Theory of Excitonic Delocalization for Robust Vibronic Dynamics in

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Biophysical Chemistry, Biomolecules, and Biomaterials; Surfactants and Membranes

On the Theory of Excitonic Delocalization for Robust Vibronic Dynamics in LH2 Felipe Caycedo-Soler, James Lim, Santiago Oviedo-Casado, Niek F. van Hulst, Susana F. Huelga, and Martin B. Plenio J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b00933 • Publication Date (Web): 04 Jun 2018 Downloaded from http://pubs.acs.org on June 5, 2018

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The Journal of Physical Chemistry Letters

On the Theory of Excitonic Delocalization for Robust Vibronic Dynamics in LH2 Felipe Caycedo-Soler,∗,† James Lim ,† Santiago Oviedo-Casado,‡ Niek F. van Hulst,¶,§ Susana F. Huelga,† and Martin B. Plenio∗,† Institute of Theoretical Physics and Integrated Quantum Science and Technology IQST, University of Ulm, Albert-Einstein-Allee 11, D - 89069 Ulm, Germany, Departmento de Física Aplicada, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain, and ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain E-mail: [email protected]; [email protected] To whom correspondence should be addressed †Institute of Theoretical Physics and Integrated Quantum Science and Technology IQST, University of Ulm, Albert-Einstein-Allee 11, D - 89069 Ulm, Germany ‡ Departmento de FÃsica Aplicada, Universidad PolitÃ©cnica de Cartagena, 30202 Cartagena, Spain ¶ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain §CREA - Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain

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ACS Paragon Plus Environment

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Abstract Nonlinear spectroscopy has revealed long-lasting oscillations in the optical response of a variety of photosynthetic complexes. Different theoretical models which involve the coherent coupling of electronic (excitonic) or electronic-vibrational (vibronic) degrees of freedom have been put forward to explain these observations. The ensuing debate concerning the relevance of either mechanism may have obscured their complementarity. To illustrate this balance, we quantify how the excitonic delocalization in the LH2 unit of Rhodopseudomonas Acidophila purple bacterium, leads to correlations of excitonic energy fluctuations, relevant coherent vi-bronic coupling and, importantly, a decrease in the excitonic dephasing rates. Combining these effects, we identify a feasible origin for the long-lasting oscillations observed in flu-orescent traces from time-delayed two-pulse single molecule experiments performed on this photosynthetic complex, and use this approach to discuss the role of this complementarity in other photosynthetic systems.

Long-lasting oscillations –ranging from hundreds of femtoseconds at room temperature to a few picoseconds at 77K– in the two dimensional electronic spectroscopy (2DES) from the FennaMatthews-Olson (FMO) complex in green sulfur bacteria, 1–5 purple bacteria light harvesting complexes, 6,7 or of reaction centers of bacteria and higher plants 8–10 have been reported. These observations are obtained by excitation and read out with four calibrated ultra-short pulses providing rich but spectrally congested data of the ensemble dynamics, with an inherent complexity that has made difficult to reconcile the observations in FMO. 1,5,11 The most promising models to explain these experiments rely on theACS generation of protected excitonic coherence due to correlated Paragon Plus Environment 8,12 fluctuations or on the existence of long-lived vibronic coherence induced and sustained by the

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spectra, as it reports oscillating fluorescence traces from single light harvesting 2 (LH2) complexes from purple bacteria, as a function of the time delay between only two pulses, or as a function of the phase difference between pulses for a given time delay. Since 2DES convolutes the excited vibronic dynamics with the ground state vibrational wave-packet motion, 4,5,17 the two pulse fluorescence traces of Hildner, et al. 24 which depend solely on the excited state dynamics, overcome this inconvenient convolution. Motivated by this experiment, in this article we provide a careful analysis of the subtleties of the excitonic delocalization in LH2 that lead to a substantial and robust vibronic interaction, which, in turn, identifies this vibronic-excitonic interaction as the possible origin of the observed oscillations. Normalized absorption

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1

B

B850 B800

0.8 0.6 0.4 0.2 0

800

850

900

wavelength [nm]

Figure 1: A The schematic representation LH2 from Rps. acidophila based on its X-ray structure. 25 The inner and outer rings of pigments correspond to the B850 and B800 structures, respectively, with Qy transition dipoles denoted by arrows. B Experimentally observed absorption of the LH2 complex 24 (blue circles) superimposed to our model calculation (thick black continuous). The spectra from excitons mainly delocalised over the B800 (≈ |αi states) and B850 (≈ |β i states) pigments are shown in green and red lines, respectively. We show averages over 2 × 104 realizations of static disorder.

High resolution X-ray crystallography of the LH2 complex of Rhodopseudomonas (Rps.) Acidophila 26,27 reveals a structure composed of 9-fold repeating apoproteins, monomeric in the B800 and dimeric in the B850 rings, with absorption maxima at about 802 and 858 nm respectively, 28 as shown in Figure 1. Several theoretical and experimental studies 6,29–41 have reached a partial consensus on quantities relevant for excitonic dynamics, such as excitonic couplings, pigment’s 3



energies and their environment-induced fluctuations. Hildner et al. 24 reported, unexpectedly, longlived oscillations in the fluorescence intensity (FI) traces from two color pulsed illumination, in a scheme intended to first excite the B800 band and then read out the B850 pigments’ population in single LH2 complexes with the second pulse. By varying the time-delay between pulses, oscillations in the FI were observed up to 400 fs with a period of ∼ 200 fs. 24 Oscillations with about the same amplitudes were also observed by varying the phase difference between pulses at a fixed time delay. The oscillations were interpreted as signatures of coherent exciton exchange between B800 and B850 rings. A similar interpretation of coherent B800-B850 interaction was called to address long-lasting oscillations in 2DES performed in full LH2s, 6 whereas, later, oscillations with the same lifetime were ascribed to a vibronic interaction within the B850 ring, however, in 2DES performed in a B800-less mutant. 7 Since the B800→B850 process is regarded as an incoherent excitonic transfer, 40,42 the contrasting interpretations prompt to a reevaluation of the excitonic dephasing rates and/or an analysis of additional degrees of freedom that may be able to explain these oscillatory features. We will reexamine the coherent B800-B850 excitonic interaction in order to provide support for a scenario in which the excitons delocalized over common pigments in the B850 ring, participate in long-lasting coherent oscillatory population exchange mediated by underdamped vibrational modes, hence, rather compatible to the assignment made in the B800-less mutant. 7 We will show that the delocalization of the excitonic wave-functions across B850 pigments fulfills the requirements of mode-mediated coherent population exchange, namely, strong vibronic coupling due to excitonic wave-functions overlap, slow excitonic dephasing, and modeexciton resonance insensitive to static disorder. We will show that the analysis of the FI traces as a function of the phase difference between pulses further supports our model and, specifically, the significance of excitonic delocalization across the B850 structure. The Qy transition dipoles d~i , shown schematically in Figure 1A, from the electronic ground state |gi to the excited state |ii of the ith pigment, present a mutual interaction, Ji j , described by

4


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the excitonic Hamiltonian N800 +N850

He =

∑

N800 +N850

Ωi |iihi| +

i

=

∑

∑

Ji, j |iih j|

i6= j

εα |αihα| + εβ |β ihβ |

α∈B800

β ∈B850

+Vα,β (|β ihα| + |αihβ |)

∑ εγ |γihγ|.

=

(1)

γ

The proximity of water molecules and protein residues introduce variations in the pigment energies Ωi in a time-scale longer than the excitonic lifetime, resulting in uncorrelated static disorder in LH2, 43 hΩi Ω j i = hΩi ihΩ j i. On the other hand, systematic deformations of the ring giving rise to correlated fluctuations of the energies Ωi explained various features of ensemble absorption and single molecule LH2 fluorescent-excitation spectra 44–46 (see SI B.1 for details). Our description considers both mechanisms to provide a realistic description of experiments performed room temperature (see SI B.1 for details). The excitons |αi and |β i are, rigorously, the eigenstates of the B800 and B850 rings, respectively, if their mutual interaction is neglected (Vα,β = 0). In agreement with previous studies 42,47 we obtained after incorporating the static disorder, that the weak inter-ring B800-B850 coupling Vα,β . 25cm−1 , is far too small in comparison to the dynamical dephasing between these excitons in order to support an appreciable delocalization across these rings (see SI B.2 for a detailed discussion). Hence, the LH2 eigenstates can be grouped into states mostly delocalised over the B850 or the B800 pigments, i.e., |γi ≈ |β i or |γi ≈ |αi, respectively, which will be called B850 and B800 states in this work. If we need to address any particular state of average wavelength k, we will use the notation |γk i, |αk i or |βk i, for LH2, B800 and B850 states, respectively. The excitation by the pulses is described by the light-matter interaction Hamiltonian H f ield (t) = ~E(t, φ ) · (∑α ~Dα |αihg| + ∑β ~Dβ |β ihg|) + h.c. where ~E(t, φ ) = E(t, φ )Eˆ gathers the polarization E, ˆ amplitude and phase φ of the laser pulses. The transformed vectors of pigment transition dipoles d~i to the exciton basis ~Dβ acquire special relevance for the description of the optical response of the 5



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B850 structure, since these pigments present nearest neighbor excitonic interactions Ji,i±1 which are comparable to the dephasing that degrades the electronic coherence of these B850 |β i states. Besides the main absorption peaking at 858 nm, the B850 ring also absorbs at about 780 nm. 7,32 Given that the absorption of this high energy transitions depends on the pigments transition dipole out-of-ring’s plane angle, 48 we slightly increased this angle from that of the available structure of Rps. acidophila, 27 in order to obtain an agreement with the experimental evidence (more details of this variation are provided in SI A.1). The excitons that constitute this absorption band at about 780 will be denoted by |β ∗ i. The interaction between a pigment’s excited state and a single vibration per pigment –with energy described by Hv = ∑i ωb†i bi – is modeled by a linear coupling term of the form √ He−v = ∑ sω b†i + bi |iihi|

(2)

i

=∑

p

† Sγγ0 ω aγγ0 + aγγ0 |γihγ0|

(3)

γ,γ0

which quantifies the vibration-pigment coupling strength through the Huang-Rhys (HR) factors s. As explained in the SI B.4, we collect all modes of identical frequency ω coupled to different pigments in Eq. (3), into a generalized coordinate with creation (annihilation) operators a†γγ0 (aγγ0 ). This coordinate is a linear function of the operators b†i (bi ) and obeys the canonical commutation relations [aγγ0 , a†γγ0 ] = 1 and the generalized HR factor Sγγ0 = s ∑i |hγ|iihi|γ0i|2 = s Pγγ0 . The quantity Pγγ0 = ∑i |hγ|iihi|γ0i|2 is the spatial overlap of the excitonic wave-functions |γi and |γ0i, which for B850 pairs of β and β ∗ states will be denoted Pβ β ∗ while for B850-B800 pairs will be presented as Pαβ . −1 ≡ P−1 , quantifies the number of pigments that participate The inverse participation ratio Pγγ γ

in exciton |γi. The diagonal HR Sγγ ≡ Sγ results in a vibrational progression, which redistributes dipole strength on side-bands (SBs) as described by the Frank-Condon principle, however, from a zero-phonon line (ZPL) associated to a delocalized exciton. Accordingly, this diagonal HR Sγ can be determined from the ratio between SB and ZPL intensities at low temperature, 49 by means of

6


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available techniques such as non-photochemical hole burning or fluorescence line-narrowing. 50–52 On the other hand, Sγγ0 mediates the excitonic population exchange between |γi and |γ0i. Notice that this coherent exchange occurs only if excitonic wave functions overlap. Moreover, this population exchange is effective if the resonance condition ω ≈ εγ − εγ0 ≡ ∆εγγ0 is met.

A (β835 ,α787 )

0 -2

0.5

Pβα /Pβ

1

B

0 -2

ζ=-0.002 0

(β835 ,β*789 )

2

δ∆ǫββ∗

2

δ∆ǫβα

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 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ζ=0.605 0

0.5

1

Pββ∗ /Pβ

Figure 2: Realizations of fluctuations of the excitonic splitting δ ∆εγγ0 and of the excitonic overlap Pγγ0 for a B800-B850 pair in A and for a B850-B850 pair in B. The correlation of the fluctuations of excitonic energies underpinned by excitonic delocalization is observed in C. Here the correlation map for all LH2 excitons shows that the large (small) correlation coefficients ζ observed in B (A), which lead to a small (large) dispersion of the energy gap fluctuations, persist between B850 (B850-B800) pairs of states. We considered 2 × 104 realizations of static disorder. Figure 2 A and B present the realizations due to the static disorder, of excitonic energy gaps and excitonic overlaps, between selected pairs of B800-B850 excitons in A or between high and low energy B850 excitons in B. Following the notation for Pγγ0 , we denote δ ∆εαβ or δ ∆εβ β ∗ to the (normalized) fluctuations of the gap between B850 and B800 states in A, or between high and low energy B850 states in B, respectively. These energy gap fluctuations are normalized to the standard deviation (SD) of this gap if the variations of the excitonic energies εγ and εγ0 were uncorrelated, i.e δ ∆εγγ0 = (∆εγγ0 − ∆εγγ0 )(σε2γ + σε2γ0 )−1/2 . Here, σεγ represents the SD of the excitonic energy εγ and ∆εγγ0 is the average of the gap ∆εγγ0 . The quantity δ ∆εγγ0 will present a ±1 SD confidence interval for uncorrelated excitonic energy fluctuations which is equal to 2, no matter the magnitude of σεγ or σεγ0 . Figure 2A shows that this is the case for the fluctuations δ ∆εαβ , since the length of the error bars, equal to the SD calculated for all these realizations, indeed equals 2. However, notice that 7



for pairs in the B850 ring represented in Figure 2B, we obtain error bars with a length smaller than two, implying that the excitonic gap ∆εβ β ∗ is more robust than the gap arising from independent fluctuations of excitonic energies. Since the SD of δ ∆εγγ0 is independent of the magnitude of the excitonic energy fluctuations, this reduction is beyond exchange narrowing, 53 namely, narrowing of excitonic energy fluctuations due to excitonic couplings Ji j . The robustness of the B850 excitonic gaps emerges, as we will see, from correlations between excitonic energy fluctuations. These correlation can be quantified by the Pearson correlation coefficient ζ ≡[hεγ εγ0 i−hεγ ihεγ0 i]/(σεγ σεγ0 ). This coefficient has a value ζ = −0.002(6) for the energies of B800-B850 states in Figure 2A contrasting with a large positive correlation of ζ = 0.605(8) for the B850 states in B. The origin of this correlation will become more transparent after determining the subtleties of the excitonic delocalization across the LH2 pigments. To that end, we also consider in Figure 2A and B the scatter plots of the excitonic overlap Pαβ /Pβ and Pβ β ∗ /Pβ , normalized by the participation ratio Pβ for reasons that will become clear in short. Unsurprisingly, the overlap between B800 and B850 excitons in A is six to seven times smaller than the overlap between B850 excitons in B. Since correlations of the excitonic gaps arise in B but not in A, these results relate a larger overlap of the excitonic wave-functions to a larger correlation among their respective energy fluctuations. Figure 2C presents a correlation map of the full LH2, in which the color scale is the value of ζ regarding pairs of B850 (ordered from 1 to 18 in ascending energy) and B800 (numbered from 19 to 27) excitons. Here we can see that a high positive correlation of fluctuations between all B850 states, contrasts with a smaller correlation between B800 states, and low and even slightly negative correlation between B850-B800 exciton pairs. We conclude that the correlations in excitonic energy variations are important and positive for overlapping excitonic wave-functions, implying that perturbations on individual pigment energies affect all excitonic energies that delocalise over that pigment, in a similar way. As a consequence of the overlap of excitonic wave-functions, the static disorder may compromise the resonance between a mode and an excitonic gap ω ' ∆εγγ0 in the B800-B850 exchange, while due to the correlations observed, this resonance is more robust for the

8


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vibronic exchange between B850 excitons. The explicit connection between the excitonic overlap and the generalized HR after Eq. (3), serves to derive the relation Sγγ0 = Sγ Pγγ0 /Pγ which allows us to determine the non-diagonal HR Sγγ0 of a specific vibrational mode, coupling the excitonic states |γi and |γ0i. Since both excitonic overlap Pγγ0 and participation ratio Pγ are determined from an accurate microscopic model of interactions and static noise 32,44–46 (see SI A-B) while the diagonal HR Sγ can be estimated experimentally from the vibrational progression with optical techniques in LH2, 50–52 we can provide a reliable estimate of the coupling between different excitons via vibrations in this complex. For modes of energy ω = 750 cm−1 which are resonant with both the B800-B850 and B850 pairs studied, these experiments determined 50,51 Sβ = 0.05, leading to averages Sαβ = 0.002 and a six times larger Sβ β ∗ = 0.012, based on results in Figure 2. The rather strong coupling of this vibrational mode to the B850 excitons compared to other discrete vibrations, has been the ground for p discussion about its role in the incoherent dynamics of LH2. 42,50,54 The interaction Sβ β ∗ ω ≈ 80 cm−1 with these modes, results in a population exchange happening with a period of about 200 fs for B850 excitons, and of about 600 fs for B800-B850 pairs. 12

A

10 8

β876

4

β857

3

B

(β835 - β784 ) (β835 - α787 )

β842

6

2

α787

4

1 2

0

0 0

0.5

0.2

1

0.4

0.6

0.8

1

Γγγ′ (Γγ + Γγ′ )−1

Pγ

Figure 3: Excitonic delocalization and dephasing. A Distribution of the participation ratio Pγ for selected states. B Distribution of the ratio of excitonic dephasing rate Γγγ0 and the sum of optical dephasing rates Γγ Γγ0 for a B800-B850 (dashed) and a B850 pair (continuous). The arrows with the same line code, show the average of the respective distributions (' 0.76 B850s and ' 1 B850B800): delocalization over common pigments reduces inter-excitonic dephasing as compared to the optical dephasing. All the distributions were obtained with 8 × 104 realizations of pigment energies. Excitonic delocalization also affects the dephasing rates of the excitons, which influence the 9



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duration of the coherent population exchange between excitons. The Redfield formalism 55,56 specifies the inter-exciton dephasing rate in terms of the pigment’s dephasing rate Γi of their groundexcited state coherence Γγγ0 = Γγ + Γγ0 − 2 ∑ |hγ|iihi|γ0i|2 Γi

(4)

i

which depends on the optical coherence |gihγ| dephasing rate 1 Γγ = ∑ |hi|γi|4 Γi + Γrelax , 2 γ i

(5)

is the total relaxation rate from the state |γi to all the other states (see SI B.3). Two where Γrelax γ reductions of dephasing rates are discernible: in Eq. (4) the inter-excitonic dephasing rate is suppressed thanks to the overlap between excitonic wave-functions, while in Eq. (5), Γγ ∝ Pγ Γi is reduced due to the excitonic participation ratio Pγ . Diagonalization of realizations of He under static disorder in Figure 3A and Figure 3B, witness this sequential reduction. First, the B850 states have an optical dephasing slower than individual pigments due to their degree of delocalization, with averages of Pβ ≈ 0.29, 0.22and0.17 for the β876 , β857 and β842 states, respectively. This leads to a slower optical dephasing with higher energies across the B850 band, as observed in single LH2 molecules. 57,58 Second, a further reduction of the inter-excitonic dephasing rate Γγγ0 between B850 states results due to their excitonic overlap Pγγ0 in Figure 3B. The B800 excitons, limited to Pα−1 ≈ 1.5 pigments (cf. in Figure 3A), do not benefit from the reduction of optical dephasing. Also, the pairs of B850-B800 states do not benefit from the reduction of inter-excitonic dephasing, as can be seen from the results in Figure 3B. From our estimate of optical dephasing from individual pigments Γi = 258 cm−1 (' 41 fs decay time) based on experimental and theoretical analysis of the B820 dimer 35,49,59,60 (further details in SI B.4), the excitonic optical dephasing lifetime slows down to about 100-150 fs and the inter-excitonic dephasing to 70-120 fs. Notice that beyond average values, we can estimate from the distribution in Figure 3B that a dephasing Γβ β ∗ slower than, e.g., 200 fs, is expected for 18% of the traces. Observe that this is already about 7 times the lifetime of optical coherences from individual pigments.

10


(3) Second pulse

B

0.8

β835 α787 β835

0.6

β*784

0.4 0.2 0

0

200

400

600

t [fs]

0.8

(β857 ,β*805 ) (β842 ,β*794 ) (β835 ,β*784 )

C

0.75 0.7 0.65

1.15

(β835 ,β*784 )

1.1

(β831 ,β*782 )

1 0.95 1.05

0.6

0.9

0.55

0.85

0.5

0.8

100

200

300

(β835 ,β*784 ) s=0

1.05

FI/

Total excitonic population

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 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Excitonic population

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400

1

D 0

0.95 0

100

Time delay [fs]

200

300

φ[rad]

400

2π

500

Time delay [fs]

Figure 4: A Schematics of a vibronic model: The first pulse (red) peaking around 780-800 nm excites the zero-phonon line of the high energy B850 excitons |β ∗ , 0i and the side-band of the bright state |β , 0i, to initialize the vibronic dynamics between these two states. The second pulse (blue) centered at around 830 nm, de-excites the phonon sideband of the bright state |β , 1i → |g, 1i. The second transition is modulated by the population exchange mediated by the vibronic coupling. B Populations of vibronic pairs of B800-B850 states (dashed) and B850 states throughout the pulse sequence (dotted lines represent the times at which pulses present maximum amplitude). C Total excited state populations from the vibronic model A, after the two pulse sequence is over. Notice ∗ i) and (|β ∗ that the two pairs (|β842 i, |β794 835 i, |β784 i) present traces with very fast oscillations modulated by a slow oscillation of period '200 fs, which is best captured in D when the resolution of time delays is increased to ∆T = 20 fs. In D we also present the trace arising from the vibronic ∗ i). Additionally, dynamics of realizations lying at 831 and 782 nm (labeled as |β831 i) and |β782 we show the case without vibronic coupling (s = 0). The inset presents FI(φ ) as a function of the relative pulses phase φ for a delay time T = 100 fs. In D all traces are normalized to their respective average, hFIi. A single mode with frequency 750 cm−1 has been used in all cases.

11



We have discussed how the excitonic delocalization affects i) the robustness of the vibronic resonance, ii) the magnitude of the vibronic coupling, and iii) the inter-excitonic dephasing rates, in LH2. To illustrate the possible relevance of these effects in other natural complex, we performed an analysis along these lines for the FMO in SI.C. For this complex we find that the excitonic overlap is not able to increase considerably the lifetime of inter-excitonic with respect to the optical coherences. Since this overlap is however, able to provide an important vibronic mixing in the excited state manifold, we postulate that solely excitonic coherence cannot explain the observations in this complex, while the vibronic mixing results in a lifetime of vibronic coherences of 286 fs, in agreement with previous theoretical models 13–16,61 which is in reasonable agreement with the latest results of non-linear spectra in FMO, 4,5 but shorter (longer) to the lifetimes observed by Engel, et al. 1 and Hayes, et al. 2 (Duan, et al. 11 ). Besides this agreement, based on the non-trivial aspects of static disorder found for the LH2, we provide a qualitative insight to the concerted motion of excitonic transition energies observed in FMO, 62 as arising from the delocalized nature of excitons in this complex (see SI.C2 for further details). In the following, we demonstrate numerically that vibronic dynamics within the B850 excitons can induce long-lasting oscillations in the FI traces with ∼ 200 fs period. To take into account the influence of laser pulses on the FI dynamics accurately, we consider the experimental laser spectrum and apply a linear phase below 820 nm, as performed in Ref., 24 in order to generate two pulses (the first with an almost flat spectrum between 780 and 800 nm; the second peaking at 828 nm) with a controllable time-delay T , which is determined by the slope of the linear phase in frequency domain (see the SI ). We have chosen the excitons in Figure 2 and Figure 3 such that their energies are on average similar to the central frequencies of the laser pulses used in Ref. 24 This condition ensures that the SB of the bright state and the chosen mode with frequency of 750 cm−1 , |β , 1i, and the ZPL of the high energy exciton, |β ∗ , 0i, can be excited by the first pulse, as we schematically show in Figure 3A. Due to the robustness of the energy gap and the low inter-excitonic dephasing, a resonant long-lasting population exchange could proceed between these two states. The second

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pulse mainly induces a population inversion of the zero-phonon transition between |β , 1i and |g, 1i concomitant with the excitation from |g, 0i to |β , 0i. The fluorescence intensity will be proportional to the total excited population after the interaction with the second pulse. Accordingly, the models we present in Figure 4 consist of a pair of excitons with the wavelengths and transition dipoles obtained from averages over static disorder (see Fig S3-C), which couple via a single mode, that represents a generalized coordinate involving the vibrations at each pigment, having a frequency of 750 cm−1 . The evaluation of the dynamics of these excitons coupled by multiple vibrational modes, plus the exciting fields probing multiple delay times, is not amenable to numerical simulations. Hence, we study the qualitative features of signals arising from selected pairs of B850 excitons and a single mode of frequency 750 cm−1 that couples them through Sβ β ∗ . These pigment modes resonantly couple several bright states (between 857 to 829 nm) whose SBs can be excited by the first pulse, and resonantly interact with several complementary high energy excitons (between 805 and 780 nm). Hence, the rather stringent vibronic resonance condition is, on average, fulfilled by multiple excitons, and, as here underlined, robust to static disorder due to overlapping excitonic wave-functions. In Figure 4B, the population dy∗ i, |γ ≈ β namics of two cases (|γ0 ≈ β784 835 i) and (|γ0 ≈ α787 i, |γ ≈ β835 i), are shown, where the

B850 states in the former case exhibit oscillatory population dynamics during T with a period of ∼ 200 fs (continuous) while the pair of B800 and B850 states does not (dashed). In this Figure we display the times at which the pulses have the largest amplitude by dotted vertical lines. We note that all the FI transients in Figure 4C show ultrafast oscillations on top of slower oscillations with a period of about 200 fs. An additional component with period ' 50 fs is observed on ∗ i) pair due to the detuning with the second pulse carrier wavelength of ' 830 nm. the (|β857 i, |β805

The ultrafast oscillations are a result of the optical coherences |gihγ| created by the first pulse (see the SI D.2) and could not be resolved in the experiment, 24 since traces were recorded with resolution of time delays ∆T ≈ 20 fs. In Figure 4D we present the total excited state population dynamics ∗ i), with resolution ∆T = 20 fs, in order to highlight that the regarding the vibronic pair (|β835 i, |β784

experiment was sensitive to the slower component with ∼ 200 fs period, while the fast component

13



could be interpreted as a noisy contribution at that resolution. To address the influence of static disorder, we calculated the FI trace for a typical realization of static noise. In this realization two ∗ i)) fulfill the vibronic resonance excitons which lie at 831 and 782 nm (labeled as |β831 i and |β782

condition ∆εβ β ≈ ω. The FI trace for this pair presents a slightly slower frequency than from the ∗ i) since the detuning between the excitonic splitting and mode’s frequency is repair (|β835 i, |β784

duced. Notice that in this case the oscillations also present a slower decay, which can be explained by the fact that our model of static noise produces realizations which have a smaller dephasing for excitons with larger energy, as discussed for Figure 3A, and observed experimentally. 57,58,63 Since the second pulse used by Hildner, et al. 24 peaked at about 830 nm, it accessed the dynamics of this sub-ensemble of excitons with slower dephasing. In Figure 4D we also present the FI trace for the ∗ i, |β (|β784 835 i) without the vibronic interaction, which does not show oscillations because in our

model, the vibronic interaction is required to coherently cycle excitonic populations and generate oscillations in FI(T ). As mentioned before, the fast oscillating component in FI(T ) at optical frequencies depends on the optical coherences produced by the first pulse (see SI D.2 for a detailed analysis). The exciting fields can imprint their relative phase φ in these optical coherences, and as a result, a modulation of FI(φ ) given a fixed time delay T . This modulation is shown in the inset of Figure 4D for the three cases of the main figure, with the same line color code. As in the experiment, 24 FI(φ ) presents a full 2π cycle, with an amplitude that depends solely on the optical dephasing. Recall from Figure 3A a trend of smaller participation ratios Pβ with increasing wavelength, which provides ∗ i and |β higher amplitude for the optical coherences on the longer wavelength states |β794 842 i) than ∗ i and |β for the pair |β784 835 i, hence a larger amplitude for FI(φ ) in the former case. Since the

dynamics of the optical coherences is insensitive to the vibronic interaction, it is natural to observe that the model where s = 0, also presents oscillations of FI(φ ) with a comparable amplitude to that of the vibronic models. Notice that different phases of the oscillations of FI(T ) and FI(φ ) are particular for every set of states, supporting the observation 24 that due to static disorder, different states give rise to the observed oscillatory traces and on average, the oscillatory component of these

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traces vanish. Our vibronic model also results in the absence of oscillating FI(T ) traces from control experiments 24 devised to only excite the 800 nm or the 830 nm spectral window (see SI E for a detailed discussion). Based upon the qualitative agreement of our model with the experimental observations, we conclude that the coherent electronic dynamics of the LH2 complex observed in experiments may originate from the complementarity between coherent vibronic dynamics and excitonic delocalization, this latter supporting an appreciable vibronic coupling, fixing a robust resonant interaction while suppressing excitonic decoherence rates. This complementarity serves to illustrate that the prospect of generalizing vibronics as the cause of oscillating features in optical traces, does not only rely on modes which resonantly couple to specific excitonic transitions, but on the subtleties of the excitonic wave-functions involved. In particular the observations of Hildner, et al., 24 are compatible with a reduced excitonic dephasing that results in the modulation of FI(φ ), and that contributes to develop a robust vibronic interaction that results in the oscillatory traces FI(T ). Single molecule experiments are therefore a promising tool to unravel the interplay between excitonic delocalisation, static disorder and dephasing in the optical response of light-harvesting systems. This work was supported by an Alexander von Humboldt Professorship, the ERC Synergy grant BioQ, and the EU STREP QUCHIP. This publication was made possible through the support of a grant from the John Templeton Foundation. Santiago Oviedo acknowledges MINECO FEDER funds FIS2015-69512-R and Fundación Séneca (Murcia, Spain) Project No. ENE2016-79282C5-5-R. Niek F. van Hulst acknowledges funding by the ERC Advanced Grant 670949-LightNet, MINECO grants SEV2015-0522 and FIS2015-69258-P, Centres de Recerca de Catalunya (CERCA) and Fundació CELLEX (Barcelona). We acknowledge Richard Hildner for facilitating the experimental traces and laser spectra.

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Theory of Excitonic Delocalization for Robust Vibronic Dynamics in

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