ARTICLE IN PRESS

Journal of Luminescence 108 (2004) 137–141

*Correspond

371-531-3143.

E-mail addr

(M. Schreiber).1Present a

Universit.at M .u

0022-2313/$ - se

doi:10.1016/j.jlu

Non-Markovian effects in the anisotropy offluorescence in LH2 units

Michael Schreibera,*, Pavel He$rmanb, Ivan Barv!ıkc, Ivan Kondova,1,Ulrich Kleinekath .ofera,d

a Institut f .ur Physik, Technische Universit .at, 09107 Chemnitz, GermanybDepartment of Physics, University of Hradec Kr !alov!e, V. Nejedl!eho 573, CZ-50003 Hradec Kr !alov!e, Czech RepubliccFaculty of Mathematics and Physics, Institute of Physics of Charles University, CZ-12116 Prague, Czech Republic

d International University Bremen, P.O. Box 750 561, 28725 Bremen, Germany

Abstract

Using the reduced density matrix formalism the time dependence of the anisotropy of fluorescence in the B850 ring of

the purple bacterium Rhodopseudomonas acidophila is calculated. Fast fluctuations of the environment are simulated by

dynamic disorder and slow fluctuations by static disorder. Both type of disorders are taken into account

simultaneously. Earlier calculations within the Markovian–Redfield theory are extended by including memory effects

within the exciton dynamics.

r 2004 Elsevier B.V. All rights reserved.

PACS: 82.39.�k; 82.53.Ps; 87.15.Aq

Keywords: Exciton transfer; Density matrix theory; Fluorescence

1. Introduction

Highly efficient light collection and excitationtransfer towards the reaction center takes place inthe so-called light-harvesting systems (LHs). In thefocus of the present contribution is the ring-shapedantenna complex LH2 from the bacterium Rho-

dopseudomonas acidophila in which there is littledoubt about the existence of non-Markovianeffects in the exciton transfer dynamics. In a ring

ing author. Tel.: +49-371-531-3142; fax: +49-

ess: schreiber@physik.tu-chemnitz.de

ddress: Theoretische Chemie, Technische

nchen, Garching 85747, Germany.

e front matter r 2004 Elsevier B.V. All rights reserve

min.2004.01.022

without static and dynamic disorder, in which allthe Qy transition dipole moments of the B850bacteriochlorophylls (BChls) lie approximately inthe plane of the ring, the entire dipole strength ofthe B850 band comes from a degenerate pair oforthogonally polarized transitions at an energyslightly higher than the transition energy of thelowest exciton state. Theoretical treatments con-sider extended Frenkel exciton states but invokemoderate static disorder in the excitation energiesof the individual BChls. This model is applied inthe present contribution, too.Time-dependent experiments made it possible to

study the long-time as well as the femtoseconddynamics of the energy transfer and relaxation[1,2]. For the B850 ring in LH2 complexes, it has

d.

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M. Schreiber et al. / Journal of Luminescence 108 (2004) 137–141138

been shown that the elementary dynamics occurson a time scale of about 100 fs [3–5]. For example,depolarization of fluorescence as a result of energytransfer has been studied already quite some timeago for a model of electronically coupled mole-cules [6,7]. Rahman et al. [6] were the first torecognize the importance of the off-diagonaldensity matrix elements (coherences) for describ-ing the fluorescence anisotropy of molecularsystems [8]. Off-diagonal density matrix elementsare missing in the well-known F .orster theory ofexciton transfer [9] and can lead to an initialanisotropy larger than the incoherent theoreticallimit of 0.4. Already some time ago substantialrelaxation on the time scale of 10–100 fs and ananomalously large initial anisotropy of 0.7 havebeen observed by Nagarajan et al. [3]. Theseresults have been interpreted in a simple model ofa homogeneous system in which excitations aredelocalized over the whole ring. The high initialanisotropy was ascribed to a coherent excitation ofa degenerate pair of states with allows opticaltransitions and then relaxation to states at lowerenergies which have forbidden transitions. Nagar-ajan et al. [4] concluded that the main features ofthe spectral relaxation and the decay of anisotropyare reproduced well by a model that considersdecay processes of electronic coherences within themanifold of the exciton states and thermalequilibration among the excitonic states. In thatcontribution, the exciton dynamics has not beencalculated explicitly.In several steps we have recently extended [10–

12] the former investigations of the time-depen-dent optical anisotropy by Kumble and Hoch-strasser [13] and Nagarajan et al. [4]. For aGaussian distribution of local energies we addedthe effect of dynamical disorder by using aquantum master equation. Since those calculationsinvolve the propagation of density matricestogether with orientational averaging, calculatingthe mean over the static disorder leads tonumerically quite demanding computations.Therefore we restricted ourselves to the Marko-vian limit, i.e. the neglect of memory effects, in theearlier studies. For this purpose we used theRedfield theory [9,14]. In the present investigation,we are now generalizing this treatment by the

inclusion of memory effects since these might be ofsome importance for the studied ultrafast pro-cesses.

2. Model

In the following we assume that only oneexcitation is present on the ring. Since the BChlsare electronically coupled the excitations, the so-called excitons, are delocalized on the ring. Let usfirst describe the Hamiltonian of an unperturbedring coupled to a bath of harmonic oscillators

H0 ¼Xm;n

Jmnawman þ

Xq

oqbwqbq

þ1ffiffiffiffiffiN

p Xm

Xq

Gmq oqaw

mamðbwq þ b�qÞ

¼H0ex þ Hph þ Hex�ph: ð1Þ

The first term in the Hamiltonian, H0ex; represents

the single exciton, i.e. the system. The operators awm

or am create or annihilate an exciton at site m: Jmn

for man are the so-called transfer integralsbetween sites m and n: The diagonal elements Jnn

are the local energies at site n: The second term inthe Hamiltonian, Hph; describes the bath ofphonons in the harmonic approximation. Thephonon creation and annihilation operators aredenoted by bw

q and bq; respectively. The last term inEq. (1), Hex2ph; represents the interaction betweenthe exciton and the bath. The exciton–phononinteraction is assumed to be site-diagonal andlinear in the bath coordinates. The term Gm

q

denotes the exciton–phonon coupling constant.The fluctuations of the local excitation energiescaused by the protein environment are assumed tobe slow. They are modeled by static disorder whichadds to the Hamiltonian of the unperturbed ring

H ¼ H0 þ Hs ¼ H0 þX

n

enaþn an: ð2Þ

A Gaussian distribution for the uncorrelated localenergy fluctuations en with a standard deviation Dis assumed.The dipole strength ~mma of eigenstate jaS of the

ring without static disorder and the dipole strength~mma of eigenstate jaS of the ring with static disorder

ARTICLE IN PRESS

M. Schreiber et al. / Journal of Luminescence 108 (2004) 137–141 139

read

~mma ¼XN

n¼1

can~mmn; ~mma ¼XN

n¼1

can~mmn; ð3Þ

where can and can are the expansion coefficients of

the eigenstates of the unperturbed and thedisordered rings in site representation. In the caseof impulsive excitation the dipole strength issimply redistributed among the exciton levels dueto disorder [13]. Thus the impulsive excitation witha pulse of sufficiently wide spectral bandwidth willalways prepare the same initial state, irrespectiveof the actual eigenstates of the real ring. Afterimpulsive excitation with polarization ~eex theexcitonic density matrix r [11] is given by [4]

rabðt ¼ 0;~eexÞ ¼1

Að~eex �~mmaÞð~mmb �~eexÞ;

A ¼Xa

ð~eex �~mmaÞð~mma �~eexÞ: ð4Þ

The usual time-dependent anisotropy of fluores-cence

rðtÞ ¼/SxxðtÞS�/SxyðtÞS/SxxðtÞSþ 2/SxyðtÞS

;

SxyðtÞ ¼Z

Pxyðo; tÞ do ð5Þ

is determined from

Pxyðo; tÞ ¼AX

a

Xa0

raa0 ðtÞð~mma0 �~eexÞð~eey �~mmaÞ

½dðo� oa00Þ þ dðo� oa0Þ�: ð6Þ

The brackets /S denote the ensemble average andthe orientational average over the sample withfixed relative directions ~eex and ~eey of the laserpulses.The crucial quantity entering the time depen-

dence of the anisotropy in Eq. (5) is the excitondensity matrix. As mentioned in the introductionwe are going to focus on memory effects in thedynamics of the relevant reduced density matrix.Let us proceed with a calculation in energyrepresentation. The details of this treatment basedon the time-convolutionless formalism are givenelsewhere [15]. The Hamiltonian Hex2ph can bewritten as

Hex2ph ¼X

m

KmFm; ð7Þ

where the system operators Km and bath operatorsFm are given by

Km ¼ awmam; Fm ¼

1ffiffiffiffiffiN

p Xq

Gmq oqðbw

q þ bqÞ:

As usual, we define the bath correlation functionCmnðtÞ ¼ /FmðtÞFnð0ÞSB assuming that the expec-tation values /FmSB vanish. For most systems,the correlations of the fluctuations decay after acertain correlation time tc [9]. In what follows, weuse a simple model for Cmn assuming that each site(i.e. each chromophore) has its own bath which iscompletely uncoupled from the baths of the othersites. Furthermore, it is assumed that these bathshave identical properties [1,9]. Then only onecorrelation function CðoÞ is needed

CmnðoÞ ¼ dmnCðoÞ

¼ dmn2p½1þ nBðoÞ�½JðoÞ � Jð�oÞ� ð8Þ

with the spectral density JðoÞ [9] and the Bose–Einstein distribution nBðoÞ: Defining

LmðtÞ ¼Z t

0

dt0Cðt0Þe�iHst0Kme

iHst0

ð9Þ

the non-Hermitian effective Hamiltonian

Heff ¼ Hs � iX

m

KmLmðtÞ ð10Þ

can be introduced. With these definitions the non-Markovian quantum master equation is given by

qrðtÞqt

¼ � iðHeffrðtÞ � rðtÞHweff Þ

þX

m

ðKmrðtÞLwmðtÞ þ LmðtÞrKmÞ: ð11Þ

The Markovian limit of these expressions caneasily be obtained by moving the upper integrationboundary in Eq. (9) to infinity [9]. For theevaluation of the last equation we use a numericaldecomposition of the spectral density [16]

JðoÞ ¼Xn

k¼1

pk

o

½ðoþ OkÞ2 þ G2

k�½ðo� OkÞ2 þ G2

k�

ð12Þ

with arbitrary real parameters pk; Ok; and Gk: Thespectral density decomposed here is chosen to be

JðoÞ ¼ YðoÞj0o2

2o3c

e�o=oc : ð13Þ

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0.2 0.4 0.6 0.8∆ /J

0

5

10

15

20

τFig. 1. Time of decay (to a value of 0.4) of the anisotropy of

fluorescence as a function of the static disorder. The crosses

show the results without dissipation, in the other cases the

spectral density is j0 ¼ 0:2: The Markovian results are denoted

by diamonds ðT=J ¼ 0:01Þ and circles ðT=J ¼ 1Þ whereas thenon-Markovian results are marked by stars ðT=J ¼ 0:01Þ andsquares ðT=J ¼ 1Þ: The symbols are connected by lines to guidethe eye.

M. Schreiber et al. / Journal of Luminescence 108 (2004) 137–141140

The advantage of this decomposition comes withthe calculation of the matrix elements of theoperator LðtÞ in energy representation

/mjLmðtÞjnS ¼/mjKmjnSZ t

0

dt0Cðt0Þe�iomnt0

¼/mjKmjnSYþðt;omnÞ: ð14Þ

Here we introduced the function

Yþðt;omnÞ ¼Xn

k¼1

pk

4OkGk

nBðOþ

k ÞiðOþ

k � omnÞ½eiðO

þk�omnÞt � 1�

�

þnBðO�

k Þ þ 1

ið�O�k � omnÞ

eið�O�k �omnÞt � 1

� ��

�2i

b

Xn0

k¼1

JðinkÞnk þ iomn

½eð�nk�iomnÞt � 1�:

ð15Þ

For the non-Markovian treatment Yþðt;omnÞ hasto be calculated at every moment in time instead ofonly once as in the Markovian limit. Fortunately,in the present approach the evaluation of thematrix elements (14) no longer contains anynumerical integration.

3. Results

The anisotropy of fluorescence has been calcu-lated using Eq. (5). The Markovian as well as thenon-Markovian dynamical equations for theexciton density matrix r have been applied toexpress the time dependence of the optical proper-ties of the model LH2 ring of BChls in thefemtosecond time range. All other details are thesame as in Ref. [11]. Fig. 1 shows the dependenceof the decay time of the anisotropy of fluorescenceas a function of the static disorder for twodifferent temperatures T : The initial decay isalways faster if dissipation is included. For weakstatic disorder, D ¼ 0:2J one can see some distinctnon-Markovian features. In the case of hightemperatures the initial decay in the first femtose-conds is much faster within the Markov approx-imation. This can easily be understood by looking

at Eq. (15) for the non-Markovian case. At t ¼ 0the function Yþðt ¼ 0;om;nÞ vanishes and thengradually increases with time. To obtain theMarkovian limit one has to set t ¼ N; whichmeans full dissipation already at t ¼ 0 in thedynamical calculations. For longer times, the non-Markovian features disappear in the anisotropy offluorescence. Looking at the curves for lowtemperatures one sees that the memory effectsare smaller around t ¼ 0 but then develop up tosome certain stage and stay for longer times thanin the high-temperature case.Increasing the static disorder D the difference

between the Markovian and the non-Markoviancalculations is similar at high temperatures. TheMarkovian decay is always somewhat faster orequal to the non-Markovian one. This is differentfor the low-temperature case. Here the non-Markovian decay is faster for D ¼ 0:2J and 0:4J:For D ¼ 0:6J the Markovian and non-Markovianresults are almost indistinguishable. For D ¼ 0:8J

the Markovian decay is faster. So there is no cleartrend of the memory effects on the anisotropy offluorescence.

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M. Schreiber et al. / Journal of Luminescence 108 (2004) 137–141 141

One of the findings of the present contribution isthat memory effects play only a minor role for thecomparison of calculated and measured initialdecay. All the calculations discussed above wereobtained with the non-Markovian theory based onthe decomposition of the spectral density and atime-convolutionless approach.

Acknowledgements

This work has been partially funded by theM $SMT $CR, the project GA $CR 202-03-0817, andby the BMBF and DFG.

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