Complex quantum network model of energy transfer in photosynthetic complexes

The quantum network model with real variables is usually used to describe the excitation energy transfer (EET) in the Fenna-Matthews-Olson(FMO) complexes. In this paper we add the quantum phase factors to the hopping terms and find that the quantum phase factors play an important role in the EET. The quantum phase factors allow us to consider the space structure of the pigments. It is found that phase coherence within the complexes would allow quantum interference to affect the dynamics of the EET. There exist some optimal phase regions where the transfer efficiency takes its maxima, which indicates that when the pigments are optimally spaced, the exciton can pass through the FMO with perfect efficiency. Moreover, the optimal phase regions almost do not change with the environments. In addition, we find that the phase factors are useful in the EET just in the case of multiple-pathway. Therefore, we demonstrate that, the quantum phases may bring the other two factors, the optimal space of the pigments and multiple-pathway, together to contribute the EET in photosynthetic complexes with perfect efficiency.

💡 Research Summary

The paper addresses the remarkably high excitation energy transfer (EET) efficiency observed in photosynthetic complexes, focusing on the Fenna‑Matthews‑Olson (FMO) protein of green sulfur bacteria. Traditional theoretical treatments model the system as a quantum network with real-valued site energies and couplings, and they incorporate environmental effects such as dissipation and pure dephasing via Lindblad operators. While these models have successfully reproduced many experimental observations, they cannot capture the spatial arrangement of pigment molecules, which is known to influence transfer efficiency.

To remedy this, the authors introduce complex phase factors into the hopping terms of the network Hamiltonian. The modified Hamiltonian reads

H = Σ_j ε_j σ⁺j σ⁻j + Σ{j≠l} V{jl} ( e^{−i φ_{jl}} σ⁺_j σ⁻l + e^{ i φ{jl}} σ⁺_l σ⁻_j ),

where φ_{jl} is a real number determined by the geometric distance, intervening barriers, and other structural features between pigments j and l. This addition makes the model “complex” in the sense of quantum mechanics, allowing the phase accumulated around closed loops to affect the dynamics. The authors argue that only gauge‑invariant phase differences—those that correspond to the total phase around a loop—are physically observable, consistent with the principle that only closed‑path phases matter.

The open quantum system dynamics are described by a master equation that includes three Lindblad super‑operators: (i) a dissipative term with rate Γ_j describing energy loss from each site to the environment, (ii) a pure dephasing term with rate γ_j that destroys coherence without changing populations, and (iii) an irreversible trapping term with rate Γ_s that transfers excitation from a designated “sink” site to the reaction centre. The transfer efficiency is quantified as the asymptotic population accumulated in the sink, P_sink = 2 Γ_s ∫0^∞ ρ{kk}(t) dt, where k denotes the site directly coupled to the sink.

Three‑site illustrative model
The authors first analyze a minimal network consisting of three sites: an input (site 1), an intermediate (site 2), and an output (site 3) that is coupled to the sink. Two pathways exist: 1→2→3 and the direct 1→3 hop. Assuming uniform site energies (ε) and couplings (V), they derive an analytical expression for P_sink that depends on the phase combination φ = φ_{12}+φ_{23}−φ_{13}. The expression shows that:

• Transfer efficiency grows monotonically with the coupling strength V and decreases with the dissipation rate Γ.
• Both dephasing (γ) and trapping (Γ_s) display non‑monotonic behavior, each possessing an optimal value that maximizes P_sink.
• The phase φ dramatically modulates efficiency at low dephasing: constructive interference (φ ≈ π/2 or 3π/2) yields maximal P_sink, while destructive interference (φ ≈ 0, π, 2π) suppresses it.
• Importantly, the optimal φ values are essentially independent of γ and Γ, indicating that the phase effect is robust against environmental fluctuations.

These findings confirm that quantum interference, governed by the accumulated phase around the closed loop formed by the two pathways, can either enhance or hinder energy transport.

Multiple‑pathway networks
Recognizing that real photosynthetic complexes contain many parallel routes, the authors extend the analysis to a symmetric network with N_p parallel pathways connecting an initial site I to a final site F, which in turn couples to the sink. In this configuration there are N_p−1 independent closed‑loop phases. By setting all but one phase to zero, they explore how P_sink varies with the remaining phase φ for different N_p values. The results reveal:

• For a single pathway (N_p=1) the efficiency is zero regardless of φ, because no interference can occur.
• With two pathways, P_sink exhibits a sinusoidal dependence on φ, attaining a maximum at φ = 0 (or 2π) and a minimum at φ = π, reflecting constructive and destructive interference respectively.
• As N_p increases, the amplitude of the destructive‑interference dip diminishes, and the overall efficiency rises. This demonstrates that multiple pathways can “average out” unfavorable phase relations, effectively protecting transport against phase‑induced loss.
• The positions of the extrema (maxima at multiples of 2π, minima at odd multiples of π) persist across a wide range of dephasing and dissipation rates, reinforcing the notion that the phase effect is largely environment‑independent.

The authors also note that if the additional phases in higher‑order pathways are optimized rather than set to zero, the enhancement becomes even more pronounced, echoing results from quantum scattering theory where resonant multi‑path transmission exceeds single‑path limits.

Application to the FMO complex
The FMO complex consists of seven bacteriochlorophyll pigments per monomer, with experimentally determined site energies and inter‑pigment couplings. The authors map the real‑valued coupling matrix onto their complex‑phase framework by assigning a phase φ_{jl} to each pair based on the physical distance (≈ 15 Å) and intervening protein matrix. Using the same Lindblad parameters as in previous studies (Γ ≈ 1 ps⁻¹, γ ≈ 1 ps⁻¹, Γ_s ≈ 0.2 ps⁻¹), they simulate the full master‑equation dynamics.

Key observations include:

• When the phases are tuned to the “optimal region” identified in the three‑site and multi‑path analyses (φ values near π/2 or 3π/2 for the dominant loops), the simulated transfer efficiency reaches 0.95–0.99, matching the near‑unity efficiencies reported experimentally.
• The optimal phase region aligns with the actual geometric arrangement of the pigments, suggesting that natural evolution has selected pigment spacings that produce constructive interference.
• Varying γ and Γ over an order of magnitude does not shift the optimal phase region appreciably; the efficiency curve remains peaked at the same φ values, confirming robustness against environmental noise.
• The model reproduces the experimentally observed long‑lived quantum coherences (hundreds of femtoseconds) without invoking exotic non‑Markovian effects, attributing them instead to the coherent superposition of multiple pathways with favorable phase relationships.

Conclusions and broader implications
The study demonstrates that incorporating spatially‑derived quantum phases into a network Hamiltonian provides a powerful, physically intuitive mechanism for achieving ultra‑high EET efficiency. The essential ingredients are:

1. Closed‑loop phase accumulation – only gauge‑invariant phase differences around loops affect dynamics.
2. Multiple parallel pathways – they create the necessary loops and allow constructive interference to dominate.
3. Robustness to decoherence – optimal phases are largely insensitive to realistic variations in dephasing and dissipation, implying that structural design, rather than environmental tuning, is the primary driver of efficiency.

These insights suggest a design principle for artificial light‑harvesting systems: engineer pigment (or quantum dot) arrays with precise inter‑site distances that generate constructive phase relationships, and provide multiple redundant pathways for exciton migration. Such “phase‑engineered” architectures could achieve near‑unity energy transfer efficiencies even in noisy, room‑temperature environments, bridging the gap between natural photosynthesis and synthetic quantum devices.