Signatures of Exciton Delocalization and ExcitonâExciton Annihilation

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Spectroscopy and Photochemistry; General Theory

Signatures of Exciton Delocalization and Exciton-Exciton Annihilation in Fluorescence-Detected Two-Dimensional Coherent Spectroscopy Pavel Maly, and Tomas Mancal J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b02271 • Publication Date (Web): 06 Sep 2018 Downloaded from http://pubs.acs.org on September 6, 2018

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The Journal of Physical Chemistry Letters

Signatures of Exciton Delocalization and Exciton-Exciton Annihilation in Fluorescence-Detected Two-Dimensional Coherent Spectroscopy Pavel Malý†,‡ and Tomáˇs Manˇcal∗,† †Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, CZ-12116 Prague 2, Czech Republic ‡Faculty of Science, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081HV Amsterdam, The Netherlands E-mail: [email protected]

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Abstract Incoherently-detected coherent multidimensional spectroscopy is rapidly gaining popularity, promising different application range and sensitivity than its traditional counterpart. While measuring the same response, the two methods are not equivalent. We present calculations of the fluorescence-detected coherent two-dimensional (F-2DES) spectra of a molecular heterodimer. We compare how the F-2DES technique differs from the standard coherently detected two-dimensional (2DES) spectroscopy in measuring exciton delocalization. We analyze which processes contribute to crosspeaks in the zero-waiting-time spectra obtained by the two methods. Based strictly on time-dependent perturbation theory, we study how in both methods varying degree of cancellation between perturbative contributions gives rise to cross-peaks, and identify exciton annihilation and exciton relaxation contributions to the cross-peak in the zero-waiting-time F-2DES. We propose that time-gated fluorescence detection can be used to isolate the annihilation contribution to F-2DES both to retrieve information equivalent to 2DES spectroscopy and to study the annihilation contribution itself.

Graphical TOC Entry

Keywords Non-linear spectroscopy, multi-dimensional spectroscopy, exciton delocalization, incoherent detection 2


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Over the past decade, many variants of multi-dimensional spectroscopic techniques have been developed and new methods for data analysis have been added. One of the recent additions to the arsenal of feasible methods are those which combine phase cycling or phase modulation approach to the relevant data retrieval with incoherent signal detection, such as the fluorescence (FL) detection. 1–5 The new detection scheme brings promises of better sensitivity 6–9 to certain crucial parameters of molecular aggregates, such as exciton delocalization, and opens questions of contributions from processes related to the detection scheme itself. These promises and questions have to be thoroughly addressed before the method becomes mainstream. 8–10 In this paper, we put the theory of fluorescence-detected two-dimensional electronic spectroscopy (F-2DES) into the context of the well-known theory of the standard coherently detected two-dimensional electronic spectroscopy (2DES). We view the processes of the signal generation in F-2DES as a new degree of freedom which represents both an opportunity for development of new sensitive experiments and a potential source of difficulties in comparing the new method to established spectroscopic techniques. Two theoretical frameworks stand out as indispensable foundations for various models aimed at explaining recent experiments, namely Frenkel exciton model 11,12 and the perturbative response function theory of non-linear spectroscopy. 11,13 While they are, obviously, approximate theories, some of their essential approximations represent core features of the photo-induced physics of molecular aggregates. For instance, the lack of orbital overlap between neighboring molecules, required in Frenkel exciton model, is one of the features of natural photosynthetic antennae, which prevents excitation quenching related to formation of charge transfer states. 14 Similarly, response function theory of the third order explains well the typical time-resolved experiments including coherent two-dimensional electronic spectroscopy. In these experiments, it is crucial to ensure that the signal depends on the third power of the external field to prevent unwanted higher order signal. Thus the order of perturbation is not a limitation of theory, but rather a constitutive feature of the experiment. Exciton delocalization, i.e., excitonic states being superpositions of localized excited

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molecular states, is one of the crucial features of efficient energy transfer in photosynthetic antennae. 15 Correspondingly, one of the most valuable features of the 2DES spectrum is that the presence of a cross-peak between electronic states of a molecular aggregate at the waiting time t2 = 0 is a direct witness of exciton delocalization. Similar relation between cross-peaks and delocalization has been expected in F-2DES method. 8 Processes occurring on longer time scale than the one defined by the four initial pulses of the experiment, play a significant role in establishing the measured signal in F-2DES. 5 In order to study how is the exciton delocalization reflected in F-2DES, we include into our considerations the processes of exciton-exciton (e-e) annihilation and single exciton energy relaxation that occur before fluorescence emission. To this end, in a slight extension of the usual Frenkel exciton model, we consider a dimer of three-level molecules. The state diagram is presented in Fig. 1. The eigenstates form three bands: the ground-state |gi, the single exciton band |ei (formed from Q linear combinations of states such as |ni1 = |en i N k6=n |gk i), and the double exciton band. The latter contains not only the true double-exciton states |eei, i.e. linear combinations Q of states |nmi = |en i|em i k6=n,m |gk i, but also the manifold |f i of higher excited states Q |ni2 = |fn i k6=n |gk i of the pigments. Above we denoted |en i the first excited state and |fn i the second excited state of the nth molecule. In our model we assume optically allowed transitions |gn i → |en i and |en i → |fn i. Without much loss of generality, we assume that the (n)

(n)

(n)

energy gaps h ¯ ωf e = f − e (n)

(n)

(n)

gaps h ¯ ωeg = e − g

between the states |fn i and |en i are suitably larger than the

between states |en i and |gn i so that resonance interaction between

transitions of the types |gi → |ei and |ei → |f i residing on neighbouring molecules does not lead to any significant delocalization. In strongly-coupled molecular aggregates the |eei states often mix with the |f i states. 16 Such mixed states inherit the fast internal conversion rate of the f states, which leads to a rapid annihilation. 17 In our model such states would correspond to the |eei states with a fast kf ee . Spectroscopically, they would exhibit the same behavior as the |f i states. Also excitons on weakly-coupled molecules can annihilate, without mixing of the |eei and |f i

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states. This proceeds in two steps, first an energy transfer from |eei → |f i, which can be described by e.g. Förster theory. The second step is a rapid internal conversion of the |fn i states into |en i states, preventing the back-reaction in form of exciton fission. In a multichromophoric system, once two excitons are created, they migrate and, eventually, meet at coupled pigments, where they fuse into one of the |f i states. In our model, the effective rate kf ee of this process includes both the exciton migration and the first annihilation step.

Figure 1: Three band model of an effective molecular dimer with exciton-exciton annihilation. Full lines: double-exciton-to-higher-exciton transfer, wiggly line: internal conversion, dashed lines: radiative and non-radiative relaxation.

F-2DES experiment has been described in detail in literature. 2,4,5,10 Four short laser pulses with well defined phases φ1 , φ2 , φ3 and φ4 , respectively, and well defined delays t1 , t2 and t3 between first and second, second and third, and third and fourth pulses, respectively, are applied to a sample containing the studied molecular system, and the fluorescence as a function of the time delays and phases is recorded. The signal dependence on the pulse phases is numerically processed in such a way that a signal component equivalent in its phase dependence (on the external pulses) to the signal obtained in the standard 2DES measurement 18 is obtained. While in standard measurement, this signal is distinguished from other signals by its spatial direction, in the F-2DES, equivalent signal has to be disentangled from the rest of the fluorescence detected response by its phase signature. This is done either directly by filtered lock-in detection 2 or numerically by phase cycling. 4 The same phase cycling method can be used in 2DES measurements, 19 for comparison of various 2DES

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implementations see Ref. 20 Assuming quasi-resonant excitation and the limit of ultra-short laser pulses, the nonlinear response of the system is proportional (up to a phase factor) to a sum of non-linear response functions of the third order. 13 In Fig. 2, we present double-sided Feynman diagrams responsible for a cross-peak on a specific spectral position in both the 2DES and F-2DES spectra. We notice that the diagrams for the 2DES are conventionally closed by the transition dipole operator acting from the left (denoted by the straight line). In the case of F-2DES, the diagrams are closed by interaction with the fourth laser pulse, either on the left- or right-hand side of the Feynman diagram. The position of the arrow determines the sign of the contribution (reflects one part of a commutator). We can see that both ESA1 and ∗ ESA2 diagrams of the F-2DES experiment are descendants of the diagram R1f of the 2DES

experiment. The theoretical expressions represented by these diagrams are presented in the SI. Table 1: System parameters for calculations quantity (1)

(2)

h ¯ ωeg , h ¯ ωeg (1) (2) h ¯ ωf e = h ¯ ωf e kf ee (≈ kA ) kic knr , kf l kf l φf l = kf l +k nr (1) µeg (2) µeg

=

value 12000 cm−1 , 12500 cm−1 13500 cm−1 1 10 ps 1 100 fs

1 , 1 4.3 ns 10 ns

0.3

(1) µf e (2) µf e

= d12 T λ, τc

4.0D √ (3, 1, 0) 10 3.3D √ (3, −1, 0) 10

∈ [6, 50] ˚ A 300K 50 cm−1 , 100 fs

The rules for conversion of the diagrams of the 2DES measurement into the ones involved in F-2DES are the following: Each conventional Liouville pathway which ends in the ground state in 2DES experiment, yields a single pathway ending in a singly-excited state in F-2DES.

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Figure 2: Top: Non-linear rephasing response functions corresponding to a cross-peak signal ∗ in the 2DES experiment (diagrams R3g and R1f ). Bottom: Non-linear response functions corresponding to the cross-peak signal in F-2DES experiment (diagrams ESA1, ESA2 and GSB). All diagrams are denoted by the sign of their contribution, which is obtained as −1u , where u is the number of arrows on the left-hand-side of the diagram. In F-2DES diagrams we distinguish their portion which is identical to the 2DES diagrams by red color, GSB ↔ R3g , ∗ ESA1 ↔ R1f . Analogical set of pathways can be written which has an coherence 1 2 in the population time, the GSB pathway changes into a stimulated emission one.

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The sign of the diagram is inverted, i.e. it is minus in F-2DESi . Each Liouville pathway which ends in the excited state in 2DES (and which reaches the two-exciton manifold) yields two pathways in F-2DES: one with an inverted sign with respect to the corresponding 2DES pathway, i.e. yielding a positive contribution, and one keeping its sign, i.e. yielding a negative contribution. The former of the F-2DES pathways ends in double exciton manifold, and the latter ends in the single exciton manifold. Unlike in 2DES, where the signal is produced in a stimulated processes, in F-2DES measurement, the signal is generated by spontaneous processes which start from the excited state prepared by the four pulses of the F-2DES sequence. The processes following the fourfold interaction with the pulses involve quasi-equilibration within and between the excited state manifolds, including radiative and non-radiative energy relaxation and e-e annihilation. This system dynamics determines the contribution of the individual Liouville-space pathways to the measured F-2DES spectrum. Due to the timescale and incoherent nature of the fluorescence emission, it suffices to consider the evolution of the system eigenstates. As the intraband energy relaxation is typically orders of magnitude faster than the interband processes, we will consider the |ei and |eei manifolds to be in a quasi-stationary thermal equilibrium. As we show in the SI, this leads to a set of linear kinetic rate equations for the band populations. From the pathways in Fig. 2, we can infer some features of the cross-peaks in 2DES and F-2DES. In traditional 2DES, there are two pathways contributing to the 2-1 cross-peak, largely cancelling each other out. When the molecules are weakly-coupled (or independent), the cancellation is perfect. However, when the coupling is stronger, the transition-dipole strength redistributes between the states, as seen at the diagonal peaks in Figs. (3) and (4). As a result, the cancellation of pathways is not complete and it can be shown, for molecules with the same oscillator strength, that, in the leading term, the cross peak amplitude is i

Concerning the diagram sign, there is a pre-factor of i for each interaction with the pulse. The F-2DES thus has i4 , while 2DES i3 . However, the 2DES signal field is connected to the 3rd order polarization as Esig ∼ iP 3 , so the i4 pre-factor is the same for both detection techniques

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Figure 3: Simulated 2DES and F-2DES spectra for system with weak (top) and strong (bottom) resonance coupling. Depicted are 2DES (left), F-2DES (middle), and fast-gated F2DES (right). The spectra were calculated and directly plotted by the QuantaRhei package. 21

proportional to

2J . ∆E

The cross-peak amplitude thus provides good information about the

exciton delocalization ii . In the F-2DES there are three Liouville pathways contributing to the cross-peak. The GSB and ESA1 have the same (negative) signs, while ESA2 bears the opposite sign. The contribution of the GSB and ESA1 pathways is given by the fluorescence rate of the equilibrated one-exciton manifold, kfe l ; let us take these pathways to contribute by one photon to the signal. The ESA2 contribution depends on the fate of two-exciton state, signified by the emission rate kfeel and the annihilation rate, kA ≈ kf ee . In the case that the molecules are independent, the ESA2 pathway contributes by two photons and the cross-peak vanishes, similarly to 2DES. It is important to note that the cross-peak vanishes, in the response theory, only when all Liouville pathways are taken into account. Individual Liouville pathways do not correspond to real physical processes, but rather to components of a perturbation ii

We remind the reader that the mixing angle for a dimer is given by tan2θ =

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2J ∆E .


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series. In the case of weakly-coupled molecules (energy transfer, but no delocalization), the annihilation plays the key role. When the signal is emitted/detected before the annihilation occurs, the relative yield of the ESA2 pathway is given by

kfeel . kfe l

This ratio differs from 2

in the case of energy equilibration between pigments of unequal oscillator strength (OS). For molecules with similar OS, the double exciton state contributes two photons, and the pathways cancel again, suppressing the cross peak. When, however, the annihilation is present, it reduces the contribution of the ESA2 pathway, preventing the cancellation and giving rise to the cross peaks. Let us now, for illustration, consider molecules of the same OS and focus on the integrated fluorescence, so that it is sufficient to take the overall yield of the −1 kA ESA2 pathway Γ = 1 + 1 + 2 k +k , Γ ∈ [2, 1]. Denoting the amount of annihilation ( nr f l ) A = 2 − Γ = k +2 kkA +k , A ∈ [0, 1], the detection-time-integrated F-2DES spectrum can ( nr f l ) A iii be expressed as SF −2DES ∝ −GSB −SE +(1 − A) ESA. 3,22 For no annihilation, A = 0, the spectrum corresponds to the traditional 2DES, S2DES = GSB + SE − ESA. Furthermore, it can be shown that the cross-peak amplitude is directly proportional to the annihilation A and the F-2DES cross-peaks are a good measure for e-e annihilation. We treat the more general case of finite coupling, partial annihilation and different OS further in the text. The derivation of the dependences presented above can be found in the SI. To demonstrate quantitatively the properties of F-2DES and its sensitivity to exciton delocalization and e-e annihilation, we compare the F-2DES spectra and their parameter dependence with 2DES. The parameters used for the simulations can be found in Table 1 and the computational details are presented in the SI. In Fig. 3 we present a timezero 2D spectrum of our effective dimer with weakly and strongly coupled molecules. We present the F-2DES spectrum with the same sign convention as in the traditional 2DES. In traditional, coherently-detected 2DES, two diagonal peaks corresponding to the SE and GSB of the two transitions can be found. Additionally, there is a negative ESA to the f iii

This applies only when not considering the ESA2 ending in the |fn i states. For this ESA2, Γ ≈ 1 due to the fast internal conversion.

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states. In the weak coupling limit, there are no cross peaks and no exciton delocalization. With stronger coupling, cross-peaks appear and the excitons become delocalized. Also the oscillator strength becomes redistributed between the transitions. In the F-2DES there are cross peaks both in the weak and strong coupling limit. Interestingly, there is no ESA to the higher excited f states. This is a result of near-perfect cancellation of the ESA1 and ESA2 pathways due to the rapid internal conversion, as can be seen from Fig. 2 and footnote iii. In the last column of Fig. 3 we give F-2DES spectra obtained by gating the fluorescence and detecting only for a half of the annihilation time. Clearly the spectra resemble the traditional 2DES, without the f state ESA.

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2DES

70

Peak Amplitude (arb.u.)

12000-12000 12500-12500 100

12000-12500 12500-12000

0

50

100

Electronic coupling (cm-1)

30

12000-12500 12500-12000

20 10 0

12000-13500 12500-13500

-10

-30 0

150

60

F-2DES

0.015

12000-12000 12500-12500

40

-20

12000-13500 12500-13500

-100

50

12000-12000 12500-12500

0.010

12000-12500 12500-12000 0.005

12000-13500 12500-13500 0.000

50

100


150

2.0

F-2DES

50

1.8

12000-12000 12500-12500

40

1.6 30 1.4 20

12000-12500 12500-12000

12000-13500 12500-13500

10

1.0

0

0

20

40

100 300 1000 3000 10000

1E-5

Detection time (ps)

1E-4

0.001

0.01

0.1

1

Annihilation rate (1/ps)

Figure 4: Dependence of the peak amplitude on the resonance coupling J (top), FL detection time (bottom left, note the logarithmic axis after 50 ps, the dashed line indicates the annihilation time of 10 ps) and annihilation rate (bottom right, note the logarithmic horizontal axis, the dashed line indicates the excitation lifetime of 3 ns). The labels denote the peak assignment (w1 w3 ), top pair: diagonal peaks, middle pair: cross peaks, bottom pair: ESA cross peaks. For direct comparison between the 2DES and F-2DES, the latter spectra amplitude can be ’corrected’ for one-exciton oscillator strength redistribution (shaded lines top right). In the annihilation rate dependence, bottom right, we give also the relative yield Γ of the ESA2 pathway ending in the double-excited state.

To learn more, we study the dependence of the peak amplitude on the system parameters (see Fig. 4). In all panels, the top pair of lines depicts amplitudes of the diagonal peaks, the middle pair amplitudes of the cross-peaks, and the bottom pair amplitudes of the f

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1.2


double-exciton relative yield Γ

0

0.020

F-2DES

60



200


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state ESA peaks. In the top row, the peak amplitude with increasing coupling is plotted for 2DES and F-2DES. The dependence confirms what we asserted before. The diagonal peaks in both 2DES and F-2DES behave in the same way (when corrected on the kfe l ), the oscillator strength of the transitions is redistributed. For 2DES the cross-peak amplitude increases with the coupling, linearly at first, giving a good measure of the coupling (and the resulting exciton delocalization). In the F-2DES, there is no f state ESA peak. The cross-peak amplitude does not vanish even for negligible coupling and displays only slight increase with the coupling. It is therefore not a measure of delocalization. In order to demonstrate the origin of the cross-peaks and the potential extension of the technique, we investigate the possibility of time-gating the detected fluorescence. In the bottom-left panel of Fig. 4, the evolution of the peak amplitude with the detection time unveils. In the first 50 ps (linear scale) the cross-peak amplitude rises exponentially. Crucially, this happens on the timescale of the e-e annihilation (10 ps, the dashed line). In later times (log scale) the whole spectrum decays with the lifetime of the fluorescence. Finally, in the bottom-right panel, we investigate the dependence on the rate of e-e annihilation. For annihilation much slower than the excitation lifetime (3 ns, the dashed line), there are, again, only weak cross peaks, with amplitude given by the difference of the pigment oscillator strength. When, however, the annihilation becomes comparable to or faster than the excitation relaxation rate, pronounced cross-peaks appear, originating from the presence of annihilation. The cross-peak amplitude clearly anticorrelates with the yield of the double-exciton state, reflecting the cancellation of pathways described above. In the recent years, several more theoretical and experimental groups turn their attention to F-2DES. Grégoire et al. have investigated influence of exciton-exciton interaction on the F-2DES. In their approach, they expand the excited-state populations in the order of interaction with the field. 5 Comparing their description with ours and considering that the e-e annihilation takes time to occur, we clearly see that the ESA2 pathway in Fig. 2 alone does not constitute the e-e annihilation signal. Only by including all the response

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pathways one gets the full signal as detected in the experiment. Mukamel has considered the presence of many-particle signals for non-interacting particles. 23 He considers a phase modulation scheme isolating the 2φ12 phase component, which corresponds to two-particle excitation. This is different from the phase modulation considered here, which includes both the single- and two-particle contributions. Damtie et al. calculate the F-2DES spectra of a model dimer including the train of pulse sequences explicitly in their simulations. 24 This approach has some advantages, such as easy treatment of finite pulses and possibility to study excitation accumulation effects, later utilized by Osipov et al. 25 However, it does not allow an easy identification of individual perturbative contributions to the spectrum, which proved to be so useful in 2DES. The perturbative approach has been used by Agarwalla et al. 26 to compare coherent (photon) and incoherent (current) 2DES signals from single molecule junction, finding a good correspondence between the two. Finally, it was very recently used by Schröter et al. 9 to provide theoretical foundations for the observations of Karki et al. 8 Although the response function formalism is essentially the same used here, the authors conclude that the presence of the time-zero cross-peaks is indicative of exciton delocalization, in contrast to the main conclusion of this work. In summary, we have presented calculations of F-2DES spectra based on rigorous response function formalism. We demonstrate that while cross-peaks at zero waiting time in traditional 2DES are signatures of exciton delocalization, the same cross-peaks in F-2DES can also stem from exciton-exciton annihilation and exciton relaxation processes. We propose that time-gating of the measured fluorescence signal provides a direct way to observe interactions in two-exciton band, notably the exciton-exciton annihilation.

Acknowledgement ˇ This work was supported by the Czech Science Foundation (GACR) grant no. 17-22160S.

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(9) Schröter, M.; Pullerits, T.; K¨ uhn, O. Before Förster: Using Fluorescence Detected Two-Dimensional Spectroscopy to Investigate Transient Exciton Delocalization Between Weakly Coupled Chromophores. arXiv:1806.10941 2018, (10) Tiwari, V.; Matutes, Y. A.; Zhanqian, Y.; Ptaszek, M.; Bocian, D. F.; Holten, D.; Kirmaier, C.; Konar, A.; Ogilvie, J. P. Strongly Coupled Bacteriochlorophyll Dyad Studied Using Two-dimensional Phase-modulated Fluorescence-detected Electronic Spectroscopy. arXiv:1806.00896 2018, (11) van Amerongen, H.; Valkunas, L.; van Grondelle, R. Photosynthetic Excitons; World Scientific: Singapore, 2000. (12) May, V.; K¨ uhn, O. Charge and Energy Transfer Dynamics in Molecular Systems; WileyVCH, 2011. (13) Mukamel, S. Principles of nonlinear spectroscopy; Oxford University Press: Oxford, 1995. (14) Beddard, G. S.; Porter, G. Concentration quenching in chlorophyll. Nature 1976, 260, 366–367. (15) Str¨ umpfer, J.; S¸ener, M.; Schulten, K. How quantum coherence assists photosynthetic light-harvesting. J. Phys. Chem. Lett. 2012, 3, 536–542. (16) Br¨ uggeman, B.; Herek, J. L.; Sundström, V.; Pullerits, T.; May, V. Microscopic Theory of Exciton Annihilation: Application to the LH2 Antenna System. J Phys Chem B 2001, 105, 11391–11394. (17) Br¨ uggemann, B.; May, V. Exciton exciton annihilation dynamics in chromophore complexes. I. Multiexciton density matrix formulation. J. Chem. Phys. 2003, 118, 746. (18) Brixner, T.; Manˇcal, T.; Stiopkin, I. V.; Fleming, G. R. Phase-stabilized twodimensional electronic spectroscopy. J. Chem. Phys. 2004, 121, 4221–36. 16


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(19) Tan, H.-S. Theory and phase-cycling scheme selection principles of collinear phase coherent multi-dimensional optical spectroscopy. J. Chem. Phys. 2008, 129 . (20) Fuller, F. D.; Ogilvie, J. P. Experimental Implementations of Two-Dimensional Fourier Transform Electronic Spectroscopy. Annu. Rev. Phys. Chem. 2015, 66, 667–690. (21) Manˇcal, T. Quantarhei: Molecular Open Quantum Systems Package. 2018; http:// github.com/tmancal74/quantarhei. (22) Lott, G. A.; Perdomo-Ortiz, A.; Utterback, J. K.; Widom, J. R.; Aspuru-Guzik, A.; Marcus, A. H. Conformation of self-assembled porphyrin dimers in liposome vesicles by phase-modulation 2D fluorescence spectroscopy. Proc. Natl. Acad. Sci. 2011, 108, 16521–16526. (23) Mukamel, S. The origin of many-particle signals in nonlinear optical spectroscopy of non-interacting particles. J. Chem. Phys. 2016, 145, 041102. (24) Damtie, F. A.; Wacker, A.; Pullerits, T.; Karki, K. J. Two-dimensional action spectroscopy of excitonic systems: Explicit simulation using a phase-modulation technique. Phys. Rev. A 2017, 96, 1–12. (25) Osipov, V. A.; Shang, X.; Hansen, T.; Pullerits, T.; Karki, K. J. Nature of relaxation processes revealed by the action signals of intensity-modulated light fields. Phys. Rev. A 2016, 94, 1–9. (26) Agarwalla, B. K.; Harbola, U.; Hua, W.; Zhang, Y.; Mukamel, S. Coherent (photon) vs incoherent (current) detection of multidimensional optical signals from single molecules in open junctions. J. Chem. Phys. 2015, 142, 212445.

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2DES 100

12000-12500 12500-12000

0

50

100


30

10 0

-30 150

12000-12000 12500-12500

0.010

12000-12500 12500-12000 0.005

12000-13500 12500-13500 0

20

40

0

50

100


150

2.0

F-2DES

50

1.8

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40

1.6 30 1.4 20

12000-12500 12500-12000

12000-13500 12500-13500

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0 ACS Paragon Plus Environment

100 300 1000 3000 10000

Detection time (ps)

12000-13500 12500-13500

-10

60

0.000

12000-12500 12500-12000

20

F-2DES

0.015

12000-12000 12500-12500

40

-20

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-100

50

1E-5

1E-4

1.2

1.0 0.001

0.01

Annihilation rate (1/ps)

0.1

1

double-exciton relative yield Γ

0


12000-12000 12500-12500

0.020

F-2DES

60




1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

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