Spectro-temporal symmetry in action-detected optical spectroscopy: Highlighting excited-state dynamics in large systems Open Access

Action-detected nonlinear spectroscopy has seen a rapid rise in popularity in recent years, with the detection of fluorescence,^1–9 photocurrent,^10,11 photoions,¹² or photoelectrons.^13,14 The experiments are typically of the pump–probe type, with the action-detected two-dimensional spectra fully resolving the third-order four-wave mixing response, probing time-dependent correlations between excited-state transitions.¹⁵ The main differences of the action-detected approach compared to the coherent detection are the detection against a dark background, natural implementation in a co-linear geometry, and acquisition of resonant signals only. From these follow advantages such as higher sensitivity that allows measurement at lower pulse intensities and of fragile samples. The resonant signal selection avoids signals from impurities, as well as cross-phase modulation and solvent response, providing access to early time dynamics. Action-detected spectroscopy can be easily combined with microscopy, adding sub-micrometer spatial resolution and decreasing the probed sample volume. Action-detected spectroscopy has been successfully used to determine the excited-state structure and dynamics of diverse systems such as molecules in solution and in gas phase,^16–18 dimers self-assembled and attached to DNA,^19,20 molecular complexes in solution^4,21 and spatially resolved in a film²² and in a membrane,²³ organic photovoltaic devices,¹¹ quantum dots,¹⁰ and recently, even single molecules.^24,25

However, in extended systems such as semiconductors or molecular aggregates, action-detected spectroscopy faces a challenge in the form of a large stationary background signal resembling ground-state linear absorption. The reason for this stationary background is the correlation of excitations by a nonlinear process after the interaction with the pulses, mixing otherwise independent excitations during the signal emission.^9,26 Accordingly, this unwanted signal generation has been termed “incoherent mixing” and presents a serious problem of action-detected spectroscopy since it obscures the resolved dynamic spectral correlations. In molecular systems, the excitation incoherent mixing process is most often exciton–exciton annihilation (EEA). Recently, EEA has been shown to decrease the contrast of excited-state dynamics to about $1N$⁠, where N is the number of molecules in the aggregate, severely limiting the applicability of action-detected spectroscopy.^27–29 

In this work, we investigate the relation between temporal and spectral symmetry in the action-detected spectra such as action-detected two-dimensional electronic spectroscopy (A-2DES), on the example of its fluorescence-based variant F-2DES. Our key finding is that stationary, time-independent signals, such as incoherent mixing, are symmetric in action-detected spectroscopy under the inversion of the time ordering of all pulses, which corresponds to an interchange of the two spectral axes in A-2DES. This provides a general way to remove the stationary signals, emphasize photoinduced dynamics, and avoid the problem of incoherent mixing.

The outline of this work is as follows: first, we introduce pulse ordering in action-detected spectroscopy and discuss the related spectral symmetries. Using the projection-slice theorem,^15,30 we relate the F-2DES spectra marginals to fluorescence-detected pump–probe (F-PP) signals, recently introduced by us.³¹ We proceed with formulation of the subtracted signal that is sensitive to time-dependent spectral features only. To outline the interpretation of the signal, we present a general expression for population dynamics in terms of excitonic states and line shapes. Comparing analytical results with numerical calculations, we verify the properties of the difference signal. Finally, we demonstrate the subtraction on both theoretical and experimental spectra of a coupled squaraine dimer and the light-harvesting complex 2 (LH2) antenna of purple bacteria.

In the following, we will focus on the most popular variant of action-detected spectroscopy with fluorescence detection. However, we expect the results to be applicable to other types of action-detected spectroscopy as well. Fluorescence-detected two-dimensional electronic spectroscopy (F-2DES) studies nonlinear response of the system encoded in the variation of its fluorescence emission intensity, depending on the mutual time delay of pulses creating the population of excited states in the sample. The F-2DES pulse sequence is illustrated in Fig. 1. In a typical F-2DES experiment, the emitted fluorescence is not time or spectrally resolved. Frequency resolution is provided by interferometric scans of two pulse pairs, the pump (ω_τ frequency) and the probe (ω_t frequency), see Fig. 2(a). The temporal resolution, on the other hand, is achieved by scanning the time delay T between the pump and probe pulse pair.

A prominent signal symmetric in T is the incoherent mixing contribution. Since the excitations interact during the signal emission only, the signal is a product of ground-state absorption of independent transitions in the first and third intervals, satisfies the condition in Eq. (9), and is eliminated by the subtraction. F-2DES_diff(ω_t, T, ω_τ) and F-PP_diff(T, ω_t) are, therefore, incoherent-mixing-free and only include signals that depend on T asymmetrically, such as energy transport.

There are two aspects of the difference signal. First, it is always free of incoherent mixing, regardless of the origin of the mixing and regardless of the type of action-detected spectroscopy. While with fluorescence detection the main source of the incoherent mixing is usually EEA, for example, the mechanism can differ in photocurrent-based techniques. Such mixing can also originate from a nonlinearity in the detection. Either way, the subtraction method removes such mixing since it is symmetric with respect to time reversal of the pulses. The second aspect of the difference signal is its interpretation. Since the stationary signals are removed, the dynamics is highlighted. Moreover, as we will illustrate, in many cases, the ground state dynamics is symmetric in the waiting time and is thus suppressed as well. We now proceed by discussing the properties and interpretation of the subtracted F-2DES and F-PP signals.

Up to now, our formulation has been general, independent of the studied system. In the following, for illustration, we will discuss the F-2DES and F-PP spectra of molecular aggregates. The primary goal of such measurements is to follow the excited-state relaxation dynamics [see Fig. 2(b)]. As previously mentioned, this is complicated by the stationary “incoherent mixing” background, which in fluorescence is typically a consequence of exciton–exciton annihilation (EEA); see Fig. 2(c). The excited-state dynamics is reflected in the stimulated emission (SE) signal, whose ratio to the stationary ground state bleach-type (GSB) signal, approaches $1N$⁠, for the system of N identical non-interacting chromophores.³⁷ This makes the excited state practically invisible in large systems, a situation only somewhat improved by the presence of excitonic delocalization.²⁸ In the language of the response pathways, the incoherent mixing is expressed by cancellation of the ESA1 and ESA2 pathways in the case of efficient annihilation (corresponding to Γ = 1). The incoherently mixed signal is then captured by a GSB-type pathway with two independent excited states being probed by the pump and probe pulse pairs. Since the signal from the two molecules is “mixed” by the EEA after the interaction with the pulses only, such a signal is necessarily symmetric to the inverted order of the pulses, satisfying Eq. (9). This applies to any type of incoherent mixing, pertinent to various modes of incoherent signal detection. The incoherent mixing, together with all other T-symmetric signals, can thus be eliminated by subtraction of the spectra acquired with opposite time ordering (or transposed spectra in case of real F-2DES). We note at this point that the time-reversal approach applies to the case of Γ > 1 as well. As we show in Sec. 2.1 of the supplementary material, in this case, the incoherent mixing is present with a relative weight of 1 − (Γ − 1) and is subtracted in the difference signal as well. For simplicity, we will take Γ = 1 from now on.

To demonstrate the properties of the difference signal, we have computed the response using fourth-order double-sided Feynman diagrams³⁸ with approximation of short pulses in the time domain.¹⁵ The case of finite pulses is further discussed in Sec. V C. Although the pulses are spectrally broad enough to justify this approximation, we do not consider them to cover the higher excited states of the molecules. According to Kasha’s rule,³⁹ excited-state absorption into these states does not lead to additional fluorescence because of their rapid relaxation. This is reflected by setting Γ = 1 for these states, leading to the cancellation of the ESA and ESA2 pathways and the absence of these states in the spectrum. We take Γ = 1 for two-exciton states as well, assuming efficient EEA as is the case in most molecular aggregates.²⁸ For simplicity, we assume short memory of the vibrational bath, leading to the factorization of the response into the ground-state absorptive line shapes $Gigg(ω)$ and excited-state stimulated-emission line shapes $Gige(ω)$⁠, and waiting time dynamics described by a propagator $U(T)$⁠. The excited state dynamics includes population transfer with rates K_ij and possibly also a part coherently oscillating in T, rapidly decaying for weakly coupled molecules. Finally, we consider a relaxation of the excited state back to the ground state with a rate K_R that is much slower than the excited state dynamics. The whole signal decays with this rate, so we will omit the factor $e−KR|T|$ in the following equations for brevity.

The fluorescence-detected pump–probe (F-PP, transient absorption) signal (τ = 0) can be acquired directly from the F-2DES expressions using the projection slice theorem³⁰ by integration over ω_τ. Since the line shapes are normalized, the F-PP expressions are obtained from F-2DES by omitting the line shapes with ω_τ.

We now illustrate the behavior of the difference signal on a model generalized dimer system of two weakly coupled groups of two-level systems with the same oscillator strengths and different transition energies, with energy transfer between the states; see Figs. 2(b) and 2(c). The F-2DES spectra of this system are shown in Fig. 3, decomposed into the GSB, SE, and ESA components. The GSB signal features two diagonal peaks at the respective transition frequencies, with cross peaks between them due to the incoherent mixing. The GSB is stationary and symmetric in T and thus cancels upon subtraction. The SE features the energy transfer from the higher energy states to the lower energy ones, manifesting as a decay of the upper diagonal peak and a rise of the lower cross peak with T > 0. At negative times, the F-2DES spectra are transposed, as derived above. The subtraction eliminates the diagonal peaks and only the population-transfer-sensitive cross peaks remain. The ESA components feature both stationary and dynamic components, but in our case, they cancel because of Γ = 1. The total difference signal thus resembles that of the subtracted SE and is a sensitive reporter on the population transfer dynamics.

The illustration in Fig. 3 is schematic and the expressions in Sec. III A above were derived with the approximation of ultrashort pulses. To verify that the difference signal behaves qualitatively the same with realistic pulses, we calculated the F-PP signal with 15 fs pulses based on the dynamics of a model system consisting of two groups of (N_blue, N_red) molecules, with energy transfer between these two groups [see Fig. 2(b)]. We numerically integrated the corresponding master equation with Lindblad dynamics (see the supplementary material for more detailed formulation).⁴² In particular, we took (N_blue, N_red) = (3, 3), with expected significant incoherent mixing and the expected dynamics contrast SE /GSB = 1/3,^27,28 energy transfer time of 100 fs, and EEA time of 20 fs. In Fig. 4(a), the F-PP time traces at the two absorption peaks are shown. At negative waiting times, the traces are constant, reflecting identical transition dipole moments (see the second row of F-2DES spectra in Fig. 3). Around T = 0, the signal features a symmetric coherent peak, oscillating at the difference frequency of 1000 cm⁻¹ and rapidly decaying. As we saw in our recent work,²⁸ both the coherence decay and pulse overlap contribute to this T = 0 peak. In the subtraction context, it is important that their contribution is symmetric in T under these conditions.⁴¹ At T > 0, the population transfer is visible in the higher energy peak decaying and lower energy peak rising on the large stationary incoherent mixing background. The traces thus agree well with Eqs. (16) and (17) integrated over the ω_τ line shape. In Fig. 4, the difference signal corresponds perfectly to Eq. (21): rising from zero at T = 0, the dynamics captures the energy transfer, by both peaks increasing in magnitude, with the sign reflecting whether the energy is transferred away from them (positive sign) or into them (negative sign). As expected, both the T = 0 peak and incoherent mixing background vanish completely upon subtraction.

The present analysis shows a great promise of the subtraction method, indicating the utility of the spectro-temporal symmetry in F-2DES in suppressing unwanted signals and highlighting the desired dynamics. The next step is to apply the procedure to real experimental data.

The data subtraction procedure does not require any modification of the experimental setups. Since the F-2DES (or other action-detected 2D spectra for that matter) already contain both T > 0 and T < 0 signals, the procedure can be applied to extract dynamics from already measured datasets. To showcase the procedure, we apply it to two systems. First, we construct the difference signal from the F-PP spectra of a coupled heterodimer, which is a relatively simple system with dynamics easily identifiable in the standard spectra as well. Then, we test the approach on energy transfer dynamics in the LH2 antenna of purple bacteria, which due to its 27 bacteriochlorophyll molecules suffers from large incoherent mixing background. In both of these systems, the EEA is highly efficient, so that Γ = 1.

To demonstrate the properties of the subtracted signal, we apply the procedure to a coupled heterodimer from our previous studies.^6,31 This particular dimer, shown in Fig. 5(a), consists of two weakly coupled squaraine molecules, with energy transfer between their first excited states. Simultaneously, each excited state also undergoes vibrational relaxation on a time scale similar to that of energy transport [Fig. 5(d)]. In Fig. 5(b), the spectral peak traces in F-PP are shown, integrated over the regions around 1.73 eV for SQB and 1.88 eV for SQA outlined in the F-PP map shown in Fig. 5(e). Apparently, the peak traces are constant for T < 0, in line with the theoretical expectation for similar transition dipole moments and broad spectra. At T > 0, the higher-energy peak amplitude decreases due to energy relaxation. The lower-energy peak simultaneously receives excitation from the higher-energy SQA, but also further relaxes by excited-state reorganization marked by a dynamic Stokes shift of the SE line shape. The subtracted F-PP data are shown in Fig. 5(c) (peak traces) and Fig. 5(f) (transient map). The subtracted signal behaves exactly as predicted by the theoretical formulas derived above, Eq. (21). The higher energy peak increases to positive values signifying the decay of the negative-signed F-PP signal due to energy relaxation. In contrast, the lower energy peak is increasing as more energy relaxes into the SQB states. In the integrated traces, faster and slower rise constants can be seen, reflecting the presence of intermolecular energy transfer and intramolecular relaxation. The SQB intramolecular relaxation can be seen by the seemingly additional positive valued feature in the subtracted F-PP map around 1.8 eV, corresponding to the relaxing initially excited vibrational mode of SQB.

As is clearly visible from the comparison of the traces shown in Figs. 5(b) and 5(c), the subtraction improves the visibility of the dynamics remarkably. Since by definition the signal has to vanish at T = 0, the contrast of dynamics is necessarily 100%. The price to pay is the lower signal-to-noise ratio, given by the increased noise due to error propagation (factor of $2$⁠), and only the dynamic signal part remaining. Nevertheless, next to the stationary ground-state background, the coherence peak around T = 0 was also removed by the subtraction. To summarize, the subtracted signal is somewhat noisier, but with clearly visible dynamics that is more straightforward to interpret.

A true test of the spectro-temporal subtraction is a system large enough for the incoherent mixing to prevent reliable measurement of excited-state dynamics. We have recently encountered exactly this situation in the LH2 light-harvesting antenna of purple bacteria.²⁸ In the LH2 antenna of Rps. acidophila, the excitation energy is transferred from a ring of 9 loosely coupled bacteriochlorophylls called B800, due to its absorption wavelength, to another ring of 18 more strongly coupled pigments called B850; see Fig. 6(a). The energy transfer is known from standard transient absorption to take about 0.8 ps,^28,44 and the EEA is highly efficient.^28,45 In F-2DES, the transport dynamics constitutes only about 5% of the signal, with the rest being the incoherently mixed ground-state response of the two rings. The F-2DES data, shown in Fig. 6(b), resemble the example in Fig. 3. The B800 → B850 transfer should be visible in the decay of the B800–B800 diagonal peak and in the rise of the B800–B850 cross peak. Leveraging the F-2DES symmetry, we integrated the data over square regions around the lower cross peak and in the symmetric location flipped along the diagonal, as depicted in Fig. 6(b), to get their evolution in the waiting time, Fig. 6(c). Clearly, it is difficult to infer the dynamics from the peak traces only. This, however, changes with the subtraction corresponding to F-2DES(ω_t, T, ω_τ) − F-2DES(ω_τ, T, ω_t), as shown in Fig. 6(d).

The difference signal in the LCP region grows in magnitude due to the excitation energy transfer from the B800 ring to the B850 ring. We include in the subtracted data our previous theoretical excitonic-model calculation²⁸ as well, showing a good agreement. To recover the time scale present in the dynamics, we fitted the calculated trace for times T > 300 fs by an exponential rise. We obtained a rise time of t₂ = 784 ± 4 fs, which agrees with the 0.8 ps value expected from theory.^28,44 The subtraction is thus clearly capable of removing the incoherent mixing and highlighting the excited-state dynamics even in large coupled molecular systems.

Interpreting the difference signal in Sec. III and in the experimental demonstration in Sec. IV, we focused on the removal of incoherent mixing and on excitation energy transfer dynamics. Another important part of the response is vibrational wavepacket dynamics. We discuss vibrational signals in F-2DES and F-PP in detail in Sec. 3 of the supplementary material. In brief, in F-2DES, the subtraction highlights excited-state vibrations and suppresses the ground-state response. Moreover, at least for individual and weakly coupled molecules, for short pulses, the non-rephasing difference F-2DES as well as the difference F-PP is completely free of the ground-state contribution and reports exclusively on the excited-state dynamics. The interpretation of the difference signal in strongly coupled vibronically mixed systems remains to be shown; the possibility of obtaining clean excited-state response is exciting and warrants a future study.

We demonstrated in this article a way of leveraging the spectro-temporal symmetry of stationary signals for their removal by subtraction of negative waiting time signals from the positive waiting time data. In F-2DES, the T < 0 and T > 0 signals are related by a spectral transposition. For similar pump and probe pulses, this allows carrying out the subtraction in post-processing from a single standard F-2DES dataset. Note that such spectral flip corresponds to the swap of the t and τ delays, together with effective phase conjugation to end up in the correct quadrant; see Fig. S1 in the supplementary material. The applicability of this approach depends on the particular experimental realization since such full 2D spectrum subtraction is in practice sensitive to line shape artifacts, such as phase twists due to imperfect phase referencing. Furthermore, since the subtraction removes the fully diagonal symmetric features anyway, most of the relevant information is in the F-PP spectra, which can be obtained as marginals of the full F-2DES dataset. The measurement of full F-2DES is thus often not necessary and the significantly simpler acquisition of F-PP suffices. In this case, the T < 0 waiting times must be acquired separately. However, this is a relatively small price to pay for the 100% contrast of excited-state dynamics. The separate measurement of T < 0 and T > 0 signals is necessary for non-overlapping pump and probe spectra, both in F-2DES and in F-PP. Furthermore, the subtraction of the negative time signal has other advantages, such as the elimination of not only all constant signals, but some coherent oscillatory signals around T = 0 and symmetric pulse-overlap artifacts as well.

The subtraction suppresses a large part of the signal and somewhat increases the noise, leading to a decreased signal to noise ratio. The visibility of the dynamics in the difference signal thus depends on its relative magnitude compared to the experimental noise. The relative contrast of the dynamics scales inversely proportional to the system size; see Ref. 28 for scaling in coupled excitonic systems. The experimental noise level can thus be understood as a limit to the maximum effective size of the system in which dynamics can be studied. Similar considerations apply for any method of incoherent signal suppression.

The negative time subtraction for the removal of incoherent mixing works with any pump and probe pulses, as formulated in Sec. II. The presented examples on excitonic systems were, however, discussed in the impulsive limit, and the experimental spectra were obtained with broadband near-identical pump and probe pulses. For different pump and probe spectra, the F-2DES spectra become asymmetric, and the symmetry of the response function can be studied directly from their symmetry only after a correction to the laser spectra and in the spectral overlap region. We demonstrate these options by additional calculations in Sec. 5.1 of the supplementary material, using the same exemplary aggregate as described in Sec. III B. For relatively narrow, spectrally distinct pump and probe pulses, we show that the F-PP difference spectra can be constructed from separately acquired F-PP(T > 0) and F-PP(T < 0) datasets. For the case of different but spectrally overlapping pump and probe pulses, we further demonstrate the spectral correction of F-2DES. This correction works well within the spectral overlap region and for waiting times after the pulse temporal overlap region.

The F-2DES correction based on Eqs. (7) and (8) works only for slowly varying signals in T. For contributions that rapidly change in T on timescale comparable with pulse length, such as oscillating vibrational and vibronic wavepackets, the influence of the pulse spectra is more complex.^30,36 A solution that works both in this case and in the case of spectrally distinct pump and probe pulses is to acquire the T > 0 and T < 0 signals independently, actually reversing the order of the pulses as shown in Fig. 1. The difference signal is then constructed using Eq. (11) for F-2DES or Eq. (15) for F-PP. Such difference signals are incoherent mixing-free and reflect only the dynamic part of the response. The interpretation of these subtracted signals in terms of, e.g., vibronic wavepackets, motivated by the vibrational response results discussed above, deserves a further discussion, which is, however, beyond the scope of this work.

Ever since the formulation of its origin, the options to eliminate incoherent mixing have been investigated.⁴⁶ The attempt to identify unique incoherent mixing phase signatures has been unsuccessful.⁴⁷ Clearly, the subtraction presented here cannot be realized by phase cycling alone, since that leaves the time ordering of the pulses and thus of the frequency axes unchanged. A recent work suggested using a special pulse polarization scheme to cancel incoherent mixing contributions in isotropically oriented samples.⁴⁸ However, the polarization removes only the isotropic incoherent mixing part, which is insufficient in large molecular systems with fixed orientations, such as light-harvesting complexes. Yet, another option to suppress incoherent mixing is short-time-gating or in general time resolution of the fluorescence emission, selecting the fluorescence before the mixing takes place.^34,49 This is, however, impractical due to the typically short EEA timescale that requires fast gating, leading to loss of signal intensity.

We have demonstrated on the example of fluorescence-detected 2D spectroscopy a general property of action-detected 2D spectra that contributions symmetric in the waiting time, probed by a reverse ordering of the excitation pulses, are symmetric in the 2D correlation spectra. This spectro-temporal symmetry can be leveraged to selectively eliminate such contributions in action-detected 2DES and pump–probe spectroscopy in a system-independent way. Among these contributions belongs the infamous “incoherent mixing,” a stationary time-independent background of ground-state signals correlated by nonlinear excitation interaction during signal emission. As we have shown in detail, subtraction of positive and negative waiting time spectra suppresses pulse-overlap and coherent signals, eliminates the incoherent mixing background, and highlights the excited state dynamics such as energy transfer. The approach can be applied to any existing action-detected spectroscopy experiment, without the need for experimental modifications. We believe that, due to its system-independent universality, the approach will find application in action-detected nonlinear spectroscopy in general and in F-PP and F-2DES in particular. Notably, it promises to open the door for fluorescence- and photocurrent-detected measurement of large coupled systems, such as photosynthetic complexes.

The supplementary material is available with the following sections: S1. Detailed derivation of the F-2DES expressions for normal and reversed pulse ordering. S2. explicit formulas for a coupled dimer. S3. Vibrational dynamics. S4. Expressions for electronic coherence signals. S5. Numerical calculation of the F-PP signal of a model aggregate. Effects of different pump and probe pulses. S6. Diagrams for pulse overlap region.

We thank Ariba Javed and Julian Lüttig from the group of Jennifer Ogilvie for providing the F-2DES data of LH2 and for enlightening discussions of F-2DES. We thank Stefan Müller for useful discussions and sharing insights into F-PP data subtraction. We further thank Tomáš Mančal for help with Quantarhei software. The authors acknowledge funding by Charles University (Grant No. PRIMUS/24/SCI/007, to P.M.).

The authors have no conflicts to disclose.

K. Charvátová: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Methodology (equal); Software (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). P. Malý: Conceptualization (equal); Funding acquisition (lead); Methodology (equal); Resources (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

All experimental data have been published in the cited works; for availability, see therein. QuantaRhei is an open source software for calculations. Remaining data that support the findings of this study are available from the corresponding author upon reasonable request.