Disentangling Dynamical Quantum Coherences in the Fenna-Matthews-Olson Complex

In the primary step of light-harvesting, the energy of a photon is captured in antenna chlorophyll as an exciton. Its efficient conversion to stored chemical potential occurs in the special pair reaction center, which has to be reached by down-hill ultrafast excited state energy transport. The interaction between the chromophores leads to spatial delocalization and quantum coherence effects, the importance of which depends on the coupling between the chlorophylls in relation to the intensity of the fluctuations and reorganization dynamics of the protein matrix, or bath. The latter induce uncorrelated modulations of the site energies, and thus quantum decoherence, and localization of the exciton. Current consensus is that under physiological conditions quantum decoherence occurs on the 10 fs time scale, and quantum coherence plays little role for the observed picosecond energy transfer dynamics. In this work, we reaffirm this from a different point of view by finding that the true onset of electronic quantum coherence only occurs at extremely low temperatures of ~20 K. We have directly determined the exciton coherence times by two-dimensional electronic spectroscopy of the Fenna-Matthew-Olson complex over an extensive temperature range with a supporting theoretical modelling. At 20 K, electronic coherences persist out to 200 fs (close to the antenna) and marginally up to 500 fs at the reaction-center side. It decays markedly faster with modest increases in temperature to become irrelevant above 150 K. This temperature dependence also allows disentangling the previously reported long-lived beatings. We show that they result from mixing vibrational coherences in the electronic ground state ...

We show that they result from mixing vibrational coherences in the electronic ground state.
We also uncover the relevant electronic coherence between excited electronic states and examine the temperature-dependent non-Markovianity of the transfer dynamics to show that the bath involves uncorrelated motions even to low temperatures. The observed temperature dependence allows a clear separation of the fragile electronic coherence from the robust vibrational coherence. The specific details of the critical bath interaction are treated through a theoretical model based on measured bath parameters that reproduces the temperature dependent dynamics. By this, we provide a complete picture of the bath interaction which places these systems in the regime of strong bath coupling. We believe this main conclusion to be generically valid for light harvesting systems. This principle makes the systems robust against otherwise fragile quantum effects as evidenced by the strong temperature dependence. We conclude that nature explicitly exploits decoherence or dissipation in engineering site energies to yield downhill energy gradients to unerringly direct energy, even on the fastest time scales of biological processes.
The question how biological function emerges from the atomic constituents of matter has intrigued scientists since the early days of quantum mechanics 1, 2 . Encouraged by the success of quantum theory to describe matter, its pioneers rapidly explored extending it to the world of chemistry and biology in the early 20th century. It was only in recent times when modern ultrafast spectroscopic tools became available that the search for nontrivial quantum effects in the primary steps of biological processes was made possible, which led to the prospect for the foundation of the field of quantum biology [3][4][5][6] . Recent experimental results obtained for the well-characterized Fenna-Mathews-Olson (FMO) protein complex have been interpreted as evidence of long-lived electronic quantum coherence in the primary steps of the energy transfer [7][8][9][10] . A functional role of long-lived quantum coherence was proposed in that it would speed up the transfer of excitation energy under ambient conditions 11 . These works triggered tremendous interest in different fields, ranging from quantum chemistry to quantum information science. A key parameter is the strength of the coupling of the exciton system to environmental fluctuations, which is related to the reorganization energy. For a conceptual understanding, initial theoretical analysis was built upon the choice of a rather small reorganization energy of 35 cm −1 to fit the reported long lifetime of electronic coherence 11 . Yet, even with this small value, Shi and coworkers found a shorter lifetime for the expected electronic coherence from more advanced calculations of the experimental 2D electronic spectra 12 . The interpretation of the long lifetime of the electronic coherence was further questioned by numerically exact results obtained within the quasi adiabatic propagator path integral method with an experimentally determined spectral density with a considerably larger reorganization energy 13 . Coker et al. and Kleinekathöfer et al. have calculated site-dependent reorganization energies with refined atomic details by advanced molecular dynamics simulations 14,15 . They found significantly larger values of the reorganization energies in the range of 150 to 200 cm −1 . With this disagreement, we have revisited the energy transfer of the FMO complex at room temperature experimentally 16 using 2D electronic spectroscopy to extract the electronic coherence time scales. Instead of a long-lived electronic coherence, the experiment, after having passed a self-consistency verification, yielded a considerably shorter coherence lifetime of 60 fs. This observed timescale for decoherence excludes any functional role for coherent energy transfer in the FMO complex, which occurs on the time scale of several picoseconds at room temperature.
Another potential key role for the electronic coherence is played by the pigment-protein host molecular vibrations [17][18][19][20] . In contrast to electronic coherence, the pigment-localized vibrations typically last for picoseconds but are not expected to enhance energy transfer in general. Yet, Plenio and coworkers have suggested the concept of vibrationally enhanced electronic coherence 21 . They reported that in a vibronic model dimer, electronic quantum coherence may be resonantly enhanced by long-lasting vibrational coherence 22 . Instead of an enhancement, Tiwari et al. alternatively suggested that nonadiabatic electronic-vibrational mixing may resonantly enhance the amplitude of particular, delocalized anticorrelated vibrational modes of the electronic ground state 23 . While in principle, this mechanism is also possible in the presence of weak electronic dephasing 24 , realistic values of the strengths of the electronic and vibrational dampings leads to a complete suppression of this mechanism 25 . Electronic-vibrational mixing was also examined in a simple dimer model 26 , but the subsequent theoretical calculations show no evidence of an enhancement of the electronic coherence 27 . More recently, the coherent exciton transfer in the FMO complex has been revisited by Zigmantas and coworkers at 77 K 28 . The long-lived oscillations have been carefully assigned to the vibrational coherence of the electronic ground state. Due to strong dissipation, the lifetime of the electronic coherence was too short to be precisely determined even at 77 K. Despite the extensive work on this problem, a complete picture of the electronic coherence and its role in the electronic-vibrational mixing for the energy transfer in the FMO complex is still elusive.
Here, we study the energy transfer process in the FMO complex with the explicit aim to ob-serve clear evidence for the onset of electronic coherence effects in energy transport by decreasing temperature. For this, we measure the 2D electronic spectra of the FMO complex in the regime of very low temperature. Specifically, we examine the electronic dephasing by directly measuring the anti-diagonal bandwidth of the main peaks, along with the decays in the cross peaks related to the inter-exciton coupling. It is only at very low temperatures (20 K) that the amplitudes and decays in these electronic coherence signatures become comparable to the energy transfer times. We provide a comprehensive analysis by a global fitting approach, and the subsequent Tukey window Fourier transform allows us to disentangle the electronic coherence from vibrational coherence.
By this, we uncover that the longest lived electronic coherence is observable up to 500 fs between the two excitons closest to the reaction centre side. Due to down-hill energy transfer, the electronic coherence of the two higher-energy excitons close to the antenna side shows a much faster decay with a lifetime <60 fs. We furthermore measure the coherent energy transfer over an extensive temperature range. Based on these temperature-dependent measurements, we are able to construct a unifying exciton model, which captures the coherent energy transfer over the entire temperature range studied. Moreover, we investigate the temperature-dependent non-Markovianity of the transfer dynamics to show that the bath fluctuations are uncorrelated even at low temperatures. By this unprecedented combination of experimental and theoretical efforts, we are able to provide a complete picture of quantum coherent effects in the FMO complex over the entire regime from high to low temperatures in one experiment and one theoretical model. Due to the generic structure of the FMO protein, we expect our observations to be extended to other more complicated photosynthetic protein complexes and even photovoltaic devices 29 .

Results
The solution of the FMO protein complex is prepared in a home-built sample cell and mounted in a cryostat (Oxford Instrument). More details of the sample preparation are given in the Materials and Methods section. Fig. 1(a) shows the structural arrangement of the bacteriochlorophyll a (Bchla) chromophores embedded in the protein matrix (data from 3ENI.pdb). The measured absorption spectrum of the FMO complex at 80 K and the laser spectrum used in this study are shown in the SI.
Two-dimensional electronic spectroscopy We measure the 2D electronic spectra of the FMO complex in the range from 20 to 150 K. The details of the 2D spectrometer are given in the Materials and Methods section. The real parts of the 2D electronic spectra at 20 K for selected waiting times T are shown in Figs. 1(b-e). The positive and negative amplitudes in the peaks represent the excitation transitions of the ground-state-bleach (GSB) and the excited-state-absorption (ESA), respectively. In Fig. 1(b), we show the measured 2D electronic spectrum at 20 K for T=30 fs. The exciton states in the FMO complex are located in the frequency range from 12120 to 12700 cm −1 , which is marked by black dashed lines. At T=30 fs, we observe a dramatic stretch of the main peaks along the diagonal, which illustrates the strong inhomogeneous broadening. In addition, one off-diagonal feature corresponding to the ESA is observed at (ω τ , ω t ) = (12300, 12580) cm −1 . At T=50 fs as shown in Fig. 1(c), we observe that the 2D spectrum has not changed significantly, except that the elongation of the main diagonal peaks is slightly reduced. However, the elongation of the main peaks along the diagonal has dramatically reduced again at T=510 fs in Fig. 1(d).
Moreover, we observe that the main peaks of the higher exciton states are replaced by one peak with ESA features. A new cross peak appears at (ω τ , ω t ) = (12340, 12120) cm −1 , which provides evidence of the down-hill energy transfer from higher exciton states to the lowest ones. Its amplitude is further increased at T=1005 fs, see Fig. 1(e). In addition to the main and cross peaks in the frequency range from 12120 to 12700 cm −1 , we observe more cross peaks appearing at the upper-left side of the 2D electronic spectrum at T=1005 fs. This provides evidence of the vibrational progression in the FMO complex. To examine the lifetime of the electronic dephasing, we analyze the anti-diagonal bandwidth of the lowest exciton peak at (ω τ , ω t ) = (12120, 12120) cm −1 for T=30 fs. To quantify the associated lifetime of the optical coherence, the broadening of the peaks is modeled by Lorentizan lineshapes. More details of the fitting procedure are described in the SI. By this, we are able to capture the electronic dephasing between the electronic ground and Energy transfer and coherent dynamics To examine the time-dependent coherent dynamics in the 2D spectra, we extract the magnitudes of the cross peaks at different waiting times. In Fig.   2(a), the trace (red line) represents the time evolution of the amplitude of the cross peak at (ω τ , ω t ) = (12340, 12120) cm −1 between exciton 1 and 3 (marked as 'CP13' in Fig. 1(d)). The underlying kinetics (black dashed line) is fitted by an exponential function and the resulting residual is shown as a black solid line in Fig. 2(b). The raw data of the oscillations are further purified by a Fourier filter with a Tukey window (<1000 cm −1 ) in Fig. 2(b). With this refined trace, we retrieve the coherent dynamics by performing a wavelet analysis. The details of the Tukey window Fourier transform and the wavelet analysis are given in the SI. The time evolution of the cross-peak coherence is shown in Fig. 2 Having analyzed the vibrational coherences, we next examine the time scales and the path-ways of the energy transfer by the global fitting approach 31 . First, we construct a 3D data set by combining a series of 2D electronic spectra with evolving waiting times. At least two exponential functions have been used to achieve a converged fitting. This yields the time constants of the en- Electronic quantum coherence To capture the signature of electronic coherence, we address the cross-peak dynamics associated to the two excitons 1 and 2 with lowest energy. We take the timedependent magnitude of the cross peak at (ω τ , ω t ) = (12120, 12270) cm −1 (marked as 'CP21' in Fig. 1(b)) to minimize contributions by the energy transfer dynamics. Again, the residual obtained after removing the kinetics and after polishing by a Tukey window Fourier transform is shown as black circles in Fig. 3(a). Here, the oscillations are induced by the electronic coherence convoluted with vibrational coherences. To disentangle both, the oscillations and decay rates are fitted to exponentially decaying sine functions to extract the oscillation frequencies and the lifetimes of the coherences. We start the fit by assuming the frequencies 68, 150, 180 and 202 cm −1 , which represent the vibrational modes found experimentally as discussed above and which agree with the known modes from the FLN experiment 30 . Moreover, we have obtained the electronic energy gap of 150 cm −1 between exciton 1 and 2 by theoretical calculations (see below), which also agrees with previous results 32 . In addition, to remove the low-frequency oscillations, one additional frequency of 17 cm −1 is included to achieve the best fit with R-square > 0.97. It represents the lowest frequency resolved associated to our basic time step. All the fitting procedures are performed using the Curve Fitting Toolbox in Matlab 2013(b); the details are given in the SI. We show the highquality fitting results by the red solid line in Fig. 3(a). The green shadow indicates the boundaries of 95% of fit confidence. This effectively allows us to separate the electronic coherence of 150 cm −1 from vibrational coherences. The oscillation related to the electronic coherence is shown in Fig. 3(b) and yields a decay time constant of 105±26 fs. With the frequency of 150 cm −1 , it is clearly observed that the electronic coherence is sustained over only two oscillation periods and disappears within 500 fs completely. More important, we observe that the identified oscillations of the electronic coherence are quite significant. They are larger than 5% of the maximum strength of the 2D spectra at 20 K. In addition, the identified vibrational coherences are shown in the SI.
Following the same procedures, we analyze the coherent dynamics of the cross peak at (ω τ , ω t ) = (12120, 12270) cm −1 (CP21) at different temperatures (50, 80 and 150 K) and plot the corresponding traces in Fig. 3(c), (e) and (g), respectively. The decay time constants for different temperatures are shown in Fig. 1 (j) marked as "decoherence". The resolved extracted electronic coherences are shown in Fig. 3(d), (f) and (h), respectively. Noticeably, at 50 K, the electronic coherence lasts less than 500 fs, with a decay time constant 96 ± 40 fs. The lifetime of the electronic coherence is significantly reduced at 80 K. We measure a decay time constant of 81 ± 26 fs. At 150 K, the red solid line in Fig. 3(h) clearly shows that the electronic coherence is strongly damped and the oscillation does not survive even a single oscillation period. Again, the associated vibrational coherences retrieved during the fitting procedures are shown in the SI.
To study the coherence of excitons in states of higher energy, we monitor the dynamics of ESA peaks at ω t = 12580 cm −1 . Our theoretical calculations (see details in the next section) retrieved energy levels of the excitons 2, 3, 5 and 7 highlighted by the corresponding ω τ -lines in Fig. 4(a). The intersection points of two marker lines are denoted as A, B, C and D, respectively, to which the excited state dynamics of the excitons 2, 3, 5 and 7 corresponds. Following the same procedure as above, we extract the time evolution of the amplitude of the cross peaks at A, B, C and D and remove the underlying kinetics by the global fitting approach. The obtained residuals of these peaks after the Fourier treatment are shown in Fig. 4(b), (c), (e) and (f), respectively, for increasing waiting times. More details of the fitting procedure are presented in the SI. By this, a high-quality fit is obtained, which is shown as the red solid line in Fig. 4(b). The extracted electronic coherence is presented as red solid line in Fig. 4(d). We find the frequency of 210 cm −1 of this coherent oscillation, which is in excellent agreement with the energy gap between exciton 2 and 5 from our theoretical calculations. Next, we examine the subsequent coherent dynamics of the ESA peak B. From our theoretical analysis, we expect this to be a peak that originates from the ESA of exciton 5. After repeating the fitting procedure described above, we obtain the residual shown in Fig. 4(c). The measured residuals and the results of the fits are shown as black dots and blue solid line, respectively. Furthermore, we show the separate electronic coherence as blue solid line in Fig. 4(d). Again, the obtained frequency of 206 cm −1 matches the energy gap between exciton 5 and 2 exactly. Interestingly, these well-resolved electronic coherences in Fig. 4(d) show evidence of anti-correlated oscillations with a slight phase offset. From our theoretical analysis, we uncover the coherence between exciton 2 and 5 is dominated by the strong electronic couplings of pigment 4 and (5, 6). Details of the transformation from the site to the exciton basis are given in the SI. The residuals of the ESA peaks C and D are shown as black dots in Fig. 4(e) and (f), respectively. The fits to the oscillations are shown as red and blue solid lines in Fig. 4(e) and (f) and the extracted electronic coherences are shown as red and blue solid lines in Fig. 4(g). We observe two electronic coherent oscillations both with the frequency of ∼310 cm −1 , which agrees perfectly with the energy gap between exciton 3 and 7. As determined by the basis transformation, this coherence is dominated by the electronic coupling between pigment 1 and 2. Moreover, compared to the lifetime of the coherence between exciton 2 and 5, these coherences show smaller time constants of 34 and 59 fs. Hence, the electronic coherence lasts shorter for higher exciton states, which is due to the down-hill energy transfer. Moreover, the larger energy gap between exciton 3 and 7 produces a shorter lifetime of the electronic coherence due to the faster energy transfer.
Theoretical calculations We construct a Frenkel-exciton model to study the coherent FMO dynamics. The electronic transitions in the pigments are approximated by optical transitions between two energy eigenstates and the electronic couplings between pigments are calculated within the dipole approximation. Moreover, to include fluctuations of the electrostatic interactions, the system is linearly coupled to a harmonic reservoir. For simplicity, we consider a standard Drude bath spectral density. Moreover, a particular localized vibrational mode of 180 cm −1 is coupled to the electronic system to investigate the role of vibrational/vibronic coherence. This vibrational mode is known to be the most relevant out of a group of 44 vibrational modes. The 2D electronic spectra are calculated by a time non-local quantum master equation 33,34 and the time evolution of the peaks are obtained by the equation-of-motion phase-matching approach 35 . More details are given in the Materials and Methods section and the SI. We initially choose the site energies from previous works 32 and then optimize them by simultaneously fitting to the experimental absorption spectra measured at different temperatures. After that, we calculate the 2D electronic spectra and refine then the system-bath interaction strength by comparing the calculated electronic dephasing lifetimes to the experimental ones at different temperatures. By this, we are able to develop a unique system-bath model with a single set of parameters. In particular, we are now able to calculate the 2D spectra. We show the calculated waiting time traces of the cross peak at (ω τ , ω t ) = (12120, . We observe that the electronic coherence lasts for two oscillation periods and has disappeared at 500 fs. More importantly, based on the wavelet analysis, the oscillation phase retrieved in (b) perfectly matches that revealed by the experimental data shown in Fig. 3(d). More details of the comparison are shown in the SI. Moreover, the electronic (black dashed line) and vibrational (black dash-dot line) coherences at 80 and 150 K are shown in Fig. 5(d) and (f). On the basis of the results at different temperatures, we can safely conclude that the lifetime of the vibrational coherence at 180 cm −1 is not significantly shortened for increasing temperature. However, the lifetime of the electronic coherence is dramatically reduced when temperature is increased. The same conclusion can be drawn for the higher-energy excitons at different temperatures; the details of the coherent lifetimes of higher excitons are shown in the SI.

Discussion
An important question related to the exciton transfer dynamics is about the nature of the bathinduced fluctuations. It may be characterized by the measure of the non-Markovianity which quantifies how strongly the dephasing and relaxation dynamics departs from ordinary Markovian, i.e., memory-less behaviour. In principle, a highly structured environment consisting of several localized vibrational modes of the FMO protein may give rise to significant non-Markovian dynamics. Early numerically exact path-integral calculations 36 on the basis of an experimentally determined spectral density have shown that the exciton dynamics is purely Markovian at ambient conditions. This expectation has been confirmed recently experimentally by comparing the decay time of optical dephasing and electronic quantum coherence in the FMO complex 16 . The equivalence of the time scales of the optical dephasing and the electronic decoherence reveals that the dynamics of energy transfer in the FMO complex at room temperature is fully Markovian.
Here, we provide a complete picture of the role of the non-Markovianity in the FMO complex at different temperatures. As discussed above, we have measured, at 20 K, the longest life time of the electronic quantum coherence for the dephasing of the two lowest-energy excitons, with the decay time of 105 fs. On the other hand, the analysis of the anti-diagonal band width yielded a decay time of 197 fs for the optical dephasing of excitons 1. The difference of almost a factor of 2 is due to the low temperature, but is still covered by a fully Markovian description of the transfer dynamics 38 . This finding is also in agreement with low-temperature calculations 36 . On the basis of these studies, we conclude that non-Markovian energy fluctuations of the pigments induced by pigment-hosted molecular vibrations do not play a role in the energy transfer of the FMO complex. Even, due to its simple structure, we believe that this conclusion can be extended to more complicated photosynthetic protein complexes.
Instead of long-lived electronic coherence, our study uncovers a long lasting beating dynamics composed of vibrational coherences in the range of 180 cm −1 in the electronic ground state.
As summarized in the Introduction, several studies have suggested a resonant enhancement of the short-lived electronic coherence by the long-lived vibrational coherence in the FMO complex.
However, our work clearly shows no such a resonant enhancement of the electronic coherence during the population transfer. In contrast, from the measured and calculated 2D electronic spectra, we have retrieved the scale of the reorganization energy of 120 cm −1 , which manifests a quite strong system-bath interaction that rapidly destroys the phase of the electronic quantum coherences between pigments in the FMO complex. Hence, instead of the energy transfer being enhanced by strongly delocalized exciton wave functions, the rather large reorganization energy sharpens the limits for delocalization and the energy pathways between pigments. Then, the efficient down-hill transfers of not too strongly delocalized excitons are determined by a simple thermal distribution of excitons which rapidly arises after an initial ultrafast non-equilibrium dynamics triggered by the photo-excitation.

Conclusions
In this paper, we provide a complete picture of the coherent contribution to energy transfer in the FMO complex by 2D electronic spectroscopy in the entire regime from low to high temperatures.
In particular, the spectroscopic measurements at low temperature of 20 K allows us to provide unambiguous evidence of the lifetime of the electronic quantum coherence and to disentangle the electronic coherence from long-lived vibrational coherence. Interestingly, due to the down-hill energy transfer, the electronic coherence between the two lowest excitons is marginally observable out to 500 fs at 20 K. However, the coherence lifetime of higher excitons is dramatically reduced by the population transfers. This analysis allows us to disentangle the previously reported long-lived beating of cross-peak signals and to show that they are composed of mixed ground state molecular Raman modes. Moreover, we uncover that the lifetime of electronic coherence is significantly modulated by temperature, while, in contrast, the resonant beatings of vibrational coherences last for picoseconds even at 150 K. A thorough analysis on the basis of a unique combination of experimental data and theoretical modelling enables us to provide a reliable estimate for the decisive parameter of the reorganization energy of the FMO complex. We find a reorganization energy of 120 cm −1 , which represents a strong system-bath interaction of the pigments with their protein environment. This coupling is sufficient to significantly reduce the lifetime of electronic coherences and leads to a rapid intermittent localization of the electronic wavefunction on a few molecular sites. Instead of a long-lived quantum coherent energy transfer, we provide a different picture of a down-hill energy transfer, in which the pathways of population transfers are dynamically constructed simply by following lower site energies of the pigments involved and by rather fast electronic damping due to significant nuclear reorganization in the excited state. The latter sharpens the funnels of the energy flow in the FMO complex. In general, we conclude that the energy transfer in the FMO complex is dominated by the thermal dynamics of weakly delocalized excitons after an initial ultrafast non-equilibrium photon-excitation. Due to common features in the bath for all light harvesting systems, despite the relative simplicity of FMO, we believe that this conclusion can be further extended to the other more complicated photosynthetic protein complexes.

Materials and Methods
Sample preparation. The FMO protein was isolated from the green sulfur bacteria C. tepidum (see the SI for more details). The sample was dissolved in a Tris buffer at PH 8.0. It was filtered with a 0.2 µm filter to reduce light scattering. The sample was then mixed 70:30 v/v in glycerol and kept in a home-built cell with an optical pass length of 500 µm. The cell was mounted in the cryostat (MicrostatHe-R) for the low-temperature measurements.
2D Electronic measurements with experimental conditions. Details of the experimental setup have already been described in earlier reports from our group 16  Phasing of the obtained 2D spectra was performed using an "invariance theorem" 37 .
The c mξ is the coupling constant between the mth pigment and ξth fluctuation mode. The bath is specified by the spectral density J m (ω) = π ξh 2 c 2 mξ δ(ω − ω mξ ). We include one overdamped mode and one underdamped mode to study the impact of vibrational coherence. The corresponding spectral density can be expressed as J(ω) = 2ΛΓω Here, Λ and Γ −1 are the damping strength and the bath relaxation time of the overdamped mode, respectively. S, ω vib and γ −1 vib are the Huang-Rhys factor, the vibrational frequency and the vibrational relaxation time of the underdamped mode, respectively. This form has been shown to describe the experimental data 16 correctly.
The nonequilibrium dynamics of the system-bath model is calculated by a time-nonlocal quantum master equation. The details of this method are described in the SI. Correlation theory is used to calculate the absorption spectrum of the FMO complex, where ρ g = |g g| and a δ-shaped laser pulse is assumed. · rot denotes the rotational average of the molecules with respect to the laser direction. Moreover, the 2D electronic spectra are obtained by calculating the third-order response function Here, τ is the delay time between the second and the first pulse, T (the so-called waiting time) is the delay time between the third and the second pulse, and t is the detection time. To evaluate 2D electronic spectra, we need the rephasing (RP) and non-rephasing (NR) contributions of the thirdorder response function, i.e., S (3) (t, T, τ ) = S NR (t, T, τ ). Assuming the impulsive limit (the δ-shaped laser pulse), one obtains The total 2D signal is the sum of the two, i.e., I(ω t , T, ω τ ) = I RP (ω t , T, ω τ ) + I NR (ω t , T, ω τ ).
The model parameters of the site energies and electronic couplings are initially taken from Ref. 32 and further refined during a simultaneous fit to the absorption spectra of the FMO complex at the considered temperature. To precisely determine the reorganization energy, the parameters are further refined by fitting of the theoretical results to the experimental anti-diagonal bandwidth of the main peak of exciton 1.