Transfer of Vibrational Coherence Through Incoherent Energy Transfer Process in Förster Limi
Tomáš Mančal, Jakub Dostál, Jakub Pšenčík, Donatas Zigmantas
aa r X i v : . [ phy s i c s . c h e m - ph ] N ov Transfer of Vibrational Coherence Through Incoherent Energy Transfer Process inFörster Limit
Tomáš Mančal , Jakub Dostál , , Jakub Pšenčík , and Donatas Zigmantas Faculty of Mathematics and Physics, Charles University in Prague,Ke Karlovu 5, CZ-121 16 Prague 2, Czech Republic and Department of Chemical Physics, Lund University, Getingevägen 60, 221 00 Lund, Sweden
We study transfer of coherent nuclear oscillations between an excitation energy donor and anacceptor in a simple dimeric electronic system coupled to an unstructured thermodynamic bath andsome pronounced vibrational intramolecular mode. Our focus is on the non-linear optical responseof such a system, i.e. we study both excited state energy transfer and the compensation of theso-called ground state bleach signal. The response function formalism enables us to investigate aheterodimer with monomers coupled strongly to the bath and by a weak resonance coupling toeach other (Förster rate limit). Our work is motivated by recent observation of various vibrationalsignatures in 2D coherent spectra of energy transferring systems including large structures witha fast energy diffusion. We find that the vibrational coherence can be transferred from donor toacceptor molecules provided the transfer rate is sufficiently fast. The ground state bleach signal ofthe acceptor molecules does not show any oscillatory signatures, and oscillations in ground statebleaching signal of the donor prevail with the amplitude which is not decreasing with the relaxationrate.Keywords: 2D coherent spectroscopy, vibrational coherence, coherence transfer, energy transfer
I. INTRODUCTION
Ultrafast time-resolved non-linear spectroscopy of elec-tronic transitions represents an indispensable tool for thestudy of photoinduced dynamic and kinetic processes inwide range of interesting molecular and solid state sys-tems [1–11]. While the degrees of freedom (DOF) of thestudied systems that are directly addressed in these ex-periments are electronic, fine details of the time resolvedspectra depend crucially on the characteristics of thesurrounding nuclear modes. It was recognized early onthat some ultrafast techniques yield unprecedented time-dependent information about the nuclear modes, even-though these might form a thermodynamic bath withbroad spectral density. For instance, the so-called pho-ton echo peakshift experiment yields almost directly theform of the energy gap correlation (also termed bathcorrelation function) [12–14]. With the advent of thetwo-dimensional (2D) coherent Fourier transformed spec-troscopy [15, 16], it was hoped shortly that one obtaineda method which would provide insight into electroniccoupling between chromophores forming molecular com-plex. Instead, the strength of the method proved to bein allowing observation of coherent oscillatory featuresof electronic and nuclear origin [4, 6, 17–23]. In partic-ular, large effort has been made to understand the vi-brational features revealed by the 2D spectroscopy andto contrast them to the electronic features [22, 24–27].Motivation for this work can be found in the ongoing de-bate on the origin of the long lived coherent oscillationsobserved in the 2D spectra of some molecular complexes[4, 6, 19, 28] for which models employing nuclear vibra-tional modes are also suggested [29–32]. Setting asidethis particular debate, the study of the vibrational fea-tures of 2D spectra has its own importance. The vibra-tional DOF are ubiquitous in molecular aggregates and almost all ultrafast time-resolved spectroscopic methodsare sensitive to them. Notable exception would be anideal frequency integrated pump-probe spectroscopy forwhich one can show that it is insensitive to the nuclear vi-brations [33]. 2D spectroscopy, however, can be shown tohave a specific sensitivity to vibrational features [34], andthe frequency resolved pump-probe spectroscopy with fi-nite pulses keeps sufficient sensitivity to vibrational fea-tures, too.Our theoretical work in this paper is motivated by ourrecent observation of coherent oscillations in low temper-ature 2D spectra of chlorosome [35]. Previous measure-ments by pump-probe technique on this system showedconvincingly that nuclear oscillations occur in the elec-tronic ground state after an ultrafast excitation [36]. Thechlorosome is the largest known bacterial photosyntheticantenna, containing ∼ aggregated chlorophylls whichare subject to large disorder in optical gaps. Correspond-ingly, a plausible explanation of the oscillations observedin 2D spectra at low temperature are nuclear oscillations,as these could survive the disorder in the electronic tran-sition energies. In our previous room temperature mea-surements of the chlorosome [11], we observed an initialultrafast time scale which could be associated with thefast diffusion of the excitation between certain coherentdomains formed in the disordered energetic landscape ofthe chlorosome. We are therefore motivated to study theinfluence of an ultrafast energy transfer on the evolutionof the nuclear oscillatory features in the 2D spectra.In order to understand the properties of nuclear os-cillations observed by 2D spectroscopy of photosyntheticenergy transferring systems, one needs to know spectro-scopic properties and time-dependent signatures of nu-clear vibrations of at least the two limiting cases in whichthese systems occur, namely the case of the Förster en-ergy transfer between spatially localized excitons and thecase of the delocalized Frenkel excitons. In photosyn-thetic energy transferring antennae, these cases never oc-cur in their pure form as many photosynthetic antennaefall in between the two limiting regimes. Nevertheless,when trying to understand experimentally observed be-haviors, it is useful to understand spectroscopic signa-tures which could be assigned these two limiting cases. Inthis paper, we start with the simpler of the two regimes,namely with the Förster energy transfer between spa-tially localized excitations. In regard to our previouswork, this corresponds to the energy transfer betweenthe coherent domains of the chlorosome [11].To simplify the theoretical treatment, we study aweakly coupled dimeric system which possesses one pro-nounced intramolecular nuclear vibrational mode withsufficient Huang-Rhys factor to be observable by elec-tronic non-linear spectroscopy, coherent 2D electronicspectroscopy in our case. We avoid the situation of a res-onance between the vibrational frequency and the donor-acceptor energy gap in which the selected intramolecu-lar nuclear vibrational mode would influence the energytransfer rate. This dimer interacts with its environment(solvent, proteins surroundings etc.). We ask the ques-tion whether the vibrational coherence excited by a se-quence of two short pulses (the first two of the threepulses of the 2D electronic spectroscopy technique) onone molecule (denoted as donor here) can be transferredto a neighboring molecule (an acceptor). We also studythe fate of the oscillations induced on the donor after itrelaxes to the electronic ground state when its excitationis transferred to the acceptor. The answers to these ques-tions could in principle shed some light on the behaviorof the coherent oscillations observed in chlorosomes.The paper is organized as follows. In the next sec-tion we introduce the model molecular Hamiltonian andwe discuss the model of system-bath interaction. Weintroduce the weak inter molecular coupling limit - theFörster limit. In Section III we derive response functionsfor all the signal components of the coherent 2D spec-tra of a dimer system with a weak resonance coupling,and we discuss their properties. In Section IV we dis-cuss particular numerical results and the dependence ofthe amplitude of the transferred vibrational coherence onthe energy transfer rates. We present our conclusions inSection V. II. MOLECULAR MODEL
In this work we consider a molecular dimer. We willdescribe unidirectional energy transfer from one moleculeto another. We therefore denote the molecule which weconsider to be the excitation donor by a letter D , whilethe other molecule (the excitation acceptor) will be de-noted by A . In actual molecular systems, both moleculescan be donors and acceptors, e.g. when they have similartransition frequency and the energy transfers in one orthe other way are close to equally probable. A general theory of excitation energy transfer has to account forthe possible back transfer of an excitation back to thedonor molecule. However, as will be shown later, theoscillation amplitude decays during the energy transfer,and it is enough to demonstrate this on a single step ofthe energy transfer. In systems in which the transitionfrequency of the acceptor molecule will be similar to thatone of the donor, such as in chlorosome, the probabilityof the back transfer will be diminished by the presenceof other possible neighbors. Most of the relevant conclu-sions about the likelihood of vibrational coherence trans-fer can therefore be reached from studying unidirectionalenergy transfer. The theory developed below applies di-rectly to a hetero dimeric system where back transfer issmall due to large energy difference between the donorand acceptor.We will describe the donor (acceptor) molecule in thedimer as a two-level system with electronic ground states | g D i ( | g A i ) and excited state | e D i ( | e A i ). To describe acomplete non-linear spectrum of such a system, it mightbe necessary to add even higher lying excited states orband of states | f D i ( | f A i ). The two molecules interactthrough a resonance coupling J so that the electronic ex-citation can be exchanged between them. This meansthat the electronic states of the individual molecules arenot eigenstates of the dimer. When, however, the interac-tion with the environment is stronger than the resonancecoupling, the identity of the molecules is approximatelypreserved. Any delocalization possibly established byresonance coupling is destroyed by fluctuations inducedby the environment. This is the limit of a strong bathinfluence, λ bath > J , where λ bath is the bath reorgani-zation energy - the parameter characterizing the system-bath coupling. In the strong system-bath coupling limit,the appropriate description of the energy transfer processis the Förster rate theory and its modifications [37–39].The Förster rate can be calculated from the known ab-sorption and fluorescence spectra of the donor and ac-ceptor molecules, respectively, and it carries a prefactorproportional to | J | . In this work we will be interestedin the influence the value of the rate has on the transferof the vibrational coherence. We will not calculate therates, rather, we will use them as a free parameter as-suming that different rates correspond to different valuesof J . The dynamics of the bath, which determines theline shape of the absorption and the fluorescence, will bespecified in terms of the bath correlation function, alsoknown as energy gap correlation function.The total Hamiltonian of the dimer reads as H = H bath + X i = A,D H (1) i + J ( | A ih D | + | D ih A | )+ H (2) , (1)where H bath is the Hamiltonian of the bath, H (1) i are theHamiltonians corresponding to the singly excited statesof non-interacting dimer H i = ( ǫ i + ∆ V i ) | i ih i | , i = A, D ,the states | A i and | D i are the singly excited states of thenon-interacting dimer | A i = | e A i| g D i , (2) | D i = | g A i| e D i , (3)and H (2) = ( ǫ A + ǫ B +∆ V A +∆ V D ) | e A i| e D ih e D |h e A | is theHamiltonian of the doubly excited state of the dimer. Ifneeded it may include the higher excited states | f D i and | f A i of the donor and acceptor, respectively. The energygap operator ∆ V A and ∆ V B describe the interaction ofthe acceptor and donor with the their surrounding envi-ronment, respectively. We set the ground state electronicenergy ǫ g to zero. We will ignore the doubly excited statein our treatment, because for weak coupling J (which isassumed in our Förster type treatment) the excited stateabsorption cancels with the ground state contributionswhich would otherwise lead to crosspeaks in 2D spectra[40].The description of the electron-phonon coupling(the coupling of the electronic states | A i and | D i with the bath of nuclear degrees of freedom) willbe done in terms of the bath correlation func-tions C A ( t ) = tr { ∆ V A ( t )∆ V A (0) W ( A ) eq } and C D ( t ) =tr { ∆ V D ( t )∆ V D (0) W ( D ) eq } which describe the fluctuationof the transition energy on the acceptor and the donor, re-spectively. The time argument on the energy gap opera-tors in the definitions of the bath correlation functions de-notes interaction picture with respect to the bath Hamil-tonian and W ( A ) eq ( W ( D ) eq ) denotes the equilibrium densityoperator of the acceptor (donor). For simplicity we willassume that their are equal, C A ( t ) = C D ( t ) ≡ C ( t ) , butthe fluctuations on different molecules remains uncorre-lated. The absorption and fluorescence spectra, as wellas non-linear optical spectra including 2D spectra can beexpressed using a double integral of the bath correlationfunction, so-called line-shape function [41] g ( t ) = 1 ~ t ˆ d τ τ ˆ d τ ′ C ( τ ′ ) . (4)We consider the bath correlation function, and corre-spondingly the line shape function, containing two com-ponents g ( t ) = g bath ( t ) + g vib ( t ) , (5)where g bath ( t ) describes the energy gap fluctuations dueto interaction with a large harmonic bath, and g vib ( t ) describes the contributions of underdamped oscillationsdue to individual intramolecular vibrational modes. Wechoose to treat the case of a single vibrational mode andwe use g osc ( t ) = λω (Θ( T )(1 − cos( ωt )) + i sin( ωt ) − iωt ) , (6)where Θ( T ) = coth( ~ ω/ k B T ) (see Ref. [41]). We willassume the bath to be characterized by Debye spectraldensity with some correlation time τ bath , and reorganiza-tion energy λ bath . The line shape function g bath ( t ) cor-responding to this bath correlation function is linear atlarge values of t > τ bath , i.e. g bath ( t ) ≈ αt . In particular D A | f D i| e D i| g D i | f A i| e A i| g A i ⊗ ⊗ | g D i h e D |h e D || e D i h e A || e A ih g D || g D i h g A || g A i| e A i h g A |h g A || g A i | g D i h e D |h e D || e D i h g D || g D i h e A || e A i h g A || g A i h e A || f A i h e A || e A i | g D i h e D |h g D || g D i h g A || g A i| e A i h g A |h g A || g A i R ( D )3 g R ( D )2 g − R ( D ) ∗ f − R ( D ) ∗ g ← e ) R ( A ← D )2 g ⊗− R ( A ← D ) ∗ f R ( A ← D )3 g | g D i h e D |h g D || g D i h g D || e D i h g D || g D i GSB | g D i h e D | | g D i h e D | | g D i h e D |h e D || e D i h e D || e D i h e D || e D i h g D || g D i h g D || e D i h g D || g D ih e D || f D i h e D || e D ih g D || e D i h g D || g D i SE ESA - GSBSE ESA GSB
Figure 1: The diagram of the levels of a weakly coupled dimersystem and the Feynman diagrams of the rephasing Liouvillepathways used in this paper. for the imaginary part we have Im g bath ( t ) ≈ − iλt at longtimes [41].The energy transfer rate K AD from the donor to ac-ceptor is in principle time dependent and it can be cal-culated with the help of correlation functions (line-shapefunctions) of the donor and acceptor [42]. We assumethat the part g vib ( t ) of the energy gap fluctuation doesnot participate on the transfer rate, and we assume thatthe rate becomes quickly time independent. Later in thispaper we study the dependence of various vibrational fea-tures in spectroscopic signals on the magnitude of therate K AD which we always assume to be constant. III. NON-LINEAR RESPONSE FUNCTIONSWITH ENERGY TRANSFER
Non-linear spectroscopic signals can be calculated con-veniently using the response function formalism [41]. Inthe following, we will use the theory developed in Ref.[43] to describe non-linear optical signal of an acceptor-donor system. In Ref. [43] the nuclear DOF are treatedvia second cumulant expansion of the response functions.For Gaussian baths, such as the bath of an infinite num-ber of harmonic oscillators coupled linearly to the elec-tronic transition, this approach leads to exact expres-sions for the response. Adding a single independent vi-brational mode, which is well pronounced in the non-linear response, allows us to model the transfer of thenuclear oscillations during energy exchange between themonomers.In Ref. [43] two two-level systems connected througha transfer rate K AD were studied. In our case we needto formally include excited state absorption to higher ex-cited state other than the two-exciton states discussedabove. However, this addition does not change the treat-ment of the energy transfer, because higher excited statesdo not contribute to the energy transfer dynamics ob-served by third order spectroscopic methods (such as 2Dcoherent spectroscopy) [41]. We will study the transferin one direction, from a donor molecule denoted as D toan acceptor molecule denoted as A . We will denote non-linear response functions corresponding to experimentalsignal by superindices ( D ) and ( A ) , or ( A ← D ) if the re-sponse contains energy transfer. We will show that thisprocess leads to a partial loss of the amplitude of theoscillations due to vibrational coherence.There are four types of double-sided Feynman dia-grams representing four general types of response func-tions. These are usually denoted by lower index n =1 , . . . , in the literature (see e.g. [41]). In addition, wedistinguish the pathways that include the ground state | g i and the excited state | e i only, and those involving thehigher excited state | f i . To the former we add a lowerindex g while to the latter we add a lower index f. Somepathways involve relaxation from the excited state | e i tothe ground state | g i and we denote them with a lowerindex ( g ← e ) . The list of rephasing pathways and theircorresponding Feynman diagrams are presented in Fig.1 In this section, without the loss of generality, we con-sider only the rephasing part of the response. All cal-culations presented later in Section IV will be based ona complete set of pathways. We assume that originallyonly the donor is excited, although in reality, the pro-cess where the role of the donor and the acceptor are ex-changed occurs simultaneously. There are seven rephas-ing Liouville pathways with the following interpretations:(i) The ground state bleach (GSB) signal from the donoris attributed to the pathway R ( D )3 g . During the populationtime, this contribution evolves due to the bath reorgani-zation and the oscillation of the vibrational mode, butit does not change due to the energy transfer process.(ii) The donor stimulated emission (SE) which exponen-tially decays with the energy transfer rate K AD is de-scribed by the response function R ( D )2 g multiplied by thecorresponding decay factor. (iii) When the donor decaysto the ground state due to energy being transferred tothe acceptor the bleach is filled with the decayed pop-ulation. This process is described by the response func-tion R ( D ) ∗ g ← e ) where the star denotes complex conjugation.This response function has an overall type of an R path-way, but it carries a minus and a complex conjugation.It rises with a factor (1 − e − K AD t ) . The minus leads tocancellation of the GSB contribution of R ( D )3 g . (iv) To-gether with the energy transfer there is a correspondingrise of the GSB of the acceptor. This is described bythe response function R ( A ← D )3 g and the rise is the sameas in pathway (iii). (v) and (vi) Similarly to the GSBalso the SE and the ESA of the acceptor rise. These pro-cesses are described by the response functions R ( A ← D )2 g and R ( A ← D )1 f , respectively. (vii) Finally, the ESA of thedonor is decaying in exactly the same way as its SE, andit is described by the response function R ( D ) ∗ f multipliedby the corresponding exponentially decaying factor. A. Transfer of the Excited State VibrationalCoherence
Using the matrix elements of the Liouville space den-sity matrix propagators for optical coherences U eg ( t ) ≡U egeg ( t ) and the ground- and excited state density matrixelements U ee ( t ) ≡ U eeee ( t ) and U gg ( t ) ≡ U gggg ( t ) , wherewe abbreviated the number of electronic indices to two,we can express the above discussed response functions inthe following way (see e.g. [40]): R ( D )2 g ≈ hU e D g D ( t ) U e D e D ( t ) U ge ( t ) i D e − K AD t , (7) R ( D )3 g ≈ hU e D g D ( t ) U g D g D ( t ) U g D e D ( t ) i D , (8) R ( D ) ∗ f ≈ −hU f D e D ( t ) U e D e D ( t ) U g D e D ( t ) i D e − K AD t , (9) R ( D ) ∗ g ← e ) ≈ − K AD t ˆ dτ hU e D g D ( t ) U g D g D ( t − τ ) × U e D e D ( τ ) U g D e D ( t ) i D e − K AD τ , (10) R ( A ← D )2 g ≈ K AD t ˆ dτ hU e A g A ( t ) U e A e A ( t − τ ) i A × hU e D e D ( τ ) U g D e D ( t ) i D e − K AD τ , (11) R ( A ← D )3 g ≈ K AD t ˆ dτ hU e A g A ( t ) U g A g A ( t − τ ) i A × hU g D g D ( τ ) U g D e D ( t ) i D e − K AD τ , (12) R ( A ← D ) ∗ f ≈ − K AD t ˆ dτ hU f A e A ( t ) U e A e A ( t − τ ) i A × hU e D e D ( τ ) U g D e D ( t ) i D e − K AD τ . (13)Here, we omitted the prefactors containing transitiondipole moments, and h . . . i D = tr bath { . . . W ( D )eq } cor-respond to the averaging over the equilibrium envi-ronmental DOF of the donor. The sign h . . . i A = tr bath { . . . W ( A )eq } represents the same averaging for theacceptor. Expressions for the standard pathways of Eqs.(7), (8) and (9) are well known from the literature, seee.g. [41]. The response of Eq. (10) will be treated inmore detail later. Let us first address the Eqs. (11) to(13). The averaging on the donor in these equations leadsto an elimination of the dependence on the variable τ inthe donor part of the response, because hU e D e D ( τ ) U g D e D ( t ) i D = tr bath {U e D e D ( τ ) U g D e D ( t ) W eq } = tr bath {U g D e D ( t ) W eq } . (14)This is because the superoperator U e D e D ( τ ) acting onan arbitrary operator A corresponds to an action of twoordinary evolution operators U e D e D ( τ ) A = U e D ( t ) AU † e D ( t ) , (15)and the operators can be reordered in a cyclic way underthe trace operation. In addition, in R ( A ← D )3 g the τ depen-dencecan be eliminated also on the acceptor part of theresponse, because U g A g A ( t − τ ) W ( A ) eq = W ( A ) eq (16)due to invariance of the equilibrium to the ground statepropagation. Correspondingly we have R ( A ← D )3 g ≈ e − g A ( t ) − g D ( t ) (1 − e − K AD t ) . (17)In the bleach signal there is therefore no transfer of anydependency of the response on t . Other transfer path-ways can now be expressed through the purely acceptorpathways which have a standard form, with the first timeargument (from the right) equal to zero R ( A ← D )2 g ≈ e − g D ( t ) K AD t ˆ dτ R ( A )2 g ( t , t − τ, e − K AD τ , (18) R ( A ← D ) ∗ f ≈ − e − g D ( t ) K AD t ˆ dτ R ( A ) ∗ f ( t , t − τ, e − K AD τ . (19)For the stimulated emission this leads to R ( A ← D )2 g ≈ e − g D ( t ) − g ∗ A ( t ) × K AD t ˆ dτ e i Im[ g A ( t − τ ) − g A ( t + t − τ )] − K AD τ , (20)which can be relatively easily evaluated. The ESA contri-bution depends on the particular assumptions we make about the higher excited states, and cannot therefore beevaluated without introducing further assumptions. Wecan consider this contribution to be similar in oscillatoryfeatures to the SE, because the source of the oscillationis the same excited state of the acceptor. B. Refilling of the Bleaching Signal
The ground state bleach contribution to the rephas-ing signal, which we denote by R ( D )3 g , appears stationary,and, at the first sight, unaffected by the energy transferprocess. However, the transfer process is accompanied bydeexcitation of the donor domain, and correspondingly,there is a signal countering the one of the R ( D )3 g pathway.Above we denoted this signal by R ( D ) ∗ g ← e ) above. Thediagram of this pathway has a form of the R diagrammirror imaged, and it is used with the minus sign. Ac-cording to Yang and Fleming, Ref. [43], this pathwayreads R ( D ) ∗ g ← e ) ( t , t , t ) = R ( D )3 g ( t , t , t ) e − i Im( g D ( t + t ) − g D ( t )) × K AD t ˆ d t ′ e − K AD t ′ e i Im( g D ( t + t − t ′ ) − g D ( t − t ′ )) . (21)The signal canceling the GSB contains the GSB responsefunction R ( D )3 g . Later in this paper, we will evaluateall the response functions numerically. Let us howeverattempt a slightly more involved analysis of Eq. (21)assuming its long t approximation. Because the g bath and the g osc components of the total donor line-shapefunction are independent, one can factorize the responsefunction into the bath- and the vibrational parts. In ad-dition, the bath line shape function g bath ( t ) is linear atits arguments larger than τ bath and correspondingly the t ′ dependence on the bath part of the line-shape func-tion vanishes. With these assumptions, we can write for t > τ bath R ( D ) ∗ g ← e ) ( t , t , t ) ≈ R ( D )3 g ( t , t , t ) e − i Im( g osc ( t + t ) − g osc ( t )) × K AD t ˆ d t ′ e − K AD t ′ e i Im( g osc ( t + t − t ′ ) − g osc ( t − t ′ )) , (22)i.e. the influence of the bath evolution is completely hid-den in the R ( D )3 g ( t , t , t ) function. The integral in Eq.(22) we see only contributions originating from the in-tramolecular vibrations.Integrating Eq. (22) by parts we obtain R ( D ) ∗ g ← e ) ( t , t , t ) ≈ R ( D )3 g ( t , t , t )(1 − e − K AD t e − i Im( g osc ( t + t ) − g osc ( t ) − g osc ( t )) ) − i λω R ( D )3 g ( t , t , t ) e − i Im( g osc ( t + t ) − g osc ( t )) × t ˆ d t ′ ω [cos ω ( t + t − t ′ ) − cos ω ( t − t ′ )] e − K AD t ′ e i Im( g osc ( t + t − t ′ ) − g osc ( t − t ′ )) . (23)Under the integral we used explicitly the form of the g osc , Eq. (6). At long times t , when e − K AD t ≈ , the donorcontribution to the overall signal consists only of the integral term S ( D ) ( t , t > K − AD , t ) = R ( D )3 g ( t , t , t ) − R ( D ) ∗ g ← e ) ( t , t , t ) ≈ − i λω R ( D )3 g ( t , t , t ) e − i Im( g osc ( t + t ) − g osc ( t )) × t ˆ d t ′ ω [cos ω ( t + t − t ′ ) − cos ω ( t − t ′ )] e − K AD t ′ e i Im( g osc ( t + t − t ′ ) − g osc ( t − t ′ )) . (24)We can see that the difference between the original GSBsignal and the signal coming from the filling of the GSBis proportional to the Huang-Rhys factor with a factor ina form of an integral over oscillating function. The wholefactor will obviously be an oscillating function of t . Itsnumerical analysis for the range of Huang-Rhys factorsbetween zero and one shows that its has a leading imag-inary contribution. Correspondingly, the GSB signal ona donor does not vanish after the excitation leaves thedonor molecule. The remaining signal is similar to theground state contribution R ( D )3 g multiplied by i and mod-ulated by oscillating real function. The numerical resultspresented in Section IV confirm this conclusion. C. Origin of the Bleach Signal
Interestingly, after an ideal impulsive excitation of amolecular system, no dynamic GBS (i.e. dependent on t ) would arise. This can be seen directly from the formof the response function while setting t = 0 , or from anintuitive picture summarized in Fig. 2. In terms of the re-sponse function, the beach corresponds to an excitationand then deexcitation of one of the sides of the Feyn-man diagram (see pathway R ( D )3 g of Fig. 1). The beat-ing which we observe in t is then the beating betweenthe static unexcited ground state wavepacket (or moreprecisely an equilibrium mixed state) and a perturbedwavepacket which was excited, evolved for a short time t on the excited state PES, and was then transferredback to the ground state into a non-equilibrium position.If the delay t = 0 , the wavepacket does not have time toevolve on the excited state PES and is returned back toits original equilibrium state. Hence no dynamical signalarises.During the excitation transfer, the GSB signal is com-pensated by an excited state population returning to theground state. In the non-linear response corresponding tothe excited states there are in principle two wavepackets(in the ket and the bra of the Feynman diagram) whichwere created in the excited state at two times occurringwith a delay t . These wavepackets are projected to theground state of the donor during the excitation transfer,and they are unlikely to have a phase as to cancel theoscillations that already occur in the ground state. It istherefore not surprising that the GSB signal will remainafter the excitation is transferred. In the next section wewill study the amplitude of these remaining oscillationsand its dependence on the value of the energy transferrate. IV. DISCUSSION
In this section, we will present results of numericalcalculations of 2D coherent Fourier transformed spectra.The definition of the 2D spectrum as well as the descrip-tion of the experimental technique can be found e.g. in[15, 44]. Throughout this section we use one parameters
D A K f K f K f E el + V | g D i| e D i | e A i| g A i Figure 2: Potential energy surfaces of the dimer model withFörster regime of energy transfer. The donor molecule D is initially excited and the excitation of is transferred withthe transfer constant K F to the acceptor molecule A . Thisprocess corresponds to a deexcitation of the donor and exci-tation of the acceptor with the same rate K F . The excitedstate wavepacket is projected to a non-equilibrium position ofthe ground state during the energy transfer to the acceptor. set for the bath correlation function, namely, λ bath = 200 cm − , τ bath = 100 fs. The temperature is assume tobe T = 300 K. These parameters ensure a realisticallybroad 2D spectrum on which a typical frequency of an in-tramolecular vibrational mode, ω = 150 cm − and an os-cillator reorganization frequency λ < ω lead to character-istic line shape modulation (see e.g. [21]). It is custom-ary to present the real part of the spectrum, which cor-responds to the absorption–absorption/stimulated emis-sion plots [40, 44]. Similarly to the situation in chloro-some, the presence of the vibrational mode does not leadhere to any discernible crosspeak, but the spectral am-plitude at its center, and the overall line shape are mod-ulated. Fig. 3 shows the characteristic oscillations ofthe amplitude of the 2D spectrum at the resonance, forthe donor molecule with no excitation energy transfer( K AD = 0 ). Fig. 4 presents the corresponding 2D spec-tra at some selected points (denoted by diamonds in Fig.3). All 2D spectra are represented with respect to the res-onant optical transition frequency, and they are thereforecentered around the (0 , point. On both figures we cannotice the initial drop of the amplitude and broadening,which occurs with the time-scale of the bath correlationtime τ bath . For delay times longer than τ bath the 2D spec-trum shows a characteristic pattern of line shape oscilla-tion which corresponds to an oscillation of the amplitude.The oscillations are more pronounced with increasing re-organization energy of the oscillator, i.e. with increasingHuang-Rhys factor.When the energy transfer from the donor to the accep-tor is allowed, the total donor signal is quickly disappear- R e l a t i v e A m p li t ude t [fs] = 15 cm -1 = 30 cm -1 = 50 cm -1 = 100 cm -1 Figure 3: Oscillations due to underdamped vibrations of nu-clear mode with frequency ω = 150 cm − for various reorgani-zation energies. The full blue diamonds denote the positionsfor which real part of the 2D spectrum is plotted in Fig. 4. -1000-500050010001500 [ c m - -1000 -500 0 500 1000 1500 [ cm -1 ] -1500 -1000 -500 0 500 1000 1500-1500-1000-500050010001500 [ cm -1 ] [ c m - ] t = 0 fs t = 100 fst = 200 fs t = 300 fs Figure 4: The time evolution of the real part of the 2D co-herent spectrum of a donor molecule in the absence of en-ergy transfer to an acceptor. The snapshots are taken at t = 0 , , and fs. The reorganization energy λ ofthe vibrational mode is cm − . The figures correspond tothe positions on a blue curve in Fig. 3 which are marked byfull blue diamonds. The figures are normalized to maximumof the 2D spectrum at t = 0 fs, and there are contoursbetween zero and the maximum. ing as the stimulated emission pathway decays exponen-tially and the ground-state bleach is compensated by thesignal from the population arriving to the ground statefrom the excited state. Fig. 5 shows, however, that thedonor signal remains oscillating around zero. The signalon the acceptor, on the other hand, rises to an amplitude R e l a t i v e A m p li t ude t [fs] T AD = 30 fs T AD = 50 fs T AD = 100 fs T AD = 150 fs T AD = 200 fs DonorAcceptor
Figure 5: Oscillations of the 2D spectrum at the optical res-onance. Amplitude of the spectrum at ω = ω is plotted forthe donor molecule and for the acceptor molecule for differentvalues of the transfer time T AD = 1 /K AD and the reorgani-zation energy λ = 100 cm − . The vibrational mode has afrequency ω = 150 cm − . The full black diamonds indicatethe positions for which the real part of the 2D spectrum isplotted in Fig. 6, and full back squares indicate the positionsfor which the real part of the 2D spectrum is plotted in Fig.7. corresponding to the one exhibited by the donor with-out energy transfer (cf. Fig. 3). One can notice by eyethat the amplitude of the oscillations decreases with theincreasing transfer time (decreasing transfer rate). Onecan also notice that this is not the case for the groundstate bleach. Characteristic 2D spectra of the donor con-tribution are presented in Fig. 6. Originally almost com-pletely positive signal quickly disappears on the timescaleof the energy transfer, and what remains is a signal witha characteristic shape of the imaginary part of the 2Dspectrum (see e.g. [21]), but with an alternating sign.The rising acceptor signal, Fig. 7, shows only the char-acteristic line shape modulations accompanied with thechange of the maximum amplitude of the spectrum.The relative amplitude of the coherent oscillationstransferred to the acceptor or left on the donor with re-spect to the amplitude expected on the non-transferringdonor is plotted in Fig. 8. We compare relative ampli-tudes for four different values of the oscillator reorgani-zation energy. We find that the transfer of oscillationsto the acceptor for all reorganization energies followsroughly the same exponential decay with the transfertime. This behavior is expected, because a slow feedingrate leads to more destructive interference between vari-ous contributions of the transferred nuclear wavepacket.For electronic coherence this effect was recently discussedin Ref. [45]. Similar behavior is observed when electroniccoherence is induced a multilevel molecular system bypulsed light with increasing duration as shown by Jiangand Brumer in Ref. [46]. This trend is however not fol- t = 50 fs -1000 -500 0 500 1000 1500 [ cm -1 ]-1500 -1000 -500 0 500 1000 1500-1500-1000-500050010001500 [ cm -1 ] [ c m - ] -1000-500050010001500 [ c m - ] t = 0 fst = 300 fs t = 400 fs Figure 6: Decay of the signal on the donor. The short timebehavior ( t = 0 and fs) corresponds to the bath dephasingand the loss of population. When the population is completelylost and the system has relaxed back to the groundstate elec-tronically, the mismatch between the returning excited statenuclear wavepacket and the groudstate wavepacket yields adispersion patern known from the imaginary part of the 2Dspectra. The dispersion patern changes sign with the periodof the oscillation. The snapshots correspond to the positionson the black curve in Fig. 5 which are marked by black di-amonds. The figures are normalized to maximum of the 2Dspectrum at t = 0 fs, and there are contours between zeroand the maximum. At times t = 50 , , and fs, we usefour times as many contours as in the t = 0 fs spectrum toenhance the small amplitude of the spectrum. lowed by the GSB signal remaining on the donor. Fig.8 demonstrates that the relative amplitude of the oscil-lations on the resonance does not decay with increasingtransfer time. It even slightly increases at short times,and it stays flat for times up to 200 fs. For longer transfertimes, the wavepacket stays for a longer time in the ex-cited state. Because the period of the oscillator is largerthan the transfer time, it apparently acquires a largeramplitude when it is projected on the ground state.Real molecular systems are studied by the non-linearspectroscopy in form of macroscopic disordered ensem-bles. The chlorosome which motivates our study of thetransfer of vibrational coherence is also a strongly disor-dered system. It is therefore worth studying the effectof the transition energy disorder on the oscillatory pat-ter observed in 2D spectra. In general, a change in thetransition energy results in a displacement of the 2D lineshape along the diagonal line of the spectrum. The dis-order thus corresponds to an additional elongation of theline shape along the diagonal line. For a line shape whichis positive everywhere, this does not mean any significantchange in the total amplitude of the spectrum (it should T = 450 fs -1000 -500 0 500 1000 1500 [cm -1 ] -1500 -1000 -500 0 500 1000 1500-1500-1000-500050010001500 [cm -1 ] [ c m - ] -1000-500050010001500 [ c m - ] t = 350 fs t = 450 fst = 550 fs t = 650 fs Figure 7: Time evolution of coherences transferred to theacceptor molecule. After the rise of the 2D signal is completed(see Fig. 5), the spectrum keeps evolving due the nuclearmotion in the excited state of the acceptor molecule. Thefigures are normalized to the maximum of 2D spectrum at t = 450 fs (largest amplitude of the four spectra), and thereare contours between zero and its maximum. get slightly diminished due to the broadening). For a lineshape corresponding to the times t = 300 fs and t = 400 fs in Fig. 6, however, the displacement may lead to ad-ditional canceling of the signal from differently displacedline shapes, because the line shapes contain both positiveand negative regions. In Fig. 9 we therefore study theinfluence of the disorder on the amplitude of the trans-ferred oscillations and on the amplitude of the remainingGSB signal. In our calculations we assume a large Gaus-sian disorder simulated by a normal distributions of thetransition energies with a full width at half maximum of cm − . In all studied cases the disorder leads to adecrease of the relative amplitude of the oscillations. Forthe oscillation transfer this decrease is less than 50 %.Similarly for the points of the 2D spectrum where theGSB has its maximum at t = 300 fs, the decay of theamplitude of the oscillations is less than 50 %. On thediagonal, i.e. near the nodal line of the 2D spectrum, theamplitude is decreased almost five times. Nevertheless,the donor signal remains oscillating despite disorder, andthe expected amplitude of the oscillations in comparableto or larger than the amplitude of the transferred oscil-lations.From the point of view of large aggregates the resultsconcerning the survival of the bleach signal are the mostinteresting. In system where each donor has a number ofacceptors, the excitation which started on a given donorwill be very rarely detected returning to the donor (afterit has passed through some other molecules). Even theprocess in which the excitation passed through its origi-
50 100 150 2000,00,10,20,30,40,50,60,70,80,91,0 R e l a t i v e A m p li t ude Transfer Time [fs] = 15 cm -1 = 30 cm -1 = 50 cm -1 = 100 cm -1 TransferRelaxation
Figure 8: Efficiency of the coherence transfer. Curves doted as“Transfer”: Ratio of the amplitude transferred to the acceptorto the original amplitude of the oscillations on the donor.Curves denoted as “Relaxation”: Ratio of the amplitude of theground state bleaching oscillations on the donor to the originalamplitude of the oscillations on the donor. All parameters aresame as in Fig. 5.
20 40 60 80 100 120 140 160 180 2000,00,10,20,30,40,50,60,70,80,91,01,1
Disorder Disorder R e l a t i v e A m p li t ude Transfer Time [fs]
GSB maximum GSB maximum, disorder GSB diagonal GSB diagonal, disorder Transfer Transfer, disorder
Disorder
Figure 9: Influence of electronic disorder on the relative am-plitude of the oscillations with frequency ω = 150 cm − andreorganization energy λ = 30 cm − . The disorder corre-sponding to a distribution of acceptor and donor transitionfrequencies with full width at half maximum (FWHM) equalto cm − was applied. The relative amplitude of thetransferred coherences (in blue) is diminished only slightly.At the diagonal of the 2D spectru, the groud state bleachingoscillations are reduced by more than a factor of (in red).However, when the amplitude is measured where the bleach-ing spectrum has its maximum (see lower panels of Fig. 6),the relative amplitude is decreased only by factor of , andthe beating on the relaxed donor remains larger than the oneon the acceptor. A. Outlook
The discussion in this paper is only a start of a moreextensive research program, which has to incorporate sev-eral important effects which were neglected here. First,one has to take into account the excitonic character ofthe states if one wants to draw some conclusions for thepossible vibrational coherence transfer in photosyntheticsystems in general. Also, various transition dipole mo-ment borrowing effects including formation of so-calledvibronic (vibrational-excitonic) excitons have to be con-sidered in case where resonance between the energy gapin the heterodimer and the vibrational frequency occurs.These two different effects amount to a study of differentmodel situation. In the present model, however, we havealso included several approximations which can be closelyinvestigated. For instance, we assumed constant energytransfer rates, i.e. we assumed certain coarse graining ofour problem in time. For fast vibrations, the relaxationrates might still be time dependent during several firstperiods of the vibrational motion after excitation. Therelaxation rates might also be dependent on the vibra-tional motion itself during this initial time interval. Such effects may change the situation both quantitatively andqualitatively in some cases, and will be studied elsewhere.
V. CONCLUSIONS
We have investigated the stimulated emission andground state bleach signals of a molecular dimeric sys-tem with pronounced vibrational modes. We have con-centrated on the transfer of the vibrational coherencebetween an excited donor molecule and its neighboringacceptor. Using response function formalism adapted forthe case of a resonant energy transfer with constant rates,we find that the nuclear oscillations can be transferred toa neighboring molecules. Their amplitude, however, de-cays with decreasing transfer rates. On the acceptor, theoscillations are solely due to the nuclear wavepacket inthe electronically excited state. Interestingly, the am-plitude of oscillations which prevail on the donor, afterit was deexcited due to energy transfer, do not decreasewith the decreasing energy transfer rate. Amplitude ofboth types of oscillations are decreasing in the presenceof electronic disorder. In systems where excitation trav-els away from the original donor, the dominating andsurviving contribution after many steps will be the oneoriginating from the electronic ground state of the donormolecule.
Acknowledgments
This work was funded by the Czech Science Foundation(GACR) grant no. 205/10/0989. [1] F. Milota, J. Sperling, A. Nemeth, T. Mančal, and H. F.Kauffmann, Acc. Chem. Res. , 1364 (2009).[2] Y.-C. Cheng and G. R. Fleming, Annu. Rev. Phys. Chem. , 241 (2009).[3] N. S. Ginsberg, Y.-C. Cheng, and G. R. Fleming, Acc.Chem. Res. , 1352 (2009).[4] E. Collini and G. D. Scholes, Science , 369 (2009).[5] D. B. Turner and K. A. Nelson, Nature , 1089 (2010).[6] E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P.Brumer, and G. D. Scholes, Nature , 644 (2010).[7] G. S. Schlau-Cohen, A. Ishizaki, and G. R. Fleming,Chem. Phys. , 1 (2011).[8] C. Y. Wong, R. M. Alvey, D. B. Turner, K. E. Wilk,D. A. Bryant, P. M. G. Curmi, R. J. Silbey, and G. D.Scholes, Nat. Chem. , 396 (2012).[9] D. B. Turner, Y. Hassan, and G. D. Scholes, Nano Lett. , 880 (2012).[10] O. Bixner, V. Lukeš, T. Mančal, J. Hauer, F. Milota, M.Fischer, I. Pugliesi, M. Bradler, W. Schmidt, E. Riedle,H. F. Kauffmann, and N. Christensson, J. Chem. Phys. , 204503 (2012). [11] J. Dostál, T. Mančal, R. Augulis, F. Vácha, J. Pšenčík,and D. Zigmantas, J. Am. Chem. Soc. , 11611 (2012).[12] M. Cho, J.-Y. Yu, T. Joo, Y. Nagasawa, S. A. Passino,and G. R. Fleming, J. Phys. Chem. , 11944 (1996).[13] M. Yang and G. R. Fleming, J. Chem. Phys. , 2983(1999).[14] T. Mančal and G. R. Fleming, J. Chem. Phys. , 10556(2004).[15] T. Brixner, T. Mančal, I. V. Stiopkin, and G. R. Fleming,J. Chem. Phys. , 4221 (2004).[16] T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E.Blankenship, and G. R. Fleming, Nature , 625 (2005).[17] P. Kjellberg, B. Bruggemann, and T. Pullerits, PhysicalReview B , 024303 (2006).[18] A. V. Pisliakov, T. Mančal, and G. R. Fleming, J. Chem.Phys. , 234505 (2006).[19] G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn,T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R.Fleming, Nature , 782 (2007).[20] D. Egorova, Chemical Physics , 166 (2008), cited By(since 1996): 8. [21] A. Nemeth, F. Milota, T. Mančal, V. Lukeš, H. F. Kauff-mann, and J. Sperling, Chem. Phys. Lett. , 94 (2008).[22] D. B. Turner, K. E. Wilk, P. M. G. Curmi, and G. D.Scholes, J. Phys. Chem. Lett. , 1904 (2011).[23] J. R. Caram, N. H. C. Lewis, A. F. Fidler, and G. S.Engel, J. Chem. Phys. , 104505 (2012).[24] T. Mancal, O. Bixner, N. Christensson, J. Hauer, F.Milota, A. Nemeth, J. Sperling, and H. Kauffmann, Pro-cedia Chemistry , 105 (2011).[25] V. Butkus, D. Zigmantas, L. Valkunas, and D. Abramav-ičius, Chem. Phys. Lett. , 40 (2012).[26] V. Butkus, L. Valkunas, and D. Abramavičius, ArXiv1205.3383v2 (2012).[27] J. Seibt and T. Pullerits, J. Phys. Chem. C , 18728(2013).[28] G. Panitchayangkoon, D. Hayes, K. A. Fransted, J. R.Caram, E. Harel, J. Wen, R. E. Blankenship, and G. S.Engel, Proc. Natl. Acad. Sci. U. S. A. , 12766 (2010).[29] N. Christensson, H. F. Kauffmann, T. Pullerits, and T.Mančal, J. Phys. Chem. B , 7449 (2012).[30] V. Tiwari, W. K. Peters, and D. M. Jonas, Proc. Natl.Acad. Sci. U. S. A. , 1203 (2013).[31] A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler,S. F. Huelga, and M. B. Plenio, Nature Physics , 113(2013).[32] A. Chenu, N. Christensson, H. F. Kauffmann, and T.Mančal, Scientific Reports , 2029 (2013).[33] J. Yuen-Zhou, J. J. Krich, and A. Aspuru-Guzik, J. Chem. Phys. , 234501 (2012).[34] T. Mančal, N. Christensson, V. Lukés, F. Milota, O.Bixner, H. F. Kauffmann, and J. Hauer, J. Phys. Chem.Lett. , 1497 (2012).[35] J. Dostál, T. Mančal, F. Vácha, J. Pšenčík, and D. Zig-mantas, submitted (2013).[36] Y. Z. Ma, J. Aschenbrücker, M. Miller, and T. Gillbro,Chem. Phys. Lett. , 465 (1999).[37] T. Förster, Annalen der Physik , 55 (1948).[38] H. Sumi, J. Phys. Chem. B , 252 (1999).[39] S. Jang, M. D. Newton, and R. J. Silbey, Phys. Rev. Lett. , 218301 (2004).[40] T. Mančal, in Quantum Effects in Biology , edited by M.Mohseni, Y. Omar, G. S. Engel, and M. B. Plenio (Cam-bridge University Press, Cambridge, 2013), Chap. Prin-ciples of Multi-Dimensional Electronic Spectroscopy.[41] S. Mukamel,
Principles of nonlinear spectroscopy (OxfordUniversity Press, Oxford, 1995).[42] S. Mukamel and V. Rupasov, Chem. Phys. Lett. , 17(1995).[43] M. Yang and G. R. Fleming, J. Chem. Phys. , 27(1999).[44] D. M. Jonas, Annu. Rev. Phys. Chem. , 425 (2003).[45] A. Chenu, P. Maly, and T. Mančal, arXiv:1306.1693(2013).[46] X.-P. Jiang and P. Brumer, J. Chem. Phys.94