Correlated Fluctuations and Intraband Dynamics of J-Aggregates Revealed by Combination of 2DES Schemes.

The intraband exciton dynamics of molecular aggregates is a crucial initial step to determine the possibly coherent nature of energy transfer and its implications for the ensuing interband relaxation pathways in strongly coupled excitonic systems. In this work, we fully characterize the intraband dynamics in linear J-aggregates of porphyrins, good model systems for multichromophoric assemblies in biological antenna complexes. Using different 2D electronic spectroscopy schemes together with Raman spectroscopy and theoretical modeling, we provide a full characterization of the inner structure of the main one-exciton band of the porphyrin aggregates. We find that the redistribution of population within the band occurs with a characteristic time of 280 fs and dominates the modulation of an electronic coherence. While we do not find that the coupling to vibrations significantly affects the dynamics of excitonic coherence, our results suggest that exciton fluctuations are nevertheless highly correlated.

Despite being the object of thorough theoretical and experimental studies for more than 60 years,[5−7] J-aggregates continue to arouse considerable interest.[1−4] Such interest is mainly due to the collective nature of their optical excitations, known as excitons, endowed with unique linear and nonlinear optical responses.[4,7−10] The attention on these systems has recently been renewed after the discovery that light-harvesting complexes in natural photosynthetic systems are governed by principles very similar to those holding in aggregates of artificial dyes.[11] It has been recognized that excitons play a fundamental role in the mechanism of electronic energy transfer, which can notably include quantum coherent dynamics, in various synthetic nanoscale and biological systems.[12−16] In light of the possible role of quantum coherence in exciton transfer, the attention is currently focused on the characterization of transport properties of excitonic systems and how they are affected by the coupling with vibrations and the environment.[2,17−22] An important aspect in this regard, still not fully investigated, is intraband dynamics (i.e., the dynamics within states building the optically active one-exciton band) in J-aggregates, foregoing the more characterized interband dynamics.

In this work, different 2D electronic spectroscopy (2DES) schemes have been applied together with Raman spectroscopy and theoretical modeling to fully characterize the intraband exciton dynamics in porphyrin J-aggregates at room temperature, with particular attention to the effects of the coupling between electronic and vibrational degrees of freedom.

Aggregates of porphyrins, in particular of the diacid form of the water-soluble tetra-(p-sulphonato)-phenyl-porphyrin (H2TPPS) (Figure a), are particularly meaningful because they have been often proposed as model systems for chlorosomes and LH2 complexes.[23,24] The aggregation properties of H2TPPS are well known, and a significant number of works have already been devoted to clarify the aggregation conditions, the aggregate geometry and structure, and the photophysical and dynamical properties in both monomeric and aggregated form. The appearance of new blue- or red-shifted bands in the absorption spectra is a typical signature of aggregate formation (Figure b).[3,4,7,25−30] The optical properties of H2TPPS aggregates are typically described using an ideal model of a linear homogeneous aggregate of N monomers with nearest-neighbor coupling only.[4] In this model the expression of eigenstates and eigenenergies can be calculated exactly. The one-exciton band is formed by N eigenstates |k⟩, characterized by a single quantum number k and an energy E = E + 2J cos(πk/(N + 1)), where E is the energy of the excited state of the monomer and J < 0 is the electronic coupling.[3,7] The state with k = 1 (|1⟩) collects the majority of the oscillator strength, and the rest is distributed among all of the odd k states with a decreasing relevance as k increases, such that the linear optical properties can be justified considering only the first two optically bright states |1⟩ and |3⟩. The description of nonlinear optical properties requires us to invoke also the two-exciton band, which is formed by states |k, k′⟩ (k ≠ k′) with energies related to the one-exciton energies by E = E + E′.[31] Exciton binding interactions leading to biexciton formation can be neglected in these systems.[28,32]

Figure 1

(a) Schematic of the linear self-assembled geometry of H2TPPS aggregates. (b) Steady-state absorption spectrum of the H2TPPS aggregate in water solution in the JB band region (black) and experimental spectrum of the exciting laser pulse used in 2D spectroscopy (green shadowed area). The band is generated by the interaction of dipole moments oriented parallel along the linear chain (head-to-tail configuration).[33] The inset compares the absorption spectra in the whole visible range for the monomer (red) and the aggregate (black), where new excitonic bands are detected at ∼23 700 cm–1 (HB), 20 400 cm–1 (JB), and 14 200 cm–1 (JQ).[7,25,33] The JB band in the aggregate is shifted by ∼2660 cm–1 to the red with respect to that of the monomer, so that E1 – E = 2J cos(π/(N + 1)) ≈ −2660 cm–1.

In this work we focus on the so-called JB band[27,33] located at 20 380 cm–1 (∼490 nm), which is characterized by the higher excitonic coupling. The exciting laser band in 2D experiments has thus been tuned to cover the above-mentioned band (Figure b). Details of 2D spectroscopy and the physical meaning of 2D signals can be found in refs (34 and 35). The optical setup and data analysis techniques specifically employed in this work are described in the SI.

Figure a shows two examples of 2D maps measured in the rephasing (R) and nonrephasing (NR) schemes, recorded at waiting time t2 = 0 fs. t2 is the time delay between the second pump and the probe pulses, during which the population and coherence dynamics take place.[35] For all times t2 investigated, the R signal is dominated by a positive diagonal peak elongated along the diagonal direction, while the NR map is dominated by a positive diagonal peak elongated in the antidiagonal direction, both attributed to ground-state bleaching and stimulated emission of the JB transition. The upper and lower off-diagonal (diagonal) negative features in R (NR) spectra are the result of the typical dispersion lineshapes. The possible contribution of an excited-state absorption from one-exciton to the two-exciton states can be invoked to explain the slight asymmetry of the upper negative peak, being more intense than the lower one, although the process cannot be fully captured in the analyzed spectral range. (see SI Figure S1)

Figure 2

(a) Examples of experimental (upper line) and simulated (lower line) 2D maps obtained in the rephasing (R), non rephasing (NR), and double-quantum (2Q) configurations for H2TPPS aggregates in solution at room temperature. The R and NR maps report the real part of the signal at t2 = 0 fs. The 2Q maps refer to t1 = 0. All of the maps are normalized to their maximum. The energies of relevant states and the coordinates where the traces shown in panels d–f are extracted are also pinpointed in the maps. (b) Simulation of the absorption spectrum. The parameters used to simulate linear and 2D response are reported in the SI. (c) Decay-associated spectrum (DAS)[36] of the purely absorptive (R+NR) maps, showing the amplitude distribution of the 280 fs time constant. A positive (negative) amplitude is recorded, where the signal is decaying (rising). (d) Comparison between beating modes in Raman and 2D spectra. Fourier transform (green) and linear prediction z-transform LPZ[37−39] (red) of the oscillating part of the NR signal extracted at coordinates where all three main oscillating components are expected to contribute. Resonant (black) and nonresonant (blue) Raman spectra. (e,f) Comparison of the experimental and theoretical decay traces as a function of t2 in R (e) and NR (f) signals extracted at relevant coordinates (ω1, ω3) ≈ (E1, E1 + ωα).

The absence of distinct features in the R and NR 2D maps does not allow the clear characterization of the internal structure of the JB band. To gain more insight, 2D experiments have been repeated in the so-called “double-quantum” scheme (2Q2D), reported in the last column of Figure a. 2Q2D is able to capture the transition toward two-exciton states in the absence of all of the other superimposed R and NR contributions. In a 2Q2D experiment, a nonnegligible signal is detected only if the system is at least a three-level system.[35,40−43] The sequence of pulses is such that the first two pulses generate a double-quantum coherence between the ground (g) and a two-exciton state (f), whose phase oscillates during t2 at the frequency of the g–f energy gap, close to twice the frequency of the ground and single exciton (g–e) gap, in resonance with the exciting pulse. The third excitation pulse then returns the system to a one-quantum coherence (g–e or e–f), and the signal is emitted. Differently from R and NR 2D maps, a 2Q2D map is obtained varying t2 for fixed values of t1, and thus it reports the signal as a function of ω2 and ω3 (see the SI). The 2Q2D technique is particularly suited in the case of J-aggregates because it allows determining the position of the two-exciton states with respect to the one-exciton transitions. Moreover, because the two-exciton state energies are related to the one-exciton state energies, a 2Q2D experiment can provide information also on the one-exciton band structure. In the absence of line-broadening effects the distance between the positive and the negative peaks along the ω3 axis Δω3 in a 2Q2D map corresponds to the difference between the two-exciton to one-exciton energy gap and the one-exciton to ground-state energy gap, that is, Δω3 = (E – E) – (E – E). Homogeneous broadening results in a slight increase in this distance as well a shift of the peaks along the ω2 axis.[41] For linear J-aggregates, the selection rules state that the transitions with the highest oscillator strength are |0⟩ → |1⟩ and |1⟩ →|1, 2⟩;[11,44] therefore, we can identify e = |1⟩ and f = |1, 2⟩ and the energy difference Δω3 can be related to the k = 2 and k = 1 exciton gap, Δω3 ≳ (E – E) – (E – E) = E2 – E1. The estimation of the E2 – E1 gap from 2Q2D measures (Δω3 ≈180 cm–1, Figure a) and the estimation of the E1 – E gap from linear absorption measures (Figure ), allow extrapolating the values of J and N that best reproduce experimental data using the expression of exciton energies. A k = 2 and k = 1 exciton gap in the range of E2 – E1 ≈ 160–200 cm–1 leads to – J ≈ 1170–1400 cm–1 and N ≈ 12–14. Using the same expression, these values have been then used to estimate the exciton gap between the states |1⟩ and |3⟩, E3 – E1 ≈ 420–520 cm–1 (see the SI).

This estimated energy gap is also consistent with the analysis of the time dependence of the R and NR 2D maps. The dynamic evolution of such 2D maps along t2 has been studied with a recently proposed global analysis methodology able to simultaneously extract coherence and population dynamics of R and NR contributions.[36] The results of this analysis are reported in Table .

Table 1

Results of the Global Multiexponential Fit of the 2D Maps (see SI and ref (36))a

componentb	DAS 1	DAS 2	CAS 1	CAS 2	CAS 3
time constant (ps)	>1	0.28	>1	>1	0.265
frequency (cm–1)			258	334	445

Standard errors of the fit are in the order of 1% for frequencies and 15% for time constants.

DAS, decay-associated spectrum; CAS, coherence-associated spectrum as defined in ref (36).

The population dynamics of the purely absorptive signal is dominated by two time constants, whose amplitude distributions in the 2D maps are shown in the form of decay-associated spectra[36] (DAS). Together with a long time constant (>1 ps) describing the overall decay of the maps, the dynamics in the investigated time window is characterized by a time constant of 280 fs. The amplitude distribution of this time constant (DAS in Figure c) shows a positive peak (i.e., signal is decaying) on the upper diagonal, whereas negative peaks (i.e., signal is raising) are present on the lower diagonal and off-diagonal positions. This witnesses a transfer of signal amplitude from higher to lower energy states with a 280 fs time constant. Other ultrafast dynamic phenomena typically characterized by similar time scales, such as spectral diffusion, would have presented significantly different amplitude distributions.[36] We therefore suggest that this time constant is associated with a population redistribution within the JB band, from states at higher energy to states at the bottom of the band.

The evolution of the 2D signal also shows the presence of oscillations, typical signatures of the coherent evolution of the quantum superpositions prepared by the exciting laser. The same global analysis allows identifying three frequency components contributing to the overall 2D map beating: 258, 334, and 445 cm–1 (Figure d).

The first two components contribute in both R and NR maps with damping times longer than the time window investigated (>1 ps) and can be interpreted as vibrational coherences. This assignment is also endorsed by the results of off- and on-resonance Raman measurements (Figure d), showing indeed a strong enhancement of two low-frequency vibrational modes at 246 and 319 cm–1.

The third frequency component does not match any relevant vibrational mode but is very close to the excitonic gap E1 – E3 estimated through the combined 2Q2D measures and model analysis. Differently from the other two components, it is damped within the first 265 fs after photoexcitation.

This evidence suggests that the 445 cm–1 oscillation has an electronic origin and corresponds to the evolution of the coherent superposition between the states |1⟩ and |3⟩, initially prepared by the laser pulse. Note that the dephasing time of this coherence is very similar to the relaxation time discussed above.

To support the interpretation of our data, the experimental optical responses have been simulated using a minimal model for the excitations and vibrations in the aggregate that can describe the most prominent features of the spectra. Given that most of the oscillator strength is on the |1⟩ and to a lower degree on |3⟩ exciton transition, we consider a four-level electronic system, consisting of the ground state (|0⟩), the two k = 1 and k = 3 one-exciton states, and the lowest two-exciton state k, k′ = 1, 2 (|1,2⟩). The vibronic coupling in this minimal model is included by considering the coupling of the electronic states to two effective vibrational modes. The total Hamiltonian of the system is given by Hs = Hel + Hel,vib + Hvib with the electronic Hamiltonian given by Hel = E1|1⟩⟨1| + E3|3⟩⟨3| + E12|1, 2⟩⟨1, 2| and the vibrational Hamiltonian Hvib = ℏωαaα†aα + ℏωβaβ†aβ, where the operators aα(β)† and aα(β) denote the creation and annihilation operators of phonon modes, respectively, and ωα(β) is the vibrational frequency. Finally, for the electron-vibrational coupling we find that an interaction of the type Hel,vib = ∑g(a† + a)(|1⟩⟨3| + |3⟩⟨1|), which takes into account the coupling of the one-exciton states by the vibrations but neglects the displacement on each excited state, best describes the experimental results. This can be justified in the light of the weak vibronic coupling, as detailed in the SI.

We include decoherence, leading to homogeneous broadening, by coupling the system to a Markovian environment that induces exponential decay of electronic coherence at rates Γ but not population transfer (for details, see the SI). In particular, the electronic decoherence rates Γ0 (k = 1, 3) describe the decay rate of the coherence between the electronic ground state and the one-exciton state k, while Γ13 represents the decay rate of the coherent superposition of the two one-exciton states and in general includes correlated fluctuation effects. Static disorder has been considered negligible in our calculations. This assumption, typically verified in the case of strongly coupled J-aggregates,[45] is also justified by the round shape of the absorptive spectra (Figure S5 of the SI).[46]

This model has been used to simulate the absorption spectrum (Figure b) and all of the 2D experiments (R, NR, and 2Q schemes). All of the parameters are given in the SI. In the simulation of the nonlinear experiments, the molecular response to laser excitation is described in the framework of the response function formalism,[47] and the finite pulse duration of the exciting pulses has been considered so that possible pulse overlap effects have been accounted for (see the SI).

Figure a shows the simulated 2D maps at t2 = 0. The position and relative strengths of the positive and negative peaks are in good agreement with the experimental data. The broadening of the positive peaks is slightly narrower in our calculations. Because we reproduce the absorption width (Figure b) it is possible that this difference is due to other experimental pulse effects that have not been captured in the simulations. Also, the dynamic behavior of the 2D maps along t2 could be properly reproduced, as shown in Figure e,f.

The values of the coupling constants g optimizing the simulations (gα = 0.24ωα and gβ = 0.2ωβ) suggest that the two vibrations are only weakly coupled to the electronic transition giving rise to the JB band. We found that the vibronic coupling does not alter in a relevant way the energies and the properties of the excited states and the aggregate. In particular, this degree of vibronic coupling does not significantly mix different electronic states, and the energies of the strongest vibronic transitions are relatively close to those of the simplistic linear chain electronic system such that the insight gained from the purely electronic chain regarding the energies of the single-exciton and two-exciton states holds.

It is also interesting to highlight the values found for the decoherence rates: Γ01 = 26 ps–1, Γ03 = 40 ps–1, and Γ13= (0.265 ps)−1 = 3.8 ps–1. The dephasing rate Γ13 resulted in being much slower than what was predicted for uncorrelated dephasing:[2] Γ13 ≪ Γ01 + Γ03 = 66 ps–1 (see the SI). This suggests that correlated fluctuations are a fundamental intraband mechanism in such strongly coupled aggregates. Correlated dephasing mechanisms can be due to excitons sharing the same pigments even when the environment-induced fluctuations at each molecule are uncorrelated (local baths) as well as molecules experiencing correlated fluctuations (shared bath).[2,48,49] Both processes are equally likely in the case of intraband relaxation processes, where the relaxation involves one-exciton states described as a combination of the same molecular states with different symmetry.

In conclusion, the synergic use of several 2D techniques, including a double-quantum experiment, allowed a full characterization of the inner structure of the one-exciton JB band of H2TPPS aggregates, including a dark state (k = 2) not detectable with conventional techniques and two weakly coupled vibrational modes (Figure ). The intraband dynamics of the most intense one-exciton band in the aggregates is characterized by an internal population relaxation toward the bottom of the band with a characteristic time of 280 fs, and it is modulated by the evolution of a coherent superposition of excitonic states dephasing on the same time scale. This time is considerably longer than what is expected only on the basis of spectral line-width considerations. The simulation of the experimental data with a theoretical model taking into account the possible coupling between vibrational modes and electronic transitions and the dephasing action of the environment lead to the conclusion that the long-living character of such electronic coherence cannot be explained with the mixing between vibrational and electronic degrees of freedom, as proposed for other multichromophoric systems.[2,50,51] This is expected in light of previous work suggesting that such mechanism requires the presence of vibrational modes having a frequency resonant with the electronic transition,[2,52] which is clearly not the case here where two weakly coupled low-frequency modes dominate the modulation of the optical response. The long dephasing time of the intraband electronic coherence captured in this work seems instead to be due to the presence of other environmentally induced correlated dephasing mechanisms. On the contrary, experimental data also suggest that the main dephasing mechanism is the population decay. The electronic coherence is indeed damped on a time scale corresponding to the relaxation of the population toward the state with the lowest energy. These findings represent an important piece of information in the debated issue of the possible relevance of correlated fluctuations[53] in energy and charge transport processes. The nature of such correlations and their influence on dynamics can indeed be crucial to begin to design environments that can be self-assembled to take advantage of these correlations as a mechanism to control energy and charge transport in fluctuating nanostructured environments. Furthermore, the characterization of the intraband dynamics is particularly important considering that other ensuing interband processes, including energy transfer, may involve an ultrafast energy equilibration as the initial step. It would be interesting to understand if this process has any effect on the overall energy-transfer rate, especially given the close resemblance of such aggregates with biological antennas.

Figure 3

Model level scheme as determined by experiments and theoretical modeling. |0⟩ is the ground state and |0, α(β)⟩ are the two main vibrational states on the ground electronic state. |k̃⟩ indicates vibronic states in the one-exciton band with predominant k electronic character (black lines) and |k̃, α′(β′)⟩ are vibronic states with predominant k electronic character and α(β) vibrational character (red lines). |2⟩ is a dark state whose energy could be approximately determined only with 2Q2D experiments (gray line).