B2)


sumr

sumn

sumq

grqω

rq

∣∣αrn(t)

∣∣2Re

[eiqnλr

nq(t)]


nm (t) given by

Sr1r2nm (t) = exp

sumq

[λr1lowast


minus1

2

∣∣λr1nq(t)

∣∣2 minus 1

2

∣∣λr2mq(t)

∣∣2] (B4)



〈δD1 (t)|δD1 (t)〉=

sumr

sumn


n(t)Rrn(t) + T r

n (t)∣∣2

+sum

r

sumαr

n(t)=0

sumq

∣∣αrn(t)r

nq(t) + rnq(t)

∣∣2 + 132D(t)

minussum

r

sumαr

n(t)=0

sumq

∣∣rnq(t)

∣∣2 minussum

r

sumn

∣∣T rn (t)

∣∣2 (B5)



αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] (B6)

and

λrnq(t) = i

[r

nq(t)

αrn(t)

+ grqradicN

ωrqe


rnq(t)

] (B7)


rnq(t) = iλr

nq(t) + grqradicN

ωrqe


rnq(t) (B8)

Rrn(t) = Re

sumq

[r

nq(t) + grqradicN

ωrqe

minusiqn

]λrlowast

nq(t) (B9)

T r1n (t) = minus

sumr2

summ

J r1r2nm αr2


r1nq(t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t)

[λr2


]

(B11)

and

132D(t) =

sumr1

sumn

[ sumr prime

2mprime

sumr2m

Jr1r

prime2

nmprime Jr1r2nm α

r prime2lowast


r prime2r2

mprimem(t)

]

(B12)




= lim13trarr0


= lim13trarr0



αrn(t) = i

[T r

n (t) + αrn(t)Rr


nprime(t) (C2)


ρ(t) = Trph[|D(t)〉〈D(t)|] (C3)


ρmn(t) = 〈m|ρ(t)|n〉 (C4)





































Eex(t) =sumr1r2

sumnm

J r1r2nm αr1lowast

n (t)αr2m (t)Sr1r2

nm (t) (B1)

Eph(t) =sum

r

sumn

∣∣αrn(t)

∣∣2 sumq

ωrq

∣∣λrnq(t)

∣∣2 (B2)


sumr

sumn

sumq

grqω

rq

∣∣αrn(t)

∣∣2Re

[eiqnλr

nq(t)]


nm (t) given by

Sr1r2nm (t) = exp

sumq

[λr1lowast


minus1

2

∣∣λr1nq(t)

∣∣2 minus 1

2

∣∣λr2mq(t)

∣∣2] (B4)



〈δD1 (t)|δD1 (t)〉=

sumr

sumn


n(t)Rrn(t) + T r

n (t)∣∣2

+sum

r

sumαr

n(t)=0

sumq

∣∣αrn(t)r

nq(t) + rnq(t)

∣∣2 + 132D(t)

minussum

r

sumαr

n(t)=0

sumq

∣∣rnq(t)

∣∣2 minussum

r

sumn

∣∣T rn (t)

∣∣2 (B5)



αrn(t) = i

[T r

n (t) + αrn(t)Rr

n(t)] (B6)

and

λrnq(t) = i

[r

nq(t)

αrn(t)

+ grqradicN

ωrqe


rnq(t)

] (B7)


rnq(t) = iλr

nq(t) + grqradicN

ωrqe


rnq(t) (B8)

Rrn(t) = Re

sumq

[r

nq(t) + grqradicN

ωrqe

minusiqn

]λrlowast

nq(t) (B9)

T r1n (t) = minus

sumr2

summ

J r1r2nm αr2


r1nq(t) = minus

sumr2

summ

J r1r2nm αr2

m (t)Sr1r2nm (t)

[λr2


]

(B11)

and

132D(t) =

sumr1

sumn

[ sumr prime

2mprime

sumr2m

Jr1r

prime2

nmprime Jr1r2nm α

r prime2lowast


r prime2r2

mprimem(t)

]

(B12)




= lim13trarr0


= lim13trarr0



αrn(t) = i

[T r

n (t) + αrn(t)Rr


nprime(t) (C2)


ρ(t) = Trph[|D(t)〉〈D(t)|] (C3)


ρmn(t) = 〈m|ρ(t)|n〉 (C4)



































excitonic energy transfer in light-harvesting complexes in

Documents