A novel construction of complex-valued Gaussian processes with arbitrary spectral densities and its application to excitation energy transfer.

The recent experimental discoveries about excitation energy transfer (EET) in light harvesting antenna (LHA) attract a lot of interest. As an open non-equilibrium quantum system, the EET demands more rigorous theoretical framework to understand the interaction between system and environment and therein the evolution of reduced density matrix. A phonon is often used to model the fluctuating environment and convolutes the reduced quantum system temporarily. In this paper, we propose a novel way to construct complex-valued Gaussian processes to describe thermal quantum phonon bath exactly by converting the convolution of influence functional into the time correlation of complex Gaussian random field. Based on the construction, we propose a rigorous and efficient computational method, the covariance decomposition and conditional propagation scheme, to simulate the temporarily entangled reduced system. The new method allows us to study the non-Markovian effect without perturbation under the influence of different spectral densities of the linear system-phonon coupling coefficients. Its application in the study of EET in the Fenna-Matthews-Olson model Hamiltonian under four different spectral densities is discussed. Since the scaling of our algorithm is linear due to its Monte Carlo nature, the future application of the method for large LHA systems is attractive. In addition, this method can be used to study the effect of correlated initial condition on the reduced dynamics in the future.

Optimal fold symmetry of LH2 rings on a photosynthetic membrane.

An intriguing observation of photosynthetic light-harvesting systems is the N-fold symmetry of light-harvesting complex 2 (LH2) of purple bacteria. We calculate the optimal rotational configuration of N-fold rings on a hexagonal lattice and establish two related mechanisms for the promotion of maximum excitation energy transfer (EET). (i) For certain fold numbers, there exist optimal basis cells with rotational symmetry, extendable to the entire lattice for the global optimization of the EET network. (ii) The type of basis cell can reduce or remove the frustration of EET rates across the photosynthetic network. We find that the existence of a basis cell and its type are directly related to the number of matching points S between the fold symmetry and the hexagonal lattice. The two complementary mechanisms provide selection criteria for the fold number and identify groups of consecutive numbers. Remarkably, one such group consists of the naturally occurring 8-, 9-, and 10-fold rings. By considering the inter-ring distance and EET rate, we demonstrate that this group can achieve minimal rotational sensitivity in addition to an optimal packing density, achieving robust and efficient EET. This corroborates our findings i and ii and, through their direct relation to S, suggests the design principle of matching the internal symmetry with the lattice order.

Molecular binoculars: how to spatially resolve environmental fluctuations by following two or more single-molecule spectral trails at a time.

We propose a novel type of spectral diffusion experiment that enables one to decouple spatial characteristics of the environmental fluctuations, such as their concentration, from the interaction with the chromophore. Traditional hole broadening experiments do not allow for such decoupling in the common case when the chromophore-environment interaction is scale invariant. Here we propose to simultaneously follow the spectral trails of a small number of nearby chromophores--two or more--which thereby sense a highly overlapping set of the fluctuations. To this end, we estimate the combined probability distribution for the frequencies of a set of chromophores contained within the same sample. The present setup introduces a new length scale, i.e., the interchromophore distance, which breaks the aforementioned scale invariance and enables one to determine independently the concentration of the environmental fluctuations and their coupling to the chromophores, by monitoring the time after which spectral diffusion of distinct chromophores becomes uncorrelated. We illustrate these results with structural excitations in low temperature glasses.

Thermally-limited exciton delocalization in superradiant molecular aggregates.

We present two-dimensional Fourier transform optical spectroscopy measurements of two types of molecular J-aggregate thin films and show that temperature-dependent dynamical effects govern exciton delocalization at all temperatures, even in the presence of significant inhomogeneity. Our results indicate that in the tested molecular aggregates, even when the static structure disorder dominates exciton dephasing dynamics, the extent of exciton delocalization may be limited by dynamical fluctuations, mainly exciton-phonon coupling. Thus inhomogeneous dephasing may mediate the exciton coherence time whereas dynamical fluctuations mediate the exciton coherence length.

Generic mechanism of optimal energy transfer efficiency: a scaling theory of the mean first-passage time in exciton systems.

An asymptotic scaling theory is presented using the conceptual basis of trapping-free subspace (i.e., orthogonal subspace) to establish the generic mechanism of optimal efficiency of excitation energy transfer in light-harvesting systems. A quantum state orthogonal to the trap will exhibit noise-assisted transfer, clarifying the significance of initial preparation. For such an initial state, the efficiency is enhanced in the weak damping limit (⟨t⟩ ∼ 1/Γ), and suppressed in the strong damping limit (⟨t⟩ ∼ Γ), analogous to Kramers turnover in classical rate theory. An interpolating expression ⟨t⟩ = A/Γ + B + CΓ quantitatively describes the trapping time over the entire range of the dissipation strength, and predicts the optimal efficiency at Γ(opt) ∼ J for homogenous systems. In the presence of static disorder, the scaling law of transfer time with respect to dephasing rate changes from linear to square root, suggesting a weaker dependence on the environment. The prediction of the scaling theory is verified in a symmetric dendrimer system by numerically exact quantum calculations. Though formulated in the context of excitation energy transfer, the analysis and conclusions apply in general to open quantum processes, including electron transfer, fluorescence emission, and heat conduction.

Efficient energy transfer in light-harvesting systems: quantum-classical comparison, flux network, and robustness analysis.

Following the calculation of optimal energy transfer in thermal environment in our first paper [J. L. Wu, F. Liu, Y. Shen, J. S. Cao, and R. J. Silbey, New J. Phys. 12, 105012 (2010)], full quantum dynamics and leading-order "classical" hopping kinetics are compared in the seven-site Fenna-Matthews-Olson (FMO) protein complex. The difference between these two dynamic descriptions is due to higher-order quantum corrections. Two thermal bath models, classical white noise (the Haken-Strobl-Reineker (HSR) model) and quantum Debye model, are considered. In the seven-site FMO model, we observe that higher-order corrections lead to negligible changes in the trapping time or in energy transfer efficiency around the optimal and physiological conditions (2% in the HSR model and 0.1% in the quantum Debye model for the initial site at BChl 1). However, using the concept of integrated flux, we can identify significant differences in branching probabilities of the energy transfer network between hopping kinetics and quantum dynamics (26% in the HSR model and 32% in the quantum Debye model for the initial site at BChl 1). This observation indicates that the quantum coherence can significantly change the distribution of energy transfer pathways in the flux network with the efficiency nearly the same. The quantum-classical comparison of the average trapping time with the removal of the bottleneck site, BChl 4, demonstrates the robustness of the efficient energy transfer by the mechanism of multi-site quantum coherence. To reconcile with the latest eight-site FMO model which is also investigated in the third paper [J. Moix, J. L. Wu, P. F. Huo, D. F. Coker, and J. S. Cao, J. Phys. Chem. Lett. 2, 3045 (2011)], the quantum-classical comparison with the flux network analysis is summarized in Appendix C. The eight-site FMO model yields similar trapping time and network structure as the seven-site FMO model but leads to a more disperse distribution of energy transfer pathways.

Phase transition in the Jarzynski estimator of free energy differences.

The transition between a regime in which thermodynamic relations apply only to ensembles of small systems coupled to a large environment and a regime in which they can be used to characterize individual macroscopic systems is analyzed in terms of the change in behavior of the Jarzynski estimator of equilibrium free energy differences from nonequilibrium work measurements. Given a fixed number of measurements, the Jarzynski estimator is unbiased for sufficiently small systems. In these systems the directionality of time is poorly defined and the configurations that dominate the empirical average, but which are in fact typical of the reverse process, are sufficiently well sampled. As the system size increases the arrow of time becomes better defined. The dominant atypical fluctuations become rare and eventually cannot be sampled with the limited resources that are available. Asymptotically, only typical work values are measured. The Jarzynski estimator becomes maximally biased and approaches the exponential of minus the average work, which is the result that is expected from standard macroscopic thermodynamics. In the proper scaling limit, this regime change has been recently described in terms of a phase transition in variants of the random energy model. In this paper this correspondence is further demonstrated in two examples of physical interest: the sudden compression of an ideal gas and adiabatic quasistatic volume changes in a dilute real gas.

Theoretical description of quantum effects in multi-chromophoric aggregates.

Recent experiments on light-harvesting complexes have shown clear indication of coherent transport of excitations in these aggregates. We discuss the theoretical models that have been used to study energy transfer in molecular aggregates, beginning with the early models of Förster and Davydov and ending with the theoretical models of the present day.

Electronic coherence lineshapes reveal hidden excitonic correlations in photosynthetic light harvesting.

The effective absorption cross-section of a molecule (acceptor) can be greatly increased by associating it with a cluster of molecules that absorb light and transfer the excitation energy to the acceptor molecule. The basic mechanism of such light harvesting by Förster resonance energy transfer (FRET) is well established, but recent experiments have revealed a new feature whereby excitation is coherently shared among donor and acceptor molecules during FRET. In the present study, two-dimensional electronic spectroscopy was used to examine energy transfer at ambient temperature in a naturally occurring light-harvesting protein (PE545 of the marine cryptophyte alga Rhodomonas sp. strain CS24). Quantum beating was observed across a range of excitation frequencies. The shapes of those features in the two-dimensional spectra were examined. Through simulations, we show that two-dimensional electronic spectroscopy provides a probe of the adiabaticity of the free energy landscape underlying light harvesting.

Correlated intermolecular coupling fluctuations in photosynthetic complexes.

The functioning and efficiency of natural photosynthetic complexes is strongly influenced by their embedding in a noisy protein environment, which can even serve to enhance the transport efficiency. Interactions with the environment induce fluctuations of the transition energies and couplings between the chlorophyll molecules, and due to the fact that different fluctuations will partially be caused by the same environmental factors, correlations between the various fluctuations will occur. We argue that fluctuations of the couplings should, in general, not be neglected, as these have a considerable impact on population transfer rates, decoherence rates, and the efficiency of photosynthetic complexes. Furthermore, while correlations between transition energy fluctuations have been studied, we provide the first quantitative study of the effect of correlations between coupling fluctuations and transition energy fluctuations, and of correlations between the various coupling fluctuations. It is shown that these additional correlations typically lead to changes in interchromophore transfer rates and population oscillations and can lead to a limited enhancement of the light harvesting efficiency.

Reaction Event Counting Statistics of Biopolymer Reaction Systems with Dynamic Heterogeneity.

We investigate the reaction event counting statistics (RECS) of an elementary biopolymer reaction in which the rate coefficient is dependent on states of the biopolymer and the surrounding environment and discover a universal kinetic phase transition in the RECS of the reaction system with dynamic heterogeneity. From an exact analysis for a general model of elementary biopolymer reactions, we find that the variance in the number of reaction events is dependent on the square of the mean number of the reaction events when the size of measurement time is small on the relaxation time scale of rate coefficient fluctuations, which does not conform to renewal statistics. On the other hand, when the size of the measurement time interval is much greater than the relaxation time of rate coefficient fluctuations, the variance becomes linearly proportional to the mean reaction number in accordance with renewal statistics. Gillespie's stochastic simulation method is generalized for the reaction system with a rate coefficient fluctuation. The simulation results confirm the correctness of the analytic results for the time dependent mean and variance of the reaction event number distribution. On the basis of the obtained results, we propose a method of quantitative analysis for the reaction event counting statistics of reaction systems with rate coefficient fluctuations, which enables one to extract information about the magnitude and the relaxation times of the fluctuating reaction rate coefficient, without a bias that can be introduced by assuming a particular kinetic model of conformational dynamics and the conformation dependent reactivity. An exact relationship is established between a higher moment of the reaction event number distribution and the multitime correlation of the reaction rate for the reaction system with a nonequilibrium initial state distribution as well as for the system with the equilibrium initial state distribution.

Is there elliptic distortion in the light harvesting complex 2 of purple bacteria?

Single molecule spectroscopy (SMS) revealed an unusually large Gap between two major exciton peaks of the B850 unit of light harvesting complex 2 (LH2), which could be explained assuming elliptic distortion or k = 2 symmetry modulation in the site excitation energy. On the basis of extensive simulation of the SMS data and ensemble line shape, we found that uniform modulation of k = 2 symmetry cannot explain the dependence of intensity ratios on the Gap of the two major peaks, which are available from SMS, nor the ensemble line shape. Alternative models of disorder with k = 1 and k = 2 symmetry correlation are shown to reproduce these data reasonably well and can even explain the Gap distribution when it is assumed that the lower major peak in the SMS line shape is an intensity weighted average of k = 1- and k = 0 states.

Quantitative interpretation of the randomness in single enzyme turnover times.

Fluctuating turnover times of a single enzyme become observable with the advent of modern cutting-edge, single enzyme experimental techniques. Although the conventional chemical kinetics and its modern generalizations could provide a good quantitative description for the mean of the enzymatic turnover times, to our knowledge there has not yet been a successful quantitative interpretation for the variance or the randomness of the enzymatic turnover times. In this review, we briefly review several theories in this field, and compare predictions of these theories to the randomness parameter data reported for ß-galactosidase enzyme. We find the recently proposed kinetics for renewal reaction processes could provide an excellent quantitative interpretation of the randomness parameter data. From the analysis of the randomness parameter data of the single enzyme reaction, one can extract quantitative information about the mean lifetime of enzyme-substrate complex; the success or the failure probability of the catalytic reaction per each formation of ES complex; and the non-Poisson character of the reaction dynamics of the ES complex (which is beyond reach of the long-standing paradigm of the conventional chemical kinetics).

Excitation energy transfer in a non-markovian dynamical disordered environment: localization, narrowing, and transfer efficiency.

The non-markovian effect of a fluctuating environment plays an important role in electronic excitation transfer in organic disordered media, such as light-harvesting systems and conjugated polymers. Stochastic Liouville equations (SLE) are used to study the interaction between excitons and the environment. We model the non-markovian environment phenomenologically with a dichotomic process. An exact approach to solve the SLE based on Shapiro and Loginov's differentiation formulas allows us to rigorously study the effect of the non-markovian environment on excitation energy transfer, such as coherence conservation and its implication for transfer efficiency. This simple SLE model goes beyond the perturbative second-order master equation valid for both the weak coupling and short time correlation conditions. In addition, we discuss why our non-markovian model is a good approximation to the SLE model driven by the stationary Gauss-Markov process (Ornstein-Uhlenbeck process) over a broad range of fluctuation strengths and correlation times. Numerical results based on our SLE model for dimeric aggregates and the Fenna-Matthews-Olson (FMO) complex reveal the important interplay of intermolecular coupling, correlation time, and fluctuation strength, and their effects on the exciton relaxation process due to the environmental phonon. The results also uncover the connection between localization and motional narrowing, and the efficiency of electronic excitation transfer, demonstrating that the non-markovian environment is critical for chromophore aggregates to achieve an optimal transfer rate in a noisy environment and to contribute to the robustness of the FMO excitation energy transfer network.

Unified treatment of coherent and incoherent electronic energy transfer dynamics using classical electrodynamics.

Recent experiments on resonance energy transfer (RET) in photosynthetic systems have found evidence of quantum coherence between the donor and the acceptor. Under these conditions, Förster's theory of RET is no longer applicable and no theory of coherent RET advanced to date rivals the intuitive simplicity of Förster's theory. Here, we develop a framework for understanding RET that is based on classical electrodynamics but still captures the essence of the quantum coherence between the molecules. Our theory requires only a knowledge of the complex polarizabilities of the two molecules participating in the transfer as well as the distance between them. We compare our results to quantum mechanical calculations and show that the results agree quantitatively.

Effect of correlation of local fluctuations on exciton coherence.

Recent experimental studies have shown both oscillations of exciton populations and long lasting coherence in multichromophoric systems such as photosynthetic light harvesting systems and conjugated polymers. It has been suggested that this quantum effect is due to correlations of the fluctuations of site energies among the closely packed chromophores in the protein environment. In addition to these, there is the strong possibility of correlations between site energies and transfer matrix elements. In order to understand the role of such correlations we generalize the Haken-Strobl-Reineker (HSR) model to include the energetic correlations and the site diagonal-off-diagonal correlations in a systematic way. The extended HSR model in the exciton basis is also constructed and allows us to study the dynamics of the exciton populations and coherences. With the extended model, we can provide insight into how these correlations affect the evolution of the populations and coherences of excitons by comparing to the original HSR model with uncorrelated fluctuating environments.

Optimization of exciton trapping in energy transfer processes.

In this paper, we establish optimal conditions for maximal energy transfer efficiency using solutions for multilevel systems and interpret these analytical solutions with more intuitive kinetic networks resulting from a systematic mapping procedure. The mapping procedure defines an effective hopping rate as the leading order picture and nonlocal kinetic couplings as the quantum correction, hence leading to a rigorous separation of thermal hopping and coherent transfer useful for visualizing pathway connectivity and interference in quantum networks. As a result of these calculations, the dissipative effects of the surrounding environments can be optimized to yield the maximal efficiency, and modulation of the efficiency can be achieved using the cumulative quantum phase along any closed loops. The optimal coupling of the system and its environments is interpreted with the generic mechanisms: (i) balancing localized trapping and delocalized coherence, (ii) reducing the effective detuning via homogeneous line-broadening, (iii) suppressing the destructive interference in nonlinear network configurations, and (iv) controlling phase modulation in closed loop configurations. Though these results are obtained for simple model systems, the physics thus derived provides insights into the working of light harvesting systems, and the approaches thus developed apply to large-scale computation.

The work-Hamiltonian connection and the usefulness of the Jarzynski equality for free energy calculations.

The connection between work and changes in the Hamiltonian for a system with a time-dependent Hamiltonian has recently been called into question, casting doubt on the usefulness of the Jarzynski equality for calculating free energy changes. In this paper, we discuss the relationship between two possible definitions of free energy and show how some recent disagreements regarding the applicability of the Jarzynski equality are the result of different authors' using different definitions of free energy. Finally, in light of the recently raised doubts, we explicitly demonstrate that it is indeed possible to obtain physically relevant free energy profiles from molecular pulling experiments by using the Jarzynski equality and the results of Hummer and Szabo.

Beyond Förster resonance energy transfer in biological and nanoscale systems.

After photoexcitation, energy absorbed by a molecule can be transferred efficiently over a distance of up to several tens of angstroms to another molecule by the process of resonance energy transfer, RET (also commonly known as electronic energy transfer, EET). Examples of where RET is observed include natural and artificial antennae for the capture and energy conversion of light, amplification of fluorescence-based sensors, optimization of organic light-emitting diodes, and the measurement of structure in biological systems (FRET). Forster theory has proven to be very successful at estimating the rate of RET in many donor-acceptor systems, but it has also been of interest to discover when this theory does not work. By identifying these cases, researchers have been able to obtain, sometimes surprising, insights into excited-state dynamics in complex systems. In this article, we consider various ways that electronic energy transfer is promoted by mechanisms beyond those explicitly considered in Forster RET theory. First, we recount the important situations when the electronic coupling is not accurately calculated by the dipole-dipole approximation. Second, we examine the related problem of how to describe solvent screening when the dipole approximation fails. Third, there are situations where we need to be careful about the separability of electronic coupling and spectral overlap factors. For example, when the donors and/or acceptors are molecular aggregates rather than individual molecules, then RET occurs between molecular exciton states and we must invoke generalized Forster theory (GFT). In even more complicated cases, involving the intermediate regime of electronic energy transfer, we should consider carefully nonequilibrium processes and coherences and how bath modes can be shared. Lastly, we discuss how information is obscured by various forms of energetic disorder in ensemble measurements and we outline how single molecule experiments continue to be important in these instances.