Enhancement of vibronic and ground-state vibrational coherences in 2D spectra of photosynthetic complexes

Abstract

A vibronic-exciton model is applied to investigate the recently proposed mechanism of enhancement of coherent oscillations due to mixing of electronic and nuclear degrees of freedom. We study a dimer system to elucidate the role of resonance coupling, site energies, vibrational frequency and energy disorder in the enhancement of vibronic-exciton and ground-state vibrational coherences, and to identify regimes where this enhancement is significant. For a heterodimer representing two coupled bachteriochloropylls of the FMO complex, long-lived vibronic coherences are found to be generated only when the frequency of the mode is in the vicinity of the electronic energy difference. Although the vibronic-exciton coherences exhibit a larger initial amplitude compared to the ground-state vibrational coherences, we conclude that, due to the dephasing of the former, both type of coherences have a similar magnitude at longer population time.

Figure 1. Illustration of the effect of transition dipole moment borrowing and excitonic mixing of excited states in a dimer without considering vibrational states (A) and considering a simplified model of vibrational presence (B).

In the case (B), a heterodimer is considered with an approximate resonance between the one-phonon state of monomer 1 and the zero-phonon state of monomer 2. Panel (C): Dimer model with a single vibrational mode per monomer. The monomers 1 and 2 can be in their respective ground states |g〉 or excited states |e〉. In each electronic state, the monomers can occupy any of the vibrational levels corresponding to their respective vibrational modes (q₁ or q₂).

Figure 2. Liouville pathways and their respective contribution to the 2D spectrum.

(A) Double-sided Feynman diagram describing the four Liouville pathways along with their dipole pre-factor, the frequency of the involved coherence and the frequency of the exciting and probing pulses. (B) Position of the signal in the 2D spectrum and its characteristics originating from the different pathways. Solid symbol denote non-oscillating contributions in t₂, open symbols denote oscillatory contributions with frequencies ω_αβ (circle), (diamond). Signals denoted by circle symbols refer to the electronic excited-state manifold and do not involve any (purely) vibrational level whereas signals on the electronic ground-state manifold, denoted by diamonds, do (for ν ≥ 1). The diads in (B) represent the eigenstates |α〉, |β〉 involved in each signal contribution. The red contours highlight the signals for which we present detailed results below.

Figure 3. Relative amplitude of the dimer dipole moment pre-factor over that of the monomer () for the non-rephasing pathway R₁ at initial time (a) involving coherence (1,2) as a function of the energy gap ΔE and coupling J, (b) involving coherence (1,β), with β = 2 (red) and β = 3 (magenta), as a function of the energy gap ΔE for ω₀ = 117 cm⁻¹.

The plain thick lines represent the amplitudes, the dashed lines denote the associated measure of the vibronic character (0.1χ_1β). In both sub-figures, the prevailingly vibrational character of the eigenstates () is highlighted with colored areas. A significant enhancement of the dipole moment pre-factor can be observed for coherence involving eigenstates 1 and 2, which is strongly vibrational for ΔE > 99 cm⁻¹.

Figure 4. Absolute value of the oscillating signal amplitudes and (Eqs. (15) and (16)) for pathways R₁ (a) and R₄ (b) as a function of the energy gap ΔE and excitation frequency ω₁.

In both pathways, the amplitude is significantly enhanced in the vicinity of the vibrational mode energy, i.e. for ΔE ~ ω₀ (ω₀ = 117 cm⁻¹ here) due to the resonance occurring.

Figure 5

(a) Relative amplitude of the signal A_r at initial time for pathways R₁ and R₄ (Eqs. (17) and (18)) involving the prevailingly vibrational coherence ρ₁₂ as a function of the excitation frequency ω₁ for ω₀ = 117 cm⁻¹. The enhancement of the vibronic coherence (pathway R₁) is up to 3 times larger than that of the ground-state vibrational coherence (pathway R₄). (b, c) Relative amplitude of the diagonal signal through pathways R₁ (b) and R₄ (c) at ω₁ = ω_opt as function of the vibrational mode frequency ω₀ for β = 2 (red, blue) and β = 3 (magenta, cyan). The shaded areas represent the domains in which the coherence is enhanced and of prevailingly vibrational character. The dotted dashed lines show the vibronic coherence life time, and the dashed lines represent the signal relative amplitude after 1 ps. The inset in (c) presents the frequency of the oscillating signal through the different pathways. The vertical black dashed line at ω₀ = 135 cm⁻¹ delimits the mode frequency at which the ordering of the eigenstates is reorganized and the vibrational character of the coherences changes. It appears that long-lived (>1 ps) vibronic coherences are created for ω₀ < 120 cm⁻¹ (b) whereas any mode generates long-lived ground-state vibrational coherences (c).

Figure 6. Time evolution of the signal involving coherences of prevailingly vibrational character originating in pathway R₁ for two mode frequencies: ω₀ = 117 cm⁻¹ (a) and 185 cm⁻¹ (b).

The dynamics of the signal has been computed with different energy gaps, namely the energy gap averaged over the energy disorder (denoted by 〈…〉_Δ), the resonant condition (ΔE = ω₀) and the reference gap energy (ΔE = ΔE₀ = 110 cm⁻¹). The signal amplitude is presented relatively to that of the monomer in the region where it is maximum (ω₁ = ω_opt). In part (a) the mode frequency in the vicinity of the energy gap (ω₀ = 117 cm⁻¹) creates long-lived oscillations of significant amplitude even with energy disorder. In part (b), i. e. for a mode frequency out of resonance with the energy gap (ω₀ = 185 cm⁻¹), the signal benefits only sightly from the enhancement mechanism, and it remains of low amplitude.