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i ≠ j, represent coherences and are complex numbers carryingphase information. Experimentally, we probe the ensemble aver-age of the populations or coherences over bath realizations.

Quantum dynamics within photosynthetic complexes aretypically modeled with second-order perturbation theory usingLindblad or Redfield master equations. Although each methodhas different strengths and weaknesses (and both reproduce thedynamics discussed here), we choose to use the Redfield masterequation for this work because it conveniently separates unitarytime evolution from dissipative dynamics and clearly illustratesthe origin of quantum transport due to coupling between popula-tions and coherences. The Redfield master equation can beexpressed as

∂!∂t

! −iℏ"H;!# − " ! ; [2]

where H is the system Hamiltonian, and " is the relaxation super-operator. In essence, all the dynamics important for functionare contained within the relaxation superoperator. Treating theinteraction perturbatively, an expansion of the Redfield equationfor the system of excitonic states yields

∂!ij∂t

! −iℏ$#i − #j%!ij −

!

kl

"ij;kl!kl: [3]

The Redfield equation shows that temporal evolution of a parti-cular element of the density matrix depends on the energy of thesystem eigenstates, #n, as well as on relaxation within the system.The relaxation dynamics are captured within the relaxation super-operator, "; elements "ij;kl determine how element !kl will contri-bute to the time evolution of the element !ij of the density matrix.The physical origin of the relaxation superoperator arises fromfluctuations from the time-averaged Hamiltonian that instanta-neously affect the eigenstates by effectively mixing them. The re-laxation superoperator elements can be divided into three differentclasses: population transfer terms, !kk → !ii ("ii;kk), representingclassical transport; coherence transfer terms, !kl → !ij for k ≠ land i ≠ j ("ij;kl); and coupling between populations and coherences,!kl → !ii for k ≠ l ("ii;kl). The "ii;kl terms represent quantum trans-port by our definition.

Simply interpreted, the Redfield equation represents a first-order differential rate law that describes the change in eachelement in the density matrix. Two different processes control therate of change of !ij: unitary dynamics, − i

ℏ $#i − #j%!ij, andnonunitary contributions (relaxation) due to coupling to otherelements of the density matrix, ∑kl "ij;kl!kl. Unitary dynamics isresponsible for the sinusoidal phase oscillation of coherences.As a result, we observe quantum coherence as an oscillation inpeak amplitude, or quantum beat, with a fixed frequency propor-tional to the energy gap. For a population term, however, therate of change depends only on relaxation dynamics because theunitary oscillation vanishes when the energy gap #i − #j is zero.For a population term, the summation notation expands to

∂!ii∂t

! −!

kl

"ii;kl ! −!

k

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|!!!!!!!!!!!!!!!!!!{z!!!!!!!!!!!!!!!!!!}Resonance Energy Transport

−!

k≠l

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: [4]

From Eq. 4, we can see that population dynamics is governedby two distinct mechanisms: The first term represents transfer toand from other populations via resonance energy transport, andthe second term causes oscillations driven by coherences. Popu-lation transfer via incoherent, resonance energy transport leadsto exponential growth or decay depending on the direction of

transport. The final term in Eq. 4 shows that populations will takeon the oscillatory character of coherences.

Fig. 1 schematically depicts the three types of transfer possibleamong elements of the density matrix. In classical resonanceenergy transfer (blue), the dynamics is mostly confined to theelectronic states of the chromophores. That is, energy is trans-ferred monotonically from one state to the next, and the popula-tion dynamics is entirely decoupled from the coherence dynamics.Next, coherence transfer (green) can accompany the populationtransfer allowing relaxation from one superposition state toanother. In quantum transport (red), coherences and populationscouple, allowing coherences to drive transfer between popula-tions. This process yields oscillatory population dynamics.

ResultsTo search for evidence of quantum transport in photosyntheticsystems, we revisit two-dimensional electronic spectroscopicdata on the FMO complex published previously (9). (A completediscussion of methods and materials can be found in ref. 9.)Phase-sensitive two-dimensional electronic spectroscopy permitsobservation and characterization of population and coherencedynamics (15). The experimental and theoretical details for thisspectroscopy have been described elsewhere (16–19). In essence,the data from 2D spectroscopy can be interpreted as a correlationmap ($%,$t) of the “input” energy, ℏ$%, and the “output” energy,ℏ$t. A series of 2D spectra as a function of waiting time revealstime-dependent evolution of a system after excitation. In rephas-ing pathways, the photon echo eliminates inhomogeneous broad-ening in the antidiagonal direction, which further improvesspectral resolution. The diagonal peak ($% ! $t) contains infor-mation on the population states and their relaxation dynamics.The existence of a cross-peak ($% ≠ $t) indicates a couplingand transfer pathway between population states that cannot beobserved directly using linear absorption spectroscopy. The spec-tral positions of the cross-peaks help to identify couplings andcoherences between excitonic states. Theoretical studies predictquantum beating due to unitary evolution of coherences willappear in cross-peaks in rephasing pathways but not in diagonalpeaks (20).

We start our investigation by first isolating the coherence beat-ing signal due to unitary evolution using the same procedurediscussed in detail in prior works (9, 21). This coherence beatingsignal is used only for comparison to determine whether coher-

Fig. 1. The Redfield relaxation superoperator contains three types oftransfer elements: transfer between populations (blue), transfer betweencoherences (green), and transfer between a population and a coherence(red). These transfer mechanisms are depicted schematically both in theFMO photosynthetic antenna complex and in the density matrix. Quantumtransport (red) occurs when populations and coherences directly couple.Each transfer scheme implies a different type of interaction with the proteinenvironment, with the most active contribution occurring in the quantumtransport regime.

Panitchayangkoon et al. PNAS ∣ December 27, 2011 ∣ vol. 108 ∣ no. 52 ∣ 20909

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Published in: Minhaeng Cho; Chem. Rev. 2008, 108, 1331-1418. DOI: 10.1021/cr078377b, Copyright © 2008 American Chemical Society

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Published in: Minhaeng Cho; Chem. Rev. 2008, 108, 1331-1418. DOI: 10.1021/cr078377b, Copyright © 2008 American Chemical Society

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…

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…

Coherent EET in many natural antennae from different organisms: LHC II (superior plants), RC (purple bacteria), FMO (green bacteria), PBP (cryptophytes)

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