Energy relaxation pathways between light-matter states revealed by coherent two-dimensional spectroscopy

Sample characteristics

A cross-sectional illustration of the exciton-microcavity samples under study is shown in Fig. 1a, where the empty cavity (blue shading) with Fock states (left|0rightrangle) and (left|1rightrangle) is coupled to N molecular excitons with ground and excited states (left|grightrangle) and (left|erightrangle) (red shading; absence of cavity), yielding the polaritonic system (purple shading). Figure 1b shows the different layers of the samples, consisting of a 100-nm silver back mirror on top of a glass substrate, a thin film layer of TDBC (see Methods for details and the inset in Fig. 1b for the molecular structure) J-aggregates in polyvinylalcohol (PVA), and a 25-nm, semi-transparent silver front mirror. The sample preparation is described in detail under Methods. All optical fields used to study the polaritonic system are incident on the sample at an angle 2φ and the absorption (A) is contained in the reflection (R) as A = 1 − R, due to the lack of transmission (Tr = 0) through the back mirror.

Fig. 1: Overview of the experiment.

a The hybridization between a (asymmetric) microcavity with Fock states (left|0rightrangle) and (left|1rightrangle) and N molecular excitons in TDBC (5,6-dichloro-2-[[5,6-dichloro-1-ethyl-3-(4-sulphobutyl)benzimidazol-2-ylidene]propenyl]-1-ethyl-3-(4-sulphobutyl)-benzimidazolium hydroxide, inner salt, sodium salt) J-aggregates with ground and excited states (left|grightrangle) and (left|erightrangle), embedded in a polyvinylalcohol (PVA) thin film, leads to the emergence of the polaritonic states P and P+, as well as N − 1 excitonic dark states. The shading highlights the empty cavity (blue), the polaritonic system (purple) and the uncoupled excitons (red). b Measuring in reflection (R) geometry and increasing the thickness of the back mirror such that the transmission (Tr) through the sample equals zero yields the absorption spectrum as A = 1 − R. The angular signal dependence is given with respect to the angle from the surface normal (φ) and the inset shows the molecular structure of TDBC. c The pulse sequence employed for two-dimensional Fourier transform spectroscopy. The laser pulses with wavevectors k1 − k3 generate a 3rd-order signal with ksig that is heterodyne detected with a local oscillator (LO) pulse kLO, which is delayed by tLO. Fourier transformation of the data along time delays τ and t yields a 2D spectrum with excitation and detection energy dimensions ωτ and ωt, respectively, at a fixed evolution time T.

2D Fourier transform spectroscopy

We use broadband visible (500–750 nm; 13,300–20,000 cm−1) 2DFT spectroscopy to simultaneously resolve the excitation energy dependence and achieve the highest possible temporal resolution (<20 fs) of the transient response of TDBC strongly coupled to a cavity. In 2DFT spectroscopy50,52,53,54,55,56 the sample is excited by a pair of compressed, broadband visible laser pulses (with wavevectors k1 and k2) and Fourier transformation along the interpulse delay τ (coherence time) yields the excitation frequency dimension (ωτ). A third laser pulse (k3) is used to monitor the transient response of the excited system after it has evolved for a time T (evolution time) by generating a signal field (ksig) that is spectrally dispersed and heterodyne detected with a local oscillator (LO) pulse (kLO) to obtain the detection frequency dimension (ωt). This last step is the optical analogue of a Fourier transformation along the time t (detection time). The pulse sequence is illustrated in Fig. 1c and the technical details of the experimental setup are described in detail under Methods and by Al Haddad et al.57. Microscopically, a signal at a spectral position ωτ/ωt is described by a 3rd-order response function in which the system coherences during τ and t oscillate with frequencies ωτ and ωt58. Since 2D spectroscopy maximally resolves the signals from different 3rd-order response functions spectrally, the experimental data is ideal to extract energy relaxation kinetics56 and can be readily simulated by quantum mechanical models59,60.

Spectral sample characterisation

Important for the discussion of the systems’ non-equilibrium dynamics is a thorough characterisation of their steady-state optical properties. The k-dependent absorption of sample 1 (2:1 mixture of 1 wt% TDBCaq:5 wt% PVAaq, spin coated at 1000 rpm, λ-mode cavity) is measured in reflection mode (Fig. 1b) and shown as a colourmap in Fig. 2a. The absorption (A) of the system is obtained directly from the specular reflection (R) as A = 1 − R − Tr = 1 − R, since the 100 nm thick back mirror ensures that the transmission (Tr) through the sample is zero. Losses due to scattering are neglected, which is a reasonable assumption considering the high quality of the mirror surfaces.

Fig. 2: Steady-state optical properties of the studied systems.

The k-dependent absorption of a sample 1 and b sample 2 shows that both cavities are red-tuned with anticrossings at 6 and 9.1 μm−1, respectively. The corresponding incidence angles φ are indicated on the top x-axis, and, in the present representation, depend on the energy (y-axis). The photon/exciton ratio for samples 1 and 2 can be read out for the P+ branch in c, d and for the P branch in e, f.

The lower (higher) energy band in Fig. 2a corresponds to the absorption into the P (P+) branch of the polariton. From their in-plane dispersion it is possible to obtain the vacuum Rabi splitting, the photon mode dispersion (red line in Fig. 2a, b), as well as the Hopfield coefficients (Fig. 2c–f), which describe the fractional photon-exciton character of the polariton branches, using a coupled-oscillator model (see Methods). The Rabi splitting of 1 is 0.28 eV and the anticrossing between the photon and exciton bands is found at k = 6 μm−1, corresponding to φ ≈ 33.5. At k = 0 μm−1 the energy of the cavity photon (red line) of 1 is ~0.06 eV below the k-independent exciton energy (black line) and P has a large photonic character, while P+ is predominantly excitonic in character. The cavity is said to be red-tuned, as the photon energy at k = 0 μm−1 is below the energy of the excitons. This tuning of the cavity allows modification of the photophysics of the cQED system by changing its energetic landscape via the cavity thickness13. Figure 2b shows the in-plane dispersion of sample 2 (1:1 mixture of 0.5 wt% TDBCaq:1 wt% PVAaq, 1000 rpm, λ/2-mode cavity), which has a Rabi-splitting of 0.36 eV, is red-tuned by 0.12 eV and shows an anticrossing of the P and P+ branches at k = 9.1 μm−1, corresponding to φ ≈ 50°. As discussed in the following paragraphs, these differences in cavity tuning and Rabi-splitting between the two samples lead to changes in the ultrafast transient 2D signal.

Signals observed by 2D Fourier transform spectroscopy

Before discussing the φ-dependent differences between the 2D spectra of samples 1 and 2 and the non-equilibrium dynamics of sample 1, it is worth introducing the observed signal. The excitation energy ωτ of the 2D spectra in Figs. 35 is plotted as the x- and the detection energy ωt as the y-dimension, such that the transient signal at a certain ωτ is read out as a vertical cut through the 2D spectrum. Here, we show the magnitude of the complex 2D data (absolute 2D spectrum), which allows to follow the non-equilibrium dynamics of the excited system via the temporal evolution of lineshapes and intensities of the on- and off-diagonal peaks (vide infra). (For an absorptive 2D spectrum, phased by comparison to data from Schwartz et al.43, see Supplementary Fig. 1 and Supplementary Note 1.) The reason why the kinetics of the individual states can be followed simply by observing a change in intensity at specific ωτ and ωt is that the 2D spectrum of TDBC J-aggregates (Fig. 2 in Finkelstein-Shapiro et al.49) is a spectrally localised and well-defined 2D peak centred at the lowest energy optical transition frequency. For the TDBC-microcavity system the on-diagonal signals of the 2D spectra in Fig. 3a report on the P, J and P+ states, where the energy of J is independent of φ at 16,870 cm−1, while the peak position of the P (~15,700 cm−1 for sample 1 at 5°) and P+ (~17,850 cm−1 for sample 1 at 5°) bands varies with φ, due to the dispersion of the photon mode. The on-diagonal signal can be compared to the linear absorption spectrum of the system, which is plotted as black lines in Fig. 3a. Note that the on-diagonal peaks for P and P+ do not perfectly line up with the steady-state absorption spectrum, due to slight differences in angle φ and cavity thickness (the latter can vary across the sample due to inhomogeneities from the spin-coating process) between the 2D and absorption measurements. Optical excitation from the ground to either of the three excited states (P, J or P+) results in transient response of the sample that is encoded as a vertical cut at the respective ωτ (follow the arrows in Fig. 3b). The resulting 2D lineshapes report on the interaction of the strongly coupled system with its environment and cross-peaks between on-diagonal signals (labelled by the combination of excited and detected species P, J and P+, as depicted schematically in Fig. 3b) are indicative of the coupling between states via the common ground state, as well as energy relaxation within the system. The former manifests itself as cross-peaks above and below the diagonal due to the ground-state bleach (GSB), while the latter appears as an increase in intensity of the below-diagonal cross-peaks, as the excited-state absorption (ESA) and stimulated emission (SE) contributions change during the excited-state dynamics and energy transfer is energetically downhill for an energy splitting between states that is >kBTRT at room temperature (TRT). This ability to resolve individual signal (cross-) peaks, together with the fact that these are spectrally well isolated, allows to disentangle the kinetics of the system more rigorously than previously possible using transient absorption spectroscopy and highlights the strength of broadband visible 2DFT spectroscopy.

Fig. 3: Sample- and angle-dependence of the observed dynamics.

a The 2D spectrum of sample 1 at T = 100 fs and φ =  5°. b A schematic 2D plot visualises the labelling of the on-diagonal- and cross-peaks, and highlights the distinction between signals: polaritonic states (purple), excitonic state (red), excitonic-polaritonic cross-peaks (white). The arrows indicate how to read out the 2D spectrum with excitation on the x-axis and detection on the y-axis. c The 2D spectra of sample 2 at T = 80 fs and φ =  5° and d at φ = 20° show that the transient behaviour depends on the cavity tuning (by comparison to sample 1 in a), as well as where along k the excitation/detection takes place. The data in a, c, and d is scaled by an arcsinh-function to visualise a larger dynamic range.

Angular dependence of the transient 2D signal

After introducing the 2D signals, we now turn to the differences that can be observed between samples 1 and 2. The 2D spectrum of 1 at T = 100 fs and φ =  5° in Fig. 3a contains all the transient features described in the previous paragraph and illustrated in Fig. 3b.

But when sample 2 is measured at identical φ at T = 80 fs (Fig. 3c), the transient signal differs strongly: The on-diagonal peaks corresponding to P, J and P+ are still present, though P is shifted to lower energy due to the slightly larger red-tuning and stronger coupling of the system, but the intensity of all cross-peaks is significantly decreased. (Note that weak cross-peak signals can be observed at P/P+ and P+/P, whose peak sub-structure, mainly visible for the P+/P cross-peak of sample 2 at 5°, most likely results from the Fourier transformation of the discrete experimental data along τ → ωτ and should be considered an experimental artefact.) This is surprising, since a priori the system is also strongly coupled, i.e. P and P+ have a common ground state. However, the excitation and detection occurs further away from the anticrossing in the in-plane photon momentum space, where the coupling between light and matter states is reduced and P and P+ acquire a larger photon and exciton character, respectively (see Fig. 2c–f)13. It is thus likely that the cross-peaks due to coupling are simply too weak (below or in the range of the detection limit of the experiment) and cannot be observed easily, as discussed in more detail during the analysis of the temporal evolution of sample 1 below. It is further possible that the larger excitonic character of P+ results in an increased lifetime and thus a delayed appearance of P+/P, according to Litinskaya et al.61 and similar to Dunkelberger et al.62. When changing the photon/exciton character of P and P+ by increasing φ to 20°, while keeping the evolution time at T = 80 fs, a strong P+/P cross-peak appears below the diagonal in the 2D spectrum of sample 2 (Fig. 3d). This is indicative of an energy relaxation process that becomes possible in 2 at larger φ and in agreement with the fact that the polariton lifetime is assumed to depend on the cavity lifetime weighted by the photonic character61. Another explanation might be the presence of a relaxation bottleneck, as observed in J-aggregate exciton-microcavities63. Specifically, the energy gap of sample 2 at 5° exceeds twice the energy (first overtone) of the highest Raman-active mode of TDBC J-aggregates, while the energy gaps of sample 1 at 5° and sample 2 at 20° match overtones of Raman-active modes64. Potential effects of the k-components of the non-collinear laser pulses have been minimised experimentally by decreasing the crossing angle between the laser beam k-vectors in the 2D experiment to <1.6° (equivalent to 0.3 μm−1).

Temporal evolution of the 2D spectrum

The evolution of the transient signal of sample 1 is measured in reflection geometry at φ = 5 and displayed as absolute value 2D spectra in Fig. 4. To ease the analysis, we will first focus on the temporal evolution of the lineshapes, before discussing the kinetics of on-diagonal signals and cross-peaks. Lineshapes in 2D spectroscopy report on the correlation between the excited and the detected transition frequency of the system, and on how it is affected by the environment10. The Lorentzian shape of the on-diagonal P signal is the result of fast, homogeneous spectral fluctuations of the polaritonic transition and is observed for the duration of the dataset (~1 ps). This indicates that the energy of P is only weakly coupled to the environment, similar to the results obtained by Takahashi and Watanabe10. The on-diagonal signal of P+ appears as an elliptical peak, characteristic of slow Gaussian spectral diffusion. Inhomogeneous disorder of the corresponding transition energy manifests itself as an elongation of the peak along the diagonal at early T and is explained by the fact that P+ is coupled to the inhomogeneously broadened exciton manifold J via the creation of a phonon27,36,65,66. Spectral diffusion leads to a loss of ωτ/ωt correlation, and is observed by the loss of diagonal elongation of the P+ peak within 150 fs. The lineshape of the strong below-diagonal cross-peak P+/P is reminiscent of P, but broadened in ωτ by the inhomogeneity of P+, as it reports on the energy relaxation from P+ to P (potentially via intermediate states).

Fig. 4: Temporal evolution of the 2D spectrum of sample 1.

The temporal evolution of the absolute value 2D spectrum of 1 at φ = 5° reveals the coupling between P, J and P+, as well as the energy transfer into J and P, via the cross-peaks. Row a The data is linearly scaled and normalised to the half-maximum intensity of the P peak and; row b scaled by an arcsinh-function and normalised to the scaled maximum, in order to visualise the large dynamic range of the data. Contour lines are superimposed at values of 0.01, 0.04, 0.08, 0.12, 0.16, 0.2, 0.4, …, 0.9 for clarity. Due to the high intensity of P, the spectra are distorted by an artefact from the Fourier transform of the discrete experimental data along τ → ωτ. This leads to peaks in the 2D spectrum that do not correspond to real transient signals of the system, mainly visible at ωt = 15,700 cm−1. Each frame shows the linear absorption spectrum as a black line for comparison.

To better assess the kinetics observed in the absolute 2D signal, time traces at combinations of excitation and detection energies ωτ and ωt corresponding to the on-diagonal peaks, as well as all cross-peaks, are displayed in Supplementary Fig. 2 and described in Supplementary Note 2. They show that the three on-diagonal signals (P, J and P+) appear instantaneously (within the temporal instrument response function of 15–20 fs) and decay on a commensurate, <100 fs time scale. Thereafter, we observe a slower, picosecond decay of the on-diagonal signals that is, unfortunately, not sampled sufficiently to extract reliable decay time constants. Cross-peaks arise due to the coupling between P, J and P+ and their more complex temporal behaviour provides detailed information about the system’s kinetics. When P is excited, the coupling with J and P+ results in a transient response of the sample at P/J and P/P+, respectively. The associated kinetic traces rise instantaneously and decay with a <100 fs time constant comparable to the three on-diagonal signals, suggesting that the same mechanism is responsible for their kinetics. Based on the predicted few 10s of femtoseconds dephasing times for both the cavity field and the excitonic coherences (discussed in the Introduction), this initial signal decay might be due to the loss of the system’s coherences, which, potentially, involves a partial recovery of the ground state via the loss of a photon from the cavity. Excitation of J results in an instantaneous cross-peak signal at J/P with a qualitatively longer decay, compared to the on-diagonal signals and P/J, hinting at an energy transfer from the excitonic states into the polaritonic state P. A corresponding cross-peak at J/P+ is absent, as evident from the time traces in Supplementary Fig. 2, as well as the 2D spectra in Fig. 4. However, its absence might simply be due to the fact that the intensity is below the detection limit of the experiment, as we do not expect a lack of coupling between J and P+. The latter assumption is based on the weak absorption cross-section of the dark states J, as well as the observation of a cross-peak at P/J, which indicates the coupling between P and J. Finally, excitation into P+ results in a weak cross-peak at P+/J and a strong one at P+/P, whose temporal kinetics are displayed in Fig. 5. The peak structure of the weak P+/J signal can be most clearly recognised at T ≈ 50 and 100 fs in Fig. 4, before it coalesces with the spectral shoulder of the strong P+/P peak. As discussed above, the transient signal at P+ (red trace in Fig. 5) shows a fast (<100 fs) decay that is also observed for all other on-diagonal signals, as well as the above-diagonal cross-peaks P/J and P/P+, and might thus be due to dephasing and the recovery of the common ground state, (left|g,0rightrangle). The fast, <100 fs rise times of P+/J and P+/P are slightly delayed from the decay of the on-diagonal and above-diagonal signals, indicating the energy relaxation from P+ to J and P via the appearance of SE and ESA signals at P+/J and P+/P. To avoid the influence of spectral shifts of P+/P on the dynamics (see Methods), we follow its maximum intensity rather than the intensity at a certain frequency combination ωτ/ωt, as shown in Supplementary Fig. 3 and described in Supplementary Note 3.

Fig. 5: Observable signal of the energy relaxation cascade from P+ to J and P.

a The 2D spectrum at T = 100 fs with spectral positions of P+ (red square), P+/J (yellow square) and P+/P (blue square) indicated and the steady-state absorption spectrum, shown as a black line for comparison. b The kinetic traces (log-scale) provide insight into the energy relaxation mechanism. For P+ and P+/P they show the evolution of the peak maxima to eliminate the influence of the peak shift on the kinetics, while for P+/J the time trace shows the evolution at ωτ = 17,850 cm−1 and ωt = 16,870 cm−1. The inset shows a sketch of the proposed energy relaxation mechanism. c For comparison, the kinetic traces at P+, P+/J and P+/P have been normalised.


An interesting question that arises in molecular cQED concerns the lifetimes of the polaritonic states and how the excitonic manifold is involved in the energy relaxation. While, at this point, we can only draw conclusions for the systems presented in this study, we allege that the methodology will provide answers to the above questions, in a systematic study with varying sample parameters. The most compelling argument for a direct P+ → P energy relaxation channel in sample 1 at 5°, is the concurrent rise of the SE/ESA signals at P+/J and P+/P within <100 fs. This suggests that relaxation from P+ into P occurs simultaneously with that from P+ into J, hinting at a branching ratio of ~50:50 based on the equal rate constants. The population of P directly from P+ can be further understood considering a rudimentary rate model (P+ →  J  → P) with and without an additional P+ → P relaxation channel (Supplementary Fig. 4 and Supplementary Note 4). This analysis assumes that the transient signals at P+, P+/J and P+/P can be associated with the population being in the respective states P+, J or P,which is reasonable considering the narrow linewidth of the transient J-aggregate signal40,49. The fact that excitation of the “bright” excitonic states results in a transient response at J/P, i.e. energy relaxes from J into the polaritonic state P, supports the hypothesis that also the energy relaxation channel P+ →  J  → P contributes to the transient population in P and thus the transient signal at P+/P. The similarity of the <100 fs energy relaxation time scale with the dephasing times of the cavity and excitonic coherences suggests that, as mentioned above, the excited-state dynamics are closely related to the dephasing processes within the system. In distinction to the downhill energy transfer described above, recent experimental10 and theoretical67 studies suggested that in similar systems population transfer happens from P to the dark, excitonic states, due to the overlapping density of states. Our results show that this effect does not happen for the present systems, since we do not observe a long-lived transient response of J after an initial population of P at P/J. This observation is rationalised by the relatively narrow energy distribution of the dark J-aggregate states and the large energy gap between P and J. The long-lived nature of P corresponds well to the observation that strong coupling can lead to the emergence of quasi-bound polaritonic states20,43,45. DelPo et al. report on strongly coupled charge-transfer transitions in 4CzIPN (1,2,3,5-tetrakis(carbazol-9-yl)-4,6-dicyanobenzene) molecules and show that the lifetime of the P+ state is significantly increased, compared to J-aggregate microcavities, due to the larger Rabi splitting68. While it is a priori not clear whether a comparison between the different systems is justified, the k-dependence observed in our study suggests that another parameter, namely the ratio of light-matter character, might need to be considered.

In conclusion, we have used 2DFT spectroscopy to observe energy relaxation channels in molecular cQED systems and show the influence of different sample parameters, e.g. varying the k-momentum or cavity tuning, on the non-equilibrium dynamics for a TDBC-microcavity system. Compared to transient absorption spectroscopy, 2DFT spectroscopy is able to resolve the transient response according to the initially excited transition, while simultaneously retaining the fastest possible temporal resolution, and allows to obtain a more detailed picture of the system kinetics. We thus find that upon excitation of the P+ transition, the energy relaxes to the P state within <100 fs, bypassing the excitonic state J initially. Our results are complementary to the recent work by Takahashi and Watanabe, who used visible 2DFT spectroscopy to investigate the interaction of molecular exciton-polaritons with the environment10. Therefore, 2DFT spectroscopy proves to be a useful methodology to study molecular cQED systems, especially with regards to systematically changing sample parameters, such as the Rabi-splitting, cavity tuning, as well as the in-plane photon momentum k.

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