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J. Phys. Chem. B 2009, 113, 9942–9947
Role of Quantum Coherence and Environmental Fluctuations in Chromophoric Energy Transport Patrick Rebentrost,*,† Masoud Mohseni,†,‡ and Ala´n Aspuru-Guzik*,† Department of Chemistry and Chemical Biology, HarVard UniVersity, 12 Oxford Street, Cambridge, Massachusetts 02138, and Research Laboratory of Electronics, Massachusetts Institute of Technology, 77 Massachusetts AVenue, Cambridge, Massachusetts 02139 ReceiVed: February 24, 2009; ReVised Manuscript ReceiVed: May 12, 2009
The role of quantum coherence in the dynamics of photosynthetic energy transfer in chromophoric complexes is not fully understood. In this work, we quantify the biological importance of fundamental physical processes, such as the excitonic Hamiltonian evolution and phonon-induced decoherence, by their contribution to the efficiency of the primary photosynthetic event. We study the effect of spatial correlations in the phonon bath and slow protein scaffold movements on the efficiency and the contributing processes. To this end, we develop two theoretical approaches based on a Green’s function method and energy transfer susceptibilities. We investigate the Fenna-Matthews-Olson protein complex, in which we find a contribution of coherent dynamics of about 10% in the presence of uncorrelated phonons and about 30% in the presence of realistically correlated ones. Introduction Exciton transfer among chlorophyll molecules is the energy transport mechanism of the initial step of the photosynthetic process. Light is captured by an antenna complex, and the exciton is subsequently transferred to a reaction center where biochemical energy storage is initiated by a charge separation event.1 Recent experiments2,3 suggest evidence of long-lived quantum coherence in the Fenna-Matthews-Olson (FMO) protein complex4,5 of the green-sulfur bacterium Chlorobium tepidum and in the reaction center of the purple bacterium Rhodobacter sphaeroides.6 Of key interest is the role of quantum coherence in the biological function of these chromophoric complexes. Additionally, not fully understood is the pigmentprotein interaction and its role in directing the exciton flow7 and protecting coherence. From a theory point of view, the energy transfer is conveniently described with a Frenkel exciton model. The interactions of the exciton with the surroundings are incorporated by an open quantum system treatment. Most commonly, (modified) Redfield/Lindblad models are employed.8-15 Cheng and Fleming16 and references therein provide a detailed review of the literature, including further aspects of modeling and spectroscopy. In studying the properties of exciton transfer, theoretical quantifiers such as the energy transfer efficiency/quantum yield, transfer time, and exciton lifetime prove useful; for example, to elucidate optimality and robustness of chromophoric networks.17-19 In this work, we investigate relevant quantum coherence effects by an in situ analysis of a success criterion for the initial step in photosynthesis, the energy transfer efficiency (ETE). We determine the percentage contribution of various physical processes to the efficiency. For closely packed multichromophoric networks, such as the FMO complex, strong intermolecular coupling and quantum coherence effects have to be taken into account. We thus want to avoid, even as a reference * E-mail:
[email protected];
[email protected]. † Harvard University. ‡ Massachusetts Institute of Technology, Cambridge, MA, 02139.
point, a semiclassical Fo¨rster approximation that leads to a classical hopping description of the exciton dynamics.20 This is in contrast to studies (e.g., in the area of quantum information) that compare the quantum dynamics to classical dynamics; for example, in the case of the comparison of a quantum walk to a classical random walk.21 The dynamics of an excitation in multichromophoric complexes can be described in terms of an environment-assisted quantum walk for the density matrix F(t):15
d F(t) ) MF(t) dt
(1)
The superoperator M comprises quantum coherent evolution driven by the excitonic Hamiltonian and decoherence effects due to coupling to the surrounding phonon environment. A highly idealized model with pure dephasing induced by Gaussian fluctuations18,22,23 already shows the assisting effect of the environment and its relation to static disorder in the Hamiltonian.24-26 Here, we include dephasing, relaxation, and spatially correlated bath fluctuations for a more realistic picture of the exciton dynamics. The main result of the present work is to quantify the role of the various physical processes involved in the energy transfer process in terms of their contribution to the ETE. Formally, we will partition the overall ETE, η, into a sum of terms
η)
∑ ηk
(2)
k
corresponding to a physical decomposition M ) ΣkM k. Each term ηk can be interpreted as a contribution to the overall efficiency originating from a particular process, M k. For example, we will split the superoperator into the major components describing the exciton dynamics: coherent evolution with the excitonic Hamiltonian, relaxation within the singleexciton manifold, and dephasing. See Figure 1 for a schematic representation. The ηk associated with the coherent part will
10.1021/jp901724d CCC: $40.75 2009 American Chemical Society Published on Web 06/24/2009
Quantum Coherence in Chromophoric Energy Transport
J. Phys. Chem. B, Vol. 113, No. 29, 2009 9943
Figure 1. Schematic representation of the exciton dynamics: the time evolution of the single-exciton density matrix is determined by a coherent part, relaxation, and dephasing. Additionally, the exciton can recombine or can be trapped by the reaction center.
then give an indication of the role of quantum evolution to the energy transfer efficiency and, hence, to the biological function within a particular chromophoric complex. We note that the exact partitioning of the ETE into a sum of terms, such as eq 2, is a nontrivial task: as will be shown below that the ETE essentially involves an exponential mapping of the complete superoperator M. A separation of the ETE into a product of terms would seem more natural but would not allow the interpretation of ηk as contributions. In the following sections, we briefly discuss the structure of the superoperator M and introduce two complementary measures of efficiency contributions: one is based on a Green’s function method, and the other is derived from energy transfer susceptibilities. We apply these two approaches to the study of the ETE in the FMO complex. We employ a standard Redfield model with the secular approximation that leads to a master equation in Lindblad form27 and time-independent superoperator. This model captures major decoherence effects, such as relaxation and dephasing. We also include spatial correlation of the fluctuations. The Markovian approximation neglects temporal correlations in the phonon bath that can be relevant in photosynthetic systems and will be treated in subsequent work. We believe that the present model and our methods can provide insight into the role of quantum coherence in photosynthetic energy transfer, a process that occurs in noisy, ambient temperature environments.
correlator can be simplified as 〈qm(t) qn(0)〉 ) Cmn〈q(t) q(0)〉. Cmn is a dimensionless, time-independent factor that takes into account the spatial correlations in the phonon bath. For spatially uncorrelated environments, it will simply be given by Cmn ) δmn. In this work, we will also take into account a phenomenological model for these correlations, as will be explained later. The time-dependent part of the correlator, 〈q(t) q(0)〉, is assumed to be the same for all sites.28 Additionally, there are two processes that lead to irreversible loss of the exciton.15,17-19 One is the excitation loss due to recombination of the electron-hole pair. The other mechanism describes the excitation transfer to the reaction center (acceptor) and subsequent trapping associated with the charge separation event. These effects are taken into account by the anti-Hermitian Hamiltonians: -iHrecomb ) -ipΓ∑mN|m〉〈m|, with Γ being the inverse lifetime of the exciton, and -iHtrap ) -ip∑mN κm|m〉〈m|, with κm being the trapping rates at site m. In summary, the dynamics of the reduced density matrix of the system can be described by the Lindblad master equation in the Born-Markov and secular approximations as27
Master Equation for Multichromophoric Systems
where {,} denotes the anticommutator. The right-hand side of eq 4 defines the superoperator, M. L is the Lindblad superoperator derived from the phonon bath coupling:
The transport dynamics of a single excitation is described by a master equation for the density matrix that includes coherent evolution, relaxation, and dephasing. Moreover, the exciton can recombine or be trapped. The Hamiltonian for an interacting N-chromophoric system in the presence of a single excitation can be written as13
HS )
N
N
m)1
n
