^{1}, Graham B. Griffin

^{1}, Stephen K. Gray

^{2}and Gregory S. Engel

^{1,a)}

### Abstract

An open question at the forefront of modern physical sciences is what role, if any, quantum effects may play in biological sensing and energy transport mechanisms. One area of such research concerns the possibility of coherent energy transport in photosynthetic systems. Spectroscopic evidence of long-lived quantum coherence in photosynthetic light-harvesting pigment protein complexes (PPCs), along with theoretical modeling of PPCs, has indicated that coherent energy transport might boost efficiency of energy transport in photosynthesis. Accurate assessment of coherence lifetimes is crucial for modeling the extent to which quantum effects participate in this energy transfer, because such quantum effects can only contribute to mechanisms proceeding on timescales over which the coherences persist. While spectroscopy is a useful way to measurecoherence lifetimes, inhomogeneity in the transition energies across the measured ensemble may lead to underestimation of coherence lifetimes from spectroscopic experiments. Theoretical models of antenna complexes generally model a single system, and direct comparison of single system models to ensemble averaged experimental data may lead to systematic underestimation of coherence lifetimes, distorting much of the current discussion. In this study, we use simulations of the Fenna-Matthews-Olson complex to model single complexes as well as averaged ensembles to demonstrate and roughly quantify the effect of averaging over an inhomogeneous ensemble on measuredcoherence lifetimes. We choose to model the Fenna-Matthews-Olson complex because that system has been a focus for much of the recent discussion of quantum effects in biology, and use an early version of the well known environment-assisted quantum transport model to facilitate straightforward comparison between the current model and past work. Although ensemble inhomogeneity is known to lead to shorter lifetimes of observed oscillations (simply inhomogeneous spectral broadening in the time domain), this important fact has been left out of recent discussions of spectroscopicmeasurements of energy transport in photosynthesis. In general, these discussions have compared single-system theoretical models to whole-ensemble laboratory measurements without addressing the effect of inhomogeneous dephasing. Our work addresses this distinction between single system and ensemble averaged observations, and shows that the ensemble averaging inherent in many experiments leads to an underestimation of coherence lifetimes in individual systems.

The authors thank M. Plenio for helpful suggestions and conversations. The authors gratefully acknowledge support from the NSF MRSEC (Grant No. DMR 08-02054), AFOSR (Grant No. FA9550-09-1-0117), DTRA (Grant No. HDTRA1-10-1-0091 P00002), and the DARPA QuBE program (Grant No. N66001-10-1-4060) for supporting portions of this work. K. Pelzer acknowledges the support of the DOE Computational Science Graduate Fellowship. Use of the Center for Nanoscale Materials was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

I. INTRODUCTION

II. METHODS

III. RESULTS AND DISCUSSION

### Key Topics

- Coherence
- 27.0
- Dephasing
- 24.0
- Excitons
- 20.0
- Energy transfer
- 16.0
- Quantum effects
- 13.0

## Figures

Location of the seven chromophores of FMO, labeled with letters A-G. For clarity, the phytyl chains have been deleted as well as pieces of the protein that obscure the view of the chromophores. We label the chromophores with letters to avoid confusion with the excitons, which we index with numbers. For comparison to literature in which the chromophores are labeled with numbers, A corresponds to 1, B corresponds to 2, etc.

Location of the seven chromophores of FMO, labeled with letters A-G. For clarity, the phytyl chains have been deleted as well as pieces of the protein that obscure the view of the chromophores. We label the chromophores with letters to avoid confusion with the excitons, which we index with numbers. For comparison to literature in which the chromophores are labeled with numbers, A corresponds to 1, B corresponds to 2, etc.

Oscillations in the ρ_{12} element over 4 ps for calculations using the Case A density matrix. The trajectory of the ρ_{12} element for each of the 2500 systems is plotted in pink. The blue line shows the values of the ρ_{12} element of the averaged density matrix, ⟨ρ_{12}⟩. The red line shows the values of ρ_{12} for a single (arbitrarily chosen) system. The red dashed line shows an exponential decay curve with a lifetime of ∼630 fs, which corresponds to the lifetime of an exponential curve fitted to the absolute value of the single trajectory shown in red (see supplementary material for details). The blue dashed line represents an exponential curve of lifetime ∼400 fs, the lifetime of a corresponding fit to ⟨ρ_{12}⟩. Comparison of the ensemble average to the individual systems shows that dephasing occurs faster than decoherence.

Oscillations in the ρ_{12} element over 4 ps for calculations using the Case A density matrix. The trajectory of the ρ_{12} element for each of the 2500 systems is plotted in pink. The blue line shows the values of the ρ_{12} element of the averaged density matrix, ⟨ρ_{12}⟩. The red line shows the values of ρ_{12} for a single (arbitrarily chosen) system. The red dashed line shows an exponential decay curve with a lifetime of ∼630 fs, which corresponds to the lifetime of an exponential curve fitted to the absolute value of the single trajectory shown in red (see supplementary material for details). The blue dashed line represents an exponential curve of lifetime ∼400 fs, the lifetime of a corresponding fit to ⟨ρ_{12}⟩. Comparison of the ensemble average to the individual systems shows that dephasing occurs faster than decoherence.

Oscillations in the ρ_{12} element over 4 ps for calculations beginning with the Case B density matrix. The trajectory of the ρ_{12} element for each of the 2500 systems is plotted in pink. The blue line shows the values of the ρ_{12} element of the averaged density matrix, ⟨ρ_{12}⟩. The red line shows the values of ρ_{12} for a single (arbitrarily chosen) system. The red dashed line shows an exponential decay curve with a lifetime of ∼560 fs, which corresponds to the lifetime of an exponential curve fitted to the absolute value of the single trajectory in shown in red (see supplementary material for details). The blue dashed line represents an exponential curve of lifetime ∼310 fs, the lifetime of a corresponding fit to ⟨ρ_{12}⟩. Comparison of the ensemble average to the individual systems shows that dephasing occurs faster than decoherence.

Oscillations in the ρ_{12} element over 4 ps for calculations beginning with the Case B density matrix. The trajectory of the ρ_{12} element for each of the 2500 systems is plotted in pink. The blue line shows the values of the ρ_{12} element of the averaged density matrix, ⟨ρ_{12}⟩. The red line shows the values of ρ_{12} for a single (arbitrarily chosen) system. The red dashed line shows an exponential decay curve with a lifetime of ∼560 fs, which corresponds to the lifetime of an exponential curve fitted to the absolute value of the single trajectory in shown in red (see supplementary material for details). The blue dashed line represents an exponential curve of lifetime ∼310 fs, the lifetime of a corresponding fit to ⟨ρ_{12}⟩. Comparison of the ensemble average to the individual systems shows that dephasing occurs faster than decoherence.

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