Cavity-modified exciton dynamics in photosynthetic units

TL;DR

This study shows that coupling a photosynthetic unit to an optical cavity can drastically speed up exciton transfer, potentially improving light-harvesting efficiency in natural and artificial systems.

Contribution

We develop a quantum master equation model to demonstrate cavity-induced acceleration of exciton dynamics in a photosynthetic complex.

Findings

Polaritons can be tuned to accelerate exciton transfer by three orders of magnitude.

Significant occupation of the LH1 complex occurs within a few femtoseconds after excitation.

Cavity coupling enables control over energy transfer pathways in photosynthetic units.

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 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 three 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.

Figures

Equations

$${\hat{H}}_{LH2}=\sum _{i=1}^{N_{LH2}}{\omega }_i{\hat{\sigma }}_i^{†}{\hat{\sigma }}_i+\sum _{i,j=1}^{N_{LH2}}{V}_{ij}\{{\hat{\sigma }}_i^{†}{\hat{\sigma }}_j+{\hat{\sigma }}_j^{†}{\hat{\sigma }}_i\},$$

$$\hat{H}$$

Full text

Light-harvesting (LH) complexes play a crucial role in the process of photosynthesis 1, 2. They are responsible for collecting, retaining, and funnelling solar energy 3, 4 into the reaction centers, where its conversion into chemical energy takes place 5. These pigment-protein compounds absorb the incident photons and convey the resulting electron-hole excitations through Förster-like, dipole-dipole interactions between neighboring molecules 6, 7. This mechanism is slower than vibrational dephasing in the system, which makes the transport process effectively incoherent 8, 9. Moreover, thanks to the extremely slow non-radiative decay inherent to bacteriochlorophyll molecules, energy transfer in photosynthetic membranes can range micrometric distances and take nanosecond times while having efficiencies approaching $100\%$ 10, 11. A paradigmatic example of phototrophic organisms, widely studied in the literature, are purple bacteria 12, 13 such as Rhodopseudomonas acidophila, in whose photosynthetic membranes two different complexes can be identified 14: LH2, which act mainly as optical antennas, and LH1, which deliver the excitation to the reaction center they enclose. Although the arrangement and distribution of both complexes within the bacterial membrane depends on the ambient and light intensity conditions, there is usually a number of LH2 in the vicinity of every LH1 and attached reaction center 10, 15.

In recent years, much research attention has been focused on exploring the opportunities that the phenomenon of exciton-photon collective strong coupling 16 brings into material science 17. The coupling between an excitonic platform and the electromagnetic modes supported by an optical cavity gives rise to polaritons, hybrid states whose formation requires that the interaction between light and matter become faster than their respective decay channels. Experimental and theoretical studies demonstrate that the appropriate tailoring of polaritonic characteristics in organic semiconductors and ensembles of organic molecules can yield a large enhancement of the efficiency and spatial range of charge and exciton conductance 18, 19, 20, 21 and energy transfer 22, 23, 24, 25, 26 in these systems. The coherent and delocalized nature of polaritons plays a crucial role in these phenomena. On the one hand, it allows energy transfer within a time scale set by the so-called Rabi frequency (collective coupling strength) 27. On the other hand, it makes the process nonlocal and robust to disorder within a length scale comparable to the optical wavelength 28.

It has been recently shown that plasmonic nanostructures can modify the optical properties of LH2 antennas 29, 30, 31, 32. Moreover, experimental evidence of collective strong coupling in ensembles of living bacteria has been reported 33, giving rise even to the concept of living polaritons 34. In Ref.  33 a Rabi splitting of around $150$ meV has been reported, implying that about one thousand 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 assesing the impact that the interaction with an optical cavity has on the efficiency of exciton transport taking place in purple bacterial photosynthetic units (PSUs) formed by several LH complexes. Using Bloch-Redfield theory 35, 36, which allows us to describe vibration-assisted incoherent interactions among bacteriochlorophyll 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 free-standing 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 neighboring 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 $\sim 10^{3}$, which leads to a considerable population of the LH1 complex within only a few femtoseconds 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 1(a) 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 final stage of the exciton transfer mechanism we analyze here. Taking this into account, we use a simplified model for the LH1 complex (red circle), valid in the low population regime. We treat it as a single two-level system with transition frequency $\omega_{\rm LH1}=1.417$ eV 38. Our attention is focused on the LH2 antennas, which we describe in more detail. They are composed of $N_{\rm LH2}=27$ pigments, distributed in a double-ring structure 39, 40, 41: while nine B800 molecules (blue dots) form 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 complex are modelled as interacting two-level systems leading to the Hamiltonian

[TABLE]

where $\hat{\sigma}^{\dagger}_{i}$ and $\hat{\sigma}_{i}$ are the creation and annihilation operators of molecular excitations, and $\omega_{i}$ their corresponding energies. The parameters in Eq. (1) are taken from Refs.  44 and  45. The freestanding-pigment energies are set to $\omega_{\rm B800}=1.549$ eV and $\omega_{\rm B850a,b}=1.520$ eV. The intra-ring nearest-neighbor (second-nearest-neighbor) couplings are $V_{\rm B800B800}=3$ meV, $V_{\rm B850aB850a}=6$ meV, $V_{\rm B850bB850b}=4$ meV, and $V_{\rm B850aB850b}=-33(-36)$ meV. Finally, the inter-ring nearest-neighbor interaction is parameterized by $V_{\rm B800B850a}=-3$ meV and $V_{\rm B800B850b}=-1$ meV.

The diagonalization of the Hamiltonian above yields the exciton energies of the LH2 complex. The small inter-ring couplings above translates into excitons strongly localized at B800 or B850(a,b) pigments, with little hybridization among them 40, 41. We check the validity of our model by calculating the absorption spectrum for a single, isolated LH2 antenna. We first compute the transition matrix element of the total dipole moment operator $\hat{M}=\sum_{i=1}^{N_{\rm LH2}}\mu_{i}\hat{\sigma}_{i}^{\dagger}$ ($\mu_{\rm B800}=\mu_{\rm B850}=6.13$ D 46) for the excitonic eigenstates of Eq. (1). The absorption spectrum is built as a sum of Lorentzian contributions centered at the excitonic energies and weighted by the square of the corresponding matrix element of the dipole moment operator. Their width was set to 15 meV for B800 and B850 excitons, in order to take into account the disorder and inhomogeneous broadening inherent to the measurements performed on ensembles of LH2 complexes 47. The spectrum obtained this way is rendered in Fig. 1(b), which reproduces the double-peaked absorption profile reported experimentally 47, 48, with maxima around 1.44 eV (860 nm) and 1.55 eV (800 nm).

We use a Bloch-Redfield master equation 6 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, $S(\omega)$. Fig. 1(c) plots $S(\omega)$ in our calculations (the same for B800 and B850 molecules), parameterized using the Franck-Condon factors in Ref.  42 and a thermal line broadening in agreement with Ref.  43. Lindblad terms of the form $\tfrac{\gamma_{0}}{2}\mathcal{L}_{\hat{\sigma}_{i}}[\hat{\rho}]=\tfrac{\gamma_{0}}{2}[2\hat{\sigma}_{i}\hat{\rho}\hat{\sigma}_{i}^{\dagger}-\{\rho\hat{\sigma}_{i}^{\dagger}\hat{\sigma}_{i}\}]$, acting on all pigment annihilation operators are also included in the master equation, weighted by a decay rate $\gamma_{0}=1$ $\mu$eV, which reflects the $\sim 1$ ns lifetime of all the molecules in the LH2 complex 4. Note that the exciton widths introduced in the cross section calculations are $\sim 10^{4}$ 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 Lindblad terms of the form $\tfrac{\gamma_{\rm B850LH1}}{2}\mathcal{L}_{\hat{\sigma}^{\dagger}_{\rm LH1}\hat{\sigma}_{n,1}}[\hat{\rho}]$. These act on the product of the annihilation operator for the B850 molecule (which we label as $i=1$) in the n-th LH2 antenna that is located next to the LH1 complex, and the LH1 creation operator. The associated decay rate is set to $\gamma_{\rm B850LH1}=2$ meV, which yields LH2-LH1 transition rates in agreement with experiments 37. This is shown in Fig. 1(d), which plots the 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 which the PSU is pumped by an ultrashort laser pulse centered around $800$ nm. We can observe that this state (blue line) decays within $\sim 3$ ps, feeding population into the LH2 B850 molecules (green line). These in turn carry the excitation to the LH1 complex (red line), whose population grows within a $\sim 20$ ps time scale after the initial excitation. The time interval in Fig. 1(d) is much shorter than $\gamma_{0}^{-1}$, and the ground state (shown in black line) is negligibly populated in the whole exciton transfer process. Note that we have taken $\gamma_{1}=0$, see Fig. 1(a), to avoid the decay of the LH1 excitations into the ground state.

In order to analyze the effect of strong coupling in the PSU exciton dynamics, we add new terms to the freestanding PSU Hamiltonian, describing the coherent interactions between pigments and cavity photons,

[TABLE]

where $\omega_{C}$ is the cavity frequency, $\hat{a}^{\dagger}$ and $\hat{a}$ are the creation and annihilation operators for the cavity mode, $g_{0}$ is the photon coupling strength to all LH2 pigments, and $\eta=3.8$ is the dipole moment of the LH1 complex normalized to $\mu_{\rm B800}$ 4. There, $\hat{H}_{{\rm LH2},n}$ is the Hamiltonian for the n-th LH2 antenna, see Eq.(1). As a result of the interaction between molecules and cavity and the symmetry of the structure, only four hybrid light-matter states arise when entering into the strong coupling regime in addition to the set of the so-called dark states. These states are linear combinations of electronic excitations within the LH1 and LH2 complexes that do not couple to the cavity mode. Figure 2(a) renders the four polaritonic frequencies, as well as the corresponding to the set of dark states, versus the cavity frequency. They are obtained directly from the diagonalization of Eq. (2). Color lines plot the dispersion of the lower (LP, yellow), middle (MP1 and MP2, green and blue) and upper (UP, violet) polariton bands. Note that, certainly, the PSU dark states (grey lines) remain uncoupled to the cavity field. In our calculations, we have taken $g_{0}=5$ meV. Using that $g_{0}=\mu_{\rm B800}\sqrt{\omega_{C}/2\epsilon_{0}V_{\rm eff}}$ 49, this value corresponds to $V_{\rm eff}=(20{\rm nm})^{3}$ at $\omega_{C}=1.6$ eV. This is attainable not only in plasmonic, but also in state-of-the-art dielectric cavities 50.

Figure 2(b) shows, from left to right, the square of the Hopfield coefficients for the LP, MP1, MP2 and UP as a function of the cavity frequency. Calculated as $|\langle i|\alpha\rangle|^{2}$, where $\alpha$ ($i$) labels the polaritonic (exciton and cavity) states, they weight the cavity (yellow), LH1 (red), B800 (blue) and B850 (green) contents of each polariton. We can observe that the polariton character can be greatly modified through $\omega_{C}$. Note that only the LP and UP present a substantial cavity content, but they do it at low and high cavity frequencies, respectively. Far from these spectral regions, LP (UP) virtually overlaps with the LH1 (B800) states. On the contrary, the MPs present a moderate cavity component, but combine excitonic contents corresponding to all PSU pigments. We anticipate that the hybrid character of these states (especially evident for MP2 at $\omega_{C}\simeq 1.5$ eV, where B800, B850 and cavity coefficients become similar) will play a fundamental role in the polariton-assisted population transfer in the PSU 26.

Having studied the formation of polaritons in the hybrid cavity-PSU system, and the tuning of their characteristics through the cavity frequency, we investigate next its population dynamics. To do so, we extend the master equation for the freestanding PSU by the inclusion of the Hamiltonian in Eq. (2), and by adding a Lindblad term describing the cavity losses. We set the cavity decay rate to $\gamma_{C}=13\,\mu$eV, which corresponds to a lifetime of 50 ps and a quality factor $Q=\omega_{C}/2\gamma_{C}\simeq 6\cdot 10^{4}$, parameters similar to those recently reported in deeply subwavelength dielectric cavities 50. Similarly to Fig. 1(d), we set the initial state as the coherent superposition of equally excited B800 molecules, and choose $\omega_{C}=1.6$ eV and $g_{0}=15$ meV. Fig. 2(b) shows that the LP is composed of LH1 excitations mostly at this cavity frequency, which allows us to set the final stage of the polariton-assisted energy transfer mechanism at the LH1 complex.

Figure 3 displays the comparison between the population dynamics for the PSU in isolation (dashed lines) and interacting with the cavity described above (solid lines). Fig. 3(a) shows that exciton-photon strong coupling in the PSU gives rise to an extremely fast occupation of the LH1 complex, which acquires a significant population ($\sim 10\%$) within only a 20 fs delay. Note that, in absence of the cavity, the LH1 population is negligible in this time scale, and becomes comparable only after a few ps, see Fig. 1(d). This is the main result in this Letter, the polariton-assisted reduction in population transfer times taking place in PSUs by three orders of magnitude. Fig. 3(b) plots the population transients for B800 (blue), and B850 (green) excitons, both exhibiting more regular Rabi oscillations than the LH1. These are especially apparent in the B800 case. Its occupation remains constant and close to unity for the freestanding PSU, but the occurrence of strong coupling gives rise to a coherent energy exchange that feeds population into the other excitonic and cavity (yellow line) states. Black lines correspond to the ground state, whose population is larger in the strong coupling regime. This is a consequence of the short lifetime of the cavity relative to the PSU pigments ($\gamma_{C}\sim\gamma_{0}/20$). This loss channel can be mitigated by using nanocavities with higher quality factors.

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: $\omega_{C}$ and $g_{0}$. Figure 4(a) displays a contourplot of the LH1 population averaged over the first 300 fs after the excitation of the B800 molecules (the time span in Fig. 3). Note that the temporal averaging naturally removes peak effects related to the irregular Rabi oscillations apparent in Fig. 3(a). We can observe that by increasing the exciton-photon coupling (reducing the cavity mode volume) the LH1 population grows, with a much weaker dependence on $\omega_{C}$. This is a surprising result, given the strong dependence of the Hopfield coefficients, and therefore the polariton character, on the cavity frequency shown in Fig. 2(b).

Figure 4(b) plots the combination of content of B800 and LH1 excitations on the various polaritons $\alpha$, $\chi_{\alpha}=|\langle\alpha|{\rm LH1}\rangle|^{2}|\langle\alpha|{\rm B800}\rangle|^{2}$, as a function of the cavity frequency and coupling strength. We can observe that, as expected, $\chi_{\alpha}$ grows with $g_{0}$ in all cases, as the light-matter hybridization increases with this parameter. However, the B800 and LH1 projections over the different polaritons varies much with $\omega_{C}$. For large couplings ($g_{0}>10$ meV), $\chi_{\rm LP}$ dominates LH1 excitations for blue-detuned cavities, whereas $\chi_{\rm MP_{2}}$ and $\chi_{\rm UP}$ are the largest contributions for red-detuned ones. On the other hand, for modest couplings ($g_{0}\lesssim 10$ meV), $\chi_{\rm LP}$ and $\chi_{\rm MP_{1}}$ are largest for red-detuned cavities, while $\chi_{\rm MP_{2}}$ and $\chi_{\rm UP}$ present a maximum within the spectral window between 1.5 and 1.6 eV. These panels indicate that the interplay among the different polaritons play a crucial role in the fast B800-to-LH1 population transfer in PSUs. This conclusion is supported by the contour lines in Fig. 4(a), which render $\sum_{\alpha}\chi_{\alpha}$, showing that this magnitude presents the same dependence on $\omega_{C}$ and $g_{0}$ as the LH1 averaged population.

To conclude, we have investigated exciton-photon strong coupling in an archetypal purple bacterial photosynthetic unit, comprising six LH2 antennas surrounding a single LH1 complex. We have developed a master equation combining Bloch-Redifield and Lindblad approaches to describe the 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 three 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 characterization of light-harvesting phenomena in natural photosynthesis, but also as a means for the design and optimization of artificial photosynthetic systems.

1 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 MINECO under contracts MAT2014-53432-C5-5-R and FIS2015-64951-R, and through the “María de Maeztu” programme for Units of Excellence in R&D (MDM-2014-0377), as well as through two Ramon y Cajal grants (JF and AIFD). 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.