Coherent and controllable enhancement of light-harvesting efficiency
Spectroscopic experiments on photosynthetic complexes have identified long-lived coherences, suggesting that coherent effects can be relevant in disordered and noisy light-harvesting systems. However, there is limited experimental evidence that light-harvesting processes can be more efficient due to a coherent effect, largely due to the difficulty of turning coherences on and off to create an experimental control. Here, we show that coherence can be used to enhance light harvesting, starting from a model system with controllable initial states. Specifically, we consider a three-site system, comprising two identical coupled donors, one of which is coupled to an acceptor. Coupling within the donor dimer results in two delocalised eigenstates that can be addressed using different light modes, allowing a coherent light source to enhance exciton populations on either donor by controlling only the phase between two exciting modes. Coherently controlling the excitation in this way can significantly enhance the light-harvesting efficiency relative to incoherent excitation. Our proposal would allow for the first unambiguous demonstration of light harvesting enhanced by intermolecular coherence, as well as demonstrate the potential for coherent control of excitonic energy transfer.
Observations of long-lasting coherences in photosynthetic pigment-protein complexes Engel et al. (2007); Panitchayangkoon et al. (2010); Collini et al. (2010); Lee et al. (2007); Hayes et al. (2013), previously thought to be too strongly coupled to their environment to support coherent effects, have raised the question of whether coherence can play a role in molecular light-harvesting processes Engel et al. (2007); Scholes (2010); Scholes et al. (2011); Kassal et al. (2013); Fassioli et al. (2013); Scholes et al. (2017); Romero et al. (2017). The question remains open, despite arguments that the observed coherences are dominantly vibrational or vibronic Christensson (2012); Tiwari et al. (2013); Thyrhaug et al. (2018) and that, in either event, they could not be induced by incoherent sunlight Jiang and Brumer (1991); Mančal and Valkunas (2010); Brumer and Shapiro (2012); Kassal et al. (2013). Nevertheless, theoretical studies have proposed that, even in those circumstances, there are mechanisms by which coherences could enable significant enhancements of light-harvesting efficiencies Dorfman et al. (2013); Svidzinsky et al. (2011); Scully (2010); Scully et al. (2011); Creatore et al. (2013); Kassal et al. (2013); León-Montiel et al. (2014); Baghbanzadeh and Kassal (2016a, b). In most of those works, as in this one, efficiency is defined as the probability of an excitation being successfully transferred to a target acceptor, which, in many cases, eventually leads to charge transfer or another means of harvesting the excitation energy. Aside from providing insight into the relevance of quantum effects in biological systems, research on this topic is also motivated by the potential application of these concepts to the design of novel artificial light-harvesting devices Scholes (2010); Romero et al. (2017).
Direct experimental evidence of an efficiency enhancement due to intermolecular coherence is lacking for two reasons. First, experiments so far have focused on observing coherences in isolated systems, with no acceptor for the excitations to be transferred to, making them unable to relate coherence to efficiency. Second, to ensure that a particular enhancement is due to coherence and not a confounding factor, it would be necessary to be able to switch coherence on and off without affecting other experimental variables. This kind of control is often not possible in existing light harvesting systems; for example, altering their molecular structures often causes significant changes to their overall energy landscape Baghbanzadeh and Kassal (2016a).
The only demonstrations of coherent enhancements have been experiments showing that the efficiency of excitation transfer from one molecule can be increased through adaptive coherent control Herek et al. (2002); Savolainen et al. (2008). These experiments targeted intramolecular (often vibrational Savolainen et al. (2008)) coherences within the donor, leaving the effect of intermolecular coherences on efficiency unobserved. Theoretical work has shown that multi-chromophoric light harvesting could also be controlled Hoyer et al. (2014); Caruso et al. (2012), but the final pulse sequences produced by sophisticated optimisation algorithms can be difficult to understand intuitively.
Here, we address the problem from the bottom up. Instead of describing existing light-harvesting systems, our goal is to design a minimally complex light-harvesting system whose efficiency can be directly monitored and whose coherence can be externally controlled. The ability to compare light-harvesting efficiencies in the presence and absence of coherence would permit the first definitive demonstration of light-harvesting enhanced by intermolecular coherence.
Our model system consisting of two identical donor sites (e.g., molecules) and an acceptor site (Figure 1a). The acceptor’s excited state is significantly red shifted compared to the donors’, ensuring that the donor-to-acceptor excitation transfer is both irreversible and spectrally resolvable. Excitonic coupling between the donors forms two eigenstates that are delocalised across the donor dimer and that can be addressed by different light modes. Using optical phase control—i.e., changing only the phases but not the intensity of the light—the system can be prepared in a wide range of coherent and incoherent initial states Shapiro and Brumer (2012); León-Montiel et al. (2014); Brüggemann and May (2004); Caruso et al. (2012). By measuring the proportion of excitations successfully transferred to the acceptor (as opposed to lost to recombination), we can compare energy-transfer efficiencies for different initial states and unambiguously demonstrate the influence of excitonic coherence on light-harvesting efficiency. In particular, certain superpositions represent excitations that are mostly localised on particular sites, allowing for efficiency enhancements if excitations are localised close to the acceptor (Figure 2).
We treat the system with a Frenkel-type (tight-binding) Hamiltonian
where represents an excitation localised on site with energy and is the coupling between sites and . We assume dipole-dipole intersite couplings
where is the dielectric constant, is the ground-to-excited-state transition dipole moment of site , is the distance between two sites, and . We denote the eigenstates of as , with energies , shown in Figure 1c.
Coupling the system to a bath gives a total Hamiltonian
where the bath consists of an independent set of harmonic oscillators on each site,
where and are the creation and annihilation operators for mode on site . The system-bath coupling is assumed to be linear,
The time evolution of the system’s reduced density operator (RDO) is given by the master equation
where the dissipator encodes the non-unitary evolution. We divide into three superoperators (illustrated in Figure 1c),
where describes the effect of on the system—and therefore the evolution of excited states due to interactions with the thermal bath—and where the additional terms and describe the relaxation of the excited states to a ground state through processes that are not accounted for in the Hamiltonian .
Excitons have a finite lifetime due to radiative and non-radiative recombination, which, if we assume an equal decay rate from each excited state, is described by the superoperator
where denotes an anticommutator.
We also define a target process that causes a decay to exclusively from a particular eigenstate at rate . This process can, for example, represent a successful separation of an exciton into charge carriers, or simply an external observation of a donor-acceptor transfer event. We model the target process as
Distinguishing between the desirable target process and the wasteful recombination leads to a time-dependent definition of efficiency as the probability of an excitation moving from to via the target process within a period of time :
If no time is specified, the efficiency refers to the long-time efficiency .
where (with eigenstates and of ) and where we ignored the Lamb shift caused by , which could instead be accounted for by redefining May and Kühn (2011); Breuer and Petruccione (2006). is the bath correlation function, with the Fourier transform May and Kühn (2011), where is the Bose-Einstein distribution for a bath with temperature and is the bath spectral density. In eq. 12 we have assumed identical spectral densities on all sites, and that the states of the baths at each site are uncorrelated at all times. It can be seen that , which implies that interaction of a system with a bath of harmonic oscillators does not cause any pure dephasing of system eigenstates May and Kühn (2011). We emphasise that only models excited state dynamics, and any transfers to are mediated by and (see Figure 1c).
It can be shown that each term in eq. 11 leads to time evolution that oscillates at a frequency , where . Often, these rates are sufficiently fast for the influence of particular terms to average to zero after sufficient time propagation. This motivates the widely used secular approximation, where all terms for which are discarded May and Kühn (2011); Breuer and Petruccione (2006), eliminating all terms that transfer populations to coherences (and vice versa) and resulting in decoupled population and coherence dynamics. However, if two levels are nearly degenerate, the energy difference between them may be sufficiently small for some terms to oscillate slowly enough to have a significant effect on population dynamics, making it unsafe to discard them Dodin et al. (2018); Tscherbul and Brumer (2014). Because the efficiency depends only on populations, the only way for coherences to influence the efficiency is if coherence-to-population transfer can occur; therefore, non-secular effects, found in the limit of nearly degenerate states, are essential for coherent efficiency enhancements.
Here, we treat a system of three sites, the left donor , the right donor , and the acceptor . We assume the two donors have degenerate excited states, so and we let , where is an energy detuning. For , by diagonalising we obtain two eigenstates that are approximately delocalised exclusively across the two donors, \cref@addtoresetequationparentequation
with energies and . In general, these two eigenstates also overlap with ; however, in this regime, this overlap is small enough that we can assume the eigenstates are contained within the donor dimer, and we refer to them as donor eigenstates. In addition, the third energy eigenstate coincides with , up to a small perturbation, and we refer to it as the acceptor eigenstate. All the eigenstates are shown in Figure 1c. Nevertheless, to ensure accurate rates in the Redfield tensor (eq. 11), the calculations below include the small overlaps of the donor eigenstates with and of with the donor sites.
To demonstrate that the efficiency can be affected by coherences between donor eigenstates, we consider cases where is smaller than the donor-to-acceptor exciton transfer rate. In this regime, non-secular terms oscillating at frequencies less than can have a significant effect on system dynamics. All other non-secular terms—namely those connecting populations to coherences involving the acceptor—oscillate quickly and can be neglected, i.e., we carry out a secular approximation on acceptor states. After this approximation, the -induced evolution of each RDO element is \cref@addtoresetequationparentequation
where the population transfer rate from to is , , and . We have also assumed that is slowly varying over the interval and can be replaced with the constant . The first term in each of these equations contains the secular incoherent rates, while the remaining, non-secular terms account for coherent effects that are non-negligible in the limit of small .
The initial state that is subject to this evolution can be generated by optical excitation. To control the initial state, the individual eigenstates should be individually addressable by different optical modes. In principle, the modes could be different light frequencies, but, in our case, the eigenstates are significantly lifetime broadened by their fast decay, which may make it impossible to resolve them spectrally. This difficulty can be overcome by also considering optical modes with different polarisation.
The initial state depends on whether the exciting light is coherent or incoherent (or, in the case of polarisation, polarised or unpolarised) Jiang and Brumer (1991); Mančal and Valkunas (2010); Brumer and Shapiro (2012). Weak coherent light prepares the excited state Breuer and Petruccione (2006); Brüggemann and May (2004); Shapiro and Brumer (2012); Jiang and Brumer (1991); León-Montiel et al. (2014)
where and are the electric field amplitude and the phase of the light mode exciting eigenstate , is the transition dipole moment for the transition and is a normalisation factor. The transition dipole moments of the eigenstates are linear combinations of the site-basis transition dipoles, .
By contrast, incoherent light excites a statistical mixture of eigenstates Jiang and Brumer (1991); Mančal and Valkunas (2010); Brumer and Shapiro (2012). Because incoherent light is stationary, it, strictly speaking, exists only as a continuous-wave process (various artefacts are present if incoherent light is suddenly switched on Tscherbul and Brumer (2015, 2014); Grinev and Brumer (2015); Dodin et al. (2016a, b); Olšina et al. (2014)). However, Jesenko and Žnidarič have shown that the efficiency of light harvesting in continuous-wave incoherent light is equal to the efficiency given a particular transient initial state Jesenko and Žnidarič (2013). In our case, the equivalent initial state is
where is the magnitude of and is the ensemble root-mean-square electric field intensity of the mode .
To maximise coherence magnitudes in the coherent excitations, we consider the case of light sources which excite equal populations in the and states. We also assume that the target process modelled by eq. 9 occurs via the acceptor eigenstate (i.e., ). Finally, we assume that there is no direct excitation of state , which could trivially contribute to the efficiency. Practically, this would correspond to an excitation by a light source with no electric field component resonant with this state. This ensures that the target process efficiency is an indicator of successful transfers from donor states to the acceptor state. In order to individually address the and states, we choose donor sites with perpendicular dipole moments of equal magnitude. This arrangement results in donor eigenstates whose dipole moments are also perpendicular and of equal magnitude, making them addressable using separate polarisation modes of the light (Figure 1b).
The initial states of the system is then
for an incoherent (unpolarised) excitation, and \cref@addtoresetequationparentequation
for a coherent (polarised) excitation with relative phase between the two light modes.
In the limit of eq. 13, exciton populations on sites and , given initial state , are \cref@addtoresetequationparentequation
By choosing the phase , the initial excitations can be significantly localised on the left or the right site, especially for small , when and are both close to . In particular, population in is maximised for , while that in is maximised at .
For concreteness, we consider the donors to have energies and all three sites to have transition dipole moments of , with geometry shown in Figure 1. The separation between the two donors was fixed at , corresponding to a donor-donor coupling and chosen to ensure that is significantly smaller than the rate of donor-acceptor transfers. The bath was assumed to have a Debye spectral density May and Kühn (2011); Pachon et al. (2017),
with reorganisation energy and Debye frequency , where is the Heavyside step function. We chose a rapid target rate and a recombination rate .
We used eq. 6, with the full Redfield tensor of eq. 11, to evolve three initial states: the coherent states and , and the incoherent state . We emphasise that the diagonal RDO elements are initially identical across the three cases, and all differences in exciton dynamics and efficiencies are caused by the coherences.
Figure 3 shows that initial coherence can profoundly affect the efficiency by comparing the time-dependent efficiency for the coherent and incoherent excitations, all computed for and with chosen so that . As shown in Figure 2, coherently increasing exciton populations in significantly increases the efficiency by starting the excitation closer to the acceptor. The observed enhancement is an example of environment-assisted single-photon coherent phase control Brumer and Shapiro (1989); Spanner et al. (2010); Arango and Brumer (2013). In this example, the difference in efficiency between the case and the incoherent case is 25 percentage points. The maximum enhancement is 50 percentage points (a doubling), because the efficiency of the incoherent excitation is always the average of the two coherent efficiencies. This is because the efficiency is a linear function of the initial RDO and .
To explore the limits of coherent efficiency enhancements, we simulated the system for a range of and , whilst holding fixed. Figure 4 compares incoherent efficiencies (Figure 4a) with those of coherent excitations with phase (Figure 4b). For simplicity, the results are shown as functions of instead of ; in all cases, is much less than and has a minor effect on the efficiency. Figure 4c shows that there is a distinct region where the coherent efficiency can exceed the incoherent one by as much as 30 percentage points. By contrast, when is small and large, Figures 4a and 4b show that donor-to-acceptor transfer is fast enough for efficiencies to be large for both excitation conditions, preventing a large enhancement. On the other hand, when is large and small, donor-to-acceptor coupling is too small for transfer rates to compete with the recombination rate, giving a low efficiency regardless of initial state.
Similar results can be obtained by propagating the approximate master equation in eq. 14, as shown in Figure 5. Across most of the parameter space, there is little difference between the estimated efficiency enhancements obtained from the full Redfield tensor and the approximate model, validating the simpler eq. 14 as a way to understand the origin and limitations of coherent efficiency enhancements.
The cause of efficiency enhancement are population transfers from the donor states to the acceptor that are mediated by the non-secular terms (those proportional to ) in eqs. 14a, 14b and 14c. In our case , so a negative causes a decrease in donor populations and an increase in acceptor populations, while a positive has the opposite effect. Since is negative when , observed donor-acceptor transfer rates are fastest when populations at are maximised. Furthermore, the sum of the additional terms is always 0, ensuring that eq. 14 is trace preserving.
In addition, because , the dephasing terms proportional to in eq. 14d matter only when has an imaginary component. Since our initial states have real coherences, in the limit , the coherence oscillates too slowly for its imaginary component to gain significant magnitude before the excitation transfers to the acceptor. Therefore, and due to the absence of pure dephasing, the coherences survive and maintain a positive real part long enough for the enhancements proportional to to be significant.
The approximate eq. 14 fails in the lower left part of Figure 5. This region is where the secular approximation with respect to the acceptor site fails. We assumed in eq. 14 that terms oscillating at frequency can be discarded due to their rapid oscillation, but this assumption fails when is small. In some cases, indicated with the white region in Figure 5, an efficiency could not be computed because the failure of the approximation led to populations becoming unphysical (either negative or greater than 1).
In summary, we have shown that excitonic coherences can significantly affect energy-transfer efficiency in a light-harvesting system. The coherences can be controlled by controlling the coherence of the exciting light; compared to incoherent excitation, engineered coherent light can double the light-harvesting efficiency for a dimeric donor. In larger systems, the enhancement could be even larger. The particular parameter regimes we explored were chosen to be realisable in engineered nanostructures, providing a platform for the development of new, quantum-inspired light-harvesting technologies.
Acknowledgements: We thank Andrew Doherty, Jacob Krich, Albert Stolow, and Joel Yuen-Zhou for valuable discussions. S.T. and I.K. were supported by the Westpac Bicentennial Foundation through a Westpac Research Fellowship, by the Australian Research Council through a Discovery Early Career Researcher Award (DE140100433), by an Australian Government Research Training Program (RTP) Scholarship, and by a scholarship from the University of Sydney Nano Institute. S.B. was supported in part by the U.S. National Science Foundation, Grant No. PHYS-1630114. S.R.-K. acknowledges financial support from the German Academic Exchange Service (DAAD) and Iran’s National Elites Foundation (INEF).
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