Abstract
A vibronicexciton model is applied to investigate the recently proposed mechanism of enhancement of coherent oscillations due to mixing of electronic and nuclear degrees of freedom. We study a dimer system to elucidate the role of resonance coupling, site energies, vibrational frequency and energy disorder in the enhancement of vibronicexciton and groundstate vibrational coherences, and to identify regimes where this enhancement is significant. For a heterodimer representing two coupled bachteriochloropylls of the FMO complex, longlived vibronic coherences are found to be generated only when the frequency of the mode is in the vicinity of the electronic energy difference. Although the vibronicexciton coherences exhibit a larger initial amplitude compared to the groundstate vibrational coherences, we conclude that, due to the dephasing of the former, both type of coherences have a similar magnitude at longer population time.
Introduction
The observation of longlived oscillations in the twodimensional spectra of FennaMattewsOlson (FMO) pigmentprotein complex^{1} has renewed the general interest in quantum effects in lightharvesting and other biological systems. Although similar observations were reported previously^{2,3}, the interpretation of the oscillations as electronic coherence and the suggestion that such dynamical coherence may play a crucial role in achieving the high efficiency of electronic energy transfer in photosynthesis^{1} have ushered a wave of research on the role of coherence in excitation energy transfer processes.
Numerous theoretical articles have been devoted to understanding and explaining the longlived oscillations in FMO. They have shown that coherent oscillations last for several hundreds of femtoseconds, but under standard assumptions about the properties of the protein environment, none of these works have been able to account for the over picosecond long dephasing times of the oscillations observed experimentally^{4,5,6}. The only proposed mechanism which predicts electronic coherence with life times >1 ps is the correlation between environmental fluctuations on different pigments^{4,5,7,8,9,10,11}. Experimental confirmation of correlated distributions of pigment energies has been claimed for FMO^{12}, but neither dynamic nor static correlation of the pigment energies have been found in molecular dynamics (MD) simulations of the FMO protein environment^{13,14}. Furthermore, dynamic correlation, if present, would lead to an effective decrease of the system bath coupling strength, and to a corresponding slowdown of the energy transfer rates. Such a decrease of the systembath coupling strength is at odds with works which find an optimal strength of the system bath interaction for transport function within the parameter range used by standard theories^{14,15}. Also, the energy transfer rates obtained by standard theory are in a good agreement with experimental data^{16}, and the standard “funnel picture” of energy transfer^{17} seems therefore to be well supported.
The assumption that the experimentally observed coherences would be relevant to the biological function of the antenna complexes has also been criticized. Such a proposal requires the excitation with coherent superpositions of states created by lasers in the laboratory to be, in some way, equivalent to the excitations under in vivo conditions^{18}, i.e. by direct sunlight or via transfer from another antenna system. However, serious objections to this view have been raised in the literature, arguing that direct excitation by light from the sun (thermal light) does not lead to such coherent excitation^{19,20,21,22,23}. A recent experiment showed that even under coherent excitation of the chlorosome antenna, i.e. with femtosecond laser pulses, no coherent oscillations could be observed^{24}. This implies that any coherence induced by the excitation would decay quickly, and that the energy is transferred to the FMO complex in an incoherent fashion independently of the excitation conditions.
The above discussion suggests that the mechanism explaining the oscillations does not need to have a strong impact on the energy transfer dynamics or be relevant to the energy transfer efficiency. The problem should therefore be studied as a problem of oscillations in experimental signals rather than a problem of oscillations of electronic excitedstate populations as it is often presented. In other words, the origin of the oscillating experimental signal has to be understood before the signal can be translated into claims about the energy transfer dynamics. Recently, Christensson et al.^{25} proposed that the excitonic interaction between electronic and vibrational states in FMO serves to create vibronic states (excitonically mixed electronic and vibrational states). Such states have a considerable vibrational character, and, at the same time, have an enhanced transition dipole moments due to intensity borrowing from the strong electronic transitions^{26}. It was also shown that coherent excitation of the vibronic states produces oscillations in the nonlinear signal that exhibit picosecond dephasing times. Moreover, the concept of vibronic excitons provides a plausible explanation for the observation of correlated distribution of site energies^{12}. The vibronicexciton states involved in the longlived coherences are to a large extent composed of different vibrational states on the same pigment. This automatically leads to a correlation in the fluctuations of the involved transitions even for a random distribution of pigment energies.
The influence of vibrational degrees of freedom (DOF) on dynamics of excitonic systems has currently become an intensely studied topic. Recent research suggests the coupling between excitons and vibrations to have a key role in the dynamics of coherent light harvesting in cryptophyte algae^{27}. Also, the vibronicexciton model has been shown to provide, compared to the purely excitonic model, more realistic relaxation rates, as verified against experimental measurements in two cyanobacterial lightharvesting proteins^{28}. In addition, recent simulations suggest the existence of modedriven coherences, where an initial excitation of an underdamped bath mode is resonantly transfered to and sustains longlived (ps) coherent oscillations between electronic eigenstates^{29}.
In a recent work, Tiwari et al.^{30} used a model similar to the vibronicexciton model used here to show that the mixing of electronic and vibrational DOF leads to an enhancement of the excitation of vibrational coherences in the electronic ground state as well, and it was also argued that this effect can explain the longlived oscillations in FMO. In order to distinguish between the coherences observed in the excitedstate manifold and the ones originating in the groundstate manifold, we will assign the term vibrational coherence strictly to the latter groundstate contribution. There, the states are of pure vibrational origin although the ability to excite them is provided by the mixing of vibrational and electronic states in the excitedstate manifold. When referring to the excitedstate coherences with a strong vibrational involvement, we will use the terms vibronic, or more specifically vibronicexciton coherences.
In this paper, we turn to a model system (an FMOinspired molecular dimer) in order to systematically investigate how the mixing of vibrational and electronic DOF leads to longlived oscillatory signal in nonlinear optical spectroscopy. The paper is organized as follows. In the next section, we explain the mechanism of coherence amplitude enhancement on a toy model. Then, we introduce the model dimer system with the associated notation, and we define a measure of the vibrational character of a coherence. A section on 2D nonlinear spectroscopy introduces the thirdorder nonlinear signal resolved in the coherent 2D Fourier transformed spectroscopy and its relation to the dynamics of molecular systems, in particular to its coherent dynamics. After determining the basic properties of the enhancement for vibronic (excitonically mixed electronic and vibrational) and the groundstate vibrational coherences on an FMOinspired heterodimer, we discuss the results in context of FMO and the low frequency vibrational spectrum of BChla.
Results
Enhancement of coherence amplitude by transition dipole moment borrowing: a toy model
To illustrate the mechanism of enhancement of the coherence amplitude that is suggested in Ref. 25 and studied in detail in this paper, let us first consider qualitatively a rather trivial example. Let us imagine a system of two molecules, one of which has a forbidden transition to the excited state. In the absence of excitonic coupling between the excited states of the two molecules, it is not possible to excite a linear combination of the collective excited (eigen)states of the dimer (i.e. the coherence observed by nonlinear spectroscopy), because one of the transitions is forbidden. If we now switch on the interaction between the monomers, it becomes possible to excite the coherence between the new eigenstates of the system (see Fig. 1A for illustration). We assume the transition dipole moment in a form , where 1〉 = e^{(1)}〉g^{(2)}〉 is the allowed collective excited state of the dimer with the monomer 1 in its excited state e〉 and monomer 2 in its ground state g〉 and 0〉 = g^{(1)}〉g^{(2)}〉 is the collective ground state (here, the upper index (n) refers to the nth monomer). Expressed in the new eigenstate basis of vectors α〉, the transition dipole moment reads The new transition dipole moment to the excited eigenstates α〉 reads . As will become clear in the following sections, the amplitude of the contribution of a coherence between two states α〉 and β〉 to the 2D spectrum is proportional to the factor . In a dimer, we have only eigenstates 1′〉 and 2′〉, i.e. only one coherence, and we obtain Because 〈1′1〉^{2} + 〈2′1〉^{2} = 1, the maximum value of is equal to when the mixing is maximum, i.e. .
In a dimer, the coefficient 〈1′1〉 is given by a sine or cosine of the mixing angle , where J is the resonance coupling and ΔE is the difference between energies of the interacting states^{31}. When J is small, the mixing can only be substantial at the resonance, i.e. when ΔE → 0. The observation of coherence in 2D spectroscopy depends crucially on the ability to excite the two involved excited states simultaneously. In the above case, this is enabled by the resonance between excited states.
In Fig. 1B, we present another interesting case. We extend the previous toy model with vibrational states at each monomer, and assume both monomers to have allowed transitions to the excited states. Each state on the molecule (n) is characterized by its electronic manifold (e or g) and its vibrational level ν, e.g. . In Fig. 1B, only the first vibrationally excited state in the electronically excited state is represented (ν ≤ 1). The onephonon state of monomer 1 interacts weakly with the zerophonon electronically excited state of monomer 2. Because the coupling is small, the interaction between the two zerophonon states of the interacting monomers can be neglected, and the borrowing effect occurs only for the collective states 2′〉 and 3′〉 of Fig. 1B. If, in addition, we assume the transition dipole moments to the onephonon state to be small (i.e. the vibrational mode has a low HuangRhys (HR) factor), we obtain the situation similar to Fig. 1A. At resonance (i.e. at maximum mixing ), the coherence amplitude in 2D spectrum reads where is the transition dipole moment between the electronic excited state in the vibrational level ν and electronic ground state in the vibrational level ν′ of molecule n. This is to be compared with the amplitude of the purely vibrational coherence on a noninteracting monomer which is and which is small due the small HR factor. The excitonic interaction thus enhances a coherence which has partially both the vibrational and the electronic characters.
The situation studied later in this paper is similar to the model illustrated in Fig. 1B. We will study coherences involving the zero and onephonon states of the same vibrational mode on a given monomer. As demonstrated, the transition from the electronic and vibrational ground state to the onephonon state can be enhanced by excitonic coupling with the allowed states of other monomers. According to Ref. 25, this is the case for FMO, where the enhanced coherences were found to be of a prevailingly vibrational character, thus exhibiting significantly prolonged life time over the purely electronic coherences.
The excitedstate mixing between the vibrational states of one monomer and the electronic (zerophonon) state of the other monomers can, however, enhance also the groundstate bleaching signal^{30}, which was not included in our toy model. The lifetime of the coherence has not been discussed within the toy model either, because it is not relevant to the illustration of the enhancement mechanism. However, it will be of crucial importance throughout the rest of the paper.
Dimer hamiltonian: full formulation
In this paper, we concentrate on the relatively strongly coupled dimer of bachteriochloropyllsa (BChlsa) 3 and 4 of the FMO complex^{32}, which we will later refer to as monomer 1 and monomer 2, respectively. Needless to say, the interaction of the other BChls in the complex with this dimer is decisive for its function as a molecular wire, and it influences the properties of the oscillations observed in 2D spectra as well. In Ref. 25, it was demonstrated that the effect of longlived coherence could plausibly originate from the interplay between the vibrational modes local to BChl 3 and the resonance interaction between BChls 3 and 4. By reducing FMO to this dimer, we expect to be able to isolate the main quantitative contributions to the observed effect, and to simplify the treatment such that the main conditions for the effect to take place can be identified. The extension of the results to the entire FMO complex is addressed in the discussion.
The studied molecular system will be represented by a dimer Hamiltonian, explicitly treating a single intramolecular vibration (with frequency ω_{0}) on each of the monomers. Unlike in Ref. 25, we will first consider the formulation with an arbitrary number of vibrationalexcited levels, which we refer to as a full formulation (full with respect to the number of vibrational levels). For practical calculations, we will then reduce the number of vibrational levels to a single excited level on each monomer. This is justified by the low HuangRhys factors of the vibrations in chlorophylls^{33}. The dimer Hamiltonian in the electronic site basis is defined as where represents the identity operator on the Hilbert space of monomer n, and is the Hamiltonian of monomer n. Here and are the dimensionless momenta and coordinate operators of the vibrational mode on site n, respectively, d is the coordinate displacement, assumed identical for all monomers, and J_{nm} denotes the intersite coupling. The corresponding level model including the vibrational states and the associated notation is presented in Fig. 1C.
This system interacts with a bath of protein DOF which is modeled as an infinite number of harmonic oscillators characterized by some continuous spectral density, e.g. . The Hamiltonian representing the coupling between the bath modes characterized by the Hamiltonian , and the dimer reads as where the mode a = 0 has been excluded since it is explicitly treated in the molecular Hamiltonian. The first term in Eq. (6) corresponds to the linear interaction between the bath and the electronic DOF, and it can be expressed through the spectral density as ^{32}. The second term in Eq. (6) describes a bilinear interaction between the intramolecular mode and the bath. On the basis of experimental measurements showing lifetimes on the order of a few ps for the low frequency vibrational coherences in BChls^{34}, we will consider this interaction to be weak, and we remove any bathinduced dephasing of the vibrational modes by setting for all n, accordingly.
Let us now define the notation used throughout this paper to describe the dimer state. As in the standard electronic molecular exciton theory, the excited states of the dimer are formed from the excited states of the monomers. We denote the electronic excited states by a diad (nm) where n (m) stands for the state (g or e) of monomer 1 (2) in correspondence with Fig. 1C. The Hilbert space in which the molecular Hamiltonian will be represented can be fully described by states n_{ν}m_{ν′}〉, where the first (second) letter describes the first (second) monomer's electronic state, and the index ν (ν′) denotes its respective vibrational quantum level, so that e.g. e_{ν}g_{ν′}〉 = e^{(1)}〉ν^{(1e)}〉g^{(2)}〉ν′^{(2g)}〉. The eigenstates of the Hamiltonian will be written as linear combinations of the states n_{ν}m_{ν′}〉, considering the singlyexcited states only, i.e. The expansion coefficients obtained by diagonalization of the Hamiltonian will report on the participation of the particular local states in the eigenstates. The eigenstates will be numbered with increasing energy , α = 1,2,….
Measure of the vibrational character
To measure the vibrational character of a coherence, we can use the square of the expansion coefficients. We will be interested in the coherence between states e_{0}g_{ν′}〉 and e_{1}g_{ν′}〉. This coherence is a vibrational coherence located on monomer 1, and we ignore the vibrational state of the second monomer. Because, in the low temperature approximation, the groundstate vibrational quanta of the second monomer cannot be excited from the thermal ground state by a single interaction with the exciting pulse in the optical domain, we can set ν′ = 0 without loss of generality. We are interested in identifying which coherences between the eigenstates have the character of this local vibrational coherence, and we want to define accordingly a measure χ_{αβ} of the local vibrational character of the coherence ρ_{αβ}. For a given coherence to be vibrational on monomer 1, the state α〉 needs to have a character of the zerophonon state on the monomer 1, i.e. it has to be predominantly composed of the state e_{0}g_{0}〉, and the state β〉 has to have predominantly the character of the state e_{1}g_{0}〉 (or vice versa). This composition is measured by the square of the corresponding expansion coefficients and , which also define the probability of finding the system in the corresponding local states, should we attempt such a measurement. The character of the coherence corresponds to the conditional probability of finding the state e_{0}g_{0}〉 by measuring on state α〉, and of simultaneously finding the state e_{1}g_{0}〉 by measuring on state β〉, or vice versa. This leads us naturally to the definition This quantity has its maximum when the eigenstates are each entirely composed of the zeroor onephonon states of the excited molecule, and thereby correspond to the site basis states (χ_{αβ} = 1 for α〉 = e_{0}g_{0}〉 and β〉 = e_{1}g_{0}〉, or vice versa). When the states α〉 and β〉 are equal mixtures of these local states, so that , then we have . Therefore, coherences ρ_{αβ} with will be considered to have prevailingly local vibrational character. We compared the local vibrational character defined by Eq. (8) with the composition of the eigenstate vectors , and we verified that, regardless of the variable used to describe it, we reach the same conclusion about the character of a given coherence. Note that the character χ_{αβ} is a timeindependent quantity. It refers to eigenstates of the Hamiltonian and their representation in the basis of states local to the chromophores. The coherences have life times dependent on the interaction of the system with the bath, but their local vibrational character χ_{αβ} only depends on the system Hamiltonian and does not evolve in time.
2D nonlinear spectroscopy and Liouville pathways
Nonlinear spectroscopy has been widely used to study the dynamics of excitonic energy transfer in lightharvesting systems, in particular because it is sensitive to the time dependent redistribution of the populations among excited states. Details about this class of measurement techniques are outside the scope of this work, and they can be found elsewhere in literature (see e.g. Ref. 35,36,37). Twodimensional (2D) coherent spectroscopy, which is one of the recent additions to this class of techniques, has enabled us to directly observe the coherent components of the excitedstate dynamics^{38,39}. In 2D coherent electronic spectroscopy, the time dependent signal (the detection time is usually denoted as t_{3} here) is generated from the interaction of the sample with three consecutive ultrafast laser pulses with wave vectors k_{1}, k_{2} and k_{3} separated by time intervals t_{1} and t_{2}^{38,40}. The signal is heterodyne detected, i.e. the generated field rather than its intensity is measured, and it is spectrally resolved in frequency ω_{3} (t_{3} and ω_{3} are correspondingly related by Fourier transform). The measured field is then numerically Fourier transformed in time t_{1} (t_{1} → ω_{1}) resulting in a signal dependent on two frequencies ω_{3} and ω_{1} and one time delay t_{2}. 2D spectra are represented as frequencyfrequency correlation plots at various time delays t_{2}. The 2D signal is related to the thirdorder polarization of the molecular system generated by the exciting laser pulses, and its theoretical description is most conveniently based on the nonlinear response function formalism in the thirdorder of perturbation theory (see e.g. Ref. 41). In this section, we will briefly present the various signal components that can be distinguished in a 2D spectrum, and we will give some of their characteristics, such as positions in the 2D spectrum, oscillation frequency, initial and timedependent amplitudes.
Depending upon the time ordering of the lightmatter interactions in the thirdorder response functions, four different types of socalled Liouville pathways (denoted R_{i}, i = 1,…,4) can be distinguished in the generated signal. They are represented in Fig. 2A by doublesided Feynman diagrams^{41}. Pathways R_{1} and R_{4}, i.e. with an experimentally fixed negative t_{1}, are of the socalled nonrephasing character, while R_{2} and R_{3} are of the rephasing character. Rephasing and nonrephasing signals can be measured separately. The Liouville pathways can also be grouped together according to the electronic band in which the coherences observed in 2D spectra originate. In R_{1} and R_{2}, the observed coherences stem from the electronically excited state, while for R_{3} and R_{4}, they come from the ground state. Thus in both rephasing and nonrephasing pairs of Liouville pathways, we observe one ground and one excitedstate coherence. It is notoriously difficult to distinguish ground and excitedstateoriginated signal experimentally, unless one can find some secondary characteristics (such as signal life time due to relaxation or frequency) which distinguishes the ground from the excitedstate contributions.
Figure 2B illustrates the position, origin (excited or groundstate manifold) and oscillating frequency of the signal generated by the different pathways in the 2D spectrum. We characterize each pathway with the superscript (αβ, g_{ν}), where α and β refer to the eigenstates, and ν denotes which state, within the electronic groundstate manifold, is involved during the optical transitions. The nonrephasing pathways and result in peaks at the position () and (), respectively, whereas the rephasing pathways and generate peaks located in similar positions, i.e. (). It is interesting to note the role of the index ν in the pathways. For ν = 0, there are no groundstate coherences, while excitedstate coherences can appear. The frequency of oscillation of the peaks coming from and is ω_{αβ}, while it is given by the groundstate vibrational frequency for and . For ν > 0, the amplitudes of R_{1} and R_{4} pathways are equal, and the same holds for R_{2} and R_{3} pathways. Thus, if we would ignore the effect of the coherence life time, there would always be the same amplitude of the ground and excitedstate oscillating contributions (ν > 0), plus one contribution from the excited state (ν = 0).
Let us now discuss the (initial) amplitudes of oscillating features in 2D spectra. The initial (t_{2} = 0) amplitude of the thirdorder response function is proportional to a transition dipole moment prefactor of the response function R_{i} averaged over an isotropic distribution of orientations of the molecules in the sample. The dipole prefactors A^{(i)} for each Liouville pathway R_{i} are defined in the Supplementary Information (SI) available online. For the nonrephasing response function R_{1}, it reads in particular where e_{i} represents the unitary orientational vector of the i^{th} laser pulse, is the eigenstate transition dipole moment and 〈…〉_{Ω} stands for the averaging over an ensemble of randomly oriented complexes. The transition dipole moment on a same molecule but with different vibrational quantum number are assumed to follow the Condon approximation, i.e. . In ultrafast spectroscopy and assuming identical polarization for the four laser pulses, the disorder averaging can be accounted for by a constant orientation factor , where θ_{αβ} is the angle between the transition dipole moments of eigenstates α and β^{42}. The amplitude prefactor then reads Here, is the length of the transition dipole moment vector.
The timedependent response function is composed of the dipole prefactor and the timedependent part representing time evolution of the state of the molecular systems during the time interval between the interactions with light. The response functions R_{i} for all pathways are presented in detail in the SI. In particular, the nonrephasing response function R_{1} reads as where and are the evolution operators of the optical coherence and of the excitedstate coherence ρ_{αβ}, respectively. Here, 〈…〉_{Δ} denotes averaging over the energy disorder. The details of the evolution operators are given in the SI. The contribution of the corresponding coherence to the signal in the 2D spectra is obtained after Fourier transformation of the evolution propagators during the time intervals t_{1} and t_{3}: with and where ω_{1}, ω_{3} are the excitation and probing frequencies, respectively. The signal for the nonrephasing pathway R_{4}, which involves the electronic groundstate coherence can be obtained in the same way. Full expressions are given in the SI.
Application: FMOinspired heterodimer in one particle approximation
In order to quantify the enhancement of coherence amplitude by borrowing of transition dipole moment, as illustrated by the toy model, we study a simplified molecular dimer model. In this section, we will first study the initial amplitude of the dipole prefactor for selected coherences as a function of the molecule intersite coupling J and the energy gap ΔE. We will include one mode of nuclear motion (ω_{0} = 117 cm^{−1}) at each of the two sites. Then, we will investigate the dynamical behavior of the signal as a function of the dimer energy gap and the vibrational mode frequency. Possible consequences of our results for the case of FMO molecular aggregate will be addressed in the discussion.
As mentioned above, the studied dimer is inspired by the FMO lowest energy BChls 3 and 4 which are relatively strongly interacting (site coupling J_{0} = −53.5 cm^{−1}) and form a heterodimer with an energy gap ΔE_{0} = 110 cm^{−1} (later referred to as the reference energy gap)^{32}. The directions of the transition dipole moments were taken from the Protein Data Bank file 3ENI^{43} (we used θ_{Bchl 3–4} = 107.2°). A HuangRhys factor of S = 0.05 has been used to represent the generally small value experimentally measured in chlorophylls^{33}. In order to assess the influence of the nuclear modes on the dimer spectrum, we studied the dependence of the spectrum on selected model parameters for a single vibrational mode. We study two cases, with frequencies ω_{0} = 117 cm^{−1} and 185 cm^{−1}. For some parameters and spectral characteristics, we study the system properties as a function of the mode frequency in an interval with a width of roughly 200 cm^{−1}.
Only one vibrational level (ν_{max} = 1) of a single mode is treated explicitly in the molecular Hamiltonian. This is motivated mainly by the small value of the HuangRhys factor. We have verified that we obtain similar results including a larger number of vibrational quanta. The excitedstate part of the total Hamiltonian, Eq. 4, in the basis reads where E is the energy of the optical transition (E = 12210 cm^{−1}), V_{νν′} = J (FC)_{ν}(FC)_{ν′}, J is the coupling between the two BChls and (FC)_{ν} = 〈g_{0}e_{ν}〉 is the FranckCondon factor characterizing the overlap of the groundstate vibrational wavefunction in the electronic ground state with that of the ν^{th} vibrational level in the electronic excited state. The restriction to the vibrational ground state for the (electronically) nonexcited molecule follows from the oneparticle approximation, detailed below.
In this work, we study the nonrephasing pathways (R_{1} and R_{4}), and more specifically, their contributions to the signal on the diagonal of the 2D spectrum (cf. red signals in Fig. 2B). In particular, we focus on and with β = 2 for ω_{0} ≤ 135 cm^{−1} and β = 3 otherwise (states numbered with increasing energy), because, as will be shown below, they both involve the zero and onephonon levels local to molecule 1, either on its electronic excited or groundstate manifold. Comparison of the results for these two particular pathways will enable us to discuss the different characteristic of the vibronic and the groundstate vibrational coherences, thus taking into account both recent theories of the origin of the longlived oscillations in the 2D spectra of photosynthetic systems^{25,30}.
Unlike in Ref. 30, we concentrate on the oscillations of the diagonal peaks in the 2D spectrum, and more specifically on pathway for the groundstate contributions (see Fig. 2B). Because the state β〉 is initially excited on the monomer 1 and originates from the ground state of the dimer, it cannot contain an excitation of groundstate vibration on the other monomer. This allows us to stay in the oneparticle approximation, where the (electronically) nonexcited molecule is always considered to be in its vibrational ground state^{44}. Unlike for the crosspeak, the oneparticle approximation holds on the diagonal cut for all states β〉 accessible from the dimer ground state. In addition, excited state absorption signal in an excitonic dimer does not contribute to the diagonal of the 2D spectrum, unless population relaxation pathways are explicitly taken into account. The population relaxation is assumed to lead to a decay of coherences only, which is described by relaxation induced dephasing rates. Correspondingly, all effects of population relaxation on relative amplitude of diagonal oscillations are contained in the stimulated emission pathway, and the excited state absorption does not have to be explicitly treated to estimate their lifetime.
Influence of coupling and energy gap on the dipole prefactor
As detailed in the spectroscopy section, the initial amplitude of the signal is directly proportional to the dipole prefactor (see Eq. 9). Here, we study the influence of the energy gap ΔE and site coupling J on the dipole prefactor, with detailed results for pathways R_{1} involving the zerophonon level of the excited state on molecule 1 (i.e. eigenstate 1, mainly localized on e_{1}g_{0}〉). Figure 3 presents the relative transition dipole moment prefactor for the dimer response function involving electronically excited eigenstate β〉 = 2〉, 3〉 for ω_{0} = 117 cm^{−1}, respective to the prefactor corresponding to the vibrational oscillations on a monomer: Figure 3a shows results for the coherence involving the two lowest excited eigenstates 1〉 and 2〉. Compared to the monomer, the amplitude of the dipole prefactor in the heterodimer is clearly enhanced (up to more than 8 times) for an energy gap comparable to the vibrational energy. It should be noted that, in the range of very small couplings the excitonic model breaks down, and the electronphonon coupling might effectively destroy the excitonic mixing. For moderate resonance coupling values and increasing its value, the region for which the amplitude is enhanced through borrowing of transition dipole moment spreads over a wider range of energy gaps corresponding to a wider mixing region. Note that the studied excitedstate coherence has a prevailingly vibrational character only in the shaded area of Fig. 3a. Fig. 3b presents the amplitude of the dipole prefactor in the coupled dimer relative to the monomeric one for coherences ρ_{1β} as a function of the energy gap ΔE, using J_{0} = −53.5 cm^{−1}. We also present the local vibrational character χ of the coherences (cf Eq. 8). We can define a characteristic energy gap ΔE_{χ} = 94 cm^{−1} for which the coherences ρ_{12} and ρ_{13} are equally delocalized, i.e. χ_{12} = χ_{13}. The highest enhancement (~ 6.5 times higher than that in a monomer) is obtained for the coherence involving the two lowest excitons (1, 2). This coherence is of prevailingly vibrational character for ΔE > 99 cm^{−1}. The coherence ρ_{13} exhibits only a low enhancement (< 2) in the region where it is prevailingly vibrational (10 < ΔE < 88 cm^{−1}) and will therefore make minor contributions to longlived coherences in the 2D spectrum.
Amplitude of the 2D signal and life time of coherences
In this section, we study the response functions R_{1} and R_{4} involving the vibrational coherence in the electronic excitedstate and in the electronic groundstate manifolds, respectively. The calculations have been performed on the FMO inspired dimer at 77 K for various vibrational frequencies, using an overdamped Brownian oscillator mode bath (see e.g. Ref. 41) with the Debye frequency Λ = 130 cm^{−1} and reorganization energy λ = 35 cm^{−1}. Pigment energies were sampled from a Gaussian distribution with 80 cm^{−1} FWHM centered at the reference energy gap (ΔE_{0} = 110 cm^{−1}). The energy transfer rates that contribute to the dephasing of the coherences were calculated from the same energy gap correlation functions as the lineshape functions (see SI) using the standard Redfield theory. The theory is identical with the one used in Ref. 25.
In Fig. 4, we present the energy gap dependence of the diagonal cut (ω_{1} = ω_{3}) through the Fourier transform of the response functions R_{1} and R_{4} before accounting for the energy disorder: The energies of eigenstates 1 and 2 depend on both ΔE and ω_{0} (ω_{0} is fixed at 117 cm^{−1} here). There is a clear resonance occurring when the energy gap ΔE is comparable to the frequency of the nuclear mode, ΔE ~ ω_{0}. This resonance demonstrates the borrowing of dipole moment from the electronic to the vibrational transition, as suggested by^{25} and illustrated by the toy model. The oscillating signal amplitude also depends on the evolution propagator which now also depends on the mixing and therefore on ΔE. This dependence leads to an even larger enhancement of the oscillation amplitude (with respect to the one observed for the vibrational oscillations on a monomer) than the transition dipole moment prefactor alone would predict.
The complete theory of enhancement has to include averaging over the distribution of energy gaps to account for the energy disorder. Figure 5 presents the maximal amplitude of the signal from pathway R_{1} involving coherence ρ_{1β} relative to that of the purely vibrational coherence in a monomer (, defined below), as well as that from , along the diagonal cut of the 2D spectrum (ω_{3} = ω_{1}). The letter β denotes the exciton level which is composed mainly of the first vibrationally excited level located on molecule 1 (β = 2 for ω_{0} ≤ 135 cm^{−1} and 3 otherwise). represents the time of the first maximum after a time t′ (t′ = 0 or 1 ps in the following) and is the time at which the signal is maximum for nondecaying coherences. Figure 5a presents results for ω_{0} = 117 cm^{−1} (β = 2) as a function of the excitation frequency ω_{1} relatively to the frequency ω_{opt}, which corresponds to the absorption maximum (for ω_{0} = 117 cm^{−1}, ω_{opt} = 12165 and 12300 cm^{−1} for R_{1} and R_{4}, respectively). For both pathways, the signal amplitude in the coupled dimer is clearly enhanced compared to that of the corresponding monomer (maximum 5 times higher for R_{4}, and more than 16 times for R_{1}).
Figure 5a also shows the life time of coherence ρ_{12} involved in R_{1}, obtained from a fit of the calculated signal – the life time of the groundstate coherence (pathway R_{4}) is not displayed because the purely vibrational coherences are assumed nondecaying as in Refs. 25,30. Using ω_{0} = 117 cm^{−1}, the observed excitedstate coherence appears to survive for slightly more than 1 ps.
Figure 5b shows the maximum enhancement obtained at initial time (plain lines) and after 1 ps (dashed lines) for different vibrational frequencies (on the diagonal cut at ω_{1} = ω_{opt} in the 2D spectrum) for pathway R_{1}. Results for pathway R_{4} are presented in Fig. 5c. In order to account for the dependency of the eigenstate composition on the mode frequency and the resulting reordering of eigenstates at ω_{0} = 135 cm^{−1}, results are presented for coherences ρ_{12} and ρ_{13} involved in both pathways R_{1} and R_{4}. For R_{1}, the relevant domain in which the vibronic coherence is of prevailingly vibrational character (χ > 0.5), is highlighted with shaded areas in Fig. 5b. For pathway R_{4}, the groundstate vibrational coherence remains purely vibrational independently of the origin of the excited state which participates in the pathway. For both pathways, the enhancement is larger when coherence ρ_{12} is involved. Comparing Figs. 5b and 5c, it can be seen that, at initial time, the enhancement for pathway R_{1} is significantly (up to 3 times) larger than that for R_{4}. Also, for both pathways, there is a broad resonance in the enhancement in the vicinity of the energy gap value ΔE_{0} = 110 cm^{−1}, which is in accordance with the similarly broad resonance for the enhancement of the groundstate contribution for a crosspeak reported in Ref. 30. Because the excitedstate coherences involve mixing of electronic and vibrational states, their life time depends, among others, on the dephasing time of the excited states. Consequently, longlived (>1 ps) coherences will only be created for ω_{0} < 120 cm^{−1} for the pathway R_{1}. In the case of the pathway R_{4}, any nuclear mode will generate longlived coherences because the purely vibrational coherences decay slowly. Therefore for higher mode frequencies, longlived coherences should originate from pathway R_{4} only. We studied in detail the R_{1} pathway involving ρ_{12} using ω_{0} = 185 cm^{−1} (not presented here). We concluded that, although at initial time the signal is of comparable amplitude with ω_{0} = 117 cm^{−1}, it is almost absent after 1 ps due to the decay. The inset in Fig. 5c shows the frequency of the oscillating signals. It is verified that the groundstate vibrational coherences oscillate at the frequency of the nuclear mode, whereas the frequency of the vibronic coherences does not exactly match it due to the excitonic splitting effect and the resulting dependence on the site coupling and excitedstate energies.
Table I presents the contributions of the local basis excitations to the vibronic states averaged over energy disorder for ω_{0} = 117 cm^{−1}. Eigenstate 1 consists of 85% excitation of the ν = 0 transition on site 1 and 15% on site 2. Eigenstate 2 corresponds to 64% of the ν = 1 transition on site 1. This composition explains the results presented above, and confirms that prevailingly vibrational coherences exhibit a prolonged life time.
Figs. 6a and 6b show the evolution of the signal (at ω_{1} = ω_{opt}) resulting from pathway R_{1} and involving the prevailingly vibrational coherence for two different nuclear mode frequencies (ω_{0} = 117 and 185 cm^{−1}), with different energy gaps, namely the reference energy gap (ΔE_{0} = 110 cm^{−1}), the resonant condition ΔE = ω_{0}, and ΔE sampled from the Gaussian distribution centered on the reference energy gap (denoted by 〈…〉_{Δ} in Fig. 6). The fitted frequencies and life time of the signal are indicated in the figures, along with the vibrational character χ of the coherence. For the mode frequency ω_{0} = 117 cm^{−1}, we can expect longlived oscillations with large amplitudes for any of the presented energy gaps, because the latter is either equal to or neighboring the vibrational mode frequency. Figure 6a confirms this expectation. A different behavior is observed for ω_{0} = 185 cm^{−1}. Here, the amplitude of the vibronic coherence is significantly increased (from 0.4 to more than 10 times that in a monomer) when the energy gap corresponds to the frequency of the nuclear mode, which illustrates the resonance interaction between the two monomers. However, because the averaged energy is sampled around 110 cm^{−1}, the total signal only slightly benefits from the enhancement mechanism and its amplitude is therefore smaller compared to that obtained using ω_{0} = 117 cm^{−1}.
To sum up, the study of a model dimer has enabled us to identify the resonance between the electronic energy gap and the vibrational frequency as a crucial condition of the observation of longlived oscillations in 2D spectra. The enhancement mechanism lies in intensity borrowing by excitonic states with strong vibrational character (the vibronic excitons) from the strongly allowed electronic states. The vibronic states thus exhibit a significant intensity despite the small HuangRhys factor of their vibrational component. This mechanism leads to an enhancement of the initial amplitude of both vibronic and groundstate vibrational coherences over a bandwidth of about 100 cm^{−1}. However, the mixing of the electronic and vibrational character of the vibronic coherences implies that these coherences experience additional dephasing. The strong dependence of the lifetime enhancement on the mode frequency means that vibronic coherences will be observable at long population times only when the frequency of the vibrational mode is close to the electronic resonance (cf Figs. 5b and 6). For the present model, this dephasing leads to similar amplitude of the vibronic and groundstate vibrational coherences on a picosecond timescale despite the stronger enhancement of the initial amplitude of the former. It does not therefore seem to be possible to exclude one or the other type of coherence a priori based on amplitude arguments. Both mechanisms will be most likely observed simultaneously in an experiment. The most useful difference between the two types of coherences is the difference in oscillation frequency (cf inset of Fig. 5c). The vibrational coherence oscillates with a frequency which is equal to that of the vibrational mode, while the frequency of the vibronicexciton coherence is shifted due to the excitonic interaction.
Discussion
The detailed investigation of the interaction between nuclear and electronic DOF in a model dimer provides more insight into the mechanism of enhancement of the amplitude and life time of the oscillations seen in 2D experiments. These results serve as a verification of the mechanism proposed by Christensson et al.^{25}, and form a basis for the discussion of this model in relation to the one proposed by Tiwari et al.^{30}. We point out that despite technical differences, such as the subtraction of the symmetric linear combination of the nuclear mode coordinates in Tiwari et al.^{30}, the Hamiltonian used in these two works is the same, and so are the basics of the enhancement mechanism. The contribution of vibrational states to the optical signal is enhanced by their interaction with electronic DOF. The models differ only in the manifold in which the oscillations take place. Christensson et al.^{25} suggested the excited state, while Tiwari et al.^{30} studied the groundstate contribution. As we have illustrated above, the amplitude of both groundstate vibrational and vibronic coherences is significantly enhanced in a system of coupled molecules as compared to the isolated monomers. The recent 2D experiments on BChls in solution did not find any significant vibrational coherences^{45}. These results are in line with what one would expect based on the HuangRhys factors of the low frequency vibrational modes determined by previous experiments. However, in aggregates, an enhancement by factor of five at 1 ps was found for both excited and groundstate contributions, strong enough to elevate the signal above the noise level.
Another mechanism of modedriven coherences has also been recently proposed, where electronic coherences are assumed to emerge from resonant coupling with robust vibrational coherences^{29}. The constant transfer from the longlasting vibrational coherences is assumed to maintain longlived oscillations on the otherwise fast dephasing electronic coherences. However, one should point out that this study uses HuangRhys factors about one order of magnitude higher than in the present paper, and that vibrational transitions are not expected to be as weak as with the parameters used here.
To distinguish ground and excitedstate contributions to nonlinear optical signals is notoriously difficult, and to identify unique signatures of electronic, vibronicexciton and vibrational coherence is less than trivial^{29,46,47,48,49}. To distinguish between electronic and vibrational coherences, a recent research suggested the use of broadbrand pump/broadband probe spectroscopy, showing that oscillations as function of the waiting time originate from electronic oscillations in the singly excited manifold^{50}. Another promising proposal has been the use of specific polarization sequences^{51}, which could further be used to perform quantum process tomography^{52}. It has been shown that a specific combination of polarizations in the four wave mixing sequence can single out coherences involving transitions with nonzero angle in the molecular frame^{53,54}. When the vibrational modes are treated as members of the bath (i.e. standard exciton picture), vibrational transitions on a single pigment are always parallel, and the orientational average for pathways representing vibrational coherences will be zero. Therefore such polarization sequences are termed “coherence specific” sequences and have been assumed to only report on electronic coherences^{51,54}. However, when the vibrational modes are treated explicitly, the mixing between vibrational and electronic DOF leads to a nonzero angle between the vibronic states, even those largely located on the same pigment. The vibronic coherences (pathway R_{1}) discussed in Ref. 25 will thus not be eliminated by the coherence specific polarization sequence. The same argument applies to the groundstate vibrational coherences. When modes are treated explicitly, there will be a nonzero angle between and , and pathways of type R_{4} will also contribute to the signal^{30}. We can thus conclude that the “coherence specific” sequence is not able to unambiguously distinguish between electronic, vibronic and groundstate vibrational coherence.
To be able to determine the origin of the oscillations observed in the experiments, additional arguments, such as the oscillation frequencies or dephasing times, are needed. The groundstate vibrational coherences will oscillate with a frequency equal to that of the relevant vibrational mode, while for the vibronic coherences, the oscillation frequency will depend on the vibrational frequency as well as on the electronic coupling and site energies. The vibronic coherences also experience additional dephasing due to the mixing of electronic and vibrational DOF, while the groundstate vibrational coherences only experience vibrational dephasing. These quantitative differences could in principle be used to distinguish the two models, if the vibrational frequencies, HuangRhys factors and dephasing times of the vibrational modes were known.
The low frequency vibrational spectrum of BChla in solutions or in different protein environments have been studied by a number of techniques. Studies of isolated BChla in frozen solution have demonstrated that the strongest vibrational mode is found around 180190 cm^{−1} ^{55,56,57,58}. These experiments have identified a mode around 160 cm^{−1}, but its HuangRhys factor was found to be significantly weaker than the one of the modes around 180–190 cm^{−1} ^{55,56}. Although the results of the experiments are rather consistent, it should be pointed out that the lowfrequency spectral range is sensitive to the local conformation of the macrocycle and the coordination of the central Mg atom. This effect was illustrated by Rätsep et al.^{55} by measuring the fluorescence line narrowing (FLN) spectra of BChla in different solvents. These experiments demonstrate that results obtained on the isolated pigment might not be a good indicator of the frequencies and HuangRhys factors observed in the protein environment.
A widely studied case of “isolated” BChla in a protein environment is the accessory BChl in the bacterial reaction center. Resonance Raman (RR) spectra of the accessory BChl (Bband) have been reported by several groups^{59,60,61}. The room temperature spectrum identified modes at 84, 117, 180 and 210 cm^{−1}, where the latter two were considerably stronger^{60}. Measurements at 95 K revealed an additional mode at 160 cm^{−1} which was ~10 times weaker than the mode at 180 cm^{−1} ^{60}. Similar results were obtained at even lower temperature, showing that modes around 180 and 220 cm^{−1} dominate the spectrum^{61}.
There is only a limited number of studies on the vibrational spectrum of the pigments in the FMO complex. A hole burning (HB) study found two modes at 120 and 160 cm^{−1} with a comparable strength, but were not able to observe any modes with higher frequencies due to the narrow absorption band in FMO^{62}. FLN measurement, on the other hand, revealed a spectrum which was similar to the RR spectrum of the Bband in the reaction center^{33,63}. The two FLN measurements on the different species are similar, showing a mode at 117 cm^{−1} and several modes between 167–202 cm^{−1}. The strongest modes in both cases were found around 190 cm^{−1}.
Taking all of the data on the low frequency vibrational spectrum of BChla, we can conclude that none of the reported spectra show the presence of a single strong mode in the frequency range corresponding to that observed in the 2D experiments (160 cm^{−1}). Rather, the spectra reveal a large number of vibrational modes with comparable HuangRhys factors in the range from 80–240 cm^{−1} corresponding to typical energy splittings in FMO^{32}. All these modes will thus, to a certain extent, experience enhancement due to the interaction between electronic and vibrational DOF. For the groundstate wavepackets, the frequency of the oscillation will match that of the vibrational mode. Based on the discussion above, we would predict that the mode at 117 cm^{−1} experiences the strongest enhancement. A rather broad amplitude enhancement curve furthermore implies that multiple oscillation frequencies should be observed. For the vibronic coherences, the observed frequency depends on the resonance coupling as well as on the vibrational frequencies, the number of pigments and probably also on the number of vibrational modes included in the Hamiltonian. For the dimer used here, we predict an oscillating frequency of 132 cm^{−1} and for the full model, the value of 140 cm^{−1} was found^{25}. Increasing the resonance coupling would blueshift the oscillation frequency further and increase the amplitude of the vibronic coherence. However, due to the increased electronic character, these coherences would also experience a stronger dephasing and a correspondingly shorter life time. Because the amplitude enhancement, oscillation frequencies and dephasing dynamics are all closely related to the electronic structure, a simulation including all pigments is required to determine whether it is possible to find a set of couplings which results in a strong enhancement, correct oscillation frequency and a long life time. Such a simulation could also determine whether vibronic or groundstate vibrational coherences dominate in the case of FMO. Furthermore, it would be possible to elucidate why a single oscillation dominates in the 2D spectrum, although multiple vibrational modes are expected to experience significant amplitude enhancement according to the reduced dimer models.
In conclusions, we have demonstrated on a model dimer how the interaction between electronic and nuclear degrees of freedom leads to enhanced amplitudes of both the excitedstate vibronicexciton coherences and the groundstate vibrational coherences in 2D spectra. The results obtained highlight the properties of the enhancement mechanism. The vibronicexciton model used here also provides, without contradiction with molecular dynamics simulations, a plausible explanation for correlated fluctuations in different spectral regions, often postulated in purely excitonic model to reproduce longlived coherences.
We find that the initial amplitude of the vibronicexciton and vibrational coherences are enhanced by up to 15 and 5 times, respectively, compared to the vibrational coherences in the isolated monomer, with a bandwidth of about 100 cm^{−1} for both mechanisms. This enhancement requires both sufficient electronic coupling and a resonance between the vibrational level on one pigment and the electronic transition on the other. Enhancement is found to be even more pronounced when the inhomogeneity of the sites is included. For the dimer model used in this work, although the vibronicexciton coherences exhibit a larger initial amplitude compared to the groundstate vibrational coherences, we find that both types of coherences have a similar magnitude at long (picosecond) population times. Whether the vibronic coherences dominate in 2D spectra or not depends on the excitedstate dephasing and population dynamics as well as the influence of inhomogeneous broadening. To decide which type of coherence dominates in FMO complex would require simulations with all seven pigments and a full spectral density including a larger spectrum of vibrational modes.
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Acknowledgements
This work was supported by the Czech Science Foundation (GACR) grant nr. 205/10/0989 and the Ministry of Education, Youth and Sports of the Czech Republic grant MEB 061107, the Austrian Science Foundation (FWF) within project P22331 and OeAD. The authors would like to thank Prof. David Jonas for providing us with the manuscript of Ref. 30 prior to publication.
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Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 5, 121 16 Prague 2, Czech Republic and
 Aurélia Chenu
 & Tomáš Mančal
Faculty of Physics, University of Vienna, Strudlhofgasse 4, 1090 Vienna, Austria
 Niklas Christensson
 & Harald F. Kauffmann
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Contributions
T.M. and N.C. created the model and detailed the theoretical methods together with A.C. who performed the calculations and interpretation. N.C. discussed the extension of results to FMO. A.C., N.C., H.F.K. and T.M. participated in writing and reviewing the paper.
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The authors declare no competing financial interests.
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Correspondence to Tomáš Mančal.
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Photosynthesis Research (2016)
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