Quantum – coherent dynamics in photosynthetic charge separation revealed by wavelet analysis

Experimental/theoretical evidence for sustained vibration-assisted electronic (vibronic) coherence in the Photosystem II Reaction Center (PSII RC) indicates that photosynthetic solar-energy conversion might be optimized through the interplay of electronic and vibrational quantum dynamics. This evidence has been obtained by investigating the primary charge separation process in the PSII RC by two-dimensional electronic spectroscopy (2DES) and Redfield modeling of the experimental data. However, while conventional Fourier transform analysis of the 2DES data allows oscillatory signatures of vibronic coherence to be identified in the frequency domain in the form of static 2D frequency maps, the real-time evolution of the coherences is lost. Here we apply for the first time wavelet analysis to the PSII RC 2DES data to obtain time-resolved 2D frequency maps. These maps allow us to demonstrate that (i) coherence between the excitons initiating the two different charge separation pathways is active for more than 500 fs, and (ii) coherence between exciton and charge-transfer states, the reactant and product of the charge separation reaction, respectively; is active for at least 1 ps. These findings imply that the PSII RC employs coherence (i) to sample competing electron transfer pathways, and ii) to perform directed, ultrafast and efficient electron transfer.

. Vibrational coherence dynamics in the PSII RC at 277 K for the 740 cm -1 2D frequency map. (a) The 2D frequency map at 730 cm -1 (this panel has been adapted from ref. 12 in the main text) (b) Experimental wavelet scalograms for the most intense 2D frequency map features.
The 730 cm -1 vibrational mode is far from all exciton splittings, however, the electronic (exciton and CTs) states have vibrational satellites separated from the zero-phonon level (ZPL) by values close to 730 cm -1 (which are affected by the exciton couplings) . As a result, immediately after excitation we observe several states oscillating in phase around 730 cm -1 which produce an intense peak at short times (within 300 fs). At longer times, these coherent oscillations decrease in amplitude due to environment-induced dephasing 1-2 .  The diagonal bands wavelet data (shown as frequency-time scalograms) shows a complex pattern where the diagonal bands oscillate at several frequencies during different time ranges. This complexity arises from the contribution of vibrational coherences associated with several electronic states (with time-dependent populations), electrochromic shifts induced by charge-transfer and charge-separated states, strong spectral overlap, and presence of a non-trivial interplay between mixed exciton-vibrational coherences 1 . Therefore, with the available 2DES experimental data and the analysis presented here, we conclude that the interpretation of the diagonal bands wavelet data is not possible at this stage and it is out of the scope of the present work. Further experiments and modeling are needed to interpret the diagonal bands wavelet data.   (665,680) nm 2D trace (after two exponential decay has been subtracted from the data) and the back FFT of the convoluted windowed FFT (windowed 2D trace).
As explained in the main text, the interference artifact is observed only when closely spaced frequencies are included in the windowed FFT: with a 20 cm -1 fwhm window there is no interference present and the 2D trace just shows a decay whereas with a 40 and 80 cm -1 fwhm window the interference artifact is clearly present.

Investigating the origins of the amplitude modulations observed in the wavelet data
Here we explore the possible origins of the amplitude modulations observed in the wavelet data further. Specifically, we compare time-frequency signals caused by interferences between different frequencies (interference artefact) with signals which rise/decay due to the creation, regeneration and/or transfer of electronic coherences (real dynamics). We show that in principle these signals cannot be distinguished. However the context in which they appear in the 2D spectra, including their position, frequency and a comparison with the known vibrational modes and electronic states in the PSII RC complex can provide further insight on how to distinguish between an interference artefact and real dynamics of coherence.

Comparison of artificial signals
Recent work has suggested that vibronic coherences might be regenerated in photosynthetic light harvesting complexes 3 . These coherences can be observed as oscillations in the 2D traces and the dynamics of such coherences are expected to modulate the amplitudes of these oscillations (real dynamics). However, since multiple vibrational modes exist in photosynthetic complexes, these modes can interfere with each other, which also leads to a modulation of the oscillation amplitudes (interference artefact).
In order to investigate the differences between these two contrasting signal origins, we consider two artificial signals, given by: where T is the population time, and ω c and ω m are the oscillation frequencies with ω c > ω m .
Eq. 1 gives the superposition of two cosine oscillations at different frequencies, ω c ± ω m and represents the case of an artefact, where two oscillation frequencies interfere. On the other hand, Eq. 2 corresponds to the case of real dynamics of a coherence: the cosine wave at frequency ω c represents the vibronic coherent oscillation frequency, whose amplitude is modulated by a cosine wave with frequency ω m to simulate the effect of regeneration or transfer of the vibronic coherence.
If A = 0, these signals are identical (up to a factor of two in the overall amplitude). They cannot therefore be distinguished from each other in a wavelet analysis. We briefly note that adding arbitrary phases to the two sinusoids in Eq.1 changes the temporal profile of the signal but adding a suitable phase to the envelope and/or carrier factors in Eq.2 can also exactly reproduce this signal (when A=0).
Given this inherent ambiguity we ignore such phases in the following. This underlines the heart of the problem: the fact that the oscillation frequency amplitude modulation caused by regeneration of coherences over the population time, T, (Eq. 2) is in principle indistinguishable from the amplitude modulation caused by interference between two closely spaced frequencies (Eq. 1). In Figure S9 we  Note that for this case, where A = 0, we could also have written a more general, complex form of the equations above as follows: Here both the real and imaginary parts of the signals are again identical (up to an arbitrary amplitude) and therefore cannot be distinguished. However, in practice it may be possible to distinguish between these types of signals by the context in which they occur, as will be explored below.

Population time at which the oscillation frequency amplitude modulation occurs
For Eq. 1, the period at which the apparent amplitude modulation occurs due to interference effects is set by the difference between the two frequencies ω c ± ω m , which is determined by ω m in our case. In practice, the frequencies at which the 2D spectral amplitude oscillates are known (they are obtained by Fourier transform of the 2D data over the population time), therefore, it is straightforward to calculate such a period (as indicated in the main text). The fact that the observed period of amplitude modulation corresponds to the difference between the two oscillation frequencies is therefore a strong indication that such modulation does not reflect real coherent dynamics but an artefact due to interference effects.
For Eq. 2, the oscillation amplitude over the population time depends on both ω m and A. Note that in the 2D frequency maps the term A in Eq. 2 could be non-zero due to an additional long-lived vibrational coherence which may appear at the same position in the 2D map as the feature corresponding to vibronic coherence. Due to the complexity of multichromophoric systems, the vibronic and vibrational features may overlap in some positions in the 2D frequency maps 1,4-5 .
However, we note that this overlap could be avoided if a specific polarisation sequence is applied in the 2DES experiment 6 . Figure S13 shows the effect of changing the amplitude, A, on the time signals and the wavelet analysis, respectively. The results show that changing A defines whether the amplitude of oscillations falls to zero, the population times at which this occurs and the overall amplitude of the amplitude modulations. In this case, it is not straightforward to distinguish between real and apparent coherence dynamics.

Changing position in 2D spectrum
Here we consider how the 2D wavelet traces vary as a function of the position (ω τ , ω t ) in the 2D frequency maps. In the 2D spectra, the diagonal peaks are expected to contain a superposition of states which oscillate at different frequencies and thus create a complicated pattern of interfering frequencies (interference artefact) overlapped with potential real coherence dynamics. In this case, advanced modelling of the experimental data is required to disentangle the different contributions in the observed signal. However, the areas above and below the diagonal are less prone to spectral congestion. For instance, the vibronic coherence between states A and B (absorbing at ω A and ω B , respectively, with ω A > ω B ) appears in the (ω A -ω B ) 2D frequency map with a maximum amplitude at position (ω A , ω B ), and the weight of this frequency decreases when moving away from position (ω A , ω B ). Using this rationale, here we investigate the effect of the varying contributions of different frequencies in the wavelet pattern (or wavelet traces) when moving away from a cross-peak maximum. For simplicity, we take a fixed ω τ and a varying ω t .
Firstly, we consider signals arising from interference between two frequencies, as in Eq. 1, which can be generalised to: where we introduce α and β, which allow us to vary the weight of contributions from different oscillation frequencies. As the weighting of the two cosine waves changes, the amplitude of the pattern observed in the wavelets will shift to different frequencies along the y-axis, as shown in Figure   S11. However the population time at which the recurrence occurs remains the same (as determined by ω m ; see above). Also notice that interference between waves at two different frequencies can cause oscillations in the amplitude of the wavelet even if one of the amplitudes α or β is significantly larger than the other and it is not obvious from the wavelet results that two different frequencies are involved. For amplitude modulation of the type given by Eq. 2, changing ω c will change the frequency (on the y axis) at which the pattern in the wavelets occurs but not the weighting of the pattern itself; see Figure   S12. Therefore shifts in the weighting of the pattern may indicate that the signal arises from interference between different frequencies (as described above).

Conclusions
In conclusion, by considering artificial signals we have shown that the time signal from two interfering oscillations with different frequencies can be identical to the signal from amplitude modulation of an oscillation which might arise from regeneration of a vibronic coherence. Therefore in principle these two different signal origins cannot be distinguished, either by a wavelet analysis or any other technique.
However, 2D spectroscopy provides a wealth of information beyond that which is contained in a single time, therefore this additional information can give further insight into the signal dynamics origin. In particular, in the case of interference between two oscillations with different frequencies, the period of the interfering envelope is determined by the frequency difference between the oscillations. This means that the period of the oscillations expected from this effect can be calculated and compared to the experimental results (as in the main text). We note that care should be taken if there are overlapping signals in the 2D spectra (for example, from ground-state bleach), since these can change the amplitude and period of the oscillations observed. In that case, it may be helpful to simulate the signals in order to make quantitative predictions. In addition, we highlight that interference between waves at two different frequencies can cause oscillations in the amplitude of the wavelet even if one of the oscillations has a significantly larger amplitude than the other. In this case it may be very difficult to discern from a visual inspection of the wavelet results that two different frequencies are interfering.
Finally, changing the position considered in the 2D spectrum is expected to have different effects depending on the signal origin. Notably, as the position in the 2D spectrum changes, for interference effects the weighting of the pattern observed in the wavelet scalograms is expected to shift, whereas for the amplitude modulation expected from electronic coherences, the period of the modulation changes.