Probing nonadiabatic dynamics with attosecond pulse trains and soft x-ray Raman spectroscopy.

Linear off-resonant x-ray Raman techniques are capable of detecting the ultrafast electronic coherences generated when a photoexcited wave packet passes through a conical intersection. A hybrid femtosecond or attosecond probe pulse is employed to excite the system and stimulate the emission of the signal photon, where both fields are components of a hybrid pulse scheme. In this paper, we investigate how attosecond pulse trains, as provided by high-harmonic generation processes, perform as probe pulses in the framework of this spectroscopic technique, instead of single Gaussian pulses. We explore different combination schemes for the probe pulse as well as the impact of parameters of the pulse trains on the signals. Furthermore, we show how Raman selection rules and symmetry consideration affect the spectroscopic signal, and we discuss the importance of vibrational contributions to the overall signal. We use two different model systems, representing molecules of different symmetries, and quantum dynamics simulations to study the difference in the spectra. The results suggest that such pulse trains are well suited to capture the key features associated with the electronic coherence.

INTRODUCTION

Conical intersections (CIs) represent fast, radiationless decay channels in electronically excited molecules [see Fig. 1(a)]. Virtually present in every molecular system, CIs play a key role in charge transfer processes, reaction mechanisms, and in the vast majority of photochemical, photophysical, and photobiological reactions such as the cis/trans isomerization of retinal. At such intersections, the Born–Oppenheimer approximation breaks down causing complex dynamics of the coupled vibronic states, which can be observed spectroscopically. As the photoexcited wave packet comes closer to the conical intersection, the energy separation between the potential energy surfaces (PESs) rapidly decreases. Thus, detection requires an unusual combination of temporal and spectral resolution that is not available via conventional femtosecond optical and infrared (IR) experiments. However, pulses in the extreme ultraviolet (XUV) to the short x-ray spectral region possess the required combination to directly detect the passage through a CI. Single attosecond pulses (SAPs) have been extensively used in pump-probe experiments. Although the availability of such pulses has seen a significant increase thanks to high-harmonic generation (HHG) and free electron laser (FEL) sources, the generation of SAPs still requires a complex setup. The HHG process in gases emits a sequence of short bursts of radiation, which are coherently driven by the generation laser, where emission events occur during each laser half cycle. Each of these short bursts is in the attosecond regime, and their interference leads to the observation of odd harmonics. Such attosecond pulse trains (APTs), unlike isolated pulses, are directly available in a HHG setup, which is now a table-top source widely found in many laboratories. Recent theoretical developments have shown the capability of APTs to probe the electronic coherence created by the CI in the context of time-resolved photoelectron spectroscopy and time-resolved electron-momentum imaging.

FIG. 1.

(a) Scheme of a CI: 1D cut of potential energy surfaces along a generic reaction coordinate, q. Following photoexcitation from the ground state (GS), the wave packet reaches the CI from the Frank–Condon (FC) region. (b) Loop diagram of the off-resonant stimulated Raman signal. The gray area represents the preparation of the system into the excited state by means of a pump pulse, temporally well separated from the detection process. After photoexcitation, the system propagates freely for a delay time T before being probed by the hybrid-shaped pulse. (c) Schematics of the pump ( ) and hybrid probe pulse ( ) setup in TRUECARS.

In this paper, we demonstrate theoretically how APTs perform as probes in the framework of the transient redistribution of ultrafast electronic coherences in the attosecond Raman signals (TRUECARS) technique to probe electronic coherences generated by a wave packet passing through a CI. We use two different model systems that represent molecules of different symmetries and use quantum dynamics simulations to study the difference in the spectrum.

SPECTROSCOPIC SIGNALS AND MODELS

The TRUECARS signal

The TRUECARS technique uses an off-resonant stimulated x-ray Raman process [see Figs. 1(b) and 1(c)], which is sensitive to coherences rather than populations. In the x-ray Raman scheme, core-hole states are involved as intermediates rather than common valence excited states. As shown in Fig. 1(b), the two pulses making up the hybrid pulse scheme, namely, and , drive the Raman process, which is in turn detected by a heterodyne detection scheme, where a local oscillator is used. The frequency and time resolved signal reads in atomic units as follows: where T is the time delay between the probe field and the preparation pulse [see Fig. 1(c)], is the time dependent expectation value of the x-ray transition polarizability, and ω is the Raman shift. For the details of the signal, see Ref. 39. The dependence on the dynamics of the system enters the TRUECARS signal through . Additional details on are given in Subsection II C.

The TRUECARS spectrum is characterized by an oscillating pattern of gain and loss features in the Stokes and anti-Stokes regime, which is only visible when a vibrational or electronic coherence is present. Assuming that both components of the probing field have the same carrier frequency, ω, the spectrum is going to be centered at a Raman shift of .

Modeling of the pulse trains

To illustrate how the pulse trains were built, we start from their definition in the time domain where G(t) is a Gaussian envelope of σ width and P(t) is an infinite train of pulses. The electric field of the single pulses inside the train is defined as with ω being the center frequency and being the period of the IR field. The expression for the electric field then reads

The pulse train employed in the calculations was built according to Eq. (5) by substitution of t with t – T and with the following parameters: and fs. For the purpose of simulating TRUECARS spectra, the center frequency ω can assume any arbitrary value.

To ease nomenclature, henceforth, single pulses in the femtosecond or attosecond regime will be broadly referred to as “Gaussian pulses.” A snapshot of the train pulse at eV is displayed in Fig. 2. Snapshots at different frequencies of the IR laser are available in the supplementary material.

FIG. 2.

Snapshot of the modeled APT shape at eV. On the left: envelope of the pulse train in the time domain. On the right: the pulse train in the frequency domain, shifted with respect to a selected central harmonic.

Models and symmetry

We use a group theory to identify Raman active transitions and the vanishing integral rule to predict whether the polarizability matrix elements such as will be zero. In particular, they will vanish if the product of the irreducible representations of the two relevant states and the operator does not contain the totally symmetric representation, that is,

The two systems studied in this work belong to the and to C point groups, respectively. According to their character tables, the polarizability tensor elements α, α, and α all transform as the totally symmetric irreducible representation, i.e., in and in . This was taken into account while modeling the x-ray polarizability operator, , that was here approximated to a 2 × 2 matrix in the electronic subspace

The full electronic polarizability matrix must be taken into account in order to be basis-independent (adiabatic vs diabatic states), and the diagonal matrix elements cannot be neglected when transforming between representations. The diabatic wave function, , is expressed in terms of the electronic states. For a system with two electronic states, the wave function reads Adiabatic and diabatic states are related by an unitary transformation, where the advantage of the diabatic basis is the absence of derivative couplings. Expanding the expectation value, , yields

According to Eq. (6), for a Raman active transition, two conditions need to be fulfilled here: (i) the diagonal and off diagonal polarizability matrix elements must all transform as the totally symmetric irreducible representation of their point group ( in and in ); (ii) the electronic states and must have the same symmetry label, or else the integral will vanish. The diagonal ( ) and off diagonal ( ) terms refer to vibrational and electronic contributions, respectively. Each vibrational normal mode has an associated irreducible representation too. Therefore, the diagonal contribution will be suppressed if the vibrational modes transform as the wrong irreducible representation. In practice, we can ensure the transition will be Raman active if , and α all fall into for and into for .

The concept of point groups is based on the approximation of a rigid molecule, that is, considering the molecule as a rigid skeleton of nuclei. However, when large amplitude motions are considered where the symmetry of the system is not conserved, the concept is inadequate. Photoisomerization processes are well known examples of large amplitude nuclear motion in molecules. To generate symmetry adapted polarizability element functions in the diabatic basis, we now introduce the complete nuclear permutation inversion (CNPI) group. A CNPI group consists of all permutations of identical nuclei and the inversion of all nuclear and electronic coordinates ( ), as well as their products. The inversion differs from the inversion operation i in point groups. The former is an operation, which results in a sign change of all nuclear and electronic coordinates in the space-fixed coordinate system. A similar application to electronic states and transition dipole moments can be found in Ref. 42. Details of the modeling of the two systems as well as the polarizability matrix are shown in Secs. II C 1–2.

Ci symmetry model

This model consists of two harmonic potential wells shifted with respect to each other (see the left panel of Fig. 3). It represents two excited electronic states with a conical intersection. An example of such a molecule could be benzene (C6H6) and its photochemistry or acetylene (C2H2). The diagonal and off diagonal elements of the polarizability have been shaped by a function of q1 and q2, which is symmetric with respect to the two normal modes. In the framework of CNPI, this translates as

FIG. 3.

Potential energy surfaces of the investigated model systems. On the left: 1D cut of the diabatic potential energy surface along in the C system. On the right: 1D cut of the diabatic potential energy surface along in the C system. The wave packet has been previously excited from the ground state to the Franck–Condon (FC) region on by means of a pump pulse.

This function behaves non-linearly with respect to the nuclear coordinates in the proximity of the CI. In this specific case, both normal modes q1 and q2 transform as the totally symmetric irreducible representation A; thus, they are all Raman active. Plots of the symmetry adapted functions of the polarizability matrix elements as well as the single contributions to are available in the supplementary material.

symmetry model

The C model consists of a Morse-like potential well and a repulsive potential with asymptotic behavior at large values of q with a conical intersection. A 1D cut of the system is shown in the right panel of Fig. 3. Hydroxylamine (NH2OH), which has symmetry, is a well-known example for the study of the effects of conical intersections in photodissociation. Similarly to the C symmetry, the diagonal and off diagonal elements of the polarizability have been here shaped by a function of q1 and q2. This time, the function is symmetric with respect to q2 and non-symmetric to q1. A fundamental transition is Raman active if a normal mode forms a basis for one or more components of the polarizability. Here, the normal mode q1 transforms as the irreducible representation , while q2 as . Thus, q2 is Raman active. Further details, including contour plots of the aforementioned functions, are available in the supplementary material.

COMPUTATIONAL DETAILS

The time evolution was simulated by solving the time-dependent (non-relativistic) Schrödinger equation numerically with the Fourier method, where the wave function is represented on an equally spaced grid of sampling points in coordinate space using the in-house software QDng. In the diabatic picture, the two-level Hamiltonian reads as where is the kinetic energy operator and given as while and are the potential energy operators, respectively, for S1 and S2, and is the diabatic coupling operator. The Arnoldi scheme was employed as the propagator for all calculations. The reduced masses, μ, the time steps, , and the number of grid points employed have all been summarized in Table I. A of 4 is equal to 96.75 as while a reduced mass of 18 000 is approximately 10 amu. For the initial state, assumed to be the result of a short-fs excitation pulse, the nuclear wave packet was approximated by a Gaussian envelope on . Finally, a perfectly matched layer was employed to absorb the wave packets at the boundary and to account for the dissociative behavior. The propagated wave packets were used for the evaluation of the expectation value of the polarizability, . Finally, the TRUECARS spectra were calculated with Eq. (1) at different values of the pump-probe delay, T.

TABLE I.

Reduced masses, time steps, and the number of grid points used in the simulations.

	μ (au)	Δt (au)	Grid p.
C_i	18 000	4	256 × 256
C_s	30 000	2	300 × 300

RESULTS

For each system, we have simulated two spectra with the conventional TRUECARS probe scheme, containing purely electronic or vibrational signals, as displayed in Fig. 4. This was achieved by simulating the vibrational contribution ( ) and the electronic contribution ( ) separately. This distinction will help us break down the interesting characteristics of the spectra and highlight the differences and similarities between the probe schemes. It should be emphasized that the distinction of vibrational and electronic degrees of freedom near a conical intersection is fictitious, as these degrees of freedom are mixed.

FIG. 4.

Comparison of electronic and vibrational contributions to the TRUECARS signals for Gaussian pulses. (a) electronic contribution in the C model; (b) vibrational contribution in the C model; (c) electronic contribution in the C model; (d) vibrational contribution in the C model. Spectra in the same row share the same Raman shift axis. The signals have been normalized with respect to the maximum value of (d).

Signals in Fig. 4 have been normalized with respect to the maximum value of the vibrational contribution in the C model [panel (d)] with the following ratios: (a) , (b) 0.78, and (c) 0.33. By comparing the overall intensities between the two models, we notice that the electronic contribution in the C is higher than the C system. The vibrational signals are, in both model systems, contained within the eV range and immediately visible in the spectrum due to vibrational coherences ( ), whereas the electronic signals spread over a broader energy range ( eV) and only appear after the wave packet has reached the CI. Hence, if the energy resolution is not sufficient, the two will overlap with each other and the electronic component may be masked by the stronger vibrational signal. A major difference between the two components can be seen in the temporal oscillations, where the electronic contribution to the signal oscillates much faster than the vibrational one.

In contrast to the conventional hybrid probe, here pulse trains have been employed instead of single Gaussian pulses. The following three combination schemes have been investigated:

an APT as and a Gaussian pulse as ,

a Gaussian pulse as and an APT as , and

two identical APTs as both fields.

In the following, we use σ0 and σ1 to indicate the Gaussian width of and , respectively. Due to the nature of the pulse trains, multiple signals are expected in the spectra simulated with schemes (I) and (III). The extra signals arise from the side peaks of the train and are expected to be symmetric to each other, but they are lower in intensity with respect to the central signal at . In scheme (II), a Gaussian pulse in the femtosecond timescale is employed as , and the spectrum will mostly consist of one central signal. This is due to the limited spectral resolution of the femtosecond narrow band pulse. Spectra similar to Fig. 4 but simulated with probe scheme (III) are given in the supplementary material.

Calculations were carried out for different values of ω at 1.55, 0.99, and 0.83 eV (i.e., 800, 1250, and 1500 nm). The IR laser frequency is an important parameter in these calculations, because it directly shapes the attosecond pulse train via the IR laser period, τ. A smaller τ implies more peaks in the time domain and, equivalently, less peaks in the frequency domain. Similarly, the σ parameter in Eq. (5) can achieve the same effect.

C symmetry model

We begin our discussion of the results starting with the model. The time evolution of the population of the excited states is plotted in the top panel of Fig. 5. Following photoexcitation, the wave packet reaches the CI in 12 fs with an overall population transfer of 45%. The electronic coherence reaches a maximum of 0.15 at 15 fs, after which starts decaying. The simulated APT TRUECARS spectra are shown in Figs. 5(c)–5(e) compared to a standard single pulse TRUECARS (b). The dashed black line in the spectra indicates the expectation value of the energy separation, , between S1 and S2. For more details, see the supplementary material of Ref. 39. Because the spectrum is antisymmetric with respect to , only signals within eV are shown. Extra signals appear above 2.5 eV with schemes (I) and (III); however, those only carry redundant information, as they are lower-intensity replicas of the central peak. Initially, the vibrational coherence is the only contribution, and it is constrained within eV, as shown in Fig. 4. Once the wave packet is in the proximity of the CI, the electronic coherence starts to build up and becomes visible in the eV region of the spectrum. As the energy separation between the states increases again, the oscillating pattern of gain and loss features in the Stokes and anti-Stokes regime can be seen in the spectrum. The oscillation period directly mirrors the coherence period: as grows, the oscillations speed up which causes the positions of the peaks in the Raman shift ω to spread apart. Due to the shape of the potential energy surfaces, the energy separation between and stays approximately constant after the CI. This can be seen from the dashed black lines in Fig. 5 as well as the positions of the peaks in the Raman shift ω.

FIG. 5.

Comparison of pulse schemes for the model system. (a) Time evolution of the population and the coherence of the two excited states in the diabatic basis. The population transfer is around 45% and occurs after about 15 fs of the wave packet propagation. The diabatic coupling is responsible for the slight oscillation in the population between 15 and 20 fs; (b) TRUECARS spectrum generated by an isolated Gaussian hybrid femtosecond/attosecond probe-pulse sequence (pulse parameters and fs). (c) TRUECARS spectrum obtained via combination (I) with Gaussian pulse parameter fs; (d) TRUECARS spectrum obtained via combination (II) with Gaussian pulse parameter fs; (e) TRUECARS spectrum obtained via a combination of two identical APTs [scheme (III)]. Each signal has been normalized with respect to its maximum value. The dashed black line represents the average time-dependent separation of the adiabatic potential energy surfaces. All spectra are simulated for eV. A snapshot of the pulse train can be found in Fig. 2.

Among the three APT probe combinations displayed in Fig. 5, only 5(c) is able to capture the CI signature at eV. By comparing panels 5(b) and 5(c), we note that scheme (I) can achieve similar energy resolution to the Gaussian/Gaussian hybrid probe, while being characterized by the presence of additional signals in the spectrum. However, probe scheme (I) still requires a single pulse in the attosecond timescale.

As the IR laser frequency decreases from 1.55 to 0.83 eV, the pulse train peaks get closer to each other in the spectral domain, and the interesting electronic coherence fingerprint becomes visible in the spectra simulated with scheme (II), as displayed in Fig. 6. This probe scheme does not require a single pulse in the attosecond regime. Moreover, it allows us to achieve better spectral resolution of the signals in the Raman shift than a standard TRUECARS [Fig. 5(b)]. In fact, the vibrational and electronic contributions are now sufficiently separated from each other to allow for an unambiguous assignment. The latter is not captured by the central harmonic of the APT but by its first side peak. This hypothesis was supported by simulations carried out varying the σ parameter, at the same ω of Fig. 6. The width of the Gaussian envelope directly shapes the width of the harmonics in the HHG spectrum. By increasing and decreasing the σ of the pulse train, we noticed a corresponding increase and decrease in the separation between harmonics and, therefore, between the vibrational and electronic contribution in the TRUECARS signal. The oscillation in Fig. 6 appears to be slightly shifted in the Raman shift and does not follow the dashed black line anymore. Nevertheless, the oscillation period is the same and can still be used to obtain information on CIs. According to the calculations, for this specific system, an IR laser frequency of 0.83 eV appears to be the most suited to resolve the time-dependent energy gap between the two PESs, as shown in Fig. 6(a).

FIG. 6.

Simulated TRUECARS spectra for increasing values of the generating IR laser frequency with probe scheme (II) for the C system. (a) ; (b) ; (c) eV. The Gaussian pulse employed as has a width of fs. The dashed black line represents the average time-dependent separation of the adiabatic potential energy surfaces.

We can extract additional insight from the oscillation period by analyzing the Wigner distribution or temporal gating spectrogram of the analytic signal, which display a time-frequency map. The Wigner distribution is defined as where is the so-called analytic signal, whose imaginary part is related to the original signal, S(T), by Hilbert transformation Here, is a temporal slice of the signal in Fig. 6(a) at a selected Raman shift. The Wigner distribution is a quadratic functional of the signal and so it will, in general, show interference between the negative and positive frequency components of the signal. However, when the analytic signal is used in the computation, no negative frequencies are present; hence, no interference will survive in the spectrogram. Figures 7(b) and 7(d) show the modulus for signal traces taken at and 0.27 eV, which are interpreted as electronic and vibrational contributions, respectively. Panels 7(a) and 7(c) capture the different temporal oscillations of the vibrational and electronic components of the TRUECARS signal. The electronic Wigner distribution spans a broader frequency window than the vibrational contribution. Such a window represents the PES splitting in proximity of the CI. More strikingly, the energy splitting of the main electronic feature is time-dependent and starts around 0.25 eV at 10 fs and converges to 1 eV at  fs. This is in good agreement with the indicated splitting (black dashed line) in Figs. 5 and 6.

FIG. 7.

Comparison between Wigner distributions of selected traces of the TRUECARS signal. (a) and (c) Signal traces at and 0.27 eV, respectively. (b) and (d) Normalized Wigner spectrograms of (a) and (c). The frequency information obtained from the spectrogram is extracted from the temporal features of the signal.

With probe scheme (III) [Fig. 5(e)], the characteristic features caused by electronic coherence are concealed in the spectrum. This happens because the small oscillation, traceable to the electronic component of the TRUECARS signal, overlaps with the dominant vibrational contribution in the same region.

The time evolution of the population of the two excited states, and as well as the electronic coherence magnitude is displayed in Fig. 8(a) for the C model system. The wave packet takes about 15 fs to reach the conical intersection, resulting in an overall population transfer of ∼45%. The electronic coherence has a maximum of 0.0015 around 15 fs, after which starts decaying because of the increasing energy splitting between the surfaces.

FIG. 8.

Comparison of pulse schemes for the model system. (a) Time evolution of the populations and coherence for states S1 and S2 in the diabatic basis. The coherence magnitude (in green) has been magnified by a factor of 200 for visual purposes. The population transfer ( ) occurs after about 15 fs when the wave packet reaches the CI. (b) TRUECARS spectrum generated by an isolated Gaussian hybrid femtosecond/attosecond probe-pulse sequence (pulse parameters and fs). (c) TRUECARS spectrum obtained via a combination (I) with Gaussian pulse parameter fs; (d) TRUECARS spectrum obtained via a combination (II) with Gaussian pulse parameter fs; (e) TRUECARS spectrum obtained via combination (III). The dashed black line represents the average time-dependent separation of the adiabatic potential energy surfaces. All spectra are calculated at eV. Each signal has been normalized with respect to its maximum value. A snapshot of the pulse train is displayed in Fig. 2.

Following the photoexcitation on S2, as the original wave packet approaches the CI, the wave packet transferred on S1 will inherit an odd symmetry from the diabatic couplings. This generates a very weak electronic coherence ( order of magnitude), because the integral gets very small. Hence, the vibronic coherence, in which TRUECARS is sensible to via the physical observable , has now a dominant vibrational component that totally conceals the interesting and characteristic features of the CI such as the time-dependent energy splitting. This is why we are not able to observe them with the TRUECARS technique for this model system, not even with the standard single Gaussian pulse probe scheme of Fig. 8(b). Decreasing the IR frequency from to 0.83 eV does not produce any significant change. The reason why this does not occur in the C model is due to the small shift of S2 in q2, breaking the symmetry and making the integral larger.

CONCLUSIONS

In this paper, we tested the suitability of attosecond pulse trains as probes for detecting electronic coherence generated at a conical intersection with the TRUECARS technique. Model systems of different symmetries and two nuclear degrees of freedom were used. The polarizability matrices were modeled to obtain Raman active vibrational modes. The inclusion of the diagonal polarizability matrix elements includes vibrational coherences, which are inevitably a part of the signal. The full vibronic polarizability matrix must be taken into account in order to be basis-independent, and the diagonal matrix elements cannot be neglected when transforming between adiabatic and diabatic representations. Although this leads to a more complex signal, we could show that it is still possible to distinguish the fast oscillating electronic feature from the vibrational contribution by analyzing the temporal gating spectrogram.

To gain additional insight into the TRUECARS signals, spectra originated from purely electronic or purely vibrational contributions were simulated by evaluating the off diagonal and diagonal contribution separately in the time-dependent expectation value of the polarizability operator. Due to symmetry and Raman selection rules, the electronic coherence appears to vanish in the C model system and concealed by the much stronger vibrational contributions.

We have explored three different schemes for the probe pulse and discussed their features in comparison to the conventional TRUECARS scheme. We found that, among the schemes reviewed, the combination of a Gaussian pulse as a narrowband pulse and an attosecond pulse train as the broad band pulse proved to be the most suitable for our purposes, offering two main advantages: first, a more clear separation between the electronic and vibrational components of the TRUECARS signal can be achieved by fine-tuning the IR laser frequency. The convolution of harmonics leads to a shift of the peaks in the spectral domain, but the energy separation can still be read off the oscillation period in the time domain. This was corroborated by the analysis of Wigner spectrograms, calculated for selected temporal traces of the signal. Second, single Gaussian pulses in the attosecond timescale are no longer required. Furthermore, this is the logical choice for the hybrid probe used in TRUECARS, since the Gaussian pulse and the APT represent a narrowband and a broad band pulse, respectively. The calculations showed the best results at eV.

The probe scheme with an APT and a Gaussian pulse, employed as a narrowband pulse and a broad band pulse, respectively, also proved to be suitable to capture the CI fingerprints. While the traditional TRUECARS spectrum is composed of one main signal centered at , multiple bands are now visible. Such bands are replicas of the main central signal with scaled intensities and are visible along the harmonic comb where the side peaks of the pulse train appear. In comparison, the two spectra contain similar information about the detected vibronic coherence. Nevertheless, such a combination for the probe still requires a single attosecond pulse acting as a broadband pulse.

The use of two identical pulse trains did not reveal most features related to the electronic coherence in the simulated spectrum. Due to the insufficient resolution, the electronic component overlaps with the dominant vibrational signal, resulting in a concealment of the electronic contribution.