Signatures and control of strong-field dynamics in a complex system

Significance

Using intense lasers to control complex molecules is a long-held dream in science. In this article, we develop a physics concept for measuring and controlling the quantum states of complex molecules by strong laser fields. We show that, in particular, the quantum-mechanical phase of excited molecular states can be manipulated by the intense laser, a key quantity for full (amplitude and phase) control of molecular quantum states. With the help of time- and intensity-resolved absorption spectroscopy experiments, we apply this idea to the dynamics of a large dye molecule in solution. The demonstrated phase-control concept thus represents a major leap toward the ultimate goal of laser chemistry.

Abstract

Controlling chemical reactions by light, i.e., the selective making and breaking of chemical bonds in a desired way with strong-field lasers, is a long-held dream in science. An essential step toward achieving this goal is to understand the interactions of atomic and molecular systems with intense laser light. The main focus of experiments that were performed thus far was on quantum-state population changes. Phase-shaped laser pulses were used to control the population of final states, also, by making use of quantum interference of different pathways. However, the quantum-mechanical phase of these final states, governing the system’s response and thus the subsequent temporal evolution and dynamics of the system, was not systematically analyzed. Here, we demonstrate a generalized phase-control concept for complex systems in the liquid phase. In this scheme, the intensity of a control laser pulse acts as a control knob to manipulate the quantum-mechanical phase evolution of excited states. This control manifests itself in the phase of the molecule’s dipole response accessible via its absorption spectrum. As reported here, the shape of a broad molecular absorption band is significantly modified for laser pulse intensities ranging from the weak perturbative to the strong-field regime. This generalized phase-control concept provides a powerful tool to interpret and understand the strong-field dynamics and control of large molecules in external pulsed laser fields.

Can we find universal concepts to understand and control the response of atoms and molecules in interactions with strong laser fields? This question is at the heart of a vast number of experiments in time-resolved spectroscopy (1–30). The wide range of light sources spanning the spectral range from the X-ray (e.g., free-electron laser sources, synchrotrons) over the visible (conventional laser systems) to the far-infrared regime and covering the temporal range from nanosecond down to attosecond time scales created a wealth of new physics insight into quantum mechanisms, however mostly of simple systems in the gas phase (1–4). In chemistry, the generation of femtosecond laser pulses enabled the investigation of wave packet dynamics in molecules, as the induced vibrations occur on these time scales. Experiments focusing on, for instance, dissociation reactions, atom transfer, isomerization, or solvation dynamics have led to a deeper understanding of chemical bonds and their breakage dynamics and have opened and established the field of femtochemistry (5, 6). The aim is not only to study the light−matter interaction, but to use the obtained understanding of the processes to control the dynamics in complex molecules and, in the future, even to be able to control chemical reactions (7–10). Shaping the amplitude and phase of femtosecond laser pulses has been used, for example, to control the shape of wavefunctions in atomic systems (11) or to control and optimize the single-photon and multiphoton fluorescence in atoms such as cesium (12) and complex systems, e.g., dye molecules (13). Shaped pulses are also used in time-resolved coherent anti-Stokes Raman scattering (14–16) or 2D spectroscopy (17, 18). Adaptive shaping of the pulses via feedback control even allows the optimization of dynamical processes, e.g., the relative photodissociation yield of organometallic molecules (19), the relative two-photon fluorescence yield of dye molecules (20), and the energy transfer in light-harvesting molecules (21).

However, the strong-field dynamics in complex systems, e.g., in the liquid phase, and its control have only recently moved into scientific focus (22–27). The dynamics of complex systems was thus far studied mainly in perturbative experiments such as transient absorption spectroscopy, which measures the evolution of a system after absorbing a single or a few photons. These experiments mainly measured population dynamics of excited states, without gaining access to the phases of the excited wavefunction coefficients. Even the phase-sensitive method of 2D/3D spectroscopy (28, 29), which evolved out of transient absorption spectroscopy, probes the perturbative third- or fifth-order response of the system and has not yet been used to systematically understand the strong-field response of a complex system. However, an important ingredient in approaching the ultimate goal of controlling chemistry is to get access to the phase of quantum-state coefficients and to analyze systematically the phase of the system’s response. In recent work, the phase of the dipole response after excitation was measured and controlled in a simple system, namely gaseous helium (31, 32). Transient absorption experiments were performed using extreme-UV attosecond pulses and 7-fs short visible to near-infrared (VIS/NIR) pulses, and the absorption was measured as a function of the time delay. The intensity of the femtosecond pulse could be varied in addition to the time delay. Thereby, the dipole response was systematically investigated as a function of the laser pulse intensity, ranging from the weak perturbative to the strong-field regime. Modifications of the absorption line shapes of helium from Fano to Lorentzian profiles and vice versa were observed with increasing VIS/NIR pulse intensity. These changes can be explained by an induced phase shift of the dipole response that is caused by the femtosecond pulse. Thus, the laser pulse intensity can be used as a control knob to modify the system’s response in a desired manner. In this work, we present the generalization of an atomic strong-field phase-control concept to complex systems in the liquid phase.

General Key Idea

We developed a toy model to explain the observed line shape modifications and to implement state-dependent interactions with a strong laser field.

In the following description, we consider the case of one single transition. A first weak (probe) laser pulse with its oscillating electric field induces a time-dependent dipole response in the system that is connected to the absorption cross section σ via its imaginary part

$$\sigma \left(\omega \right)\propto \mathrm{\Im }\left\{d\left(\omega \right)\right\}\propto \mathrm{\Im }\left\{\mathrm{ℱ}\left[\tilde{d}\left(t\right)\right]\right\},$$ [1]

where the dipole $d\left(\omega \right)$ is the Fourier transform of the temporal dipole response $\tilde{d}\left(t\right)$. The presence of a subsequent strong-field (pump) laser pulse causes a time-dependent energy $E\left(t\right)$ of the excited states. In the following, the pump pulse will be called control pulse, as we consider its action to be more general than just a single- or few-photon ”pump” or excitation step. Starting from the time-dependent Schrödinger equation

$$i\hslash \frac{\partial \psi \left(t\right)}{\partial t}=E\left(t\right)\psi \left(t\right)$$ [2]

and using the ansatz

$$\psi \left(t\right)\propto \exp \left[-i\left(\frac{E_0}{\hslash }t-\mathrm{\Delta }\phi \left(t\right)\right)\right]$$ [3]

for the wavefunction (with the unperturbed energy $E_0$), it can be easily shown that the phase shift $\mathrm{\Delta }\phi$ can be described by

$$\mathrm{\Delta }\phi \left(t\right)\propto \frac{1}{\hslash }{\displaystyle \underset{0}{\overset{t}{\int }}dt\prime \mathrm{\Delta }E(t\prime )}.$$ [4]

The energy shift $\mathrm{\Delta }E\left(t\right)$ can be induced by a short intense laser pulse that follows a short excitation pulse and leads to a transient energy shift of the excited states, e.g., due to the dynamic Stark effect. The dipole response can generally be expressed by

$$\tilde{d}\left(t\right)\propto \exp \left[-\frac{\mathrm{\Gamma }}{2}t-i\left({\omega }_{0}t-\phi \right)\right]$$ [5]

with the decay constant $\mathrm{\Gamma }$, the angular frequency ${\omega }_0$ of the resonance, and an arbitrary phase φ, which is 0 for a Lorentzian spectral line shape. Taking Eq. 1 into account, the following result is obtained for the absorption cross section:

$$\sigma \left(\omega \right)\propto \mathrm{\Im }\left\{-\frac{\mathit{ϵ}}{1+{\mathit{ϵ}}^2}{e}^{i\phi }+i\frac{1}{1+{\mathit{ϵ}}^2}{e}^{i\phi }\right\}$$ [6]

with $\mathit{ϵ}=\left(\omega -{\omega }_0\right)/\left(\mathrm{\Gamma }/2\right)$. It can be clearly seen that a variation of φ changes the imaginary part (and at the same time the real part, i.e., the dispersion) of the dipole response. Thus, shifting the phase of the temporal dipole response leads to a modification of the spectral absorption profile.

A single, isolated transition, decaying exponentially in time, corresponds to a Lorentzian absorption line profile (Fig. 1A, black lines). If the phase of this dipole oscillation is shifted immediately after excitation, the absorption profile changes, for instance, for a phase shift of $\pi /2$, an asymmetric Fano profile is obtained (Fig. 1A, red lines). This was demonstrated recently for the helium atom (31). In complex molecular systems in the condensed phase, however, additional vibrational (in the gas phase also rotational) degrees of freedom superimpose the electronic transition, yielding a set of so-called (ro-)vibronic transitions (compare Fig. 1C). This set of energetically dense transitions ultimately determines the shape of the (UV/VIS) absorption spectrum. In addition, due to intramolecular (and intermolecular, in solution) interactions, the excited states are typically short-lived, causing the dense absorption lines to overlap spectrally, forming broad absorption bands. Bearing in mind the results obtained for single atoms, the question arising here is: How will strong pulsed laser fields affect the band shape of broad, molecular absorption bands composed of a set of vibronic transitions?

Fig. 1.

Absorption line profiles and corresponding dipole response functions. (A) The absorption profile of a single, exponentially decaying excitation corresponds to a Lorentzian line (black lines). If the phase of the temporal dipole response is shifted, the absorption line is modified to an asymmetric, Fano-like profile, which is indicated here for a phase shift of $\pi /2$ as example. This phase shift can be caused after excitation of the system by a laser-induced energy shift of the excited states. (B) Complex systems (e.g., molecules) typically exhibit broad absorption bands. The case of an absorption maximum (upper part, black line) consisting of four single transitions is depicted to illustrate the mechanism. In the temporal domain, this case corresponds to four dipole responses that coherently add up to an overall decaying dipole response. A phase shift of one of the dipole oscillations by $\pi /2$ modifies the overall dipole response (lower part, red line) significantly. In the spectrum, a minimum occurs due to this phase shift. (C) Schablonski diagram representing the vibronic transitions between the ground state G and excited states E. In the presence of an additional excited state E′, one of the vibrational levels E might couple to E′, which could be more sensitive to external light fields. Thus, as a consequence, the vibronic state E would strongly shift in energy ($\mathrm{\Delta }E$) under the influence of a laser field, causing a $\mathrm{\Delta }\phi ={\displaystyle \int \mathrm{\Delta }E\left(t\right)\hspace{0.17em}dt}$ phase shift after integration over the laser pulse.

To illustrate absorption-profile modifications in complex molecules, we consider the following scenario (compare Fig. 1B): Let a broad absorption maximum consist of four individual Lorentzian resonances. In the temporal domain, this situation corresponds to four exponentially decaying dipole oscillations that coherently add up to an overall quickly decaying dipole response. Now, if the phase of only one transition’s dipole response is shifted by $\pi /2$, for example, the overall dipole oscillation changes significantly, and a clear minimum is visible in the spectral domain. As different excited states have different coupling strengths (e.g., different Stark shifts) to an external electric field, for instance due to coupling to additional excited states enabling a dynamic polarization (compare Fig. 1C), this model assumption of a selective coupling of one or just a few states out of many is realistic. It has to be pointed out again that we use strong laser fields on purpose to strongly modify the quantum states.

In the following, we present transient absorption measurements in a dye molecule to provide clear evidence for this generalized phase-control formalism in complex systems.

Experiment

We performed transient absorption measurements with 7-fs short laser pulses in the VIS/NIR spectral range acting as both control (i.e., pump) and probe pulse. Further information about the characterization of the pulses is provided in Characterization of Laser Pulses. In our experiment, we used a 0.125-mM solution of the dye IR144 in methanol as the sample. The absorption was measured as a function of the time delay τ between control and probe pulse. As a second parameter, we varied the fluence of the control pulse and repeated the time-delay scans for different fluence settings. In our measurements, we specifically focused on negative time delays, that is, the probe pulse precedes the control pulse, which is also referred to in the literature as perturbed polarization decay. It also corresponds to the above-described situation of interest, where a strong laser pulse can have different coupling strengths for individual excited states and can transiently modify their energies.

The experimental setup is displayed in Fig. 2A. The control and probe pulses are cut out from the incident 7-fs laser pulse by a spatial mask. The pulses are reflected off a split mirror, which is used to introduce the time delay τ between the two pulses. Then, the pulses are focused into the sample, which is a cuvette filled with the IR144 solution. To mitigate spatial-volume averaging in the sample, the probe pulse is focused to a smaller spot size than the control pulse. The dye molecule has a broad absorption maximum at a center wavelength of about 750 nm. Its structure is depicted in Fig. 2B. The energy of the probe pulse is kept constant at a low level of 21 nJ, which corresponds to a fluence of about $\left(0.5\pm 0.1\right)\cdot {10}^{-4}$ J/cm². The spectrum of the transmitted probe pulse is recorded as a function of the time delay τ and also for varying control-pulse energies or fluences.

Fig. 2.

Ingredients of the transient absorption spectroscopy measurements. (A) Experimental setup. The control and probe pulses are cut out of the incident 7-fs VIS/NIR laser pulses by a mask. The neutral and variable density filters are used to adjust the intensities of the pulses. The time delay τ is introduced by the split mirror. The two pulses are focused into the sample, which is a solution of the dye IR144 in methanol filled into a standard cuvette (SpecVette CSV500 from Ocean Optics, path length 500 μm). The transmitted spectrum of the probe pulse is detected by a spectrometer as a function of the time delay and the control-pulse energy. The online monitoring allows checking of the temporal and spatial overlap of the two pulses during the course of the measurement. (B) Molecular structure of the dye IR144. The dye exhibits a broad absorption maximum centered around a wavelength of 750 nm.

Fig. 3 shows time-delay scans for two different control-pulse fluences. It can be clearly seen that for time delays close to 0 fs, the absorption spectrum is strongly modified, even for low fluences. These modifications become more prominent, and, most importantly, change the overall spectral shape, for higher fluences. Note that OD (optical density) instead of $\mathrm{\Delta }$OD is shown, which means that the complete absorption spectrum, in particular on the negative time-delay side, is massively reshaped by the interaction with a strong laser field. Since we focus on such strong modifications in the absorption spectra, OD is chosen to represent the absolute absorption spectrum instead of $\mathrm{\Delta }$OD, which is typically used to represent small changes in perturbative (e.g., single-photon pump, single-photon probe) transient absorption spectroscopy at lower intensities of pump and probe. At positive time delays, the signal recovers on a timescale of roughly 50 fs, which can be attributed to the lifetime of the excitation. This is in agreement with a number of experiments in which the dye IR144 was already studied, e.g., three-pulse photon echo spectroscopy, transient grating, and transient absorption measurements (33, 34). However, these measurements, like most of the transient absorption measurements in complex systems, focused on positive time delays, i.e., probing the dynamics after the system was excited by a perturbative (single-photon) control pulse, neglecting the perturbed polarization decay at negative time delays. The perturbed polarization decay (35) is typically regarded as a nuisance that is removed in data analysis (36). In contrast, we focus on this negative time-delay axis in the following.

Fig. 3.

Time-delay scans for a control-pulse fluence of about $4.4\cdot {10}^{-4}$ J/cm² (A) and $18.2\cdot {10}^{-4}$ J/cm² (B). The positive time-delay range corresponds to the typical pump–probe scenario. Negative time delays correspond to the perturbed polarization decay, where the system is first excited by the weak pulse (i.e., probe) and, afterward, the control pulse (i.e., pump) perturbs the absorption process. Significant modifications of the absorption spectra are clearly visible in the region around a time delay of 0 fs. For higher fluences, these modifications become even stronger.

To observe the strong-field response of the molecular system, we now systematically analyze the perturbed polariation decay as a function of laser fluence in Fig. 4 for fixed time delays τ. The negative time delays correspond to the case where the molecules are first excited by the weak probe pulse and then exposed to the influence of the stronger control pulse. Significant modifications of the absorption spectra occur in a range of time delays of −20 fs to 0 fs. The maximum of the absorption band shifts to higher frequencies. For smaller time delays, this shift becomes stronger, and a minimum occurs around 2.5 fs⁻¹. To rule out the possibility that the observed modifications are caused by the solvent, measurements in pure methanol are performed. The transient absorption measurements performed in the pure solvent and their results are presented in Measurements in Pure Methanol.

Fig. 4.

Measured absorption spectra showing optical density (OD) as a function of the control-pulse fluence plotted for fixed time delays $\tau =$ −5 fs, −10 fs, −20 fs, and −30 fs. At low time delays, strong features can be seen. First, the maximum shifts to higher frequencies. Secondly, at an angular frequency of about 2.5 fs⁻¹, a minimum occurs with increasing fluence. These modifications remain visible even for larger time delays. The control pulse, i.e., the pump, that follows the excitation of the dye molecules leads to a significant perturbation of the absorption process with increasing intensity.

Numerical Model and Discussion

The question is whether the experimentally observed modifications of the absorption spectra as a function of increasing intensity of the control pulse can be explained by the generalized phase-control mechanism as described above. To answer this question, we modify the numerical toy model to implement state-dependent interactions with a strong laser field. As explained above, for the case of a molecule, there is not only one single transition to be taken into account as in an atomic gas (31) but many possible transitions, which are overlapping in the spectrum. Thus, a single-excited-state dipole response has to be replaced by a sum over many dipole responses ${\displaystyle {\sum }_k{\tilde{d}}_k\left(t\right)}$, one for each excited state. In contrast to the atomic case, where the laser-induced phase shift was considered instantaneous compared with the lifetime of the excited state, in the molecular system, it cannot be safely assumed that the lifetime of the states is much longer than the exciting laser pulse. Corresponding to Eq. 4, the induced phase shift is given by

$$\mathrm{\Delta }\phi \left(t\right)\propto \frac{1}{\hslash }{\displaystyle \underset{0}{\overset{t}{\int }}dt\prime }\mathrm{\Delta }E(t\prime )={\displaystyle \underset{0}{\overset{t}{\int }}dt\prime }c{I}_{\text{control}}(t\prime ).$$ [7]

Here, we consider the energy shift $\mathrm{\Delta }E\left(t\right)$ induced by the dynamical Stark effect to be directly proportional to the control-pulse intensity $I_{\text{control}}\left(t\right)$, and the constant c represents a generalized coupling strength (e.g., proportional to the state’s polarizability α). The phase shift thus increases linearly with laser pulse fluence (${\displaystyle \int I\left(t\right)dt}$). It is explicitly time-dependent and accumulates during the entire presence of the control pulse and while the dipole oscillation decays.

To mimic the experiment, we estimated the molecular transitions based on a particle-in-a-box model, as we could not find more quantitative information about the exact energy structure of the dye molecule IR144 in the literature. We model the absorption spectra by 22 equidistant transitions ranging from 2.34 fs⁻¹ to 2.76 fs⁻¹, spaced by 0.02 fs⁻¹ and equal in transition strength. Modeling the absorption spectra by equidistant energy transitions corresponds to the harmonic approximation that is usually used to describe molecular vibrations. The assumed spacing of 0.02 fs⁻¹ matches a typical large-scale vibration of π-conjugated systems (i.e., about 100 cm⁻¹). The chosen spacing of the vibronic transitions was derived from calculations of the normal-mode vibrations in the electronic ground and first excited state and also agrees with previous measurements of fluorescence and absorption spectra (37). A common lifetime is assumed, which is expressed by a decay constant of the dipole oscillation of 0.1 fs⁻¹. This configuration of transitions approximately reproduces the measured experimental spectrum. The laser pulses are described by a Gaussian spectrum with an angular center frequency of 2.6 fs⁻¹. In analogy to the measurements, we model the absorption spectra as a function of the control-pulse intensity for four different time delays as displayed in Fig. 5. It turns out that only four laser-coupled (phase-shifted) transitions, namely at frequencies 2.34 fs⁻¹, 2.36 fs⁻¹, 2.46 fs⁻¹, and 2.48 fs⁻¹, are enough to reproduce the experimental observations. To find the best agreement, for each of these four transitions, we used a coupling constant c of $2\times {10}^3$ rad⋅cm²⋅J⁻¹, corresponding to a polarizability of about 3,400 a.u., which agrees in order of magnitude with an isotropic polarizability on the order of 1,100 a.u. that was obtained from calculations. With these assumptions, the toy model mimics the observed features of the experiment, namely, the shift of the absorption maximum to larger frequencies and the occurring minimum. A more detailed analysis of the quality of the chosen parameters is provided in Choice of Parameters and Their Impact on the Numerical Model. The model based on the generalized phase-control formalism is thus capable of qualitatively reproducing the experimental observations. The energy shifts and phase shifts in our model can, in reality, be caused by a number of coupling mechanisms that exist in molecules, e.g., Stark effect, strong coupling of near-resonant electronic states, or Raman coupling of vibrational states (38–45). Typically, the higher the pulse energy, the stronger are the energy shifts and, hence, the larger are the phase shifts of the dipole responses (see Eq. 4). This model assumption thus suffices to explain the strong intensity and time-dependent spectral reshaping effects observed in the strong-field-controlled absorption spectra. The influence of possible effects like cross-phase modulation and ionization of the medium can be excluded. The details of the corresponding analysis are presented in Further Possible Mechanisms: Cross-Phase Modulation and Ionization. Therefore, we have provided evidence that the concept of controlling the phase of quantum states can be generalized to large molecules in the liquid phase. The question of why only specific resonances experience the phase shift cannot be answered at this point due to the lack of detailed information about the dye molecule. With more such knowledge becoming available, e.g., from quantum-chemical calculations, the general model developed here should help to test our understanding of molecules in strong fields, and thus pave a route to comprehensive control of molecular excited states by intense laser fields (20, 46–49).

Fig. 5.

Simulation of the absorption spectra as a function of the control-pulse fluence based on the developed phase-control concept. The absorption band is modeled by 22 transitions, equally spaced by 0.02 fs⁻¹ and a common decay width of 0.1 fs⁻¹. To mimic the measured spectra, only four transitions, namely at 2.34 fs⁻¹, 2.36 fs⁻¹, 2.46 fs⁻¹, and 2.48 fs⁻¹, have to be taken into account to couple to the control laser pulse, i.e., only these dipole responses experience the laser-induced phase shift. With these assumptions keeping the model as simple as possible, the measured observations can be reconstructed qualitatively extremely well. The shift of the absorption maximum and the arising minimum at about 2.5 fs⁻¹ are reproduced by the described toy model.

The presented strong-field control mechanism goes beyond previous weak-field experimental approaches used in the liquid phase, exerted, e.g., by controlling the initial excitation step (20, 21, 30). While the previous schemes used shaped pulses (20, 21) or a pulse sequence (30, 50) to control the population and vibrational coherence in excited states, here, strong fields are explicitly applied to transiently shift energy levels and thereby the phase evolution of the excited states after their excitation. In the future, such strong-field control may allow the time-dependent control of the entire potential-energy landscape, even for complex molecules in solution, directing molecular wave packets through light-induced transition states to desired final product states.

Conclusions and Perspectives

By performing transient absorption experiments of the dye molecule IR144 in the liquid phase, we have demonstrated that the phase-control model developed for isolated helium atoms (31) can be generalized to more complex systems. This enables measurements of the intensity-dependent strong-field response of systems ranging from simple atoms in the gas phase all the way to condensed-phase systems, where decay and dephasing times of excited states can be on the same order as the excitation and control pulse durations. The presented model is completely general, and its applicability certainly goes beyond the herein used dye molecule IR144, which was only used as a representative sample system. It can be adapted to any complex system that provides many resonances that spectrally overlap and form broad absorption bands. In case the energy structure of the system is known in detail, quantitative studies come within reach.

In addition, having a closer look at Fig. 1B reveals that, in the temporal domain, the overall dipole response lasts for a longer time if a single transition is perturbed and phase-shifted. The amplitude of the perturbed total dipole oscillation (red curve) is larger at later times than in the unperturbed case (black curve). This means that by a simple phase control of a subset of transitions, as is described here, the overall coherence time of the dipole response is extended. Manipulating such temporally extended dipole responses by strong-field control of specific quantum states may thus allow completely new routes toward coherent control of larger molecules such as proteins or enzymes in their natural aqueous environments.

Materials and Methods

The optical density of the absorption spectra shown in Figs. 3−5 is determined by $\text{OD}=-\log \left(S_p/S_0\right)$, where $S_p$ is the measured probe pulse spectrum transmitted through the sample and $S_0$ is the reference laser spectrum, i.e., the unperturbed probe pulse spectrum without pump pulse and without sample. The concentration of the solution was chosen such that a transmission in the range of 7–10% was obtained, corresponding to an optical density of about 1–1.2.

Characterization of Laser Pulses

For the experiments, the duration of the laser pulses was adjusted to be minimal by optimizing the plasma generation in air with a pair of fused silica wedges. Thus, the pulses can be regarded as close to bandwidth limited. The measured laser spectrum is presented in Fig. S1A and corresponds to a bandwidth-limited pulse duration of about 7 fs. To cross-check the pulse duration, the time-delay scan for the lowest control-pulse fluence of ∼0.33 mJ/cm² is used. Such low pulse fluences act only perturbatively on the sample, and, hence, the region of the pulse overlap directly reveals information about the pulse duration. The projection of the scan onto the time axis for a small range around the pulse overlap is shown in Fig. S1B. The change in the amplitude can be fitted with an error function. An FWHM (intensity) pulse duration of about 6 fs ± 1 fs can be derived from the fit.

Fig. S1.

Characterization of the laser pulses. (A) Measured spectrum of the laser pulses used in the transient absorption measurements. The bandwidth limit corresponds to a pulse duration of about 7 fs. (B) The time delay scan for the lowest control-pulse fluence of ∼0.33 mJ/cm² is projected onto the time-delay axis. The region of temporal overlap of the two laser pulses is fitted by an error function that directly includes the pulse duration. The fit gives a pulse duration of 6 fs ± 1 fs (FWHM).

Measurements in Pure Methanol

The sample used in the transient absorption measurements is a solution of the dye IR144 in methanol as explained in Experiment. As the solvent itself might have some impact on the measured absorption spectra, it is necessary to investigate the possible influences. For this purpose, we performed transient absorption measurements in pure methanol. The time-delay scan for a control-pulse fluence of about 4.4 mJ/cm² is shown in Fig. S2A. For comparison, the time-delay scan in the dye IR144 for the maximum control-pulse fluence used in the experiments of about 2.8 mJ/cm² is presented in Fig. S2B. Although the chosen fluence is significantly higher than the maximum fluence used for the measurements in the dye, there are no significant effects visible in pure methanol. Hence, other potential influences on the measured absorption spectra caused by the pure solvent can be ruled out.

Fig. S2.

Effects of the pure solvent, i.e., methanol. (A) Transient absorption measurement in methanol for a control-pulse fluence of ∼4.4 mJ/cm². (B) Time-delay scan in a solution of the dye IR144 in methanol for a control-pulse fluence of ∼2.8 mJ/cm². In the case of the dye, the modifications of the spectrum are strongly visible. However, the effects in pure methanol are minor and, thus, negligible.

Further Possible Mechanisms: Cross-Phase Modulation and Ionization

The impact of cross-phase modulation or ionization of the medium is investigated in numerical simulations and presented in the following. In both simulations, the laser pulses are modeled in the same manner as in the toy model, namely by Gaussian envelopes with a temporal duration of 8 fs (FWHM) and an angular center frequency of 2.6 fs⁻¹. In the experiment, the spectrum of the weak (probe) pulse is detected as a function of the control-pulse fluence/intensity. For the case of cross-phase modulation or ionization, the refractive index changes as a function of time, and, thus, the control pulse induces a phase shift that leads to the modifications in the absorption spectra. In the simulations presented here, the absorption spectra are calculated by taking the Fourier transform of the electric field of the probe pulse, E_probe(t). Thereby, the temporal phase of E_probe(t) is modulated by the presence of the control pulse, and the modulation strength depends on the control-pulse intensity.

For the case of cross-phase modulation, i.e., an intensity-dependent refractive index, the temporal phase is proportional to the intensity sum of both laser pulses, $I\left(t\right)=I_{probe}\left(t\right)+I_{control}\left(t\right)$, i.e., we model the phase by $\varphi \left(t\right)=C\cdot I\left(t\right)$, with C as a constant. It has to be noted that self-phase modulation is automatically included in this model. In Fig. S3, the obtained laser spectra are shown for a time delay τ = −10 fs as a function of the relative intensity between control and probe pulse, $I_{control}\left(t\right)/I_{probe}\left(t\right)$. The absorption spectra are calculated for three chosen constants C. The constant C = n₃⋅k⋅l contains the nonlinear refractive index n₃, the vacuum wave vector k, and the medium length l. For that reason, the overall scale of the intensity-dependent self-phase modulation is linearly proportional to n₃ for a given medium thickness and wavelength and thus allows a comparison with the experimentally observed intensity dependence. First of all, the effect of cross- and self-phase modulation leads to a red shift of the absorption maximum. If the phase shift becomes larger (compare Fig. S3 B and C), an oscillation of the absorption maximum occurs. In addition, it has to be pointed out that there are areas of negative OD, indicating emission instead of absorption. In the experiment, only absorption was observed. As the calculated absorption spectra considering the effect of cross-phase modulation do not even qualitatively resemble the measured spectra, we can state that cross-phase modulation does not cause the observed modifications.

Fig. S3.

Influence of cross-phase modulation. Calculated absorption spectra as a function of the relative intensity $I_{control}\left(t\right)/I_{probe}\left(t\right)$ for a time delay τ = −10 fs. The phase is modeled by $\varphi \left(t\right)=C\cdot I\left(t\right)$ with chosen constant C = 0.02 π (A), 0.05 π (B), and 0.1 π (C). In the experiment, the relative intensity ranges from about 6 to 55. Note the areas of negative OD, i.e., emission, which were not observed in the experiment.

In the case of ionization of the medium, the control pulse ionizes the molecules leading to a change of the refractive index n(t) and, hence, inducing a time-dependent phase $\varphi \left(t\right)=n\left(t\right)\cdot k\cdot \mathrm{\Delta }z$, with the wave vector k and the thickness Δz of the sample. Here, small ionization probabilities are assumed because, then, the change of the refractive index Δn(t) can be regarded to be approximately proportional to the density of the neutral atoms N(t). The time-dependent change of the density of neutral atoms can be described by a rate equation

$$\frac{dN\left(t\right)}{dt}=-\gamma \left(t\right)\cdot N\left(t\right)dt,$$ [S1]

yielding

$$N\left(t\right)=N_0\cdot \exp \left[-{\displaystyle \underset{0}{\overset{t}{\int }}\gamma (t\prime )\hspace{0.17em}dt\prime }\right],$$ [S2]

with γ(t) parameterizing the ionization rate. As the refractive index scales with N(t), the induced phase shift is proportional to N(t): $\varphi \left(t\right)=C\cdot N\left(t\right)$. The ionization rate is given by

$$\gamma \left(t\right)\propto I_{control}^n\left(t\right),$$ [S3]

where n is the number of photons required to ionize the molecule. According to Koopmans' Theorem, in which the energy of the highest occupied molecular orbital of a Hartree−Fock calculation (here HF/def2-TZVP) equals the ionization potential, the latter amounts to ∼6.7 eV. The assumed angular center frequency of 2.6 fs⁻¹ corresponds to an energy of about 1.7 eV, thus resulting in a required photon number of n = 4. As no plasma formation was observed in the experiment, the amplitude of γ(t) is adjusted such that only ∼10% of the dye molecules are ionized at maximum control-pulse intensity. The obtained absorption spectrum is presented in Fig. S4A for a time delay τ = −10 fs as a function of the relative intensity between control and probe pulse. For comparison, the absorption spectrum is calculated assuming that a fraction of ∼90% of the molecules is ionized (compare Fig. S4B). The resulting spectra differ significantly from the measured absorption spectra. At both sides of the calculated spectra, emission (corresponding to the areas of negative OD) appears, even for a small fraction of ionization. In case of large ionization rates, the induced emission becomes even stronger. Since no such features were observed in the transient absorption measurements, we can rule out that ionization of the medium has any influence on the experimental results.

Fig. S4.

Influence of the ionization of the medium. Calculated absorption spectra as a function of the relative intensity $I_{control}\left(t\right)/I_{probe}\left(t\right)$ for a time delay τ = −10 fs. The phase is modeled by $\varphi \left(t\right)=C\cdot N\left(t\right)$ with the neutral density N(t) given by Eq. S2. As constant C, the value 128 π was chosen. In addition, an ionization of about (A) 10% and (B) 90% of the dye molecules is assumed, corresponding to an absolute change in phase of about 13 π (A) and 115 π (B). In the experiment, the relative intensity ranges from about 6 to 55. Note the areas of negative OD, i.e., emission, which were not observed in the experiment.

Choice of Parameters and Their Impact on the Numerical Model

To interpret the experimental results, i.e., the observed modifications in the absorption spectra, we developed a toy model based on the phase-control concept as described in General Key Idea. Since there was no quantitative information about the energy structure, i.e., transitions, of the molecule IR144 available in the literature, we mimicked the absorption spectrum by 22 transitions at equidistant frequencies. In total, 66 parameters arise if the frequencies, the lifetimes (i.e., decay rates), and the coupling strengths of each transition to the external laser field are taken into account. A common lifetime and coupling strength are assumed to keep the model as simple as possible. In addition, the parameters are chosen to be physically reasonable to model the relevant features in the measured absorption spectra.

In the following, we will justify our choice of parameters. This is done by varying the parameters in a physically reasonable range and analyzing the root-mean-square (rms) deviation between experiment and calculation to determine the quality of the chosen parameters:

$$rms=\sqrt{\frac{{\sum }_i{\left(f\left({\omega }_i\right)-g\left({\omega }_i\right)\right)}^2}{n}}.$$ [S4]

The measurement is represented by f(ω), and the simulation is represented by g(ω). The number of intervals into which the frequency axis is divided is given by n. For a fixed time delay τ and laser intensity, the sum of the squared differences between measurement f(ω) and simulation g(ω) is calculated in the frequency interval between 2.2 fs⁻¹ and 2.8 fs⁻¹.

The rms deviations for the different control-pulse intensities are averaged for a fixed time delay τ. The rms values of the simulation presented in Fig. 5 are 0.37 arbitrary units (arb.u.) (τ = −5 fs), 0.304 arb.u. (τ = −10 fs), 0.304 arb.u. (τ = −20 fs), and 0.325 arb.u. (τ = −30 fs).

To evaluate the impact of the coupling constant on the numerical results, the coupling strengths are chosen randomly between 0 and 2 × 10³ rad⋅cm²⋅J⁻¹ for each transition line. The numerical simulation is run to generate 600 data sets, for time delays τ = −10 fs, −20 fs, and −30 fs. The obtained rms values of the 600 data sets are displayed in Fig. S5. The rms values of the simulation presented in Fig. 5 (with a chosen coupling constant of 2 × 10³ rad⋅cm²⋅J⁻¹) are smaller than the mean rms deviation of the stochastically chosen coupling parameters. Hence, our guess of the physically reasonable parameters used to model the experimental data can be considered to be well suitable.

Fig. S5.

Randomly chosen coupling strengths and their quality. The calculated rms deviation of 600 data sets for the time delays −10 fs, −20 fs, and −30 fs, generated by randomly chosen coupling constants between 0 and 2 × 10³ rad⋅cm²⋅J⁻¹ for each transition. The data sets consist of 22 transition lines with equally spaced angular frequencies in the range from 2.34 fs⁻¹ to 2.76 fs⁻¹ and a common decay rate of 0.1 fs⁻¹. In comparison, the rms deviations of the simulation shown in Fig. 5 amount to 0.304 arb.u. (τ = −10 fs), 0.304 arb.u. (τ = −20 fs), and 0.325 arb.u. (τ = −30 fs).

In Fig. S6, the mean coupling constants of the six data sets providing the smallest sum of the rms deviations at time delays τ = −10 fs, −20 fs, and −30 fs are displayed. The manually chosen coupling strengths used in our primary simulation (compare Fig. 5) are depicted in comparison. It confirms that our simple guess of parameters points into the right direction. In the frequency ranges 2.34–2.38 fs⁻¹ and 2.46–2.50 fs⁻¹, the data sets with minimum rms values show a clear tendency: The coupling strengths are higher compared with the coupling constants of the neighboring transitions. The results of the frequency region 2.7 fs⁻¹ and above have to be evaluated with caution. In the simulation, transitions up to 2.76 fs⁻¹ are taken into account. A careful look at the measured spectra reveals that they extend a bit farther to about 2.85–2.9 fs⁻¹. Therefore, the deviation between experiment and our primary simulation is larger. In case of the data sets with minimum rms deviation, the randomly chosen coupling strengths in the specified frequency range have values such that the simulation result fits the experimental data better. This finding suggests that the selected frequency range should have been extended in our simulation. However, this does not have any impact on modeling the observed significant modifications in the frequency regime below 2.7 fs⁻¹.

Fig. S6.

Comparison of chosen coupling strengths. The mean coupling constants (black squares) of the six best rms values of Fig. S5 are shown (with SD error bars). The coupling constants used in Fig. 5 are also displayed (red triangles).

A further issue to be considered is: How does the choice of the maximum coupling strength influence the quality of the numerical model? To answer this question, the same stochastic algorithm is used, but the maximum coupling strength is set to a value twice as large, i.e., c_max = 4 × 10³ rad⋅cm²⋅J⁻¹. By comparing the histograms obtained for a maximum coupling strength of 2 × 10³ rad⋅cm²⋅J⁻¹ and of 4 × 10³ rad⋅cm²⋅J⁻¹ (compare Fig. S7), it becomes apparent that the mean rms deviation in the case of c_max = 2 × 10³ rad⋅cm²⋅J⁻¹ is smaller. Therefore, it is more appropriate to choose a maximum coupling strength of 2 × 10³ rad⋅cm²⋅J⁻¹.

Fig. S7.

Comparing the results for different maximum coupling strengths c_max. The histograms with a maximum coupling strength of 2 × 10³ rad⋅cm²⋅J⁻¹ (red) and 4 × 10³ rad⋅cm²⋅J⁻¹ (blue) are displayed. The sum of the rms deviations of the simulation shown in Fig. 5 amounts to 0.933 arb.u. Again, 600 data sets were generated based on the toy model, i.e., 22 equally spaced angular transition frequencies in the range from 2.34 fs⁻¹ to 2.76 fs⁻¹. A common decay rate of 0.1 fs⁻¹ was assigned to each transition frequency, and the coupling was randomly chosen in the range from 0 to 2 × 10³ rad⋅cm²⋅J⁻¹ (red) and from 0 to 4 × 10³ rad⋅cm²⋅J⁻¹ (blue).

For a more detailed investigation, an evolutionary algorithm searching for the best set of parameters is applied. The algorithm minimizes the sum of the rms deviations between experiment and calculation for the fixed time delays τ = −5 fs, −10 fs, −20 fs, and −30 fs. The coupling constants of the 22 transition frequencies can be freely selected in the range from 0 to 2 × 10³ rad⋅cm²⋅J⁻¹. Each transition frequency can also be varied in a range of ±0.02 fs⁻¹ around its center frequency that was used in the simulation presented in Numerical Model and Discussion. The decay rates are fixed to a common value of 0.1 fs⁻¹. The coupling constants and angular transition frequencies obtained within the optimization process of the evolutionary algorithm are displayed in Fig. S8. In the frequency range from 2.3 fs⁻¹ to 2.6 fs⁻¹, the same trend can be observed as in the case of stochastically selected coupling constants. The first, second, seventh, and eighth transition lines exhibit a large coupling strength, whereas a small coupling constant is assigned to the other dipole transitions. For higher frequencies (above 2.6 fs⁻¹), the coupling constants used in the simulation displayed in Fig. 5 and the coupling constants obtained by the evolutionary algorithm deviate. In the case of the evolutionary algorithm, smaller rms values can be achieved in the range of higher frequencies. However, the results for the range of higher frequencies have to be considered with caution due to the same reasons already mentioned above. A 2D representation of the absorption spectra derived from the evolutionary algorithm is provided in Fig. S9. The excellent agreement between the primary simulation (compare Fig. 5) and the results of the evolutionary algorithm justifies our original guess of parameters. Although our toy model is based on very little quantitative knowledge about the dye molecule IR144, the chosen set of parameters turns out to be reasonable and well suitable.

Fig. S8.

Results of the evolutionary algorithm. The coupling constants versus transition frequencies found by the evolutionary algorithm (black rectangles) are compared with the parameters of the simulation depicted in Fig. 5 (red circles). The decay rate was fixed to 0.1 fs⁻¹ for each transition line. The result of the evolutionary algorithm additionally validates the choice of parameters in our primary simulation.

Fig. S9.

Transient absorption spectra as function of control-pulse fluence derived from the evolutionary algorithm. The absorption spectra are calculated using the parameters determined by the evolutionary algorithm (compare Fig. S7). The result is in very good agreement with the primary simulation shown in Fig. 5.