1 Introduction

Two-dimensional electronic spectroscopy (2DES) can be defined as the “ultimate” time-resolved nonlinear optical experiment, since it measures both the real (absorptive) and imaginary (dispersive) parts of the complex third-order nonlinear polarization response [9]. 2DES has shown to be more powerful than one-dimensional four-wave-mixing measurements like transient absorption (often called pump–probe spectroscopy). In a formal description of transient absorption measurements based on the framework of nonlinear optics, the pump beam electric field interacts twice with the sample to photo-excite a fraction of the chromophores into electronic excited states. These two interactions simply represent the product of the transition dipole connecting the ground state to an excited state with its complex conjugate to give the probability of pump absorption. A probe pulse arrives after a variable delay T to stimulate a signal field that is detected as a change in transmission of the probe.

2DES experiments provide simultaneously high temporal and spectral resolution [10]. The excitation frequency is resolved in 2DES by delaying the two ultrafast excitation pulses with an interferometric resolution. Any other third-order nonlinear spectroscopy (i.e., transient absorption) is contained in the 2DES map (Fig. 2) where the emission detection is followed along the waiting time. The main breakthrough of 2DES resides in the disentanglement of processes in systems with multiple interacting components [3, 4, 11,12,13,14].

Fig. 2
figure 2

Third-order 3D spectrum of the isomer Pr of the chromoprotein Cph1 represented in a 3D isosurface, the transient absorption spectra for an excitation centered at 650 nm (left panel), and the absorptive 2DES map for a waiting time of 100 fs (right panel) are extracted. The data are from the Ref. [15]

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As excitation and emission frequencies are resolved, 2DES data are represented in 2DES maps for a waiting time (T). In a 2DES map, the peaks on the diagonal, i.e., at equal excitation and emission frequencies, are the signature of the populations of the individual transitions; whereas the peaks out of the diagonal, i.e., at different excitation and emission frequencies, probe couplings and relaxation dynamics between electronic states. With respect to other nonlinear spectroscopic techniques, 2DES has the following advantages:

  1. (a)

    It is possible to separate, and thus distinguish, contributions to the nonlinear signal that are spectrally overlapped in one-dimensional experiments. Analysis of cross-peaks reveals whether the different transitions seen in the sample absorption spectrum arise from the same or different molecular species and can quantify electronic couplings and correlations between different excited states [3, 4, 11].

  2. (b)

    2DES distinguishes inhomogeneous and homogeneous broadening by the analysis of the lineshapes of the spectroscopic signals, enabling the individual levels to be singled out in strongly congested spectra [3, 15,16,17,18,19].

  3. (c)

    2DES can follow the parallel pathways by which the coupled electronic dynamics evolve after photoexcitation in real time. This makes the 2DES technique a particularly powerful tool for tracking energy and electron transfer processes [4, 13, 20].

Our aim here is to provide examples highlighting the real advantages of using 2DES. As physical chemists, we know that to understand a photo-initiated reaction, one should initially assign spectral signatures to reactants, products—eventually intermediates—and then propose a kinetic model to explain the evolution of the species, based on the observed spectral changes. However, in complex condensed-phase systems, spectral signatures are often overlapped and difficult to untangle. For example, steady-state measurements (e.g., linear absorption, fluorescence, anisotropy, and circular dichroism) supply information on the electronic structure. However, homogeneous line broadening can significantly obscure physical insights such as the distinction between homogeneous and inhomogeneous broadening and sometimes cryogenic temperature experiments are not able to reach the desirable spectral resolution. Further insights can be gained by time-resolved techniques, such as transient absorption and pump–probe spectroscopies, which can also track photo-induced dynamics, as energy and electron transfers. However, these techniques have intrinsic resolution limitations on either high temporal or high-frequency resolved-capability. In this context, 2DES has emerged as an optical technique that can accomplish many of the goals of conventional spectroscopies, also overcoming all the limitations mentioned above.

2 Implementation

2DES implementation faces two main technical challenges: (1) it requires a careful control of the interferometric stability between pulse pairs [21], which need to be phase-locked within a small fraction of their carrier wavelengths (i.e., a precision of few nanometers is required for ~ λ/100 at 550 nm) [3, 22, 23]; (2) to fully exploit its benefits, 2DES calls for ultra-broadband pulses, with the duration of just a few optical cycles, which are challenging to generate and control. Thus, the phase stability in a pair of broadband pulses has been the limiting factor that explains why 2DES was developed after 2D-IR spectroscopy [24]. However, thanks to the technological advancement of ultrafast optics, 2DES techniques are rapidly evolving and have today various applications, providing access to systems that contain electronic transitions spanning from the ultraviolet to the near-infrared and beyond. Technically, one can build a 2DES apparatus following three different types of geometries: (1) the non-collinear so-called “box-car” geometry (Fig. 3a), and (2) the collinear, so-called “pump-probe” geometry (Fig. 3b) and (3) the fully collinear geometry (Fig. 3c).

Fig. 3
figure 3

Adapted from [41], with the permission of AIP Publishing

a Non-collinear heterodyne and b collinear heterodyne and c fully collinear homodyne 2DES implementation. Rephasing (\(P_{\text{R}}^{(3)}\)) and non-rephasing (\(P_{\text{NR}}^{(3)}\)) signals are emitted in a new direction or in the same direction for the non-collinear and the collinear geometry, respectively. The signal is heterodyned by a fourth pulse, called local oscillator (LO).

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The box-car geometry is based on diffractive optics to generate four identical pulses and a set of crystals with variable thickness (wedges), placed in each individual path beam to control their relative delays [25, 26]. Depending on the arrival order of the two pump beams, the rephasing and non-rephasing signals are detected. The rephasing signal is recorded when the first pump arrives first, whereas the non-rephasing signal is measured when the second pump beam arrives first. The absorptive 2DES map is the sum of the rephasing and non-rephasing signals. Further box-car implementations substitute the diffractive optics with beam-splitters pairs (as for 2D-IR) to overcome the limit of the spatial chirp bandwidth introduced by the diffractive optics [27,28,29]. Even if homodyne detection is possible, heterodyne detection is preferred in most of 2DES setups [21, 30]. In a heterodyne detection, a weak signal is interfering with a strong “local oscillator” field to amplify the signal and enable the extraction of the complex signal field. The frequency of the mixing product is the sum or the difference of the frequencies of the signal and the local oscillator. The main advantages of the non-collinear geometry are the possibility to measure separately the non-rephasing and the rephasing signals and the possibility to reach high signal-to-noise ratios.

The collinear geometry exploits adaptive optics, as pulse shapers, to turn a conventional pump-probe setup into a 2DES apparatus by creating the first two phase-locked pulses from a single pump pulse that propagate in the same direction, in contrast to the box-car geometry [31,32,33]. Several groups have shown the potential and the capability to perform “two-color” kind of experiments by exploiting the collinear geometry [34]. In these experiments, the excitation axis is given by a first identical pulse pair resonant with the electronic transition of the system under study, while the detection axis can be spectrally tuned and depends solely on the bandwidth of the third pulse [35,36,37]. The generation of the first pulse pair is usually obtained by pulse-shaping techniques [38, 39], however, recently a new compact Translating-Wedge-Based Identical Pulses eNcoding System TWINS [40, 41] device has been shown to have capability of creating intrinsically phase-locked pulses exploiting the birefringence of non-linear crystals. The main advantages of the collinear geometry are the phase stability and the possibility to carry two-colors experiments.

The fully collinear geometry uses only one beam, which is pulse shaped into phase coherent pulse trains [31, 42]. It is based on a homodyne detection. The main advantage of this technique is to provide a simplified experimental design.

A summary of the various implementations of 2DES, together with benefits and drawbacks, bandwidth limitations, and typical bandwidths used has been nicely reviewed by Ogilvie et al. in a recent review [43]. A detailed discussion of the several technical aspects on how to perform 2DES experiments goes beyond the scope of this book, however review papers and books are available describing various 2DES implementations, as can be seen in references [11, 38, 44].

3 Heterogeneity

One of the main challenges in condensed-phase experiments is the heterogeneity of the samples, which present very broad absorption spectra at room temperature. 2DES allows unraveling the presence of multi-chromophoric systems and sometimes disentangles their dynamics. Furthermore, the temporal evolution of 2D line shape allows distinguishing a dynamic to a static contribution of a molecular dipole transition. The inhomogeneous broadening of a system is interpretable from the shape of diagonal peaks on a 2DES map (Fig. 4).

Fig. 4
figure 4

Schematic 1D linear absorption a and the corresponding 2D absolute value line shapes b from the homogeneous to the inhomogeneous limit. c Schematic illustration of the temporal evolution (along the waiting time) of the 2D absolute value line shape in the inhomogeneous limit case

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Firstly, at the waiting time T = 0, any particular transition excited will not yet have undergone dynamical processes that could change its resonant frequency; thus, its resonant frequency will be detected at its excitation frequency. In the homogeneous limit, the absorptive signal along the diagonal is expected to have a 2D-Lorentzian shape, because the rephasing and non-rephasing signals are identical [45]. As the inhomogeneous broadening increases, the 2D line shape broadens along the diagonal axis while the antidiagonal remains unchanged, i.e., the rephasing signal is larger than the non-rephasing signal at early time. Thus, it is possible to quantify the linewidth broadening along the diagonal to determine how heterogeneous the system is [3, 46, 47]. The lineshape along the diagonal will have an elliptical shape whose profile along the diagonal frequency axis is proportional to the traditional 1D absorption spectrum [13]. The width perpendicular to the diagonal represents the homogeneous linewidth. The ellipticity of 2D line shape quantifies how much the system is inhomogeneous.

Secondly, at longer waiting time, spectral diffusion might occur [3]. In the so-called dynamic disorder case, the correlation between excitation and detection frequency will be lost and 2D line shape will become symmetric. Whereas in the static disorder case, 2D line shape will remain elongated along the diagonal [46, 47]. The time evolution of an asymmetric 2D line shape is discriminating between static and dynamic disorder.

The first example describes how 2DES probes the dynamic disorder of a system by tracking the time evolution of an asymmetric diagonal peak. Chlorophyll a is the smallest chromophore involved in photosynthesis [48]. Chlorophyll a is well studied for bio-mimetic applications [49]. The main goal is to understand how the complex interactions underlying the efficient energy conversion of absorbed photons into chemical energy to adapt this to biomimetic energy transduction devices. As shown on Fig. 5a, the broadband excitation spectrum spans over the entire absorption spectrum of chlorophyll a.

Fig. 5
figure 5

Adapted with permission from [18]. Copyright 2015 American Chemical Society

a 2DES spectra of chlorophyll a in acetone. The red numbers indicate the waiting time in picoseconds. On the top panels, the absorption spectra and the laser spectrum are recalled in red and blue, respectively. b Molecular structure of chlorophyll a. c Ellipticity of the main peak as function of waiting time of chlorophyll a in cyclohexane (red) and acetone (blue)

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The monomeric chlorophyll a was probed in different solvents where an inhomogeneous broadening is always observed (Fig. 5) [18]. The elongated 2D line shape evolves towards a symmetric one in different time-scale depending on the solvent. This clearly unravels a spectral diffusion attributed to solvation. The slowest spectral diffusion was observed for H-bonding and/or viscous solvent. It was attributed to a spectral diffusion. Understanding these phenomena is a key parameter to optimize the biomimetic devices.

In Fig. 5a, there are two positive bands on the 2DES map of chlorophyll a in acetone at 50 fs: one elongated along the diagonal and another one out of the diagonal. The first one is attributed to the photobleaching and stimulated emission of the molecule and the second one to its stimulated emission from a vibronic energy level. From T = 50 fs to T = 600 ps, the photobleaching band gets more symmetric. This is clear evidence of a spectral diffusion. To quantify this inhomogeneity, the ellipticity of the photobleaching band was measured and plotted in Fig. 5c. The ellipticity is the ratio:

$$e = \frac{{(D^{2} - A^{2} )}}{{(D^{2} + A^{2} )}}$$

with D and A the 2DES spectrum diagonal and antidiagonal full width at half maximum, respectively [46]. An ellipticity close to 1 corresponds to the inhomogeneous limit, whereas e = 0 to a more homogeneous system. In all of the solvents tested, an initial inhomogeneous broadening was observed. In an apolar solvent like cyclohexane, the system evolves in a homogenous system in less than 1 ps, whereas in polar and even more in protic solvent, it takes nanoseconds to reach this homogeneity. This inhomogeneous broadening was interpreted as a spectral diffusion with different regimes:

  1. 1.

    Typical solvation response behavior in the fs/ps time-scale

  2. 2.

    Solvent dynamics modified by interactions with the solute

  3. 3.

    A strong function of solvent, being greater in H-bonding and viscous media.

Thus, it was possible to probe solvation and assign its dynamics directly by 2DES.

The second example illustrates the conformational heterogeneity in the ground state of chromoproteins: phytochromes (Fig. 6a). Phytochromes are red (Pr) and far-red (Pfr) light photoreceptors of plants and some cyanobacteria, fungi, and algae that regulate germination and flowering [50]. Phytochrome Cph1 from the cyanobacterium Synechocystis sp. PCC6803 has become the ubiquitous model of plant phytochrome [51]. More specifically, the photodynamics of the Pr isomer of phytochrome Cph1 has been controversial for years with two kinetics models describing multiphasic excited-state decay kinetics: (1) heterogeneous model, i.e., coexistence of ground state conformational subpopulations [52, 53], or (2) homogeneous model, i.e., a single ground state conformation, which undergoes an excited-state photoisomerization process—either branching on the excited state or relaxing through multiple sequential intermediates [54, 55]. Numerous time-resolved 1D studies support one of each model but none were able to clearly distinguish which one is correct.

Fig. 6
figure 6

Adapted with permission from [15]. Copyright 2017 American Chemical Society

a Protein structure of the phytochrome Cph1 and its bilin chromophore in the inset. b Steady-state absorption (dot) and fluorescence (dash) spectra of Cph1 in a PBS buffer. The spectrum of the laser is represented by a black line. c 2DES maps at a waiting time of 100 fs and 9 ps of Cph1. The bottom panel shows the ellipticity of peak A during the primary photoisomerization.

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2DES is an ideal probe of heterogeneous line broadening and was able to clearly answer to this controversy. The 2DES maps (Fig. 6c) present one main positive peak on the diagonal (photobleaching) and one negative peak out of the diagonal (excited state absorption). As described before, the ellipticity of the diagonal peak is an indicator of the inhomogeneous broadening. Thus, the ellipticity of the positive peak was followed along the waiting time T. It remains constant and small (~ 0.2) from 100 fs to 100 ps (Fig. 6c) [15]. This is clear evidence that the ground state is homogeneous and the heterogeneous model can be discarded.

It is worth noticing that the rephasing of the data is crucial to address the heterogeneity question. Indeed, the line shape of the rephasing and non-rephasing data alone has a meaningless lineshape due to an artifact known as phase twist [1, 43, 56, 57]. It is important to probe the ellipticity on the 2DES maps of the absorptive signal, i.e., after summing the rephasing and non-rephasing signals.

Due to the high degree of information contained within 2DES maps, it was possible to resolve a controversy that lasted for years in the literature. The photoisomerization of Pr isomer of the phytochrome Cph1 involves one homogeneous ground state. The photoisomerization takes place in the excited state. 2DES shows how it is easy to unravel this mechanism, which was impossible to interpret from 1D data.

The third example demonstrates how 2DES measurements can reveal insights into spectra that are completely dominated by inhomogeneous broadening. In this case, measuring non-rephasing and rephasing signals was crucial. It is a study on cubic CdSe nanocrystals where the nanocrystal size variation makes this system highly heterogeneous (Fig. 7) [12]. CdSe nanocrystals have applications in photovoltaics devices, electronics, and others depend on the efficiency of the dynamics of each states and how they interact to each other [58]. 2DES can bring some clues on these phenomena.

Fig. 7
figure 7

Adapted with permission from [12]. Copyright 2012 American Chemical Society

a Representation of colloidal cadmium selenide (CdSe) nanocrystals in cubic phase. b Steady-state absorption spectrum of CdSe nanocrystals in toluene (line) and the excitation laser spectrum (dash). c Rephasing, non-rephasing and absorptive 2DES map in magnitudes of CdSe nanocrystals at a waiting time of 120 fs.

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The absorption spectrum of CdSe nanocrystals is broad with two low-energy exciton bands. During those experiments, both bands were excited and probed (Fig. 7b). The 2DES spectra show two diagonal and two anti-diagonal peaks corresponding to the two exciton bands and the two coupled exciton states, respectively. In comparison to the simulation, those 2DES spectra are really blurry because of the inhomogeneous broadening. This inhomogeneous broadening takes its origin in the size distribution of the nanocrystals in solution.

Figure 7 shows that analyzing the 2DES spectra by their rephasing, non-rephasing and absorptive signals give some insights into the inhomogeneous broadening. Indeed, photon-echo (rephasing signal) can improve the spectroscopic resolution by removing inhomogeneous line broadening [17]. After this analysis, the two diagonal and anti-diagonal peaks appear clearly on the spectra. This allows to determine that the binding energies of the (\(|X_{1} X_{2}\rangle\)) and the (\(\left| {X_{1} } \right.X_{1}\rangle\)) biexcitons are similar. A value of 25 meV for this energy reproduces the peaks very well in simulations. The non-rephasing spectra do not contain photon-echo signal and therefore contain inhomogeneous contributions to both the diagonal and antidiagonal linewidths. We will see in the section photoreactivity that the disentanglement of those state dynamics is rendered possible by 2DES.

4 Excitation Energy Transfer (EET)

Excitation energy transfer (EET) in complex electronic systems has been successfully studied by 2DES. Among the advantages of 2DES over standard transient absorption measurement is the capability of distinguishing between weakly or strongly coupled systems (Fig. 8). In the case of a weakly coupled donor-acceptor system (Fig. 8a), EET can be tracked in a 2DES experiment as the appearance of a cross-peak at the donor (B)/ acceptor (A) frequencies in the 2DES map as a function of the waiting time T. In the strong coupling regime (Fig. 8b), the donor and acceptor share a common ground state and the excitation becomes delocalized over two excitonic states, α and β. Thus, at T = 0 the 2DES map reveals the existence of separate cross-peaks at the αβ/βα positions, which evolve along T. In transient absorption experiments, due to the lack of resolution over the excitation axis, these cases will be indistinguishable. In particular, in photosynthetic complexes, the inter-chromophore electrostatic interaction gives rise to Coulombic couplings, which in turn re-define energetically shifted delocalized and coupled states (namely exciton [59]) that ultimately determine the photosynthetic optical properties. Accordingly, the electronic absorption spectra are—in most cases—broad and congested, which is beneficial for maximizing light absorption, but makes the analysis of these systems extremely difficult.

Fig. 8
figure 8

Schematic illustration of energy transfer in absorptive 2DES spectra for the case of weak or strong coupling regime. a In the case of a weak coupling system, the correlation spectrum at T = 0 fs reveals two absorption species along the diagonal, while at T > 0 fs, a cross-peak appears (AB) and indicates energy transfer from B to A. b In the case of strong coupling, at T = 0 fs two instantaneous cross-peaks appear, at the αβ/βα positions, indicating the existence of a shared common ground state

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2DES is the perfect tool to identify the chromophore couplings and track the energy transfer pathways in protein-pigment complexes. In addition to spectrally resolve excitation and emission frequencies with femtosecond resolution, this technique has enabled discoveries about the structure and dynamics of various photosynthetic light-harvesting complexes. A complete picture on how the energy transfer cascade evolves from the light harvesting antenna complex to the reaction center has been recently shown in Fig. 9 for the photosynthetic apparatus of a green sulfur bacterium [60]. This study was made possible due to 2DES developments, which enable measurements in highly scattering environments as cells [60, 61], finally allowing determination of the full transfer network between electronic transitions. The steady-state spectrum of the apparatus (Fig. 9a) at 77 K reveals clear distinguishable features [48]: the high-energetic strongest band, which corresponds to the assembly of light harvesting antenna complex (chlorosome), three absorption peaks around 800 nm corresponding to the Fenna–Matthews–Olson (FMO) proteins that mediate the energy transfer from the chlorosome to the reaction center (RC), which shows a weak absorption band at the lowest energy (833 nm). The EET processes have been extensively studied in the isolated photosynthetic FMO subunits [4, 5, 62] (Fig. 9b), and we show here the main interchromophoric EET pathways measured in the isolated FMO complex (Fig. 9b, c) [62]. The 2DES absorptive maps reveal cross-peaks below and above the diagonal. While the cross-peak above the diagonal do not exhibit any strong evolution, a clear signature of coupling among the chromophores, the ones below the diagonal, show a clear increase of the signal intensity, associated with the downhill EET.

Fig. 9
figure 9

Adapted with permission from [62]. Copyright 2016 American Chemical Society. And adapted by permission from Macmillan Publishers Ltd: Nature Chemistry [60], copyright 2016

a Steady-state absorption spectrum of the photosynthetic apparatus of the green sulfur bacterium and laser spectrum used in the 2DES experiment reported in d. b Arrangement of BChla pigments within the FMO sub-units. c Absorptive 2DES maps measured in the FMO complex at 77 K for increasing waiting times T with dashed lines indicating excitonic transition energies. d Absorptive 2DES maps on the entire apparatus described in b for increasing waiting times T. The purple dashed lines indicate the spectrum measured on the isolated FMO complex described in a.

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In Fig. 9d we show a series of 2DES maps recorded at different T times for the whole photosynthetic cell [60]. The early map at T = 30 fs shows a series of diagonal peaks attributed to the different transition in the steady-state absorption spectrum. At later T times, one can clearly observe the formation and decay of cross-peaks below the diagonal, revealing how energy flows within and between the individual complexes in the apparatus. By examining the connectivity between different complexes and applying a global fitting method [60], it was possible to track the step-by-step energy flow through the entire unit (Fig. 9d) and observed for the first time that the FMO complex serves as energy conduit between the chlorosome and the RC.

A second example of EET process resolved by 2DES is reported in Fig. 10 [63]. In this work, the isolated light-harvesting antenna complex (namely LH1) of a purple bacterium is studied with a two-color 2DES apparatus, where both transition in the near-infrared and in the visible spectra region were covered with two different femtosecond broadband pulses. The absorption spectrum of the LH1 complex is constituted of two chromophores (Fig. 10a), a carotenoid—namely, spirilloxanthin (Spx)—whose 0–0 first optically allowed S0 → S2 transition peaks at 540 nm, and a bacterio-chlorophyll (BChl) named B890 due to its Qy band peaking at 881 nm (the higher energy Qx band of the B890 peaks at 585 nm). By exciting in the visible range and detecting over both visible and near-IR ranges, it was possible to follow all the photoinduced processes, namely, i) the Spx internal conversion, ii) the BChl Qx → Qy internal conversion, and iii) the Spx → B890 EET process by tracking the formation of several cross-peaks in the 2DES maps. In the degenerate 2DES experiment (Fig. 10b), the internal conversion for the bright S2 state of the Spx to the dark S1 state is observed as the appearance of a negative cross-peak (color-coded in blue) on within the first few hundreds of femtoseconds, assigned to the formation of an excited state absorption from the S1 state, at the 545/620 nm excitation/detection cross-peak. Moreover, a second negative feature is assigned to the formation of a parallel long-lived state, named S*, at the 545/570 nm excitation/detection cross-peak [64]. The degenerate 2DES map also shows that the diagonal peak of the Qx of the B890 (585/585 nm excitation/detection) has become less pronounced after T = 65 fs, indicating that population in the B890 moiety, has started the internal conversion process to Qy, reducing the stimulated emission contribution to the positive signal on the diagonal.

Fig. 10
figure 10

Adapted from [63], with the permission of AIP Publishing

a Arrangement of the LH1 complex and supramolecular architecture of the BChl B890 (purple) and the Car Spx (orange) together with the absorption spectrum of LH1 complex extracted from Rsp. rubrum (black line) and pulse spectra used in the 2DES experiments in the visible (NOPA1, green line) and near-IR (NOPA2, red line) wavelength regions. b Degenerate 2DES maps of the LH1 complex from Rsp. rubrum, for different waiting times T, following excitation by a sub-10-fs visible pulse, resonant with the S0 → S2 transition of the Car and the Qx transition of the BChl. c Two-color 2DES maps of the LH1 complex for different waiting times T, with same excitation axis as in B but detection axis in the near-IR.

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The two-color 2DES maps (excitation axis in the visible range and detection in the near-IR) in Fig. 10c show two positive cross-peaks at 875-nm detection wavelength that appear at the excitation wavelengths of 545 and 585 nm, corresponding to resonances of S2 and Qx, respectively. The center wavelength of the positive bands at 875 nm identifies these features as pure photobleaching contributions from Qy, showing that at these early T times, the population is still entirely on Qx and the optical probes which are specific for population in Qy. At later T times, the further evolution of the 2DES maps consists in the growth of the positive cross-peaks at 875 nm and the appearance of two excited-state absorption peaks on the short wavelength side, around 845 nm. Both the features at 875 and 845 nm are specific optical probes for population on Qy, showing that both the Spx → B890 EET and the internal conversion from Qx to Qy are occurring on this time scale. The observation of energy transfers and internal conversion parallel pathways was only possible thanks to the two-color 2DES experiment.

In the third example (Fig. 11), a pulse generated via continuum filamentation spanning from 500 to 1300 nm is used to collect 2DES-WL (white light) spectra to resolve energy transfer within a network of semiconducting carbon nanotubes (CNTs) [36]. CNTs thin films are being explored for energy harvesting and optoelectronic devices because of their exceptional transport and optical properties. In this study, the nanotubes in the film are in close contact, allowing the energy to flow through the films. The film is composed primarily of four different diameter nanotubes, defined by their chiral indices: (7,5), (7,6), (8,6) and (8,7). Each tube exhibits an S1 transition in the near-infrared (corresponding to the optical bandgap) and an S2 transition in the visible. The 2DES-WL map described in Fig. 11b, exhibits several peaks, both on the diagonal as off diagonal. Each of the electronic transitions in the absorption spectrum creates a pair of diagonal peaks in the 2DES-WL spectrum. The negative peak on the diagonal corresponds to photobleaching (color-coded in blue) of the direct bandgap transition, while the blue-shifted positive peak along the detection axis is from an excited state absorption (color-coded in red). Moreover, the cross-peaks appearance correlates the different tubes.

Fig. 11
figure 11

Adapted by permission from Macmillan Publishers Ltd: Nature Communications [36], copyright 2015

a Absorption spectrum of films of CNTs, where the S0–S1 and the S0–S2 transitions are highlighted for the different tubes, according to their chirality. b 2DES map covering four quadrants generated by the combination of two pulses, spanning the visible and the near-IR range. c Zoom of the 2DES map on the top left quadrant in b and progression of three 2DES maps at various waiting times T. d Energy level scheme of the CNTs, highlighting the pathways observed in c.

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We discuss as an example, only a sub-quadrant of the 2DES-WL map. We choose the S1/S1 quadrant. Excitation of S1 lies below the ionization threshold and so no charges, but only excitons are generated. The cross-peaks appear on the lower half of the diagonal, indicating that energy transfer occurs downhill from larger to smaller bandgap tubes. The cross-peaks raise simultaneously, indicating that exciton transfer is equally probable between any downhill combination of nanotubes. Moreover, the cross-peaks are round, whereas the diagonal peaks are elongated, which imply uncorrelated energy transfer. This means that a photoexcited exciton created on a random nanotube has equal probability of transferring to another nanotube anywhere within the acceptor’s distribution of energies. Thus, the inhomogeneity of the bandgap transitions does not impact energy transfer. Thus, it is equally likely for an S1 exciton to relax to the next smallest bandgap nanotube as it is to relax directly to the smallest bandgap nanotube.

Based on the similarity of the transfer rates and a lack of correlation, the transfer was found to be independent of the spectral overlap, ruling out Förster resonance energy transfer between bright states as the rate limiting mechanism. These observations were only possible thanks to the accurate analysis of cross-peaks and it provided a basis for understanding the photo-physics of energy flow in CNT-based devices.

5 Vibronic Coupling

The study of electronic coupling in complex systems benefits enormously from 2DES, due to the observation of coherent oscillations detected on the cross-peaks of the 2DES maps [5, 6, 65] generated via broadband pulses. These oscillations can be observed as an amplitude modulation of the dynamics along the waiting time T measured at specific point on the excitation/detection 2DES map. The analysis of such oscillations requires the acquisition of various 2DES maps at different waiting times T and a subsequent Fourier transform along this axis of the residual oscillations after removing the slow-varying component associated to the specific cross-peaks trace [66] (Fig. 12). Understanding coherent oscillations in the 2DES maps gives profound insight of the system under study at a quantum mechanical level. Here we do not aim at describing the quantum mechanical formalism, but give some intuitive insight into how coherent oscillations can be observed and interpreted.

Fig. 12
figure 12

Progression of the 2DES map of a model dimer system of Fig. 8 along the waiting time T. The amplitudes of the cross-peaks oscillate at the difference frequency of the two states. A cut through different waiting time T maps at the cross-peak position shows the residual signal amplitude modulation as a function of T cross-peak. An oscillation of the cross peak amplitude is observed at the difference frequency between the two states, and the Fourier transform of the signal amplitude gives a peak at the measured oscillation frequency

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Coherent oscillations were observed early on 2DES measurements performed on the photosynthetic Fenna–Matthews–Olson (FMO) complex (described previously in Fig. 9). Interestingly, the oscillatory beating lasted for a time scale longer than the energy transfer rate (> 1 ps) [5]. This initial observation sparked interest in coherent phenomena in photosynthetic complexes and how such oscillations could have a role in the energy transfer processes. Accordingly, there has been a growing interest in investigating the origin of these long-lived oscillations and their relation to energy transfer in the system [67]. 2DES was used to observe robust oscillations in the FMO complex [68], antenna complexes of marine algae [6], the major light-harvesting antenna LHCII [69], and the reaction centers (RCs) of higher plants [70, 71], at low temperature and even at room temperature (Fig. 13).

Fig. 13
figure 13

Reprinted by permission from Macmillan Publishers Ltd: Nature Chemistry [70], copyright 2014. Reprinted by permission from Macmillan Publishers Ltd: Nature Physics [71], copyright 2014

Examples of 2DES studies where coherent oscillations have been observed in photosynthetic complexes, specifically in: a the FMO complex. Reprinted from Copyright 2010 National Academy of Sciences [68]; b LHCII complex. Reprinted with permission from Copyright 2009 American Chemical Society [69]; c–d reaction center RC of PSII.

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These oscillations have been attributed to electronic, vibrational, and vibronic origins in various systems. Gradually, the consensus has been reached that the long lasting oscillations and their frequencies must be explained by mixing of electronic and vibrational coherences [72,73,74]. In particular, vibrational modes with frequencies resonant with electronic energy gaps were suggested to be important for both spectroscopic signals and energy-transfer dynamics [75].

The following example (Fig. 14) shows a recent experimental and theoretical verification of coherent electronic–vibrational (vibronic) coupling as the origin of long-lasting coherence, measured in a model system for artificial light harvesting, a molecular J-aggregate [76]. In this tubular system (Fig. 14a, b), a polarization-controlled 2DES approach offered a great tool to distinguish the specific optical responses, leading to fewer region of the 2DES map, which show oscillatory components. In this system, a quasi-resonance exists between the vibrational frequencies ν1 and ν2 and the energy splitting of the electronic transition, which, in principle, makes it complicated to distinguish the contribution from electronic or vibrational origin. The resonance promotes an electronic–vibrational coupling that gives origin to vibronic (excited states) and vibrational (ground state) coherences, which can both lead to long-lived beating signals in 2DES maps.

Fig. 14
figure 14

Adapted from Nature communications [76]

a Molecular structure of the tubular J aggregated, where the main electronic transition dipole moments are drawn along their directions. b Absorption spectrum of a measured in parallel and perpendicular configuration. c The Fourier-transform (FT) amplitude maps of non-rephasing (non-reph.) and rephasing (reph.) at ν1 = 700 cm−1, which reveal the presence of a non-rephasing diagonal peak N11 and a rephasing cross-peak R31. d The time traces of N11 (red) and R31 (blue) in normalized intensity against waiting time T: experimental results shown as circles and theoretical simulation shown as a full line.

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By retrieving the 2D Fourier-transform amplitude maps of selected oscillation at frequency ν1 close to 700 cm−1, the authors observed a very defined coherence pattern for the non-rephasing and rephasing spectra. Specifically, the oscillations appear on the diagonal peak for the non-rephasing map and as a cross-peak on the rephasing map (Fig. 14c). To describe these features, a vibronic model is employed that describes the coupling of the two electronic bands (specifically 1 and 3 in Fig. 14b) with a quasi-resonant vibrational mode with frequency ν1. The model predicts an initial fast decoherence, sustained by a long-lived vibrational character of the underdamped vibration (Fig. 14d). The polarization-implemented sequence of this 2DES result provides a strong foundation for understating vibronic coupling and its implication in energy transfer processes.

Coherent oscillations are clearly observed and assigned in inorganic complexes as well, which are characterized by narrower spectral lineshapes and reduction of ensemble disorder. A clear example of electronic coherence is described in Fig. 15 for the case of colloidal semiconductor nanoplatelets (NPLs) [77], where the electronic oscillations are not hidden by vibrational coherences or ensemble dephasing (Fig. 15). The absorption spectra of the CdSe and CdSe/CdZnS NPLs are shown in Fig. 15a, b together with the excitation laser spectra used for the 2DES experiments. Figure 15c, d, show the plot of the room-temperature 2DES map of the two samples for a waiting time of T = 52 fs.

Fig. 15
figure 15

Adapted by permission from Macmillan Publishers Ltd: Nature Communications [77], copyright 2015

Absorption spectra of CdSe (a) and CdSe/CdZnS (b) NPLs showing the HX and LX exciton transitions together with the laser spectra used in the 2DES. c, d 2DES maps (total and magnitude signal) for a waiting time T = 52 fs. e Coherent oscillations along the waiting time T of the lower cross-peak of the rephasing 2DES map (population relaxation removed) of CdSe NPL (top), CdSe/CdZnS NPL (middle) and CdSe/CdZnS NPL (bottom) are shown as black dots.

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Diagonal and cross-peaks are clearly resolved at the electronic transitions involving different hole bands and labeled heavy-hole and light-hole excitons (HX and LX, respectively). The presence of the cross-peaks is assigned to the strong coupling between the two transitions expected because they share the same electronic state.

Monitoring the evolution of the amplitude of the peaks appearing in the 2DES maps as a function of time allowed the study of coherent superpositions of exciton states. Oscillations are evident and visible on both the diagonal and the lower cross-peaks in the CdSe NPL and in the CdSe/ZnS NPL heterostructure. The authors reported the results for the oscillations observed at the lower cross-peak position in three different samples (CdSe, heterostructure CdSe/ZnS and CdSe/ZnS with thinner core) (Fig. 15e). The oscillations last for about 150 fs. The frequency of the oscillations in these two peaks matches the HX–LX frequency difference measured by the absorption spectrum and they are assigned to electronic coherences. The HX–LX electronic coherence dephases for the three different samples with different time constants. From these results, it is evident that the pure homogeneous line broadening of the colloidal NPLs enables clear observation of the electronic coherence.

6 Photoreactivity

2DES provides a key advantage over methods such as pump-probe spectroscopy for unraveling ultrafast kinetic processes because it decouples the frequency of excitation with the time resolution of the experiment. This renders possible since the excitation is resolved by an interferometric measurement between the two pump beams. Thus the discrimination between a sequential and a parallel mechanism is achievable, even if the states are too entangled and/or their dynamics are too fast [7].

For a sequential mechanism (Fig. 16a), a loss of population (photobleaching) is expected for the species A and a cross-peak will appear at the excitation frequency of A and emission of B. The gain of population (photoproduct) will result in a signal with the opposite sign [78, 79]. In the case of an equilibrium between A and C (Fig. 16b), two diagonal peaks will be detected as a photobleach for A and C and two off-diagonal peaks will be of the opposite sign at the excitation of A and emission of C and excitation of C and emission of A. For instance, if A gives B and C with A and C being in an equilibrium (Fig. 16c), two diagonal peaks are expected in A and C, and three off-diagonal peaks are expected in excitation of A and emission of B and C, and excitation of C and emission of A. Those cases are some extreme cases to describe a 2DES map and will be useful to interpret the data.

Fig. 16
figure 16

Schematic photoreaction pathways and their corresponding 2DES maps for A becoming B (a), A and C are in equilibrium (b) and A becoming B and A is in equilibrium with C (c)

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The first example is the electron and hole relaxation dynamics in cadmium telluride (CdTe) nanorods (NRs) in toluene (Fig. 17a inset) [80]. Indeed, conventional transient absorption spectroscopy cannot disentangle those phenomena because of the overlap of the transitions and their very fast relaxation time scales. We saw in the Heterogeneity paragraph that those systems are highly heterogeneous. Semiconductor nanocrystals are very interesting materials because the quantum confinement effect leads to: i) discrete energy levels, ii) tunable energy bandgap depending on their size and iii) enhancement of the Coulomb interaction [81]. Their properties can be easily tuned by advanced synthetic methods. Therefore, they have been applied to a variety of fields: optics, photovoltaics, sensing, and electronics [82]. For instance, in photovoltaic and photocatalytic applications, the electron and hole dynamics contribute separately to the photon-to-electron conversion efficiency. It is crucial to understand and disentangle the dynamics of those species in order to improve the final materials.

Fig. 17
figure 17

Adapted with permission from [80]. Copyright 2017 American Chemical Society

a Steady-state absorption spectrum of cadmium telluride (CdTe) nanorods (NRs) in toluene (black line) and the pump laser spectrum (green dashed line). The three lowest excitonic states are indicated by colored lines (S1 in pink, S2 in green and S3 in purple). The inset shows a TEM image of the NRs (average dimensions of 21 nm length and 5 nm width). b Summary of the identified relaxation pathways and respective time scales. c 2DES maps of CdTe NRs in toluene at different waiting times (30, 60, 130, and 500 fs).

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Femtosecond transient absorption spectroscopy is a powerful technique to follow the electron/hole dynamics in real time, but it is very difficult to disentangle them. Indeed, the states are very close in energy and a broadband excitation (femtoseconds pulses) leads to an overlap of their transient signals, while narrowband pulses allow state selectivity but the temporal resolution is lost. 2DES is capable of disentangling the excitonic states and to distinguish between electron and hole relaxation dynamics.

Figure 17c shows 2DES maps of the differential transmission (∆T/T) of CdTe NRs at different waiting times [80]. At T = 30 fs, diagonal and cross-peaks are observed: i) three distinctive diagonal peaks are assigned to the photobleaching of S1, S2, and S3 excitons, ii) positive S2/S1 (excitation/emission) and S3/S1 cross-peaks along the vertical at 1.8-eV emission energy are the signature of an ultrafast relaxation of S3 and S2 into S1 in 30 fs, and iii) a positive signal at the S1/S2 cross-peak due to the shared electron level of the two excitons. The other 2DES maps allow tracking the photoreaction dynamics. At T = 60 and 130 fs, the relaxation from S3 and S2 to S1 keeps happening (growth of the amplitude of the S3/S1 and S2/S1 cross-peaks). The amplitude of the S2/S2 peak decreases and the one of the S3/S3 stays stable. The dynamics of hot hole and electrons is faster than 500 fs, because the 2DES maps at 130 and 500 fs are similar.

By monitoring the decay dynamics of the diagonal and cross-peaks, a proposed mechanism of the ultrafast hole and electron dynamics and their relaxation pathways for the first three excitons of CdTe NRs is given in Fig. 17b. A hot hole thermalization from 1Σ3/2 to 1Σ1/2 in the range of 30 fs, while higher energetic electrons after S3 pumping relax from Σ′ to Σ within 50 fs time constant. Those results were unraveled only by 2DES and can design complex hybrid nanostructures that aim to separately control the electron and hole dynamics.

The second example is a new approach described by Ruetzel et al. [7] that allows tracking of the putative mechanisms involved in complexes photochemical reactions. The main advantage of 2DES is to be able to probe the reactants axis (excitation axis) at the same time that the products one (emission axis). The ring-open-6-nitro-BIPS (Fig. 18a) in acetonitrile exists in a mixture of two stable merocyanine isomers, which photoconvert through a photoisomerization process. The photoisomerization was reported but the identification of the mechanism remained unknown because of the spectral overlap of those isomers [83, 84]. Figure 18a shows the absorption spectrum of the different isomers of 6-nitro-BIPS: the dominant trans–trans–cis isomer (TTC) and only 10% of merocyanine molecules exist in the trans–trans–trans isomer (TTT). The TTC isomer has an absorption peak at 557 nm, whereas the absorbance peak of TTT isomer is centered at 595 nm.

Fig. 18
figure 18

Reprinted from: Copyright 2014 National Academy of Sciences [7]

a Steady-state absorption spectrum of ring-open 6-nitro-BIPS in acetonitrile. The predicted absorption spectra of the two isomers trans–trans–cis configuration (TTC) and trans–trans–trans configuration (TTT) are represented in red dots and blue dashes, respectively. b 3D isosurface representation of the 3D spectrum of the 6-nitro-BIPS sample. c Slices of the 3D spectrum in the (νpump,νprobe) plane for ν T  = 164 (left) and ν T  = 360 cm−1 (right) and the phase for ν T  = 164 cm−1 at a vertical cut at the TTC excitation wavenumber (center).

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The photoisomerization was recorded for a long waiting time. In order to be able to disentangle the mechanism, the 2DES data are Fourier-transformed along the waiting time, T, to obtain a third-order 3D spectrum, represented on Fig. 18b. The main contribution is observed in a plane at ν T  = 0 cm−1. By contrast, the observed vibrational motion leads to separated cross-peaks in the 3D spectrum at 176 cm−1, which corresponds to a period of 190 fs centered at the crossing of TTC excitation and TTT photobleaching emission. A second isolated cross-peak appears around ν T  = 363 cm−1 (period of 90 fs) and is slightly shifted away from the crossing point toward higher excitation frequencies, indicating that more excess energy is required to initiate it. The location in the 3D spectrum reveals that oscillations of cross-peaks connecting the two isomers give rise to these signatures. This analysis was possible only by measuring 2DES data.

Figure 18c shows slices of the 3D spectrum in the excitation/emission plane for the two main vibrational modes together with the phase. The main contributions are situated at TTC excitation and TTT emission wavenumbers. A vibrational wavepacket motion in the excited state absorption band of TTT is enhanced in the rectangle IV. Both modes are observable at the red edge of the stimulated emission (marks V and VI). Those oscillations take place in the excited state. Moreover, a clear phase shift is observed at the minimum of the potential energy surface according to the coherent wavepacket evolution. Thus, the formation of the TTT involves wavepacket motions on the excited potential energy surface S1. The photoproduct is formed on the timescale of a few vibrational periods in the S1 state.

This analysis clearly probes an ultrafast photoisomerization dynamics where the main reaction coordinates are a torsional angle and a second not-further-specified vibrational coordinate. Those two modes were clearly observed by wave-packet dynamics on the excited state. After TTC excitation, most molecules relax back to the ground state. However, a small portion undergoes a photoisomerization cis-trans on the excited state. 2DES can isolate molecular reactive modes involved in an ultrafast photochemical process.

7 Conclusions and Perspectives

As we have shown in this chapter, 2DES allows to study i) heterogeneity in the ground state, ii) electronic coupling between chromophores during excitation energy transfer, iii) coherent oscillations in photoinduced dynamics, iv) intermediate or dark states in a photoreactivity cycle.

Furthermore, lots of experimental efforts have been recently done in pushing the limit of  2DES, expanding the ways one can excite and detect the non linear emitted signals (Fig. 19). For example, the third-order signal can be read via fluorescence [85–87] or photocurrent [65, 88] detection schemes. 2DES technique has been also implemented  in the ultraviolet spectral range [89] or  adding external perturbations such as an electric field (2DES Stark [90]). Moreover, some attempts have been made to study the coherent effects observed in 2DES experiment using incoherent light as excitation source [91,92,93,94].

Fig. 19
figure 19

Developments of 2DES either in the excitation or in the detection sides. Each technique is based on a more complex setup and bring more insight into the studied dynamics

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7.1 2DES-UV

2DES was successfully implemented in the deep-UV region in the collinear [95] and non-collinear [89] geometry. Those developments were not possible until recently, because the control of the dispersion of UV pulses and the stability of their phase (stability of 2–3 nm) are more challenging in UV.

Various biological samples absorbed in UV, especially DNA or proteins. For instance, it is well known that the photodeactivation of DNA nucleobases is faster than the one of DNA polynucleotides [96, 97]. However, the nature of this fast photodeactivation remains controversial: the most accepted explanation is an ultrafast deactivation through a conical intersection between the excited state and ground state potential energy surfaces [98]. 2DES-UV experiments clearly show that the deactivation pathway is universal for all the nucleobases and is a two-step mechanism: the initial bright state ππ* is depopulated to a dark state nπ* and then to the ground state through two conical intersections [89].

Biological applications will benefit from this new development. Indeed, all the advantages of 2DES can be retrieved in UV region: heterogeneity, excitation energy transfer, coherent oscillation, and photoreactivity.

7.2 Incoherent 2DES

The large majority of 2DES experiments involved the use of coherent femtosecond pulses. Recently, some attempts have been made to perform 2DES using quasi-incoherent light—I(4) 2DES, where I(4) refers to four incoherent excitation fields. Incoherent excitation light combined with time-domain interferometry can produce a strong four-wave mixing signal in the phase-matched direction and this signal can be resolved in phase and amplitude using a local oscillator field. The light used in this series of experiments is referred to as noisy light, and can be continuous light, but often is constituted by pulses on the order of hundreds of nanoseconds, still “continuous” with respect to the femto-to-picosecond dynamics. Moreover, the coherent oscillations can be measured and then Fourier transformed to produce 2DES maps. The 2DES maps measured with this technique have similar information as conventional 2DES maps [91,92,93,94].

7.3 2DES Stark

Recently, a new method combining 2DES and Stark spectroscopy has been proposed and implemented successfully [90]. This method has the main advantages of distinguishing kinetic processes as energy and charge transfer, which is difficult in complex systems due to lack of clear spectral signatures of charge transfer states. Stark spectroscopy enables the identification of charge transfer states, their coupling to other charge transfer and exciton states, and their involvement in charge separation processes. Conventional Stark spectroscopy will benefit from the implementation of Stark-2DES for the following reasons:

  1. 1.

    the correlation excitation/detection map obtained by 2DES should be able to push the information about the change in dipole moment and polarizability of transitions available from conventional Stark spectroscopy. 2DES should thus reveal coupling of charge transfer states to other electronic transitions of the system.

  2. 2.

    the additional waiting time dimension of 2DES may enable identification of charge separation processes and the states that drive them.

7.4 Fluorescence 2DES

Phase-modulation two-dimensional fluorescence spectroscopy was developed by detecting the fluorescence out of a 2DES experiment [99]. Four phase-modulated collinear pulses have controlled delays and the signal is retrieve by phase cycling procedure where the phase-modulation fluorescence 2DES is based on a phase selective detection scheme of the nonlinear signals.

This technique have been used for the characterization of the excitonic coupling of dimer probes, and to determine dimer conformation. For instance, it was applied to an electronically coupled porphyrin dimer in a biological membrane [85, 100] or to probe the conformation-dependent electronic coupling of a dinucleotide of 2-aminopurine [101].

Fluorescence 2DES is a powerful technique to disentangle the dynamics of excited states and the electronic couplings involved in dimers.

7.5 2DES Photo-Current

An interesting approach to conventional optical heterodyne-detection techniques employs the detection of the 2DES through photocurrent [65, 88, 102]. A sequence of four collinear pulses creates the fourth-order population, which is detected by utilizing acousto-optic phase modulation of the first two excitation pulses in combination with phase-synchronous detection using a lock-in amplifier. This approach offers several advantages compared to conventional 2DES methods:

  1. 1.

    simultaneous detection of the rephasing, non-rephasing, and two-quantum signals,

  2. 2.

    possibility to obtain diffraction-limited spatial resolution due to fully collinear geometry.

2DES photo-current is particularly useful for the study of photovoltaic materials [103] and other photo-devices, since it provides direct access to ultrafast dynamics of the device under typical operating conditions.