This is an open access article published under a Creative Commons Non-Commercial No Derivative Works (CC-BY-NC-ND) Attribution License, which permits copying and redistribution of the article, and creation of adaptations, all for non-commercial purposes. Letter Cite This: J. Phys. Chem. Lett. 2019, 10, 4252−4258 pubs.acs.org/JPCL Cavity-Modified Exciton Dynamics in Photosynthetic Units Rocío Sáez-Blázquez,† Johannes Feist,† Elisabet Romero,‡ Antonio I. Fernández-Domínguez,*,† and Francisco J. García-Vidal*,†,§ † Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain ‡ Institute of Chemical Research of Catalonia (ICIQ), Barcelona Institute of Science and Technology (BIST), E-43007 Tarragona, Spain § Donostia International Physics Center (DIPC), E-20018 Donostia−San Sebastián, Spain ABSTRACT: Recently, exciton−photon strong coupling has been proposed as a means to control and enhance energy transfer in ensembles of organic molecules. Here, we demonstrate that the exciton dynamics in an archetypal purple bacterial photosynthetic unit, composed of six LH2 antennas surrounding a single LH1 complex, is greatly modified by its interaction with an optical cavity. We develop a Bloch−Redfield master equation approach that accounts for the interplay between the B800 and B850 bacteriochlorophyll molecules within each LH2 antenna, as well as their interactions with the central LH1 complex. Using a realistic parametrization of both the photosynthetic unit and optical cavity, we investigate the formation of polaritons in the system, revealing that these can be tuned to accelerate its exciton dynamics by 3 orders of magnitude. This yields a significant occupation of the LH1 complex, the stage immediately prior to the reaction center, with only a few-femtosecond delay after the initial excitation of the LH2 B800 pigments. Our theoretical findings unveil polaritonic phenomena as a promising route for the characterization, tailoring, and optimization of light-harvesting mechanisms in natural and artificial photosynthetic processes. L ight-harvesting (LH) complexes play a crucial role in the process of photosynthesis.1,2 They are responsible for collecting, retaining, and funnelling solar energy3,4 into the exciton−photon collective strong coupling16 brings into material science.17 The coupling between an excitonic platform and the electromagnetic modes supported by an optical cavity reaction centers, where its conversion into chemical energy gives rise to polaritons, hybrid states whose formation requires takes place.5 These pigment−protein compounds absorb the that the interaction between light and matter become faster incident photons and convey the resulting electron−hole than their respective decay channels. Experimental and excitations through Förster-like, dipole−dipole interactions theoretical studies demonstrate that the appropriate tailoring between neighboring molecules.6,7 This mechanism is slower of polaritonic characteristics in organic semiconductors and than vibrational dephasing in the system, which makes the ensembles of organic molecules can yield a large enhancement transport process effectively incoherent.8,9 Moreover, thanks to of the efficiency and spatial range of charge and exciton the extremely slow nonradiative decay inherent to bacterio- conductance18−21 and energy transfer22−26 in these systems. chlorophyll molecules, energy transfer in photosynthetic The coherent and delocalized nature of polaritons plays a membranes can range micrometric distances and take crucial role in these phenomena. On one hand, it allows energy nanoseconds while having efficiencies approaching 100%.10,11 transfer within a time scale set by the so-called Rabi frequency A paradigmatic example of phototrophic organisms, widely (collective coupling strength).27 On the other hand, it makes studied in the literature, is purple bacteria12,13 such as the process nonlocal and robust to disorder within a length Rhodopseudomonas acidophila, in whose photosynthetic mem- branes two different complexes can be identified:14 LH2, which scale comparable to the optical wavelength.28 act mainly as optical antennas, and LH1, which deliver the It has been recently shown that plasmonic nanostructures excitation to the reaction center they enclose. Although the can modify the optical properties of LH2 antennas.29−32 arrangement and distribution of both complexes within the Moreover, experimental evidence of collective strong coupling bacterial membrane depend on the ambient and light intensity in ensembles of living bacteria has been reported,33 giving rise conditions, there is usually a number of LH2 in the vicinity of every LH1 and attached reaction center.10,15 Received: May 24, 2019 In recent years, much research attention has been focused Accepted: July 10, 2019 on exploring the opportunities that the phenomenon of Published: July 10, 2019 © 2019 American Chemical Society 4252 DOI: 10.1021/acs.jpclett.9b01495 J. Phys. Chem. Lett. 2019, 10, 4252−4258 The Journal of Physical Chemistry Letters Letter even to the concept of living polaritons.34 In ref 33, a Rabi splitting of around 150 meV has been reported, implying that about 1000 chlorosomes present in green sulfur bacteria are coherently coupled to a cavity photon. In the absence of a cavity, the study of exciton transport in photosynthetic materials has been triggered by the prospect of transferring this knowledge to human-made energy-harvesting structures. In this Letter, we go a step further by assessing the impact that the interaction with an optical cavity has on the efficiency of exciton transport taking place in purple bacterial photo- synthetic units (PSUs) formed by several LH complexes. Using Bloch−Redfield theory,35,36 which allows us to describe vibration-assisted incoherent interactions among bacteriochlor- ophyll pigments,6,9 we construct first a quantum master equation describing a single LH2 antenna, involving 27 interacting pigments of three different families (B800, B850a, and B850b). Our model reproduces experimental absorption spectra of freestanding LH2. Next, we consider an archetypal PSU configuration:10,37 a ring of six LH2 antennas surrounding a single LH1 complex. We extend our master equation to the whole PSU, including incoherent interactions among neighbor- ing pigments within different LH complexes. By introducing pigment−photon coupling terms in the Hamiltonian, we study the formation of polaritons in the system, with special emphasis on the cavity characteristics. We find that strong coupling in realistic cavities can accelerate PSU exciton dynamics by a factor ∼103, which leads to a considerable population of the LH1 complex within only a few femto- seconds after the initial excitation of LH2 B800 pigments. Our model also reveals how the contribution of the different polaritonic states to this fast population transfer depends on the frequency of the cavity mode and its effective volume (or pigment−photon coupling strength). Figure 1a sketches the PSU configuration under study: six LH2 antennas arranged around a single LH1 complex. The latter is formed by a number of B875 pigments,4 which are the Figure 1. (a) Sketch of the PSU considered in this work: 6 LH2 final stage of the exciton transfer mechanism we analyze here. antennas, comprising 9 B800 and 18 B850 molecules each, Taking this into account, we use a simplified model for the surrounding a single LH1 complex. The insets show the two-level LH1 complex (red circle), valid in the low population regime. system exciton model of B-molecules and LH1, which experience We treat it as a single two-level system with transition both radiative and vibrational decay. (b) Absorption spectrum of a single LH2 complex, including disorder and inhomogeneous broad- frequency ωLH1 = 1.417 eV.38 Our attention is focused on the ening. (c) Vibrational spectral density, S(ω), for all LH2 pigments. LH2 antennas, which we describe in more detail. They are (d) Exciton population dynamics for the PSU in panel a in an initial composed of NLH2 = 27 pigments, distributed in a double-ring state given by the superposition of excited B800 molecules in the six structure:39−41 while nine B800 molecules (blue dots) form LH2 antennas. one of the rings, the other comprises nine pigment dimers made up of a B850a and a B850b molecule each (light and dark green dots). In our model, the pigments in each LH2 The diagonalization of the Hamiltonian above yields the complex are modeled as interacting two-level systems leading exciton energies of the LH2 complex. The small inter-ring to the Hamiltonian couplings above translate into excitons strongly localized at B800 or B850(a,b) pigments, with little hybridization among NLH2 NLH2 them.40,41 We check the validity of our model by calculating ĤLH2 = ∑ ωiσî†σî + ∑ Vij{σî†σĵ + σĵ†σî} the absorption spectrum for a single, isolated LH2 antenna. We i=1 i,j=1 (1) first compute the transition matrix element of the total dipole moment operator M̂ = ∑Ni=1LH2 μiσ̂ †i (μB800 = μB850 = 6.13 D46) where σ̂ †i and σ̂ i are the creation and annihilation operators of for the excitonic eigenstates of eq 1. The absorption spectrum molecular excitations, and ωi is their corresponding energies. is built as a sum of Lorentzian contributions centered at the The parameters in eq 1 are taken from refs 44 and 45. The excitonic energies and weighted by the square of the freestanding-pigment energies are set to ωB800 = 1.549 eV and corresponding matrix element of the dipole moment operator. ωB850a,b = 1.520 eV. The intraring nearest-neighbor (second- Their width was set to 15 meV for B800 and B850 excitons, to nearest-neighbor) couplings are VB800B800 = 3 meV, VB850aB850a take into account the disorder and inhomogeneous broadening = 6 meV, VB850bB850b = 4 meV, and VB850aB850b = −33(−36) inherent to the measurements performed on ensembles of LH2 meV. Finally, the inter-ring nearest-neighbor interaction is complexes.47 The spectrum obtained this way is rendered in parametrized by VB800B850a = −3 meV and VB800B850b = −1 meV. Figure 1b, which reproduces the double-peaked absorption 4253 DOI: 10.1021/acs.jpclett.9b01495 J. Phys. Chem. Lett. 2019, 10, 4252−4258 The Journal of Physical Chemistry Letters Letter profile reported experimentally,47,48 with maxima around 1.44 couple to the cavity mode. Figure 2a renders the four eV (860 nm) and 1.55 eV (800 nm). polaritonic frequencies, as well as those corresponding to the We use a Bloch−Redfield master equation6 to describe the vibrational dissipation and incoherent interactions experienced by B800 and B850 molecules. This requires the inclusion of the vibronic spectral density of the pigments, S(ω). Figure 1c plots S(ω) in our calculations (the same for B800 and B850 molecules), parametrized using the Franck−Condon factors in ref 42 and a thermal line broadening in agreement with ref 43. γ γ † † Lindblad terms of the form 20 3 σî [ρ ̂] = 20 [2σρσ î ̂ ̂ i − {ρσî σî }], acting on all pigment annihilation operators, are also included in the master equation, weighted by a decay rate γ0 = 1 μeV, which reflects the ∼1 ns lifetime of all the molecules in the LH2 complex.4 Note that the exciton widths introduced in the cross-section calculations are ∼104 times larger than this value. The master equation for the whole PSU is built next. It is composed by blocks, corresponding to the six LH2 antennas and the central LH1 complex, only connected through γ Lindblad terms of the form B850LH1 2 3 σLH1 ̂ † σn̂ ,1[ρ ]̂ . These act on the product of the annihilation operator for the B850 molecule (which we label as i = 1) in the nth LH2 antenna that is Figure 2. (a) Energies of the lower (LP, yellow), middle (MP1, green, located next to the LH1 complex, and the LH1 creation and MP2, blue), and upper (UP, violet) polaritons versus the cavity operator. The associated decay rate is set to γB850LH1 = 2 meV, frequency, ωC, and for g0 = 9 meV. (b) Coefficients representing the which yields LH2−LH1 transition rates in agreement with cavity (yellow), LH1 (red), B800 (blue), and B850 (green) content of experiments.37 This is shown in Figure 1d, which plots the the four polaritons in panel a as a function of ωC. population transients for a freestanding PSU. The initial state corresponds to the coherent superposition of excitations in all the B800 pigments, which mimics an experimental setup in set of dark states, versus the cavity frequency. They are which the PSU is pumped by an ultrashort laser pulse centered obtained directly from the diagonalization of eq 2. Color lines around 800 nm. We can observe that this state (blue line) plot the dispersion of the lower (LP, yellow), middle (MP1 and decays within ∼3 ps, feeding population into the LH2 B850 MP2, green and blue), and upper (UP, violet) polariton bands. molecules (green line). These in turn carry the excitation to Note that, certainly, the PSU dark states (gray lines) remain the LH1 complex (red line), whose population grows within a uncoupled to the cavity field. In our calculations, we have ∼20 ps time scale after the initial excitation. The time interval taken g0 = 9 meV. Using g0 = μB800 ωC/2ϵ0Veff ,49 this value in Figure 1d is much shorter than γ−1 0 , and the ground state corresponds to Veff = (15 nm)3 at ωC = 1.6 eV. This is (shown in black line) is negligibly populated in the whole attainable not only in plasmonic but also in state-of-the-art exciton transfer process. Note that we have taken γ1 = 0, see dielectric cavities.50 Figure 1a, to avoid the decay of the LH1 excitations into the Figure 2b shows, from left to right, the square of the ground state. Hopfield coefficients for the LP, MP1, MP2, and UP as a To analyze the effect of strong coupling in the PSU exciton function of the cavity frequency. Calculated as |⟨i|α⟩|2, where dynamics, we add new terms to the freestanding PSU α(i) labels the polaritonic (exciton and cavity) states, they Hamiltonian, describing the coherent interactions between weight the cavity (yellow), LH1 (red), B800 (blue), and B850 pigments and cavity photons (green) contents of each polariton. We can observe that the 6 polariton character can be greatly modified through ωC. Note Ĥ = ωCa†̂ a ̂ + ωLH1σLH1 ̂ † σLH1 ̂ + ∑ ĤLH2,n that only the LP and UP present a substantial cavity content, ÄÅ 6 ÉÑ ÅÅ ÑÑ but they do it at low and high cavity frequencies, respectively. Å Ñ n=1 + g0ÅÅÅ∑ ̂ a ̂ )ÑÑÑ Far from these spectral regions, LP (UP) virtually overlaps ÅÅ ÑÑ NLH2 ÅÇ n = 1 ÑÖ with the LH1 (B800) states. On the contrary, the MPs present ∑ (σn̂†, ia ̂ † + σn̂ , ia ̂ ) + ̂† â η(σLH1 + σLH1 † a moderate cavity component but combine excitonic contents i=1 corresponding to all PSU pigments. We anticipate that the (2) hybrid character of these states (especially evident for MP2 at † where ωC is the cavity frequency, â and â are the creation and ωC ≃ 1.45 eV, where B800 and B850 coefficients become annihilation operators for the cavity mode, g0 is the photon similar) will play a fundamental role in the polariton-assisted coupling strength to all LH2 pigments, and η = 3.8 is the population transfer in the PSU.26 dipole moment of the LH1 complex normalized to μB800.4 Having studied the formation of polaritons in the hybrid There, Ĥ LH2,n is the Hamiltonian for the nth LH2 antenna (see cavity−PSU system, and the tuning of their characteristics eq 1). As a result of the interaction between molecules and through the cavity frequency, we investigate next its population cavity and the symmetry of the structure, only four hybrid dynamics. To do so, we extend the master equation for the light−matter states arise when entering into the strong- freestanding PSU by the inclusion of the Hamiltonian in eq 2, coupling regime, in addition to the set of the so-called dark and by adding a Lindblad term describing the cavity losses. We states. These states are linear combinations of electronic set the cavity decay rate to γC = 13 μeV, which corresponds to excitations within the LH1 and LH2 complexes that do not a lifetime of 50 ps and a quality factor Q = ωC/2γC ≃ 6 × 104, 4254 DOI: 10.1021/acs.jpclett.9b01495 J. Phys. Chem. Lett. 2019, 10, 4252−4258 The Journal of Physical Chemistry Letters Letter parameters similar to those recently reported in deeply B800 case. Its occupation remains constant and close to unity subwavelength dielectric cavities.50 Similarly to Figure 1d for the freestanding PSU, but the occurrence of strong and to perform a meaningful comparison against the coupling gives rise to a coherent energy exchange that feeds freestanding PSU, we set the initial state as the coherent population into the other excitonic and cavity (yellow line) superposition of equally excited B800 molecules, and choose states. Black lines correspond to the ground state, whose ωC = 1.6 eV and g0 = 9 meV. Figure 2b shows that the LP is population is larger in the strong-coupling regime. This is a composed of LH1 excitations mostly at this cavity frequency, consequence of the short lifetime of the cavity relative to the which allows us to set the final stage of the polariton-assisted PSU pigments (γC ∼ γ0/20). This loss channel can be energy transfer mechanism at the LH1 complex. mitigated by using nanocavities with higher-quality factors. Figure 3 displays the comparison between the population Note that all the time traces in Figure 3 were calculated dynamics for the PSU in isolation (dashed lines) and assuming the same coupling strength for all the B800 and B850 pigments in the PSU (see eq 2). Importantly, our findings hold beyond this approximation, as long as the PSU−cavity Rabi frequency remains the same as in the uniform description.26 Up to this point, we have demonstrated that the exciton dynamics in PSUs is greatly modified due to the interaction with a particular optical cavity configuration. We have also shown that this phenomenon is mediated by the polaritons that emerge in the system, whose character varies strongly with the frequency of the cavity. In the following, we shed insights into both findings by investigating the dependence of the B800-to-LH1 population transfer and the polaritonic content of B800 and LH1 excitations on the two parameters set by the optical cavity: ωC and g0. Figure 4a displays a contour plot of the LH1 population averaged over the first 300 fs after the excitation of the B800 molecules (the time span in Figure 3). Note that the temporal averaging naturally removes peak effects related to the irregular Rabi oscillations apparent in Figure 3a. We can observe that, by increasing the exciton− photon coupling (reducing the cavity mode volume), the LH1 population grows, although not in a purely monotonic fashion (small oscillations are apparent). On the contrary, the dependence on ωC is much weaker. This is a surprising result, given the strong dependence of the Hopfield coefficients, and therefore the polariton character, on the cavity frequency shown in Figure 2b. Figure 4b plots the combination of content of B800 and Figure 3. Exciton dynamics for the PSU in Figure 1. (a) LH1 LH1 excitations on the various polaritons α, χα = |⟨α|LH1⟩|2 population versus time after the initial excitation of the B800 |⟨α|B800⟩|2, as a function of the cavity frequency and coupling molecules for the PSU isolated (dashed line) and coupled to an optical cavity with ωC = 1.6 eV and g0 = 9 meV (solid line). The thin strength. We can observe that, as expected, χα grows with g0 in solid line plots the LH1 population for an initial excitation of the all cases, as the light−matter hybridization increases with this cavity mode. (b) Temporal evolution of the ground state (black), parameter. However, the B800 and LH1 projections over the B800 (blue), and B850 (green) populations with (solid line) and different polaritons vary much with ωC. For large couplings (g0 without (dashed line) cavity. The cavity population is shown in the > 10 meV), χLP dominates LH1 excitations for blue-detuned yellow solid line. The differences between populations have been cavities, whereas χMP2 and χUP are the largest contributions for shaded in all cases to highlight the effect of strong coupling. red-detuned ones. On the other hand, for modest couplings (g0 ≲ 10 meV), χLP and χMP1 are largest for red-detuned cavities, interacting with the cavity described above (solid lines). Figure 3a shows that exciton−photon strong coupling in the while χMP2 and χUP present a maximum within the spectral PSU gives rise to an extremely fast occupation of the LH1 window between 1.5 and 1.6 eV. These panels indicate that the complex, which acquires a significant population (∼10%) interplay among the different polaritons plays a crucial role in within only a 20 fs delay. In absence of the cavity, the LH1 the fast B800-to-LH1 population transfer in PSUs. This population is negligible in this time scale and becomes conclusion is supported by the contour lines in Figure 4a, comparable only after a few ps (see Figure 1d). This is the which render ∑αχα, showing that this magnitude presents the main result in this Letter, the polariton-assisted reduction in same dependence on ωC and g0 as the LH1 averaged population transfer times taking place in PSUs by 3 orders of population. magnitude. The thin solid line renders the LH1 population Finally, Figure 4c displays the LH1 population evaluated 40 when the cavity, instead of the B800 pigments, is initially ps after the initial B800 excitation. We can observe that, for the excited, proving that this phenomenon also takes place in this cavity parameters in Figure 3 (which maximize its population configuration. at short times), the LH1 occupation at long times is ∼50%, Figure 3b plots the population transients for B800 (blue) lower than for the freestanding PSU (see Figure 1d). The and B850 (green) excitons, both exhibiting more regular Rabi origin of this low LH1 occupation at long times can be found oscillations than the LH1. These are especially apparent in the in Figure 2b. Note that the Hopfield coefficients for the LP 4255 DOI: 10.1021/acs.jpclett.9b01495 J. Phys. Chem. Lett. 2019, 10, 4252−4258 The Journal of Physical Chemistry Letters Letter vibration-assisted incoherent interactions among the B800, B850, and LH1 excitons, as well as their coherent coupling to the electromagnetic mode supported by an optical cavity. Using this tool, we have explored the formation of polaritons in the system, analyzing their dependence on the cavity configuration. We have revealed that these hybrid light− matter states yield a 3 orders of magnitude reduction in the B800-to-LH1 population transfer times, leading to a significant LH1 occupation in a few-femtosecond time scale. We believe that our theoretical findings demonstrate the potential of exciton−photon strong coupling not only for the character- ization of light-harvesting phenomena in natural photosyn- thesis but also as a means for the design and optimization of artificial photosynthetic systems. ■ AUTHOR INFORMATION Corresponding Authors *E-mail: a.fernandez-dominguez@uam.es. *E-mail: fj.garcia@uam.es. ORCID Johannes Feist: 0000-0002-7972-0646 Antonio I. Fernández-Domínguez: 0000-0002-8082-395X Francisco J. García-Vidal: 0000-0003-4354-0982 Notes The authors declare no competing financial interest. ■ ACKNOWLEDGMENTS This work has been funded by the European Research Council under Grant Agreements ERC-2011-AdG 290981 and ERC- 2016-STG-714870, the EU Seventh Framework Programme (FP7-PEOPLE-2013-CIG-630996), and the Spanish MINE- CO under contracts MAT2014-53432-C5-5-R and FIS2015- 64951-R, and through the “Marı ́a de Maeztu” programme for Units of Excellence in R&D (MDM-2014-0377), as well as through two Ramon y Cajal grants (J.F. and A.I.F.-D.). We also acknowledge support by the QuantERA program of the European Commission with funding by the Spanish AEI through project PCI2018-093145. E.R. thanks the ICIQ Foundation for the Starting Career Programme and the Generalitat de Catalunya for the CERCA Programme. ■ REFERENCES (1) Scholes, G. D.; Fleming, G. R.; Olaya-Castro, A.; van Grondelle, Figure 4. (a) LH1 population averaged over the first 300 fs after the R. 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