Energy migration in molecular aggregates induced by stochastic coupling

PHYSICAL REVIEW A VOLUME 47, NUMBER 6 JUNE 1993

Energy migration in molecular aggregates induced by stochastic coupling

A. S. Cordan, A. J. Boeglin, and A. A. VillaeysInstitut de Physique et Chirnie des Materiaux de Strasbourg, Groupe d'Optique 1Vonlineaire et d'Optoelectronique,

5 rue de l'Universite, 67084 Strasbourg CEDEX, France(Received 28 December 1992)

In the present paper, we report a theoretical description of the internal dynamics of one-dimensionalmolecular aggregates, subject to Auctuations of the excitation energy and site-site coupling. The roleplayed by these diagonal and nondiagonal stochastic interactions on the nonlinear optical response ofthe aggregate is studied. As an example, the inhuence of these processes on the third-order hyperpolari-zability is evaluated.

PACS number(s): 42.65.—k

I. INTRODUCTION

Over the last few years, considerable interest has beendirected toward understanding the optical properties ofmolecular aggregates involved in media as different asmolecular beams [1,2], solutions [3], or surfaces [4], andhas spurred intense activity. Since the observation of astrong narrowing and redshift in the absorption spectrumof certain dye molecules [5], often termed the J band[6,7], we have a signature of the existence of small aggre-gates. This observation has been attributed to motionalnarrowing of the inhomogeneous broadening of an exci-tonlike state [8]. In fact, among all the theoreticaldescriptions, the one-dimensional Frenkel exciton Inodelhas provided the simplest description to the internal dy-namics of molecular aggregates [9].

Some recent spectroscopic studies, performed withhole-burning and accumulated-photon-echo experiments[10],have shown evidence that in strongly coupled aggre-gates the electronic excitation is delocalized, while forweakly coupled aggregates there is no J band formation,but instead an excitation trapping and polaron formation.In optically thin samples, the excited-state dynamics of Jaggregates having intermolecular dipole-dipole couplingand a varying inhomogeneous broadening have firmly es-tablished that the enhanced fluorescence rate results frommicroscopic superradiance and that the cooperativity isdetermined by a coherence length which can be equal toor smaller than the aggregate size depending on theamount of dephasing [11]. As the homogeneous or inho-mogeneous broadening is increased, the coherence lengthdecreases inducing in turn the disappearance of the su-perradiant decay.

Further studies have shown an enhanced radiative de-cay rate that depends on aggregate size [10,12]. Thisenhanced radiative rate may result from the N'' scalingof the transition dipole moment, which in turn, generatesan N scaling of the radiative decay rate with aggregatesize N. In fact, it has been shown that pure dephasingcompetes with the radiative damping [13],and this com-petition depends on the relative value of the pure dephas-ing with respect to the enhanced radiative rate. For in-creasing values of the pure dephasing, the enhanced radi-

ative rate decreases up to the value of the radiative rateof the monomer which is recovered for very large puredephasings.

The investigation of third-order nonlinear optical hy-perpolarizability in molecular aggregates and its scalingwith aggregate size has been very active in recent years.Since the volume-squared scaling of the third-order hy-perpolarizability for semiconductor microcrystallites orquantum dots predicted by Hanamura [14], manytheoretical works concerning the electronic behavior ofmolecules in restricted geometries have been published.

The size dependence of the third-order hyperpolariza-bility in linear molecular aggregates incorporating theone- and two-exciton states has been considered byIshahara and Cho [15],as well as by Spano and Mukamel[16,17]. In addition, by taking advantage of a modelbased on a coupled set of anharmonic oscillators, Spanoand Mukamel were able to establish the influence on thenonlinear optical properties when the N-molecule systemranges from the small-aggregate limit with an N scalingto the bulk-crystal limit giving an N scaling [17].

Other works have emphasized the exciton motion inmolecular aggregates. Their main goal has been thestudy of the coherence loss inherent to the exciton migra-tion. To this end, microscopic models, including aquantum-mechanical description of the interaction be-tween Frenkel excitons and the phonons of the surround-ing heat bath, are required [18,19]. Because these modelsare very difticult to solve, stochastic descriptions havebeen introduced where the influence of the phonons is ac-counted for by correlated Gaussian stochastic processes.As long as the stochastic variables describing the fluctua-tions of the local excitation energy are 6 correlated, themodel can be handled. As an example, this approach hasenabled the study of the coupled coherent and incoherentmotions of excitons [20]. The inhuence of the fluctua-tions of both excitation energy and transfer matrix ele-ment on the line shape of optical absorption has also beenevaluated under the same assumption [21]. This studyhas enabled the evaluation of the contributions to thelinewidth resulting from the local and nonlocal fluctua-tions. An exactly solvable model for coherent and in-coherent motion of exciton in organic crystals has also

1050-2947/93/47(6)/5041(15)/$06. 00 47 5041 1993 The American Physical Society

A. S. CORDAN, A. J. BOEGLIN, AND A. A. VILLAEYS 47

been developed [22]; it gives explicit results for themean-square excitonic displacement [23], as well as forthe diffusion constant [24].

Besides the linear optical properties of stochasticmolecular systems, other works have been developed con-cerning the nonlinear optical responses, such as thethird-order hyperpolarizability. However, to ourknowledge, only the cases of diagonal stochastic interac-tions [25,26] or of correlated diagonal stochastic interac-tions between near-neighboring sites [27] have been dis-cussed. Finally, some recent improvements have beenachieved which go beyond the white-noise treatment ofthe stochastic processes. Among them, we can mentionthe study of the optical line shapes of one-dimensional ex-citons under the inhuence of heat bath with colored noise[28]. It has clearly been established that upon increasingthe rate of the excitation-energy fluctuations a transitionfrom static to dynamic disorder is observed. Also, thecorresponding optical absorption line shapes which areasymmetric for slow fluctuations become Lorentzian forfast fluctuations. With a diferent approach, Sato andShibata [29—31] have developed a theory based on thetime-convolution equation formalism. In their work,only the linear optical response of the material systemhas been considered. Assuming a medium subject to Auc-tuations generated by a Gaussian process, the cases of di-agonal and nondiagonal randomness have been studied.In a first step, the problem has been handled perturba-tively up to second order [29], while higher-order contri-butions have been accounted for by using the partial cu-mulant expansion to calculate absorption spectra anddensity of states [30,31]. It is worth mentioning thattheir approach is not restricted to the Markovian limit.

It is our intent to study the inhuence of both the Auc-tuations of the local excitation energies and the site-siteinteractions. Because of the difficulties inherent to thecumulant method when nondiagonal stochastic interac-tions are present, we will just be concerned in this workwith the 5-correlated case. In Sec. II we introduce thedescription of the aggregates. It is done in the local basisset as well as in terms of the exciton basis set. Next, wedescribe the general dynamical evolution in Sec. III anddiscuss the inhuence of the stochasticity on the one- andtwo-exciton states. In Sec. IV we derive a forrnal expres-sion of y' ', the third-order susceptibility, which is explic-

itly evaluated in Sec. V. Finally, in Sec. VI we presentthe results concerning the role of the diagonal and nondi-agonal Auctuations on the third-order hyperpolarizabili-ties, and we summarize our findings in the last section.

II. DESCRIPTION OF THE AGGREGATES

N

g, & le„ & ,

njWn, j=1

(2.1)

ll, m)=I&j&m, j=1

Igj ) le( ) Ie ) ,

where Ig. ) and Ie ) are the ground and excited states ofmolecule j. Therefore we have one ground state, X singlyexcited states, and N(N 1)/2 dou—bly excited statessince, for our model, two excitations cannot reside on thesame site. Of course, higher excited states must be intro-duced in the study of hyperpolarizability of order higherthan three. Also, we will assume periodic boundary con-ditions, say, Ip ) = Ip+N ), and Ip,p+n ) = Ip+n, p )= Ip+n, p+N ). If the intersite coupling RV is restrictedto nearest-neighboring molecules, the correspondingHamiltonian takes the simple form

Ho= g RQbpbp+ g AV[bpbp+, +b +,b ], (2.2)p=1 p=1

or in terms of its spectral decomposition

In order to give a theoretical description of the roleplayed by intersite stochastic coupling on the third-orderhyperpolarizability, a basis for the representation of theaggregates must be chosen. The model consists of a set ofnoninteracting molecular aggregates, each of them hav-ing N identical molecules with two electronicconfigurations, an electronic energy gap AQ, and paralleltransition dipole moments. As long as the temperature issufficiently low, phonon interactions will be neglected.

For the problem at hand, two types of representationare of interest. The first one consists of a set of localizedstates which are given by

N (N —1)/2

H, = y &IHIP&&pl+ & &V[IP&&p+ll+Ip+1&&pl]+ g»&flip, p+n &&p,p+nlp=1 p=1 p=1 n =1

N N —1

+ y y &V[lp,p+n &&P+l,p+nl+lp+l, p+n &&P,P+nl] .p=1 n=2

(2.3)

Now, because of the coupling, it is convenient to intro-duce a delocalized basis set of states which diagonalizesHo. In such a way, we may introduce the one- and two-exciton states, denoted

I Ak ) and IBz ), respectively,and defined by

H, l A„)=fico„I A„),(2.4)

H, Ia, & =en~, Ia~, &,where excitonic states and their corresponding energiesare given by

47 ENERGY MIGRATION IN MOLECULAR AGGREGATES INDUCED. . . 5043

N

) y ei2nkP/NIp )~N ~

ficok =AQ+2A'Vcos- 2mk

iYk=O, . . . , X—1,

{N—1)/2(2p+„)/x . (2e —1)~n

e sin

x lp, p+n), (2.5)

X —1K=O, . . . , X—1, q=1, . . . , 2

2qrE (2q —1)qrAQz =2 AQ+2A V cos . cos '

K, q X

Here, for the sake of simplicity, N is taken to be odd.Notice that only linear molecular aggregates of arbitrarysize and free ends admit fermionic eigenfunctions. There-fore the one- and two-exciton states

I Ak ) and l&x q ),which are eigenfunctions of Ho, cannot be eigenfunctionsof a fermion number operator [32]. However, in theinfinite limit, the fermion character is recovered. This isbecause in this limit, the eigenenergies of the two-excitonstates are the sum of two different one-exciton energiesand their corresponding two-exciton eigenfunctions areSlater determinants built up from the same one-excitoneigenfunctions [33].

From these representations, the transition dipole ma-trix elements can be evaluated, depending on the problemat hand. The spectral decomposition of the aggregate-field interaction takes the form

{N—1)/2

HAF(t)= —d E(r, t) po( Ao) &gl+Ig) & Aol)+ y p, .oq(l&, q && A, I+ I A, &&&oql)q=1

N —1 (N —1)/2+ y y p,x ~q(l&~q&&A, I, +IA,~&&&&ql)

k=1 q=l(2.6)

In the previous expression, we have introduced the nota-tions p,„ for the matrix element of the monomer dipolemoment, and

pa= N' p,„ (2.7)

because only the totally symmetric stateI Ao) carries os-

cillator strength from the ground state, and

I

first established by Lehmberg [34], and later extensivelystudied by Gross and Haroche [35]. Here, p(t) is the den-

sity matrix of the aggregate, and the prime on the sum-mation means mWn The .applied electric fields E(r„,t)and the Pauli creation and annihilation operators b„andb„are defined at site r„. These operators obey the an-ticommutation rules

(N —I)/2 (2q 1 )qrnpk. x =4p,„N ' g sm .

n=1

2~AnXcos ' 5k px. , (2.8)

[bt, b„]+=5 „+2bt b„(1—5 „),[b„,b ]+=(1—5 „)2btb

(3.2)

Notice that if the excitations are limited to a single site,

which gives the selection rules for optical excitation fromthe other states of the one-exciton band. Finally, dstands for the unit vector characterizing the orientationof the dipole moments p,„. The redistribution of thetransition dipole moments in the delocalized basis set isschematically depicted in Fig. 1. Bo,q {Bk,q1

III. DYNAMICAL EVOLUTION

The dynamical evolution of a one-dimensional cyclicaggregate, consisting of X coupled two-level systems in-teracting with the applied electromagnetic fields, is con-veniently described, in the rotating-wave approximation,by the superradiant master equation

iQ[p(t), b„b„]+ g' iQ „[p(t),bt b„]n=1 m, n =1

+ g y „[b p(t)b„m, n=1

,' [b t b„p(t)+p—(r—)bt b„]]

B0,3

Bo,z

Bo,y

k gO

{Bk,3){Bk,2)

{Bk,t}

{Ak)

N

E(r„,t ) [p( t), b„~+b„],n=1

(3.1) FIG. 1. Redistribution of the transition dipole moments inthe one- and two-exciton basis set.


they behave like fermions, while for different sites theyare bosons. Finally, p,„ is the monomer transition di-pole moment, 0 the electronic energy gap, and the con-stants 0 „and y „account for the excitation transferand damping including spontaneous emission and super-radiance. These quantities, respectively, reduce to V and

y in the long-wavelength limit and near-neighbor interac-tions. Notice that y is the monomer spontaneous emis-sion rate. Their explicit expressions have been estab-lished for the particular case of parallel dipole momentsand can be found in the literature [11,35]. In addition, ifwe assume that the aggregate is small compared to an op-tical wavelength, its corresponding wave vector k satisfiesthe inequality k r „«1, and consequently E(r„,t) willbe approximated by E(r, t).

If we omit the interaction of the external fields with theaggregates, the evolution of the density matrix as givenby Eq. (3.1) can be accounted for by the Liouvillian Lsdefined by

N N

L~p(r)= —g AQ[p(t), btb„]—g AV[p(r), btb„]n=1 mWn

+i g fiy I b p(t)b„m, n =1

,'[—bt—b„p(t)+p(r)btb„]I .

(3.3)

Besides the free evolution of the aggregates, we must in-troduce the description of the surrounding medium.Despite the fact that a stochastic description does not al-low us to calculate the temperature dependence, it will beconvenient for our purpose to construct a dampingoperator which accounts for the relaxation and dephasingprocesses induced by the surrounding medium. To thisend, we introduce the stochastic part of the LiouvillianL(t) = [H(t), ], where the Hamiltonian is defined by

N N N (N —1)/2H(t)= & E~(t) p &&pl+ g a„(t)[lp&&p+II+Ip+1&&pl]+ g & [e~(t)+8~+„(t)]pip +n &&p p +nl

p=1 p=1 p=1 n =1N N —1

+ g g 8~(t)[ p,p+n &&p+ 1,p+n I+ Ip+1,p+n &&p p+n I] . (3.4)p=l n=2

Here, the fluctuations are defined in the local basis. Thequantity E~.(t) represents the fluctuations of the local exci-tation energy for the mono-excited and doubly excitedstates, while 8 (t) describes the fluctuations of the inter-molecular coupling between nearest-neighboring sites jand j+1, as depicted in Fig. 2. When the Auctuationtime scale is very short with respect to all the othercharacteristic times of the aggregates, the correlationfunctions of the stochastic variables associated to theGaussian Markov processes are 6 correlated and satisfythe relations

mal zero-order solution with respect to L ~F ( t ) corre-sponds to

pl '(t)=Texp ——J dt, LI(t, ) pl(t, ), . (3.8)

(i /fi)L~ tp, (r) =e ' p(r),

(i IA)L~t — —(i /A)L~t(3.9)

where T represents the chronological ordering operatorand

for the diagonal fluctuations, and

(3.5)

(3.6)

For Gaussian Markov processes with 5-correlated vari-ables, the cumulant expansion reduces the expression ofthe averaged density matrix in the Schrodinger represen-tation to the simple form

for the nondiagonal ones, where yo and y, stand for theAuctuation amplitudes. Because we are dealing with 6-correlated variables, the cumulant expansion methodworks and a stochastic damping operator can be defined[36]. Therefore the Liouville equation which governs theevolution is given by

V

[Amon

V V V V

P,anon

1)

p-1

Bp(t) i [Ls+L(t)+L~F(t—)]p(t), (3.7)

if Li „(t)= [H~„(t), ] stands for the interaction with theexciting fields. In the interaction representation, the for-

FIG. 2. Diagrammatic representation of the stochastic in-teractions acting on the aggregate. The broken line correspondsto the diagonal and the wavy lines to the nondiagonal stochasticcouplings. Also shown are the coupling constant V and the di-

pole transition moments p,„.

ENERGY MIGRATION IN MOLECULAR AGGREGATES INDUCED. . .

(p' '(t)) e=e (Texp ——I dttLt(tt) )e *'p(t;)1

—(i /A)Ls=e ' exp f dt( f dt2(LI(t, )LI(t2)) e 'p(t;) . (3.10)

From time derivation of (p' '(t) ), the Liouville equation of the averaged density matrix

()(p"'(&) & (0)a~

[L li)ir(r)](p (r) )

is straightforwardly established with1,(

— —(elf')Lq(t t )—'—, )lA)L~(t —t')

)fi

(3.1 1)

(3.12)

Consequently, in the Markovian limit, the stochastic damping operator I" corresponding to the stochastic processes isgiven by

1 ~, , — (i lh)Lz(t ——t ) —', (i ll)L&(t —t')&

fi2

Because we assume the processes uncorrelated,

r=r, +r,

(3.13)

(3.14)

is the sum of two independent contributions, j. 0 for the diagonal part and I 1 for the nondiagonal part. However, inboth cases their dynamical influence depends on the mixing induced by the averaged Liouvillian—(i lh' Lz(t —t') —,(i lt))Lz(t —t')(L(t)e L(t')e ) on the excitonic states. As shown in the Appendix, we have no mixing in theI Ak, g )) Liouvillian subspace, because the relations

rol ~k g && =To ~k g &&

are satisfied. On the contrary, the diagonality is lost in theI Ak, Ak. )) subspace, and the mixing results in the form

y0 N —1

rol&k ~k'&&=2rol&k, &k &&—2

(3.15)

2~, 2mr)leak, ak »=4r)leak ak » —4 y s —,— «s (k —k') +«s (k+k2) leak Ak )) .

kl, k2=0

(3.16)

Similarly, the mixing induced by r in the Ig, Bz q )) sub-

space is described by the relations

N —1 (N —1)/2 N —1

y O(K, q, k;K, q, k )K'=0 q'=1 k'=0

rolg B~,, )& =2rolg, B~, &), ,

rilg»x. q )& =4y) 1 ——sin (2q —1 )qr

(3.17)

Ig, Bx, )&

K'=0 q'=1 k'=0

x IBx

xlB~, „wk, )&

N —1 (N —1)/2 N —1

r, lB „w„»= g g y r(K, q, k;K', q, k )

y1 (N —1)/2—8q'Wq, q'= 1

(2q —1)qrsin

(2q' —1)qrX sin ~

x lg, B

Finally, the action of 1 on the IBz q, Ak )) states has beendeveloped in the Appendix and is given by

where the functions O(K, q, k;K', q', k')Y(K,q, k; K', q', k') are given by relations (A3) and (AS).

From the previous expressions, it appears that after anoptical excitation satisfying strict selection rules, the sto-chastic damping operator induces a redistribution of en-ergy. Due to the nature of the coupling, we have no mix-ing between the subspaces corresponding to the Liouvilli-an states Ig, g)), IAk, g)), Ig, Bxq», and IB&q, Ak)).All these relations will be of interest, in the following, toevaluate the nonlinear optical properties of the molecularaggregate.


IV. INFLUENCE OF THE STOCHASTICITYON THE NONLINEAR OPTICAL RESPONSE

As the dynamical evolution of the molecular aggregateinteracting with its surrounding is known, the nonlinearresponse of the system to an optical excitation can bedetermined. We are interested here by the third-orderhyperpolarizability which characterizes any four-wave-mixing process.

For a steady-state experiment, the electric fields haveconstant amplitudes E, and take the form

E(r, t ) = g [e E~e. ' . ' +c.c.], (4.1)j=a, b, c

P' )(r, t)=Tr[(p")(r, t))p], (4.2)

where the symbol Tr stands for the trace and the aver-aged density matrix of the molecular aggregate initiallyat equilibrium is given by

where c.c. denotes the complex-conjugate part, if co, k,and e are the frequency, wave vector, and unit polariza-tion vector of the jth field, respectively. As usual, thethird-order polarization is deduced from the third-orderterm of the time-dependent perturbation expansion of thedensity matrix. It can be expressed as

t3(p"'(r, t)) =

3 dt3 dt2 dt)Gp(t t3)L~F(t3)Gp(t3 t2)L~F(tp)Gp(t2 t) )L~F(t) )p(t ) (4.3)

because, in the Markovian limit, the factorization assumption is valid. Also, it is assumed that the aggregate is initiallyat equilibrium. Notice here that the I.iouvillian Gp(t —t ) is defined by

(4.4)

if 6(t t ) stand—s for the Heaviside function. If the initial conditions are defined at t; = —~, with a change of variableswe get

(p' '(r, t))=—,g g g (d e )(d e )(d e„)p =a, b, cq=a, b, c, r=a, b, c

X J dt3 J dt2 J dt, G()(t3)L„G()(t2)L„G()(t,)L„p(t, )0 0 0

[k -r —co (t —t3)] i[k .r —co (t —t2 —t3)]

i[k„r—co„(t—t&

—t2 —t3 ) ] (4.5)

where we have introduced the notation

L~F(t) = —d.E(r, t )L„, (4.6)

with L„=[@,]. The formal development of the third-order polarization can be written in terms of its Fourier com-ponents as

P"'(r, t)= g [P'"(co )e +c.c.], (4.7)

where all the combinations

CO~—

COp COq CO q=+ + +

(4.8)

are present and the indices p, q, and r run over a, b, and c. If we consider the particular combination k =k, +kb+ k,and co =p), +cub+ co„ from the identification of Eqs. (4.2) and (4.7), we obtain

P' )(co )= — d J dt3 f dt2 J dt (()~)G(((tp)L3„G (tp~)L„G (tp)L)„~g,g ))o o o

perm

(4.9)

if the molecular aggregate is in its ground state at the initial time, so that p(t;) = ~g, g )). Also, g „stands for the 3.permutations over the fields since any ordering, involving the three different fields, participates only once.

At this stage, we can introduce the third-order hyperpolarizability from the formal expression

P~ '(m~) =y' '(m~;m, ~wham~ )e~E~ebEbe~E

where

(4.10)

47 ENERGY MIGRATION IN MOLECULAR AGGREGATES INDUCED. . .

and

y' '(co~;co„cob, co, )= g R '(co, +cob+co„co, +cob, co, )ddddperm

(4.11)

R' '(co, +cob+co„co, +cob, co, ) = —((pl Go(co, +cob+co, )L„GO(co, +co„)L„GO(co,)L„Ig, g » . (4.12)

Notice that of the four d's, three of them are to be ap-plied separately on the unit vector e. of the threedifferent exciting fields, and the remaining one defines thepolarization component. Also, we have introduced thedefinition of the frequency-dependent Green function

and the functions g(K, q) and g'(K, q, q ') are given by

g(K )y (2q 1 )+2K

2N 2N2

Go(co)= ——f dt e' 'Go(t)= (4.13)(2q —1)—2K+cot 7T

2N

Ls Lo+L, +L2, (4.14)

To get an explicit evaluation of the third-order hyper-polarizability, we have to evaluate R ' '(co, + cob

+co„co,+cob, co, ). This calculation can only be done ifwe know the various couplings induced, in the Liouvilli-an space, by I and L&. Concerning I, the couplingshave been calculated previously and are given by Eqs.(3.15)—(3.18).

We still require the couplings induced by Lz in theLiouvillian space. To this end, we introduce a partitionof Ls given by

~,(K, )2y ~' . (2q —1)mmN „, N

(4.18)

(2q' —1 )urnX sin

2qrK(m n)—X cos

N

Finally, the last couplings of interest in the present modelare induced by the Liouvillian L2. They are described bythe following relations:

mWnn=1

where the various terms correspond toN N

L,p(t)= —g R&[p(t), btb„]—g AV[p(t), bt b„],

L, la„w, »=tayNlg, g »,

L2 I~ »oo &q& =2i&y &ot lg, Ao &&

(2q —1)qr(4.19)

L,p(t)= i g ,'Ry—[b b„p(t—)+p(t)b b„],m, n =1N

L~p(t) =i g fiyb p(t)b„.m, n =1

(4.15) L2IBO q, Ao » =2ifiy cotI Ao, g » .

2N

From the knowledge of these various couplings, the func-tion R ' '(co, + cob +co„co,+cob, co, ) can be evaluated.

L)l~k, g &&= — y(k)l~, g &&,

L, I Ak, A„.» = —' [y(k)+y(k')]I A„, A„.»,

2(4.16)

L)IB,„g»= thy(K, q)IB~„g—&&

(N —1)l2i A g g'(K,—q, q') Btc q, g »,

q'= 1,q'Wq

L ( I Blc, A k » = i fi[g( K, q ) + ,' y ( k )—] I B~,A I, &&—

(,N —1)/2i' g g'(K—, q, q')IBtc ., Ak »,q'= 1,q'Wq

where

y(k) =Ny5k o, (4.17)

The first term Lo is purely diagonal, as it can be seenfrom the eigenvalue problem of Ho described by Eq. (2.4).However, this is not the case of the two other terms. Themixing induced by L, can be described by the followingrelations:

V. EVALUATIONOF THE THIRD-ORDER HYPERPOLARIZABILITY

For the sake of convenience, we introduce in the expli-cit evaluation of the third-order hyperpolarizability somesimplifying assumptions. While Lo and L2 will be treat-ed exactly, we will neglect in the following some nondiag-onal couplings introduced by L1 and I . With respect toL„the range of validity of this assumption has been dis-cussed previously [25] and is given by the conditionN'y «V~'. The second assumption is on the stochasticdamping operator I. Here, we must distinguish twotypes of assumptions. The nondiagonal terms can beneglected in the subspace IBz,g », as long as the condi-K, q&

tion Ny, «Vm is satisfied. However, rejecting the non-diagonal terms in the subspace IBz, Ak » is more res-trictive because, here, the condition 8N (yo+5y, ) ((Visrequired. Notice that all the previous conditions havebeen obtained by assuming that the various nondiagonalmatrix elements are small compared to the correspondingenergy levels.


The last difficulty, which needs to be overcome toevaluate the third-order hyperpolarizability, is the partic-ular role played by that part of the stochastic operatorcorresponding to the nondiagonal stochastic interactionI 1. Its introduction in the complete dynamics of the ag-gregate is quite tedious. Because the ratio of y1 over yp isgenerally small [22], a first-order perturbational treat-ment of I, on the subspace A)„A), » can be introduced.The iterative development of Go(co) given by

G,(~)= y G,")(~)[—iver, G,")(~)]" (5.1)

G(o)(~)— 1

A'~ —L,+iar, '

the matrix element

(5.2)

is obtained from Eq. (4.13). Therefore, in this subspace,we get successively for the zero-order term

« ~g, WHIG('& )(co)lag, Ag »= —„.'. +&k, p (1—5),o)

fi co+iNy+2l pp co+2l pp

2iyo (co+2iyo)(co+2iyo+iNy )~k, k'+

(co+i y ~ )(ca+i y )

&),o (1—&),, o)

co+iXy+2iyp cu+2i yp(5.3)

where we have introduced the notation

y+= ,'[&y—+2yo]+,'[(&-y+2yo)' gy—yo]'" .

For the first order-term, the expression is similarly deduced from the relation

« A)„A), IGo"(co)I do, Ao»= ih g—« A)„A )I6(') '(co)l A), , Ai, »

k, , k,

x « a, , a, lr I a, , ~, &&&& &,, &,IG,")(~)l~o, w, &&,

(5.4)

(5.5)

where we have used the property that I 1 and Gp ' couplepopulations to populations and coherences to coherencesonly. The expressions of the matrix elements of interestin the other subspaces are easily obtained by using the as-sumptions previously discussed. For the states I A)„g »we have

r, , =«B&,„glrlB&,„g»

4y) . p(2q —1)qr=2 (yo+2y, ) — sin ', (5.8)

still valid for I 0 and 1,. Finally, the last type of termscorresponding to the states IBx q, A), » are given by theexpression

1

A'[co —co), ]+if&[ ,'y(k)+yo+—2y, ]

In the IBx,g » subspace, we obtain similarly

«B,„g IGo( )IB,„g»

(5.6)q & ~g I Go(co) Bx q Aq &&

1

Pi[co Qx q+co), ]+i%[—I x q. ), +g(K, q)+ —,'y(k)]

(5.9)

if we introduce the notationwhere the restriction of I to its diagonal part is describedby the complementary notation

gy) . z (2q —1)qr=3(y +2y ) — sin0 1

6gp (X 1 )/2

x' n=1

(2q —1)norsin

71—16 cos .x~

I

2qr(K —k) 2qr(K+k)'+cos '

(,N —1)/2

n=1

(2q —1)qrn . (2q —1)m(n+ 1) . . (2q —1)qr(n —1)si - sin ~ +sm ' (5.10)

The last required terms correspond to the nondiagonal matrix elements included by L2. Notice that the evaluation ofR ' '(co, +cob+co„co, +coi„co,) implies the diff'erence of the following matrix elements:

ENERGY MIGRATION IN MOLECULAR AGGREGATES INDUCED. . . 5049

«g, glGp(co)lg, g »= —,1

(co+iy+ )(co+iy )—2yy, (N —1)+2i y p(co+2iy p)

((g,glG."'( )l~„~,&&= ——6) (co+iNy+2iyp)(co+iy~)(co+iy )

(5.11)

In fact, the divergence due to the term 1/co is just an artifact since it can be seen from the previous relation that thisdependence cancels. %'e still require the matrix element

«& ~G ( )~& & &&2iy cot[(2q —1)n/2N]

(5.12)&[co cop—+iyp+2iy, +(i /2)Ny][co —IIp q+cop+iI p q. p+(i/2)Ny+ig(O, q)]

The same expression holds for ((g, Ap ~Gp(co) ~ Ap, 8p && provided that cop and Qp are replaced by —cop and —Qp

Of course, the complete expression of the third-order hyperpolarizability is quite cumbersome. Nevertheless, becauseit constitutes the starting point of the numerical simulations presented in the next section, we give its explicit form, thatis to say

R' '(co, +cob+co„co, +cob, co, ) R=)+Rq+R3+R4,

where the various terms take the form

Po 1 1

coo+cop+(i/2)Ny+ iy p+2i y, coo cop+(i/2)Ny+iy p+2'y

1 1X .cog +cob +cog cop+ (1 /2)Ny+iyp+21y $ cog +cob +cog +cop+(i /2)Ny+iyp+2ly ]

r

X . (( Ap, Ap~Gp' '(co, +cob)+Gp" (co, +cob)i Ap, Ap &&

(5.13)

(co, +cob+iy+)(co, +cob+)y ) —2yyp(N —I )+2iyp(co+2iyp)+-Pi (co, +cob+2iyp+iNy )(co, +cob+iy+ )(co, +cob+i y )

p (X—1)/2

Po;o, qq=l

X 1

[co +cob + co cop + (i /2 )N y +i y p +2i y, ][co + cob Qp &+ i g( 0, q ) + & I p g ]

1

cop+(i /2 )Ny+iyp+2iy

1

+cob +ceo+ co+p(i/2)Ny+iyp+2iyi][ co+acob+Qpq+4(0 q)+il p & g]

xaco+cop+(i/2)Ny+iyp+2iy,

1

co, +cob+co, —Qp +co +(i/2)Ny+iI . +g(O, q)

(5.14)

X fi(( Ap, Ap~G' '(co, +co„)+G"'(co,+cob)~ Ap, Ap&&

1 1

co, +cop+(i/2)Ny+iyp+2iy, co, —cop+(i/2)Ny+iyp+2iy,

1

[co + cob Qp &+ i g( 0 q ) + i I p &

.g ][co —cop + ( i /2 )N y +i y p +2i y & ]

+ 1

co, +cob+co, +Op —cop+(i/2)Ny+il p q.p+g(O, q)

X t« a„ap~Gp'"(~. +~b)+Gp"'(~. +~b)l ~p, ~p &&

5050 A. S. CORDAN, A. J. BOEGLIN, AND A. A. VILLAEYS 47

1 1

u, +cop+ (i /2)Ny+iy p+2iy, co, —cop+(i /2)Ny+iy p+2i y,

R3=—

+ 1

[M +cog +Op&+ if(0 q )+i? p &.g ][co +cop+(i /2)Ny +iyp+2iy, ]

p X —1 (X —1)/20 1

co, + cob + co, +n g—

cop + i g( K, q ) + i r /X»K~q(5.15)

co, +cob+co, —n~ +co~~+lg(K, q)+I r~

X (( A ~~ ) 4 ~~ ~G p ( co + co b ) + G p ( co g + co ~ )

~r4 p ~ A p ))

p0 (N —1)/2R 4 ~3 X Ppp q

q=1

co, +cop+ (i /2 )Ny + i y p+ 2i y ) co, —cop+ (i /2 )N y +i y p+ 2i y )

(5.16)

(2q —1)mX —2iy cot2N

1

[co, + cob +co, cop+ (i /2)N—y + i y p+ 2i y ) ]

[co, +cob+co, —Qp +cop+(i/2)Ny+ig(O, q)+iI p ~.p]

X A(( Ap, dp~G(') ~(co, +co„)+Gp"(co, +coq)~ Ap, 2p))

1 1

co +cop + (i /2 )Ny +i y p+ 2i y &co, cop + (

—i /2 )N y +i y p +2i y &

+ 1

[co, —cop+(i/2)Ny+iyp+2iy, ][co, +co& —Qp +P(0,q)+il p

(2q —1)m.+2iy cot

2iV

1

[cog + cob +co, +cop+ (i /2 )N y +i y p+ 2i y ) ]

[co, +cob+co, +Op —cop+(i/2)Ny+i g(O, q )+ i 1 p q. p]

X A(( 2p, Ap~G(') '(co, +cob)+Gp" (co, +cob)~ Ap, 3p))

1 1

co, +cop+(i /2)Ny+i yp+2i y ) co, —cop+(i /2)Ny+i yp+2iy )

+ 1

[co, + +co(pi/2)Ny+iyp+2iy, ][co,+cob+Op +i((O, q)+iI p . ](5.17)

At this point, we have the explicit expression of thethird-order hyperpolarizability which is required for thenumerical calculations.

VI. NUMERICAL CALCULATIONS AND DISCUSSION

In this section, we present and discuss the role of thediagonal and nondiagonal fluctuations on the third-orderhyperpolarizabilities. We want to emphasize the role of

the nondiagonal fluctuations y& and compare their con-tribution to the one of the diagonal Auctuations ya.

We treat two cases: the resonant case through a degen-erate forward four-wave-mixing experiment, and the non-resonant case through a phase-conjugate four-wave-mixing experiment. By "resonant" or "nonresonant" wemean that the frequencies of the electric fields are close toor far off the monoexcitonic resonance co0= 0+2 V.

The nonlinear susceptibility has been studied as a func-


tion of the parameters N, yo, and y&. However, their rel-ative magnitudes are subject to restrictions because of theapproximations y &

& yo and more importantly8N (go+ 5y, ) ~ V, made in the evaluation of y' '.2

In the following, we are not interested in the superradi-ant limit, largely discussed by Spano and Mukamel [25],and yo is taken to be much larger than Ny, with y beingthe monomer spontaneous emission rate. Of course, thisalso means that for fixed values of y, yo, and y &, the ap-proximations made put a limit on the aggregate size. No-tice that as is the case of J aggregates, the nearest-neighbor coupling V is considered to be negative. Forour numerical simulations, we have chosen V= —600—1 —3 —1cm, y = 10 cm, and the monomer electronic ener-

gy gap 0 equal to 16000 cm

A. Resonant case

In a degenerate forward four-wave-mixing experiment,three nearly collinear laser beams with the same frequen-cy co, and wave vectors k„k2, and k3, respectively, propa-gate in a nonlinear medium. A signal with frequency co isgenerated in the direction k&

—k2+k3. However, we arenot concerned with the propagation of the fields throughthe medium, and focus instead on the response of an ag-gregate. g' ' can be written as

g' '(co;co, co, —co) =R '"(co,O, co)+R "'(co,O, —co)(3)

+R (co, 2co, co), (6.1)

where we have expressed the permutations over the fieldsof Sec. IV as a sum of three terms. The frequency co isscanned over a frequency interval centered atcoo=Q+2V, to be close to the monoexcitonic resonance.In this case, g can be accurately simplified by(3)

(co;co, co, —co)=R, (co, O, co)+R, (co, O, —co), (6.2)(3)

keeping only the two terms being triply resonant forco =coo as seen in Eq. (5.14).

In Fig. 3, we first represent ly' '(co;co, co, —co)l for

N=5, go=1 cm ', and y] varying from 0 to 0.3 cmThe curves are normalized to a peak height of unity tocompare the different linewidths. As expected, thesewidths are increasing with y, . The full width at halfmaximum (FWHM) can be easily derived from Eqs. (6.2)and (5.14), and varies linearly with (yo+2yi). Thereforethe amplitude of the nondiagonal fluctuations, althoughsmaller than the amplitude of the diagonal ones, contrib-utes twice as much to the overall linewidth.

Next, in Fig. 4, we compare the linewidths of themonomer and the aggregate. Notice that the monoexci-tonic resonance of the chain is shifted by 2 V with respectto the monomer, reflecting the intermolecular couplingV. Therefore the line shapes are plotted against the de-tuning Aco=co —0 for the monomer, and Am=~ —coo forthe aggregate. Here, we have set y() equal to 0.5 cmand y &

to 0.05 cm '. First of all, notice that thelinewidths are independent of the aggregate size. Numer-ical calculations for %=5,7, 9 show that the line shapesare indistinguishable. This result is characteristic of thenonsuperradiant behavior considered here, with therelevant parameter (yp+2) i) much larger than Ny. Thesame observation remains true for y&=0, where we re-cover the result discussed by Spano and Mukamel [25] faroff the superradiant limit.

The second feature is the difference between the mono-mer and the aggregate's linewidth. This difference stemsfrom the parameter y &, increasing the width of the aggre-gate compared to the monomer's, the latter lacking forobvious reasons any nondiagonal fluctuations. Indeed,when y& is set equal to zero, the aggregate's linewidth ex-actly coincides with the monomer's.

In order to further characterize the effect of y& and 1V

on the nonlinear response of the aggregate, we integrateover the frequency co,

~(N, yp, yi)= f y("(co;co,co, —co)l'dco,

so as to recover a quantity proportional to the total inten-

1.001.00

0.750.75

0.500.50

0.250.25

0.00-10 -5 0 5 10

Frequency Detuning I - coo ( cm ")

FIG. 3. Third-order hyperpolarizability ly' '(co;co, co, —co)l

vsdetuning (co—co0) with N=5, y =1 cm ', and y&=0, 0.1, 0.2,and 0.3 cm '. The curves are normalized to peak height of uni-ty. Curves with increasing widths correspond to increasingvalues of y&.

0.00-3 -2 -1 0 1 2 3

Frequency Detuning 6 ( cm )

FICx. 4. Third-order hyperpolarizability y'"(co;co, co, —co)l

vsdetuning hen for the monomer and the aggregate, with y0=0. 5cm ' and y&=0.05 cm '. The wider curve corresponds to theaggregate and the narrower one to the monomer. Only fory &

=0 does the aggregate line shape coincide with themonomer's.

505'

3.0

A. S. CORDAN , A. J. BOEQLIN, AND A. A. VILLAEYS

18

47

2.415

&A

-c:l 1.8

V)

12

CO

1.2

Z'.0.6

N 9C)

6

&C

0.1

(10 cm )

I

900 3136 8100 17424

[N(N+1)]233124

FIG. 5. A(N, y0, ) vs, y0, y, ) vs y&, close to resondf 1

corres oaggregate sizes N=5 7 9.

1 ggregate.

FIG. 6. g(N, yo y&)» [&(N+1)]', clos~ose to resonance, h

nal o d to 1o a arger value of y, .

ly symmetric state~A0). Again asgain, as was shown in Fi . 5

a ecreasing functio fno y&.

srty of the record de signal.In Fig. 5, A(N , yo, y, ) is plotted ve

values of X with y bein hei, c ose to resonance, A(N, yo yl)

of h 1

n is case, it turns olng

igna times (y +20 yl

The dependence on N can b 'g. th

1 es o y&, with eor

, yo, y, ) is proportional to

ther y is finit o h thment c

r w et er it vanishes.

strrength between the fe rough the

'g increased oscillato

e undamental statei aor

a e g and the total-

B. onresonant case

We now consider hexpenment with two u

er a p ase-con'uj gate four-wave-mixin

the

p oe wo pump beams and minear medium. Th

quency is m . The&, with k =—I= —k;, and their fr-

e wave vector andre-

probe are k and co

n the frequency of thco, respectivel . A

e

1 'th fr 2

d 'th thgain notice t

propagation of the field the pro a at& s rough the

This time, we write

(6.4)

as

(2CO; CO. CO.(3) J,CO;, CO, COl ) —R (2CO. CO. ,—.. . )+R"'(22CO; COJ, CO; COi, CO )+R; &, , CO.2CO CO; &, J (2CO; —CO&, 2CO;, CO )7 1

~

The freequency ~, is held fixnarrow frequency interval

e xed and co is sscanned over a

far os resonance w hva centered at co

e ave chosen co

co;. In order t bo e

turns out that th 1'e inewidth of ~

co; equal to co +10V. It

h""not because the FWHM isweter

y on y, in the limit yo))N andis proportional to

, yo, y ) is 1 or several

is obtained by integratin

A(N lX"'(2—oo

CO;—

CO, ;CO;, —COJ, COl )I dCO J

(6.5)

In contrast to the resone resonant case, A (Not 1 f

The dependence on N c1'

can be seen in Fi

any ~s depicted ver oversus fo

cm . The straight lines scaled as

3.0

2.5

2.0

C)

1.5

2 1.0

0.5 I

2

I

4 6

FIG. 7. A (N,(10 cm )

~ g g o P

47 ENERGY MIGRATION IN MOLECULAR AGGREGATES INDUCED. . . 50S3

8.0

7.2

6.4

Q 5.6

04.8

g 40

3.2

2.4

1.6

0.825 49 121 169

FIG. 8. A{N, y0, yl) vs N, far off resonance, with y0=0. 2

cm ' and for y, =0, 0.01, and 0.02 cm '. A higher signal cor-responds to a larger value of y&.

VII. CONCLUSION

In this work, we have studied the internal dynamics ofsmall molecular aggregates (k r „«1) subject to sto-chastic perturbations, and how they affect the nonlinearoptical responses of the aggregates. This work, initiallydeveloped in the limit of diagonal stochastic perturba-tions, has been extended to include diagonal as well asnondiagonal coupling s. Only the far-off-superradiantcase has been considered here, with pure dephasing ratesmuch larger than spontaneous emission rates.

For degenerate forward four-wave mixing, the total in-tensity of the signal displays an important enhancement

I

show that there is no enhancement with chain size, aswas the case in the resonant process. As expected, fory, =0, one recovers N times the monomer result. Again,we observe that A(N, yo, y&) is an increasing function ofV1

with the aggregate size close to resonance. On the otherhand, fluctuations in nondiagonal couplings efficientlyreduce the integrated intensity.

The difference between the linewidths of an aggregateand a monomer stems from the nondiagonal parametery1, pertaining to the chain only. The two widths coincidein absence of nondiagonal fluctuations.

For far-off-resonance phase-conjugate four-wave mix-ing, the total signal does not show any enhancement at allwith size, but this time the intensity is an increasing func-tion of y1 ~ When y1 vanishes, one recovers N times themonomer result.

In this paper, we have shown that despite the fact thatthe nondiagonal amplitude y, is smaller than the diago-nal one yo, nondiagonal Auctuating couplings have a siz-able effect on the nonlinear responses and optical lineshapes.

So far, we have analyzed the contributions of indepen-dent diagonal and independent nondiagonal fluctuationsto the nonlinear optical susceptibilities. In a forthcomingpaper, we plan to compare the predictions of our modelto those resulting from a model based on diagonal corre-lated stochastic fluctuations only.

Notice that only the Markovian limit has been con-sidered here. It will be of interest, in the future, to de-scribe the internal dynamics in the non-Markovian limit,for both diagonal and nondiagonal Auctuations, and toevaluate the resulting nonlinear responses of the aggre-gates in this case.

APPENDIX

As an example, we give here the evaluation of the mix-

ing induced by the stochastic damping operator I . Forthe sake of simplicity, only the coupling generated by(L(t, )L(tz ) ) on the Liouvillian states lBz q, Ak ))=

l Bz ) ( A k lis explicitly calculated. From the

definition of the Liouvillian, we have

&L(t, )L(t, ) &IB.,„~k»=l[&H(t, )H(t, ) &, IB&q && ~& ]+» —21&H(t&)IB&q && ~klH(t2)»& .

The evaluation of the first term [(H(t, )H(t2) ), Bx ) ( Ak l]+ can be obtained from the expression

(A 1)

N (N —1)/2&H(t, )H(t, )&=Pi'5(t, —t, ) ~ y, g lp)(pl+2y, g g lp, p+n)(p, p+ l

—n2y, g lp, p+1)(p,p+llp =1 p=l n=1 p =1

(A2)

where the notation y, =yo+2y, has been introduced. Using relation (2.5), we get the result

y1 N N

[(H(t, )H(t2)), lBx )( Aql]+=A' 5(t, t2) ~ 3y, lB~ )( Akl —4— g g sin ~ .e'p =1p'=1

~ e i2nkP'/Nl p+ 1 )—

(p~l

Next, we must calculate the second term (H(t& )lBx ) ( Ak lH(t2) ) of relation (Al). It can be obtained from the evalu-ation of the separate factors


2 N (N (—)/2 . (2q 1)qrnH(t, ) BK )=—g g [e (t))+e +„(t,)]sin.p=1 n=1

+ 2 ~( ~) ~ ( ). (Zq —1)qrn

/znK(zp+n )/Nl + )P~P

and

+ [eiznK(zp+2+n)/Nlp p+n + 1 ) +(1 $ )ei2nK(2P+2 —n)/Nlp p+++ 1 n )

+e/2'(zp —n)/Nlp+ 1 p++ n ) +(1 Q )e/2~K(zp+n)/Nlp + 1 p +n ) ]

NH(t )lA ) = g [E (t )e' P/ lp)+8 (t )[e' "'P+" lp)+e' "P lp+1)]]

p =1

(A4)

(A5)

From these expressions, their product and statistical average must be performed. Therefore the mixing due to the diag-onal fluctuations is given by the expressions

N —1 (N —1)/2 N —1

I ()lBK, Ak )) = g g g 8(K,q, k;K', q', k')lBK. , Ak. )),K'=0 q'= 1 k'=0

where the function 8(K,q, k;K', q', k') is defined by

8(K,q, k'K', q', k')

{A6)

VO(N —1)/2

=31o&KK &qq &kk 162 &2(K —Ic')k —/' X (2q —1)qrn . (2q' —1)qrn

sin ' 'sin(K K')27rn-

'cos '

(A7)

Similarly, the mixing induced by the nondiagonal Auctuations corresponds to

N —1 (N —1)/2 N —1

r, lBK,„&&»= g g g Y'(K, q, k;K', q', k')lBK, „&k », (AS)K'=0 q'=1 k'=0

where the function f(K, q, k;K', q', k') is defined by

(2q —1)qr . (2q' —1)7rsin ~K K'~k k'

V1&(K,q, k;K', q', k')=6@)I'iKK &q, q &/, , /,S—

{2 —1)~nz ~2(K —K'), k —k'

N(2q' —1)qr(n + 1)

N

X cos [n(K K')+(K' k')]- —2&N

+cos [n {K K') +(K'+ k )]-2m

N

(2q —1 )qrn+sinN

sin(2q' —1) (nqr—1)

277X cos [n(K K') (K' k')]- — —N

(A9)+cos [n (K K') (K'+ k')]- —27r

N

All the other types of mixing can be deduced similarly. Because their evaluations follow the same lines, they are notdeveloped in this appendix.


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Energy migration in molecular aggregates induced by stochastic coupling