^{1}, Alex W. Chin

^{1}, Javier Almeida

^{1}, Susana F. Huelga

^{1}and Martin B. Plenio

^{1}

### Abstract

Based entirely upon actual experimental observations on electron-phonon coupling, we develop a theoretical framework to show that the lowest energy band of the Fenna-Matthews-Olson complex exhibits observable features due to the quantum nature of the vibrational manifolds present in its chromophores. The study of linear spectra provides us with the basis to understand the dynamical features arising from the vibronic structure in nonlinear spectra in a progressive fashion, starting from a microscopic model to finally performing an inhomogeneous average. We show that the discreteness of the vibronic structure can be witnessed by probing the diagonal peaks of the nonlinear spectra by means of a relative phase shift in the waiting time resolved signal. Moreover, we demonstrate that the photon-echo and non-rephasing paths are sensitive to different harmonics in the vibrational manifold when static disorder is taken into account. Supported by analytical and numerical calculations, we show that non-diagonal resonances in the 2D spectra in the waiting time, further capture the discreteness of vibrations through a modulation of the amplitude without any effect in the signal intrinsic frequency. This fact generates a signal that is highly sensitive to correlations in the static disorder of the excitonic energy albeit protected against dephasing due to inhomogeneities of the vibrational ensemble.

We acknowledge J. Caram, D. Hayes, and G. S. Engel for providing access to experimental data and insightful discussions, and Financial support of the EU integrated project Q-ESSENCE, the EU STREP CORNER, and the Alexander von Humboldt Foundation.

I. INTRODUCTION

A. Brief history of the FMO low energy band B825

II. THE MODEL

III. RESULTS

A. Linear spectra

B. Nonlinear 2D spectroscopy

1. Background

2. Vibronic features in 2D spectra: Diagonal peaks

3. Vibronic features in 2D spectra: Non-diagonal peaks

C. Ensemble inhomogeneities: On the static nature of the difference among site and excitonic coherence decay

IV. CONCLUSIONS AND PERSPECTIVES

##### C08

## Figures

In (a) fluorescence intensity from solution of stationary state in Eq. (4). Dashed and continuous lines correspond to Huang-Rhys factors *s* = 0.5 and *s* = 0.12, at T = 4 K, drawn with the same scale. The arrows point the frequency of the zero-phonon (0 cm^{−1}), phono sideband (36 cm^{−1}), and overtone (72 cm^{−1}). The inset shows the fluorescence intensity at T = 77 K, *s* = 0.12. In (b) *A*(ω) is normalized for the three resonances and correspond in continuous, dashed, and dotted lines, to the Δ = 0, 36, and 72 cm^{−1} resonances, that have a FWHM of {γ_{0}, γ_{1}, γ_{2}} = {1.6, 3.5, 5.6} cm^{−1} at 4 K, from a fit with lorentzian functions. Inset (b) shows the variation of the lorentzian widths with temperature for the ZPL (γ_{0}, circles), sideband at 36 (γ_{1}, boxes) and overtone 72 cm^{−1} (γ_{2}, diamonds), that yields to values of {γ_{0}, γ_{1}, γ_{2}} = {6.2, 8.1, 11.2} cm^{−1} at 77 K. If otherwise not stated Γ = 7.9 × 10^{−2} cm^{−1}, γ = 2.36 cm^{−1}, γ_{ z0} = 3 × 10^{−3} cm^{−1}, *s* = 0.12 , ω = 36 cm^{−1}, Ω = 10^{−3} cm^{−1}.

In (a) fluorescence intensity from solution of stationary state in Eq. (4). Dashed and continuous lines correspond to Huang-Rhys factors *s* = 0.5 and *s* = 0.12, at T = 4 K, drawn with the same scale. The arrows point the frequency of the zero-phonon (0 cm^{−1}), phono sideband (36 cm^{−1}), and overtone (72 cm^{−1}). The inset shows the fluorescence intensity at T = 77 K, *s* = 0.12. In (b) *A*(ω) is normalized for the three resonances and correspond in continuous, dashed, and dotted lines, to the Δ = 0, 36, and 72 cm^{−1} resonances, that have a FWHM of {γ_{0}, γ_{1}, γ_{2}} = {1.6, 3.5, 5.6} cm^{−1} at 4 K, from a fit with lorentzian functions. Inset (b) shows the variation of the lorentzian widths with temperature for the ZPL (γ_{0}, circles), sideband at 36 (γ_{1}, boxes) and overtone 72 cm^{−1} (γ_{2}, diamonds), that yields to values of {γ_{0}, γ_{1}, γ_{2}} = {6.2, 8.1, 11.2} cm^{−1} at 77 K. If otherwise not stated Γ = 7.9 × 10^{−2} cm^{−1}, γ = 2.36 cm^{−1}, γ_{ z0} = 3 × 10^{−3} cm^{−1}, *s* = 0.12 , ω = 36 cm^{−1}, Ω = 10^{−3} cm^{−1}.

In (a) and (b) we present the second order expansion for rephasing and non-rephasing paths, respectively. (c) The addition of rephasing and non-rephasing contributions, while (d) subtraction FT[*R* _{2} + *R* _{3} − (*R* _{1} + *R* _{4})]. In plots (a)–(b) an arcsinh scale has been used in order to discuss the sideband features, while (c)–(d) use a linear scale for purposes of comparison and *t* _{2} = 450 fs. In all plots the zero phonon emission line is shown in continuous line to guide the eye. Exciton 1 energy 12 121 cm^{−1},^{37} vibrational mode energy ω = 36 cm^{−1}, and Huang-Rhys factor *s* = 0.12.

In (a) and (b) we present the second order expansion for rephasing and non-rephasing paths, respectively. (c) The addition of rephasing and non-rephasing contributions, while (d) subtraction FT[*R* _{2} + *R* _{3} − (*R* _{1} + *R* _{4})]. In plots (a)–(b) an arcsinh scale has been used in order to discuss the sideband features, while (c)–(d) use a linear scale for purposes of comparison and *t* _{2} = 450 fs. In all plots the zero phonon emission line is shown in continuous line to guide the eye. Exciton 1 energy 12 121 cm^{−1},^{37} vibrational mode energy ω = 36 cm^{−1}, and Huang-Rhys factor *s* = 0.12.

In (a) and (b), rephasing and non-rephasing 2D spectra of exciton 1 diagonal peak of FMO complex, broadened by excitonic energy inhomogeneities. In (c), (d), and (e) are presented the real (continuous, blue) and imaginary (dashed, red) contributions to the inhomogeneous signal at points , and respectively, for complexes whose energy differ from the ensemble average as explained in the text. In (f) and (g), rephasing and non-rephasing 2D spectra of exciton 1 diagonal peak of FMO complex, broadened by both excitonic and vibrational mode energy inhomogeneities. σ_{ E } = 102 cm^{−1}, σ_{ω} = 10 cm^{−1}, *t* _{2} = 2 ps.

In (a) and (b), rephasing and non-rephasing 2D spectra of exciton 1 diagonal peak of FMO complex, broadened by excitonic energy inhomogeneities. In (c), (d), and (e) are presented the real (continuous, blue) and imaginary (dashed, red) contributions to the inhomogeneous signal at points , and respectively, for complexes whose energy differ from the ensemble average as explained in the text. In (f) and (g), rephasing and non-rephasing 2D spectra of exciton 1 diagonal peak of FMO complex, broadened by both excitonic and vibrational mode energy inhomogeneities. σ_{ E } = 102 cm^{−1}, σ_{ω} = 10 cm^{−1}, *t* _{2} = 2 ps.

In (a), Feynman diagrams involving the phonon sideband. Letters label electronic states, the superscript the boson state. In (b) are shown the waiting time domain signals at the homogeneously broadened peak at , for the set {γ_{0}, γ_{1}, γ_{2}} = {6.2, 8.1, 11.2} (left panel) and {γ_{0}, γ_{1}γ_{2}} = {50, 136, 156} cm ^{−1} (right panel). In (c) are presented the results for inhomogeneous broadened peaks with σ_{ E } = 102 cm^{−1}, σ_{ω} = 10 cm^{−1}: in the left panel are shown the signals resolved in waiting time of the ZPL (top), sideband (middle) and overtone (bottom) peaks; in the right panel the slope (long enough to avoid overlap with the second pulse *t* _{2} ⩾ 16 fs, the equality fulfilled for the pulses duration achieved in Ref. 38) is presented; arrows highlight the positions of the ZPL, sideband and overtone. In all plots non-rephasing (continuous, black) and rephasing (dashed, blue) path signals.

In (a), Feynman diagrams involving the phonon sideband. Letters label electronic states, the superscript the boson state. In (b) are shown the waiting time domain signals at the homogeneously broadened peak at , for the set {γ_{0}, γ_{1}, γ_{2}} = {6.2, 8.1, 11.2} (left panel) and {γ_{0}, γ_{1}γ_{2}} = {50, 136, 156} cm ^{−1} (right panel). In (c) are presented the results for inhomogeneous broadened peaks with σ_{ E } = 102 cm^{−1}, σ_{ω} = 10 cm^{−1}: in the left panel are shown the signals resolved in waiting time of the ZPL (top), sideband (middle) and overtone (bottom) peaks; in the right panel the slope (long enough to avoid overlap with the second pulse *t* _{2} ⩾ 16 fs, the equality fulfilled for the pulses duration achieved in Ref. 38) is presented; arrows highlight the positions of the ZPL, sideband and overtone. In all plots non-rephasing (continuous, black) and rephasing (dashed, blue) path signals.

Excitonic coherence ρ_{2, 1}(*t*) = Tr{|ψ_{1}〉〈ψ_{2}|ρ(*t*)}. In (a) full numerical calculation (continuous) and electronic coherence analytical result (dotted). In (b) are shown the Feynman diagrams that contribute to oscillating terms at frequencies concerning the excited electronic states on the response function. In (c) the result of the homogeneously broadened signal is presented for the addition of both paths, at a waiting time *t* _{2} = 20 fs. In (d)–(e), the waiting time resolved signal at points and respectively, is presented. Thick continuous line represent the real part of these oscillatory contributions. For comparison purposes, we show the absolute value of the rephasing signal (dotted) and the amplitude envelope from electronic coherence solution Eq. (11) (thin, continuous) scaled in amplitude and shifted in phase as described in the text.

Excitonic coherence ρ_{2, 1}(*t*) = Tr{|ψ_{1}〉〈ψ_{2}|ρ(*t*)}. In (a) full numerical calculation (continuous) and electronic coherence analytical result (dotted). In (b) are shown the Feynman diagrams that contribute to oscillating terms at frequencies concerning the excited electronic states on the response function. In (c) the result of the homogeneously broadened signal is presented for the addition of both paths, at a waiting time *t* _{2} = 20 fs. In (d)–(e), the waiting time resolved signal at points and respectively, is presented. Thick continuous line represent the real part of these oscillatory contributions. For comparison purposes, we show the absolute value of the rephasing signal (dotted) and the amplitude envelope from electronic coherence solution Eq. (11) (thin, continuous) scaled in amplitude and shifted in phase as described in the text.

In (a), the result of averaging homogeneous contributions with independent energy variations equal to those reported^{17} of cm^{−1} (dashed) and with excitonic energy variation with cm^{−1} (real part, continuous; absolute value dotted). In (b) the additional inhomogeneity arising from the vibronic ensemble is included, and the result of the averaging using gaussian distributions with standard deviations σ_{ω} = {1, 10} cm^{−1} are presented in continuous and dashed lines, respectively. Dotted line is the absolute value for σ_{ω} = 10 cm^{−1}. In all plots *s* = 0.24 (see text).

In (a), the result of averaging homogeneous contributions with independent energy variations equal to those reported^{17} of cm^{−1} (dashed) and with excitonic energy variation with cm^{−1} (real part, continuous; absolute value dotted). In (b) the additional inhomogeneity arising from the vibronic ensemble is included, and the result of the averaging using gaussian distributions with standard deviations σ_{ω} = {1, 10} cm^{−1} are presented in continuous and dashed lines, respectively. Dotted line is the absolute value for σ_{ω} = 10 cm^{−1}. In all plots *s* = 0.24 (see text).

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