Results and Discussion

The variation of the dissociative ionization yield with time delay is shown in Fig. 2A. The total ionization yield exhibits a similar behavior (Fig. S1). This yield exhibits a rapidly oscillating pattern superimposed on a much slower one. The slow oscillations have a maximum at zero delay, when the two pulses overlap in time, and every 28–30 fs. To understand the origin of these dynamics and extract the relevant phases, we use a model in which the population of the initial state is assumed to be unity at all times and the probability to absorb three or more photons is null (which are excellent approximations for the short wavelengths and relatively low intensities of the XUV pulses). Hence, the ionization probability at time , differential in both the electron energy and the nuclear energy , for a final total energy , can be approximately written aswhere is the complex amplitude corresponding to direct two-photon ionization (Fig. 1A); is the amplitude for the sequential process, i.e., the amplitude describing excitation from the ground to the mth vibronic (vibrational-electronic) state and, after free evolution of the WP, from the latter state to the ionization continuum (Fig. 1C, right arrow); ; and . Note that the formula includes a coherent sum over all m states contributing to the intermediate WP and that the energy differences and involve total energies, i.e., electronic plus nuclear energies, for the final state (), the ground state (), and all m intermediate states () of the molecule. We also note that, in general, m runs over vibronic states associated with various electronic states.

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Fig. 2.

(A) Dissociative ionization yields as a function of time delay. (B) Contributions to the dissociative ionization yields from each of the three different terms defined in Eq. 3. (C) Short-time Fourier transform of the dissociative ionization yield (see text).

To better visualize the phases involved in the process, we decompose Eq. 1 aswhere is thus associated with the direct ionization process and its time-delayed replica [i.e., ], is associated with the sequential process (i.e., ), and is the cross term. The total ionization yield, = , is obtained by integrating over all electronic continuum energies and all vibrational and dissociative energies associated with that electronic continuum. The contribution of the , , and terms to the dissociative ionization yield is shown in Fig. 2B. Similar results are obtained for the dissociative and nondissociative ionization yields. As can be seen, the slow oscillations (28–30 fs period) come from the sequential process ( term), whereas the fast ones (∼0.35 fs period) come from the cross term (). The contribution from the direct process ( term) is nearly independent of time delay.

The observed beatings can be easily understood by assuming that the and amplitudes are almost independent of electron and nuclear energy in the region of interest (which excludes the case of autoionization). Under these assumptions and for not too short time delays, integration of Eqs. 3a, 3b, and 3c leads towhere , , , and are constants. It is therefore clear that the phases , which control the free propagation of the pumped WP, are responsible for the slow beating observed in the ionization yield. The maxima in the yield correspond to quantum revivals of the nuclear component of the wave packet in the inner turning point of electronic state (the only one significantly populated at 12.2 eV), where the Franck–Condon overlap is maximum for absorption of the second photon. Therefore, the slow beating represents the average oscillation period of the nuclear WP created by the pump pulse in the state. The fast beating is controlled by the phases , which involve the energies of the ground and the vibronic states. Therefore, it results from the coherent superposition of the direct and sequential two-photon ionization processes. The above equations also explain why barely depends on time delay: the phase is practically washed out by integration over and .

The relative phases imprinted in the excited states by the pump pulse are more easily extracted when performing a short-time Fourier transform (STFT) of the ionization yield. The resulting energy/time interferogram, obtained with a Gaussian shape window of 2 fs, is shown in Fig. 2C. The interferogram provides a clean picture of the pumped electronic state at around 12 eV revealed as a maximum every 28–30 fs, which is the averaged periodicity of the nuclear component of the WP associated with that electronic state. The structures are tilted due to the anharmonicity of the electronic potential: the most energetic part of the WP (containing the higher m vibrational states, hence more affected by anharmonicity) is logically associated with larger periods. They also stretch with time delay as a result of the WP spreading. Such a clean picture of maxima associated with revivals of the WP can only be obtained by using a time window shorter than the natural time intervals (periods) associated with the motion of the nuclear WP components. If, in addition, one is interested in obtaining energy resolution (y axis), the chosen time window must be considerably larger than the periods associated with the beatings encoded in the WP that one seeks to resolve. For the example shown in Fig. 2C, the chosen time window (2 fs) reveals both the electronic beatings (∼12 eV) and the period of the nuclear component of the molecular wave packet (∼30 fs) because this time window is larger than the periods associated with the beatings (∼300–400 as) and smaller than the period of the nuclear component of the WP in the state.

The above analysis tells us how to decouple the relevant electronic and nuclear motions in more complex situations. As an illustration, we present in Fig. 3A the dissociative ionization probabilities that result from a similar pump–probe scheme but with XUV pulses of 14 eV central energy. Pulse duration (2 fs) and intensities are kept as in the previous example. At this energy, two electronic states are populated: and B′ (see the corresponding potential energy curves in Fig. 1). As a consequence, the variation of the ionization yield with time delay is much more complicated than that shown previously (Fig. 3A). Nevertheless, a window Fourier analysis similar to that presented above unambiguously reveals two nuclear components in the molecular WP, each one associated with a different electronic state. In Fig. 3B one can see features at around 12 eV similar to those already seen in Fig. 2C for 12.2-eV pulses: they reflect the motion of a nuclear WP component in the state. Additional features are observed at around 14 eV. They reflect the motion of another nuclear WP component in the B′ state. The period in the B′ state is shorter than in the B state because at 14 eV, only the lower vibrational states of the B′ state are populated. In addition, the curvature of the B′ state is larger. This result implies that by simply choosing a particular range of time delays, one can selectively catch a particular electronic state. For example, at τ ∼ 20 and 50 fs, the WP is mainly composed of the B′ state, whereas at τ ∼ 35 and 65 fs, the WP is mainly composed of the B state. This control is possible because the periods of the nuclear WP components in different electronic states are different.

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Fig. 3.

(A) Dissociative ionization probability as a function of time delay using 2-fs pulses centered at 14 eV. (B) STFT of the ionization probability in A.

At this point, it is important to emphasize that short pulses, i.e., pulses with rather large bandwidths, are strictly necessary to make a coherent superposition of molecular states, hence to produce a molecular (electronic plus nuclear) wave packet. The requirement is that the bandwidth is larger than the energy separation between the molecular states that are populated. Although the use of short pulses is in principle at the cost of energy resolution, the use of the present pump–probe scheme (the molecular interferometer), in which the time delay between pulses is varied, allows one to retrieve the relative energies (beatings) of all of the states that are efficiently populated by the pump pulse. This can be done, e.g., by just looking at the Fourier transform (FT) of the total dissociative ionization spectrum (see Fig. 4 for the two photon energies considered above). As can be seen, the observed peaks reveal all of the and beatings in the molecular WP. The beatings appear between 0 and 2.5 eV. Because only the B and B′ electronic states are efficiently populated, one can distinguish three kinds of beatings: those involving (i) two vibrational states in the B state, (ii) two vibrational states in the B′ state, and (iii) one vibrational state in the B state and one vibrational state in the B′ state. Beatings i and ii are associated with the nuclear components of the WP in the B and B′ states, respectively. Beating iii is associated with the electronic component of the WP, i.e., to electron dynamics arising from the B–B′ beating. A blowup of the low-energy region of the FT (Fig. S2) shows that the B–B′ beatings appear between 1.5 and 2.5 eV and that for photon energies of 14 eV, these electronic beatings have a larger probability than the nuclear beatings in the B′ state. This result proves that the present pump–probe scheme gives access to both electron and nuclear dynamics.

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Fig. 4.

Fourier transform of the dissociative ionization probabilities shown in Figs. 2A and 3A for 12.2- and 14-eV pulses, respectively. Along the upper x axis we also plot the position of the vibronic intermediate states: the vibrational states associated with the electronic state (in black) and the vibrational states associated with the B′ one (in red).

We focus now on a different observable: the ionization yield differential in both the electron and nuclear kinetic energy (for short, doubly differential ionization yield). This is relatively easy to measure in the case of dissociative ionization by detecting electrons and protons in coincidence. The calculated 2D coincidence spectra for 12.2-eV pulses are presented in Fig. 5 (second row, labeled as “total”). Fig. 5 only shows results for short time delays because, as for the total yield, the main patterns repeat with a periodicity of 28–30 fs. For analysis purposes, the first row shows the position of the nuclear component of the WP in the state when the second photon is absorbed from the probe pulse. As can be seen, for short time delays, ionization can only proceed through the 1sσ_g channel, whereas for larger time delays, ionization through the 2pσ_u channel is also possible. Fig. 5 also shows the contribution of the different terms defined in Eq. 3: , , and .

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Fig. 5.

The first row shows diagrams showing the relevant potential energy curves, an illustration of the direct and sequential ionization processes, and the evolution of the different WPs at different time delays (see Fig. 2 for notations). The second row shows dissociative ionization probabilities as a function of electron energy (x axis) and nuclear kinetic energy (y axis) in eV. Each column corresponds to a given time delay between the pulses. The third through fifth rows show , , and contributions defined in Eq. 3.

We first notice that the doubly differential yields exhibit an increasing number of oscillations as time delay increases. Most of these oscillations come from the term (Fig. 5, third row). Therefore, they are associated with the direct ionization process and, in particular, with the interference between the WP generated by the pump pulse and its exact replica generated by the probe pulse in the dissociative ionization continuum. According to Eq. 3a, the interferences are described by the expression , which is the molecular analog of the so-called Ramsey interferences in atoms (17, 35, 37, 44, 45). The oscillatory pattern is 2D because the energy is shared between the escaping electron and the remaining molecular ion (, where DI = 18.15 eV is the dissociative ionization energy of H₂). As shown above, these oscillations, as well as those related to the phases, are wiped out when the doubly differential yield is integrated over both electron and nuclear energies. These are also difficult to see in the singly differential yields, i.e., when integration is performed over either the electron (Fig. S3) or the nuclear energy, because the corresponding electronic or nuclear phases are averaged. Thus, the Fourier transform of singly differential probabilities vs. time delay does not provide a clear picture of the beating frequencies involving the final state; hence, it is not more informative than total ionization yields (Analysis of Singly Differential Ionization Yields).

In the doubly differential yields (plotted in Fig. 5 and Fig. S4), one can also see the appearance of a peak for time delays larger than 4 fs, a peak that moves toward low proton kinetic energies as time delay increases. This feature comes entirely from the term (Fig. 5, fourth row), and therefore, it is due to the sequential ionization process. After 4 fs, the nuclear WP component in the state has reached the region of internuclear distances ( a.u.) where absorption of the probe photon can lead to ionization above the 2p threshold. Due to the pure dissociative character of the 2p potential energy curve, the contribution from the 2p channel provides a direct mapping of the intermediate nuclear WP component. Indeed, the position of that peak in the 2D spectrum can be easily inferred by projecting this component into the proton energy axis through the 2p potential energy curve [this is equivalent to the well-known reflection approximation for the description of Coulomb explosion in molecules (46)]. This position is indicated by red arrows in Fig. 5, first row. According to this picture, the larger the time delay, the larger the internuclear distance that the intermediate WP can reach, and consequently, the lower the proton kinetic energy at which the 2p appears. When the intermediate nuclear WP component turns back and moves toward smaller internuclear distances, the motion of the 2p peak is reversed. This motion repeats with a periodicity of 28–30 fs, i.e., that of the nuclear WP motion in the state.

All remaining features in the term are due to autoionization from the DESs of the molecule. These are metastable states embedded in the electronic continuum that may decay by emitting one electron as a result of electron correlation. The typical decay time is much longer than the typical time for direct ionization and is comparable to the time needed by the nuclei to move significantly (several fs). Their signature in the 2D coincidence spectra covers a wide range of electron and proton energies (47). Fig. 5, first row, shows the potential energy curves of the Q₁ DESs of symmetry, which, according to the dipole selection rule, are the only ones that can be populated by absorption of two photons. One can see that for very short time delays, fs, only the lowest Q₁ state is accessible. By progressively increasing the time delay up to 4–5 fs, one has access to the whole Q₁ series due to the motion of the nuclear WP component in the state. Beyond 6 fs, the DESs are no longer accessible by the probe pulse, until the nuclear WP component turns back in its periodic motion in the state and reaches again the region of internuclear distances at which transition to the DESs is possible. Fig. 5 shows that the term takes positive and negative values, but these are significantly smaller than those of the and terms. Furthermore, as for the total yield, the observed pattern oscillates very rapidly with time delay [due to the large value of the phases in Eq. 3c] and always around a mean value of zero. Therefore, the physical information contained in is not relevant regarding autoionization features.

These results suggest that one can extract the signature of autoionization by simply removing, from the doubly differential yield, the contributions from the 2p channel (Fig. S5; by using, e.g., the reflection approximation) and the Ramsey interferences (by using, e.g., the corresponding analytical formula in Eq. 3a). The 2D spectra that result after removing these contributions are shown in Fig. S6. From these modified spectra, the dynamics of autoionization, i.e., the lifetimes, can be extracted by using a semiclassical model based on the classical paths followed by the nuclei in a given potential energy curve (48, 49). The latter approach has been successfully used in previous work on one-photon ionization of H₂ (49), and its application to the present case is given in Extracting the Dynamics of Autoionizing States (Fig. S7). Therefore, the present analysis also opens the door to the visualization of the autoionization dynamics.