Polarons, localization, and excitonic coherence in superradiance of biological antenna complexes T. Meier, Y. Zhao, V. Chernyak, and S. Mukamel Department of Chemistry and Rochester Theory Center for Optical Science and Engineering, University of Rochester, Rochester, New York 14627 ͑Received 24 February 1997; accepted 3 June 1997͒ A real-space formulation of time-resolved fluorescence of molecular aggregates is developed using the one-exciton density matrix (t) of the optically driven system. A direct relationship is established between the superradiance enhancement factor Ls and the exciton coherence size L associated with the off-diagonal density matrix elements in the molecular representation. Various factors which affect the latter, including finite temperature, energetic disorder, coupling with phonons, and polaron formation are explored. The theory is applied for the interpretation of recent measurements in the B850 system of the LH2 photosynthetic complexes. © 1997 American Institute of Physics. ͓S0021-9606͑97͒50634-9͔
I. INTRODUCTION showed coherence sizes of 2–3 at room temperature, whereas at low temperatures Lsϭ3 for LH2 antenna com- 22 Optical properties of photosynthetic antenna complexes plexes and Lsϭ9 for LH1. Significant decrease of fluores- have become an object of intensive experimental and theo- cence lifetimes in J-aggregates of pseudoisocyanine ͑PIC͒ 1,2 retical studies over the last years. Time and frequency re- dyes compared to single chromophores has been observed29 solved measurements including fluorescence depolariz- with the fluorescence lifetime varying between 70 ps at 1.5 K 3,4 5–8 9–12 ation, hole burning, pump-probe, and photon to 450 ps at 200 K. PIC J-aggregates are one-dimensional 13,14 echoes provide direct information regarding the dynam- clusters of dye molecules with collective superradiant emis- ics of excitons, energy transfer, and electronic and nuclear sion and optical nonlinearities. For example, in PIC-Br relaxation in these systems. In this paper we focus on time- J-band, a Strickler-Berg analysis gives the monomer radia- resolved fluorescence and cooperative spontaneous emission tive lifetime 3700 ps30 and 70 ps for the aggregate. This ͑superradiance͒ which is one of the most interesting elemen- implies a superradiance coherence size Lsϭ50 which is tary signatures of intermolecular coherence. Molecular ag- much smaller than the number of molecules in the aggregate: gregates are often characterized by ultrafast radiative decays. Ϸ5ϫ104 obtained from a study of exciton-exciton This effect has a simple classical interpretation: When a col- annihilation.21 Coherence sizes of 2–3 were found in mix- lection of dipoles oscillate in phase, their amplitudes add up tures of isocyanine dyes adsorbed on silver halide substrates coherently to form a large effective dipole. The oscillator- at room temperature,28 whereas L ϭ70 has been reported for strength and consequently the radiative decay rate is then s PIC aggregates in a low temperature glass.27 proportional to the number of dipoles. The superradiance co- The finite coherence size L in large aggregates is related herence size L of a molecular aggregate is defined as the s s to the breakdown of translational symmetry of excitons due ratio of its radiative decay rate to that of a single molecule. to static disorder18,20,24,25 or to exciton-phonon interactions.23 ͑For simplicity we assume in this definition that all mol- ecules are identical and have the same radiative decay rate. In the former case, the coherence size Ls has been attributed to exciton localization length18,20,24,25 which exists in one- This definition can be generalized to aggregates made of dif- 31 ferent molecules.͒ The concept of a coherence size was in- dimensional systems even for infinitely weak disorder. Ef- voked by Mobius and Kuhn15 in analyzing the dependence of fects of weak disorder on emission of two-dimensional ag- 32 fluorescence quenching on the acceptor surface density for a gregates have been studied using perturbative methods. system consisting of an acceptor monolayer on top of a Phonon-induced exciton dephasing has been shown to de- 23 J-aggregate monolayer. Molecules separated by more than stroy excitonic coherence and control the superradiance. the optical wavelength may not emit coherently. (/a)d is In this paper we show how the time-resolved fluores- 16,17 cence from aggregates may be calculated and analyzed by therefore the fundamental upper bound for Ls , a being the lattice constant, and d the dimensionality ͑see Ref. 18 for exploring exciton dynamics in real-space using the single- dϭ1 and Ref. 19 for dϭ2). In practice, however, the coher- exciton density matrix. The superradiance size Ls is shown ence size is usually determined by other dephasing mecha- to be primarily determined by the exciton coherence size L . nisms such as exciton-phonon interactions and static A precise definition of the latter in term of the ‘‘anti- disorder18,20–26 and is typically much smaller than both the diagonal’’ section of the density matrix is given. We study aggregate physical size and the optical wavelength.20,27 Simi- the effects of static disorder and strong exciton-phonon- lar types of aggregates can have very different coherence coupling on L . The density matrix approach further allows sizes in different environments.28 us to express the time-resolved fluorescence signal as an Recent work on photosynthetic antenna complexes overlap of a doorway and a window excitonic wave packets.
3876 J. Chem. Phys. 107 (10), 8 September 1997 0021-9606/97/107(10)/3876/18/$10.00 © 1997 American Institute of Physics
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The doorway represents the reduced density matrix of the a unit vector. Otherwise its magnitude will vary with n. exciton driven by the external pump whereas the window Un(q) are 3ϫ3 orthogonal rotation matrices describing mo- accounts for the system’s geometry. lecular orientations. In Sec. II we develop the doorway-window ͑DW͒ pic- The time- and frequency-resolved fluorescence can be ij ture and define the density matrix coherence size L . The related to the dipole-dipole tensor S (,): influence of temperature and static disorder on L in cyclic aggregates is then explored in Sec. III. An additional mecha- ␥ ϱ t t Sij͑,͒ϵ dteit Pi ϩ Pj Ϫ , ͑4͒ nism for L resulting from strong exciton-phonon coupling 2͵Ϫϱ ͳ ͩ 2ͪ ͩ 2ͪʹ and the formation of polarons is discussed in Sec. IV. A variational calculation of the polaron wave function using where ␥ϭ42¯3/3បc3 is the radiative decay rate of a single Toyozawa’s Ansatz allows us to compute the reduced exci- molecule with optical frequency ¯, and the time evolution of ton density matrix by tracing the joint ͑exciton plus phonon͒ Bn() is given by the molecular Hamiltonian with the exter- density matrix over the phonon variables. When nuclear mo- nal driving field included (Hˆ ϩHˆ ). Here i, jϭx,y,z denote tions are slow, we may adopt the simplified adiabatic polaron int the tensor components of S in a molecular frame. To elimi- model commonly used in the description of exciton self- nate propagation and polariton effects which are not essential trapping in low-dimensional systems.33,34 The DW approach for the present discussion, we assume that the aggregate’s is utilized in Sec. V to calculate the dependence of the su- size is much smaller than the optical wavelength.35 perradiance enhancement factor L on static disorder, diago- s The model introduced by Eq. ͑1͒ conserves the number nal and off-diagonal exciton-phonon-coupling using the of excitons N ϵ͚ ͗B†B ͘. If exciton annihilation processes models of Secs. III and IV. Finally in Sec. VI we discuss and n n n ͑radiative or nonradiative͒ are taken into account, N be- summarize our results. comes time-dependent; however, the timescale of N is typi- cally long compared with exciton equilibration. In either case it is convenient to represent the time-resolved signals as N II. REAL-SPACE PICTURE OF TIME-RESOLVED times the signal per exciton. FLUORESCENCE The time-resolved polarized signal Sn() is given by In this section we develop a real-space description of 3 ij i j time-resolved polarized and depolarized fluorescence from Sn͑͒ϭ␥N ͑͒ ͚ P ͑͒n n , ͑5͒ molecular aggregates. We show how this signal may be re- i, jϭ1 cast in terms of a doorway function which contains all rel- or using operator notation evant information about the electronic state of the system ͑expressed through the exciton density matrix͒, and a win- Sn͑͒ϭ␥N ͑͒n•P ͑͒•n. ͑6͒ dow function related to the system’s geometry. To set the stage, we introduce a Frenkel-exciton model Hamiltonian Here n is a unit vector denoting the polarization direction of which represents an aggregate made out of two-level mol- the emitted light, and ecules interacting with a bath consisting of nuclear ͑intramo- ϱ d lecular, intermolecular, and solvent͒ degrees of freedom: P ij͑͒ϭ͓␥N ͔͑͒Ϫ1 Sij͑,͒ ͵Ϫϱ 2 mÞn ˆ † † ˆ ph 1 Hϭ͚ ⍀n͑q͒BnBnϩ ͚ Jmn͑q͒BmBnϩH . ͑1͒ i j n m,n ϭ ͗P ͑͒P ͑͒͘ ͑7͒ 2N ͑͒ † Here Bn (Bn) are exciton annihilation ͑creation͒ operators is the polarization tensor. for the nth molecule, Hˆ ph is the bath ͑phonon͒ Hamiltonian, The total time- and frequency-resolved fluorescence sig- and q represents the complete set of nuclear coordinates. nal S(,) ͑summed over directions of polarizations͒ can be Exciton-phonon interactions and static disorder are described represented in a form through the dependence of molecular frequencies ⍀n and the intermolecular couplings J on nuclear coordinates q. mn ␥ ϱ t t it The dipole interaction with the optical field is S͑,͒ϭ dte P ϩ •P Ϫ 2 ͵Ϫϱ ͳ ͩ 2ͪ ͩ 2ͪʹ ˆ HintϭϪE͑t͒•P, ͑2͒ ϭ Sii͑,͒. ͑8͒ where the polarization operator is given by ͚i
† The corresponding time-resolved signal is Pϭ Un͑q͒dn͑BnϩBn͒. ͑3͒ ͚n ϱ d S͑͒ϵ S͑,͒ϭ␥N ͑͒Ls͑͒, ͑9͒ Here dn is the transition dipole of the n-th molecule, is ͵Ϫϱ 2 an average transition dipole and dn a dimensionless vector. When all molecular dipole moments are the same, dn will be where Ls() is given by the trace of the polarization matrix
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3 with the window given by Eq. ͑13͒, and the doorway ii Ls͑͒ϭTr͓P͔͑͒ϭ͚ P ͑͒. ͑10͒ iϭ1 ¯ij,lk Ϫ1 il jk † Nmn ͑͒ϵN ͑͒͗Um͑q͒Un ͑q͒BmBn͘. ͑19͒ We shall now introduce the DW representation for the The doorway function of Eq. ͑14͒ is related to the doorway time-resolved signal S() by relating L () to the time- s function of Eq. ͑19͒ by dependent doorway (N) and window (M) functions 3 ij ij ij ¯kk,ij Ls͑͒ϵTr͓MN͔͑͒ϭ MmnNmn͑͒, ͑11͒ Nmn͑͒ϭ͚ Nmn ͑͒. ͑20͒ mnij͚ kϭ1 where we use the tensor notation of Eq. ͑8͒. M and N() are Assuming fixed molecular orientations and neglecting reori- thus direct products of L0ϫL0 matrices in the exciton space entational disorder, the DW representation adopts the form (L0 is the number of molecules͒ and 3ϫ3 matrices in real space. Hereafter we invoke the rotating-wave approximation ij ij † † P ͑͒ϭ͚ Mmnmn͑͒, ͑21͒ for the signal by omitting the terms ͗BmBn͘, ͗BmBn͘, and mn ͗B B†͘ occurring in correlation functions of polarization op- m n ij erators in Eqs. ͑4͒, ͑7͒, and ͑8͒. The doorway function N() where M mn is given by Eq. ͑13͒ and mn is the reduced is then given by: exciton density matrix defined by Eq. ͑17͒. The window function M introduced in this section ij Ϫ1 † ij † mn Nmn͑͒ϵN ͓͑͒͗Um͑q͒Un͑q͔͒ BmBn͘, ͑12͒ carries all the information regarding dipole orientations nec- where the expectation value in the right hand side ͑r.h.s.͒ of essary for calculating the fluorescence. We note that both M Eq. ͑12͒ is taken with respect to the full density matrix of the and are L0ϫL0 Hermitian matrices, L0 being the number system at time . The window function is defined by: of one-exciton states. They satisfy ij i j M mnϵdmdn . ͑13͒ Tr͓͔ϭ1, Tr͓M͔ϭf, ͑22͒ Equations ͑11͒, ͑12͒, and ͑13͒ constitute the DW representa- where f is the total aggregate oscillator strength tion of time-resolved spontaneous light emission. Assuming that the averaging in the r.h.s. of Eq. ͑12͒ can be factorized, 2 f ϵ f n , f nϵ͉dn͉ , ͑23͒ we obtain ͚n
ij Ϫ1 † ij † Nmn͑͒ϵN ͓͑͒͗Um͑q͒Un͑q͔͒ ͗͘BmBn͘. ͑14͒ f n being the oscillator strength of the nth molecule. Equation ͑14͒ holds in three typical situations: ͑i͒ static dis- The matrices M mn and mn can be diagonalized using order, when orientational disorder is not correlated with en- basis sets denoted ␣ and ⌿␣ , respectively, with ergetic disorder; ͑ii͒ weak exciton-phonon-coupling, ␣ϭ0,1,..., L0Ϫ1. These basis sets are generally different. whereby the full density matrix can be factorized into a prod- However, if the aggregate has some symmetry, they may uct of electronic and equilibrated nuclear parts, which makes coincide. It is shown in Appendix A that M has no more than the first factor in the r.h.s. of Eq. ͑14͒ time independent; ͑iii͒ three nonzero eigenvalues ͑superradiant states͒ denoted by ¯ when nuclear motions do not affect dipole orientations. f ␣ , ␣ϭ1, 2, 3; the corresponding three eigenstates ␣ will Neglecting orientational disorder and assuming that all be hereafter referred to as the superradiant states. We further molecules have fixed directions in space, the doorway and introduce three orthogonal vectors d␣ which represent the window functions become scalars in real space: transition dipoles between the ground state and the superra- diant states:
Ls͑͒ϵTr͓MN͔͑͒ϭ Mmnmn͑͒, ͑15͒ 3 ͚mn dmϭ ͚ d␣␣͑m͒, ͑24͒ with ␣ϭ1 ¯ 2 M mnϵdn•dm . ͑16͒ and f ␣ϭ͉d␣͉ . Equation ͑24͒ implies that each superradiant state is responsible for the components of all molecular di- The doorway function now coincides with the reduced exci- poles along one of the orthogonal directions in the laboratory ton density matrix: frame; these directions are uniquely determined by the ag- Ϫ1 † gregate geometry and are not arbitrary. When all molecular Nmn͑͒ϭmn͑͒ϵN ͑͒͗Bm͑͒Bn͑͒͘, ͑17͒ dipoles are in the plane determined by d␣ with ␣ϭ1,2 we normalized to a unit trace Tr͓͔ϭ͚ ϭ1. n nn have ¯f ϭ0, while ¯f ϭ¯f ϭ0 when all dipoles are oriented These results can be readily extended to the polarized 3 2 3 ¯ signal ͓cf. Eq. ͑11͔͒. The DW representation for the polariza- along d1. Denoting ␣ϵ͗␣͉͉␣͘, we obtain for the su- tion matrix becomes perradiance factor 3 ij lk ¯ij,kl ¯ ¯ P ͑͒ϭ MmnNmn ͑͒, ͑18͒ Lsϭ f ␣ ␣ , ͑25͒ ͚mn ͚lk ␣͚ϭ1
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L0Ϫ1¯ ¯ ing the coherence length L and then relating the superradi- and since ͚␣ϭ0 ␣ϭ1 we have Lsрmax␣( f ␣), with 3 ¯ ance factor Ls to L and to geometry through the window ͚␣ϭ1 f ␣ϭ f , which gives an upper bound for the superradi- ance factor for a given geometry. function. The polarization matrix can be also represented in terms In the following sections we study various mechanisms of the superradiant states (␣ϭ1,2,3): which determine the coherence size L : finite temperatures, ␣ disorder-induced localization of excitons, and exciton self- 3 trapping induced by exciton-phonon-coupling. Characteristic ij i j ¯ P ϭ d␣d␣␣ , ͑26͒ sizes associated with these mechanisms will be introduced, ␣͚ϭ1 and their relations to Ls will be discussed in Sec. V. We shall i restrict the following analysis to the case where the exciton where d␣ are components of the superradiant dipole d␣ . The DW representation of time-resolved spontaneous equilibration timescale in the excited state manifold is short emission given by Eqs. ͑15͒–͑17͒ will be applied in the com- compared with the radiative decay, and the fluorescence ing sections to explore the roles of temperature, disorder and comes from the fully-relaxed exciton density matrix. This exciton-phonon coupling on the time-resolved fluorescence case, which always represents the fluorescence at long times, and superradiance of aggregates. To that end we shall intro- allows us to introduce all the relevant coherence sizes and dephasing mechanisms. duce the coherence size L which characterizes the size of a domain where the molecules emit coherently. Within the ap- proximations adopted here, all information about the state of the system relevant for superradiance is given by the reduced III. TEMPERATURE- AND DISORDER-CONTROL OF EXCITONIC COHERENCE IN CYCLIC exciton density matrix mn() whose time dependence re- flects the role of relaxation processes on the superradiance AGGREGATES factor Ls(). This allows us to introduce a formal definition The Ϸ2.5 Å resolution structure of the LH2 antenna of the characteristic coherence size L in terms of the density complex of purple bacteria shows two rings: an inner ring matrix. By applying the inverse participation ratio concept 36 ͑the B850 system͒ made of 18 chlorophyll molecules and an commonly used in the theory of quantum localization outer ring ͑the B800 system͒ with 9 chlorophylls.41 Cyclic Ϫ1 2 complexes appear also in other systems such as LH1 which 2 Lϵ L0͚ mn ͚ mn . ͑27͒ has 32 chlorophylls. All calculations in this article were per- ͫ mn ͯ ͯ ͬ ͫͩ mn ͯ ͯͪ ͬ formed on the B850 system. This aggregate is slightly dimer- ized and in Appendix B we present the window functions for This quantity gives the length-scale on which the density circular aggregates with two molecules in a unit cell. The matrix decays along the ‘‘anti-diagonal’’ direction, i.e., as a values of the coupling J between adjacent chlorophyll mol- function of nϪm. Similar measures have been successfully ecules in the B850 band used in Ref. 42 are Ϫ273 cmϪ1 and used in the analysis of off resonant polarizabilities of Ϫ291 cmϪ1. However, we do not expect this dimerization to aggregates,37,38 conjugated polymers,39 and semiconductor significantly affect the superradiance and for simplicity we nanocrystals.40 In the absence of any coherence have used an average nearest neighbor coupling parameter of ϭL Ϫ1␦ and we have L ϭ1. For a completely coher- nm 0 nm JϭϪ280 cmϪ1. ent distribution ϭL Ϫ1, and L ϭL . If the molecular nm 0 0 In the absence of disorder, the exciton eigenstates of a dipoles are oriented in the same direction, then the superra- ring and their energies are determined by the mo- diance factor is related to the number of molecules in this ⌿␣ ⑀␣ menta k ϭ2 /L , ϭ0,1,..., L Ϫ1 domain and loss of coherence which leads to the decrease of ␣ ␣ 0 ␣ 0 L reduces the superradiance. The picture is very different in 1 ik␣m the opposite situation when the total dipole of an aggregate ⌿␣ϭ e ; ⑀␣ϭ2J cos͑k␣͒. ͑28͒ ͑i.e., the vector sum of the molecular dipoles͒ vanishes due ͱL0 to cancellation of contributions from individual molecules. Since J is negative, the k0ϭ0 state has the lowest energy. In this case the emission is induced by the loss of coherence, In this section we shall consider two mechanisms for the and should increase with the decrease of L . It is important loss of excitonic coherence ͑dephasing͒: finite temperature, to note that the size L depends on the state of the aggregate and diagonal energetic disorder. A third mechanism: strong and is changed when exciton relaxation takes places. This exciton-phonon-coupling resulting in polaron formation will length has only an implicit dependence on the dipole orien- be discussed in the next section. tation, which comes from the process of preparation of the For weak exciton-phonon-coupling, electronic and initially excited state by the exciting optical field. These ar- nuclear degrees of freedom are decoupled in the equilibrium guments show that superradiance is a combined effect of density matrix ͑the only role of phonons is then to equilibrate aggregate geometry ͑dipole orientations͒ and its state, and at the electronic degrees of freedom͒, and the reduced exciton least qualitatively all information about the aggregate state density matrix assumes the canonical form: relevant for superradiance is contained in the coherence size Ϫ1 ␣ϭ␦␣Z exp͑Ϫ⑀␣ /T͒, ͑29͒ L , irrespective of the underlying physical dephasing mechanisms. Superradiance can then be analyzed by first where we set Boltzmann’s constant to 1 and studying the role of different physical mechanisms in affect- Zϵ͚␣exp(Ϫ⑀␣ /T), ⑀␣ were introduced in Eq. ͑28͒ and T is
J. Chem. Phys., Vol. 107, No. 10, 8 September 1997
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Ϫ1 mnϭZ ⌿␣*͑m͒⌿␣͑n͒exp͑Ϫ⑀␣ /T͒. ͑30͒ ͚␣
At temperatures lower than the spacing between exciton levels, the aggregate is in the lowest, completely delocalized, Ϫ1 exciton state, mnϭL0 and LϷL0. As the temperature is raised, higher-momentum excitons become occupied, which decreases L . We define the characteristic thermal size lT as the size of a cyclic J-aggregate with nearest-neighbor inter- action J in which the splitting between the lowest exciton levels is equal to the temperature T
2 2 Ϫ1 lTϭ4 ͉J͉T . ͑31͒ In a translationally-invariant system the density matrix co- herence length L is related to lT . This relation contains a numerical factor which can be calculated using Eq. ͑30͒. For example, when the temperature is much smaller than the ex- citon bandwidth, but much larger than the splitting between the exciton levels at the bottom of the band: 2 2 4 J/L0ӶTӶ4J, the numerical factor can be evaluated analytically using Eq. ͑30͒ and turns out to be close to 1: FIG. 1. Thermally relaxed density matrices. ͑a͒ Purely exciton systems ͑no Lϭͱ2/lT . disorder and phonons͒, for different temperatures, solid: 50 K, L ϭ14.55; Disorder can be easily incorporated by averaging Eq. dashed: 100 K, Lϭ10.47; dotted: 300 K, Lϭ5.95. ͑b͒ For different dis- Ϫ1 Ϫ1 ͑30͒ over realizations of disorder using Monte Carlo sam- order strengths at 4 K, solid: 330 cm , Lϭ14.55; dashed: 565 cm , Ϫ1 pling, resulting in . Static disorder breaks translational Lϭ10.64; dotted: 1130 cm , Lϭ6.16. ͑c͒ For different disorder ͗ ͘ Ϫ1 Ϫ1 strengths at 100 K, solid: 330 cm , Lϭ9.40; dashed: 565 cm , symmetry. However, this symmetry is restored upon averag- Ϫ1 Lϭ8.044; dotted: 1130 cm , Lϭ5.53. ͑d͒ Diagonal exciton-phonon ing over disorder, which implies that ͗͘ becomes diagonal Ϫ3 coupling, JRϭ8, ϭ0.5 and three reduced temperatures, solid: TRϭ10 , in the momentum representation. The coherence size L is Lϭ15.5; dashed: TRϭ0.1, Lϭ9.58; dotted: TRϭ0.2, Lϭ7.26. ͑e͒ Diag- Ϫ3 determined by the population of higher-momentum excitons onal exciton-phonon coupling, TRϭ10 , JRϭ8 and three values of , even at low temperatures. Since in one-dimensional systems solid: ϭ0.125, Lϭ17.9; dashed: ϭ0.5, Lϭ15.5; dotted: ϭ0.781, Lϭ3.52. static disorder leads to localization of exciton states, L can be related to the exciton Anderson localization length l ,as can be seen from Eq. 30 . ͑ ͒ pendent Gaussian distributions with the same mean ⍀¯ ͑inde- Using these definitions we now discuss the influence of pendent of n) and FWHM . Calculations were made using temperature and disorder on the density matrix mn . Figures a Monte Carlo sampling over different realizations. As 1 a –1 c show the ‘‘anti-diagonal’’ sections of the density ͑ ͒ ͑ ͒ shown in Fig. 1͑b͒, this localization becomes stronger with matrices calculated using Eq. 30 for different tempera- mn ͑ ͒ increasing . The corresponding values of L at 4 K are tures and disorder strengths. Neglecting disorder and assum- 14.6, 10.6, and 6.16 for ϭ330 cmϪ1, 565 cmϪ1, and 1130 ing a temperature lower than the splitting between the lowest cmϪ1, respectively. In the limit of very strong disorder the excitons, only the lowest exciton is populated in thermal density matrix becomes completely localized ͑diagonal͒ and equilibrium. Due to the symmetry, the corresponding density we have L ϭ1. The combined influence of disorder and matrix will be delocalized over the entire ring, and L will be temperature is illustrated in Fig. 1͑c͒ where equilibrium den- equal to the system size L0ϭ18. sity matrices are shown at Tϭ100 K for different values of With increasing temperature, higher exciton states are . The values of L at 100 K are 9.40, 8.04, and 5.53 for populated and becomes more localized. This effect is ϭ330 cmϪ, 565 cmϪ1, and 1130 cmϪ1, respectively. The shown in Fig. 1͑a͒, where the canonical density matrices for density matrices become more localized with increasing dis- temperatures 50 K solid , 100 K dashed , and 300 K dot- ͑ ͒ ͑ ͒ ͑ order and temperature, as is evident by comparing Fig. 1͑c͒ ted are displayed. The temperature-induced localization can ͒ with Fig. 1͑b͒. be clearly seen. The corresponding values of L are 14.6, 10.5, and 5.95, respectively. In the high temperature limit all IV. POLARON CONTROL OF EXCITONIC COHERENCE exciton states are equally populated and the density matrix becomes completely localized ͑diagonal͒ resulting in Lϭ1. When exciton-phonon-coupling is weak, the phonon de- Figure 1͑b͒ illustrates how disorder leads to localization grees of freedom may be eliminated and their effects incor- of the density matrix at low temperatures. Energetic diagonal porated through relaxation superoperators calculated pertur- 43,44 disorder has been included by assuming that ⍀n have inde- batively in the exciton-phonon-coupling strength. The
J. Chem. Phys., Vol. 107, No. 10, 8 September 1997
Downloaded 08 Mar 2001 to 128.151.176.185. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html Meier et al.: Antenna complexes 3881 detailed-balance condition satisfied by the relaxation super- mÞn ˆ GH † † ˆ ph operator guarantees that at long times the system evolves H ϭ͚ ⍀n͑qϭ0͒BnBnϩ ͚ Jmn͑qϭ0͒BmBnϩH into a thermal distribution of bare excitons. The correspond- n mn ing coherence size L then does not depend on the nature of ϩHˆ diagϩHˆ o.d.. ͑37͒ the phonons and their coupling with excitons. The analysis of equilibrated excitons in Sec. III therefore implicitly as- There are two competing energy scales in the Holstein 2 sumes weak exciton-phonon coupling. This state of affairs Hamiltonian, namely, the lattice relaxation energy g 0 and changes drastically as the coupling strength is increased. In the bare exciton bandwidth 4J. Their ratio will be denoted this section we consider the effects of strong exciton- the coupling strength phonon-coupling on the coherence size. It is shown in Ap- ϭg2 /4J ͑38͒ pendix C how higher order effects of exciton-phonon cou- 0 which determines the size of the polaron as well as exciton- pling on the reduced exciton density matrix mn can be taken 2 into account perturbatively. The following analysis is non- phonon correlations. In typical molecular crystals, g Շ1, in 2 perturbative and is based on the polaron model.33 ionic crystals g is large compared to unity, and in semicon- 2 We adopt the Hamiltonian Eq. ͑1͒ with the Einstein pho- ductors g is between the former two. In anthracene, for 47 non Hamiltonian Hˆ ph, example, is about 0.4, and in pyrene, about 0.8ϳ1.6. The other two important parameters of the model are the reduced bandwidth ͑the ratio between the bare exciton bandwidth 4J ˆ ph † H ϭ ប0bnbn , ͑32͒ and the characteristic phonon frequency 0) ͚n JRϭ4J/0 , ͑39͒ where b† creates a phonon of frequency on site n, and we n 0 and the reduced temperature have one Einstein oscillator per molecule. Exciton-phonon interactions enter through the nuclear coordinate influence on TRϭT/0 . ͑40͒ both molecular frequencies ͑diagonal coupling͒ and intermo- For strong exciton-phonon-coupling (ӷ1), solutions of the lecular interactions ͑off-diagonal coupling͒. Expanding Holstein Hamiltonian are known as small polarons because ⍀ (q) to first order in phonon coordinate q, the first term of n the exciton-induced lattice distortion is confined to essen- Eq. ͑1͒ reads tially a single exciton site.48 For weak exciton-phonon cou- pling (Ӷ1), the spatial extent of the lattice distortion is † † ˆ diag significantly increased and the resulting phonon-dressed ex- ͚ ⍀n͑q͒BnBnϭ͚ ⍀n͑qϭ0͒BnBnϩH , ͑33͒ n n citon is called a large polaron. The crossover from a large polaron to a small polaron with increasing exciton-phonon with the diagonal exciton-phonon-coupling term coupling ͑often called the self-trapping transition͒ is rather abrupt for large intermolecular coupling J. In the limit of ˆ diag † † slow lattice motions (JRӷ1), adiabatic polaron theories ad- H ϭgប0͚ BnBn͑bnϩbn͒, ͑34͒ n mit approximate solutions in the form of solitons. The off-diagonal coupling in Eq. ͑36͒ has positive ͑nega- and g is a dimensionless diagonal coupling constant. tive͒ signs for electronic transfer to the left ͑right͒ of the Expanding Jmn(q) to first order in phonon coordinates, distorted lattice site. While exact solutions exist for diagonal we write the second term of Eq. 1 as ͑ ͒ coupling in the JR 0 limit ͑the small polaron͒, the Hamil- tonian with off-diagonal→ coupling has not been diagonalized mÞn mÞn analytically. Adding intermolecular interactions and diagonal † † ˆ o.d. Jmn͑q͒BmBnϭ Jmn͑qϭ0͒BmBnϩH ͑35͒ ͚m,n ͚m,n coupling complicates the problem even further. Due to diffi- culties in obtaining simple solutions,49 off-diagonal coupling with the off-diagonal coupling term45,46 is less frequently used compared with its diagonal counter- part. To calculate mn we assume that exciton-phonon cou- ˆ o.d. † † H ϭ ប0 ͓BnBnϩ1͑bl ϩbl͒͑␦nϩ1,lϪ␦nl͒ pling leads to the formation of bands of collective exciton- 2 ͚nl phonon states, and the many-body polaron wave function is † † 50 ϩBnBnϪ1͑bl ϩbl͒͑␦nlϪ␦nϪ1,l͔͒. ͑36͒ given by the Toyozawa Ansatz ͑see Appendix D͒:
The second term of ͑36͒ is the Hermitian conjugate of the Ϫ1 iKn K K † ͉K͘ϭL0 ͚ e ͉⌳n ͚͘ mϪnBm͉0͘ex . ͑41͒ first. Here we have assumed nearest-neighbor coupling, and n m is a dimensionless parameter controlling the off-diagonal Here ͉K͘ is the lowest polaron state with momentum K, coupling strength. Equation ͑34͒ and Eq. ͑36͒, together with K ͉0͘ex is the exciton vacuum state, and ͉⌳n ͘ are phonon wave Hˆ ph and the zeroth-order intermolecular coupling term, re- functions centered at site n containing a coherent state on GH K K sult in the generalized Holstein Hamiltonian Hˆ ͑the origi- each site l with a displacement l ͓cf. Eq. ͑D6͔͒. ͉⌳n ͘ is nal Holstein Hamiltonian contains diagonal coupling only͒:34 different from K only by a shift of nϪn lattice con- ͉⌳nЈ͘ Ј
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K K stants. The parameters l , l , and the polaron energy band EK , are obtained variationally. The phonon wave functions K ͉⌳n ͘ represent a lattice distortion forming a potential well centered at n and trapping the exciton with an amplitude K distribution of l . At sufficiently low temperatures (TRӶ1), only the low- est polaron band is populated, and the many-body exciton- phonon density matrix M is given by
Ϫ1 Ϫ1 ϪEK /T MϭZ ͗K͉K͘ ͉K͘e ͗K͉, ͑42͒ ͚K where EK are the polaron energies and Zϵ͚Kexp(ϪEK /T)is the partition function. It should be emphasized that the polaron state is delocal- ized over the entire aggregate even though both exciton am- K K plitudes l and phonon displacements l are localized. We K shall demonstrate that the size of l , hereafter referred to as the polaron size l p , which can be formally defined by K 4 l pϵ1/͉͚ll ͉ , shows up as the localization length of mn ͑i.e., as the coherence size L). lp can therefore be directly FIG. 2. The Debye-Waller factor FK . a Weak coupling, ϭ0.0125, observed in superradiance measurements. To that end we nЈn ͑ ͒ † narrow band, JRϭ0.8, two crystal momenta are shown: Kϭ0 ͑solid͒ and substitute Eqs. ͑42͒ and ͑41͒ into mmЈϭTr (MBmBmЈ) yielding Kϭ ͑dashed͒; ͑b͒ weak coupling, ϭ0.125, broad band, JRϭ8, three crystal momenta are shown: Kϭ0 ͑solid͒, Kϭ ͑dashed͒ and Kϭ2/3 ͑dotted͒; ͑c͒ intermediate coupling, ϭ0.5, JRϭ8, two crystal momenta are Ϫ2 Ϫ1 ϪEK /T selected: Kϭ0 solid and Kϭ dashed . mm ϭL0 ͗K͉K͘ e ͑ ͒ ͑ ͒ Ј ͚K
ϫ eiK͑nЈϪn͒ K K* FK , 43 polaron-induced localized structure at small ͉mϪn͉. To trace ͚ mϪnmЈϪnЈ nЈn ͑ ͒ nnЈ the origin of this behavior, let us examine more carefully the Debye-Waller factor. FK decreases from 1 at nϭn to a with the Debye-Waller factor nЈn Ј constant at large ͉nϪnЈ͉ over a characteristic lengthscale K K K Ϫ1 K 2 iq͑nϪnЈ͒ which will be denoted by lDW . A formal definition of lDW Fn nϵ͗⌳n ͉⌳n ͘ϭexp L0 ͚ ͉q ͉ ͑e Ϫ1͒ , Ј Ј ͫ q ͬ can be obtained by subtracting its value ¯F for large ͉nϪnЈ͉ ͑44͒ K ¯ and substituting Fn nϪF into Eq. ͑27͒ instead of mn . The K K Ј and q is the Fourier transform of n ͑see Appendix D͒: Debye-Waller factor can be recast in the form FK ϭexp K Ϫ K , 47 K Ϫiqn K nЈn ͑nϪnЈ 0 ͒ ͑ ͒ q ϭ͚ e n . ͑45͒ n where K is the Fourier transform of K 2. Because nϪnЈ ͉q ͉ is determined by the combined ef- K Localization of mmЈ nϪn decays rapidly to zero with increasing ͉nϪnЈ͉, the K Ј K K fect of electronic confinement ͑that of mϪn) and the nuclear Debye-Waller factor at large nϪn equals ͗⌳nЈ͉⌳n ͘ ͉ Ј͉ factor FK which represents the overlap ͑scalar product͒ K 2 nЈn exp(Ϫ0 ) which becomes exp(Ϫg )intheJR 0 limit. K K K → ͗⌳ ͉⌳ ͘ of two coherent states. In Fig. 2, the Debye-Waller factor K is displayed nЈ n ͗⌳nЈ͉⌳n ͘ The reduced density matrix mϪn is shown in Fig. 1͑d͒ as a function of nϪnЈ for different polaron momenta K. for JRϭ8, ϭ0.5 and three reduced temperatures Two weak-coupling cases are shown, ϭ0.0125, JRϭ0.8 Ϫ3 TRϭ10 , 0.1, 0.2. L for the three reduced temperatures (JRϽ1, narrow band͒ and ϭ0.125, JRϭ8(JRϾ1, broad- are 15.5, 9.58, and 7.26, respectively. Comparison with the band . For the first case, FK hardly deviates from 1 for both ͒ nЈn other panels shows that at finite temperatures, polaron- Kϭ0 and Kϭ, as shown in Fig. 2͑a͒. For the second case, induced and disorder-induced localization of mϪn are very the Debye-Waller factor reflects the one-phonon plane-wave similar. At zero temperature, only the Kϭ0 polaron state is of high momentum states. For JRϭ8, ϭ0.5, the Debye- populated, and the reduced density matrix becomes K Waller factor has a constant component exp(Ϫ0 ) which de- creases with increasing crystal momentum as is evident from Kϭ0 Kϭ0* Kϭ0 Kϭ0 . 46 mmЈϰ͚ mϪnmЈϪnЈ͗⌳nЈ ͉⌳n ͘ ͑ ͒ comparing the two crystal momenta, Kϭ and Kϭ2/3, nnЈ displayed in Fig. 2͑c͒. For intermediate to strong couplings, In Fig. 1͑e͒, we display the reduced density matrix mϪn for it is safe to assume that the Debye-Waller factor size lDW is Ϫ3 TRϭ10 , JRϭ8, and ϭ0.125, 0.5 and 0.78. The corre- much smaller than l p . This implies that only those terms K sponding L are 17.9, 15.5, and 3.52, respectively. mϪn has with ͉nϪnЈ͉Ӷl p contributes to Eq. ͑43͒, and since l has a a constant finite component at large ͉mϪn͉ in addition to the size of l p the reduced density matrix is localized on the l p
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FIG. 3. Reduced exciton density matrices for the polaron model. ͑a͒ For two reduced temperatures TRϭ0.27 ͑solid, Lϭ13.24), and TRϭ0.49 ͑dashed, Lϭ14.27), JRϭ8,ϭ0.125. The corresponding polaron band is shown in Fig. 4͑a͒. ͑b͒ For two temperatures TRϭ0 ͑solid, Lϭ5.95) and TRϭ0.2 ͑dotted, Lϭ2.93), ϭ1, JRϭgϭ0. Also shown for comparison is the reduced density matrix for ϭgϭ1, JRϭTRϭ0 ͑dashed, Lϭ3.71). The corresponding polaron band is shown in Fig. 4͑b͒. ͑c͒ ϭ1, JRϭ2 and FIG. 4. Dispersion curves (EK vs K) for the lowest polaron band. ͑a͒ Di- Ϫ3 Ϫ2 TRϭ10 ͑solid, Lϭ13.9), TRϭ10 ͑dashed, Lϭ8.9). The correspond- agonal coupling only, JRϭ8, ϭ0.125. ͑b͒ Diagonal and off-diagonal cou- ing polaron band is shown in Fig. 4͑c͒. ͑d͒ For soliton profiles ͓Eqs. ͑48͒ and pling, JRϭ0, ϭgϭ1. ͑c͒ Off-diagonal coupling only, ϭ1, JRϭ2. ͑50͔͒ for three strengths of nonlinearity: a0ϭ3 ͑solid͒, a0ϭ4 ͑dashed͒ and a0ϭ5 ͑dotted͒. Lϭ17.9, 17.7, and 16.8, respectively.
exciton wave function, whereas lDW shows a typical separa- lengthscale as a function of ͉nϪnЈ͉. The constant compo- tion between the traps related to the left ͑bra͒ and right ͑ket͒ nent of the reduced density matrix mϪn for large nϪm components of the density matrix. The coherence size de- shown in Fig. 1͑d͒ reflects the similar behavior of the Debye- pends on both l p and lDW , however when lDWӶlp the length Waller factor. ¯ lDW is irrelevant. The constant tail F of FnnЈ which does not In the weak coupling (Ӷ1) limit of the broadband vanish at large values of ͉nϪnЈ͉ may be important in large (JRϾ1) case, the Debye-Waller factor Eq. ͑44͒ for high aggregates and at low temperatures. Representing the Debye- crystal momentum states has a one-phonon plane-wave com- ¯ ¯ Waller factor as Fmnϭ(FmnϪF)ϩF in Eq. ͑43͒ we recast ponent due to strong interactions with the one-phonon scat- (0) (1) the density matrix mmЈ in a form mmЈϭmm ϩmm .By tering continuum, as shown in Fig. 2͑b͒. As a result, for (0) (1) Ј Ј substituting and into Eq. ͑27͒ we obtain L which some temperature range, may become more delocal- mmЈ mmЈ mmЈ saturates and scales like L , respectively, for large L . Since ized at higher temperatures. This counterintuitive behavior is 0 0 ¯ related to the increase in population of higher-momentum in the case of strong and intermediate coupling F is usually states which have the one-phonon plane-wave as the phonon small at intermediate sizes L0, L does not depend on L0 wave function, as the temperature is raised. As shown in Fig. while at large L0 we have L ϰ L0. The size lc which determines the crossover region can be defined as the value 3͑a͒, mmЈ at TRϭ0.49 has a longer off-diagonal tail than of L when contributions of (0) and (1) are equal. The mmЈ at TRϭ0.27. For small ͉mϪmЈ͉, though, mmЈ at 0 mmЈ mmЈ ¯ TRϭ0.49 remains slightly more localized than mmЈ at smaller F, the larger is lc . TRϭ0.27. L for the two cases are 14.27 and 13.24, respec- We now turn to off-diagonal coupling. LH2 has signifi- tively. Figure 4͑a͒ shows the corresponding polaron band. cant dipole-dipole interactions up to the third neighbor.42 It is The flat regions of the polaron band at higher crystal mo- reasonable to assume that these interactions are affected by menta are responsible for the counterintuitive behavior of the various phonon modes. The two sets of variational param- reduced exciton density matrix shown in Fig. 3͑a͒. eters l and l of the Toyozawa Ansatz provide an adequate In summary, we have shown that in the absence of static description of off-diagonal coupling as well, and the result- disorder and at low temperatures, the coherence size L is ing exciton-phonon correlations and polaron energy bands related to the characteristic sizes l p and lDW , even though compare favorably with the Munn-Silbey approach at zero the excitons are not localized, in contrast to the case of static temperature.45,46 disorder. The polaron size l p reflects the size of the trapped In Fig. 3͑b͒ we show the influence of off-diagonal cou-
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Downloaded 08 Mar 2001 to 128.151.176.185. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html 3884 Meier et al.: Antenna complexes pling on the reduced density matrix mn . For JRϭgϭ0 ͑off- by the Holstein Hamiltonian, while in the Fro¨hlich Hamil- diagonal coupling only͒, the polaron band is symmetric with tonian only the large polaron is physically meaningful. If the respect to KϭϮ/2 as demonstrated in Fig. 4͑b͒.Asare- characteristic phonon frequency is small compared to the sult, mϪn vanishes for odd mϪn. For even mϪn, mϪn has splitting between exciton levels the lattice kinetic energy can alternating signs reflecting the dynamic dimerization intro- be neglected in the zero-order adiabatic approximation.53 duced by off-diagonal coupling. Increase in temperature re- For the diagonal coupling we get ͑see Appendix E for duces the amplitudes of mϪn at nonzero even mϪn. Diag- the derivation͒ onal coupling has a similar effect on mϪn as the 1 temperature, i.e., it suppresses for nonzero even mϪn. mϪn mnϭ ͚ ⌿*͑mϩs͒⌿͑nϩs͒, ͑48͒ However, as is evident from Fig. 3͑b͒, diagonal coupling L0 s reduces equally mϪn for all nonzero even mϪn. This is where the exciton wave function satisfies the discrete nonlin- 33,34,54 clearly reflected in the sizes of L for the three cases dis- ear Schro¨dinger equation played in Fig. 3͑b͒: 5.95 (T ϭ 0, ϭ 1, J ϭ g ϭ 0), R R Ϫ ϩ ϩ Ϫ Ϫ 2 ϭ 2.93 (T ϭ 0.2, ϭ 1, J ϭ g ϭ 0) and 3.71 (T J͓⌿͑n 1͒ ⌿͑n 1͔͒ 4͉⌿͑n͉͒ ⌿͑n͒ E⌿͑n͒, R R R ͑49͒ ϭ 0, g ϭ ϭ 1, JR ϭ 0). With increasing temperature, 2 mϪn for m Ϫ nϭϮ4, Ϯ6, Ϯ8 vanishes much faster than where 2 is the Stokes shift of a single molecule (2ϭg 0 mϪnϭϮ2. For diagonal coupling only, mϪn is completely for the Holstein model͒ and E is the polaron energy. For localized on mϪnϭ0. Off-diagonal coupling thus induces large polarons ͑i.e., when the polaron size l p is much larger localization as well as assists transport of electronic excita- than the lattice constant but still smaller than the system’s tions. size͒, one can apply the one-dimensional infinite-size con- Addition of the transfer integral J destroys the folded tinuum model, which yields the following ͑soliton͒ solution 33,34 polaron band structure by lowering the zone center energy of Eq. ͑49͒: Kϭ0 E ͓cf. Fig. 4͑b͔͒. Consequently, the structure of mϪn is 1 n expanded while retaining the oscillatory feature in Fig. 3͑b͒. ⌿͑n͒ϭ sech , ͑50͒ ͱ2a ͩ a ͪ 0 0 As the zone center energy EKϭ0 is lowered to the level of EKϭϮ/2 , the polaron band has a rather flat structure at the with a0ϵJ/. Solutions of the continuum version of Eq. zone center, as illustrated in Fig. 4͑c͒ for the case of ͑49͒ on a ring, which generalize Eq. ͑50͒ to finite systems ϭ 1. JR ϭ 2, causing the reduced density matrix to change have been obtained recently in Ref. 55. The connection be- abruptly as the temperature is raised from zero. In Fig. 3͑c͒ tween the soliton solutions and the Toyozawa Ansatz is dis- Ϫ3 we compare mϪn for two low temperatures TRϭ10 and cussed in Appendix D. In Fig. 3͑d͒, we show the reduced Ϫ2 10 for JRϭ8, ϭ1. A temperature increase of one hun- density matrices calculated using Eqs. ͑48͒ and ͑50͒. For the dredth of the phonon frequency drastically affects the profile three strengths of nonlinearity shown a0 ϭ 3, 4, 5, the cor- of mϪn due to a flat band bottom at the zone center, and L responding values of L are 17.9, 17.7 and 16.8, respec- Ϫ3 Ϫ2 also drops from 13.9 (TRϭ10 ) to 8.94 (TRϭ10 ). For tively. sufficiently large transfer integrals, the polaron band acquires Equation ͑48͒ allows us to treat effects of disorder and a similar shape as the weak coupling band of diagonal phonon coupling on exciton coherence size in the same fash- exciton-phonon-coupling, and so does the reduced density ion. In the case of static disorder ⌿(m) is a localized exciton matrix mn . wave function, whereas in the polaron picture represented by So far our calculations were based on the assumption the Toyozawa Ansatz, ⌿(m) is the self-trapped exciton K that the temperature is much lower than the phonon fre- wave function represented by m in Eq. ͑41͒. In the case of quency TRӶ1, which is typically the case for optical static disorder, Eq. ͑48͒ should be averaged over disorder phonons even at room temperature. Aggregates may have realizations with the canonical distribution ͓see Eq. ͑30͔͒. strong coupling to low-frequency solvent modes. Nuclear For strong or intermediate exciton-phonon-coupling the av- spectral densities representing coupling to low-frequency eraging of Eq. ͑48͒ is given by Eq. ͑43͒ by setting K K modes have been used to fit photon-echo measurements in ͗⌳ ͉⌳n ͘ϭ␦n n . In the adiabatic polaron approach ⌿(m)is 14 nЈ Ј light-harvesting complexes. This corresponds to the oppo- the solution of Eq. ͑49͒. site limit TRϵT/0ӷ1 ͑here 0 denotes a typical frequency of low-energy solvent modes͒ and the adiabatic polaron theory can be applied. The calculation of the polaron wave V. SUPERRADIANCE IN CYCLIC AGGREGATES functions is greatly simplified when nuclear motions are much slower than the electronic ones. In this case we can It was shown in Sec. II that the superradiance enhance- adopt the Born-Oppenheimer ͑adiabatic͒ approximation. The ment factor depends on two ingredients: the exciton density 51 adiabatic polaron theory was first developed by Pekar for matrix and the window function M ij which carries informa- an analogous continuum Hamiltonian—the Fro¨hlich tion regarding the aggregate geometry. In this section we Hamiltonian—describing a slow electron moving in an ionic consider various models for dipole orientations and combine crystal.52 Later the Pekar-type adiabatic solution was gener- them with the calculation of the density matrix presented in alized to the Holstein Hamiltonian for molecular crystals.33,34 the last two sections to explore the superradiant behavior of Both the large and small adiabatic polaron can be captured the LH2 complexes.
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displayed for comparison decreases with increasing tempera- ture ͓see Fig. 1͑a͔͒. For model III the temperature depen- dence of Ls is close to that of L : both decrease from 18 for low temperatures to 1 in the high temperature limit. How- ever, for model I, Ls shows a very different temperature dependence: at low temperatures the density matrix is com- pletely delocalized, and the summation over all the inner products of the dipole vectors in the calculation of Ls yields Lsϭ0. With increasing temperature the coherence size L decreases and Ls becomes finite and eventually reaches a maximum when L is somewhat larger than half the system size. At high temperatures, all three models have similar su- perradiance factors which eventually approach 1. It follows from Eq. ͑54͒ that the superradiance factor for model II is simply the average of models I and III. It is therefore easy to calculate Ls for all angles ⌰, using the curves for models I and III. Figures 5͑b͒–5͑d͒ display the temperature-dependence of Ls and L due to diagonal and off-diagonal exciton- phonon-coupling. Within the polaron model as outlined Sec. IV, localization of mmЈ may have several origins. Increased temperature results in smaller L . This is clearly shown when the bandwidth 4J is less than the Einstein phonon fre- quency 0 (JRϽ1) so that the bare exciton band does not cross into the one-phonon scattering continuum. For J Ͼ1, FIG. 5. Variation of Ls with temperature. Solid: model III; dashed: model R II; dotted: model I; dashed-dotted: L . ͑a͒ Without disorder and exciton- the weak coupling limit is nonperturbative in the sense that phonon-coupling. ͑b͒ Diagonal coupling, the weak coupling limit 0 and the 0ϩ polaron band is different from the bare exciton Ϫ1 → JRϭ0.8. ͑c͒ Diagonal coupling, JRϭ8, ϭ0.5. 0ϭ1253 cm . ͑d͒ Off- → 56–59 Ϫ1 band due to level repulsion from the phonon continuum. diagonal coupling, ϭ1 and JRϭ5. 0ϭ1253 cm . In Fig. 5͑b͒, we display L and the superradiance length Ls for our three models and for JRϭ0.8 in the weak coupling The window function for a cyclic aggregate can be limit ( 0). At zero temperature, both L and the superra- → readily calculated. Denoting the angle between the molecular diance length Ls for parallel dipoles ͑model III͒ are equal to dipole and the aggregate plane by ⌰ we obtain: the size of the system, 18. Due to the momentum selection rule, L for the in-plane dipole configuration ͑model I͒ 2͑mϪn͒ s M ϭd2 sin2 ⌰ϩcos2 ⌰ cos . ͑51͒ reaches a maximum at a temperature whereby the Kϭ2/18 mn ͫ ͩ L ͪͬ 0 state is most populated. L and the superradiance length Ls It follows from the circular symmetry, that and M are for ⌰ϭ0°, 45°, 90° are are shown in Fig. 5͑c͒ for interme- diagonal in the same basis set and the superradiant states are diate exciton-phonon coupling ϭ0.5, JRϭ8. At zero tem- perature the superradiance length Ls for parallel dipoles is 1 1 only half the size of the ring, and L is also less than 18. For m ϭ eϮi2m/L0, m ϭ , 52 1,2͑ ͒ 3͑ ͒ ͑ ͒ strong exciton-phonon coupling, the superradiance size L is ͱL0 ͱL0 s independent of temperature. In Fig. 5͑d͒ we show L and the with the corresponding oscillator strengths superradiance length Ls for models I, II and III as a function 1 of the temperature for JRϭ4, ϭ1 and gϭ0. The depen- ¯f ϭ¯f ϭ d2L cos2⌰, ¯f ϭd2L sin2⌰. ͑53͒ 1 2 2 0 3 0 dence of the superradiance length Ls on temperature is very similar to the diagonal coupling case. The state 3 has a transition dipole orthogonal to the aggre- In Fig. 6͑a͒ we display Ls and L versus the static dis- gate’s plane, whereas 1 and 2 have an in-plane dipole. order parameter at 4 K. The curves are slightly different, Combining Eqs. ͑15͒ and ͑51͒–͑53͒ we have for the super- but their general appearance and limits are similar to Fig. radiance factor: 5͑a͒. Figure 6͑b͒ repeats the calculations of Fig. 6͑a͒ for a higher temperature ͑100 K͒. Due to the temperature-induced 1ϩ2 L ϭ cos2 ⌰ ϩsin2 ⌰ L , ͑54͒ loss of coherence, L and L for model III are smaller than s ͩ 2 3 ͪ 0 s the system size for weak disorder, and Ls for model I no with 1ϩ2ϩ3р1. longer vanishes. Analogous to Fig. 6͑a͒, with increasing dis- We first examine the effect of finite temperature. Figure order Ls becomes similar for all models. For very strong 5͑a͒ shows the variation of Ls with temperature for three disorder, L as well as Ls approach 1. orientations of the dipoles with respect to the ring: model I The correlations between Ls and L are clearly shown in 0 0 0 (⌰ϭ0 ), model II (⌰ϭ45 ), and model III (⌰ϭ90 ). L Fig. 7͑a͒, where we display them for the different localiza-
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FIG. 6. Variation Ls with disorder strength, in the absence of exciton- phonon coupling. ͑a͒ Tϭ4 K and ͑b͒ Tϭ100 K. Solid: model III; dashed: model II; dotted: model I; dashed-dotted: L . tion mechanisms and dipole orientations. For the two local- ization mechanisms considered in Fig. 7͑a͒͑temperature- and disorder-induced dephasing͒ we find a very similar cor- relation between Ls and L in models I and III, which dem- onstrates that the superradiance enhancement factor is in both cases primarily determined by the off-diagonal spread of the density matrix L , and that both mechanisms influ- ence the superradiative enhancement factor in a similar fash- ion. In Fig. 7͑b͒, Ls from model III is plotted vs L within the polaron approach for different temperatures, intermolecu- lar couplings and diagonal exciton-phonon couplings. This again shows the strong correlation between Ls and L is not sensitive to the underlying mechanisms. We have identified the density matrix size L as the primary factor in determining the superradiance size Ls . This allows us to predict Ls using L , regardless of the spe- cific microscopic mechanism that determines L . A com- monly used measure of the exciton localization length is pro- FIG. 7. Correlation between the superradiance enhancement factor Ls and vided by the inverse participation ratio of a wave function the density matrix size L . ͑a͒ Squares: model III for different temperatures, 36 triangles up: model III for different disorder strengths at Tϭ4 K; dashed ⌿␣(m) line: model III for different disorder strengths at Tϭ100 K, circles: model I 1 for different temperatures; triangles down: model I for different disorder l ϵ . ͑55͒ strengths at Tϭ4 K; dotted line: model I for different disorder strengths at ␣ 4 ͚m͉⌿␣͑m͉͒ Tϭ100 K; ͑b͒ model III for different temperatures T, intermolecular cou- plings J and diagonal exciton-phonon coupling strength g. The thermally averaged size associated with the wave func- tion is defined by:
Ϫ1 3 lϵZ ͚ l␣exp͑Ϫ⑀␣ /T͒. ͑56͒ 3l L0 ␣ L ϭ , for l р ; 2 2 2lϩ1 We shall now discuss the connection between l and L .At ͑57͒ 4 low temperatures L depends on exciton localization and is 3l L0 related to l . The numerical factor which relates the two Lϭ , for lϾ . l ͑2l2 ϩ1͒ϩ͑2l ϪL ͒3Ϫ͑2l ϪL ͒ 2 depends on the form of the wave function and the system 0 0 size L0. To illustrate this point we plot in Fig. 8͑a͒ the de- In particular Lϭl when lϭL0. Therefore in the absence pendence of L on l for different models. Assuming a uni- of disorder and at low temperatures (Tϭ4K͒both measures form wave function ⌿(m)ϭ(m)(lϪm)/ͱl where give LϭlϭL0ϭ18, since the lowest exciton is completely (m) is the Heaviside function, we obtain delocalized. With increasing disorder however, they start to
J. Chem. Phys., Vol. 107, No. 10, 8 September 1997
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Therefore L is determined by localization of the amplitude and is related to l . In the absence of disorder exciton states are delocalized over the entire aggregate and L is deter- mined by oscillations of wave functions corresponding to higher-energy excitons which are populated when the tem- perature is raised. This implies that the characteristic length lT reflects the phase oscillations of the exciton wave func- tions. In a disordered aggregate at low temperatures when lӶlT , phase oscillations are negligible and L is related to l . As the temperature is raised and higher-energy excitons become populated, l increases since localization is weaker at higher energies whereas phase oscillations become more important, as reflected in the decrease of lT . L then de- creases while l increases. In this intermediate region lϳlT , L is influenced by the competition of the amplitude and the phase-induced mechanisms. At higher temperatures lTӶl , exciton localization becomes irrelevant and L is determined by the thermal size lT . This can be illustrated as follows: our numerical simulations of the thermal- and disorder-averaged density matrix ͗͘, with ϭ377 cmϪ1 gives Lϭ13.7. Using Eq. ͑56͒ we then obtain lϭ4.8, so FIG. 8. ͑a͒ L versus l of the lowest exciton wave function for different models. Dashed-dotted: uniform wave function; dashed: Gaussian wave that L /lϷ3. We recall that for these parameters we find a function; dotted: sech wave function ͓Eq. ͑50͔͒; solid: Tϭ4 K with varying superradiance enhancement factor of Lsϭ3.2 for model I. disorder. ͑b͒ Difference between the solid curve of ͑a͒ and the Pade´ approx- With increasing disorder both measures become more local- imant Eq. ͑58͒. Ϫ1 ized. For ϭ754 cm and T ϭ 4 K we obtain Lϭ8.6 and lϭ2.6, and their ratio is 3.3. With increasing temperature, due to the contribution of energetically higher exciton states differ, since the exciton wave function is no longer uniform. which are more delocalized, l increases, while L de- For L ӷl ӷ1, we have L Ϸ 1.5l as shown in Fig. 8͑a͒. 0 creases. For Tϭ 300 K we have L ϭ5.8 and l ϭ6.6. These For the Gaussian wave function ⌿(m), we have L ϭ2l if examples illustrate an interplay of the characteristic sizes l L ӷl ӷ1. The soliton wave functions Eq. ͑50͒ show a simi- 0 and l in determining the coherence size L . We reiterate lar behavior. Monte Carlo averaging over diagonal disorders T that superradiance is strongly affected by geometry and L is yields the solid curve in Fig. 8͑a͒. An excellent fit is obtained s not necessarily proportional to l , i.e., for weak disorder L using the Pade´ approximant ͓see Fig. 8͑b͔͒ s is determined by the loss of coherence L0ϪL rather than 1ϩ7.88͑lϪ1͒ l . Usually one can divide the aggregate into domains Lg (g Lϭ . ͑58͒ stands for geometry͒ within which molecular dipoles are ori- 1ϩ0.316 l Ϫ1͒ϩ0.00393 l Ϫ1͒2 ͑ ͑ ented in more or less the same direction. These directions are These examples illustrate that the coherence length L different for different domains and the total dipole of the representing the number of coherently coupled molecules is aggregate vanishes due to cancellation of contributions from closely related to the size of the corresponding wave func- different domains. At long times, when excitons are equili- tions which is usually defined through the inverse participa- brated, L is independent on the system’s geometry. In con- tion ratio l given by Eq. ͑55͒ where lϭl for the model of trast, the size Lg depends only on the dipole orientations ͑and static disorder and lϭl p for the case of exciton-phonon cou- the system’s symmetry͒ and is independent on the state of pling. However, the relations between these quantities con- the aggregate. It is easy to see that for a given geometry ͑and tain important numerical factors representing the wave func- Lg) there is an optimal value of the coherence size L , 60 tion shape and the system size. All models show a very which maximizes the superradiance, namely LϷLg . similar behavior with the numerical ratio of l and L vary- The dipole orientations in LH1 and LH2 imply that the ing from 1.0 to 3.3 depending on the model and the value of fluorescence should vanish if all monomers emit coherently 61 l . This has a very important implication for analyzing ex- ͑i.e., LϭL0). The factor Ls can then be small in two perimental data on the superradiance enhancement factor Ls cases: either LӶL0 where we have LsϷL ͑since the di- 22 in light-harvesting complexes. poles emitting coherently are almost parallel͒ or L0ϪLӶL0 In all four cases shown in Fig. 8͑a͒, the wave functions where we have LsϷL0ϪL and the signal is proportional to are nonoscillatory and l is the only factor which determines the measure of coherence destruction. This means that Ls is L . This situation is changed as the temperature is increased. not necessarily proportional to l . For example, in calcula- 61 It follows from Eq. ͑30͒ that both the amplitude and the tions performed for LH2 lϷ6, LϷ15 and LsϷ3 which phase of thermally populated exciton wave functions affect satisfies LsϷL0ϪL . This could explain the difference be- L . At low temperatures only excitons at the bottom of the tween low-temperature superradiance of LH1 (LsϷ9) and band with nonoscillatory wave functions are populated. LH2 (LsϷ3) observed in Ref. 22 despite close values of
J. Chem. Phys., Vol. 107, No. 10, 8 September 1997
Downloaded 08 Mar 2001 to 128.151.176.185. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html 3888 Meier et al.: Antenna complexes parameters for both complexes. The differences in Ls could motions exciton localization affects transport properties by be attributed to different geometric factors ͑and window increasing the depolarization timescale. functions͒ resulting from the different number of pigments Static localization of excitons also affects the tempera- 24,63 (L0ϭ32 versus L0ϭ18) in these complexes. ture dependence of fluorescence decay and pump-probe signals9,25,27,61,62 in a more delicate way. The effect may be rationalized using the Mott picture of Anderson localization VI. DISCUSSION which implies that localized states with close energies must be spatially separated.66 This leads to a qualitative picture Spectroscopic properties of J-aggregates24,25,27,62,63 and whereby the aggregate can be divided into noninteracting light-harvesting complexes3,4,9,14,22,64 are often related to a segments whose sizes are equal to the localization length. coherence size known as the exciton delocalization length The signals from a large disordered aggregate can therefore ͓cf. Eq. ͑56͔͒. In the absence of disorder and exciton-phonon be related to the signal from a homogeneous aggregate interaction, the exciton states are delocalized over the entire whose physical size is given by the localization length. aggregate, and the exciton delocalization length is equal to It has been shown in Ref. 61 that polaron formation the aggregate size. Since in infinite one-dimensional disor- ͑self-trapping͒ and static localization lead to similar features dered systems all exciton states become localized,31 static in the pump-probe signal for long time delays. In both cases disorder induces a finite ͑Anderson-type͒ exciton delocaliza- the shift between positive ͑induced-absorption͒ and negative tion length. Effects of exciton localization on optical signals ͑bleaching͒ peaks in the differential absorption reflects the of disordered aggregates can be rationalized by examining energy difference between the lowest and the next lowest properties of individual states and their contributions to l . excitons. This is in turn related to the polaron binding energy Effects of nuclear motions on the exciton localization E p ͑i.e., the binding energy of the self-trapped exciton͒ in the length have been denoted dynamical localization.14,62 Dy- dynamical case, and is given by the minimal energy splitting namical localization is much more complex than its static between exciton states located in the same region in space.67 counterpart, since excitons are coupled to a large ͑often mac- For weak exciton-phonon coupling61 the pump-probe signal roscopic͒ number of nuclear degrees of freedom. The only was related to the one-exciton Green function, and the shift practical way to introduce an exciton delocalization length in in the pump-probe signal was related to the exciton mean- this case is through observables which involve exciton vari- free path which plays the role of the coherence size for the ables. Such observables can always be represented in terms pump-probe signal. of Green functions of exciton operators36,49,65 and in some The exciton delocalization length can show up directly cases in terms of reduced exciton density matrices ͑which in transport processes. Although aggregate sizes are usually constitute a particular case of Green functions͒. Despite this smaller than the optical wavelength, nonuniform population formal difficulty which can be overcome by applying the distributions are obtained by optical excitation of light- Green function formalism, it is clear that there is a substan- harvesting complexes with polarized light due to dipole ori- tial physical difference between static and dynamical local- entations. Population relaxation can be observed by depolar- ization: in aggregates with exciton-phonon coupling but ization measurements.3,4 A model in which an excitation is without static disorder excitons are self-trapped33,34 rather localized on a dimer due to static disorder has been used3,4 to than localized. This implies that a natural measure for the fit energy transfer experiments. The possibility of dynamical exciton dynamical delocalization will be the polaron size l p contributions to exciton localization in the context of trans- defined in Sec. IV as the size of the self-trapped exciton port phenomena has been discussed.14 A theory of energy K wave function ⌿mϪn which enters the expression for the transfer in molecular aggregates with strong exciton-phonon polaron wave function ͓Eq. ͑41͔͒. The main difference be- coupling using canonical transformations in the joint tween static localization and dynamic ͑self-trapping͒ local- exciton-phonon space has been developed.45,68 The role of ization is that a localized exciton is confined in a certain polaron effects on the energy transfer has been discussed for region in space whereas the exciton density in the polaron the cases of small and large polarons.45,69 An application of wave function is spread over the entire aggregate. However, transport theories will be to connect the exciton delocaliza- K since the wave function of the self-trapped exciton mϪn is tion size due to coupling to nuclear motions with the depo- localized, self-trapping and localization have common fea- larization timescale observed in Refs. 3 and 4. An interesting tures and similar effects on some spectroscopic measure- attempt to introduce a coherence size related to transport by ments. means of the Green function techniques in Ref. 64 for a In the remainder of this section we discuss how static system with static disorder and exciton-phonon coupling has and dynamical localization show up in various spectroscopic been treated within the Redfield theory44,70 with the Redfield techniques. Exciton localization naturally leads to a decrease relaxation operator taken in the Haken-Strobl form.71,72 (0) of the superradiance factor in the time-resolved emis- However, the Green function Gii,ii used in Eq. ͑39͒ of Ref. sion20,22,24,61 and affects energy transfer which affects in de- 64 to define the participation ratio is related to a model with polarization measurements.3,4,64 In particular, in the absence a finite lifetime rather than homogeneous dephasing, for of nuclear motions no transport should take place in a system which most calculations have been made. This can be clearly with localized excitons, and the fluorescence will retain its seen from Eq. ͑40͒ of Ref. 64 which shows the same rate ⌫ initial polarization at all times. In the presence of nuclear for the decay of populations and coherences. Thus the coher-
J. Chem. Phys., Vol. 107, No. 10, 8 September 1997
Downloaded 08 Mar 2001 to 128.151.176.185. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html Meier et al.: Antenna complexes 3889 ence size defined in Ref. 64 is related to exciton lifetime and can be viewed as a bilinear form on V0 and all eigenstates of not to pure dephasing processes due to exciton-phonon cou- M viewed as a linear operator M:V V with nonzero eigen- → pling. values belong to V0. M can be diagonalized on V0 in an In this paper, utilizing the DW representation we have orthogonal basis set represented by ␣ , ␣ϭ1, 2, 3. We then related the time-resolved fluorescence signal to the reduced denote for ␣ϭ1, 2, 3 exciton density matrix. Superradiance is usually attributed to coherent emission of a set of electronic dipoles confined d␣ϭ͚ dm␣͑m͒. ͑A3͒ within a region of a size much smaller than the wavelength m of the emitted light. This coherence can be partially or com- In the notation of Sec. II, ␣ are the superradiant states with pletely eroded by interactions with additional degrees of transition dipoles d␣ . The relations d␣•dϭ0 for ␣ Þ  freedom such as nuclear motions and static disorder. The 2 and f ␣ϭ͉d␣͉ follows from the fact that M is diagonal in the density matrix further allows us to visualize various mecha- basis set of superradiant states and using Eqs. ͑A1͒ and ͑A2͒. nisms by staying in the exciton phase space ͑as opposed to the joint exciton-phonon space͒. This is impossible using a APPENDIX B: WINDOW FUNCTION FOR DIMERIZED wave function description. The calculations presented here CIRCULAR AGGREGATES clearly show that the coherence size of the density matrix is the natural lengthscale which controls cooperativity in spon- In this Appendix we present expressions for the window taneous emission. Calculations based on eigenstates are nu- functions for a circular aggregate with two molecules in a merically expensive and often do not lead to a clear insight. unit cell. Let ⌰ j , jϭ0,1 be the angles which molecular di- Since properties of individual eigenstates are usually highly poles in a unit cell form with the aggregate plane and j are averaged and lost in observables. For example, the thermal the angles between their projections into the plane and the density matrix at high temperatures is localized, although the circle. A simple calculation yields: individual exciton states are fully delocalized! The destruc- M tion of superradiance is determined by the density matrix. It 2mϩ j,2nϩk is hard to envision this effect in a unified fashion for various 2 models by examining individual eigenstates. Our calcula- ϭd ͭ sin ⌰ j sin ⌰kϩcos ⌰ j cos ⌰k tions demonstrate that the exciton localization length l due to static disorder and the size of self-trapped exciton l in the 2 p ϫcos ͑2mϪ2nϩ jϪkϩ Ϫ ͒ , ͑B1͒ case of strong exciton-phonon-coupling have similar signa- ͫ L j k ͬ 0 ͮ tures in the reduced exciton density matrix and superradi- with m, nϭ0,1,...,(L Ϫ2)/2 and j,kϭ0, 1. ance. 0 The superradiant states can be evaluated using the ap- proach of Appendix A and have the following form: ACKNOWLEDGMENTS L Ϫ1/2 0 2 2 We wish to thank R. van Grondelle, A. A. Muenter and ⌽3͑2mϩ j͒ϭ ͑sin ⌰0ϩsin ⌰1͒ sin ⌰ j , E. I. Rashba for useful discussions. The support of the Air ͫ 2 ͬ Force office of scientific research, the National Science Ϫ1/2 L0 ⌽ ͑2mϩ j͒ϭ ͑cos2 ⌰ ϩcos2 ⌰ ͒ Foundation Center for Photoinduced Charge Transfer, and 1,2 ͫ 2 0 1 ͬ the National Science Foundation through Grants No. CHE- 9526125 and No. PHY94-15583 is gratefully acknowledged. 2 ϫcos ⌰ exp Ϯi ͑2mϩ j͒ϩ , ͑B2͒ j ͫ L jͬ ͭ 0 ͮ APPENDIX A: THE SUPERRADIANT STATES with the corresponding oscillator-strengths In this Appendix we prove the statements in Sec. III L made with regard to the eigenstates of the window function ¯ 2 0 2 2 f 3ϭd ͑sin ⌰0ϩsin ⌰1͒, M. To that end we define a linear map D:V R3 from the 2 → L0 dimensional space of real one-exciton functions to the L0 space of real 3-dimensional vectors by ¯f ϭ¯f ϭd2 ͑cos2 ⌰ ϩcos2 ⌰ ͒. ͑B3͒ 1 2 4 0 1
Dϭ dm͑m͒. ͑A1͒ ͚m APPENDIX C: DENSITY MATRIX OF EXCITONS COUPLED TO PHONONS The matrix M viewed as a bilinear form of V can be written in a form: In this Appendix we discuss perturbative treatments of exciton-phonon coupling on the reduced exciton density ma- M ,⌿ ϭD D ⌿ . ͑A2͒ ͑ ͒ ͑ ͒• ͑ ͒ trix. For weak exciton-phonon-coupling the reduced exciton 3 Since R is 3-dimensional and V is L0-dimensional there is density matrix adopts the Boltzmann form in the exciton a(L0Ϫ3) dimensional subspace V1ʚV with D(V1)ϭ0. Let basis set ͓cf. Eqs. ͑29͒ and ͑30͔͒ at long time. This is no V0 be the 3-dimensional orthogonal component to V1, then longer the case as the exciton-phonon coupling is increased. M(,⌿) Þ 0 only if both ,⌿V0 , which means that M This can be shown using perturbative arguments. We con-
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Downloaded 08 Mar 2001 to 128.151.176.185. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html 3890 Meier et al.: Antenna complexes sider the generalized master equation ͑GME͒ for the evolu- ˆ ex † tion of the reduced exciton density matrix which can be H ϭϪJ Bn͑Bnϩ1ϩBnϪ1͒, ͑D2͒ mn ͚n derived using the projection operator techniques.38,43 The kernel of the GME can be explicitly calculated perturbatively ˆ ph † in the exciton-phonon-coupling. The long-time value of mn H ϭប0͚ bnbn , ͑D3͒ is then the stationary point of the GME n
ˆ exϪph † † Ϫi͓h,͔mnϪ Rmn,klklϭ0, ͑C1͒ H ϭgប0͚ BnBn͑bnϩbn͒ ͚kl n where the matrix hijϭ⍀i␦ijϩJij(1Ϫ␦ij) represents one- † † 44,70 ϩ ប0 ͓BnBnϩ1͑bl ϩbl͒͑␦nϩ1,lϪ␦nl͒ exciton Hamiltonian and Rmn,kl is the Redfield tensor, 2 ͚nl given by the zero-frequency value of the GME kernel. When † † the Redfield tensor is calculated to lowest ͑second͒ order in ϩBnBnϪ1͑bl ϩbl͒͑␦nlϪ␦nϪ1,l͔͒. ͑D4͒ exciton-phonon coupling, Eq. ͑C1͒ yields the Boltzmann dis- 70 Here ͉0͘ is the vacuum state for both the exciton and the tribution of bare excitons given by Eqs. ͑29͒ and ͑30͒ for † † phonon degrees of freedom, and Bn (bn) creates an exciton mn . Finding mn from Eq. ͑C1͒ with the Redfield tensor R calculated in the next ͑fourth͒ order in coupling ͑phonon͒ on site n. 0 is the Einstein phonon frequency, J is mn,kl the exciton transfer integral between nearest-neighbor sites, strength will show a deviation of mn from the Boltzmann distribution. and g () is the diagonal ͑off-diagonal͒ exciton-phonon cou- Another perturbative argument can be made as follows: pling strength. The original Holstein model contains only diagonal exciton-phonon-coupling. The Hamiltonian Eq. assuming that at long times the full density matrix M of the joint exciton-phonon system adopts the canonical form, the ͑D1͒ can be obtained from Eq. ͑1͒ by expanding molecular reduced exciton density matrix can be represented as frequencies ⍀n(q) and intermolecular couplings Jmn(q)up mn to first order in nuclear coordinates q. We also assume the † mnϭTr͑BmBnM͒. ͑C2͒ nearest neighbor form of intermolecular couplings Jmn(q) and Einstein phonons. The r.h.s. of Eq. ͑C2͒ can be evaluated perturbatively in the Following Toyozawa,50 we construct the one-exciton exciton-phonon coupling strength, e.g., by applying the Mat- wave function subara Green’s function techniques.65 The zero-order result yields Eq. ͑30͒ whereas the second-order contribution results in deviations of from the bare Boltzmann distribution. ͉K͘ϭ eiKn K B† exp Ϫ ͑K b† mn ͚ ͚ n1Ϫn n1 ͚ n2Ϫn n2 Deviation from the Boltzmann distribution of bare exciton at n n1 ͫ n2 long times for weak exciton-phonon-coupling has been dem- K onstrated in Ref. 23. However, the non-Boltzmann Ϫn *Ϫnbn ͒ ͉0͘. ͑D5͒ 2 2 ͬ asymptotic behavior of the exciton distribution originates from neglecting environmental e.g., solvent nuclear modes K ͑ ͒ The variationally optimized parameters n describe the pho- in the model: The system does not have enough phonons to K non distortion of site n, and n are the corresponding exci- equilibrate. Adding solvent modes with a continuous spec- ton amplitudes. Equation ͑D5͒ is equivalent to Eq. ͑41͒ trum to that model will lead to a Boltzmann distribution of K where the phonon wave functions ͉⌳n ͘ are explicitly written bare excitons at long times within the formalism of Ref. 23, in terms of phonon coherent states which is based on perturbative in exciton-phonon coupling 43,73 calculation of the GME with memory. This implies that K K † K ͉⌳ ͘ϭexp Ϫ ͑ b Ϫ * bn ͒ ͉0͘ph . ͑D6͒ the results of Ref. 23 describe a situation with weak coupling n ͚ n2Ϫn n2 n2Ϫn 2 ͫ n2 ͬ to solvent modes for intermediate timescales when coupling to solvent modes is unimportant and exciton dephasing is Equation ͑D5͒ can be alternatively represented in induced by the aggregate intermolecular and intramolecular momentum-space as modes. Ϫ1/2 i͑KϪk͒n K † ͉K͘ϭL0 e k Bk ͚nk
APPENDIX D: TOYOZAWA’S ANSATZ FOR THE Ϫ1/2 K Ϫiqn † K iqn POLARON WAVE FUNCTION ϫexp ϪL0 ͚ ͑q e bqϪq *e bq͒ ͉0͘, ͫ q ͬ In this Appendix we present the details of the Toyozawa ͑D7͒
Ansatz for polaron wave functions. We start with generaliz- † † K K † K where Bk , bq , k and q are Fourier transforms of Bn , bn , ing the Holstein Hamiltonian, also known as the molecular K K crystal model, by adding the linear off-diagonal exciton- n and n : phonon coupling: † Ϫ1/2 Ϫikn † ex ph exϪph Bk ϭL0 ͚ e Bn , ͑D8͒ Hˆ ϭHˆ ϩHˆ ϩHˆ , ͑D1͒ n
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Taking into account the translational symmetry, we may la- † Ϫ1/2 Ϫiqn † bqϭL0 e bn , ͑D9͒ bel with one index, i.e., ϭ . ͚n mn mϪn mn We now discuss the connection between adiabatic po- laron theory and variational Ansa¨tze. Equation ͑49͒ can also K ikn K k ϭ e n , ͑D10͒ be deduced using a time-dependent variational principle as ͚n soliton theory is usually implemented. Localized polaron Ansa¨tze, such as the Davydov Ansatz, K iqn K q ϭ e n . ͑D11͒ ͚n † † ͚ n͑t͒Bn͉0͘exexp Ϫ͚ ͑n͑t͒bnϪn*͑t͒bn͒ ͉0͘ph K n ͫ n ͬ The exciton amplitudes n serve as weights in a linear superposition of phonon coherent states if Toyozawa’s An- ͑D20͒ 74 satz ͑D5͒ is viewed in a different form: are often employed, and equations of motion are derived for the time-dependent parameters characterizing the exciton Ϫ1/2 iKn † ϪiKn1 K ͉K͘ϭL0 e Bn e n wave function and the phonon fields. Neglecting the time- ͚n ͚n 1 1 derivative of the phonon fields n as the adiabatic condition requires, Eq. ͑E8͒ is recovered, and the exciton wave func- ϫexp Ϫ ͑K b† ϪK* b ͒ ͉0͘. tion is found to follow Eq. ͑49͒. For large polarons where the ͚ n2ϩn1Ϫn n2 n2ϩn1Ϫn n2 ͫ n2 ͬ continuum approximation is justified, the static solution for ͑D12͒ the exciton wave function again assumes the soliton profile ͑50͒. The physical picture is that the exciton creates a static Equation ͑D12͒ can be obtained from Eq. ͑D5͒ by relabeling lattice deformation and then become self-trapped in the n as nϪn , and n as n. Defining the phonon part of the n 1 1 potential well that is established by the deformation. wave function as ͉⌽K͘ ͓the summation over n in Eq. n 1 The delocalized Toyozawa Ansatz employed in this ͑D12͔͒, work can be obtained by applying a delocalization projection operator K ϪiKn1 K K ͉⌽n ͘ϭ e n ͉⌳nϪn ͘, ͑D13͒ ͚n 1 1 1 ˆ Ϫ1 † † PϭL0 ͚ exp in KϪ͚ kBkBkϪ͚ qbqbq ͉K͘ is written in the compact form n ͫ ͩ k q ͪͬ ͑D21͒ Ϫ1/2 iKn K † ͉K͘ϭL0 e ͉⌽n ͘Bn͉0͘ex . ͑D14͒ to the Davydov Ansatz ͑D20͒. Within the Toyozawa Ansatz, ͚n K K a ‘‘locking’’ relation similar to ͑E8͒ is found for n and n We approximate the density matrix by Eq. ͑42͒, which is in the adiabatic regimes of the phase diagram for both small 46 well-justified in the low temperature regime ͑especially for and large polarons, although for weak exciton-phonon- intermediate and strong exciton-phonon-couplings a finite coupling nonadiabaticity emerges as the one-phonon plane- gap is formed between the one-exciton ground state and the wave from the background of the large adiabatic polaron as high-lying phonon continuum͒. Carrying out the trace over the coupling is decreased. It is worth noting that extended 75 the phonon bath, one arrives at states yield lower energies than the localized parent states. K K Ansatz ͑D5͒ with optimized n and n , and its generaliza- K Ϫ1 ϪEK iK͑mϪn͒ K K tion with the replacement of n Ϫn in ͑D5͒ by a two- mnϭ͚ ͗K͉K͘ e e ͗⌽n ͉⌽m͘, ͑D15͒ 2 K parameter sum ␣K ϩK , constitute the most sophisti- n2Ϫn n2Ϫn1 where cated variational wave functions applied to the Holstein Hamiltonian. We note that the delocalization procedure K K iK͑n Ϫn ͒ K K which transforms D20 into D5 preserves the locality of ͗⌽ ͉⌽ ͘ϭ e 1 2 * ͑ ͒ ͑ ͒ n m ͚ n1 n2 n1 ,n2 exciton-phonon correlations.
Ϫ1 K 2 iq͑n2Ϫn1͒ϩiq͑nϪm͒ ϫexp L0 ͚ ͉q ͉ ͑e Ϫ1͒ . ͫ q ͬ APPENDIX E: ADIABATIC POLARONS ͑D16͒ In this Appendix we evaluate the reduced exciton den- Noting that sity matrix mn using the adiabatic polaron model. To that end we represent the phonon Hamiltonian which enters Eq. ⌽K ⌽K ϭLϪ1 K K , ͑D17͒ ͗ n ͉ n ͘ 0 ͗ ͉ ͘ ͑1͒ in a form: we have H phϭTphϩV͑q͒, ͑E1͒ Ϫ1 nnϭL0 . ͑D18͒ where T ph is the nuclear kinetic energy and V(q) is the po- tential energy. We adopt the Born-Oppenheimer approach to Also is found to be real so that mn the polaron problem60,76,77 by representing the eigenfunc- mnϭnm . ͑D19͒ tions of the joint system in a form
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ϱ † d C͑͒ ⌽ϭ͚ ⌿␣͑n;q͒Bn͉0͘⌳␣͑q͒, ͑E2͒ ϭ . ͑E9͒ n ͵Ϫϱ 2 where ⌿␣(n;q) is the ␣th exciton wave function with en- Solutions of Eq. ͑49͒ are commonly referred to as solitons. ergy E␣(q). ⌳␣(q) is an eigenfunction of the phonon effec- The nonlinearity induced by the exciton-phonon coupling is eff tive adiabatic Hamiltonian H␣ defined as the mechanism supporting the soliton excitations. For the eff eff continuum model with the solution given by Eq. ͑50͒ the H␣ ϵTphϩV␣ ͑q͒, ͑E3͒ energy splitting between the lowest trapped exciton and the 2 34 with next one is given by /͉J͉. This implies that Eq. ͑E5͒ which is valid in the adiabatic limit is derived under the eff 2 V␣ ͑q͒ϵV͑q͒ϩE␣͑q͒. ͑E4͒ condition 0Ӷ /͉J͉. This equation leads to Eq. ͑E6͒ in the TӶ2/͉J͉ limit. Equation ͑E7͒ which follows from Eq. ͑E6͒ The adiabatic ansatz for the wave functions represented by 2 when Tӷ0 is hence valid when 0ӶTӶ /͉J͉. Eqs. ͑E2͒–͑E4͒ allows to represent the reduced exciton den- Neglecting fluctuations of collective bath coordinates in sity matrix in a form the vicinity of the minimum of Veff(q) we obtain a simple expression for mn given by Eq. 48. Equation. ͑48͒ can also ϭ dq * m;q n;q q mn ͚ ͵ ⌿␣͑ ͒⌿␣͑ ͒͗ ͉ be obtained using the Toyozawa ansatz and setting the ␣ K K Debye-Waller factor ͗⌳n ͉⌳n ͘ϭ␦nnЈ, which means that in eff Ј ϫexp͑ϪH␣ /T͉͒q͘. ͑E5͒ the adiabatic polaron theory lDWϭ0 ͑this reflects the fact that the traps related to left and right components of the If the energy splitting between trapped excitons is larger than density matrix have the same positions͒ and l p is the only the temperature, only the ␣ϭ0 term related to the lowest lengthscale which determines L . exciton needs to be retained in Eq. ͑E5͒ which yields
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