Exciton Dynamics in Photosynthetic Complexes: Excitation by Coherent and Incoherent Light

In this paper we consider dynamics of a molecular system subjected to external pumping by a light source. Within a completely quantum mechanical treatment, we derive a general formula, which enables to asses effects of different light properties on the photo-induced dynamics of a molecular system. We show that once the properties of light are known in terms of certain two-point correlation function, the only information needed to reconstruct the system dynamics is the reduced evolution superoperator. The later quantity is in principle accessible through ultrafast non-linear spectroscopy. Considering a direct excitation of a small molecular antenna by incoherent light we find that excitation of coherences is possible due to overlap of homogeneous line shapes associated with different excitonic states. In Markov and secular approximations, the amount of coherence is significant only under fast relaxation, and both the populations and coherences between exciton states become static at long time. We also study the case when the excitation of a photosynthetic complex is mediated by a mesoscopic system. We find that such case can be treated by the same formalism with a special correlation function characterizing ultrafast fluctuations of the mesoscopic system. We discuss bacterial chlorosom as an example of such a mesoscopic mediator and propose that the properties of energy transferring chromophore-protein complexes might be specially tuned for the fluctuation properties of their associated antennae.


I. INTRODUCTION
In recent years, primary processes in photosynthesis have received a renewed interest from a broader physical community thanks to experimental observation of coherent energy transfer in some photosynthetic systems. The ground breaking coherent two-dimensional electronic spectroscopy (2D-ES) experiment of Engel et al. [1] has led to new appreciations of the role that may be played by coherent dynamics in excitation energy transfer (EET), and of the quantum mechanical nature of photosynthetic systems in general [2]. Special theoretical effort has been made to understand the role of noise [3][4][5][6][7] in the dynamics of excitation energy transfer, and the role of coherence [8][9][10][11][12] in excitonicaly coupled systems. On the experimental front, the method of coherent 2D-ES [13,14] has established itself as a tool opening new window into the details of energy transfer dynamics in photosynthetic [15][16][17][18][19][20], and other molecular systems [3,21,22]. Coherent effects have been now reported in different molecular systems, often biologically relevant [22,23] -a generality that asks for a search of the possible evolutionary advantage underlying their abundance in photosynthetic pigment-protein complexes.
The principle pigment molecules responsible for the primary processes of photosynthesis are chlorophylls (Chls) and bacteriochlorophylls (BChls) [24,25]. They are involved in accumulation of light energy via the excitation energy transfer to specific pigment-protein complexes -reaction centers. Spectral variability of photosynthetic light-harvesting pigment-protein complexes arises either from excitonic interactions between pigment molecules or from their interactions with protein surrounding. Both these interactions are the main factors determining the excitation dynamics in light-harvesting [26]. Excitonic aggregates are subject to interaction with two types of environments, and they provide means of transferring energy from one environment to another. First of these environments, the radiation, is under natural conditions at much higher temperature than the second environment, the protein scaffold and indeed the photosynthetic chemical machinery as a whole. The excess of photons on suitable wavelength in the radiational environment is used to excite spatially extended antenna systems that concentrate excitation energy to the reaction center, which in turn drives charge transfer processes across cellular membranes to create the transmembrane potential and the pH gradient [24].
Non-equilibrium processes occurring in photosynthetic systems during light harvesting are conveniently described by reduced density matrix (RDM) theory [26][27][28] which has an advantage of being applicable to disordered statistical ensembles that the experiments often deal with. However, with recent 2D experiments that enable us to distinguish the homogeneous and inhomogeneous spectral broadening, and with the progress in single molecular spectroscopy [29] we can gain insight into the time evolution characteristic to single molecules interacting with their environment [30,31]. This fact enables us to return to the wavefunction formalism and to look at light harvesting from the point of view which takes the superposition principle of quantum mechanics seriously. It has been show that such an approach yields many interesting insights into the emergence of the classical prop-erties of molecular system from their underlying quantum mechanical nature [32,33]. As the light-harvesting processes seem to operate on the interface between classical and quantum worlds it seems appropriate to look at them from the point of view of the decoherence program of Zeh, Zurek and others [34,35].
The process of light harvesting could then be describes as follows. First, the system is in an "equilibrium" initial state |Ψ 0 characterized by the excitonic ground state |g , the state of protein (phonon) environment |Φ P corresponding to this electronic ground state and some state of light |Ξ 0 , i.e.
The light-harvesting occurs when the state of light is such that the time evolution of the system leads to population of higher excited states |e n of photosynthetic antenna. These states are formed from excited states of Chls and other chromophores, such as carotenoids [26]. We denote these combined excited states as excitons. In the first approximation, photosynthetic antenna remains in the excited state until the excitation energy is transferred to the reaction center. This happens much faster than competing process of spontaneous emission which can therefore be neglected in our discussion. When the interaction of the antenna with light is switched on, the change occurring in the ground state portion of the total state vector after the passage of time ∆t is The subsequent time evolution of the excited state portion of the state vector is independent of the ground state part, and we can thus look at it separately. Because we neglect spontaneous emission, any excitation to higher excited state, as well as transitions between exciton states due to the light, the state vector |Ξ ′ remains approximately unentangled with excitons and the protein bath for the rest of the energy transfer process. It can therefore be omitted. The initial state for the energy transfer process thus reads where we omitted the lower index ∆t. If the basis of the states |e n is chosen so that the molecular Hamiltonian is diagonal, the energy transfer occurs only due to interaction of excitons with their surrounding environment. This interaction leads to an entanglement of excitons and the environment |Ψ e (t) = n β (n) (t)|e n |Φ (n) After a sufficiently long time the environment state vectors corresponding to different electronic state diverge maximally and the reduced density matrix becomes diagonal in some basis, i.e.
Often, to a good approximation, such preferred basis is the one in which the electronic Hamiltonian is diagonal, the so-called excitonic basis. However, notable corrections to this rule are predicted even for weak system-bath coupling [12,36]. The final state of the energy transfer is the one in which just reaction centers are populated The last step of the energy transfer, from the antenna to the reaction center is often slower than typical transfer times between antenna complexes, and so the final state is well localized on the reaction center, and coherences between individual reaction centers are unlikely to survive. It is clear from the above discussion, that decoherence during the energy transfer in the antenna is determined by the evolution of the environmental degrees of freedom (DOF). The decoherence from the rest of the system might be required for the localization of the energy in the reaction center, but there is no obvious reason for fast decoherence during the initial steps of energy transfer in the antenna, apart from the fact that a bath formed by a completely random disordered environment would lead to just such fast decoherence. It has been suggested before that the protein environment might play a more active role in steering and protecting electronic excitation [1,23] and controlling the decoherence might be one of the possible pathways to more robust EET.
There is however one important caveat in the above scheme. The initial condition, Eq. (3), has been introduced artificially into Eq. (2) as a result of an interaction occurring during some short time interval ∆t. If the system is continuously pumped, individual contributions similar to Eq. (3) will interfere, possibly disabling any effect of cooperative involvement of the bath. It is even more important to consider the question what are the effects of natural sun light [37], i.e. whether the coherent scenario outlined above is plausible for the photosynthetic system in vivo, or not. This depends strongly on the nature of the excitation process, whether it occurs in discrete independent jumps of the kind described by Eq.
(2), or continuously over a long period of uncertainty interval of the photon arrival. The former view is usually held in support of the relevance of ultrafast spectroscopic finding for in vivo function of the photosynthetic systems [11]. Below we derive a general formula which enables us to describe all these regimes by a unified formalism, and also enables us to place the observables of ultrafast coherent spectroscopy in perspective with the dynamics under natural conditions. In a somewhat extended form our result is also applied to another case cited in support of the utility of coherent dynamics in photosynthetic systems, a case where a small photosynthetic complex is excited through another, possibly mesoscopic, antenna [11].
The paper is organized as follows. Next section introduces a rather general model of photosynthetic aggregate. In Section III we discuss the dynamics of a system excited by coherent pulsed light and the observables of the ultra-fast non-linear spectroscopy. Section IV is concerned with the excitation of a photosynthetic system by the light from a general source. Implications of the theory for excitation by thermal and coherent light, as well as excitation mediated by mesoscopic system are discussed in Section V.

II. HAMILTONIAN OF A MODEL PHOTOSYNTHETIC SYSTEM
In this section we briefly review the excitonic model that was very successfully applied to model the spectroscopic properties of Chl-and Bchl-based light harvesting chromophore-protein complexes (see e.g. [16]). We assume N monomers with ground states |g n , excited states |ẽ n , n = 1, . . . , N , and with electronic transition energiesε n . These monomers are interacting with the phonon bath of protein DOF so that the Hamiltonian of the monomer reads H n = (T + V g )|g n g n | + (ε n + T + V e )|ẽ n ẽ n |. (7) Here, T is the kinetic energy operator of the bath, and V g and V e are the potential energy operators of the bath when the system is in the electronic ground-and excited states, respectively. We set the ground state electronic energy to zero for conveniency. The Hamiltonian, Eq. (7) can be split into the pure bath, pure electronic and the interaction terms so that Here, I B is the unity operator on the bath Hilbert space and I M is the unity operator on the Hilbert space of the electronic states. The equlibrium average V e − V g eq of the potential energy operators was added to the electronic energy so that the interaction term is zero for the system in equilibrium.
In chromophore-protein complexes many such monomers are coupled by resonance coupling. The whole complex can be described by means of collective states including the ground state one excitation states and states containing higher number of excitations. For the sake of brevity we now stop writing the symbol of the direct product ⊗ and the unity operators I n B etc. explicitely. The total Hamiltonian of the complex including resonance interaction is then defined as If the system-bath interaction with bath is weak, the referred basis into which the electronic system relaxes due to interaction with the bath is, to a good approximation, the one in which the electronic part of the Hamiltonian is diagonal. Let us denote these states as |e n . They are usually termed excitons and they represent certain linear combination of the collective states |ē n where excitations are localized on individual chromophore molecules. One of the most important characteristics of this model is that it does not include direct relaxation of the electronic excited states to the ground states due to electron-phonon coupling. This is well satisfied by Chls and BChls on the ultrafast time scale of which light harvesting processes occur.

III. EXCITATION BY COHERENT PULSED LIGHT AND NON-LINEAR SPECTROSCOPY
Let us now consider experimental methods which provide information about time evolution of excited states of photosynthetic systems. Because of the timescale of EET processes, spectroscopy with ultrashort time resolution is a necessary tool. The interaction of the pulsed coherent light with the photosynthetic system is well described in semi-classical approximation [38]. Electric field of the light is then considered as an external parameter of the system Hamiltonian. Electronic DOF can be prepared very fast in an excited state, not affecting, to a good approximation, the bath DOF. Thus, in an experiment with an ideal time resolution, we would have the system prepared in the excited state, Eq. (3). The time evolution of the system is governed by the Schrödinger equation with initial condition |Ψ e (t) = 0 for t < t 0 . The last term in Eq. (12) describes the ultrafast event of the molecule-radiation interaction. Formal solution of this where we defined evolution operators U B (t), U M (t) of the bath and the molecule, respectively, as and the remaining interaction evolution operator as After excitation, the process of energy transfer proceeds according to the description presented in Introduction and can be experimentally monitored.

A. Evolution superoperator
Matrix elements of the RDM of the molecule, which holds the information about the population probabilities and the amount of coherence between electronic states are given by expectation value of projectors |e n e m |, This can be rewritten by defining an evolution superoperator U(t) which acts on initial density matrix ρ 0 W eq , i.e.
The matrix elements of the superoperator read where the dots . . . denote where an operator on which U (e) (t) acts has to be inserted. The reduced evolution operatorŪ (e) (t) defined as contains information about the evolution of the RDM only.

B. Non-linear spectroscopy
In non-linear spectroscopy, coherent laser light is used to investigate the dynamics of molecular systems by applying special sequences of pulses. Some pulses act to induce non-equilibrium dynamics (pump), and other pulses act to monitor (probe) the evolution after the pump. One of the most advanced of these methods, coherent 2D-ES [13,39] measures the response of a system to three pulses traveling in different directions k 1 , k 2 and k 3 . The detection is arranged in such a way (measuring in the direction −k 1 + k 2 + k 3 ) that the signal is predominantly of the third order, with contributions of one order per pulse [38]. Let us denote delays between the first two pulses by τ and the delay between the second and the third pulse by T . If the pulses are ideally short, the signal is composed of two kinds of contribution. First, contribution that involves population of the excited state corresponding to the density operator and second, contribution that involves evolution in the ground state Here, we denote the pulses acting on the state vector by their corresponding wave vector in the upper index, and the excited state or ground state bands by the lower index g and e, respectively. For these statistical operators we can define evolution superoperators U (e) (t), U (g) (t), U (eg) (t) and U (ge) (t) in analogy with Eqs. (17) and (18), so that and The superoperator U (eg) (t) is the evolution superoperator of a coherence projector n |e n g| and analogically for U (ge) (t). After a delay T the third ultrafast pulse is applied and the non-linear signal is recorded. The signal corresponds to indirectly to non-linear polarization of the sample, and is usually measured in frequency domain Here, we denoted and ρ g = |g g|. In 2D coherent spectroscopy, the signal is in addition Fourier transformed along the time delay τ , so that the spectrum is defined as The spectrum defined in this way has a suitable interpretation of an absorption -absorption and absorption -stimulated emission correlation plot, with a different waiting times T between the two events. The 2D spectrum is in practice measured with finite pulses, and the measured time domain signal is thus a triple convolution of the responses to a delta pulse excitation, with the actual finite pulses [40]. From this rough sketch of the principles and the information content of the coherent 2D spectroscopy it should be clear that 2D spectroscopy is aimed at disentangling the dynamics of the system during the time delay T . In the so-called Markov approximation, when the dynamics in time intervals τ , T and t is assumed separable, and the bath is assumed stationary, the ground state evolution during interval T can be neglected. Then 2D measurement essentially accesses the reduced evolution superoperator, Eq. (19) and possibly also the more general superoperator We will show below that this superoperator, together with the light properties, determines the way in which the molecule is excited in a general case, even at illumination by natural light.

IV. EXCITATION BY LIGHT
In order to account for general light properties we will consider the problem fully quantum mechanically, and assume only deterministic evolution of the system wavefunction. The Hamiltonian of the system reads We have divided the system into a molecule (H M ), its environment or bath (H B ), the radiation (H R ) and the light emitting body (LEB) which produces it, e.g. Sun or laser medium (H S ). It seems reasonable to neglect a direct interaction between the molecule (together with its environment) and the molecules of the LEB. Consequently, the terms H M−S and H B−S can be disregarded. To make the treatment simpler we can also neglect the interaction between radiation and the molecular environment, H B−R . The assumption is that the energy of the molecular transition that is used to harvest light for photosynthetic purposes is much larger than any of the transitions in this environment and the two regions of the light spectrum can thus be treated separately. One can also assume that the part of radiation spectrum which would interact with the bath is simply filtered out, and the environment is kept at certain temperature by other means.

A. Radiation entangled with the light emitting body
An important special case is the one in which the radiation and the LEB is in equilibrium with each other so that the radiation is described by the canonical equilibrium density matrix Here, |N λq is the N -photon state of the radiation mode with polarization vector e λ and wave-vector q. As we have already noted above, the statistical concept of density matrix will be replaced here with the concept of entangled states, so that we can describe the whole system by its state vector. Thus, we introduce a state vector in which the light is fully entangled with the states |φ N λq (t) of the LEB . The LEB states have to fulfill the condition so that when the total density matrix of the LEB and radiation is averaged over the states of the body, we obtain Eq. (29). W (eq) R is recovered provided that In the absence of the light absorbing body, the evolution of the state |Ξ(t) is governed by the Hamiltonian and is the corresponding evolution operator.

B. Equation of motion
For the subsequent treatment of the system dynamics, we introduce the interaction picture with respect to Hamiltonian operators H M , H B and H L , where and Equation of motion for the total state vector in the interaction picture thus reads The solution of Eq. (39) can be found formally as We will assume that the system is initially in the state |Ψ 0 of Eq. (1). With this choice we have Further in this paper, we will assume weak interaction with the radiation, so that it can be described by linear theory. Thus, we need to collect all terms in the expansion of Eq.
Now we introduce a projector P e that excludes the excitonic ground state |g P e = n |e n e n |.
Applying this projector to Eq. (42) has only the effect of eliminating the first term of the series. Introducing abbreviations and It is possible to verify easily that this series is a solution of the equation with initial condition |Ψ Hamiltonian H M−R will be assumed in the dipole approximation, i.e.
where µ is the transition dipole moment operator of the aggregate µ = n d n |e n g| + h.c., and E T is the operator of the (transversal) electric field of the radiation with Here, Ω is a quantization volume. We consider a molecule much smaller than the wavelength of the light, so that e iq·r is constant in the volume of the molecule. The origin of the coordinates can thus be conveniently put into the molecule yielding e iq·r ≈ 1.

The interaction Hamiltonian in Eq. (48) then reads
where the creation and annihilation operators of the field are in the interaction picture with respect to Hamiltonian The transition dipole moment operator projected on the polarization vector of a mode λq appears in the interaction picture with respect to Hamiltonian H M , The evolution operator U L (t), Eq. (34), can be rewritten as where Since Hamiltonian H S commutes with the radiation operators and we have and Here, we introduced slow oscillating envelops andã Inserting these expressions into Eq. (53) we can distinguish two terms associated with the transition from the ground state |g to an excited state |e a with respective phase factors e i(ωag −ωq )t and e i(ωag +ωq )t . While the first one will lead to a resonance excitation around ω q ≈ ω ag , the later term is oscillating fast and will therefore contribute very little compared to the former one. Thus we drop the fast oscillating part, and obtain the source term in the form Using this form of the source term, we can find state into which the molecule is weakly driven by any type of light.

D. Excited state dynamics under pumping
So far we have treated the problem systematically using the wavefunction approach. The time evolution of the system wavefunction is governed by Eq. (39). To find the probabilities of creating population on and coherence between certain excitonic levels |e a we solve Eq. (39) formally, Here, we used the fact that |Ψ e (t 0 ) = 0. Now let us evaluate matrix element P ab (t) = Ψ e (t)|P ab |Ψ e (t) of a projector which gives the probability of finding the molecule in state |e a if a = b, or characterizes the amount of coherence between states |e a and |e b if a = b. Note that we have removed the interaction picture, Eq. (38). We have where the evolution superoperatorŪ (e) (t − τ, τ − τ ′ ) has been defined in Eq. (27). In Eq. (67), the light is represented by a first order correlation function (see e. g. Ref. [41]), which comprises all its relevant properties. We also denoted The quantities P ab (t) are the matrix elements of the RDM (P ab (t) = e b |ρ(t)|e a ) of the system which reads For a weakly driven system, Eq. (70) has a very wide range of applicability. We will discuss its application to thermal light and pulsed coherent light in the following section.

V. DISCUSSION
Thorough discussion of excitation dynamics of molecular systems excited by incoherent light was made in Ref. [37]. Molecular systems were considered without the bath effect which is however significant for light harvesting. Eq. (70) contains reduced evolution superoperator of the molecular system so that the state of the system created by the incident light depends on its reduced dynamics. It is not possible to consider a general case of such dynamics analytically, and we will therefore commit ourselves to some simple cases.
In so called secular and Markov approximations (see e.g. Ref. [27]) matrix elements of the evolution superoperator governing the coherences take a very simple form. First, it is possible to separate the two time arguments in the superoperatorŪ (e) (t, τ ) so that Since each coherence is independent of the population dynamics and of other coherences, the one-argument superoperator elements read andŪ (eg) Here the dephasing rate comprises the pure dephasing rate γ p and the rate K a of depopulation, i.e. the sum of transition rates from state |e a to other states. A simplified treatment of the populations is possible for the states that are only depopulated, i.e. no contributions to the population can be attributed to the transfer from other levels. They are found at the top of the energetic funnel of the antenna. For these states we havē Eqs. (72) to (75) neglect all coherence transfer effects, as well as possible coupling between the dynamics of population and coherence.

A. Excitation of coherences by thermal light
For an equilibrium thermal light the correlation function I (1) λq,λ ′ q ′ (τ, τ ′ ) depends only on the difference of the times τ and τ ′ . As discussed above, |Ξ 0 represents the equilibrium of the system described by Hamiltonian H L . The equilibrium density matrix is stationary, i.e.
so we can write It can be shown that Assuming some simple form of a light correlation function, e.g.Ĩ we obtain for the populations ρ aa (t) = 2Re Here, [ρ] ab ≡ e a |ρ|e b . We utilized Eq. (79) and the fact that, by definition (see Eqs. (68) and (69)), the time τ corresponds to the action of the dipole moment operator from left, whereas time τ ′ corresponds to the same action from the right. At long times t − t 0 → ∞ this yields However, neglecting the influence of environment as in Ref. [37] yields which grows linearly with time. For coherences we have which turns into Eq. (81) for a = b (with additional assumption K a = 2Γ a ). In case of no dephasing, the first fraction in Eq. (83) yields a delta function δ(ω ab ) [37]. Thus, for slow or non-existent relaxation due to interaction with environment, the system is excited predominantly into a state represented by a diagonal RDM, as all coherence terms are negligible compared to the linearly growing population. For fast relaxation, the coherences may be of the same order of magnitude as the populations.
The case of very fast relaxation is particularly interesting. It was suggested previously that coherent dynamics can be relevant for the in vivo case, because the fluctuating light from the Sun corresponds to a train of ultrafast spikes [11]. The relaxation of the antenna must be in such a case fast enough to prevent averaging over many such spikes. Eqs. (81) and (83) with large K a describe just such a situation. The RDM created by incoherent light resembles in certain sense the one created by ultrafast pulses; it represents a linear combination of excitons. The coherences in Eq. (83) are however static at long times.
In our demonstration we concentrated on a simple model assuming both Markov and secular approximations to be valid. The presence or absence of coherences has no significance in such a case, and more involved theories of the RDM dynamics [8,9,12] have to be used to investigate the role of coherences in energy transfer processes by Eq. (70).

B. Coherent pulsed light
In derivation of Eq. (70) we assumed certain initial state |Ξ 0 of the system composed of the light and its source. The condition that the light is in a stationary state, fully entangled with its source, has only been used to simplify the correlation function I (1) λq,λ ′ q ′ (τ, τ ′ ) for the case of the thermal light. In a general case |Ξ 0 will not represent an equilibrium state. It can indeed describe even systems such as a laser producing coherent Gaussian light pulses with some carrier frequency ω 0 and a width parameter ∆. If we in addition assume that the light is described by a single polarization, and that we consider the dynamics after one such pulse centered at time t = τ 0 , the light is described as Coherence element created by such light reads where ρ 0 ba = 1 2 e b |µ|g g|µ|e a . In the limit of ultrashort pulses when e − (τ −τ 0 ) 2 ∆ 2 → αδ(τ − τ 0 ) the pulse creates a pure state at τ 0 , which then dephases as In case of a finite pulse and no dephasing our results coincides with those found in Ref. [37].

C. Mediated excitation
The major difference between excitation by the thermal light and a coherence pulse is in the occurrence of a sudden event which populates a nearly pure state of the excited state band. Clearly, a single molecule interacting with an ideal continuum of radiation modes in equilibrium does not experience such sudden events. Rather, its interaction with light corresponds to a continuous pumping, and the suddenness of the photon arrival is the consequence of our ability to register only classical outcomes. In order to register them we have to interact with the system and become entangled with it. Our experience is that macroscopic systems interacting with low intensity light can be used to detect single photons, and certain more or less definite times can be attributed to their arrivals. Interaction of a photon with a macroscopic detector yields a temporal localization of the arrival event.
A mesoscopic system may play a role of such a detector (mediator) that provides its fluctuations to be harvested by dedicated nano sized antenna. Green photosynthetic bacteria, from which the photosynthetic complex FMO was isolated, collect light mainly by means of so-called chlorosoms [24,42]. The chlorosom is a self-assembled aggregate of ∼ 10 5 BChls and carotenoids with very little protein. The typical dimensions the chlorosom are of the order of 100 nm [42]. It does not seem to be organized as an energy funnel [43,44], and the energy transfer time between its main body and the base plate to which FMO complexes are attached is of the order of 120 ps [45], i.e rather slow. The excitation in such a mesoscopic system may have enough time to become localized through interaction with the large number of the systems DOF and arrive at the FMO complex in a particle like, i.e. also temporally localized fashion.
In this section, we will generalize our result, Eq. (70), for a case when the excitation of the photosynthetic systems occurs by transfer from another system. We will therefore assume that our molecule does not interact directly with light, but is pumped in a similar fashion by another system. The source term, Eq. (64), is then generalized as Here, A = α,n |e n |ξ g ξ α | g| + h.c. is the moleculemediator interaction Hamiltonian and the time dependence results from the interaction picture We denoted the ground and excited states of the mediator by |ξ g and |ξ n , respectively. The state of the molecule at long times is in analogy with Eq. (70) The complicated two-point correlation function in Eq.
(89) results from the pumping of the mediator similarly to the direct pumping of the molecule in Eq. (70). A mesoscopic system especially when excited will, however, always exhibit fluctuations which will prevent the correlation function from having a simple smooth dependence without recurrences. Such recurrences can temporally localize the excitation events of the molecule. In such an excitation regime, when coherent dynamics from different excitation times do not interfere, optimization of the FMO's energy channeling capability for case of initially coherent states would be an advantage.

D. Outlook
More research into specific forms of both the light correlation function for different situation that may occur in vivo, and the analogical interaction of systems like FMO with mesoscopic antennae is clearly needed. Ultrafast spectroscopic experiments play a pivotal role in this research by yielding information about the system response to the light. To conclude on the utility of coherent dynamics for the function of the photosynthetic system is, however, only possible by taking into account the properties of light at the natural conditions, for which the results of this paper provide means. If the coherent dynamics observed in some photosynthetic chromophore-protein complexes has a significance for their light-harvesting efficiency, and these systems evolved to optimize it for their corresponding ecological situation, it can be expected that the properties of at least some parts of the photosynthetic machinery would be tuned to the fluctuation properties of their source of excitation. For plants and some bacteria this may be the Sun light, others like FMO complexes could be expected to be tuned to the properties of their associated chlorosoms.

VI. CONCLUSIONS
In this paper we have discussed dynamics of a molecular system subject to external pumping by a light source. In particular we have considered excitation by thermal light, by coherent pulsed light and an excitation through a mesoscopic antenna. With a completely quantum mechanical treatment, we have derived a general formula which enables us to study the effect of different light properties on the photo-induced dynamics of a molecular systems. This formula naturally contains the systemenvironment interaction contribution to the excitation process which enters via appearance of the reduced density matrix dynamics. We show that once the properties of light are known in terms of a certain two-point correlation function, the only information needed to reconstruct the systems dynamics is the reduced evolution superoperator, which is in principle accessible through ultrafast non-linear spectroscopy. This conclusion applies to any type of light and makes thus the results of ultrafast spectroscopic experiments universally relevant. Considering a direct excitation of a small molecular antenna we found that excitation of coherences is possible due to overlap of homogeneous line shapes associated with different excitonic states. These coherences are however static and correspond to a change of the preferred basis set into which the system relaxes from the one defined by the bath only, to the one defined by the action of both the light and the bath. When an excitation of a photosynthetic complex mediated by a larger, possibly mesoscopic, system is considered, the complex can harvest fluctua-tions originating from the non-equilibrium state of the mediator. Fluctuations of the mesoscopic system such as chlorosoms may time localize excitation events of the energy channeling complex, and to excite adjacent energy channeling complex coherently. It is likely that in such a case the properties of energy channeling complexes like the well-know Fenna-Mathews-Olson complex would be specially tuned to the fluctuation properties of their associated chlorosoms.