Perspective: Detecting and measuring exciton delocalization in photosynthetic light harvesting Gregory D. Scholes and Cathal Smyth Citation: The Journal of Chemical Physics 140, 110901 (2014); doi: 10.1063/1.4869329 View online: http://dx.doi.org/10.1063/1.4869329 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of strong electron correlation on the efficiency of photosynthetic light harvesting J. Chem. Phys. 137, 074117 (2012); 10.1063/1.4746244 Excitonic energy transfer in light-harvesting complexes in purple bacteria J. Chem. Phys. 136, 245104 (2012); 10.1063/1.4729786 Iterative linearized density matrix propagation for modeling coherent excitation energy transfer in photosynthetic light harvesting J. Chem. Phys. 133, 184108 (2010); 10.1063/1.3498901 Quantum oscillatory exciton migration in photosynthetic reaction centers J. Chem. Phys. 133, 064510 (2010); 10.1063/1.3458824 Effects of excited state mixing on transient absorption spectra in dimers: Application to photosynthetic lightharvesting complex II J. Chem. Phys. 111, 3121 (1999); 10.1063/1.479593
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THE JOURNAL OF CHEMICAL PHYSICS 140, 110901 (2014)
Perspective: Detecting and measuring exciton delocalization in photosynthetic light harvesting Gregory D. Scholesa) and Cathal Smyth Department of Chemistry, University of Toronto, 80 St. George St., Toronto, Ontario M5S 3H6, Canada
(Received 12 February 2014; accepted 11 March 2014; published online 20 March 2014) Photosynthetic units perform energy transfer remarkably well under a diverse range of demanding conditions. However, the mechanism of energy transfer, from excitation to conversion, is still not fully understood. Of particular interest is the possible role that coherence plays in this process. In this perspective, we overview photosynthetic light harvesting and discuss consequences of excitons for energy transfer and how delocalization can be assessed. We focus on challenges such as decoherence and nuclear-coordinate dependent delocalization. These approaches complement conventional spectroscopy and delocalization measurement techniques. New broadband transient absorption data may help uncover the difference between electronic and vibrational coherences present in twodimensional electronic spectroscopy data. We describe how multipartite entanglement from quantum information theory allows us to formulate measures that elucidate the delocalization length of excitation and the details of that delocalization even from highly averaged information such as the density matrix. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4869329] I. INTRODUCTION
Photosynthetic solar energy conversion occurs on an immense scale across the earth, influencing our biosphere from climate to oceanic food webs.1–3 Energy from sunlight is absorbed by chromophores, like chlorophyll, embedded in proteins comprising the photosynthetic unit. Emerson and Arnold4 defined the photosynthetic unit to explain the abundance of chlorophyll molecules that are employed to produce a single molecule of oxygen. Later it was shown by Duysens that while the chain of reactions involved in photosynthesis starts at a reaction center, hundreds of chromophores are used to harvest sunlight and “concentrate” (in the steady-state) electronic excitation at the reaction centers.5–7 Thus these light-harvesting complexes compensate for the mismatch between solar irradiance and the optimal rate of reaction center operation.8–12 The photosynthetic solar energy conversion process is incredible, not for its performance as gauged by engineering metrics,13, 14 but because such diverse examples abound. Fronds in kelp forests (a brown algae) can grow 15 cm in a single day.15 Crustose coralline red algae can dwell in conditions on the threshold of net primary production. They have been documented to grow on the sea floor, 20 m under water with greater than 1 m of ice cover.16 Unusual purple bacteria have been discovered in the South Andros Black Hole, Bahamas, where they dominate their environment by shortcircuiting photosynthetic light harvesting complexes to generate heat and warm the surrounding water.17 All of these examples are fascinating case studies, particularly in chemical physics, with experiments and theories revealing the mechanisms involved in the ultrafast energy transfer processes of light harvesting.10, 18–25 a) [email protected]
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A big question has been how to discover the mechanisms promoting electronic energy transfer, since there are various contributions to electronic coupling as well as regimes of dynamics that come into play according to how chromophores are arranged.21, 26–30 Femtosecond spectroscopy typically times population changes by probing their spectral signatures. This determines the time scale of ultrafast processes, but not the underlying mechanism. For example, researchers have long wondered whether the Fermi Golden Rule basis of the Förster energy transfer model is a good approximation for photosynthetic light harvesting, or whether a more sophisticated theory is needed.18, 20, 31–37 Can an experiment directly address this question? That is what was especially exciting about the report in 2007 from the Fleming group;38 it was shown that two-dimensional electronic spectroscopy (2DES) could directly address the question of how rapidly excitonic coherence is lost. Subsequent work has concluded the coherence related to excitonic delocalization does indeed affect energy transfer dynamics.29, 39–41 The goals of this paper are to discuss: (a) What are the demands for photosynthetic light harvesting and therefore possible role of coherence in the biological context? (b) How can we quantify excitonic delocalization in complex systems? (c) Are there different ways to think about the coherent oscillations seen in 2D spectra? Are new experimental measures required?
II. DISCUSSION A. The layout and function of light harvesting systems
The layout of light-absorbing molecules in the photosynthetic unit, Fig. 1, may appear like sprawling labyrinths of chromophores with little order. Nevertheless, short-lived 140, 110901-1
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FIG. 1. Model of the photosystem II (PSII) supercomplex C2S2M2 from higher plants. (a) The protein organization showing LHCII trimers (red) on the periphery, proteins in the core (magenta). CP24 is colored yellow; CP29 orange; CP26, cyan. (B) A model showing only the organization of chlorophylls (Chls) bound in these proteins. Chls a, green; Chls b, blue. The special pair Chls are colored red. Reprinted with permission from Croce and van Amerongen, J. Photochem. Photobiol. B. Biol. 104, 142–153 (2011). Copyright 2011 Elsevier.
electronic excitations (lifetimes of 1–5 ns) traverse these molecularly vast landscapes with high efficiency to sensitize reaction centers and initiate photosynthetic energy transduction. The typical photosynthetic unit is an excellent example of finely tuned adaption to limiting conditions. Chromophores are organized so there is optimal harvesting of light across a relatively broad spectral range. Optimal in this case means within possible constraints of resources available—light, nutrients, etc. The photosynthetic unit typically comprises about 200– 400 light harvesting chromophores associated with each reaction center, depending on organism and growth conditions.42 The upper limit to the size of a photosynthetic unit is dictated by interplay of two primary factors: the rate of excitation transfer and the excited state lifetime.14 The minimum size of a photosynthetic unit is controlled by a delicate balance of energy absorption, conversion, and dissipation of excess excitations. The physical size and layout of a supercomplex of the photosynthetic unit in the vicinity of photosystem II in higher plants is shown in Fig. 1. Experiments have estimated there are about 200 chlorophylls per reaction center under normal growth conditions.43 In purple bacteria, a combination of siteenergy tuning of chromophores by the protein and excitonic effects creates a natural energy gradient directing excitations towards the reaction center (specifically to the special pair). Imaging of photosynthetic membranes of purple bacteria by atomic force microscopy reveals the arrangement of light harvesting complexes (LH2) and reaction centers clearly,44–46 while the ratio of LH2 to reaction centers can also be measured spectroscopically.47 It is found that 2–7 LH2 complexes are associated with each reaction center, meaning there are ∼100–200 bacteriochlorophyll molecules per reaction center. Despite this low ratio of antenna molecules to reaction centers, Vos et al. have shown that excitation migrates over about 1000 bacteriochlorophylls48, 49 —in other words, about 25 photosynthetic units are connected.
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The size of the photosynthetic unit can change depending on whether or not growth is light limited. The essential principle is that the reaction centers should be photo-excited at a rate balanced with the rate of linear electron flow from H2 O to NADP+ . The right balance inhibits back reactions, and thus losses, downstream from the photo-initiation processes. A large photosynthetic unit helps maintain the rate of photosynthetic production under low light conditions because it provides the reaction center with an increased absorption cross section. Light can be limiting, for example in microbial mats,50–52 turbid water,53 competition in aquatic communities,54 or at great depths in the sea.55 A further interesting complexity is that not only the size, but also the organization of light harvesting complexes with reaction centers has been documented to change depending on light conditions.56 It is generally thought that excitation energy moves more or less by a random walk, historically based on early models for energy hopping7, 57 and Pearlstein’s hypothesis in the 1960s that coherent transfer should not dramatically change the limits of excitation diffusion length.32 However, such conclusions are subject to revision, especially as we develop this field of research in terms of measurement, theory, and experimental probes of the mechanisms of energy transfer. Recently, research has suggested that a balance between coherence (quantum effects) and decoherence (loss of quantum effects) is optimal for excitation transport.40, 58, 59 In that case, the random walk model must be modified60 and more sophisticated theories of energy transfer are required.21, 61 Likewise any quantum measurements made must be appropriate for open systems; delocalization measures such as the Inverse Participation Ratio are not appropriate when decoherence is taken into account. In terms of scale, we believe it is useful to delineate excitation energy transfer in light harvesting complexes from excitation migration and trapping in the photosynthetic unit. We suggest that the designation “excitation energy transport” is preferably reserved for the excitation migration throughout the photosynthetic unit, while the rapid excitation energy equilibration and flow in an isolated complex is excitation energy transfer. Calculations at present have mainly focused on energy transfer in isolated light harvesting complexes because there exists the opportunity to examine deeply energy transfer mechanisms. The consequences of more sophisticated energy transfer theories and the deviations they will predict from random walk models on longer length scales, for example, for photosynthetic units (>100 chromophores) are unknown. As we discussed previously,14 the average effect is likely difficult to discern, but mechanisms at play in tails of the distributions of trapping times will be interesting. We can summarize this viewpoint by stating that an outstanding challenge is to connect details of energy transfer at the isolated light harvesting protein level to the collective function of many proteins.62
B. Exciton delocalization
At the foundation of the most interesting coherence effects in light harvesting systems are molecular excitons. Molecular exciton states (which is what we mean by “exci-
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ton” in this context) are electronic excited states where the wavefunction is shared by two or more molecules. The electron density remains localized on the molecules individually, but the transition density (see Ref. 63) for the exciton is a well-defined superposition of molecule-localized transition densities. In the kinds of exciton coherence that has interested researchers recently, this delocalization evolves in time. From a definition point of view, that means we cannot find a basis rotation among localized and delocalized representations that consistently diagonalizes the density matrix as the dynamics evolve after photo-excitation. Thus we find off-diagonal density matrix elements that signal quantum mechanical coherence. Exciton delocalization, if strong, can be indicated by spectral features that are perturbed, shifted, or split.64 Scaling of transition strength or nonlinear susceptibility is useful when comparative measurements can be made.65, 66 Similarly, changes in the radiative rate (“superradiance”) are quite sensitive to delocalization.67–69 Coherent superpositions of states that contribute oscillating cross-peaks to two-dimensional electronic spectroscopy (2DES) data have recently been proposed to reveal delocalization and coherences—and particularly how they dephase.20, 21, 27, 38, 70–83 Another probe of exciton delocalization can be excited state absorption in pump-probe and 2DES experiments.36, 84, 85 Two-exciton states, lying at approximately twice the exciton energy, necessarily accompany exciton states. While exciton states are defined in the basis of permutations of excitation among the interacting molecules, two-exciton states are defined as permutations of two excitations.86, 87 Excited state absorption reveals the exciton to two-exciton electronic transition(s). From a physical point of view, reasons behind changes in delocalization include exciton localization—the interplay of the Stokes shift in the site basis and exciton delocalization with a lesser nuclear reorganization88–90 —or an explicit dependence of energy resonance on various nuclear coordinates (vide infra). It is clear that experiments that can follow or characterize delocalization, with femtosecond resolution, after photoexcitation will give insights into some interesting photophysics. Perhaps such experiments can even reveal how exciton wavepackets evolve in time, although achieving state or process tomography for disordered condensed phase systems will be challenging.91–93 Another approach is to devise and measure metrics of exciton delocalization. The 2007 report by Fleming and co-workers ignited research about quantum coherence in light harvesting complexes.38 The foundation for this new experimental insight was a body of literature examining exciton delocalization in photosynthesis. A European Science Foundation workshop (organized by Leonas Valkunas and Rienk van Grondelle) held in Birstonas, Lithuania in 1996 highlighted the excitement about coherent delocalization of excitons. The book of abstracts notes that “Major questions concern. . . the time over which the excitation must be considered as coherent. . . .” The topic of discussion was largely Cogdell and coworkers’ report94 of an atomic resolution crystal structure for the peripheral light-harvesting complex LH2 from the purple anoxygenic bacterium Rhodopseudomonas (Rps.) acidophila strain 10050. Papers associated with this workshop highlight
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studies of delocalization in LH2 and the measurement of exciton delocalization. They are collected in volume 101, issue 37 (1997) of the Journal of Physical Chemistry B. At the time the Inverse Participation Ratio (IPR), defined by Thouless95 was frequently employed as a measure of delocalization.85, 96, 97 This measure, also known as a second order statistical moment, looks at the variance of probabilities within a wavefunction ψ = a1 ϕ 1 + a2 ϕ 2 + . . . an ϕ n delocalized over n sites: n |aj |4 . (1) IPR = j =1
The IPR can only be used, however, in calculations based on wavefunctions. For example, when exciton wavefunctions are calculated as a function of static disorder by Monte Carlo averaging.98 The crucial distinction here is that each wavefunction in the ensemble is calculated and the IPR is calculated for each wavefunction and statistically averaged. That is different than first averaging the wavefunctions themselves, by constructing a density matrix, then calculating the delocalization length. Consider an experiment that measures the evolution of a wavefunction:99 after many iterations, a clear picture of the population dynamics is formed; however, the coherences between these populations get washed out due to the phase differences in averaging the slightly different states. Representing these populations and coherences in a density matrix ρ, one can see how much the populations differ from their coherences using a measure called the purity. The purity is defined as the trace of the square of a density matrix, Purity ≡ Tr(ρ 2 ).
(2)
A pure state has a purity value of 1 and represents a fully coherent state that can also be written as a wavefunction. A mixed state represents a statistical mixture of different coherent states and will have a smaller value as low as 1/n, in a system of size n, if the state is fully mixed. In information terms, a decrease in purity leads to an increase in entropy. Therefore the IPR, as strictly a measure of state populations, is unsuitable in its application to mixed states. As such, it is unable to reveal the evolution of coherence within these light harvesting complexes when we are concerned with including the effects of homogeneous line broadening. In parallel to research on delocalization in LH2, the field of quantum computation and quantum information was being established, and with it a whole new field dedicated to measuring entanglement in mixed states. The emergence of this field has been a great asset, and the suitability of these measures to light harvesting complexes has recently been demonstrated. Measures of bipartite entanglement such as the tangle,100 the negativity,101 and the relative entropy102 have all been employed to chart the evolution of coherence in light harvesting systems.41, 103, 104 Interestingly, these measures, the IPR, the tangle, and the relative entropy, can be related using the purity measure.105 For example, the total two-body entanglement, or bipartite entanglement, E2 (ρ), in a system with one excitation can be written as the difference between the purity and the IPR, E2 (ρ) = Tr(ρ)2 − M2 (ρ),
(3)
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where M2 (ρ) = nj=1 (ρjj )2 is the statistical moment of order two of the system populations, which is exactly equivalent to the IPR. This result could prove essential in the experimental measurement of coherence in light harvest complexes; by gathering time dependent population statistics and employing a judicious assessment of entropy within a system, it could be possible to gain a strong understanding of the time dependent coherence within a system, without having to do full state tomography. For example, one can fully measure the purity of a state with a number of measurements that scale linearly with system size.106 Though such entanglement measures provide excellent information on the coherence within these systems, they only do so using two-body correlations. Recently, measures of many-body entanglement for pure states, using higher order statistical moments than the IPR, were applied to model light harvesting complexes.107 Adapting these measures to mixed states, which is significantly harder, finally delivers the true “delocalization length” in the systems: a precise number of sites that share quantum correlations during every step of energy transfer, despite decoherence. A computationally intense numerical method derived by Levi et al.108 would provide such a measure. We have recently demonstrated that it is also possible to measure multipartite entanglement in mixed states analytically, once more by exploiting the relationship between purity, entanglement, and the statistical moments.109 Measures that are a function of the statistical moments107 can be applied to the mixed state in question, as well as to a reference state σ with the same level of purity. In order to measure the k-partite entanglement in a mixed state one can use the equation (4) Ek (ρ) = k τk (ρ) − τk (σ ), where τ k is a function of the statistical moments up to order k as described in Ref. 109. The challenge is in deriving the reference state σ , which should have the maximum level of (k–1)-partite entanglement for that level of purity. This measurement returns values of statistical correlation that are not possible for states with lower orders of entanglement at that level of entropy. Effectively, one can remove the noise from their measurement in order to process the signal. Much like the bipartite measure, knowledge of population statistics and purity within a system is all that is needed to fully measure the delocalization length, at any level of decoherence. These measures enable us to detect how entanglement among multiple excitons diminishes during dynamics evolution, for example, four-site entanglement decays quickly, followed by three-site entanglement, and so on.109 We can thereby map out in considerable detail how excitation is delocalized in a multichromophore aggregate. C. Intramolecular vibrations, delocalization, and coherence
Intramolecular vibrations play a marked role in molecular spectroscopy. High frequency modes (compared to kT/h) are clearly evident as vibronic progressions in absorption and emission spectra. Torsional modes that have a frequency
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change from ground to excited state can broaden absorption spectra substantially.110 Förster theory for electronic energy transfer employs a separation between electronic and nuclear factors and thereby naturally includes vibronic details of molecular spectroscopy in the Förster spectral overlap.26 That is of profound quantitative importance because it is often vibronic overlap of the red tail of the donor fluorescence spectrum with the vibronic progression of the acceptor absorption spectrum that enables energy conservation and therefore decides the rate of energy transfer.111 Recent work has highlighted the importance of understanding the role of intramolecular vibrations in energy transfer (exciton relaxation) in the intermediate to strong electronic coupling regimes, especially for interpretation of coherent oscillations observed in two-dimensional electronic spectroscopy, see Refs. 76 and 112–116, and references cited therein. In a striking recent report, O’Reilly and Olaya-Castro show that it is the intramolecular nuclear coherences in combination with electronic coherences that ensure the coherences are definitively quantum in nature.117 A common theoretical framework for going beyond Förster theory is to diagonalize the electronic Hamiltonian, then add coupling to low frequency bath nuclear degrees of freedom perturbatively (e.g., Redfield theory) or nonperturbatively. Despite the recent leaps in theoretical models for energy transfer, it is not obvious that roles of high frequency intramolecular vibrations—that are so important in the Förster spectral overlap—are appropriately captured in present theories for intermediate or strong coupling. We would like to stimulate discussion about the limitations of the electronic exciton basis, or the polaron basis, versus a vibronic basis. Leutwyler and co-workers have studied model molecular dimers in the gas phase.118–120 They measured high resolution spectra, free from an “environment” that would cause line broadening and contribute to exciton relaxation. Therefore these experiments reveal, in essence, a zeroth-order description of the vibronic exciton levels that may be a good basis for theories of exciton transfer in the condensed phase. The main finding of these studies is that it is essential to include vibrations in the basic model because the electronic transition strength is distributed over the entire vibronic manifold, leading to a substantial reduction in exciton interactions. The electronic coupling is adjusted by Franck-Condon factors, so that exciton splittings—even of the origin bands—are significantly smaller than predicted from electronic structure calculations. Leutwyler and co-workers call this striking reduction of electronic coupling as vibronic quenching. The vibronic ladder of states measured for a molecular aggregate is thus very different from that calculated under the crude assumption that nuclei are frozen. The breakdown in the Born-Oppenheimer approximation produces an interesting coordinate-dependence to exciton delocalization with respect to intramolecular geometry distortions, Fig. 2. This effect nicely shows why including intramolecular modes in theories for energy transfer beyond the Förster limit is difficult. Owing to the opposite way, the potentials are displaced in the example shown in Fig. 2, typical of an antisymmetric normal mode of a molecular dimer, degeneracy, and near degeneracy of the diabatic potentials
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FIG. 2. Potential energy of the lowest energy exciton states of a molecular dimer along the normal coordinate of an antisymmetric intramolecular vibrational mode. The black curves are the diabatic potentials of excited states of molecules A and B in isolation (note that the sign of the displacement is arbitrary so we can draw them crossing at coordinate Q = 0). The red curves show how the adiabatic potentials deviate once electronic coupling is “turned on.” The arrows highlight the coordinate-dependence of exciton delocalization.
occurs only near the ground state equilibrium geometry, where the diabatic curves cross. At these coordinates, therefore, the exciton is delocalized. At larger displacements in either direction along the x-axis, the diabatic curves have an energy gap much greater than the electronic coupling, so the lowest energy excited state is localized on just one molecule. The coordinate dependence of the exciton delocalization is both the challenge for theory—it directly reflects the breakdown of the Born-Oppenheimer approximation—and a fascinating target for experiments to probe. Vertical transitions probed by linear spectroscopy are useful starting points, particularly when vibronic transitions are well resolved. On the other hand, it would be very revealing if delocalization as a function of the nuclear coordinate axis could be traced out. A possible approach is to use nuclear wavepackets generated by femtosecond laser pulses121 as a localized spectroscopic probe. Considering that excited state absorption (ESA) indicates exciton delocalization, by detecting the exciton to two-exciton transitions, it could be used as an interesting probe for coherence in 2DES. Figure 3 shows a selected 2DES spectrum recorded for PC645 (Chroomonas sp. CCMP270). Partly connected with the excitonic dimer located in the center of the PC645 complex, the 2D spectra show an off-diagonal cross-peak that oscillates remarkably as a function of waiting time.75 This oscillation cannot be explained solely by vibrational coherences.77, 78 Note also that the cross-peak coincides with an ESA, although we cannot distinguish whether the oscillations are present only in the ESA and/or the underlying bleach. It is interesting to hypothesize a scenario where the ESA oscillates—essentially blinks on and off—as a nuclear wavepacket travels to and fro along a nuclear coordinate for which the electronic wavefunction varies strongly. Specifically, the exciton delocalization depends on this nuclear coordinate like the example shown in Fig. 2. To illustrate, consider the coordinate of an intramolecular vibrational mode sketched
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FIG. 3. (a) A 2DES spectrum recorded for PC645 (Chroomonas sp. CCMP270) with population time of 55 fs. The absorption spectrum with approximate absorption frequencies of the 8 exciton transitions are indicated by colored lines (see Refs. 75 and 76 for more detail). (b) Hypothesized potential energy curve, analogous to 2, over an intramolecular nuclear coordinate involving the DBV and MBV chromophores. The solid blue arrow indicates the coordinate where excited state absorption is strong, the dashed blue arrow is the coordinate where the excited state absorption is negligible (see text for discussion).
in Fig. 3(b). This particular example has some specific features: two interacting molecules have electronic absorption bands offset by approximately the energy of the vibration and the electronic coupling between the molecules is smaller than the energy difference between the absorption bands. We imagine that if the short laser pulse generates a wavepacket starting at a coordinate in the region indicated as “localized,” the probe will not detect ESA because the electronic coupling is small compared to the zeroth-order energy difference between electronic transitions on the two molecules at this geometry. Now imagine the wavepacket evolving along the coordinate axis until it reaches the region indicated as “delocalized.” At this point, the zeroth-order electronic transitions of the two molecules are almost degenerate (because the electronic energy gap is matched by the vibrational energy) and exciton states will prevail. The probe will now detect ESA as an exciton to two-exciton transition. The oscillating nuclear wavepacket will produce a modulation in the probe because the ESA is modulated by exciton delocalization that depends on nuclear coordinate. A modulation of the transition moments would also be evident and detected in anisotropy measurements, which might be a more sensitive probe. If such a scenario can be identified it will help to map out the intramolecular coordinate dependence of exciton delocalization. Perhaps this is what we have detected for PC645? Quantitative approaches are needed to disentangle the effects of many vibrational degrees of freedom. In recent work,122 we used broad-band femtosecond pump-probe spectroscopy to study a photosynthetic cryptophyte antenna complex, PC577 isolated from Hemiselmis pacifica (CCMP 706), Fig. 4. This light harvesting complex has a peculiar open structure, so unlike PC645 it does not incorporate a strong excitonic dimer. Analysis of vibrational wave-packet dynamics showed the oscillations are contributed by superpositions of levels in the excited electronic state. The formalism developed in the past for coherent superpositions of vibrational modes (wave packets) was useful for interpreting how the amplitude and phase of the oscillations vary with probe wavelength (λprobe ) in the transient absorption data.123 Direct
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The reason for the node is that the wavepacket passes through this point twice per period of oscillation rather than just once like at the other values of λprobe . We conclude that beyond detecting solely the presence of coherent oscillations, it will be worthwhile to analyze also their spectral dependence, amplitude, and phase. In addition, polarization-dependence probes have been shown to be very useful, for example, by Moran and co-workers.126 These properties will give insight into the nature of the potential-energy surfaces of the sample being studied and will thereby help to expose more details of the exciton states and the origin of various coherences. Broadband transient absorption data complements two-dimensional electronic spectroscopy, and employing both techniques in unison may be useful for differentiating between electronic and vibrational coherences in future studies.
III. CONCLUDING REMARKS
FIG. 4. (a) The linear absorption spectrum of the light-harvesting complex PC577 from the cryptophyte Hemiselmis pacifica (CCMP 706). (b) The transient absorption signal of PC577 showing the coherent oscillations during the first picosecond after excitation. The background decay features were independently removed at each probe wavelength to highlight the coherence oscillations as a function of probe wavelength and pump-probe delay time. A sharp node is seen around 640 nm. The oscillations on either side of this node are nearly out-of-phase relative to each other. The dashed black line indicates the peak of the linear absorption spectrum. The dashed green line shows a biexponential fit to the wavelength of the node in the oscillations as a function of pump-probe time delay. It reveals the dynamic Stokes shift and asymptotically approaches the peak of the fluorescence spectrum. (c) The steady-state fluorescence spectrum of PC577 after excitation at 550 nm (see Ref. 122 for details).
Franck-Condon excitation of the ground-state equilibrium population with coherent, broadband light generates a superposition of multiple, closely spaced vibrational levels along different vibrational coordinates of the molecule. This superposition of vibrational levels is non-stationary under the Hamiltonian of the system and launches a wave packet that explores the multi-dimensional phase space of the excited vibrational coordinates. The wave packet oscillates about the minimum of the multi-dimensional potential-energy surface and, based on the energy spacing between the ground-state and excited-state surfaces, the position of the wave packet as a function of pump-probe time delay is evident in the transient absorption spectra,124 resolved by λprobe . A key feature is that the phase of the oscillations undergoes a π radians shift at a probe wavelength (λprobe ) corresponding to the global minimum of the potential-energy surface.125 For example, in Fig. 4(b), the oscillations clearly decrease in amplitude and abruptly change phase around λprobe = 640 nm, which corresponds to the maximum of the fluorescence spectrum. That shows that the wavepacket was produced on the excited electronic state.
The observation of coherent oscillations in 2DES with attributes partially due to electronic coherence has been startling because of the unanticipated long dephasing times of the coherences. At first it was thought that these crosspeaks were incisive probes of electronic coherences, but subsequent work uncovered a more complicated picture. Consequently, several groups are seeking alternative explanations for the coherences, and typically pose the question of whether the beats are electronic or vibrational or some kind of mixed electronic-vibrational state. We discussed experimental directions to help elucidate the nature and properties of coherent oscillations in femtosecond spectroscopy. We foresee the importance of elucidating from experiment data that can be analyzed to retrieve information on delocalization, in which case it is important to choose the measures carefully. Quantum information science has established a foundation for measurement that can be fruitfully transferred to the field of molecular excitons.
ACKNOWLEDGMENTS
This work was supported by the Natural Sciences and Engineering Research Council of Canada, DARPA (QuBE) and the United States Air Force Office of Scientific Research (FA9550-13-1-0005). 1 J.
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THE JOURNAL OF CHEMICAL PHYSICS 140, 110901 (2014)
Perspective: Detecting and measuring exciton delocalization in photosynthetic light harvesting Gregory D. Scholesa) and Cathal Smyth Department of Chemistry, University of Toronto, 80 St. George St., Toronto, Ontario M5S 3H6, Canada
(Received 12 February 2014; accepted 11 March 2014; published online 20 March 2014) Photosynthetic units perform energy transfer remarkably well under a diverse range of demanding conditions. However, the mechanism of energy transfer, from excitation to conversion, is still not fully understood. Of particular interest is the possible role that coherence plays in this process. In this perspective, we overview photosynthetic light harvesting and discuss consequences of excitons for energy transfer and how delocalization can be assessed. We focus on challenges such as decoherence and nuclear-coordinate dependent delocalization. These approaches complement conventional spectroscopy and delocalization measurement techniques. New broadband transient absorption data may help uncover the difference between electronic and vibrational coherences present in twodimensional electronic spectroscopy data. We describe how multipartite entanglement from quantum information theory allows us to formulate measures that elucidate the delocalization length of excitation and the details of that delocalization even from highly averaged information such as the density matrix. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4869329] I. INTRODUCTION
Photosynthetic solar energy conversion occurs on an immense scale across the earth, influencing our biosphere from climate to oceanic food webs.1–3 Energy from sunlight is absorbed by chromophores, like chlorophyll, embedded in proteins comprising the photosynthetic unit. Emerson and Arnold4 defined the photosynthetic unit to explain the abundance of chlorophyll molecules that are employed to produce a single molecule of oxygen. Later it was shown by Duysens that while the chain of reactions involved in photosynthesis starts at a reaction center, hundreds of chromophores are used to harvest sunlight and “concentrate” (in the steady-state) electronic excitation at the reaction centers.5–7 Thus these light-harvesting complexes compensate for the mismatch between solar irradiance and the optimal rate of reaction center operation.8–12 The photosynthetic solar energy conversion process is incredible, not for its performance as gauged by engineering metrics,13, 14 but because such diverse examples abound. Fronds in kelp forests (a brown algae) can grow 15 cm in a single day.15 Crustose coralline red algae can dwell in conditions on the threshold of net primary production. They have been documented to grow on the sea floor, 20 m under water with greater than 1 m of ice cover.16 Unusual purple bacteria have been discovered in the South Andros Black Hole, Bahamas, where they dominate their environment by shortcircuiting photosynthetic light harvesting complexes to generate heat and warm the surrounding water.17 All of these examples are fascinating case studies, particularly in chemical physics, with experiments and theories revealing the mechanisms involved in the ultrafast energy transfer processes of light harvesting.10, 18–25 a) [email protected]
0021-9606/2014/140(11)/110901/8/$30.00
A big question has been how to discover the mechanisms promoting electronic energy transfer, since there are various contributions to electronic coupling as well as regimes of dynamics that come into play according to how chromophores are arranged.21, 26–30 Femtosecond spectroscopy typically times population changes by probing their spectral signatures. This determines the time scale of ultrafast processes, but not the underlying mechanism. For example, researchers have long wondered whether the Fermi Golden Rule basis of the Förster energy transfer model is a good approximation for photosynthetic light harvesting, or whether a more sophisticated theory is needed.18, 20, 31–37 Can an experiment directly address this question? That is what was especially exciting about the report in 2007 from the Fleming group;38 it was shown that two-dimensional electronic spectroscopy (2DES) could directly address the question of how rapidly excitonic coherence is lost. Subsequent work has concluded the coherence related to excitonic delocalization does indeed affect energy transfer dynamics.29, 39–41 The goals of this paper are to discuss: (a) What are the demands for photosynthetic light harvesting and therefore possible role of coherence in the biological context? (b) How can we quantify excitonic delocalization in complex systems? (c) Are there different ways to think about the coherent oscillations seen in 2D spectra? Are new experimental measures required?
II. DISCUSSION A. The layout and function of light harvesting systems
The layout of light-absorbing molecules in the photosynthetic unit, Fig. 1, may appear like sprawling labyrinths of chromophores with little order. Nevertheless, short-lived 140, 110901-1
© 2014 AIP Publishing LLC
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FIG. 1. Model of the photosystem II (PSII) supercomplex C2S2M2 from higher plants. (a) The protein organization showing LHCII trimers (red) on the periphery, proteins in the core (magenta). CP24 is colored yellow; CP29 orange; CP26, cyan. (B) A model showing only the organization of chlorophylls (Chls) bound in these proteins. Chls a, green; Chls b, blue. The special pair Chls are colored red. Reprinted with permission from Croce and van Amerongen, J. Photochem. Photobiol. B. Biol. 104, 142–153 (2011). Copyright 2011 Elsevier.
electronic excitations (lifetimes of 1–5 ns) traverse these molecularly vast landscapes with high efficiency to sensitize reaction centers and initiate photosynthetic energy transduction. The typical photosynthetic unit is an excellent example of finely tuned adaption to limiting conditions. Chromophores are organized so there is optimal harvesting of light across a relatively broad spectral range. Optimal in this case means within possible constraints of resources available—light, nutrients, etc. The photosynthetic unit typically comprises about 200– 400 light harvesting chromophores associated with each reaction center, depending on organism and growth conditions.42 The upper limit to the size of a photosynthetic unit is dictated by interplay of two primary factors: the rate of excitation transfer and the excited state lifetime.14 The minimum size of a photosynthetic unit is controlled by a delicate balance of energy absorption, conversion, and dissipation of excess excitations. The physical size and layout of a supercomplex of the photosynthetic unit in the vicinity of photosystem II in higher plants is shown in Fig. 1. Experiments have estimated there are about 200 chlorophylls per reaction center under normal growth conditions.43 In purple bacteria, a combination of siteenergy tuning of chromophores by the protein and excitonic effects creates a natural energy gradient directing excitations towards the reaction center (specifically to the special pair). Imaging of photosynthetic membranes of purple bacteria by atomic force microscopy reveals the arrangement of light harvesting complexes (LH2) and reaction centers clearly,44–46 while the ratio of LH2 to reaction centers can also be measured spectroscopically.47 It is found that 2–7 LH2 complexes are associated with each reaction center, meaning there are ∼100–200 bacteriochlorophyll molecules per reaction center. Despite this low ratio of antenna molecules to reaction centers, Vos et al. have shown that excitation migrates over about 1000 bacteriochlorophylls48, 49 —in other words, about 25 photosynthetic units are connected.
J. Chem. Phys. 140, 110901 (2014)
The size of the photosynthetic unit can change depending on whether or not growth is light limited. The essential principle is that the reaction centers should be photo-excited at a rate balanced with the rate of linear electron flow from H2 O to NADP+ . The right balance inhibits back reactions, and thus losses, downstream from the photo-initiation processes. A large photosynthetic unit helps maintain the rate of photosynthetic production under low light conditions because it provides the reaction center with an increased absorption cross section. Light can be limiting, for example in microbial mats,50–52 turbid water,53 competition in aquatic communities,54 or at great depths in the sea.55 A further interesting complexity is that not only the size, but also the organization of light harvesting complexes with reaction centers has been documented to change depending on light conditions.56 It is generally thought that excitation energy moves more or less by a random walk, historically based on early models for energy hopping7, 57 and Pearlstein’s hypothesis in the 1960s that coherent transfer should not dramatically change the limits of excitation diffusion length.32 However, such conclusions are subject to revision, especially as we develop this field of research in terms of measurement, theory, and experimental probes of the mechanisms of energy transfer. Recently, research has suggested that a balance between coherence (quantum effects) and decoherence (loss of quantum effects) is optimal for excitation transport.40, 58, 59 In that case, the random walk model must be modified60 and more sophisticated theories of energy transfer are required.21, 61 Likewise any quantum measurements made must be appropriate for open systems; delocalization measures such as the Inverse Participation Ratio are not appropriate when decoherence is taken into account. In terms of scale, we believe it is useful to delineate excitation energy transfer in light harvesting complexes from excitation migration and trapping in the photosynthetic unit. We suggest that the designation “excitation energy transport” is preferably reserved for the excitation migration throughout the photosynthetic unit, while the rapid excitation energy equilibration and flow in an isolated complex is excitation energy transfer. Calculations at present have mainly focused on energy transfer in isolated light harvesting complexes because there exists the opportunity to examine deeply energy transfer mechanisms. The consequences of more sophisticated energy transfer theories and the deviations they will predict from random walk models on longer length scales, for example, for photosynthetic units (>100 chromophores) are unknown. As we discussed previously,14 the average effect is likely difficult to discern, but mechanisms at play in tails of the distributions of trapping times will be interesting. We can summarize this viewpoint by stating that an outstanding challenge is to connect details of energy transfer at the isolated light harvesting protein level to the collective function of many proteins.62
B. Exciton delocalization
At the foundation of the most interesting coherence effects in light harvesting systems are molecular excitons. Molecular exciton states (which is what we mean by “exci-
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ton” in this context) are electronic excited states where the wavefunction is shared by two or more molecules. The electron density remains localized on the molecules individually, but the transition density (see Ref. 63) for the exciton is a well-defined superposition of molecule-localized transition densities. In the kinds of exciton coherence that has interested researchers recently, this delocalization evolves in time. From a definition point of view, that means we cannot find a basis rotation among localized and delocalized representations that consistently diagonalizes the density matrix as the dynamics evolve after photo-excitation. Thus we find off-diagonal density matrix elements that signal quantum mechanical coherence. Exciton delocalization, if strong, can be indicated by spectral features that are perturbed, shifted, or split.64 Scaling of transition strength or nonlinear susceptibility is useful when comparative measurements can be made.65, 66 Similarly, changes in the radiative rate (“superradiance”) are quite sensitive to delocalization.67–69 Coherent superpositions of states that contribute oscillating cross-peaks to two-dimensional electronic spectroscopy (2DES) data have recently been proposed to reveal delocalization and coherences—and particularly how they dephase.20, 21, 27, 38, 70–83 Another probe of exciton delocalization can be excited state absorption in pump-probe and 2DES experiments.36, 84, 85 Two-exciton states, lying at approximately twice the exciton energy, necessarily accompany exciton states. While exciton states are defined in the basis of permutations of excitation among the interacting molecules, two-exciton states are defined as permutations of two excitations.86, 87 Excited state absorption reveals the exciton to two-exciton electronic transition(s). From a physical point of view, reasons behind changes in delocalization include exciton localization—the interplay of the Stokes shift in the site basis and exciton delocalization with a lesser nuclear reorganization88–90 —or an explicit dependence of energy resonance on various nuclear coordinates (vide infra). It is clear that experiments that can follow or characterize delocalization, with femtosecond resolution, after photoexcitation will give insights into some interesting photophysics. Perhaps such experiments can even reveal how exciton wavepackets evolve in time, although achieving state or process tomography for disordered condensed phase systems will be challenging.91–93 Another approach is to devise and measure metrics of exciton delocalization. The 2007 report by Fleming and co-workers ignited research about quantum coherence in light harvesting complexes.38 The foundation for this new experimental insight was a body of literature examining exciton delocalization in photosynthesis. A European Science Foundation workshop (organized by Leonas Valkunas and Rienk van Grondelle) held in Birstonas, Lithuania in 1996 highlighted the excitement about coherent delocalization of excitons. The book of abstracts notes that “Major questions concern. . . the time over which the excitation must be considered as coherent. . . .” The topic of discussion was largely Cogdell and coworkers’ report94 of an atomic resolution crystal structure for the peripheral light-harvesting complex LH2 from the purple anoxygenic bacterium Rhodopseudomonas (Rps.) acidophila strain 10050. Papers associated with this workshop highlight
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studies of delocalization in LH2 and the measurement of exciton delocalization. They are collected in volume 101, issue 37 (1997) of the Journal of Physical Chemistry B. At the time the Inverse Participation Ratio (IPR), defined by Thouless95 was frequently employed as a measure of delocalization.85, 96, 97 This measure, also known as a second order statistical moment, looks at the variance of probabilities within a wavefunction ψ = a1 ϕ 1 + a2 ϕ 2 + . . . an ϕ n delocalized over n sites: n |aj |4 . (1) IPR = j =1
The IPR can only be used, however, in calculations based on wavefunctions. For example, when exciton wavefunctions are calculated as a function of static disorder by Monte Carlo averaging.98 The crucial distinction here is that each wavefunction in the ensemble is calculated and the IPR is calculated for each wavefunction and statistically averaged. That is different than first averaging the wavefunctions themselves, by constructing a density matrix, then calculating the delocalization length. Consider an experiment that measures the evolution of a wavefunction:99 after many iterations, a clear picture of the population dynamics is formed; however, the coherences between these populations get washed out due to the phase differences in averaging the slightly different states. Representing these populations and coherences in a density matrix ρ, one can see how much the populations differ from their coherences using a measure called the purity. The purity is defined as the trace of the square of a density matrix, Purity ≡ Tr(ρ 2 ).
(2)
A pure state has a purity value of 1 and represents a fully coherent state that can also be written as a wavefunction. A mixed state represents a statistical mixture of different coherent states and will have a smaller value as low as 1/n, in a system of size n, if the state is fully mixed. In information terms, a decrease in purity leads to an increase in entropy. Therefore the IPR, as strictly a measure of state populations, is unsuitable in its application to mixed states. As such, it is unable to reveal the evolution of coherence within these light harvesting complexes when we are concerned with including the effects of homogeneous line broadening. In parallel to research on delocalization in LH2, the field of quantum computation and quantum information was being established, and with it a whole new field dedicated to measuring entanglement in mixed states. The emergence of this field has been a great asset, and the suitability of these measures to light harvesting complexes has recently been demonstrated. Measures of bipartite entanglement such as the tangle,100 the negativity,101 and the relative entropy102 have all been employed to chart the evolution of coherence in light harvesting systems.41, 103, 104 Interestingly, these measures, the IPR, the tangle, and the relative entropy, can be related using the purity measure.105 For example, the total two-body entanglement, or bipartite entanglement, E2 (ρ), in a system with one excitation can be written as the difference between the purity and the IPR, E2 (ρ) = Tr(ρ)2 − M2 (ρ),
(3)
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where M2 (ρ) = nj=1 (ρjj )2 is the statistical moment of order two of the system populations, which is exactly equivalent to the IPR. This result could prove essential in the experimental measurement of coherence in light harvest complexes; by gathering time dependent population statistics and employing a judicious assessment of entropy within a system, it could be possible to gain a strong understanding of the time dependent coherence within a system, without having to do full state tomography. For example, one can fully measure the purity of a state with a number of measurements that scale linearly with system size.106 Though such entanglement measures provide excellent information on the coherence within these systems, they only do so using two-body correlations. Recently, measures of many-body entanglement for pure states, using higher order statistical moments than the IPR, were applied to model light harvesting complexes.107 Adapting these measures to mixed states, which is significantly harder, finally delivers the true “delocalization length” in the systems: a precise number of sites that share quantum correlations during every step of energy transfer, despite decoherence. A computationally intense numerical method derived by Levi et al.108 would provide such a measure. We have recently demonstrated that it is also possible to measure multipartite entanglement in mixed states analytically, once more by exploiting the relationship between purity, entanglement, and the statistical moments.109 Measures that are a function of the statistical moments107 can be applied to the mixed state in question, as well as to a reference state σ with the same level of purity. In order to measure the k-partite entanglement in a mixed state one can use the equation (4) Ek (ρ) = k τk (ρ) − τk (σ ), where τ k is a function of the statistical moments up to order k as described in Ref. 109. The challenge is in deriving the reference state σ , which should have the maximum level of (k–1)-partite entanglement for that level of purity. This measurement returns values of statistical correlation that are not possible for states with lower orders of entanglement at that level of entropy. Effectively, one can remove the noise from their measurement in order to process the signal. Much like the bipartite measure, knowledge of population statistics and purity within a system is all that is needed to fully measure the delocalization length, at any level of decoherence. These measures enable us to detect how entanglement among multiple excitons diminishes during dynamics evolution, for example, four-site entanglement decays quickly, followed by three-site entanglement, and so on.109 We can thereby map out in considerable detail how excitation is delocalized in a multichromophore aggregate. C. Intramolecular vibrations, delocalization, and coherence
Intramolecular vibrations play a marked role in molecular spectroscopy. High frequency modes (compared to kT/h) are clearly evident as vibronic progressions in absorption and emission spectra. Torsional modes that have a frequency
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change from ground to excited state can broaden absorption spectra substantially.110 Förster theory for electronic energy transfer employs a separation between electronic and nuclear factors and thereby naturally includes vibronic details of molecular spectroscopy in the Förster spectral overlap.26 That is of profound quantitative importance because it is often vibronic overlap of the red tail of the donor fluorescence spectrum with the vibronic progression of the acceptor absorption spectrum that enables energy conservation and therefore decides the rate of energy transfer.111 Recent work has highlighted the importance of understanding the role of intramolecular vibrations in energy transfer (exciton relaxation) in the intermediate to strong electronic coupling regimes, especially for interpretation of coherent oscillations observed in two-dimensional electronic spectroscopy, see Refs. 76 and 112–116, and references cited therein. In a striking recent report, O’Reilly and Olaya-Castro show that it is the intramolecular nuclear coherences in combination with electronic coherences that ensure the coherences are definitively quantum in nature.117 A common theoretical framework for going beyond Förster theory is to diagonalize the electronic Hamiltonian, then add coupling to low frequency bath nuclear degrees of freedom perturbatively (e.g., Redfield theory) or nonperturbatively. Despite the recent leaps in theoretical models for energy transfer, it is not obvious that roles of high frequency intramolecular vibrations—that are so important in the Förster spectral overlap—are appropriately captured in present theories for intermediate or strong coupling. We would like to stimulate discussion about the limitations of the electronic exciton basis, or the polaron basis, versus a vibronic basis. Leutwyler and co-workers have studied model molecular dimers in the gas phase.118–120 They measured high resolution spectra, free from an “environment” that would cause line broadening and contribute to exciton relaxation. Therefore these experiments reveal, in essence, a zeroth-order description of the vibronic exciton levels that may be a good basis for theories of exciton transfer in the condensed phase. The main finding of these studies is that it is essential to include vibrations in the basic model because the electronic transition strength is distributed over the entire vibronic manifold, leading to a substantial reduction in exciton interactions. The electronic coupling is adjusted by Franck-Condon factors, so that exciton splittings—even of the origin bands—are significantly smaller than predicted from electronic structure calculations. Leutwyler and co-workers call this striking reduction of electronic coupling as vibronic quenching. The vibronic ladder of states measured for a molecular aggregate is thus very different from that calculated under the crude assumption that nuclei are frozen. The breakdown in the Born-Oppenheimer approximation produces an interesting coordinate-dependence to exciton delocalization with respect to intramolecular geometry distortions, Fig. 2. This effect nicely shows why including intramolecular modes in theories for energy transfer beyond the Förster limit is difficult. Owing to the opposite way, the potentials are displaced in the example shown in Fig. 2, typical of an antisymmetric normal mode of a molecular dimer, degeneracy, and near degeneracy of the diabatic potentials
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FIG. 2. Potential energy of the lowest energy exciton states of a molecular dimer along the normal coordinate of an antisymmetric intramolecular vibrational mode. The black curves are the diabatic potentials of excited states of molecules A and B in isolation (note that the sign of the displacement is arbitrary so we can draw them crossing at coordinate Q = 0). The red curves show how the adiabatic potentials deviate once electronic coupling is “turned on.” The arrows highlight the coordinate-dependence of exciton delocalization.
occurs only near the ground state equilibrium geometry, where the diabatic curves cross. At these coordinates, therefore, the exciton is delocalized. At larger displacements in either direction along the x-axis, the diabatic curves have an energy gap much greater than the electronic coupling, so the lowest energy excited state is localized on just one molecule. The coordinate dependence of the exciton delocalization is both the challenge for theory—it directly reflects the breakdown of the Born-Oppenheimer approximation—and a fascinating target for experiments to probe. Vertical transitions probed by linear spectroscopy are useful starting points, particularly when vibronic transitions are well resolved. On the other hand, it would be very revealing if delocalization as a function of the nuclear coordinate axis could be traced out. A possible approach is to use nuclear wavepackets generated by femtosecond laser pulses121 as a localized spectroscopic probe. Considering that excited state absorption (ESA) indicates exciton delocalization, by detecting the exciton to two-exciton transitions, it could be used as an interesting probe for coherence in 2DES. Figure 3 shows a selected 2DES spectrum recorded for PC645 (Chroomonas sp. CCMP270). Partly connected with the excitonic dimer located in the center of the PC645 complex, the 2D spectra show an off-diagonal cross-peak that oscillates remarkably as a function of waiting time.75 This oscillation cannot be explained solely by vibrational coherences.77, 78 Note also that the cross-peak coincides with an ESA, although we cannot distinguish whether the oscillations are present only in the ESA and/or the underlying bleach. It is interesting to hypothesize a scenario where the ESA oscillates—essentially blinks on and off—as a nuclear wavepacket travels to and fro along a nuclear coordinate for which the electronic wavefunction varies strongly. Specifically, the exciton delocalization depends on this nuclear coordinate like the example shown in Fig. 2. To illustrate, consider the coordinate of an intramolecular vibrational mode sketched
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FIG. 3. (a) A 2DES spectrum recorded for PC645 (Chroomonas sp. CCMP270) with population time of 55 fs. The absorption spectrum with approximate absorption frequencies of the 8 exciton transitions are indicated by colored lines (see Refs. 75 and 76 for more detail). (b) Hypothesized potential energy curve, analogous to 2, over an intramolecular nuclear coordinate involving the DBV and MBV chromophores. The solid blue arrow indicates the coordinate where excited state absorption is strong, the dashed blue arrow is the coordinate where the excited state absorption is negligible (see text for discussion).
in Fig. 3(b). This particular example has some specific features: two interacting molecules have electronic absorption bands offset by approximately the energy of the vibration and the electronic coupling between the molecules is smaller than the energy difference between the absorption bands. We imagine that if the short laser pulse generates a wavepacket starting at a coordinate in the region indicated as “localized,” the probe will not detect ESA because the electronic coupling is small compared to the zeroth-order energy difference between electronic transitions on the two molecules at this geometry. Now imagine the wavepacket evolving along the coordinate axis until it reaches the region indicated as “delocalized.” At this point, the zeroth-order electronic transitions of the two molecules are almost degenerate (because the electronic energy gap is matched by the vibrational energy) and exciton states will prevail. The probe will now detect ESA as an exciton to two-exciton transition. The oscillating nuclear wavepacket will produce a modulation in the probe because the ESA is modulated by exciton delocalization that depends on nuclear coordinate. A modulation of the transition moments would also be evident and detected in anisotropy measurements, which might be a more sensitive probe. If such a scenario can be identified it will help to map out the intramolecular coordinate dependence of exciton delocalization. Perhaps this is what we have detected for PC645? Quantitative approaches are needed to disentangle the effects of many vibrational degrees of freedom. In recent work,122 we used broad-band femtosecond pump-probe spectroscopy to study a photosynthetic cryptophyte antenna complex, PC577 isolated from Hemiselmis pacifica (CCMP 706), Fig. 4. This light harvesting complex has a peculiar open structure, so unlike PC645 it does not incorporate a strong excitonic dimer. Analysis of vibrational wave-packet dynamics showed the oscillations are contributed by superpositions of levels in the excited electronic state. The formalism developed in the past for coherent superpositions of vibrational modes (wave packets) was useful for interpreting how the amplitude and phase of the oscillations vary with probe wavelength (λprobe ) in the transient absorption data.123 Direct
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The reason for the node is that the wavepacket passes through this point twice per period of oscillation rather than just once like at the other values of λprobe . We conclude that beyond detecting solely the presence of coherent oscillations, it will be worthwhile to analyze also their spectral dependence, amplitude, and phase. In addition, polarization-dependence probes have been shown to be very useful, for example, by Moran and co-workers.126 These properties will give insight into the nature of the potential-energy surfaces of the sample being studied and will thereby help to expose more details of the exciton states and the origin of various coherences. Broadband transient absorption data complements two-dimensional electronic spectroscopy, and employing both techniques in unison may be useful for differentiating between electronic and vibrational coherences in future studies.
III. CONCLUDING REMARKS
FIG. 4. (a) The linear absorption spectrum of the light-harvesting complex PC577 from the cryptophyte Hemiselmis pacifica (CCMP 706). (b) The transient absorption signal of PC577 showing the coherent oscillations during the first picosecond after excitation. The background decay features were independently removed at each probe wavelength to highlight the coherence oscillations as a function of probe wavelength and pump-probe delay time. A sharp node is seen around 640 nm. The oscillations on either side of this node are nearly out-of-phase relative to each other. The dashed black line indicates the peak of the linear absorption spectrum. The dashed green line shows a biexponential fit to the wavelength of the node in the oscillations as a function of pump-probe time delay. It reveals the dynamic Stokes shift and asymptotically approaches the peak of the fluorescence spectrum. (c) The steady-state fluorescence spectrum of PC577 after excitation at 550 nm (see Ref. 122 for details).
Franck-Condon excitation of the ground-state equilibrium population with coherent, broadband light generates a superposition of multiple, closely spaced vibrational levels along different vibrational coordinates of the molecule. This superposition of vibrational levels is non-stationary under the Hamiltonian of the system and launches a wave packet that explores the multi-dimensional phase space of the excited vibrational coordinates. The wave packet oscillates about the minimum of the multi-dimensional potential-energy surface and, based on the energy spacing between the ground-state and excited-state surfaces, the position of the wave packet as a function of pump-probe time delay is evident in the transient absorption spectra,124 resolved by λprobe . A key feature is that the phase of the oscillations undergoes a π radians shift at a probe wavelength (λprobe ) corresponding to the global minimum of the potential-energy surface.125 For example, in Fig. 4(b), the oscillations clearly decrease in amplitude and abruptly change phase around λprobe = 640 nm, which corresponds to the maximum of the fluorescence spectrum. That shows that the wavepacket was produced on the excited electronic state.
The observation of coherent oscillations in 2DES with attributes partially due to electronic coherence has been startling because of the unanticipated long dephasing times of the coherences. At first it was thought that these crosspeaks were incisive probes of electronic coherences, but subsequent work uncovered a more complicated picture. Consequently, several groups are seeking alternative explanations for the coherences, and typically pose the question of whether the beats are electronic or vibrational or some kind of mixed electronic-vibrational state. We discussed experimental directions to help elucidate the nature and properties of coherent oscillations in femtosecond spectroscopy. We foresee the importance of elucidating from experiment data that can be analyzed to retrieve information on delocalization, in which case it is important to choose the measures carefully. Quantum information science has established a foundation for measurement that can be fruitfully transferred to the field of molecular excitons.
ACKNOWLEDGMENTS
This work was supported by the Natural Sciences and Engineering Research Council of Canada, DARPA (QuBE) and the United States Air Force Office of Scientific Research (FA9550-13-1-0005). 1 J.
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