Estimation of damped oscillation associated spectra from ultrafast transient absorption spectra

When exciting a complex molecular system with a short optical pulse, all chromophores present in the system can be excited. The resulting superposition of electronically and vibrationally excited states evolves in time, which is monitored with transient absorption spectroscopy. We present a methodology to resolve simultaneously the contributions of the different electronically and vibrationally excited states from the complete data. The evolution of the excited states is described with a superposition of damped oscillations. The amplitude of a damped oscillation cos(ωnt)exp(-γnt) as a function of the detection wavelength constitutes a damped oscillation associated spectrum DOASn(λ) with an accompanying phase characteristic φn(λ). In a case study, the cryptophyte photosynthetic antenna complex PC612 which contains eight bilin chromophores was excited by a broadband optical pulse. Difference absorption spectra from 525 to 715 nm were measured until 1 ns. The population dynamics is described by four lifetimes, with interchromophore equilibration in 0.8 and 7.5 ps. We have resolved 24 DOAS with frequencies between 130 and 1649 cm-1 and with damping rates between 0.9 and 12 ps-1. In addition, 11 more DOAS with faster damping rates were necessary to describe the "coherent artefact." The DOAS contains both ground and excited state features. Their interpretation is aided by DOAS analysis of simulated transient absorption signals resulting from stimulated emission and ground state bleach.


INTRODUCTION
][17][18][19] In this regime, we expect superimposed population and coherence dynamics in transient absorption spectra.Whether vibrational wavepackets are generated on the ground or excited state potential energy surfaces depends on the properties (spectrum, duration, and chirp) of the exciting pulse 4,[20][21][22][23][24] and ground state vibrations can be suppressed by a sufficiently short broadband pulse.25][26][27][28][29] Vibronic coherences are relevant to the study of photosynthetic energy transfer, 25,27,[30][31][32][33][34][35][36][37][38] photosynthetic charge separation, 39 exciton coherence, 40 electa) Electronic mail: i.h.m.van.stokkum@vu.nl.Telephone: +31205987868.][46] In order to investigate the dynamics revealed in transient absorption spectra, populations and oscillations are currently treated separately.First the population dynamics is fitted and subtracted from the data.Then the residual signals well after the instrument response function (IRF) duration are analyzed.Time traces at single detection wavelengths are either Fourier transformed, or fitted with damped cosines. 1,2,4,25,27,41,47Next, for a few dominant frequencies the amplitude and phase can be plotted as function of the detection wavelength.No attempt has been made yet to globally analyze the complete data.In this paper, we present a methodology to resolve simultaneously the contributions of the different electronically and vibrationally excited states from the complete dataset in order to extend the applicability of global analysis to datasets comprising both features.
In a case study, new high quality data from the cryptophyte photosynthetic antenna complex PC612 25,48 will be analyzed.When structural information is available, the dynamics of complex molecular systems can be studied with quantum mechanical methods, and time resolved spectra can be predicted, which has been done for the photosynthetic antenna complexes PE545 49,50 and PC577. 26Here no structural information is used, nor quantum mechanical methods.Instead, the data are described by a parametric model, employing the global and target analysis methodology [51][52][53] to describe the spectral evolution of electronically excited states.
In target analysis, the inverse problem is to determine the number of electronically excited states (N states ) present in the system, and to estimate their spectral properties SADS l (λ) (species associated difference spectra) and their populations c S l (t) (superscript S stands for species).The time resolved spectra TRS(t, λ) are described by a parameterized superposition model , where the populations are determined by an unknown compartmental model that depends upon the unknown kinetic parameters θ.In the target analysis, constraints on the SADS are needed to estimate all parameters θ and SADS l (λ).We here describe the evolution of the vibrationally excited state wavepackets created by the short laser pulse with a superposition of damped oscillations.The amplitude of a damped oscillation cos(ω n t) exp(−γ n t) as a function of the detection wavelength constitutes a damped oscillation associated spectrum DOAS n (λ) with an accompanying phase characteristic ϕ n (λ).When the vibrational evolution can be considered independently from the electronic evolution (Born-Oppenheimer approximation), we arrive at a superposition of the electronic and vibrational contributions to the TRS(t, λ), 70 where t ′ indicates that the actual model function still has to take into account the IRF (vide infra).Thus the inverse problem is now extended: to determine additionally the number of vibrationally excited states N osc , and to estimate their parameters, the eigenfrequency ω n and damping rate γ n , and the DOAS n (λ) and ϕ n (λ).Note that it is now the system that reveals its eigenfrequencies from the complete dataset.The amount of vibrationally excited states that can reliably be resolved will critically depend upon the signal to noise ratio of the measurements TRS(t, λ).We will demonstrate the applicability of the methodology and compare with results from published high-resolution measurements of light harvesting proteins. 25,27

Modelling and parameter estimation
The population of the l-th compartment is c S l (t).The concentrations of all compartments are collated in a vector c  T , which obeys the differential equation where the transfer matrix K contains off-diagonal elements k pq , representing the microscopic rate constant from compartment q to compartment p.The diagonal elements contain the total decay rates of each compartment.The input to the compartments is j(t) = IRF(t) x 1 . . .x n comp T , with x l the absorption of the l-th compartment.
The impulse response of the system, which is a sum of exponential decays, has to be convolved with the IRF.Typically, a Gaussian shaped IRF is adequate, with parameters µ for the location of the IRF maximum and ∆ for the full width at half maximum (FWHM) of the IRF, where ∆ = ∆/(2  2 log(2)).The convolution (indicated by an *) of this IRF with an exponential decay (with decay rate k) yields an analytical expression, which facilitates the estimation of the decay rate k and the IRF parameters µ and ∆, When the compartmental model consists of independently decaying species, with populations c D l (t, k l , µ, ∆) (superscript D stands for decay) their spectra are termed DADS l (λ) (decay associated difference spectra).The solution of the general compartmental model described by the K matrix consists of exponential decays with decay rates equal to the eigenvalues of the K matrix.The interrelation between the DADS and SADS is expressed in the following matrix equation: Here the matrix C D (θ, µ, ∆) contains in its l-th column the decay c D l (t, k l , µ, ∆) and the matrix C S (θ, µ, ∆) contains in its columns the populations c S l (t) of the general compartmental model.
The superposition model for the TRS(t, λ) is given by the matrix formula Here the matrices Cos(ω, γ, µ, ∆) and Sin(ω, γ, µ, ∆) contain the damped oscillations, which will be detailed below, and the matrices A and B comprise their amplitudes.
To improve the precision of the estimated parameters, the experiment can be repeated, and the set of N exp experiments can be analyzed simultaneously.For each additional dataset TRS e only one scaling parameter α e and one time shift parameter µ e must be added The dimensions of the matrices are collated in Table I.Assuming wavelength independence of the eigenfrequency ω n and of the damping rate γ n is necessary to limit the amount of free parameters.The model function for the n-th damped oscillation is given by Note that we have chosen this approximate product expression in order to avoid the complications, which arise from the convolution of c D (t, γ n , µ, ∆) with a sine or cosine function.
Using these two functions, the matrices Cos(ω, γ, µ, ∆) and Sin(ω, γ, µ, ∆) can be computed as a function of all ω n , γ n .The intrinsically nonlinear parameters of this model are the vector θ (containing the microscopic decay rates and inputs to the compartments x l ), the IRF parameters (µ, ∆), scaling parameter α e , and time shift parameter µ e for each additional dataset, and the N osc -vectors ω, γ.The conditionally linear parameters are the matrices SADS, A, and B. Typically, there are nλ times more conditionally linear parameters than intrinsically nonlinear parameters, see Table I.The variable projection algorithm 52,[54][55][56] which minimizes the nonlinear least squares criterion as a function of the intrinsically nonlinear parameters only, and implicitly solves for the conditionally linear parameters, makes the parameter estimation feasible.
The DOAS and phases cannot be fitted directly, but must be computed from the estimated cosine and sine amplitude matrices A and B. The n-th DOAS n at wavelength λ j is computed as The reconstruction of the phase ϕ n is more cumbersome.From A j n and B j n the phase ϕ j n modulo 2π can be computed.Then undesired 2π jumps between subsequent phases ϕ j n and ϕ j+1, n can be corrected by applying appropriate counterjumps of 2π.

Transient absorption measurements
Transient absorption measurements of cryptophyte light harvesting proteins have been described in Refs. 25 and 27.Different from those measurements, detection here was at the magic angle, instead of parallel, which precludes depolarization effects.A typical experiment consists of nt = 959 time gated spectra measured at nλ = 650 wavelengths (with wavelength step 0.293 nm).In the estimation of spectra there is always a balance between variance and bias.When smoothing the DADS or DOAS by averaging four consecutive wavelength points (corresponding to 1.172 nm), a small bias is introduced but the variance is expected to halve.In the main text, we present results with averaging four consecutive wavelength points.In the supporting information, we compare with the unaveraged data, cf., Figure S 1 of the supplementary material.Time-gated difference spectra were taken at equidistant time points in the first two ps (with time step of 3 fs), and with gradually increasing time steps until 1000 ps.The estimated FWHM of the IRF was 21 fs.Each experiment was repeated at least 5 times on 3 different days to confirm reproducibility.
Here we analyse data recorded for the PC612 light harvesting complex from the cryptophyte Hemiselmis virescens CCAC 1635 B, which contains eight covalently bound chromophores: two dihydrobiliverdin (DBV) and six phycocyanobilin (PCB), indicated in red and blue in Figure 1.PC 612 is a phycobiliprotein.Phycobiliproteins are found FIG. 2. Broadband excitatory pulses were tuned centrally to 580 nm with an asymmetric Lorentzian extending past 650 nm (cyan), overlapping well with the entire PC612 absorption (black) and emission spectrum (magenta).The wavelength dependence of the root mean square error (rmse) of the fit (green), scaled for comparison (rmse maximum 1.36 mOD at 525 nm, minimum 0.22 mOD at 582 nm).The reciprocal of the rmse (red) is used as the weight in the nonlinear least squares fit. in cyanobacteria (blue-green algae), red algae, and the cryptomonads. 57roadband excitation pulses were tuned centrally to 580 nm with an asymmetric Lorentzian tail extending past 650 nm, overlapping well with the entire PC612 absorption spectrum (indicated by, respectively, cyan and black lines in Figure 2), most strongly exciting the higher-energy DBV chromophores.The red curve represents the wavelength dependence of the reciprocal of the root mean square error (rmse) of the fit, which is used as a weighting factor in the nonlinear least squares fit.This weight is determined iteratively, and the curve shown here represents the best fit.The repetition rate of the excitatory pulses was 5 kHz.Thus a long lived species (e.g., a triplet or a charge-transfer state) can accumulate if its lifetime is longer than 50 µs.If its population becomes sizeable, possible annihilation processes have to be taken into account as well in the target analysis.The spectrum of this long lived species is depicted in grey in Figure S 4. It will be discussed below.

Residual analysis
Following a successfully converged fit, the matrix of residuals is analyzed with the help of a singular value decomposition (SVD).Formally, the residual matrix can be decomposed as where u l and w l are the left and right singular vectors, s l the sorted singular values, and m is the minimum of the number of rows and columns of the matrix.The singular vectors are orthogonal and provide an optimal least squares approximation of the matrix.The SVD of the matrix of residuals is useful to diagnose shortcomings of the model used, or systematic errors in the data.A power spectrum P l (ω) of each of the most important left singular vectors u res l (t) can be computed with the help of the FFT (for the part of the data measured at equidistant time points) or of the discrete Fourier transform (DFT), with nonequidistant time points.Typically, a peak in P l (ω) coincides with an obvious trend in u res l (t).The frequency of the peak can then provide a starting value for the frequency of an additional damped oscillation.N osc can be decided upon visual inspection, from the absence of obvious trends in u res l (t) and of obvious peaks in P l (ω).When the w res l (λ) of all N exp experiments are similar, consideration of the u res l (t) and P l (ω) strengthens this procedure, since a putative additional oscillation should be present in all experiments.Thus a reliable N osc can be determined.Figure S 2 demonstrates that after a global analysis with exponential decays, but without fitting the damped oscillations, the residual matrices show a common structure for all three experiments.The scree plot on the top shows an elbow around l = 17, which indicates that for higher values the noise will dominate.The first and second pairs of left and right singular vectors are dominated by the coherent artifact phenomena straddling time zero, showing very broad power spectra.The third and higher left singular vectors depict the long lived oscillations.Their power spectra show distinct peaks at many frequencies.Those peak frequencies can be used as starting values for the ω n parameters of the damped oscillations.Note that the common structure for all three experiments is present until about the 17th pair of left and right singular vectors.Above that the noise that differs for each experiment starts to interfere more and more, and less common structure is visible.Thus we infer that at least 17 components that are linearly independent in both time and wavelength will be needed to describe the oscillations in the data.Note that it is impossible to infer the number of resolvable damped oscillations from this residual analysis, because the SVD cannot take into account any phase characteristic.

RESULTS
The excellent reproducibility of three independent measurements is demonstrated in Figure S 1.The problem is: what is the signal and what is the noise?Reproducible oscillations are clearly present until about 1 ps, after that a noise level of 0.51 mOD is present.Averaging over four consecutive wavelength points reduces this to a noise level of 0.29 mOD.Thus our modelling should aim to describe the signal up to this noise level.The quality of the fit is demonstrated in Figure 3 for a small selection of the data, emphasizing the early times.The residuals will be discussed below.The spectral evolution of the electronically excited states is described by four lifetimes of 0.84, 7.5, 109, and 1587 ps.The estimated DADS are depicted in Figure 4.The first DADS of 0.84 ps can be interpreted as equilibration between the more blue absorbing DBVs and the more red absorbing PCBs.This downhill energy transfer is followed by two slower steps of 7.5 and 109 ps which are less easily interpretable.Some loss of excited state as well as some further equilibration appears to be present.The final equilibrated state decays with a lifetime of 1587 ps.A target kinetic scheme with four consecutive compartments and three equilibria is depicted in Figure 5(a).By manipulating the inputs to each of the four compartments, and by adjusting the free energy differences, we arrived at the populations and SADS  and ESA at ≈680 nm.They represent the six PCB chromophores.Thirty-five damped oscillations were needed to fit the vibrational evolution up to the noise level, cf., Figures S 5 and S 6.The first left and right singular vectors of the residual matrix are depicted in Figures S 7 and S 8.No obvious structure that is consistent for the three experiments is discernible.Thus we conclude that the maximal number of resolvable damped oscillations with this signal to noise level is 35.The parameter estimation is still a time consuming trial and error process, which could probably be automated.
A complete overview of all estimated DOAS and phases is given in Figure S 9, and a selection thereof in Figure 6.
The averaging over four consecutive wavelengths results in a variance reduction, compare Figure 6 and Figure S 10.The estimated eigenfrequency and damping parameters are collated in Table II.Eight DOAS with damping rates above 20 ps −1 were needed to fit the coherent artefact or cross-phase  26,59 In the data, these are obvious as low frequency oscillations extending before time zero (Figure 3).They are mainly described by the black DOAS32, with a frequency of 406 cm −1 , and damping rate of 39 ps −1 , which has its largest amplitude above 670 nm (black in Figure 6(w)).The origin of DOAS25 is unclear (black in Figure S 9P-R).Thus the oscillatory signal well after the IRF and XPM is described by 24 DOAS.Of these 24, four selected DOAS are depicted in Figures 6(c)-6(i).
The most slowly decaying (0.9 ps −1 ) is DOAS15 (green in Figures 6(c for DOAS 5-12 and DOAS 14-24, with corresponding phase changes in the accompanying phase profiles. Comparing to the population profiles, with the fastest equilibration rate constant of 0.82 ps −1 (cf., Figures 5(a) and 6(a)), and initial populations of all four species, it is not easily possible to associate the DOAS to species.Due to the ubiquitous presence of the node at the emission frequency of the terminally emitting PCB molecule, these oscillations can at least be attributed to that chromophore.Though there is likely additional amplitude from contributions of the other PCB molecules in the region.The DOAS amplitudes are relatively small near 580 nm, the DBV absorption maximum that corresponds to the bleach of the black species.This is probably caused by the pump pulse not having enough excess energy (cf., Figure 2) to excite many higher vibrational states of this species.The dominating contribution to this signal is then the ground state bleach, which does not exhibit pronounced oscillations (see Figure 3, Figure S 6, and below).

SIMULATIONS
In order to connect the fitted DOAS parameters with the actual system coherent evolution, the dynamics of a vibrational wavepacket was simulated and analyzed in the same way as the experimental data.In the past, many efficient theoretical methods were developed for the description of transient absorption experiments.1][62] In this work, we choose for the sake of simplicity a somewhat more direct approach in the form of response functions.Although impractical for the calculation of larger systems and/or finite pulses, the advantage of our approach is that it enables to directly follow the state of the system responding to the interaction with the light.As we will see further on, it also enables to correlate the waiting-time oscillation frequency with the emission frequency.The employed theoretical description is detailed in Ref. 63.Briefly, one selected strong vibrational mode of interest is quantized and explicitly included into the system, while all the other vibrations of either the pigments or the protein/solvent are included in the vibrational bath.In this description, both the explicitly quantized vibrational mode and the electronic degrees of freedom are interacting with the bath and the interaction is treated within second-order perturbation theory.The evolution of optical coherences, yielding lineshapes, is expressed by cumulant expansion.
The excited-and ground-state dynamics is calculated by time-independent Redfield theory.The transient absorption signals were evaluated in an impulsive limit of ultrashort pulses. 62In this limit, the two interactions with the pump pulse create an initial condition for the waiting time evolution in the excited and ground state.The interaction of the probe then causes emission of the signal in the probe direction.The double-sided Feynman diagrams used for the calculation of the simulated transient absorption signals are depicted in Figure 7.
The goal of the simulations is not to quantitatively reproduce the experimental data, but rather to qualitatively describe features present in the fitted DOAS and connect them to the physical properties of the system studied.To achieve this, we calculate separately the stimulated emission (SE) and ground state bleach (GSB) signals for a single pigment.To ease the comparison with the experiment, we choose the zero-phonon line at 16 450 cm −1 (608 nm) and the explicit vibrational frequency 662 cm −1 (about 25 nm in the spectral range considered).We use an overdamped Brownian oscillator to describe the bath spectral density, 62 with the bath correlation time of 50 fs and the coupling to the bath in order to obtain vibrational relaxation on the order of 100 fs and a lineshape width of 200 cm −1 (about 8 nm).The calculation is at room temperature and the initial equilibrium state before interaction with the pump is a canonical Boltzmann distribution in the ground state.The simulated transient absorption signals can be found in Figure S 12.Note that because of the 10 fs sampling interval of the emission time t these data are periodic in the frequency domain with 3333 cm −1 .Already from the unprocessed data, we can clearly see that in the SE signal the excited state dynamics including vibrational relaxation and coherent vibrational wavepacket oscillations is dominating.In the GSB signal, however, the vibrational wavepacket is not displaced by the two interactions with the ultrashort pump.Therefore there is no vibrational relaxation and in the impulsive limit also no coherent oscillations.Mathematically, this can be seen by using perturbation theory in the interaction with the light and realizing that in the Condon approximation the transition dipole moment operator is independent of the vibrational degrees of freedom.The two interactions with the light from one side, see Fig. 7 right, then leave the system in the ground state equilibrium. 64In reality, this strictly applies when the duration of the pulse is shorter than the oscillation period of the vibration.In a real experiment with ultrashort pulses, the amplitude of the GSB oscillations will be strongly suppressed.The global SADS and DOAS of the simulated SE signals can be found in Figure 8.In the SE data, the intramolecular vibrational energy relaxation is described by a sequential kinetic scheme with increasing lifetimes, the populations of the species are shown in Figure 8(a).In the shape of the SADS the vibrational progression is clearly visible (Figures 8(a Let us now focus on the origin of the oscillations.Since the energy relaxation dynamics is described by the sequential model and the SADS, the remaining DOAS evolution should reflect the coherent dynamics of the wavepacket.A vibrational wavepacket is a coherent superposition of the vibrational states; the fixed phase relation between its individual components holds it together.The observed oscillations, fitted by the damped cosines, are caused by the evolution of the coherences of this superposition.These coherences oscillate with a frequency equal to the energy difference between the constituent states, that is, with a multiple of the vibrational frequency.This can also clearly be seen from the double-sided Feynman diagrams in Fig. 7.The oscillation decay rate is given both by decoherence and by the wavepacket relaxation.At this point, we would like to note that because of the bath memory and coherence transfer, the decay of the coherences is not necessarily exponential.Then a single coherence decay will be fitted with multiple exponential components with the same DOAS shape, oscillation frequency, and different damping rates.This is probably the case of DOAS #6 and #7 in the SE, see Figures 8(g)-8(i).Because of this the number of resolved DOAS does not correspond to the number of coherences present.Generally the extracted frequencies indeed correspond (within the fitting error) to the vibrational frequency and its second and third harmonic, cf., Figures 8(d)-8(f), demonstrating the power of the DOAS to resolve the vibrations of the molecular system.
Considering the shape and phase of the DOAS, in the SE the DOAS peaks are found both at the SADS peak position of the transitions and around these peaks.In the latter case, in the DOAS shape there are sharp nodes present at the position of the SADS peaks (e.g., indicated by the grey vertical line at 17 135 cm −1 in Figure 8(h) for DOAS #6 and #7, green and magenta).The phase exhibits abrupt changes, with π jumps at the position of the sharp nodes (e.g., indicated by the grey vertical line at 17 135 cm −1 in Figure 8(i) of the phase of DOAS #6 and #7, green and magenta).][27] Our results correspond to those in Fig. 4 Finally, let us compare the DOAS features in the experimental, Figure 6, and simulated, Figure 8, data.In the experimental DOAS shapes and phases, we can see both of the features described above.In DOAS #15 (green in Figures 6(c)-6(f)) the peaks correspond to the vibrational progression, being about 662 cm −1 apart.This DOAS thus has the same features as the simulated DOAS #8 in the SE (black in Figures 8(c)-8(f)).In contrast, in experimental DOAS #2 (blue in Figures 6(g)-6(i)) the peaks are not separated by the vibrational frequency and there is a sharp node present at the position of the transition at 625 nm.The phase exhibits a rapid jump at the position of the node.This DOAS thus corresponds to the simulated DOAS #6 and #7 (green and magenta in Figures 8(c)-8(f)), with the node and phase jump at 17 135 cm −1 .Similar correspondences can be found for other experimental and simulated DOAS.This provides the desired connection between the DOAS and actual vibrational wavepacket dynamics.The presence of sharp nodes and phase jumps, together with the large oscillation amplitude (relative to the absolute signal), indicates that the oscillations originate from the excited state.

DISCUSSION
The long lived species that was mentioned in the Transient absorption measurements section is depicted in grey in Figure S 4. It shows a bleach at ≈650 nm, which corresponds to the red edge of the absorption spectrum.Thus it could correspond to a long-lived charge-transfer state that is directly excited by the broadband pulse.The putative bleach extends until the end of our measurement range, 715 nm.From 590 to 650 nm the shape of the grey SADS resembles the shape of the three PCB SADS.Below 590 nm absorption is present.Alternatively, the long lived species could correspond to accumulated triplets.In the 7.5 ps DADS (red in Figure 4) there is a net loss of bleach of excited states near 625 nm.Probably, in a small fraction of the complexes (≈9%), singlets annihilate with the long lived states.In the target analysis this loss is now visible as a smaller amplitude of the blue SADS relative to the red SADS (Figure 5(c)).Thus the timescale of this annihilation (in ≈9% of the complexes) is comparable to that of the 7.5 ps equilibration taking place in all PC612 complexes.
The frequencies of the DOAS with the damping rates smaller than 15 ps −1 most probably correspond to properties of the molecular system.DOAS with faster damping rates were required to fit XPM and IRF properties until about 50 fs after the maximum of the IRF.Compared to earlier studies 25,27 more than twice as many frequencies have been resolved.This is possible through the model based analysis of these very high quality data.The IRF was 21 fs FWHM, and the pulse possessed a large power until 715 nm (Figure 2).Thus we could easily resolve frequencies up to ≈1600 cm −1 .The highest frequencies resolved in earlier studies were only ≈800 cm −1 .
The DOAS are analogous to DADS and can be directly interpreted as the amplitude of a given coherent oscillation, with a defined phase and dephasing lifetime, as a function of detection wavelength.While these are global lifetimes and cannot necessarily be assigned to a particular species, excluding the possibility of vibrational coherence across multiple chromophores, it is likely that each DOAS resolves oscillations fixed to a particular chromophore.The DOAS reveal clearly visible nodes, with corresponding phase change, that flag the minimum of the potential energy surface.This location is where, due to interference effects, oscillations cancel. 26reviously, Fourier transform maps of the dataset well after the IRF were used to identify oscillations and their phase character.In addition to resolving significantly more oscillations, at the frequencies that characterize the system, the DOAS method very clearly reveals the unique nodal features, phase profiles, and dephasing times for each of them.This represents the most detailed and complete analysis of vibrational coherence in a phycobiliprotein, fit simultaneously with its population dynamics, in the time domain from complete transient absorption datasets.
The placement of nodes in the data allows certain assignments of the DOAS in the excited state potential energy surface of individual chromophores.Nodes present at the emission wavelength of the complete complex indicate that the DOAS likely belongs to the terminally emitting PCB chromophore.In a complex with eight chromophores, this may not be possible to resolve.However, for those DOAS that have nodes to the blue of 640 nm, we can rely on the amplitude in the ESA region of the spectrum to assign them to the excited state.Overall we can see that excited state coherences are exclusively generated, as expected from impulsive excitation with a broadband pulse that covers the entire absorption spectrum of the complex. 22,24he comparison of our DOAS global fit with the previous results and the theoretical simulations well demonstrates the power of our analysis to resolve the frequencies present in the system.6][67][68][69] A particular emphasis is being put on distinguishing purely vibrational, excitonic, and mixed (vibronic) coherences and on assigning the oscillating spectral features to specific dynamic events, for instance, in photosynthetic energy transfer and charge separation. 39he DOAS analysis can prove to be a valuable tool, as the estimated eigenfrequencies can be compared to known vibrations, elucidating the character of the coherences, while the unique spectral features can attribute them to the electronic excited state.
By global fit alone of the experimental data it is difficult, if not impossible, to interpret in detail the underlying dynamics.At the same time by a theoretical simulation it is unfeasible to directly fit the data already for moderately complex systems.
Our DOAS analysis thus provides a useful "middle ground" where the theoretical description and fit of experimental data can meet.

CONCLUSIONS
The methodology presented here provides a complete description of the time evolution of the electronically and vibrationally excited states.No data are left out, and the IRF is fully taken into account.Thus from a huge amount of data interpretable parameters are estimated, describing the equilibration of electronically excited states (Figure 5) and the damped oscillations (cf., Figure 6 and Table II).The DOAS are expressing the most important vibrational frequencies present in the molecular system estimated from a simultaneous, global and target analysis of three complete data sets.The simultaneous analysis of multiple datasets is mandatory to distinguish signal and noise, and with the help of residual analysis establish the amount of oscillations needed to describe the complete data (Figure S 7 and  Figure S 8).The DOAS methodology is applicable to broadband transient absorption of both simple and complex systems.

SUPPLEMENTARY MATERIAL
See supplementary material for thirteen figures demonstrating the reproducibility of the data and the quality of the fit, the analysis of the residuals, and the complete DOAS analysis of the data and of the simulations.In addition, the effect of averaging over four consecutive wavelengths is demonstrated.

FIG. 5 . 6 vanFIG. 6 .
FIG. 5. (a) Kinetic scheme with microscopic rate constants (in ns −1 ).Input fractions (red) are on top of each of the four compartments.(b) Population profiles of the four species, note that the time axis is linear until 1 ps (after the maximum of the IRF), and logarithmic thereafter.(c) Estimated SADS.

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FIG.7.Double-sided Feynman diagrams used for calculating the simulated transient absorption signal, left: SE and right: GSB.The system starts in canonical equilibrium given by the Boltzmann distribution of the population of the vibrational states in the ground state, g k .The two interactions with the pump take the system out of the equilibrium.The coherence dynamics in the waiting time T results in the observed oscillations, and the oscillation frequency is given by the energy difference of the constituent states.The optical coherence evolution in emission time t determines the spectral position of the pathway in the transient absorption spectrum.
) and 8(b)), the peaks being a vibrational frequency apart.

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FIG. 8. Global fits of the simulated SE signals.(a) Evolution of species populations, (b) associated SADS, (c) five selected DOAS.(d) and (g) Cosine oscillationswith frequencies νn (in cm −1 ) (where n is the DOAS number) and damping rates γ (in 1 ps −1 ) written in the legend at the left, using the appropriate color.(e) and (h) Estimated DOAS, color coded as in (d) and (g).(f) and (i) Accompanying phase profiles.The grey vertical lines at 17 135 cm −1 in panels (h) and (i) are discussed in the text.

TABLE I .
Dimensions of the matrices.

TABLE II .
Estimated DOAS parameters of PC612: area (divided by γ n ), eigenfrequency ω n (in cm −1 ), and damping rate γ n (in ps −1 ) and provisional assignment.The errors estimated from the nonlinear least squares fit are a lower bound.These errors are huge for the coherent artefact related parameters.A zero error indicates that the parameter was kept fixed.
(c) from Ref.27and in Fig.7from Ref.25and are consistent with Ref.26.Most interestingly, in accordance with the simulation results, these features can be ascribed to the excited state.The presence of such features can thus be regarded as a signature of the excited state dynamics.