Vibration-assisted resonance in photosynthetic excitation-energy transfer

Understanding how the effectiveness of natural photosynthetic energy-harvesting systems arises from the interplay between quantum coherence and environmental noise represents a significant challenge for quantum theory. Recently it has begun to be appreciated that discrete molecular vibrational modes may play an important role in the dynamics of such systems. Here we present a microscopic mechanism by which intramolecular vibrations may be able to contribute to the efficiency and directionality of energy transfer. Excited vibrational states create resonant pathways through the system, supporting fast and efficient energy transport. Vibrational damping together with the natural downhill arrangement of molecular energy levels gives intrinsic directionality to the energy flow. Analytical and numerical results demonstrate a significant enhancement of the efficiency and directionality of energy transport that can be directly related to the existence of resonances between vibrational and excitonic levels.

Figure 1

Diagram of the three-site model, with site 3 coupled to a vibrational mode that is restricted to either 0 or 1 excitation(s). Solid arrows indicate coherent couplings, dashed arrows indicate irreversible decay pathways, and dotted arrows indicate dephasing pathways.

Figure 2

Simplified model illustrating the resonance mechanism within a perturbative framework. The delocalized excitonic states $|\pm \rangle$ diagonalize the $\left\{\right|1\rangle ,|2\rangle \}$ subspace. The transformed couplings $J_{\pm 3}$ are of similar magnitude to $\lambda$, which is much smaller than ${\epsilon }_{\pm }-{\epsilon }_3$. When $\omega \simeq {\epsilon }_{\pm }-{\epsilon }_3$, a resonance occurs and the exciton state $|\pm \rangle$ becomes strongly mixed with the vibrationally excited state $|3,1\rangle$ even though the states are only coupled to second order (via the intermediate state $|3,0\rangle$) and both $J_{\pm 3}$ and $\lambda$ are small.

Figure 3

Dynamics of site populations in the absence of dephasing: blue, $|1\rangle$; cyan, $|2\rangle$; magenta, $|3,0\rangle$; red, $|3,1\rangle$. The initial state is $|1\rangle$. (a) Fully coherent dynamics without vibrational coupling ($\lambda =0$). Disorder-induced localization prevents $|3,0\rangle$ from ever becoming significantly populated. (b) Fully coherent dynamics with a vibrational mode at $147{\mathrm{cm}}^{-1}$ coupled to site 3, showing resonant oscillations between exciton state $|-\rangle$ and the excited vibrational level $|3,1\rangle$. The absence of vibrational decay means that $|3,0\rangle$ is still not significantly populated. (c) Dynamics including decay of vibrational excitations at rate ${\gamma }_{\mathrm{vib}}=7.5{\mathrm{cm}}^{-1}$. The initial population in $|-\rangle$ is transferred to $|3,0\rangle$ in roughly $6\mathrm{ps}$.

Figure 4

Occupation probability of $|3,0\rangle$ following initial excitation of $|1\rangle$, for various values of the vibrational decay rate; from bottom to top, ${\gamma }_{\mathrm{vib}}=\{2.5$ (magenta), 5.0 (red), 7.5 (black), 10.0 (cyan), 12.5 (blue)$\}{\mathrm{cm}}^{-1}$. For clarity, successive curves have been vertically offset by $0.05$; the corresponding horizontal lines indicate $P=0.45$ for each curve.

Figure 5

Trap-state population $P_{\mathrm{t}}$ at 5 ps for an initial state on site 1 as a function of vibrational frequency $\omega$ and dephasing rate ${\gamma }_{\mathrm{deph}}$ (both measured in ${\mathrm{cm}}^{-1}$). The trapping rate has been set to ${\gamma }_{\mathrm{trap}}=1{\mathrm{cm}}^{-1}$. Peaks at $\omega \approx 147{\mathrm{cm}}^{-1}$ and $341{\mathrm{cm}}^{-1}$ correspond to resonances of $|-\rangle$ and $|+\rangle$ with $|3,1\rangle$.

Figure 6

Population of $|3,0\rangle$ ($P_{|3,0\rangle })$ in the absence of trapping for an initial excitation on site 1, both (a) with and (b) without vibrational coupling. Dephasing values (in ${\mathrm{cm}}^{-1}$) are ${\gamma }_{\mathrm{deph}}=0$ (solid), 2 (dashed), 10 (dotted), and 50 (dot dashed). A 25 ps timescale has been chosen to show the steady-state behavior of the population.