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

Hong-Guang Duan1,2,3,∗ Ajay Jha1,4,∗ Lipeng Chen5,∗ Vandana Tiwari1,6 Richard
J. Cogdell7 Khuram Ashraf7 Valentyn I. Prokhorenko1 Michael Thorwart2,3 & R.
J. Dwayne Miller1,3,8

###### Abstract

In the primary step of natural light-harvesting, the energy of a solar photon
is captured in antenna chlorophyll as a photoexcited electron-hole pair, or 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. Key to this process is the degree of
interaction between the chlorophyll chromophores that can lead to spatial
delocalization and quantum coherence effects. The importance of quantum
contributions to energy transport depends on the relative coupling between the
chlorophylls in relation to the intensity of the fluctuations and
reorganization dynamics of the surrounding protein matrix, or bath. The latter
induce uncorrelated modulations of the site energies, resulting in quantum
decoherence, and localization of the spatial extent of the exciton. The
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
important electronic quantum coherence only occurs at extremely low
temperatures of $\sim$20 K. We have directly determined the exciton coherence
times using two-dimensional (2D) electronic spectroscopy of the Fenna-Matthew-
Olson (FMO) complex over an extensive temperature range. At 20 K, we show that
electronic coherences persist out to 200 fs (close to the antenna) and
marginally up to 500 fs at the reaction-center side. The electronic coherence
is found to decay 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 thought to be
evidence for electronic quantum coherence contributions. 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.

Max Planck Institute for the Structure and Dynamics of Matter, Luruper
Chaussee 149, 22761, Hamburg, Germany

I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9,
20355 Hamburg, Germany

The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg,
Germany

The Rosalind Franklin Institute, Rutherford Appleton Laboratory, Harwell
Campus, Didcot, Oxfordshire, OX11 OFA, United Kingdom

Department of Chemistry, Technische Universität München, D-85747 Garching,
Germany

Department of Chemistry, Universität Hamburg, Martin-Luther-King Platz 6,
20146 Hamburg, Germany

Institute of Molecular, Cell, and Systems Biology, College of Medical,
Veterinary, and Life Science, University of Glasgow, Glasgow G12 8QQ, United
Kingdom

The Departments of Chemistry and Physics, University of Toronto, 80 St. George
Street, Toronto Canada M5S 3H6
∗These authors contributed equally to this work.

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 observe 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.

###  0.1 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 ($\omega_{\tau}$,
$\omega_{t}$) = (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 ($\omega_{\tau}$, $\omega_{t}$) = (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 ($\omega_{\tau}$, $\omega_{t}$) = (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 excited states for
different temperatures, see Figs. 1(f)-(i). We find the lifetimes of the
electronic dephasing of 197, 181, 147 and 75 fs for the temperatures 20, 50,
80 and 150 K, respectively, which are shown in Fig. 1(j), marked by
“dephasing”.

###  0.2 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 ($\omega_{\tau}$, $\omega_{t}$) = (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
($<$1000 cm-1) in Fig. 2(b). With this refined trace, we retrieve the coherent
dynamics by performing a wavelet analysis. The details of the Tukey window
Fourier transform and the wavelet analysis are given in the SI. The time
evolution of the cross-peak coherence is shown in Fig. 2(c). We are able to
resolve oscillations with frequencies in the range from 160 to 200 cm-1.
Clearly, these identified mode frequencies are in excellent agreement with the
vibrational modes in the FMO complex identified by fluorescence-line-narrowing
(FLN) studies [30]. More specifically, we recover the modes at 167, 180, 191
and 202 cm-1 on the electronic ground state, which are highlighted by black
dashed lines in Fig. 2(c). Since the frequencies lie close-by, we observe
clear evidence of beating of vibrational oscillations at waiting times between
1000 and 1700 fs. Our analysis clearly reveals that the beatings originate
from the resonant enhancement of individual vibrational (electronic ground
state) modes at close-by frequencies. These vibrational beatings have been
incorrectly assigned to the enhancement of electronic coherence by long-
lasting vibrational coherences. However, our analysis now uncovers that this
enhancement is purely caused by resonant beating of vibrational molecular
modes in the electronic ground state. Moreover, we show the traces of the
cross peak amplitudes (red solid line) at ($\omega_{\tau}$, $\omega_{t}$) =
(12570, 12480) cm-1, CP56 (marked in Fig. 1(e)), the associated fits (black
dashed line) in Fig. 2(d)., and the polished residuals (black solid line)
stemming from the Tukey window Fourier transforms in Fig. 2(e). We plot the
obtained results of the wavelet analysis, which yield the coherent dynamics of
the residuals in Fig. 2(f). The resolved modes are marked as black dashed
lines. We find the mode frequencies of 46, 68, 117, 202, 243 cm-1, which
perfectly agree with those revealed by FLN experiments [30].

Having analyzed the vibrational coherences, we next examine the time scales
and the pathways 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 energy transfer of 160 fs and 8.8 ps, respectively. The corresponding two
decay-associated spectra are shown in Figs. S6 and S7 of the SI. It is
interesting that the fastest component of the energy transfer is 160 fs and
thus close to the retrieved life time of the electronic coherence (described
in the next section). Hence, at this very low temperature, this dynamical
component of the energy transfer may largely be mediated by electronic quantum
coherence. The ultrafast timescales and the multiple pathways of the energy
transfer are described in detail in the SI. Moreover, we also show in the SI
the other decay-associated spectrum related to the time constant of 8.8 ps. In
particular, this shows clear evidence of a down-hill energy transfer from
higher-energy excitons to the lowest-energy exciton.

###  0.3 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 time-dependent magnitude of the cross peak at ($\omega_{\tau}$,
$\omega_{t}$) = (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 high-quality 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$\pm$26 fs. With the frequency of 150 cm-1, it is clearly
observed that the electronic coherence is sustained over only two oscillation
periods and disappears within 500 fs completely. More important, we observe
that the identified oscillations of the electronic coherence are quite
significant. They are larger than 5$\%$ 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 ($\omega_{\tau}$, $\omega_{t}$) = (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 $\pm$ 40 fs. The lifetime of the
electronic coherence is significantly reduced at 80 K. We measure a decay time
constant of 81 $\pm$ 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 $\omega_{t}$ = 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 $\omega_{\tau}$-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 $\sim$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.

###  0.4 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 ($\omega_{\tau}$, $\omega_{t}$) = (12120, 12270) cm-1
(CP21) in Fig. 5 which reveal the coherent dynamics between exciton 1 and 2.
The obtained red solid lines at 50, 80 and 150 K are plotted in Fig. 5(a), (c)
and (e), respectively. As above, we analyze the calculated data again by the
global fitting approach. The so obtained fits of the traces are plotted as
black dashed lines and the retrieved residuals are shown as black solid lines
in Fig. 5(a), (c) and (e). To calculate the time-dependent cross-peak
dynamics, we use the wavelet analysis aiming to separate the electronic and
vibrational coherences. The results are shown in Fig. 5(b), (d) and (f). We
find, at 50 K, clear evidence of long-lived vibrational coherence at 180 cm-1,
as marked by the dashed-dotted line in (b). In addition to the vibrational
mode frequency, clear evidence of electronic coherence is highlighted in the
wavelet spectrum by the black dashed line in (b). 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 bath-induced 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

###  0.5 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 $\mu$m 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 $\mu$m. The cell was mounted in the cryostat
(MicrostatHe-R) for the low-temperature measurements.

###  0.6 2D Electronic measurements with experimental conditions.

Details of the experimental setup have already been described in earlier
reports from our group [16]. Briefly, the measurements have been performed on
a diffractive optics based, all-reflective 2D spectrometer, with a phase
stability of $\lambda/160$. The laser beam from a home-built nonlinear optical
parametric amplifier (NOPA, pumped by a commercial femtosecond Pharos laser
from Light Conversion) is compressed to $\sim$16 fs using the combination of a
deformable mirror (OKO Technologies) and a prism pair. Frequency-resolved
optical grating (FROG) measurement is used to characterize the temporal
profile of the compressed beam and the obtained FROG traces are evaluated
using a commercial program FROG3 (Femtosecond Technologies). A broadband
spectrum so obtained carried a linewidth of $\sim$100 nm (FWHM) centered at
750nm. Three pulses are focused on the sample with the spot size of $\sim$100
$\mu$m and the photon echo signal is generated in the phase-matching
direction. The photon-echo signals are collected using a Sciencetech
spectrometer model 9055 which is coupled to a CCD linear array camera
(Entwicklungsbüro Stresing). The 2D spectra for each waiting time T were
collected by scanning the delay time $\tau=t_{1}-t_{2}$ in the range of [-350
fs, 200 fs] with a delay time step of 2 fs. At each delay step, 100 spectra
were averaged to reduce the noise. The waiting time $T=t_{3}-t_{2}$ was
linearly scanned in the range of 2.0 ps in steps of 15 fs. For all
measurements, the energy of the excitation pulse is attenuated to 8 nJ with 1
kHz repetition rates. Phasing of the obtained 2D spectra was performed using
an “invariance theorem” [37].

###  0.7 Theoretical calculations.

A Frenkel-exciton model is constructed to calculate the coherent energy
transfer and the 2D electronic spectra of the FMO complex. The total
Hamiltonian is constructed in the form of the system, bath and system-bath
interaction terms, $H=H_{S}+H_{B}+H_{SB}$. The system Hamiltonian is given as
$H_{S}=\epsilon_{g}|g\rangle\langle{g}|+\sum_{m}^{N}\epsilon_{m}|m\rangle\langle{m}|+\sum_{m\neq{l}}^{N}J_{ml}|m\rangle\langle{l}|$,
where $\epsilon_{g}$ and $\epsilon_{m}$ are the site energies of ground and
$m$th excited pigment, respectively. $J_{ml}$ is the electronic interaction
between the $m$th and $l$th pigments. $N=7$ pigments are included in the
monomeric FMO complex. Moreover, each pigment is coupled to its own individual
bath. The bath Hamiltonian can be written as
$H_{B}=\sum_{m}^{N}\sum_{\xi}\hbar\omega_{m\xi}b_{m\xi}^{\dagger}b_{m\xi}$,
where, $b_{m\xi}^{\dagger}(b_{m\xi})$ is the creation (annihilation) operator
of $\xi$th fluctuation mode associated with site $m$, and its angular
frequency $\omega_{m\xi}$. $H_{SB}=\sum_{m}V_{m}W_{m}$ describes the
interaction of the system with the bath, where, we have defined
$V_{m}=|m\rangle\langle{m}|$ and
$W_{m}=-\sum_{\xi}c_{m\xi}(b_{m\xi}^{\dagger}+b_{m\xi})$. The $c_{m\xi}$ is
the coupling constant between the $m$th pigment and $\xi$th fluctuation mode.
The bath is specified by the spectral density
$J_{m}(\omega)=\pi\sum_{\xi}\hbar^{2}c_{m\xi}^{2}\delta(\omega-\omega_{m\xi})$.
We include one overdamped mode and one underdamped mode to study the impact of
vibrational coherence. The corresponding spectral density can be expressed as
$J(\omega)=\frac{2\Lambda\Gamma\omega}{\omega^{2}+\Gamma^{2}}+\frac{4S\gamma_{\mathrm{vib}}\omega_{\mathrm{vib}}^{3}\omega}{\left(\omega^{2}-\omega_{\mathrm{vib}}^{2}\right)^{2}+4\gamma_{\mathrm{vib}}^{2}\omega^{2}}$.
Here, $\Lambda$ and $\Gamma^{-1}$ are the damping strength and the bath
relaxation time of the overdamped mode, respectively. $S$,
$\omega_{\mathrm{vib}}$ and $\gamma_{\mathrm{vib}}^{-1}$ 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,
$I(\omega)=\left\langle\int_{0}^{\infty}dte^{i\omega{t}}\mathrm{tr}(\bm{\mu}(t)\bm{\mu}(0)\rho_{g})\right\rangle_{\mathrm{rot}}$,
where $\rho_{g}=|g\rangle\langle{g}|$ and a $\delta$-shaped laser pulse is
assumed. $\langle\cdot\rangle_{\mathrm{rot}}$ 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

$S^{(3)}(t,T,\tau)=\left(\frac{i}{\hbar}\right)^{3}\Theta(t)\Theta(T)\Theta(\tau)\mathrm{tr}\left(\bm{\mu}(t+T+\tau)\left[\bm{\mu}(T+\tau),\left[\bm{\mu}(\tau),\left[\bm{\mathrm{\mu}}(0),\rho_{g}\right]\right]\right]\right).$
(1)

Here, $\tau$ 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 third-order response function, i.e.,
$S^{(3)}(t,T,\tau)=S_{\mathrm{RP}}^{(3)}(t,T,\tau)+S_{\mathrm{NR}}^{(3)}(t,T,\tau)$.
Assuming the impulsive limit (the $\delta$-shaped laser pulse), one obtains

$\displaystyle I_{\mathrm{RP}}(\omega_{t},T,\omega_{\tau})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}d\tau\int_{-\infty}^{\infty}dte^{i\omega_{t}t-i\omega_{\tau}\tau}S_{\mathrm{RP}}^{(3)}(t,T,\tau),$
(2) $\displaystyle I_{\mathrm{NR}}(\omega_{t},T,\omega_{\tau})$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}d\tau\int_{-\infty}^{\infty}dte^{i\omega_{t}t+i\omega_{\tau}\tau}S_{\mathrm{NR}}^{(3)}(t,T,\tau).$
(3)

The total 2D signal is the sum of the two, i.e.,
$I(\omega_{t},T,\omega_{\tau})=I_{\mathrm{RP}}(\omega_{t},T,\omega_{\tau})+I_{\mathrm{NR}}(\omega_{t},T,\omega_{\tau})$.

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.

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This work was supported by the Max Planck Society and the Excellence Cluster
’CUI: Advanced Imaging of Matter’ of the Deutsche Forschungsgemeinschaft (DFG)
- EXC 2056 - project ID 390715994. K. Ashraf and R. Cogdell acknowledge
funding by “The Photosynthetic Antenna Research Center” under the US DoE
Energy Frontier Research Center grant number DE-SC 0001035.

The Supplementary Information includes the 2D electronic spectra measured at
the temperatures 50, 80 and 150 K and the description of the global fitting
approach and of the fitting procedures to retrieve electronic and vibrational
coherences. The calculated 2D electronic spectra are presented from which the
lifetimes of the electronic quantum coherences of higher exciton states are
extracted.

The authors declare that they have no competing financial interests.

Correspondence of theoretical part should be addressed to M.T. (email:
michael.thorwart@physik.uni-hamburg.de), Correspondence and requests for
experimental details should be addressed to R.J.D.M. (email:
dmiller@lphys.chem.utoronto.ca).

Figure 1:  (a) The structural arrangement of the pigments in the FMO protein
complex. (b-e) Real parts of measured 2D electronic spectra of the FMO complex
for the waiting times 30, 50, 510 and 1005 fs, respectively. (f-i) Anti-
diagonal bandwidth of the diagonal peak at ($\omega_{\tau}$, $\omega_{t}$) =
(12120, 12120) cm-1 at 20, 50, 80 and 150 K. The decay times of the electronic
dephasing between the ground and excited states are obtained by fitting to a
Lorentzian lineshape (red dashed line). We find 197, 181, 147 and 75 fs,
respectively. These values (“dephasing”) are shown in (j), together with the
decay times associated with the cross-peak decay for growing waiting times.

Figure 2:  (a) Kinetics (red line) and fitting trace (black dashed line) of
the cross peak measured at ($\omega_{\tau}$, $\omega_{t}$) = (12340, 12120)
cm-1. (b) The time trace (black solid line) after removing high-frequency
noise by a Tukey window Fourier analysis. (c) Time evolution of the
vibrational coherences in a wavelet analysis. Close-by lying frequencies of
the ground-state vibrational modes at 167, 180, 191 and 202 cm-1 result in an
oscillatory resonant beating at the waiting times T = 1000 and 1700 fs. (d)
Kinetics (red line) and fitting curve (black dashed line) of the cross peak at
($\omega_{\tau}$, $\omega_{t}$) = (12570, 12480) cm-1. (e) The time trace
after Tukey-window treatment (black solid line). (f) Wavelet spectrum with the
coherent dynamics of the vibrational modes at 46, 68, 117, 202 and 243 cm-1.

Figure 3:  Lifetime of the measured electronic quantum coherence between
exciton 1 and 2 at different temperatures. (a) Residual (black circles) of the
cross peak (20 K) at ($\omega_{\tau}$, $\omega_{t}$) = (12120, 12270) cm-1 and
fit to a multi-exponential function (red line). The green shadow shows the
boundaries of 95$\%$ confidence. (b) Resulting electronic coherence with a
decay time constant of 105$\pm$26 fs. The oscillation frequency of 150 cm-1
perfectly matches the energy gap between exciton 1 and 2. (c), (e) and (g)
Residuals and fitting traces at 50, 80 and 150 K, obtained following the same
procedure. (d), (f) and (h) The obtained electronic coherences with decay time
constants of 96$\pm$40, 81$\pm$26 and 45$\pm$60 fs.

Figure 4:  (a) Measured 2D electronic spectrum at T=1005 fs. The ESA peaks of
exciton 2, 3, 5 and 7 are labeled as A, C, B and D, respectively. (b) Residual
of the ESA peak (black circles) of exciton 2 (red marker A,
$\omega_{\tau}=12270$ cm-1) after removing the kinetics by exponential fits.
The residual is further analyzed by fitting functions (red line, for details
of the fitting procedure, see the Supporting Information). Following the same
procedure, the residual (black circles) and the fitting trace (blue line) for
exciton 5 are shown in (c) (blue marker B in (a), $\omega_{\tau}=12480$ cm-1).
(d) The obtained oscillations of the electronic coherence between exciton 2
and 5 with the frequencies of 206 and 210 cm-1. The decay constants are 91 fs.
The frequencies agree perfectly with the energy gap between exciton 2 and 5.
(e) and (f) Residuals and fitting traces of exciton 3 and 7. (g) The obtained
electronic coherences of exciton 3 and 7 with the decay time constants of 34
and 59 fs, respectively. The resolved oscillations show the frequencies of 312
and 310 cm-1, which matches the energy difference between exciton 3 and 7.

Figure 5:  (a) Time trace (red solid line) of the cross peak between exciton 1
and 2 (CP21) in the calculated 2D electronic spectra. The residuals (magnified
by 3 times) are plotted as black solid line after removing the kinetics (black
dashed line) by an exponential fit. (b) Wavelet analysis of the residuals at
50 K. The frequencies of the coherent dynamics of 150 and 180 cm-1 are
highlighted by black dashed and black dashed-dotted lines, respectively. (c)
Trace (red solid line) of the cross peak between exciton 1 and 2 at 80K. The
subsequent fit and the residuals are shown as black dashed and solid lines.
The frequencies of the coherent dynamics of 150 and 180 cm-1 are marked in
(d). (e) Trace of the cross peak between exciton 1 and 2 at 150 K. The fit and
residuals are shown as black dashed and solid lines, the wavelet spectrum is
shown in (f).