1 Simulation of Femtosecond Phase-Locked Double-Pump Signals of2 Individual Light-Harvesting Complexes LH23 Lipeng Chen,† Maxim F. Gelin,‡ Wolfgang Domcke,‡ and Yang Zhao*,†

4†Division of Materials Science, Nanyang Technological University, Singapore 639798, Singapore

5‡Department of Chemistry, Technische Universitat Munchen, D-85747 Garching, Germany

6 *S Supporting Information

7 ABSTRACT: Recent phase-locked femtosecond double-pump experi-8 ments on individual light-harvesting complexes LH2 of purple bacteria at9 ambient temperature revealed undamped oscillatory responses on a time10 scale of at least 400 fs [Hildner et al. Science 2013, 340, 1448]. Using an11 excitonic Hamiltonian for LH2 available in the literature, we simulate12 these signals numerically by a method that treats excitonic couplings and13 exciton−phonon couplings in a nonperturbative manner. The simulations14 provide novel insights into the origin of coherent dynamics in individual15 LH2 complexes.

16 Deciphering the numerous intriguing facets of photosyn-17 thesis, one of the most important biochemical processes18 occurring on Earth, has been an active area of research over19 many decades.1−4 This research has provided illuminating20 insights in the cellular level architecture and the mechanisms of21 the machinery used by green plants, algae, and photosynthetic22 bacteria to carry out the extremely complex process of23 absorption of solar energy and its conversion to chemical24 energy.5−7

25 Our understanding of the photophysics of light harvesting26 has been largely shaped by data delivered by femtosecond27 nonlinear ensemble spectroscopy1−11 and single-molecule28 (SM) spectroscopy.12,13 However, both of these spectroscopies29 have significant intrinsic limitations. Ensemble spectroscopy30 provides the responses averaged over an ensemble of species,31 while the excited-state dynamics of photosynthetic systems32 vary dramatically among individual complexes and even within33 a single complex over time because of the high degree of34 (conformational and electronic) disorder. SM spectroscopy, on35 the other hand, monitors signals of individual species and is36 free of inhomogeneous broadening. However, it usually relies37 on the detection of fluorescence, which is emitted on a38 nanosecond time scale. Therefore, significant dynamic39 processes occurring on femtosecond to picosecond time scales40 cannot be resolved.41 Recently, van Hulst and co-workers combined the best of42 both worlds, extending SM spectroscopy into the femtosecond43 time domain.14−18 In this technique, individual chromophores44 are interrogated by two14−17 or three18 femtosecond phase-45 locked pulses and the time resolution is achieved through the46 detection of the SM fluorescence as a function of the interpulse47 delay(s). After the demonstration for single chromo-

48phores,14−17 double-pump femtosecond SM spectroscopy has49been applied to individual LH2 complexes of purple bacteria50under physiological conditions.19 It was found that signals of51individual LH2 complexes are oscillatory and, unexpectedly, do52not exhibit decay on a time scale of ∼400 fs.19

53Previously, we have developed a computational method to54simulate femtosecond double-pump SM signals of molecular55aggregates and applied it to light-harvesting complexes II56(LHCII) of higher plants.20 In the present work, we extend the57analysis to account for environment-induced fluctuations and58employ it to the simulation and interpretation of femtosecond59responses of individual LH2 complexes.60The LH2 complex of the purple bacterium Rhodopseudomo-61nas acidophila (PDB ID: 1kzu21) consists of 27 bacteria-62chlorophyll a (Bchla) molecules arranged in two concentric63B800 and B850 rings labeled according to their central64absorption wavelength. In the present work, we adopt a65description in which all relevant excitonic and vibrational66degrees of freedom of LH2 are included into the system67Hamiltonian H.22−24 This Hamiltonian, which comprises 2768excitonic states and 17 high-frequency vibrational modes, has69been well-characterized in previous studies21,25−28 and can be70written as

H H H Hex vib ex ph= + + − 71(1)

72Here, Hex is the Frenkel-exciton Hamiltonian; Hvib describes73molecular vibrations; Hex−vib is responsible for the exciton−

Received: June 15, 2018Accepted: July 23, 2018Published: July 23, 2018

Letter

pubs.acs.org/JPCL

© XXXX American Chemical Society A DOI: 10.1021/acs.jpclett.8b01887J. Phys. Chem. Lett. XXXX, XXX, XXX−XXX

ajs00 | ACSJCA | JCA11.1.4300/W Library-x64 | research.3f (R4.0.i9 HF05:4883 | 2.1) 2018/07/18 12:44:00 | PROD-WS-118 | rq_78118 | 7/24/2018 14:32:57 | 7 | JCA-DEFAULT

74 vibrational coupling, which is taken as diagonal in the site75 representation and linear in the vibrational modes:

H g g m m J n m

H b b

H b b m m( )

gm

mn m

nm

qq q q

q mqm q q q

ex1

27 27

vib1

17

ex vib1

17

1

27

∑ ∑

∑

∑ ∑

ε ε

ω

χ ω

= | ⟩⟨ | + | ⟩⟨ | + | ⟩⟨ |

= ℏ

= ℏ + | ⟩⟨ |

= ≠

=

†

−= =

†

76 (2)

77 Here, |g⟩ denotes the state in which all Bchla molecules are in78 the electronic ground state; |m⟩ denotes the state in which the79 mth Bchla is excited, εm is the electronic energy; Jnm is the80 electronic coupling (transfer integral) between Bchla n and m;81 bq

†(bq) is the creation (annihilation) operator of the qth82 vibrational mode with frequency ωq; χqm is the exciton−83 vibrational coupling which is quantified by the Huang−Rhys84 factor Sqm = χqm

2.85 The interaction of LH2 with two phase-locked laser pulses is86 described in the dipole approximation and in the rotating wave87 approximation as

H t E t X E t X( ) ( ( ) ( ) )F = − + *†88 (3)

89 where

E t E f t f t

X g me

( ) ( ( )e ( )e )

( )

t t

mm

0 1i

2i( )

1

27

1 2

∑ μ

τ= + −

= | ⟩⟨ |

ω ϕ ω− −

=90 (4)

91 E0 is the amplitude of the pulses; f1(t) and f 2(t) are their92 dimensionless temporal envelopes; ω1 and ω2 are the pulse93 carrier frequencies; τ and ϕ are the time delay and phase94 difference of the pulses, respectively; e is the unit vector of the95 pulse polarization; μm is the transition dipole moment of the96 mth Bchla.97 In the femtosecond double-pump SM experiments of ref 19,98 individual LH2 complexes are embedded in a polymer matrix99 at ambient temperature. This environment is highly heteroge-100 neous and exhibits thermal fluctuations, which modulate the101 parameters specifying H and HF(t) (see refs 29 and 30 for a102 detailed discussion). In a typical SM experiment, the signal as a103 function of the time delay τ is detected with a certain time step104 Δτ

j j, 0, 1, 2, ...τ = Δ =τ105 (5)

106 In ref 19, for example, Δτ ≈ 25 fs. The time interval between107 the measurements j and j + 1 is much longer than any relevant108 microscopic time interval specifying dynamics and fluorescence109 detection of the individual LH2. It is therefore safe to assume110 that there is no correlation between the values of the111 parameters of the Hamiltonians 1 and 3 in any two consecutive112 measurements. Such a measurement protocol can be simulated113 by introducing a stochastic modulation of the parameters.29,30

114 For LH2, as for other pigment−protein complexes, the115 dominant contribution to the static disorder comes from116 electronic energies.31−34 Hence, we adopt the following117 modulation law:

r( ) ( )m m m mε τ ε δ τ= +118 (6)

119Here, εm(τ) is a stochastic realization of the site energy at a120specific time delay τ; εm represents its mean value; δm controls121the amplitude of modulations; rm(τ) is a random Gaussian122variable with unit dispersion. Because of these modulations, H123depends on τ parametrically and can be regarded as a snapshot124Hamiltonian.125The double-pump SM signal is defined as the total (relaxed)126fluorescence of a single LH2 complex measured as a function127of the interpulse delay τ:20,29

I m m t( ) Tr ( )m

f1

27lmoonoo

|}ooo~oo∑τ ρ= | ⟩⟨ |

= 128(7)

129Here, tf is any time moment after the passing of the second130pump pulse, and the density matrix ρ(t) satisfies the Liouville131equation

t H H t t( )i

( ), ( )t Fρ ρ∂ = −ℏ

[ + ]132(8)

133Because all vibrational modes with significant exciton−134vibrational coupling are included in the system Hamiltonian H,135eq 8 adequately describes evolution of ρ(t) on the time scale of136τ ≈ 400 fs typical for the experiments of ref 19. There exist137several inter- and intramolecular relaxation processes that are138not described by eq 8. However, as argued in ref 29, these139processes do not affect the total population ρ(tf) and need not140be accounted for. The impact of a liquid solvent, which is141usually described by the coupling of the system to an142overdamped harmonic bath,35 should not be considered in143the present case, because the individual LH2s are embedded in144a polymer matrix and are not in direct contact with a solvent.145The signal of eq 7 can be expressed as an expansion in the146system−field coupling20,29

I I( ) ( )k

k2,4,6,...

∑τ τ== 147(9)

148where Ik(τ) ∼ E02k. In the SM experiment of ref 19, the signal

149was controlled to scale linearly with the pump-pulse intensity,150I(τ) ∼ E0

2. In this case, the SM signal can be associated with151the lowest-order contribution I2(τ), which can be evaluated152as20

I ( ) ( )a b

ab2, 1

2

∑τ σ τ==

153where

t t E t E t t R t( ) 2Re d d ( ) ( ) ( )ab a b0

1 1 1∫ ∫σ τ = * −−∞

∞ ∞

154(10)

155and R(t) is the linear response function. Because the156characteristic energies of the high-frequency molecular157modes are much higher than kBT (kB is the Boltzmann158constant, and T is the bath temperature), R(t) can be evaluated159as

R t C n n0 0( ) en n

N

nnH t

, 1vib

ivib

S∑= ⟨ |⟨ | | ′⟩| ⟩′=

′−

160(11)

161where Cnn′ = (eμn)(eμn′) are geometrical factors and |0vib⟩ is162the vacuum state for the high-frequency vibrational modes.163In the simulations, we assume that the pulses have Gaussian164envelopes

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f t t( ) expa a2 2= {−Γ }

165 (12)

166 where 2 ln 2 /a aτΓ = and τa are the pulse durations (full167 width at half-maximum). In this case, the integration over t in168 eq 10 can be performed analytically, yielding

I t A t R t( ) Re d ( , ) ( )0

1 1 1∫τ τ∼∞

169 (13)

170 Here

A t c c

c

( , ) e e e e

e e (e e e e )

t t t t

t t t

1 1/2 i

2/2 i

12/ i ( ) /2 i( ) ( ) /2 i( )

12

12

1 1 22

22

2 1

2 21

21

2 21

2

τ = +

+ +

ω ω

ω γ τ ϕ ξωτ τ ϕ ξωτ

−Γ −Γ

− Ω −Γ − − − −Γ + −

171 (14)

172 where

,( )

, 2212

22 2 1 2

2

222

2γγ

ξγ

= Γ + Γ Γ =ΓΓ

=Γ

173 (15)

( )/2,2 122

1 12

22ω ω ω

ω ωγ

= − Ω =Γ + Γ

174 (16)

175 and

c c c2

,2

,112 2

22 12 2

π π πγ

=Γ

=Γ

=176 (17)

177 The parameters of the LH2 Hamiltonian are taken from178 previous studies21,25−28 and are collected in section I of the179 Supporting Information. The amplitudes δm of the modulation180 of the site energies εm(τ) are retrieved from ref 33. The carrier181 frequencies of the two pump pulses are set in resonance with182 the B800 and B850 bands of the linear absorption spectrum of183 LH2, respectively, that is, ω1 = 12468 cm−1(802 nm) and ω2 =184 11655 cm−1(858 nm).19 The pump pulses are assumed to be185 polarized along the X axis of the molecular frame, e∥X. The186 effect of other choices of e on the signals is discussed in section187 V of the Supporting Information.188 For specific values of the model parameters, the propagator189 in the linear response function R(t) of eq 11 has been190 evaluated by using the Davydov D1 ansatz

36−38

t n

t n t b

( ) e 0

( ) exp ( ) H.c. 0

DH t

nn

qnq q

i/vib

1

27

1

17

vib

S1

lmooonooo

|}ooo~ooo

∑ ∑α λ

|Ψ ⟩ = | ⟩| ⟩

= | ⟩ [ − ] | ⟩

− ℏ

= =

†

191 (18)

192 The application of the Dirac−Frenkel variational principle193 yields the equations of motion for the time-dependent

194parameters αn(t) and λnq(t) (see section II of the Supporting195Information). The derivation of the equations of motion as196well as the means to control the accuracy of the ansatz are197described in detail in refs 36−38. For the diagonal form of the198exciton−vibrational coupling (eq 2), the Davydov D1 ansatz199provides a numerically accurate evaluation of the linear200response function (eq 11). It should be noted that the201response function R(t) represents nonadiabatic nuclear202dynamics on 27 coupled electronic potential-energy surfaces203with 17 inseparable vibrational degrees of fredom.204Equation 14 reveals that the SM signal I(τ) is composed of a205τ-independent background (the aggregate interacts twice with206the same pulse) and a τ-dependent contribution (the aggregate207interacts once with pulse 1 and once with pulse 2).208Furthermore, I(τ) does not monitor vibronic wavepackets,209which are associated with populations of the aggregate density210matrix ρnn(τ) in the manifold of singly excited excitonic states |211n⟩. The signal rather monitors the evolution of excitonic212coherences ρgn(τ), which are described by the linear response213function (eq 11). Furthermore, eq 14 predicts that the signal214can be represented as

I( ) ( ) cos ( )τ θ τ ϕ φ τ ωτ= [ − − ] 215(19)

216where the explicit form of θ(τ) and φ(τ) depends on R(t) and217the pulse shapes. Hence, the SM signal exhibits a universal218“cosine” dependence on ϕ for fixed τ. This is a direct219consequence of the weak system−field coupling approximation220and is consistent with the experimental data of ref 19 and with221the simulations of ref 34. On the other hand, ϕ determines the222initial (at τ = 0) value of the signal.29,30 When the two pump223pulses overlap (τ = 0) and have relative phase ϕ = π, they224cancel each other (destructive interference) and do not excite225the LH2 complex at all, yielding I(0) = 0. Overlapping pulses226with ϕ = 0 or 2π, on the other hand, reinforce each other227(constructive interference), and I(0) attains its maximum228value. In general, ϕ determines the relative phases of229oscillations in SM signals. That is why different experimental230realizations of SM signals exhibit a variety of oscillatory231transients that are shifted with respect to each other (see232Figure 1 of the Supporting Information), as predicted by eq 19.233Keeping these considerations in mind, we set ϕ = 0 in the234subsequent simulations. This implies I(τ) = I(−τ).235To establish a reference picture as well as to uncover how236the dynamics generated by the Hamiltonian (eq 1) manifests237itself in SM signals, we first assume that modulations of the site238 f1energies can be neglected. Figure 1 shows the SM signals239evaluated for pump pulses of duration τ1 = τ2 = 15 fs (a) and τ1240= 15 fs, τ2 = 60 fs (b). The signal in Figure 1a corresponds to

Figure 1. SM signals I(τ) simulated for pulse durations (a) τ1 = τ2 = 15 fs and (b) τ1 = 15 fs and τ2 = 60 fs.

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241 short pulses, which provide good temporal resolution. The242 signal exhibits fast high-frequency oscillations with a period of243 τfast ≈ 39 fs superimposed on slow low-frequency beatings with244 a period of τslow ≈ 287 fs. Fast oscillations and slow beatings245 have also been found in the signals simulated in ref 34. The246 oscillations and beatings in Figure 1a are of a vibronic247 character: Their periods depend on the excitonic couplings Jnm248 as well as on the exciton−vibrational couplings Sqm (polaron249 effect) decreasing with the decrease of Sqm (see section IV of250 the Supporting Information). The modulation of Sqm may thus251 be responsible for the broad distribution of τslow in the signals252 of ref 19. The amplitude of the low-frequency beatings slightly253 decreases with τ, owing to incomplete rephasing of the254 vibronic dynamics in the excited-state manifold. The signal in255 Figure 1b has been calculated for the pulse durations of ref 19.256 Because τ2 > τfast, the high-frequency oscillations cannot be257 detected in the signal of Figure 1b and in the signals of ref 19.258 On the other hand, the low-frequency beatings in Figure 1a,b259 look the same, because τ2 = 60 fs is much shorter than τslow.

260These are the beatings which were detected in the SM double-261pump signals reported in ref 19 (cf. Figure 1 of Supporting262Information).263 f2To clarify the origin of the low-frequency beatings, Figure 2264shows SM signals simulated without coupling between the265B800 and B850 rings of LH2 (Jnm are artificially set to zero if266Bchla m and n belong to different rings). The signal in panel a267is simulated for short pulses (τ 1 = τ 2 = 15 fs), while the signal268in panel b is simulated for the experimental values of the pulse269durations19 (τ 1 = 15 fs, τ 2 = 60 fs). Let us first analyze the270low-frequency beatings which are seen in both panels of Figure2712. The period of these beatings is ∼106 fs, which is almost272three times shorter than the τslow ≈ 287 fs obtained with the273coupling between the rings. The ∼106 fs beatings manifest274coherent vibronic dynamics inside the B800 and B850 rings.275Because a small fraction of SM signals exhibiting ∼100 fs276oscillations has been detected in ref 19, these signals may277reveal responses of individual B800 and B850 rings. On the278other hand, Figure 2 demonstrates that intraring excitonic

Figure 2. SM signals I(τ) simulated for pulse durations (a) τ1 = τ2 = 15 fs and (b) τ1 = 15 fs and τ2 = 60 fs. The electronic coupling between theB800 and B850 rings is set to zero.

Figure 3. SM signals I(τ) simulated for excitonic energy modulation amplitudes δB800 = 60 cm−1 and δB850 = 130 cm−1. The discretization step Δτ is5 fs (upper panels) and 25 fs (lower panels). Panels a and c correspond to τ1 = τ2 = 15 fs, while panels b and d correspond to τ1 = 15 fs and τ2 = 60fs.

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279 couplings alone cannot produce ∼200 fs oscillations in the SM280 signals, contrary to what has been found in ref 34. The value of281 ∼200 fs dominates the distribution over the oscillation periods282 extracted from the SM signals of ref 19. One thus concludes283 that the majority of these signals depend on the B800−B850284 inter-ring coupling but are not exclusively determined by this285 coupling, as was assumed in ref 19. As for the high-frequency286 oscillations, they are clearly seen in Figure 2a (short pulses)287 but cannot be resolved in panel Figure 2b (experimental288 pulses), as expected. The period of these oscillations, ∼33 fs, is289 shorter than that obtained with the B800−B850 inter-ring290 coupling.291 We now discuss how the modulation of the site energies292 affects SM signals. We assume that modulations of εm(τ) are293 uncorrelated for different m and τ, but the modulation294 amplitudes δB800 = 60 cm−1 and δB850 = 130 cm−1 are different295 for the Bchla molecules belonging to the B800 and B850 rings

f3 296 (see ref 33 and references therein). Figure 3 shows the SM297 signals simulated with a fine discretization step Δτ = 5 fs298 (upper panels) and with the experimental discretization step299 Δτ = 25 fs19 (lower panels). The signals in panels a and c300 correspond to short pulses (τ1 = τ2 = 15 fs), while the signals in301 panels b and d correspond to the experimental pulse durations302 (τ1 = 15 fs, τ2 = 60 fs).19

303 Let us first consider the signal in panel a, which corresponds304 to optimal conditions for information acquisition (short pulses305 and fine discretization step). For τ ≤ 100 fs, the level of noise306 is rather low and the high-frequency oscillations are clearly307 seen in the signal. For τ > 100 fs, not only are the high-308 frequency oscillations buried in noise but also the second309 maximum due to the low-frequency beating is not clearly310 visible. Apparently, the level of noise in SM signals increases311 with τ. This phenomenon can be understood by the following312 considerations. For overlapping short pump pulses, the signal313 is independent of modulations of any parameters of H because314 no dynamic evolution occurs. For nonoverlapping short pulses,315 the effect of the modulation is small if τ is shorter than the time316 scale at which the SM signal changes significantly, τ ≲ τslow/2,317 but the effect becomes substantial if τ ≳ τslow/2. Furthermore,318 the second pump pulse is in resonance with the B850 ring and319 δB850 > δB800, which renders the contribution from the B850320 ring noisier than that from the B800 ring.321 The signal in panel b (experimental pulses and fine322 discretization step) does not resolve the high-frequency323 oscillations (due to the reasons explained above), while the324 first maximum in I(τ) induced by the low-frequency beating is325 visible reasonably well. The lower panels (c and d) of Figure 3326 depict the same signals as in the upper panels but are simulated327 with the experimental discretization step, Δτ = 25 fs. Because328 this discretization step almost coincides with the period of the329 high-frequency oscillations, it comes as no surprise that these330 oscillations are not seen in the lower panels of Figure 3 and331 could not be detected in the SM signals of ref 19. This332 argument has also been noted in ref 34. On the other hand,333 both the first and second maximums due to the low-frequency334 beating can be imagined in the signals in the lower panels.335 On the other hand, traces of both the first and the second336 maximum due to the low-frequency beating can be found in337 the signals in the lower panels. Despite being calculated for338 different pulse durations, the signals in the lower panels look339 qualitatively the same, which are also very similar to the340 experimental ones (cf. Figure 1 of Supporting Information).

341An interesting question is whether one can uncover the342modulation law (eq 6) by analyzing experimental SM signals.343We suggest that this can be done if the number of detected344signals I(τ,n) (n 1, 2, . . . ,= ) is sufficiently large. In this345case, the ensemble double-pump signal is given by

I I n( ) ( , )n1τ τ = ∑− . Furthermore, one can also determine

346the variance I n I( , ) ( )n1 2 2τ τ∑ − − as well as higher-order

347moments, which are not accessible in ensemble spectroscopy.348This may provide insight into stochastic properties of the349environment-induced modulations.350To summarize, we have simulated phase-locked double-351pump SM signals of LH2 complexes in the limit of weak352system−field coupling. The simulated signals exhibit un-353damped small-amplitude high-frequency oscillations with a354period of ∼30 fs, superimposed on high-amplitude low-355frequency beatings with a period of ∼200−300 fs. The356oscillations and beatings are of vibronic origin because their357periods depend on excitonic interstate couplings as well as on358exciton−vibrational coupling (polaron effect), which supports359the current consensus on the vibronic character of long-lived360coherences in light-harvesting complexes.11,34,39 The double-361pump SM signals monitor the evolution of the coherences362between the excitonic ground state and the manifold of single-363exciton states. Hence, the signals cannot be directly associated364with the dynamics of energy transfer between Bchla molecules365within individual LH2 complexes.366The comparison of the simulated signals with the367experimental signals of ref 19 reveals the following: (i) The368low-frequency beatings in the LH2 signals depend on the369coupling between the B800 and B850 rings but are not solely370determined by this coupling. (ii) The absence of the high-371frequency oscillations in the SM signals of ref 19 can be372explained by the (relatively) long duration of the second pump373pulse and by a too coarse discretization in the interpulse time374delay. An optimization of the experiment along these two375directions (shorter pulses and a smaller discretization step)376would increase the information content of the double-pump377SM signals. It should also be noted that the heterogeneity of378excited-state energies of LH2 is larger than for most pigment−379protein complexes.32 One can thus expect that femtosecond380double-pump SM experiments on other pigment−protein381complexes may reveal more informative signals.382The experiment of ref 19 was performed with weak laser383pulses, because the SM signals were controlled to scale linearly384with the pulse intensity. However, an earlier variant of SM385pump−probe experiments by van Hulst and co-workers386employed strong lasers pulses and saturation effects,40 and387the same experimental method has been applied recently to388individual LH2s.41 Very recently, the influence of the pump-389pulse amplitude on the responses of individual LH2 complexes390has been theoretically studied in ref 34. A recent analysis of391strong-field responses of individual chromophores30 suggests392that the adjustment of the strength of the system−field393coupling (by changing the laser pulse intensity) can be an394efficient tool for the enhancement of information content of395femtosecond double-pump SM signals of light-harvesting396complexes. The use of stronger laser pulses can also improve397the temporal resolution.42

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398 ■ ASSOCIATED CONTENT399 *S Supporting Information400 The Supporting Information is available free of charge on the401 ACS Publications website at DOI: 10.1021/acs.jp-402 clett.8b01887.

403 Parameters of the system Hamiltonian, time-dependent404 variational approach on the basis of Davydov D1 ansatz,405 different SM signals extracted from experimental406 measurement, a brief analysis of polaron effects, and407 effect of polarization of the pump pulses on the SM408 signals (PDF)

409 ■ AUTHOR INFORMATION410 Corresponding Author411 *E-mail: YZhao@ntu.edu.sg.412 ORCID413 Wolfgang Domcke: 0000-0001-6523-1246414 Yang Zhao: 0000-0002-7916-8687415 Notes416 The authors declare no competing financial interest.

417 ■ ACKNOWLEDGMENTS418 Support from the Singapore Ministry of Education Academic419 Research Fund Tier 1 (Grant No. RG106/15) is gratefully420 acknowledged. L.C. acknowledges support from a postdoctoral421 fellowship of the Alexander von Humboldt-Foundation.422 M.F.G. and W.D. acknowledge support from the Deutsche423 Forschungsgemeinschaft through a research grant and through424 the DFG Cluster of Excellence Munich-Centre for Advanced425 Photonics (http://www.munich-photonics.de). We are grateful426 to Fulu Zheng for providing numerical values of the electronic427 coupling coefficients and to Martin Plenio for useful428 discussions and for providing his manuscript34 prior to429 publication.

430 ■ REFERENCES(1)431 Van Amerongen, H.; Valkunas, L.; van Grondelle, R.

432 Photosynthetic Excitons; World Science: Singapore, 2000.(2)433 Ishizaki, A.; Fleming, G. R. Quantum Coherence in Photo-

434 synthetic Light Harvesting. Annu. Rev. Condens. Matter Phys. 2012, 3,435 333−361.

(3)436 Chen, L. P.; Shenai, P.; Zheng, F. L.; Somoza, A.; Zhao, Y.437 Optimal Energy Transfer in Light-Harvesting Systems. Molecules438 2015, 20, 15224−15272.

(4)439 Blankenship, R. E.Molecular Mechanisms of Photosynthesis; World440 Scientific: London, U.K., 2002.

(5)441 Croce, R.; van Amerongen, H. Natural strategies for photo-442 synthetic light harvesting. Nat. Chem. Biol. 2014, 10, 492−501.

(6)443 Romero, E.; Novoderezhkin, V. I.; van Grondelle, R. Quantum444 design of photosynthesis for bio-inspired solar-energy conversion.445 Nature 2017, 543, 355−365.

(7)446 Scholes, G. D.; Fleming, G. R.; Chen, L. X.; Aspuru-Guzik, A.;447 Buchleitner, A.; Coker, D. F.; Engel, G. S.; van Grondelle, R.; Ishizaki,448 A.; Jonas, D. M.; Lundeen, J. S.; McCusker, J. K.; Mukamel, S.;449 Ogilvie, J. P.; Olaya-Castro, A.; Ratner, M. A.; Spano, F. C.; Whaley,450 K. B.; Zhu, X. Using coherence to enhance function in chemical and451 biophysical systems. Nature 2017, 543, 647−656.

(8)452 Abramavicius, D.; Palmieri, B.; Voronine, D. V.; Sanda, F.;453 Mukamel, S. Coherent Multidimensional Optical Spectroscopy of454 Excitons in Molecular Aggregates: Quasiparticle versus Supermolecule455 Perspectives. Chem. Rev. 2009, 109, 2350−2408.

(9)456 Segale, D.; Karavitis, M.; Fredj, E.; Apkarian, V. A. Quantum457 coherent dissipation: a glimpse of the “cat. J. Chem. Phys. 2005, 122,458 111104.

(10) 459Jang, S. J.; Mennucci, B. S. Delocalized excitons in natural light460harvesting complexes. 2018, arXiv:1804.09711 [physics.bio-ph].

(11) 461Mirkovic, T.; Ostroumov, E. E.; Anna, J. M.; van Grondelle, R.;462Govindjee; Scholes, G. D. Light Absorption and Energy Transfer in463the Antenna Complexes of Photosynthetic Organisms. Chem. Rev.4642017, 117, 249−293.

(12) 465Kondo, T.; Chen, W. J.; Schlau-Cohen, G. S. Single-Molecule466Fluorescence Spectroscopy of Photosynthetic Systems. Chem. Rev.4672017, 117, 860−898.

(13) 468Gruber, J. M.; Maly, P.; Kruger, T. P. J.; van Grondelle, R.469From isolated light-harvesting complexes to the thylakoid membrane:470a single-molecule perspective. Nanophotonics 2018, 7, 81−92.

(14) 471Brinks, D.; Stefani, F. D.; Kulzer, F.; Hildner, R.; Taminiau, T.472H.; Avlasevich, Y.; van Hulst, N. F. Visualizing and controlling473vibrational wave packets of single molecules. Nature 2010, 465, 905−474908.

(15) 475Hildner, R.; Brinks, D.; Stefani, F. D.; van Hulst, N. F.476Electronic coherences and vibrational wave-packets in single477molecules studied with femtosecond phase-controlled spectroscopy.478Phys. Chem. Chem. Phys. 2011, 13, 1888−1894.

(16) 479Brinks, D.; Hildner, R.; van Dijk, E. M. H. P.; Stefani, F. D.;480Nieder, J. B.; Hernando, J.; van Hulst, N. F. Ultrafast dynamics of481single molecules. Chem. Soc. Rev. 2014, 43, 2476−2491.

(17) 482Piatkowski, L.; Accanto, N.; van Hulst, N. F. Ultrafast meets483ultrasmall: controlling nanoantennas and molecules. ACS Photonics4842016, 3, 1401−1414.

(18) 485Liebel, M.; Toninelli, C.; van Hulst, N. F. Room-temperature486ultrafast nonlinear spectroscopy of a single molecule. Nat. Photonics4872018, 12, 45−49.

(19) 488Hildner, R.; Brinks, D.; Nieder, J. B.; Cogdell, R. J.; van Hulst,489N. F. Quantum coherent energy transfer over varying pathways in490single light-harvesting complexes. Science 2013, 340, 1448−1451.

(20) 491Chen, L. P.; Gelin, M. F.; Domcke, W.; Zhao, Y. Theory of492femtosecond coherent double-pump single-molecule spectroscopy:493Application to light harvesting complexes. J. Chem. Phys. 2015, 142,494164106−164118.

(21) 495Prince, S. M.; Papiz, M. Z.; Freer, A. A.; McDermott, G.;496Hawthornthwaite-Lawless, A. M.; Cogdell, R. J.; Isaacs, N. W.497Apoprotein structure in the LH2 complex from Rhodopseudomonas498acidophila strain 10050: modular assembly and protein pigment499interactions. J. Mol. Biol. 1997, 268, 412−423.

(22) 500Huynh, T. D.; Sun, K. W.; Gelin, M. F.; Zhao, Y. Polaron501dynamics in two-dimensional photon-echo spectroscopy of molecular502rings. J. Chem. Phys. 2013, 139, 104103−104113.

(23) 503Chernyak, V.; Mukamel, S. Collective coordinates for nuclear504spectral densities in energy transfer and femtosecond spectroscopy of505molecular aggregates. J. Chem. Phys. 1996, 105, 4565−4583.

(24) 506Mukamel, S.; Abramavicius, D. Many-Body Approaches for507Simulating Coherent Nonlinear Spectroscopies of Electronic and508Vibrational Excitons. Chem. Rev. 2004, 104, 2073−2098.

(25) 509Anda, A.; Hansen, T.; De Vico, L. Multireference Excitation510Energies for Bacteriochlorophylls A within Light Harvesting System 2.511J. Chem. Theory Comput. 2016, 12, 1305−1313.

(26) 512Madjet, M. E.; Abdurahman, A.; Renger, T. Intermolecular513Coulomb Couplings from Ab Initio Electrostatic Potentials:514Application to Optical Transitions of Strongly Coupled Pigments in515Photosynthetic Antennae and Reaction Centers. J. Phys. Chem. B5162006, 110, 17268−17281.

(27) 517Renger, T.; Muh, F. Understanding photosynthetic light-518harvesting: a bottom up theoretical approach. Phys. Chem. Chem. Phys.5192013, 15, 3348−3371.

(28) 520Cherepy, N. J.; Shreve, A. P.; Moore, L. J.; Boxer, S. G.;521Mathies, R. A. Electronic and Nuclear Dynamics of the Accessory522Bacteriochlorophylls in Bacterial Photosynthetic Reaction Centers523from Resonance Raman Intensities. J. Phys. Chem. B 1997, 101,5243250−3260.

(29) 525Palacino-Gonzalez, E.; Gelin, M. F.; Domcke, W. Theoretical526aspects of femtosecond double-pump single-molecule spectroscopy. I.527Weak-field regime. Phys. Chem. Chem. Phys. 2017, 19, 32296−32306.

The Journal of Physical Chemistry Letters Letter

DOI: 10.1021/acs.jpclett.8b01887J. Phys. Chem. Lett. XXXX, XXX, XXX−XXX

F

(30)528 Palacino-Gonzalez, E.; Gelin, M. F.; Domcke, W. Theoretical529 aspects of femtosecond double-pump single-molecule spectroscopy.530 II. Strong-field regime. Phys. Chem. Chem. Phys. 2017, 19, 32307−531 32319.

(31)532 Novoderezhkin, V. I.; Rutkauskas, D.; van Grondelle, R.533 Dynamics of the Emission Spectrum of a Single LH2 Complex:534 Interplay of Slow and Fast Nuclear Motions. Biophys. J. 2006, 90,535 2890−2902.

(32)536 Schlau-Cohen, G. S.; Wang, Q.; Southall, J.; Cogdell, R. J.;537 Moerner, W. E. Single-molecule spectroscopy reveals photosynthetic538 LH2 complexes switch between emissive states. Proc. Natl. Acad. Sci.539 U. S. A. 2013, 110, 10899−10903.

(33)540 Rancova, O.; Abramavicius, D. Static and Dynamic Disorder in541 Bacterial Light-Harvesting Complex LH2: A 2DES Simulation Study.542 J. Phys. Chem. B 2014, 118, 7533−7540.

(34)543 Caycedo-Soler, F.; Lim, J.; Oviedo-Casado, S.; van Hulst, N. F.;544 Huelga, S. F.; Plenio, M. B. Theory of Excitonic Delocalization for545 Robust Vibronic Dynamics in LH2. J. Phys. Chem. Lett. 2018, 9, 3446.

(35)546 Novoderezhkin, V. I.; Cohen Stuart, T. A.; van Grondelle, R.547 Dynamics of exciton relaxation in LH2 antenna probed by multipulse548 nonlinear spectroscopy. J. Phys. Chem. A 2011, 115, 3834−3844.

(36)549 Sun, J.; Luo, B.; Zhao, Y. Dynamics of a one-dimensional550 Holstein polaron with the Davydov ansatze. Phys. Rev. B: Condens.551 Matter Mater. Phys. 2010, 82, 014305.

(37)552 Luo, B.; Ye, J.; Guan, C. B.; Zhao, Y. Validity of time-553 dependent trial states for the Holstein polaron. Phys. Chem. Chem.554 Phys. 2010, 12, 15073−15084.

(38)555 Ye, J.; Sun, K. W.; Zhao, Y.; Yu, Y. J.; Lee, C. K.; Cao, J. S.556 Excitonic energy transfer in light-harvesting complexes in purple557 bacteria. J. Chem. Phys. 2012, 136, 245104.

(39)558 Maly, P.; Somsen, O. J. G.; Novoderezhkin, V. I.; Mancal, T.;559 van Grondelle, R. The Role of Resonant Vibrations in Electronic560 Energy Transfer. ChemPhysChem 2016, 17, 1356−1368.

(40)561 van Dijk, E. M. H. P.; Hernando, J.; Garcia-Parajo, M. F.; van562 Hulst, N. F. Single-molecule pump-probe experiments reveal563 variations in ultrafast. J. Chem. Phys. 2005, 123, 064703−064710.

(41)564 Maly, P.; Gruber, J. M.; Cogdell, R. J.; Mancal, T.; van565 Grondelle, R. Ultrafast energy relaxation in single light-harvesting566 complexes. Proc. Natl. Acad. Sci. U. S. A. 2016, 113, 2934−2942.

(42)567 Gelin, M. F.; Egorova, D.; Domcke, W. Strong and Long Makes568 Short: Strong-Pump Strong-Probe Spectroscopy. J. Phys. Chem. Lett.569 2011, 2, 114−119.

The Journal of Physical Chemistry Letters Letter

DOI: 10.1021/acs.jpclett.8b01887J. Phys. Chem. Lett. XXXX, XXX, XXX−XXX

G