Dynamics of electronic dephasing in the Fenna–Matthews–Olson complex

Electronic coherence has been shown to persist in the Fenna–Matthews–Olson (FMO) antenna complex from green sulfur bacteria at 77 K for at least 660 fs, several times longer than the typical lifetime of a coherence in a dynamic environment at this temperature. Such long-lived coherence was proposed to improve energy transfer efficiency in photosynthetic systems by allowing an excitation to follow a quantum random walk as it approaches the reaction centre. Here we present a model for bath-induced electronic transitions, demonstrating that the protein matrix protects coherences by globally correlating fluctuations in transition energies. We also quantify the dephasing rates for two particular electronic coherences in the FMO complex at 77 K using two-dimensional Fourier transform electronic spectroscopy and find that the lifetimes of individual coherences are distinct. Within the framework of noise-assisted transport, this result suggests that the FMO complex has been locally tuned by natural selection to optimize transfer efficiency by exploiting quantum coherence.


Introduction
Photosynthetic organisms harvest solar power by converting energy from electronic excitations into chemical energy stored in the bonds of organic compounds. The general scheme for photosynthesis begins with absorption of a photon by an antenna complex, a three-dimensional (3D) array of chromophores embedded in a protein matrix. Transfer of the excitation to the reaction centre then initiates the light reactions that produce ATP and NADPH. An antenna complex effectively broadens and increases the absorption cross section of the associated reaction centre chromophores, but it also introduces the challenge of transferring energy through a dynamic, disordered network of excitons [1]. Remarkably, this transfer process can occur with a quantum efficiency approaching unity.
In a strictly classical description of electronic energy transfer, an incoherent excitation 'hops' between chromophores, as described by Förster theory. Recent experiments have shown, however, that quantum coherence may exert an important influence on energy transfer within antenna complexes and may contribute to the near-perfect quantum efficiency of these systems [2]- [5]. Coherent energy transfer improves transport efficiency because excitations follow a quantum random walk during transfer, sampling different pathways simultaneously. In 2007, Engel et al demonstrated that quantum coherence persists in the Fenna-Matthews-Olson complex (FMO) at 77 K for at least 660 fs after excitation, more than twice as long as the average population transfer time [2]. Not only does this finding demand a description of electronic energy transfer that includes both population and coherence dynamics, but it also suggests that the protein matrix promotes coherence transfer and protects coherences from the rapid dephasing typically associated with environmental fluctuations.
The role of coherent energy transfer in FMO has since been the subject of many theoretical studies. Rebentrost et al [6,7] and Plenio and Huelga [8] independently demonstrate that, within a certain regime, thermal dephasing arising from fluctuations in site energies can actually increase the efficiency of transfer by reducing the gaps between energetically isolated excitons. In this noise-assisted transport model, efficiency is optimized when the dephasing frequency is of the order of the excitonic energy gaps, because dephasing can, if necessary, assist transfer between excitons, but does not collapse a coherent excitation before it completes at least a period of quantum beating. By modelling the FMO protein as a harmonic oscillator bath with an Ohmic spectral density, Rebentrost et al predict that dephasing is optimal near room temperature, 3 suggesting that this complex has been tuned by natural selection to promote highly efficient transfer under natural conditions [7]. More recently, Ishizaki and Fleming present a rugged yet overall downhill energy landscape for FMO, proposing that coherence allows an excitation to overcome initial energy barriers, but subsequent dephasing leads to trapping of the excitation at the target site, thereby preventing energy loss due to exciton recombination and non-radiative decay [9].
These models illustrate the otherwise counterintuitive role of dephasing in the transport process and also indicate that transport efficiency is strongly and unimodally dependent on the dephasing rate. Previously, this rate has been estimated using spatially homogeneous phonon baths with uncorrelated fluctuations in the protein matrix, but experimentally determined sitespecific dephasing rates could be used to improve theoretical models of transport dynamics in FMO. Here, we present a simple method for determining the dephasing rates of individual coherences by analysing quantum beating in the cross-peaks of 2D spectra. We find that the lifetimes of two particular zero-quantum coherences are both of the order of a picosecond but are also distinct, suggesting that transport efficiency has been optimized within the FMO complex by the site-specific tuning of dephasing.

Bath-induced dynamics
The FMO antenna complex from green sulfur bacteria acts as a biological energy conduit connecting the larger light-harvesting chlorosome to the reaction centre [1,10]. Each of the three monomers that comprise FMO contains seven electronically coupled bacteriochlorophyll-a (BChl) chromophores arranged asymmetrically [11]. Due to the relatively simple electronic structure of the complex, FMO has become a model system for studying energy transport in photosynthetic antennae [2], [12]- [15]. The linear absorption spectrum of FMO at 77 K shows only three peaks at 805, 815 and 825 nm, but a number of theoretical studies have been undertaken to calculate the transition energies of each of the seven excitons by fitting spectroscopic data and modelling electrostatic interactions from crystallographic data [16]- [18].
2D Fourier transform electronic spectroscopy has proven a valuable tool in the study of FMO by offering additional resolution and, more recently, information on transport dynamics [2,13,15]. By probing the third-order polarization of the system under study, this technique allows electronic couplings between excitons to be directly observed as cross-peaks in a 2D frequency-frequency spectrum. Monitoring the evolution of these cross-peaks along an additional time dimension provides information on coherence and population dynamics during that time [19]. Beating in the shape and amplitude of spectroscopic peaks due to wavefunction interference indicates the existence of electronic coherence. Accordingly, measuring the damping of these beating signatures that occurs as coherences dephase provides a measure of coherence lifetimes.
While noise-assisted transport models have shown that a dephasing rate of the order of a few hundred wavenumbers improves the efficiency of energy transfer in FMO [7], a microscopic understanding of how zero-quantum coherence dephasing could be so slow at physiological temperature remains unclear. Even at 77 K, the dephasing of one-quantum coherences in FMO occurs at a rate of over 300 cm −1 , corresponding to a lifetime of approximately 100 fs. Zero-quantum coherences, however, are found to survive for at least 660 fs under the same conditions [2]. Lee et al observe a similar difference between coherence lifetimes in a bacterial reaction centre and suggest that the protein surrounding the chromophores protects zero-quantum coherences by correlating transition energy fluctuations across the complex [3]. Because the phase of a coherence evolves as exp[−i(E 1 − E 2 )t/h], introducing the same fluctuation in both transition energies E 1 and E 2 for a zero-quantum coherence does not affect the time propagation of the relevant density matrix elements. This is not true for a one-quantum coherence, of course, where the difference frequency is equal to the fluctuating transition energy.
The requirement that fluctuations in transition energies be correlated to preserve coherence can be derived by considering bath-induced electronic transitions. The model developed below is analogous to the theory for collision-induced intersystem crossing in small molecules developed by Freed and co-workers in the 1970s [20]- [22]. Figure 1 depicts an energy level diagram for a system of two non-degenerate excitons denoted as A and B. The following analysis considers transfer within this model system, but the theory can easily be extended to systems of more excitons such as FMO. Because the excitons are non-resonant, direct transfer between the pure electronic states |A and |B cannot occur. This restriction is lifted, however, as vibrational motions of the bath introduce small fluctuations in the system. While the time-averaged Hamiltonian is diagonal in the excitonic basis, the time-dependent off-diagonal (coupling) terms fluctuate about zero due to coupling with the bath and are therefore likely to be non-zero at any given moment. We construct the mixed excitonic eigenstates |+ and |− of the system in terms of the instantaneous coupling parameter λ 1 according to Because the system is coupled to a phonon bath or to the polarization modes of the bath, fluctuations in site energies also arise from interactions between the system and the bath as given by the Hamiltonian H s−b . The transition between mixed states induced by the bath is given by the coupling matrix element, 5 Assuming that H s−b is electronically quasi-elastic but inelastic in other modes in |A and |B (vibrational, rotational, bath, etc) [23], the final two terms vanish, leaving only the diagonal elements of H s−b , which have equal and opposite coefficients. The resultant coupling matrix element also appears in the expression for the pure dephasing rate obtained from the relaxation superoperator expanded to second order in the system-bath interaction [24], where |α and |β are eigenstates of the bath and is the energy difference between the |+ and |− states. When fluctuations in |A and |B are not perfectly correlated, the diagonal terms do not cancel, and both the transition probability and the dephasing rate are nonzero. Because |+ is primarily localized on exciton A and |− on exciton B, the transition from |+ to |− results in population transfer between the two excitonic states, accompanied by dephasing. As fluctuations become increasingly correlated, however, the difference between diagonal terms diminishes, and dephasing of the excitation becomes slower. We note that while the population transfer rate given by (3) is fast relative to typical rates of a few ps and the pure dephasing rate given by (4) is slow relative to typical rates of tens of fs, both rates are of the order of a few hundred fs.
In addition to showing that correlated fluctuations lead to long-lived electronic coherence, this model demonstrates how bath dynamics can facilitate the transfer between weakly coupled excitons by a mechanism fundamentally different from Förster resonance energy transfer (FRET) [25]. While FRET requires the donor and acceptor states to be momentarily isoenergetic, bath-induced transitions only require uncorrelated transition energy fluctuations, which do not necessarily require energetic coincidence between the excitonic states. This mechanism produces significantly higher transition rates in cases where the overlap integral between donor and acceptor is small, demonstrating how slow dephasing processes generally improve transport efficiency in accordance with the noise-assisted transport model.

Two-dimensional spectroscopy
In a four-wave mixing experiment such as 2D electronic spectroscopy, three weak-field laser pulses interact with a sample to generate a third-order polarization. We follow the time evolution of the system using the nth-order perturbation to the Liouville-von Neumann equation, given byρ This expression is integrated iteratively starting from a stationary ρ (0) to evaluate higher orders of ρ(t), where each order of perturbation corresponds to an interaction with an electromagnetic field according to H int . For a third-order response, each of the three pulses can act on either the ket or the bra of the density operator, and the order in which the interactions take place may vary, producing a total of 48 different terms that contribute to the total nonlinear polarization. All but six of these terms, however, are eliminated from the detected signal within the k s = −k 1 + k 2 + k 3 phase-matching condition and the pulse ordering enforced in traditional 2D electronic spectroscopy. These terms, called Liouville pathways, can be represented graphically using double-sided Feynman diagrams, as exhibited in figure 2. This representation greatly facilitates the assignment of terms to individual spectral features [24].  The left and right vertical lines represent the ket and bra of the density operator, respectively, and the curved arrows indicate interactions with an electromagnetic field. The ground state of the system is given by |g , single-exciton states by |e j and double-exciton states by |f . Diagrams 1-3 represent rephasing (photon echo) pathways, in which the coherence and rephasing frequencies are of opposite sign, while diagrams 4-6 represent non-rephasing (free induction decay) pathways. For a = b, diagrams 1 and 4 represent pathways that give rise to quantum beating off and on the diagonal, respectively [19]. The sequential excitation pathways given by diagrams 2 and 5 do not give rise to quantum beating during the waiting time because the difference frequency is zero. Diagrams 3 and 6 show excitedstate absorption pathways, which appear as negative features in the absorption spectrum [24]. Population and coherence transfer during the waiting time are shown by diagrams 7 and 8, respectively.
The theoretical and experimental details for 2D electronic spectroscopy have been reported previously [26]- [29]. Briefly, two independently variable time delays are introduced between the three noncollinear laser pulses before they reach the sample. Chronologically, these delays 7 are termed the coherence time, τ , and the waiting time, T . Uniformly stepping τ for a fixed waiting time and recording the spectrally resolved, heterodyne detected signal at each step gives a frequency-time 2D spectrum, and a Fourier transform along the coherence time axis gives a frequency-frequency spectrum. This process is then repeated for a series of uniformly spaced waiting times, which provides temporal resolution of spectroscopic features. It is worth noting that while the pulses used in this experiment are necessarily of finite duration, analysis of the third-order response in the impulsive limit greatly simplifies the interpretation of spectra and is satisfactory for a qualitative discussion.
As illustrated by the Feynman diagrams depicted in figure 2, the system is in a onequantum coherence during the coherence time and evolves phase before the second pulse creates either a population or a zero-quantum coherence. Both population and coherence transfer occur during this time, as shown in diagrams 7 and 8, respectively. The third pulse generates a second one-quantum coherence, which evolves phase during the rephasing time t until the signal pulse is emitted, returning the system to a population. Because we are interested in measuring the lifetime of coherences, we now focus only on pathways for which the system is in a coherence during the waiting time. The periodic phase evolution of such a coherence results in quantum beating in the signal at the corresponding spectral position. In diagram 1, the coherence frequency ω τ = (E a − E g )/h is not equal to the rephasing frequency ω t = (E g − E b )/h, so the contribution to the signal from this pathway appears as a cross-peak in the 2D spectrum. Diagram 4, however, describes a pathway for which the frequencies are equal, so the contribution appears on the diagonal. Because the beating observed in a particular diagonal peak contains frequency contributions from all possible zero-quantum coherences involving the exciton located at that peak, it is much simpler to analyse the beating in crosspeaks, which should contain only a single frequency. Furthermore, the coherence and rephasing frequencies of pathways that beat in off-diagonal positions (diagrams 1-3) are of opposite sign, resulting in rephasing of the ensemble during t and emission of a photon echo. This signal is significantly stronger than the free induction decay emitted from non-rephasing pathways (diagrams 4-6), providing additional motivation for analysing cross-peaks. The non-collinear beam arrangement allows us to record the rephasing and non-rephasing signals separately within the k s = −k 1 + k 2 + k 3 phase-matching condition simply by switching the order of pulses 1 and 2 [30].

Experimental methods
In our experimental setup, the output of a self-mode-locking Ti:sapphire oscillator (Coherent Micra) was regeneratively amplified (Coherent Legend Elite) to produce a 5.0 kHz pulse train of 38 fs pulses centred at 806 nm with a spectral bandwidth of 35 nm. The beam was split using a 50 : 50 beam splitter (CVI), and a time delay (the waiting time) was introduced between the beams using a retroreflector mounted on a translation stage (Aerotech). The beams were focused onto a diffractive optic to give two pairs of phase-locked beams arranged in a box geometry. Another time delay (the coherence time) was introduced between one pair of beams by sending each beam through a pair of one-degree fused silica wedges (Almaz Optics) mounted on translation stages (Aerotech). The local oscillator (LO) beam was attenuated using neutral density filters with a total optical density of 3.1 at 809 nm to prevent significant interaction with the sample. The beams were focused to a spot size of approximately 70 µm in the sample, where the total incident power was 4.8 nJ (1.6 nJ per pulse). The LO was aligned 8 into a 0.3 m spectrometer (Andor Shamrock) and focused on a 1600 × 5 pixel region centred on a back-illuminated CCD (Andor Newton). Delay calibration was performed using spectral interferometry, as reported previously [26,31].
2D spectra were acquired for waiting times from 0 to 1800 fs in steps of 20 fs by varying the coherence time from −500 to 500 fs in steps of 4 fs. Multiple spectra taken at T = 0 were acquired throughout the experiment to monitor sample integrity. The pump-probe signal was acquired for the same series of waiting times to recover the absolute phase of the 2D spectra and separate the signal into absorptive (real) and dispersive (imaginary) parts [26]. In the following analysis, however, we sacrifice spectral resolution by using the absolute value rather than the absorptive part to avoid introducing phase errors in amplitude beating. Scatter subtraction, Fourier windowing and transformation to frequency-frequency space were performed, as reported elsewhere [26].
The FMO sample was isolated from Chlorobium tepidum, as described previously [32], and stored in 800 mM tris/HCl buffer (pH 8.0). This solution was mixed 65 : 35 v/v in glycerol with 0.1% lauryldimethylamine oxide detergent and loaded into a 200 µm quartz cell (Starna). The sample was cooled to 77 K in a cryostat (Oxford Instruments), and the optical density at 809 nm was measured to be 0.32.

Results
The amplitude of the cross-peak corresponding to excitons 1 and 3 was evaluated for each waiting time by integrating the absolute value of the rephasing signal over a 20 cm −1 × 20 cm −1 area of the spectrum centred at ω τ = 12 344 cm −1 and ω t = 12 164 cm −1 , as shown in figure 3(b). This position was chosen because it shows the strongest amplitude beating and the spectral coordinates are near those deduced from the Hamiltonian calculated by Adolphs and Renger (ω 3 = 12 373 cm −1 and ω 1 = 12 175 cm −1 ) [18]. The integrated signal decays as a function of T due to population transfer during the waiting time (see diagram 7 in figure 2), but subtraction of a multi-exponential fit with constant offset yields the beating signal presented in figure 3(c). A Fourier transform of the extracted beating signal shows primarily two frequency components, ω T = 160 and 207 cm −1 . The higher frequency closely corresponds to the predicted energy gap between excitons 1 and 3, so the lower frequency must correspond to the energy gap between excitons 1 and 2. Adolphs and Renger predict this frequency to be 135 cm −1 . A visual inspection of the resolved cross-peak clearly indicates that it is broadened along the coherence frequency axis, suggesting that the observed feature is in fact the sum of overlapping peaks. Furthermore, the relative contribution of the low-frequency component in the power spectrum increases as the area under analysis is red-shifted along the coherence frequency axis toward the maximum of the 1-2 cross-peaks.
In order to determine the dephasing rates for the 1-2 and 1-3 coherences, the beating signal was fit to a function of the form S(T ) = A 21 e −γ 21 T sin(ω 21 T + φ 21 ) + A 31 e −γ 31 T sin(ω 31 T + φ 31 ), using the frequencies obtained from the Fourier transform of the beating as initial guesses for the fitting parameters ω 21 and ω 31 . Data points for which T < 80 fs were not included in the fit to avoid pulse overlap effects. As displayed in figure 3(c), the fit successfully captures the non-uniform spacing between successive local maxima and the multimodality of the beating envelope that cannot be fit with a single frequency. The fit gives a lowfrequency component ω 21 = 160 ± 1 cm −1 with a dephasing rate of γ 21 = 29 ± 3 cm −1 and a high-frequency component ω 31 = 198 ± 2 cm −1 with a dephasing rate of γ 31 = 47 ± 9 cm −1 .  shows the spectrum acquired at T = 960 fs, when the amplitude is at a maximum. The spectra were normalized to their respective maxima and plotted using an arcsinh colour scale to highlight cross-peaks. In panel (c), the integrated amplitude of the cross-peak after removal of exponential population decay (blue line) is plotted against waiting time. The beating signal was fit to the sum of two independently decaying sine waves (red line) to find lifetimes for the two zero-quantum coherences that give rise to the beating.
These dephasing rates correspond to lifetimes of 1100 and 700 fs, respectively, although beating can still be clearly observed at 1800 fs. A similar analysis can be performed on one-quantum coherences by following a particular rephasing frequency over positive coherence time from the scatter-subtracted frequency-time spectrum taken at T = 300 fs. The dephasing rate of the exciton 3 coherence was found to be 320 ± 8 cm −1 , corresponding to a lifetime of 100 fs, in good agreement with the results of Adolphs and Renger. This fit is shown in the supplementary data, available from stacks.iop.org/NJP/12/065042/mmedia.

Discussion
The lifetimes of zero-quantum coherences in FMO are approximately an order of magnitude greater than those of one-quantum coherences, in agreement with the findings of Lee et al for a bacterial reaction centre [3]. As demonstrated by our model for bath-induced electronic transitions, this striking disparity between coherence lifetimes suggests that transition energy fluctuations are correlated across the complex, perhaps due to the relative spatial uniformity of protein dielectric fluctuations. The protection of coherences imparted by the protein environment facilitates coherent transfer during the initial stages of transfer toward the reaction centre, thereby improving the overall quantum efficiency. The degree to which the protein has been tuned by natural selection to improve efficiency, however, remains unclear. The significant error associated with the dephasing rates calculated for the 1-2 and 1-3 coherences prevents us from unambiguously determining whether the two rates are distinct or not, but we can state with 92% statistical confidence that the 1-2 coherence persists longer despite an approximately equivalent spacial overlap between exciton 1 and excitons 2 and 3. The present results, in conjunction with the two site-basis transfer pathways proposed by Ishizaki and Fleming [9], suggest evolutionary fine-tuning. In one pathway, the excitation is initially transferred from the chlorosome baseplate to BChl 1 before moving energetically uphill to BChl 2. These chromophores are the primary components of exciton 3 [16], so coherence between excitons 1 and 3 would be important in assisting this thermodynamically unfavourable process in the early stages of transport. Both pathways terminate at BChl 3, the chromophore coupled to the reaction centre [33] and the only significant component of exciton 1. The next lowest energy chromophore is BChl 4, a significant component of exciton 2, so coherence between excitons 1 and 2 could be important at longer times. Cheng et al [34] have suggested that the oscillatory site populations that arise from excitonic coherence can improve transfer efficiency beyond the thermodynamic limit when coupled to an energy trap, providing a specific role for long-lived 1-2 coherence. Coherence between excitons 1 and 3, however, would be detrimental at long times because it would facilitate transfer backwards along pathway 1, so we expect it to dephase faster in an optimized system. While the dephasing rates calculated for the 1-2 and 1-3 coherences have considerable error, the frequencies found to contribute to the cross-peak beating are quite precise and in good agreement with those found by Fourier decomposition. The 1-2 energy gap is inconsistent with the prediction of Adolphs and Renger, but fits the beating signal significantly better than the predicted value as shown in the supplementary data (available from stacks.iop.org/NJP/12/065042/mmedia). This method may therefore be useful in refining theoretical Hamiltonians, especially if all energy gaps can be determined. In this experiment, only one cross-peak had sufficient resolution to permit this level of analysis, but selectively probing cross-peaks using polarized pulses [15] or narrower bandwidth could be quite useful in this effort. Greater precision in dephasing rates could also provide insight into the microscopic mechanism of system-bath interactions in coherence transfer in this complex.

Conclusions
The observation of long-lived quantum coherence in FMO at 77 K [2] was of great interest not only to photosynthesis researchers, but also to the quantum computing community, where controlling decoherence in a dynamic environment remains a major challenge. Since that time, there have been reports of long-lived quantum coherence in many other photosynthetic pigment-protein complexes, including the reaction centre of Rhodobacter sphaeroides [3], LHCII from Arabidopsis thaliana [4] and most recently the phycobilisomes of marine algae at room temperature [5]. This growing body of work indicates that coherent energy transfer is not only possible under physiological conditions, but also prevalent among photosynthetic organisms. Here, we have calculated explicit dephasing rates for two excitonic coherences in FMO at 77 K and have demonstrated that the lifetimes of these coherences are indeed longer than population transfer times in this complex [35]. We also present a model for bath-induced electronic transitions, illustrating the role of the protein matrix in protecting coherences and facilitating incoherent energy transfer. Further studies refining these lifetimes and measuring the lifetimes of other coherences could provide insight into how we can control and optimize energy transfer efficiency in vitro by mimicking the manner in which nature has employed quantum coherence effects to permit autotrophy even under extremely low-light conditions.