Two-dimensional fluorescence excitation spectroscopy: A novel technique for monitoring excited-state photophysics of molecular species with high time and frequency resolution

Electronic two-dimensional (2D) spectroscopy, being the optical analog of nuclear magnetic resonance (NMR) spectroscopy,^1–3 has emerged as a powerful technique, both in IR^4–6 and UV/Vis.^7–12 The 2D signal S^(σ)(τ, T, τ_t) can be envisaged as a variant of the third-order heterodyne-detected four-wave-mixing signal. As sketched in Fig. 1(a), this signal can be recorded in the rephasing (σ = R, k_S = k₁ − k₂ + k₃) or non-rephasing (σ = NR, k_S = −k₁ + k₂ + k₃) phase-matching direction as a function of the delay times between the first two pulses (coherence time τ), the last two pulses (detection time τ_t), as well as the second and the third pulses (population time T). The first applications of electronic 2D spectroscopy for multi-chromophore systems were mainly targeted toward energy transfer in biological light-harvesting complexes.^13,14 Nowadays, electronic 2D spectroscopy is applied to a great variety of systems, from atoms¹⁵ and polyatomic molecules¹⁶ to molecular aggregates and solids.^17–19 

Researchers are frequently interested in the real-time monitoring of the photophysical and photochemical processes related to the lower-lying excited electronic states.^19,31 In this context, the SE contribution to the 2D spectrum, $SSE(σ)(ωτ,T,ωt)$⁠, delivers direct access to the desired dynamics.³² However, this contribution has to be extracted from the experimental 2D signals, which are often highly spectrally congested and very difficult to interpret for nontrivial material systems. Methods designed to mitigate this problem have been devised, e.g., polarization-sensitive detection of 2D signals.^33,34 Yet, unambiguous discrimination between ground- and excited-state contributions to the 2D spectra of biological light harvesters has triggered a controversy on the origin of the transients oscillating as a function of the population time T^35–42 and required painstaking analyses and careful theoretical simulations and interpretations.^43–45 

The following question naturally arises: Can the SE contribution be measured exclusively, i.e., without overlapping GSB and ESA contributions? In this work, we show that the answer is positive if the investigated signal is based on the detection of spontaneously emitted photons. The spectral profiles of stimulated and spontaneous emissions are similar, and one can rely on time-resolved detection of fluorescence to select the deexcitation pathways from the lower-lying electronic states for measurement of the desired signals. However, the necessary fluorescence-detection scheme has to be selected and designed carefully.

This work is structured as follows: we outline the theoretical foundations of 2D-FLEX spectroscopy, clarify its similarities and differences with electronic 2D spectroscopy, and discuss the experimental feasibility and realizability of 2D-FLEX (Sec. II). To clarify how 2D-FLEX works, we contrast 2D-FLEX and electronic 2D spectra simulated for a model four-level system (Sec. III A) and the Fenna–Matthews–Olson (FMO) complex (Sec. III B).

For clarity of presentation, we assume that the polarization vectors of all laser pulses are identical; and we do not explicitly consider orientational averaging. Polarizations of laser pulses and vectorial properties of transition dipole moments can be readily reinstalled in all formulas if necessary, and orientational averaging can be performed by using standard methods.^48–50 

It is noteworthy that the fluorescence signal of Eq. (7) is equivalent to the SE part of the so-called integral transient-absorption pump–probe signal,⁵⁵ provided E_t(t) and ω_t specify the envelope and the carrier frequency of the probe pulse.

We can briefly summarize the above discussion as follows: It is expected that the vast experience gained in the interpretation of 2D spectra can be helpful for the analysis of 2D-FLEX spectra. Nevertheless, Fourier-limited $(δTδωt∼1)$ 2D-FLEX spectra $S2DF(σ)(ωτ,T,ωt)$ are not identical to 2D SE spectra $SSE(σ)(ωτ,T,ωt)$ [hence ≈ in Eq. (4)], meaning that the intensities and shapes of the peaks in 2D-FLEX and 2D SE spectra are different, in general.

From an experimental perspective, 2D-FLEX requires time-gated fluorescence detection after double-pulse excitation. The delay between the excitation pulses corresponds to coherence time τ [see, e.g., Eq. (12)]. The most widespread technique to resolve fluorescence signals on femtosecond timescales is fluorescence upconversion.^64–69 Briefly, fluorescence from a photo-excited sample is mixed with a gating pulse in a non-linear medium such as a β-barium borate (BBO)-crystal to create a sum-frequency or upconverted signal, as schematically depicted in Fig. 1(b). An intense gate pulse can therefore amplify the potentially weak fluorescence signal. Alternatively, transient gratings induced in an isotropic medium by ultrashort pulses are used to scatter and produce time-resolved fluorescence.⁷⁰ This method also allows for broadband detection of emission spectra with femtosecond time resolution, defined by the duration of the grating-forming pulses. However, in the absence of an upconverting gating pulse, this technique relies solely on the efficiency of diffraction of the transient grating, which is in the single digit percent range and makes this technique best suited for fluorophores with a strong emission yield. Kerr-gating experiments⁷¹ also allow for broadband and time-resolved detection of fluorescence. As a disadvantage, this method relies on the high extinction ratio of the employed crossed polarizers to suppress unwanted, slowly decaying signal backgrounds. This issue is avoided in fluorescence upconversion by spatial separation of the upconverted signal, which explains the high signal-to-noise levels in this technique.⁶⁹ This makes fluorescence upconversion the method of choice for recording small variations on a large signal background, which is crucial for 2D-FLEX: excitation frequency dependence is encoded as potentially weak modulations on the fluorescence spectra recorded at different coherence times [see Fig. 1(c)]. In terms of time resolution, Joo and co-workers have demonstrated that sub 30 fs is possible and 50 fs is routinely achievable.^72,73 Based on the 2D spectroscopy toolbox, there are several ways of producing phase stable double-pulses with variable femtosecond delays. In this context, the most straightforward implementation of 2D-FLEX is realized by excitation with a collinear double pulse and subsequent detection of the total 2D-FLEX signal $S2DF(ωτ,T,ωt)=S2DF(R)(ωτ,T,ωt)+S2DF(NR)(ωτ,T,ωt)$⁠. Devices for collinear double pulses are readily inserted into standard fluorescence upconversion experiments. This speaks for either acousto-optic modulators⁷⁴ or interferometers based on birefringent wedges.⁷⁵ In essence, standard fluorescence upconversion experiments can be turned into 2D-FLEX experiments with the same effort it takes to upgrade transient absorption to electronic 2D experiments.⁷⁶ 

The signals are calculated by using the simple dynamical model of Refs. 80 and 81, in which the environment is characterized by three parameters, namely vibrational relaxation rates in the ground (ν_g) and excited (ν_e) states (responsible for the recovery of Boltzmann’s equilibrium distribution in the manifolds of ground and excited states) and electronic dephasing rate ξ_eg. We set $νe−1=200$ fs, $νe−1=100$ fs, $ξeg−1=300$ fs. For clarity, we adopt zero temperature, which ensures that the system relaxes to the lowest ground state |1⟩⟨1| after $T>νg−1$ and to the lowest excited state |3⟩⟨3| after $T>νe−1$⁠.

The present model does not contain higher-lying excited electronic states, and ESA is therefore absent. GSB (first column), SE (second column), and GSB+SE (third column) contributions to the 2D spectrum as well as the 2D-FLEX spectrum (fourth column) for this four-level system at T = 0 fs (first row), T = 50 fs (second row), T = 100 fs (third row), and T = 500 fs (fourth row) are plotted in Fig. 2. The system reveals three different transition frequencies: ω₃₁ = ω₄₂ = E₀, ω₃₂ = E₀ − E_v, and ω₄₁ = E₀ + E_v [ω_kk′ = (E_k − E_k′)/ℏ]. As the system is initially prepared in the lowest state |1⟩⟨1|, the spectra show at most 2 × 3 = 6 peaks and exhibit only two peaks along ω_τ. corresponding to transitions from the ground state |1⟩ to the upper states |3⟩ and |4⟩.

2D spectra of such simple models are well understood (see, e.g., Ref. 82). The GSB and SE spectra consist of incoherent and coherent contributions. For the former, the system is in the population state (|1⟩⟨1| for GSB and |3⟩⟨3| or |4⟩⟨4| for SE) after interaction with the first two pulses. For the latter, the system is in a coherent state (|1⟩⟨2| or |2⟩⟨1| for GSB and |3⟩⟨4| or |4⟩⟨3| for SE) after interaction with the first two pulses. Coherent contributions are responsible for vibrational beatings determined by the phase factors exp(±iE_vT), while the T-dependence of incoherent contributions is exclusively caused by vibrational relaxation. Furthermore, relaxation brings the system to the lowest ground and excited states and kills coherent contributions. This is why the GSB spectra show a sextet of peaks exhibiting vibrational beatings at $T<νg−1$ (three upper panels), which turns into a static quartet of peaks revealing ω₁₃ and ω₁₄ frequencies at $T>νg−1$⁠. Similarly, the SE spectra also display a sextet of peaks exhibiting vibrational beatings at $T<νe−1$ (three upper panels), which turns into a static quartet, revealing transitions from the lowest excited state |3⟩ to the ground states |1⟩ and |2⟩ at $T>νe−1$⁠. The total, GSB+SE, 2D spectrum has a multi-peak structure even at $T>νg−1,νe−1$⁠. The shapes and widths of all peaks in impulsive 2D spectra are exclusively determined by the electronic dephasing rate ξ_eg.

2D-FLEX spectra (fourth column) have to be compared with their SE counterparts. Three differences are worth mentioning. First, 2D-FLEX spectra are predominantly determined by incoherent contributions, which is the direct consequence of the finite gating/upconversion pulse duration (Γ_t < E_v). Hence, the 2D-FLEX spectrum at T = 0 exhibits a quartet of peaks corresponding to ω_τ = ω₃₁ and ω_t = ω₃₁, ω₃₂, as well as ω_τ = ω₄₁ and ω_t = ω₄₁, ω₄₂. Second, due to the same reason, 2D-FLEX peaks do not exhibit visible vibrational beatings. As T increases, the two rightmost 2D-FLEX peaks move down, mirroring vibrational relaxation in the excited states. At $T>νe−1$⁠, the entire population moves to the lowest excite state |3⟩⟨3|. Hence the 2D-FLEX spectrum reveals a quartet of peaks at ω_τ = ω₃₁, = ω₄₁, ω_t = ω₃₁, ω₃₂, similar to the SE spectrum. Third, peak shapes of 2D-FLEX spectra along ω_τ are determined by the electronic dephasing rate ξ_eg, while peak shapes along ω_t are specified by both ξ_eg and the gate-pulse duration $Γt−1$⁠. This is why all 2D-FLEX peaks have characteristic elongation along the ω_t-axis. It becomes stronger with shorter gating pulses, leading to poorer frequency resolution in ω_t.

The differences between SE and 2D-FLEX signals can also be discussed through the use of double-sided Feynman diagrams. Figure 3(a) shows a sketch of the investigated four-level system, where the ground state (excited state) energy levels are marked as γ (ɛ), respectively. The SE diagrams in Fig. 3(b) and the fluorescence diagrams in Fig. 3(c) are both determined by the same four-point correlation functions^55,83 with strong similarities as discussed above. The main difference lies in the signal emission, which may occur in the rephasing/non-rephasing phase-matching direction specified by the wavevector, k_S for SE [see Fig. 3(b)]. For fluorescence, the signal is emitted in all directions, has no well-defined wavevector and two light–matter interactions are required to bring the system from the excited-state population |ɛ⟩⟨ɛ| to the ground-state population |γ⟩⟨γ|. The important difference between SE and 2D-FLEX lies in the fact that the last two interactions in the fluorescence-diagrams in Fig. 3(c) stem from spontaneous emission, meaning that there is no experimental control over the time at which the two inherently different diagrams in Fig. 3(c), corresponding to R_R(t₃, t₂, t₁) and $RNR*(t3,t2,t1)$⁠, contribute to the signal. This effect—in combination with the dependence of the ω_t peak shape on the gate-pulse duration as discussed above—describes the observed differences in lineshapes and relative peak intensities between SE and 2D-FLEX; see Fig. 2.

In summary, the time-dependent behavior of all electronic 2D and 2D-FLEX peaks in the presented model system is easy to comprehend. However, the mere comparison of 2D-FLEX spectra (fourth column in Fig. 2) with their electronic 2D counterparts (third column in Fig. 2) demonstrates the advantages of 2D-FLEX. In particular, the evolution of 2D-FLEX spectra from T = 0 to $T>νg−1,νe−1$ can be qualitatively understood in terms of downhill population transfers among excited levels, which requires little a priori information about the system under study. Transformations of electronic 2D spectra, on the other hand, cannot be explained without information about the level structure of the system. First, beyond two-level chromophores, ESA alters 2D spectra significantly. Second, even when ESA is spectrally separated or entirely vanishes, such as in the present model, GSB renders simulations essential for interpreting the observed peak structures and dynamics.

The FMO antenna complex was scrutinized by femtosecond transient-absorption pump–probe spectroscopy^84,85 and electronic 2D spectroscopy^13,14,43,44 (see also recent reviews^86–88). To our knowledge, femtosecond time- and frequency-resolved fluorescence spectra of FMO were not detected (nanosecond spectra were measured, although⁸⁹) and were simulated only recently.⁹⁰ Here, we present the simulated 2D-FLEX spectra of FMO and contrast them with the corresponding electronic 2D spectra. All simulations are performed at 80 K with the time non-local master equation derived for the bath featuring one overdamped mode and one underdamped mode. The parametrizations and details of the model are described in Ref. 44. These simulations, which are performed with a full account of orientational averaging and static-disorder, reproduce the 2D spectra reported in the same work.

The results are depicted in Fig. 4, showing 2D-FLEX (left column), the SE-contribution to electronic 2D (middle column), and the complete electronic 2D (right column) spectra of FMO at T = 30 fs (upper row), T = 510 fs (middle row), and T = 1005 fs (lower row). Since the photophysics and energy transfer in FMO are well understood,^{13,14,43,44,91,92} the interpretation of the 2D-FLEX and 2D SE spectra presents no difficulty. They reveal population transfer from the initially excited higher-energy excitons 4–7 or BChls 2, 5, 6, 7 (upper panels) to the lower-energy excitons 1, 2 or BChls 3, 4 (lower panels). The peak shapes in the first and second columns are quite different, although. Due to the Fourier-limited resolution $(δTδωt∼1)$⁠, peaks in 2D-FLEX spectra are broader along the ω_t-axis, and certain spectral features merge. However, 2D SE spectra cannot be detected separately, as only the total spectrum of Eq. (1), being the sum of the GSB, SE, and ESA contributions, is experimentally accessible. The total electronic 2D spectra, which are presented in the rightmost column, have much higher complexity than the 2D SE spectra and are, therefore, harder to interpret. The FMO complex has been comprehensively studied, and its electronic 2D spectra have been deciphered in great detail.^13,14,43,44 However, 2D-FLEX spectra, which are free of GSB and ESA contributions, can be interpreted and qualitatively understood directly: they monitor population transfer from the higher-energy excitons to the lower-energy excitons.

We have proposed a novel nonlinear femtosecond technique, 2D-FLEX spectroscopy, which allows the direct monitoring of SE contributions. In 2D-FLEX, the material system is excited with a pair of phase-locked laser pulses with an adjustable relative delay. Subsequently, time- and frequency-resolved fluorescence spectra are detected. We discussed the experimental feasibility and realizability of 2D-FLEX, developed the theoretical framework for the description of 2D-FLEX spectra S_2DF(ω_τ, T, ω_t), and discussed similarities and differences between S_2DF(ω_τ, T, ω_t) and SE contributions to electronic 2D spectra S_SE(ω_τ, T, ω_t). The high information content and straightforward interpretation of 2D-FLEX were demonstrated by simulations of a model four-level system and the FMO antenna complex.

Electronic 2D spectroscopy represents the most sophisticated and insightful third-order technique. However, electronic 2D spectra consist of GSB, SE, and ESA contributions, hindering their facile interpretation. This problem was addressed by advanced methodologies such as polarization-sensitive detection in combination with beating map analysis,⁴³ but this approach requires extensive theoretical support and simulations.^13,14,43,44 The main advantage of 2D-FLEX spectra is the absence of GSB and ESA contributions. This means they can be interpreted—at the qualitative level at least—without an in-depth theoretical treatment of the material system under study. These attractive features make 2D-FLEX a potentially highly useful new addition to the family of multidimensional femtosecond spectroscopic techniques. A polarization-sensitive detection of 2D-FLEX (for example, detection of the 2D-FLEX anisotropy) may also be useful for studying multi-chromophore systems, as recently shown for polarization-controlled transient absorption spectroscopy.⁹³ 

M.F.G. acknowledges support from Hangzhou Dianzi University through startup funding. L.C. is supported by the Key Research Project of Zhejiang Lab (Grant No. 2021PE0AC02). J.H. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project No. 514636421. F.S. thanks the Czech Science Foundation (GACR) for financial support through Grant No. 22-26376S.

The authors have no conflicts to disclose.

Jianmin Yang: Data curation (equal); Formal analysis (equal); Software (equal); Writing – original draft (equal). Maxim F. Gelin: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Lipeng Chen: Data curation (equal); Formal analysis (equal); Software (equal). František Šanda: Data curation (equal); Writing – original draft (equal); Writing – review & editing (equal). Erling Thyrhaug: Data curation (equal); Formal analysis (equal); Writing – review & editing (equal). Jürgen Hauer: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.