Interplay between structural hierarchy and exciton diffusion in artificial light harvesting

Unraveling the nature of energy transport in multi-chromophoric photosynthetic complexes is essential to extract valuable design blueprints for light-harvesting applications. Long-range exciton transport in such systems is facilitated by a combination of delocalized excitation wavefunctions (excitons) and exciton diffusion. The unambiguous identification of the exciton transport is intrinsically challenging due to the system’s sheer complexity. Here we address this challenge by employing a spectroscopic lab-on-a-chip approach: ultrafast coherent two-dimensional spectroscopy and microfluidics working in tandem with theoretical modeling. We show that at low excitation fluences, the outer layer acts as an exciton antenna supplying excitons to the inner tube, while under high excitation fluences the former converts its functionality into an exciton annihilator which depletes the exciton population prior to any exciton transfer. Our findings shed light on the excitonic trajectories across different sub-units of a multi-layered artificial light-harvesting complex and underpin their great potential for directional excitation energy transport.


Supplementary Note 1: Calculation of Exciton Densities
The exciton densities, i.e., the number of excitons ( e ) normalized by the number of molecules ( m ) in the focal volume, were computed as outlined elsewhere 1 . The formula that was used for computation is given as:  Optical density Monomer extinction coefficient:

S4
The second estimate for the molar concentration is based on the fact that about 60% and 40% of the molecules reside in the outer and inner layer, respectively 2 . Therefore, in case of perfect flash-dilution, where all inner tubes stay intact, one expects a monomer concentration of = 6.67 × 10 −5 M, which would lead to an OD ≈ 1 at 520 nm. This has to be considered as a lower The low-energy main transition of the isolated inner tubes appears blue-shifted by ~50 cm -1 relative to the corresponding transition in case of complete nanotubes, which is consistent with earlier findings from bulk flash-dilution experiments reported in literature 2 . It has previously been shown that the nanotubes' absorption spectrum depends critically on the tube radius 3 so that we hypothesize that stripping of the outer layer leads to slight inflation of the inner tubes' radius, which in turn causes the blue-shift.

Supplementary Note 2: C8S3 Monomers Signal via One-and Two-Photon Absorption
In this section we verify that the 2D spectra of the inner tube do not contain any contribution from dissolved C8S3 monomers left after flash-dilution. For that, we examine the spectral regions where signals from the monomers are expected following absorption of one photon in the small overlap region between monomer absorption and excitation spectrum ( Figure 1b in the main text), or following two-photon absorption via the second electronic excited or high-lying vibronic states. Along the excitation axis, absorption of a single photon in the small overlap region of the excitation spectrum and monomer absorption spectrum would give rise to a signal around ~18200 cm -1 , whereas for two-photon absorption we consider the full bandwidth supported by the excitation pulse spectrum, i.e., from 32000 cm -1 to 36000 cm -1 . In Supplementary Figure 2b this is schematically depicted as 2 × ~590nm, which marks the center of the excitation pulse spectrum, although theoretically any frequency combination may contribute a signal. Jablonski diagram for the excitation of C8S3 monomers via single photon at the red edge of C8S3 absorption (~550 nm; green arrow) or two-photon absorption at the peak of the excitation spectrum (2 × ~590 nm; red arrows). The latter is followed by ultrafast relaxation (wiggly arrow

S7
Examination of these two spectral regions of interest for the monomer response in EEI2D experiments reveals that in neither case we observe any distinct signal originating from the monomers at the background of low-amplitude noise. Since any monomer signal would be strongest in these spectral windows, we conclude that the spectral region along the detection axis around ~600 nm (~16670 cm -1 ) which is relevant to the inner tube response, is free of any monomer signal.
We further support this observation by estimating the exciton density of monomers. The small overlap region of the low-energy tail of the monomer absorption spectrum and the laser excitation spectrum at around 550 nm ( Figure 1b in the main text) could lead to weak excitation of C8S3 monomers. For the calculation of the monomer exciton density, we use an (average) monomer concentration of = 9.06 × 10 −5 M after flash-dilution as estimated in the preceding section and find an exciton density of one excitation per ~3300 C8S3 monomers under the highest excitation fluence in the experiment. Simultaneously, the exciton density for the inner tubes in the same experiment is about one exciton per ~20 molecules, which is a factor of ~165 higher than for the monomers. Therefore, we conclude that excitation of monomers via absorption of one photon in the low-energy tail of the monomer absorption is negligible given the small spectral overlap with the excitation spectrum. The fact that the two-photon absorption cross section of C8S3 monomers is not known prevents such an estimate for two-photon absorption. However, as the two-photon excitation proceeds via a non-resonant state, its cross-section is expected to be even lower. These estimates support the absence of the monomer signals in the absorptive and EEI2D spectra.

Supplementary Note 3: Integration of the Absorptive and EEI Signals
In order to retrieve the absorptive and EEI transients for isolated inner tubes as well as complete nanotubes (Figures 3, Figure 4 and Figure 5 in the main text), the 2D spectra were integrated in the rectangular regions of interest as depicted in Figure 2 in the main text. In practice, vertical slices of the 2D spectra were averaged over 250 cm -1 (corresponding to three data points) along the excitation axis. Next, the baseline was subtracted from these vertical slices and the respective signal of interest was averaged along the detection axis over 100 cm -1 (corresponding to 10 data points). Due to the increased number of features in the absorptive 2D and EEI2D spectra in the case of complete nanotubes, individual contributions from GSB/SE and ESA with opposite signs are more likely to spectrally overlap and, hence, partially compensate S9 each other. At the highest exciton density and, hence, the strongest signals we found this partial compensation to lead to peak shifts, which we accounted for by slightly adjusting the integration area (specified in parenthesis in Supplementary Table 1) in order to avoid simultaneous integration over negative and positive signals.
One of the dominant sources of uncertainty of the extracted signal amplitudes were fluctuations of the background due to unsuppressed scattering of the pump and probe pulses. We determine the standard error of these background fluctuations during each measurement (i.e., at a given exciton density) for the respective spectral regions of interest for the absorptive and EEI signals. The same excitation frequency limits (Supplementary Table 1) are used as before from which the background signal is extracted for each waiting time in the spectral interval from 16000 cm -1 to 16200 cm -1 along the detection axis. The error bars are identical for all waiting times within the same scan, but may be slightly different for the absorptive and EEI signals.

Relaxation
In the case of isolated inner tubes, weak cross peaks can be identified in the absorptive 2D and EEI2D spectra at the detection frequency of the inner tubes (ωinner) at higher excitation frequencies. The appearance of these cross peaks is linked to one of the blue-shifted transitions of the nanotube absorption spectrum ( Figure 1b in the main text), which originates from the complex molecular packing with two molecules per unit cell 2 . In fact, each molecule in the unit cell gives rise to two excitonic transitions, one of which is polarized parallel and the other orthogonal to the nanotubes' long axis 4 . As a result, the absorption spectrum of the inner tubes comprises a total of four transitions, out of which only the parallel polarized transitions at ~16750 cm -1 and ~17500 cm -1 are relevant for 2D spectroscopy due to polarization-selective excitation. The latter was facilitated by the polarization of the excitation pulses set parallel to the sample flow along which the nanotubes preferentially align due to their large aspect ratio. The states corresponding to the strong transition at ωinner ~16750 cm -1 are situated at the bottom of the exciton band, i.e., the super-radiant states 5 , for which an extensive analysis is presented in the main part of the paper. In contrast, the high-frequency transition corresponds to states that lie deep within the exciton band (Supplementary Figure 5c), which we denote as |e〉 with the corresponding frequency ωe and transition dipole moment e . Excitation of this transition is followed by ultrafast intra-band relaxation on a sub-100 fs timescale 6 , which leads to additional population of the bottom states of the exciton band encoded in a rapidly in-growing S14 cross peak in the absorptive 2D spectra (ωe → ωinner). Note that in our experiments the corresponding diagonal peak could hardly be detected because of its short-lived nature and sparse sampling of the waiting time. However, previously published transient absorption (TA) data revealed a decay time as short as ~60 fs for this transition 7 . An additional complication in measuring the diagonal peak arises from the fact that its amplitude scales with the already small dipole moment (| e | 4 ), whereas the cross peak involves the stronger transition dipole moment of the inner tube (| e | 2 | inner | 2 ) and is therefore easier to detect. In comparison, the same cross peak (ωe → ωinner) is present in the absorptive 2D spectra of complete nanotubes, but only visible as peak elongations towards higher excitation frequencies ( Figure 2b in the main text), as they partially overlap with the much stronger cross peak due to the outer layer. For the EEI diagonal peak of isolated inner tubes we consider diagrams that give rise to a signal at an excitation frequency of 2ωinner and detection at ωinner. In the diagrams for the isolated inner tubes |g〉 represents the electronic ground-state, |i〉 and |j〉 the one-exciton states of the inner tube, |ii〉 and |jj〉 bi-exciton states, and analogously |jjj〉 for the tri-exciton states (see the level diagram in Supplementary Figure 6. Note that we formally distinguish between the states |i〉 and |j〉 (and |ii〉 and |jj〉) of the two neighboring excitons to include the fact that the exciton state after the waiting time T is not necessarily identical to the exciton state prepared by the pump pulses.

S16
All diagrams in Supplementary Figure

S17
In absence of exciton transfer (ET) between the tubes, the description of the outer tube diagonal peak is identical to the isolated inner tube with exception of the notation of the states.
Hence, the former can be obtained by renaming the states according to |i〉 → |o〉, |ii〉 → |oo〉, etc.
Nevertheless, we explicitly include these diagrams in Supplementary Figure 7 here, as they form the basis for the discussion on the EEI cross peak in the next section.

Supplementary Figure 7.
Rephasing double-sided Feynman diagrams, which contribute to the EEI diagonal peak of the outer tube. The level diagram for the outer tube assuming that the absence of any inter-tube ET, is shown in the upper left corner. S18

EEI Cross Peak
In order to understand the character of the EEI cross peak we discuss the double-sided EEA on the outer tube in absence of any ET leads to a modified set of diagrams for the EEI cross peak, where the third diagram is replaced by one that contains EEA (Supplementary Figure   8, orange box; diagram on the right). However, under the premise of no ET and assuming weak coupling, EEA on the outer tube does not influence the exciton dynamics on the inner tube and, thus, does not alter the interaction of the probe pulse with the latter. Therefore, the diagrams at zero waiting time still mutually cancel, as it is the case in upper panel of Supplementary Figure 8 and, thus, no EEI cross peak emerges.

S20
For finite waiting times T, inter-tube ET has to be considered explicitly. As a result, the condition that excitons on the outer tube will not influence exciton processes on the inner tube does not hold any longer. The corresponding double-sided Feynman diagrams, which contain both contributions, i.e., ET (shaded in blue) and EEA (shaded in orange), are shown in Supplementary Figure 9. As shown in literature the EEI signal is dominated by pathways that include EEA 1,10 , although these diagrams co-exist with a number of diagrams that contain only ET, which formally also give rise to a signal at 2ωouter → ωinner. The fact that the EEI cross peak dynamics in experiment (see Figure 5 in the main text) exhibit a dependence on the excitation intensity corroborates the fact that the diagrams containing both ET and EEA are most relevant to the EEI cross peak, as for the diagrams with ET alone no intensity dependence is expected.
Supplementary Figure 9. Rephasing double-sided Feynman diagrams which contribute to the cross peak of the EEI signal. Additional to EEA (orange) population transfer is included (blue).

Seventh-Order Signals
As discussed in the main text, higher-order, namely at least seventh order, effects occur at high exciton densities, which were indeed observed experimentally at three times of the fundamental frequency (Supplementary Note 7). Although the seventh-order signal can be glimpsed spectroscopically isolated at triple the fundamental frequency, it also contributes to the

S22
Possible contributions to the seventh-order signal include a sequential cascade of annihilations, in which excitons participate in multiple EEA events during the waiting time T.
For example, after a bi-exciton state relaxes to a one-exciton state, it is excited one more time to the bi-exciton state that subsequently relaxes. However, such processes are not very likely in our experiments because of the short (~15 fs) pulse duration, the low intensity of the probe pulse and a single-pump-beam geometry, which prevents the second excitation to occur with a photon from the same pump pulse.

Simulation Grid
The molecular grid for Monte-Carlo (MC) simulations was set up to match the size known from cryo-TEM measurements 2,12,13 and previously published theoretical models 2 . Тhe boundary conditions for the grid are given by the radii and the molecular surface densities of both tubes.
For simplicity we assume identical square grids for the inner and outer tube with a single molecule on each grid site, although more sophisticated models for the molecular packing have been proposed including brickwork models [14][15][16]   However, it is important to realize that assuming a simple square grid for the molecular packing yields a single effective lattice constant, which averages the actual separation between individual molecules in different directions for more sophisticated packing motifs. In reality, the molecules are expected to stack in one direction with their chromophores closely aligned at distances on the order of 0.4 nm (as reported for a structurally similar molecule 17 ), while the molecular separation in the "lateral" direction would roughly correspond to the molecular size of 1.7 nm.
Combined this yields a unit cell area of 1.7 nm × 0.4 nm = 0.68 nm 2 , which agrees well with the unit cell area derived from the square grid. The remaining molecular dimension (~1.7 nm) sticks out perpendicular from the planes considered here and contributes to the wall thickness of the double-walled assembly. According to the inner and outer radii the thickness amounts to ~3 nm, which is in good agreement with two times the molecular size in this direction.

Extraction of the Absorptive and EEI Signal
Supplementary Table 3 summarizes the criteria after which the absorptive and EEI signals for spectra were extracted from the MC simulations. In order to obtain the amplitude of a specific peak at time t in the simulations only excitons fulfilling the below criteria were counted.

Definition of the Annihilation Radius
We use the annihilation radius as a quantity to characterize the distance dependence of the interactions of two approaching excitons that ultimately results in annihilation of one of the excitons. If we assume a Förster-type exchange for an exciton-exciton annihilation event [18][19][20] , the probability of the event scales with distance as (1 + 6 0 6 ⁄ ) −1 , where 0 is the Förster radius.
In order to ease the computations, we approximate this dependence by a step function: the probability of annihilation within 0 is unity, otherwise it is null (Supplementary Figure 11). In S26 other words, within the cut-off distance 0 the annihilation event outcompetes any other relevant rate in the system (e.g., the exciton decay rate and the exciton diffusion rate).

Exciton displacement
In order to compute the mean (square) exciton displacement Monte-Carlo simulations were conducted in an annihilation-free setting, i.e., with EEA switched off, and at a low exciton density. The latter was important to not hinder the exciton motion by having too many occupied sites. All excitons were labelled with their initial [Xi, Yi, Zi] and final [Xf, Yf, Zf] position on the S27 grid, where the X and Y coordinates refer to sites along and across the molecular grid, respectively. The Z coordinate encodes whether an exciton resides on the inner or outer layer for which Z can take values of 0 or 1. As inter-layer exciton transfer is constrained to occur vertically, i.e., between corresponding sites on the inner and outer layer, the Z-coordinate can be neglected for the calculation of the (square) displacement. Instead, the displacement and square displacement 2 of the n th exciton are computed via: Here

Exciton-Exciton Annihilation Statistics: Isolated Inner Tubes
In the Monte-Carlo simulations the path of each individual exciton is recorded, during which it could either naturally decay due to its finite lifetime or undergo EEA with another exciton. In  increasing the exciton density. The upper panel does not refer to any exciton density in experiment, but was added to illustrate 20% probability of EEA even at exciton densities several times lower than experimentally used.

S31
Supplementary Figure 13 depicts histograms for the number of annihilation events that excitons accumulated during the simulation of isolated inner tubes for a range of exciton densities. Increasing the exciton density leads to more prominent EEA. As a result, there is a lower number of excitons that decay naturally, which is reflected in a decreasing number of excitons that did not participate in any annihilation event, i.e., the number of accumulated EEA events of zero. Simultaneously, the distribution shifts to higher numbers of EEA events, as excitons are more likely to encounter another exciton and, thus, engage in another EEA event.
Expectedly, the mean number of EEA participations increases from 1.01 at one exciton per ~600 molecules up to 1.91 at one exciton per ~20 molecules for increasing exciton densities evidencing the importance of multi-annihilation events accumulated by individual excitons. Even for the lowest exciton density in experiment, MC simulations show that a considerable share of the excitons has accumulated two EEA events by the time of their death, which is the primary requirement for the observation of multi-exciton processes encoded in seventh and higher-order signals.
In order to elucidate the fate of the excitons that were originally planted on the outer tube, we extract the fraction of these excitons that (1) decay naturally, (2) decayed due to EEA on the inner tube or (3)

Supplementary Note 7: Observation of the Seventh-Order Signal
In the Monte-Carlo simulations, we implicitly included the occurrence of multi-exciton processes, where excitons could participate in multiple exciton-exciton annihilation events. However, due to back folding at the Nyquist limit the seventh-order signal appeared at (44000 -6000) cm -1 = 38000 cm -1 along the excitation axis 8 . For high exciton densities this signal was indeed resolved in the EEI signal (Supplementary Figure 16), which corroborates the influence of multi-exciton processes for the experimentally observed signals as well as in our modelling.
The seventh-order signals are observed for both complete nanotubes as well as isolated inner tubes, but significantly stronger for the former. The structure of the seventh-order signals is identical to the lower-order signals with diagonal peaks for the outer and inner tube as well as a cross peak. The only difference is that the peak signs are again inverted compared to the EEI signal and, thus, identical to the absorptive signal, where the ground-state bleach (GSB) shows up negative and excited state absorption (ESA) positive. This again originates from the two additional interactions of the sample with the incident light fields in the perturbative expansion.
A quantitative analysis of these signals, however, is hindered due to the low signal amplitudes as well as the multitude of involved processes and, therefore, is beyond the scope of this paper.

Supplementary Note 8: Thermal Heating Induced by Exciton-Exciton Annihilation
In the absorptive 2D spectra of complete nanotubes an interesting side effect of excitonexciton annihilation was observed, which was reflected in transient heating of the nanotubes and a few surrounding solvent layers. The mechanism is the following: The energy of the annihilated exciton is transferred via a number of (vibrational) relaxation steps to low-frequency modes thereby creating a quasi-equilibrium Boltzmann distribution at elevated temperature. The whole relaxation process takes only a few ps which might be related to ultrafast cooling in liquid water 22,23 . The increased temperature leads to small but detectable modifications of the nanotubes' absorption spectrum, which causes a TA signal offset without detectable temporal variation on the time scale up to 100 ps especially evident at high exciton densities. The signal offset vanishes before the arrival of the next laser pulse, i.e., after 1 ms.
In order to investigate this effect in greater depth we have performed a series of transient absorption (TA) measurements with an extended scanning range of the delay time T. Here we indeed find a prominent signal offset for high exciton densities (a representative TA map is shown in Supplementary Figure 17a). In order to analyze the signal offset, we average the TA signal for delay times between 90 ps and 100 ps (Supplementary Figure 17a, side panel) and over 50 cm -1 along the detection axis. Specifically, the intervals for averaging are 16660 cm -1 to 16710 cm -1 and 16950 cm -1 to 17000 cm -1 for the inner and outer tube, respectively. As a next step, the signal amplitudes are plotted as a function of the respective exciton density in experiment (Supplementary Figure 17b, circles). For comparison, the signal at zero delay time is According to the one-exciton lifetime of 33 ps for complete nanotubes, the signal is expected to decay to about 6 mOD (i.e., 5% of its initial amplitude ~130 mOD) after 100 ps, which is significantly lower than the measured value of ~17 mOD.
As a further test of this hypothesis we collected absorption spectra induced by a temperature jump, i.e., a change in temperature (black and orange curves in Supplementary Figure 18a). The spectra were measured by taking consecutive absorption spectra between which the sample was heated by Δ = 2 K and then allowed to slowly cool down to room temperature (RT) again.
Throughout the measurement the sample temperature was monitored using a thermocouple Overall, the agreement of the TA offset and the difference spectra due to temperature regarding the spectral shape and peak positions is good with the exception of the magnitude being slightly underestimated so that a temperature change of 4 K would have likely been a better estimate. Nevertheless, these results strongly suggest that the offset observed in the absorptive 2D and TA experiments is likely to originate from transient heating of the sample.
A temperature change of 4 K corresponds to the heating due to a single laser shot (ΔE = 2.5 nJ) from which 15% are absorbed and subsequently converted into heat. To reach this estimate a number assumptions have to be made: (1) the heated volume is confined to the nanotube volume, with the supplied heat , absorbed fraction of the laser pulse , pulse energy Δ , and the heat capacity . The corresponding mass of water was calculated from that fraction of the focal volume Vfoc that is actually occupied by nanotubes. Therefore, the latter were treated as simple hollow cylinders with an inner and outer radius of inner = 3.551 nm and outer = 6.465 nm, respectively. The focal volume is assumed cylindrical with a radius of 50 µm and thickness of 50 µm yielding for the water mass

S44
Despite the low exciton density, we find that the transients exhibit a small degree of nonexponentiality. Therefore, we fit the transients with a convolution of two exponential decays and the instrument response function (IRF), which can be approximated by a Gaussian with standard deviation width of ~3 ps. From these fits we extract the weighted averaged lifetime of a single exciton for the isolated inner tubes and complete nanotubes as 58 ps and 33 ps, respectively. In either case the lifetime is shorter than for C8S3 monomers dissolved in methanol (~100 ps) due to the formation of a super-radiant state 5 .
Here, ε corresponds to the excitation energy of molecule n, which is taken from a Gaussian distribution with mean ε o = 18868 cm −1 and standard deviation = 250 cm −1 in order to account for static disorder. These energetic parameters are the same as previously reported in ref.

Diffusion Tensor Elements
As a result of the calculation, we obtain the diffusion tensor elements shown in Supplementary Figure 20 as a function of the dephasing rate .

Exciton Dynamics
Here we test a scenario in which the exciton-exciton interaction radius is nullified, i.e., two excitons only annihilate in case they occupy the same site after a hopping event, which is compensated by an increased hopping rate to obtain a diffusion constant of 100 nm 2 ps -1 in accordance with previously published results 27 . All other conditions as outlined in the main text, stayed unaltered.  Figure 3 in the main paper.
The solid lines depict transients from MC simulations.

S49
In the regime of high exciton densities (Supplementary Figure 21, gray), the dynamics at early waiting times are captured reasonably well, as a faster exciton diffusion can compensate the lack of an extended annihilation radius and vice versa. However, towards longer waiting times the increased diffusion constant leads to unsatisfactory fit of the data, as the calculated dynamics are generally too fast. In particular, this trend becomes apparent at intermediate exciton densities, where the simulations predicts the maximum EEI signal to occur at ~600 fs (Supplementary Figure   21, red) and ~200 fs (Supplementary Figure 21, blue), although the experimental data reach the maximum amplitude at ~6 ps and ~1 ps, respectively. Therefore, we conclude that exciton diffusion alone cannot account for the experimental observations, but an extended radius for exciton-exciton interactions is required to describe the data adequately.

Supplementary Note 12: EEI2D Spectra on Laser Dye Sulforhodamine 101
As a control experiment, absorptive 2D and EEI2D spectra of the laser dye sulforhodamine 101 (SR101, Radiant Dyes) diluted in water at concentration of 10 -4 mol L -1 were recorded (Supplementary Figure 23) at which the average distance between individual molecules amounts to ~26 nm. SR101 was chosen as its absorption peak is located in the same spectral range as the nanotube absorption spectrum investigated here. The response from diluted SR101 molecules is expected to be annihilation-free, as the individual molecules are spaced far apart and, thus, energetically uncoupled. Therefore, any photo-excitation remains localized on a single molecule, which prevents exciton-exciton annihilation. The optical density was set at OD ≈ 0.08 at 586 nm, which is similar to the OD of the nanotubes sample; the excitation energy was set at 40 nJ per pulse. This resulted in excitation of approximately 10% of the SR101 molecules in the focal volume, or, in terms of the main text, one excitation per 10 molecules. This exceeds the highest exciton density used for 2D spectroscopy of the nanotubes by a factor of ~2.  depicts the full range of excitation frequencies from 15000 cm -1 to 36000 cm -1 in order to prove that the signal at intermediate frequencies is free of any artifacts or spurious signals. These results confirm that the spectral range, where EEI signals are expected, is free from artifacts from the experimental apparatus. Importantly, it also supports our assignment of the EEI signal arising from exciton-exciton annihilation, because an even higher excitation density for dissolved sulforhodamine 101 molecules does not result in any observable EEI signal, which in turn justifies our theoretical approach.

Supplementary Note 13: Inter-Wall Excitation Transfer Rate
In order to determine the transfer rate from the outer to the inner tube for complete nanotubes, absorptive 2D spectroscopy was used with fine sampling of the waiting time in steps of 5 fs. To minimize the influence of exciton-exciton annihilation, the exciton density was set to only one exciton per ~500 molecules, which caused the increased noise level of the transient.
Supplementary Figure 24 depicts the ground-state bleach (GSB) cross peak amplitude as a function of waiting time.