Photosynthetic antenna complexes harvest sunlight and efficiently transport energy to the reaction center where charge separation powers biochemical energy storage. The discovery of existence of long lived quantum coherence during energy transfer has sparked the discussion on the role of quantum coherence on the energy transfer efficiency. Early works assigned observed coherences to electronic states, and theoretical studies showed that electronic coherences could affect energy transfer efficiency—by either enhancing or suppressing transfer. However, the nature of coherences has been fiercely debated as coherences only report the energy gap between the states that generate coherence signals. Recent works have suggested that either the coherences observed in photosynthetic antenna complexes arise from vibrational wave packets on the ground state or, alternatively, coherences arise from mixed electronic and vibrational states. Understanding origin of coherences is important for designing molecules for efficient light harvesting. Here, we give a direct experimental observation from a mutant of LH2, which does not have B800 chromophores, to distinguish between electronic, vibrational, and vibronic coherence. We also present a minimal theoretical model to characterize the coherences both in the two limiting cases of purely vibrational and purely electronic coherence as well as in the intermediate, vibronic regime.

The remarkable quantum efficiency of energy transfer from light harvesting antenna complex to the reaction center (RC) has attracted immense experimental and theoretical studies.1–4 While incoherent (or hopping) dynamics has been found to be the dominant mechanism of energy transfer, it is not the only mechanism.5 Coherent dynamics involves ballistic energy flow between sites. It has been suggested that energy transfer is characterized by interplay of the two regimes.5,6 The microscopic distinction between the regimes arises from how the bath interacts with the electronic states. While four-wave mixing experiments had been employed to understand coherent and incoherent nuclear motion and energy transfer dynamics in biological systems,7,8 the development of two-dimensional electronic spectroscopy (2DES) has facilitated detailed analysis of four-wave mixing signals by resolving absorption and emission frequencies.9–17 Recent observations of long lived coherences in FMO and reaction center were attributed to electronic states,8,15 and it was hypothesized that the protein scaffold of the antenna complex protects coherences, through correlated bath fluctuation, to enhance the quantum efficiency in energy transfer.16 Theoretical works by Aspuru-Guzik et al.18 and Plenio and Huelga19 support the claim that electronic coherences can indeed enhance quantum efficiency in energy transfer. These theories involve electronic states coupled weakly to a vibrational bath. However, alternative interpretations, that coherences induced by laser excitation are vibronic in nature, comply with experimental findings as well.20–22 These theories take some vibrational modes out of the bath and allow them to mix explicitly with the electronic states.23 In this vibronic model, electronic coherences borrow coherence lifetime from vibrational coherences by mixing electronic and vibrational states of the chromophores that make up the photosynthetic complex. These models are palatable because the vibronic model considers Franck-Condon allowed vibrational states of the chromophores and does away with the hypothesis of correlated bath fluctuations, invoked to explain long-lived electronic coherences. In this manuscript, we present experimental results that directly characterize the nature of coherences in a mutant of LH2 (R26.1 LH2) from Rhodobacter (Rb.) sphaeriodes, which does not have B800 chromophores.24 We also present a minimal theoretical model to estimate the extent of vibronic coupling and characterize the coherences between the two limiting cases of vibrational coherence and electronic coherence. We find the coherences to have some vibronic character, but a rather small mixing angle of about 13°-17°.

The strength of the signal in a 2DES experiment depends on the strength of the four system-field interactions, each given by μ E where μ is the transition dipole moment of the system under study and E is the electric field strength of the excitation pulse. The direction of E is determined by the polarization of the excitation pulse. Because the pulses’ polarizations are experimentally controlled, the relative angle between the four transition dipoles directly governs the signal amplitude.25 The signal’s amplitude dependence on the polarization of the electric fields has been used to determine peptide structure in proteins by determining the angle between transition dipoles, resolve 2D spectra, and study coherent dynamics in LH2.26–30 In this experiment, we select a pulse polarization scheme to distinguish between electronic and vibrational coherence which are characterized by different angles between the transition dipoles that give rise to the coherence signal.

The details of our GRAPES optical apparatus are described elsewhere.31,32 Briefly, a Coherent Micra Ti:sapphire oscillator seeds a Coherent Legend Elite USP-HE regenerative amplifier to generate 30 fs, transform-limited pulses centered at 805 nm (30 nm FWHM) with a 5 kHz repetition rate. Additional bandwidth is achieved by focusing the pulse in argon gas (∼2 psi) to generate ∼90 nm FWHM pulse with 0.5% power stability (10 Hz measurement, 15 min duration). A 50:50 beam splitter and two wedged optics are used to create four pulses that are focused to a line in a homogeneous, flowing sample. The pulse is compressed at the sample using the multiphoton intrapulse interference phase scan method (Biophotonics Solution, Inc.) to get ∼15 fs pulses.33 The resulting fluence is 14 μJ/cm2 per pulse. The optical density (OD) of R26.1 measured in a 1 mm Starna cell is 1.4 at 850 nm (see supplementary material).34 The R26.1 LH2 sample is placed in a 200 μm Starna cell, and 2DES experiments are performed with canonical 2D and coherence-specific polarization sequences. The planes of pulse-polarizations are altered using half wave plates (ThorLabs), and calibrated using a Glan-Thompson polarizer (attenuation factor 10−5). First, the plane of polarization of beam 4 was determined using a Glan-Thompson polarizer. To calibrate the polarization to beam 1, the polarizer was turned by 45° ± 0.5° (the error set by the least count on the dial of the polarizer’s mount) and the half-wave plate in the path of beam 1 was turned to achieve an attenuation of 10−5, thus, effectively calibrating beam 1 to 45° ± 0.5° with respect to beam 4. Similar procedures were followed to characterize pulses 2 and 3, thus, enabling us to achieve the polarization states +45° ± 0.5°, −45° ± 0.5°, +90° ± 0.5°, and 0° for beams 1, 2, 3, and 4. To verify polarizations and check for any ellipticity, we confirm the absence of a measurable interferogram from perpendicularly polarized pulses (e.g., beam 3-4 pair) in a thermally cooled CCD camera (Andor Instruments).

The sample preparation was, as far as practically possible, carried out in the dark. Approximately ∼5 g of pelleted R26.1 cells were suspended in ∼30 ml of 20 mM Tris.Cl pH 8.0 and a few grains of MgCl2 and DNase were added to degrade the released DNA during cellular disruption. The sample was homogenised and passed three times through a French Press cell (at 15 000 psi) to ensure complete disruption. The resulting suspension was centrifuged at low-speed (Sorvall SS-34 rotor) at 3000 × g for 10 min at 4 °C. The pellet was discarded and the supernatant further centrifuged (Beckman Ti70 rotor) at 180 000 × g for 2 h at 4 °C. The chromatophore pellet was resuspended in 20 mM Tris.Cl pH 8.0 at the NIR maximum to an OD of 50 cm−1. The membranes were then solubilised by adding 4% decylmaltoside (DM) (w/v) and the sample stirred for 60 min at room temperature in the dark. The sample was then centrifuged for 30 min at 27 000 × g (Sorvall SS-34 rotor) to remove denatured protein and any non-solubilised material. The supernatant was then loaded onto a discontinuous sucrose gradient using 0.2 M steps from 1.8 to 1.0 M sucrose, 0.15% DM, 20 mM Tris.Cl pH 8.0 and centrifuged (Beckman Ti70 rotor) 150 000 × g for 14 h at 4 °C. This gradient effectively separates the LH2 complex from the LH1-RC. The sucrose solution band containing the LH2 complex was carefully removed from each tube and pooled. The LH2 complex was then purified further by anion exchange chromatography using DE52 resin (Whatman Scientific) and then by size exclusion chromatography on a Superdex-200 (GE Healthcare) column. LH2 fractions were collected, assayed, the best fractions pooled. The LH2 was concentrated and flash-frozen in suitable aliquots until required.

We present results from 2DES studies of a mutant of LH2 (R26.1 LH2) from Rb. sphaeriodes, that does not have B800 chromophores, using two pulse polarization schemes. The first is the “canonical polarization” sequence routinely used in 2DES experiments. In this pulse polarization sequence, all the pulses have the same (parallel) linear polarization. The second pulse polarization sequence is the “coherence-specific” polarization sequence in which the first, second, and third pulse make +45°, −45°, and +90° with respect to the local oscillator (LO). Here, “+” and “−” refer to clockwise and anti-clockwise direction when looking in the direction of pulse propagation.

Figure 1 (top row) shows the 2D spectra of rephasing signal from R26.1 LH2 acquired at room temperature using the canonical 2D polarization sequence. The spectra are for T = 50, 150, 250, 350, 450, and 550 fs. The coherence frequency, λτ, can be interpreted as absorption frequency and the rephasing frequency, λt, can be interpreted as emission frequency. The intensity of the diagonal peaks represents populations. In Figure 2(a), the diagonal peak centered at λτ = 780 nm and λt = 780 nm, marked in the white box, represents the population of B850* states. In Figure 2(b), we show the B850* peak on a different colormap. Off-diagonal peaks arise from energy transfer between the two bands. The intensity of the lower off-diagonal peak increases with waiting time due to energy transfer from B850* states to B850 states. However, the intensity of the lower off-diagonal starts to decrease after ∼150 fs. The decrease in strength of the lower off-diagonal feature can be attributed to out of band relaxation of B850 states to its fluorescence state, and localization of exciton in the B850 ring (see supplementary material for more details).34 The exciton dynamics of B850* states is obtained by observing the trace of the diagonal feature at 780 nm as it evolves in waiting time T. In Figure 2(c), we show the trace of B850* states (grey) averaged over λτ = [775, 785] and λt = [775, 785]. The trace was fit to a single exponential decay (black) and gave population relaxation time constant of about 160 ± 30 fs for the B850* states. The fast B850* to B850 energy transfer time scale is consistent with the prediction made by Novoderezhkin et al. for Rhodospirillum molischianum, which has exciton structure similar to LH2 of Rb. sphaeriodes.35 

FIG. 1.

Two-dimensional spectra of rephasing signals acquired with the canonical 2D (top row) and coherence-specific (bottom row) polarization scheme for waiting times, T = 50, 150, 250, 350, 450, and 550 fs.

FIG. 1.

Two-dimensional spectra of rephasing signals acquired with the canonical 2D (top row) and coherence-specific (bottom row) polarization scheme for waiting times, T = 50, 150, 250, 350, 450, and 550 fs.

Close modal
FIG. 2.

(a) 2D spectrum of rephasing signal acquired with canonical 2D polarization scheme at T = 50 fs. The B850* states appear in the section marked by white box. (b) The B850* diagonal peak, shown on a different color scheme. (c) The average trace of B850* population dynamics in the region λτ = [775, 785] and λt = [775, 785] is shown in grey. The black trace is a mono-exponential fit the B850* population dynamics with a time constant of 160 fs.

FIG. 2.

(a) 2D spectrum of rephasing signal acquired with canonical 2D polarization scheme at T = 50 fs. The B850* states appear in the section marked by white box. (b) The B850* diagonal peak, shown on a different color scheme. (c) The average trace of B850* population dynamics in the region λτ = [775, 785] and λt = [775, 785] is shown in grey. The black trace is a mono-exponential fit the B850* population dynamics with a time constant of 160 fs.

Close modal

Coherent dynamics of the R26.1 LH2 is studied by employing canonical and coherence-specific pulse polarization sequences. The 2D spectrum of the rephasing signal of R26.1 LH2 acquired with coherence-specific sequence is shown in the bottom row of Figure 1 for waiting times T = 50, 150, 250, 350, 450, and 550 fs. Results, concerning coherence signals, are summarized in Figures 3(a)3(d). The top left panel (Figure 3(a)) shows the 2D spectrum of rephasing signal acquired at T = 70 fs using canonical 2D polarization sequence. We look at the evolution of points in the upper cross peak marked by the white box. The trace obtained by observing the evolution, in the waiting time T, of a particular point was fitted to a sum of a Gaussian (representing inertial terms) and an exponential (representing relaxation) decay. The fit was subtracted from the trace and the residue was fit to an exponentially decaying sinusoidal function to estimate coherence lifetime and coherence frequency. The average lifetime and frequency of coherences observed with canonical 2D polarization sequence is found to be 137 ± 35 fs and 687 ± 16 cm−1, respectively. Six representative coherence signals for points marked in yellow are shown alongside in Figure 3(b) in blue with the corresponding fits in black.

FIG. 3.

(a) 2D spectra of rephasing signal acquired with canonical 2D sequence at T = 70 fs. (b) Oscillatory features (blue) present in the traces of points marked in yellow in the upper cross peak of the 2D plot in (a). The fits to the oscillatory features are in black. (c) 2D spectra of rephasing signal acquired with coherence-specific sequence at T = 70 fs. (d) Oscillatory features (red) present in the traces of points marked in yellow in the upper cross peak of the 2D plot in (c). The fits to the oscillatory features are in black.

FIG. 3.

(a) 2D spectra of rephasing signal acquired with canonical 2D sequence at T = 70 fs. (b) Oscillatory features (blue) present in the traces of points marked in yellow in the upper cross peak of the 2D plot in (a). The fits to the oscillatory features are in black. (c) 2D spectra of rephasing signal acquired with coherence-specific sequence at T = 70 fs. (d) Oscillatory features (red) present in the traces of points marked in yellow in the upper cross peak of the 2D plot in (c). The fits to the oscillatory features are in black.

Close modal

To investigate the nature of coherences, we employ the coherence-specific pulse polarization sequence. Results are summarized in Figures 3(c) and 3(d). The bottom left panel (Figure 3(c)) shows the 2D spectrum of rephasing signal acquired at T = 70 fs using coherence-specific polarization sequence. The data acquired using the coherence-specific sequence are analyzed and interpreted in the same fashion as for the canonical 2D sequence. On the 2D spectrum, the diagonal feature corresponding to stimulated emission from B850* states does not show up, as expected for signals with parallel transition dipoles in coherence-specific experiments. The diagonal feature for B850 state at early waiting times corresponds to decoherence of k = + 1 k = 1 state during positive waiting times. See supplementary material34 for more details on the nature of signals for positive and negative waiting times in the coherence-specific experiment. A coherence signal, visible in this dataset, is shown in red and the corresponding fits are shown in black. The average lifetime and oscillation frequency of the coherence signal is found to be 88 ± 8 fs and 695 ± 30 cm−1. The frequency of this oscillation is similar to the average coherence frequency observed in the canonical 2D polarization, but the decay time differs markedly.

Within the Condon approximation, the angle between the transition dipoles, giving rise to vibrational coherence during the waiting time, is zero. Calculations have shown that a coherence signal with parallel transition dipoles will not survive the coherence-specific pulse polarization sequence.25,28 In the case of purely vibrational coherence, no coherences should be observed with coherence-specific sequence. In the case of purely electronic coherence involving non-parallel transition dipoles, the observed coherence lifetime should be the same with both the pulse polarization sequences. The difference in lifetime of coherences observed with the two pulse polarization sequences excludes the possibility of the two limiting cases of purely electronic or purely vibrational coherence. Our data are consistent with a simple model of weak non-adiabatic coupling between dark vibrational states and higher excitonic states. This intermediate regime suggests coherences arise from mixed vibrational and electronic, or vibronic states. In Sec. IV, we present a simplified vibronic model to corroborate the hypothesis of vibronic coherence giving rise to different lifetimes when probed by canonical and coherence-specific pulse polarization sequence in a 2DES experiment. Alternatively, the quantum beating observed in the canonical experiment could conceivably arise from a sum of purely vibrational and purely electronic coherence with nearly identical frequency, similar dephasing rate, and similar strength. We discount this hypothesis because it requires a vibrational coherence to dephase on a time scale much faster than population transfer or typical vibrational dephasing times.

A standard vibronic model is often adopted to study mixing of vibrational and electronic states to understand the nature of coherences in photosynthetic systems.21–23,36,37 In the vibronic model, vibrational states are attributed to electronic potential energy surfaces of the individual BChla’s solved within the Born-Oppenheimer (BO) approximation. Franck-Condon factor weighted Coulombic interactions between the electronic and vibrational states of the chromophores generate vibronic states of the entire photosynthetic complex.36 Consequently, the exciton states of the photosynthetic complex in the vibronic model have mixed vibrational and electronic properties and are neither purely electronic nor purely vibrational in character. Such vibronic mixing frustrates examination of the two limiting cases: purely electronic coherence and purely vibrational coherence.

Here, we adopt an alternative, phenomenological approach to LH2 exciton structure to analyze coherences in the two limiting cases of purely electronic coherence and purely vibrational coherence. This formalism is similar in concept to that derived by Spano for excitons in molecular aggregates.38 Contrasting with the approach pursued in vibronic model where vibrational states are attributed to electronic states of individual chromophores, we envisage the exciton structure where first the electronic potential energy surface of the entire photosynthetic complex is solved within the BO approximation and then decorated by vibrational (phonon) states. We refer to these states as BO exciton-states. The BO exciton-states do not couple to each other but rather couple to the bath which is responsible for system relaxation from one BO exciton-state to the other. However, accidental degeneracy or near degeneracy between the BO exciton-states can lead to non-zero, non-adiabatic coupling between BO exciton states. We refer to the states resulting from non-adiabatic coupling between BO exciton-states as non-Born-Oppenheimer (NBO) exciton states. This coupling is identical in nature to a Fermi resonance between a combination band (exciton plus phonon) and a fundamental mode (exciton).39 

The energy levels in our simplified model of R26.1 LH2 are shown in Figure 4(a). The yellow and the cyan potential energy surfaces are the B850 and B850* exciton states, respectively. Resonance Raman and surface enhanced resonance Raman studies of LH2 from Rb. sphaeriodes reveal vibrational modes with frequency similar to the gap between B850 and B850* states.40 We posit that such a vibrational mode on the B850 state can interact with B850* states though non-adiabatic coupling and give rise to NBO exciton-states shown in green, though the resonance Raman data do not directly measure this quantity. The extent of non-adiabatic coupling governs coherence properties and, as we will show, gives accessible explanation to the difference in the coherence lifetime observed with canonical 2D and coherence-specific polarization sequence.

FIG. 4.

(a) The two different potential energy surfaces for B850 (yellow) and B850* (cyan) exciton states resulting from Born-Oppenheimer (BO) approximation. a 0 and a 1 are first harmonic and second harmonic of a vibrational mode on B850 states. b 0 are the ground vibrational states on B850* states. Near degeneracy between a 1 and b 0 can result in non-adiabatic coupling of a 1 and b 0 (shown as red). (b) Non-adiabatic coupling mixes BO states a 1 and b 0 to give NBO states 2 and 3 . (c) Non-adiabatic coupling alters transition rates ka1a0 and kb0a0 between BO states to give new transition rates k2→1 and k3→1 between NBO states.

FIG. 4.

(a) The two different potential energy surfaces for B850 (yellow) and B850* (cyan) exciton states resulting from Born-Oppenheimer (BO) approximation. a 0 and a 1 are first harmonic and second harmonic of a vibrational mode on B850 states. b 0 are the ground vibrational states on B850* states. Near degeneracy between a 1 and b 0 can result in non-adiabatic coupling of a 1 and b 0 (shown as red). (b) Non-adiabatic coupling mixes BO states a 1 and b 0 to give NBO states 2 and 3 . (c) Non-adiabatic coupling alters transition rates ka1a0 and kb0a0 between BO states to give new transition rates k2→1 and k3→1 between NBO states.

Close modal

We describe the effect of non-adiabatic coupling on BO states to give NBO states in Figures 4(a)4(c). BO exciton-states are represented by alphabets such as a 0 and NBO exciton states are represented by numbers such as 1 . For our purposes, B850 states are considered as one state; the lower potential energy surface shown in Figure 4(a) (yellow). a 0 and a 1 represent zero and one quanta of vibrational excitation on the B850 exciton state of R26.1 LH2. Similarly, B850* states are considered as one state; the higher energy exciton state shown in cyan (Figure 4(a)). b 0 represents the ground vibrational state on B850* exciton state of R26.1 LH2. Given our laser bandwidth, we consider only the ground vibrational state on B850* state. The adiabatic approximation breaks when the energy states are close to each other as in the case of a 1 and b 0 (shown in red in Figure 4(a)).41 The resulting non-adiabatic coupling mixes a 1 and b 0 to give 2 and 3 , shown in green in Figure 4(b). The state a 0 is unaffected by non-adiabatic coupling but, in the context of NBO states, a 0 is referred to as 1 .

The introduction of non-adiabatic coupling between BO states alters relaxation time scales within the system. Below, we present a method to calculate relaxation rates between NBO states that arise from non-adiabatic coupling induced mixing of BO states. The unitary transformation U mixes BO states a 1 and b 0 to give NBO states 2 and 3 as

2 = U a 1 2 a 1 + U b 0 2 b 0 ,
(1)
3 = U a 1 3 a 1 + U b 0 3 b 0 .
(2)

In the absence of non-adiabatic coupling, transitions from a 1 to a 0 and from b 0 to a 0 take place at the rate ka1a0 and kb0a0, respectively (Figure 4(a)). The transition rate k2→1 from state 2 to 1 , and k3→1 from state 3 to 1 after introducing the non-adiabatic coupling can be calculated as (see supplementary material for detailed derivation)34 

k 2 1 = ( U a 1 2 ) 2 k a 1 a 0 1 ω 21 ω a 1 a 0 ω a 1 a 0 + ( U b 0 2 ) 2 k b 0 a 0 1 ω 21 ω b 0 a 0 ω b 0 a 0 ,
(3)
k 3 1 = ( U a 1 3 ) 2 k a 1 a 0 1 ω 31 ω a 1 a 0 ω a 1 a 0 + ( U b 0 3 ) 2 k b 0 a 0 1 ω 31 ω b 0 a 0 ω b 0 a 0 ,
(4)

where ωa1a0 = ωa1ωa0, etc. Similarly, the pure dephasing rates k a 1 a 0 pd and k b 0 a 0 pd between pairs of states a 1 and a 0 , and between b 0 and a 0 , respectively, can be calculated within the secular approximation to obtain the pure dephasing rates k 21 pd and k 31 pd between pairs of states 2 and 1 , and between 3 and 1 , respectively, as (see supplementary material)34 

k 21 pd = ( U a 1 2 ) 2 k a 1 a 0 pd + ( U b 0 2 ) 2 k b 0 a 0 pd ,
(5)
k 31 pd = ( U a 1 3 ) 2 k a 1 a 0 pd + ( U b 0 3 ) 2 k b 0 a 0 pd .
(6)

This reductionist approach has a number of advantages. First, for a system with many chromophores, such as R26.1 LH2, the complexity of exciton structure in the vibronic model compounds if vibrational states on the chromophores are included to study mixing of vibrational and electronic states. The approach presented above greatly simplifies calculations and makes the picture clearer. Second, the approach presented above has the advantage of incorporating non-adiabatic coupling in a simplified manner and distinguishes clearly between limits of purely vibrational coherence and purely electronic coherences.

Compared to the standard vibronic model, our model takes a fundamentally different approach to describe excitonic properties. The standard vibronic model takes a bottom-up approach to describe the coherence properties of photosynthetic complexes, while the approach of our model can be described as a top-down approach. In the standard vibronic model, properties of individual components of photosynthetic complexes, i.e., chromophores, are described in great detail. These properties include site energy, chromophore-bath coupling (Huang-Rhys factor), and vibrational frequencies with associated Franck-Condon factors. Starting with such a description, Coulombic interaction between chromophores results in the exciton structure. Our model does not elucidate such microscopic details, but rather considers phonons atop Born-Oppenheimer excitonic states.

In our model, vibrational mode a 1 is added, not to the chromophores, but to the complex’s exciton state a 0 , and made to interact with the b 0 exciton state in a non-adiabatic fashion. Such an addition to the electronic state can be justified on the basis of resonance Raman spectrum. The treatment of coherences in the model is analogous to the approach adopted to explain Fermi resonance where a vibrational overtone is taken into consideration only to explain the doublet.42 

Calculation of rate constants between NBO states from Eqs. (3)–(6) requires as input energies, detuning (Δ), non-adiabatic coupling (V NAC), and relaxation rates among BO states. In addition, the orientation of transition dipoles of BO states is required for ultimately calculating the coherence signals. Given our bandwidth (see supplementary material),34 we can only excite the blue side of the B850 band. We therefore assign states a 0 and b 0 within our model to have energies of 12 000 cm−1 and 12 700 cm−1, respectively. ka1a0 and kb0a0 being vibrational and electronic transition rates between BO exciton-states are assumed to be 1/2 ps−1 and 1/100 fs−1, respectively. The pure dephasing rates k a 1 a 0 pd and k b 0 a 0 pd for the vibrational and electronic coherences are assumed to be 1/2 ps−1 and 1/100 fs−1, respectively. In analogy to the observed difference in the average coherence frequencies between the canonical and coherence-specific 2D experiments, we set the energy difference between states 2 and 3 , ΔE23, to be 10 cm−1 in our model. The energy gap between states 2 and 3 results from mixing of the BO exciton-states a 1 and b 0 (see supplementary material).34 However, this gap alone is insufficient to describe the mixing because the mixing angle depends on two parameters: the detuning (Δ) and coupling (V NAC) between a 1 and b 0 . Here, we use the measurable difference in experimentally measured coherence lifetimes to constrain the mixing angle between the states. Ultimately, it is this difference in coherence lifetimes rather than the small difference in the experimentally observed coherence frequency that confirms mixing between vibrational and electronic states. The dipole strength | μ a 0 | 2 of state a 0 is taken as the reference and set to one, i.e., | μ a 0 | 2 = 1 . We estimate the dipole strength | μ a 1 | 2 , of state a 1 , and | μ b 0 | 2 , of state b 0 , to be 0.04 and 0.09, respectively.40,43 The pair of transition dipoles, μ a 0 and μ a 1 , belong to the same potential energy surface and are parallel to each other within the Condon approximation. For simplicity, the direction of μ b 0 is set at 45° with respect to μ a 0 and μ a 1 based on a symmetry of the ring-like structure of R26.1 LH2 (see supplementary material for explanation).34 

Using the input parameters specified above, we calculate the amplitude of the coherence signal with canonical 2D and coherence-specific pulse polarization sequence as described by Hochstrasser.25 The calculated signal was analyzed in the same way as experimentally obtained coherence signals. In Figure 5(a), we show the coherence lifetime from the calculated coherence signals using canonical 2D (blue) and coherence-specific (red) pulse polarization sequence as a function of the mixing angle between the BO exciton states a 1 and b 0 . The blue and the red stripes in Figure 5(a) are experimentally observed coherence lifetimes in the canonical and coherence-specific experiments, respectively. In Figures 5(b) and 5(c), we show the mixing between BO exciton states a 1 and b 0 , and the calculated coherence signals with canonical and coherence-specific sequence for a particular case of mixing angle, θmix = 12.5°. In Figure 5(b), the transition dipole vectors μ a 0 , μ a 1 , and μ b 0 of BO states a 0 , a 1 , and b 0 are shown in dashed yellow, solid yellow, and cyan, respectively. While μ a 1 is parallel to μ a 0 , μ b 0 makes a 45° angle with μ a 0 as mentioned above. The non-adiabatic coupling between BO states mixes μ a 1 and μ b 0 to give μ 2 and μ 3 , both of which are shown in green. μ a 0 is unaffected by non-adiabatic coupling but, in the context of NBO states, is referred to as μ 1 . The mixed states, μ 2 and μ 3 , make an angle θ12 and θ13, respectively, with respect to μ 1 .

FIG. 5.

(a) Coherence lifetimes from calculated coherence signals using canonical 2D (blue) and coherence-specific (red) pulse polarization sequence as a function of the mixing angle between the BO exciton states a 1 and b 0 . The mixing angle θmix = 0° and θmix = 45° are excluded. (b) Mixing between BO exciton states a 1 and b 0 for a particular case of mixing angle, θmix = 12.5°. The vectors shown in dotted yellow, solid yellow, and blue are transition dipole vectors for BO states a 0 , a 1 , and b 0 , respectively. The non-adiabatic coupling between BO states a 1 and b 0 results in redistribution of oscillator strength between vectors μ a 1 (solid yellow) and μ b 0 (blue) to give μ 2 and μ 3 (both shown in green). μ 2 makes an angle θ12 = 17° with respect to μ 1 , and μ 3 makes an angle θ13 = 40° with respect to μ 1 . (c) Calculated dephasing of coherences with canonical 2D (blue) and coherence-specific (red) polarization sequence, and the corresponding fits (dashed black).

FIG. 5.

(a) Coherence lifetimes from calculated coherence signals using canonical 2D (blue) and coherence-specific (red) pulse polarization sequence as a function of the mixing angle between the BO exciton states a 1 and b 0 . The mixing angle θmix = 0° and θmix = 45° are excluded. (b) Mixing between BO exciton states a 1 and b 0 for a particular case of mixing angle, θmix = 12.5°. The vectors shown in dotted yellow, solid yellow, and blue are transition dipole vectors for BO states a 0 , a 1 , and b 0 , respectively. The non-adiabatic coupling between BO states a 1 and b 0 results in redistribution of oscillator strength between vectors μ a 1 (solid yellow) and μ b 0 (blue) to give μ 2 and μ 3 (both shown in green). μ 2 makes an angle θ12 = 17° with respect to μ 1 , and μ 3 makes an angle θ13 = 40° with respect to μ 1 . (c) Calculated dephasing of coherences with canonical 2D (blue) and coherence-specific (red) polarization sequence, and the corresponding fits (dashed black).

Close modal

In our data and in our model, we observe different coherence lifetimes in the canonical and coherence-specific pulse polarization 2DES experiment. The observed oscillatory signal is a sum of two coherences: one from each pair 1 , 2 and 1 , 3 . The lifetime of the observed signal is determined by the relative amplitudes, A12 and A13, of the underlying coherences. To explain the relative contributions and lifetimes of these two coherences in each experiment, we start from a limit of zero non-adiabatic coupling and then introduce non-adiabatic coupling to determine the window in which our model is consistent with our data. In the limiting case of no coupling, coherence from the pair 1 , 2 is purely vibrational and coherence from the pair 1 , 3 is purely electronic. We expect that the lifetime τ12 of the vibrational coherence is greater than the lifetime τ13 of the electronic coherence. Experimentally, we observe that coherence lifetime τCanon2D in the canonical 2D experiment is greater than the coherence lifetime τCohSpec in the coherence-specific experiment. Therefore, we argue that the amplitude of coherence from the pair 1 , 2 , which in the limit of weak coupling represents a transition of primarily vibrational character, contributes more to the signal in the canonical 2D experiment than in the coherence-specific 2D experiment.

The relative strength of the contribution of a coherence signal to the coherence-specific experiment compared to the canonical 2D experiment is determined entirely by the angle between the transition dipole moments of the states involved in the coherence. The orientational prefactors vary between the different polarization-sequences depending on the angle between the transition dipoles in the molecular frame. From these orientation factors, we infer that the angle θ12 between the transition dipoles in the pair 1 , 2 is smaller than the angle θ13 between the dipoles in the pair 1 , 3 , i.e., θ12 < θ13 (see supplementary material for further explanation).34 The inference θ12 < θ13 helps us to deduce the extent of non-adiabatic coupling between the BO states a 1 and b 0 . In the limiting case of absence of non-adiabatic coupling, the transition dipoles are identical to those of a 1 and b 0 , namely, θ 12 = 0 = θ a 0 a 1 and θ 13 = 45 ° = θ a 0 b 0 . As the non-adiabatic coupling increases, transition dipoles μ a 0 and μ a 1 mix to redistribute dipole strengths and give new transition dipoles μ 2 and μ 3 . While μ 2 points away from μ 1 to make an angle θ12 (>0°) with respect to μ 1 , μ 3 points in a direction closer to μ 1 , to make an angle θ13 (<45°) with respect to μ 1 . As an example, see Figure 5(b). As the strength of non-adiabatic coupling increases, θ12 increases while θ13 decreases. However, as deduced above, θ12 < θ13 and this condition sets the upper bound to the strength of non-adiabatic coupling at θmix ∼ 22.5° (shown by the dotted vertical line in Figure 5(a)). Our model additionally provides the opportunity to directly compare extracted coherence lifetimes with experiment. We find that our data are consistent with our model only within the narrow range of mixing angles between 13° and 17° (Figure 5(a)).

We have directly observed the B850* to B850 transfer rate of 160 fs by employing the canonical 2D sequence on the LH2 mutant of the Rb. sphaeriodes, R26.1 LH2. The fast dynamics is consistent with the prediction by Novoderezhkin.35 With the canonical 2D sequence, we observe coherences between B850* and B850 states that persist for 137 ± 35 fs. To understand the origin of coherences, we employed the coherence-specific pulse polarization sequence to suppress signals from states with parallel transition dipoles. With the coherence-specific sequence, we observe coherences with an average lifetime of 88 ± 8 fs; a noticeable reduction in lifetime. From these data, we deduce that the states involved in the coherence are vibronic states. We present a simplified theoretical approach to calculate relaxation rates between NBO states in terms of non-adiabatic coupling and relaxation between BO states. Our calculations suggest weak to mild coupling between vibrational and electronic states and extract a mixing angle θmix between 13° and 17° between the vibrational and excitonic states within this model.

The authors would like to thank AFOSR (Grant No. FA9550-09-1-0117), DARPA QuBE (Grant No. N66001-10-1-4060), NSF MRSEC Program (Grant No. DMR 14-20709), the Alfred P. Sloan Foundation, and the Camille and Henry Dreyfus Foundation. P.D.D. acknowledges support from the NSF GRFP program and the NIH (Grant No. T32 Eb009412). M.W. acknowledges support from the Fulbright Program. R.J.C. and A.T.G. acknowledge support from the Biotechnology and Biological Sciences Research Council (BBSRC).

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Supplementary Material