Molecular interferometer to decode attosecond electron–nuclear dynamics Alicia Palaciosa, Alberto González-Castrilloa, and Fernando Martína,b,1 a Departamento de Química, Universidad Autónoma de Madrid, 28049 Madrid, Spain; and bInstituto Madrileño de Estudios Avanzados en Nanociencia, Campus de Cantoblanco, 28049 Madrid, Spain
Edited by Philip H. Bucksbaum, Stanford University, Menlo Park, CA, and approved January 30, 2014 (received for review September 5, 2013)
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attosecond molecular dynamics free electron lasers generation XUV pump-probe spectroscopy
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dvances in optical technology in the last 2 decades have enabled a variety of applications in which ultrashort laser pulses can be used to probe, manipulate, and eventually control electron and fast nuclear dynamics in molecules. A successful approach, inspired by femtochemistry (1), is to use a combination of ultrashort pump and probe pulses, in which the dynamics are launched by the pump pulse and the imaging of these dynamics is obtained by scanning the time delay between both pulses. By controlling optical coherence, early experimental works made use of intense infrared (IR) pulses with durations of the order of femtoseconds (fs) to steer electronic motion in atoms and molecules and to probe it, in a subfemtosecond time scale, by analyzing either the high harmonic emission, the photoelectron spectrum resulting from rescattered electrons, or the kinetic energy of the generated ions (2–4). More recent experimental approaches have developed pump–probe schemes in which attosecond extreme UV (XUV) pulses have been used to ionize and induce some dynamics in the target, and phase-locked IR probe pulses have been used to trace it, both in atoms (5–7) and molecules (8, 9). A step forward in pump–probe schemes has been the advent of new detection and analysis strategies, such as interferometric methods to reconstruct electron wave packets in atoms (10) or the attosecond streak camera (7, 11, 12, 13) to retrieve phases and time-varying intensity envelopes of attosecond XUV pulses from their imprint in photoelectron spectra (14). The optical characterization of attosecond pulses is thus directly linked to the understanding of photoionization (13, 15, 16). In all of the above schemes, however, the dynamics launched by the pump pulse are usually hidden by that induced by the IR probe field, which is strong enough to significantly distort the molecular potential. A possible way out to explore field-free molecular dynamics is the use of “weak” XUV pulses for both www.pnas.org/cgi/doi/10.1073/pnas.1316762111
the pump and the probe because their short wavelengths ensure that the ponderomotive energy, i.e., the energy acquired by an electron due to its interaction with the external field, is negligible in comparison with the electron–electron and electron–nucleus interactions. Very recent experimental work (17) has succeeded in implementing such a pump–probe scheme and in providing ionization yields for Xe (17) and H2 (18) with a time resolution of the order of 1 fs. The key point in these experiments is the use of XUV pulses generated from high harmonic generation (HHG), which are short enough (∼1 fs) to ensure high temporal resolution. Still, XUV pulses generated in this way are so weak that measurements are generally affected by rather low statistics, which can eventually blur the electron dynamics. An alternative to HHG is the use of XUV pulses generated by free electron lasers (FEL). Here intensities are substantially higher (although still low enough to avoid distortions of the molecular potential), but pulse durations are usually too high to achieve the necessary temporal resolution for electron dynamics. Significant efforts are progressively reducing the duration of FEL pulses, and in fact, proof-of-principle experiments have already been carried out with time resolutions of tens of fs (19, 20). In either HHG or FEL approaches, pump–probe schemes can strongly benefit from existing multicoincidence (21) and velocity map imaging (22) detection methods, so efforts along this direction are also in progress (19, 20, 23, 24). It is thus timely to guide these experimental efforts with the help of accurate theoretical methods that provide realistic information on different experimental observables and on possible ways to extract the molecular dynamics that are sought. Nearly exact theoretical methods able to describe the coupled electron and nuclear dynamics in molecules are limited, even nowadays, to the hydrogen molecule H2. Because of this, H2 (and D2) has Significance Current attosecond technologies designed to study fast electron and nuclear motion in molecules make use, at some stage, of infrared laser pulses that strongly perturb the molecular potential, thus modifying the dynamics inherent to the system. To access the actual dynamics of the molecule, gentler laser sources must be used. Very recently, extreme UV pulses have been successfully combined to provide a picture of the intrinsic electron dynamics in atoms with subfemtosecond time resolution. Here we show that by using two identical extreme UV pulses, one can also obtain a complete description of the coupled electronic and nuclear dynamics in molecules. Visualization of such dynamics is possible by varying the wavelength and/or the time delay between the two pulses. Author contributions: A.P. and F.M. designed research; A.P., A.G.-C., and F.M. performed research; A.P., A.G.-C., and F.M. contributed new reagents/analytic tools; A.P., A.G.-C., and F.M. analyzed data; and A.P. and F.M. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. 1
To whom correspondence should be addressed. E-mail: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1316762111/-/DCSupplemental.
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Understanding the coupled electronic and nuclear dynamics in molecules by using pump–probe schemes requires not only the use of short enough laser pulses but also wavelengths and intensities that do not modify the intrinsic behavior of the system. In this respect, extreme UV pulses of few-femtosecond and attosecond durations have been recognized as the ideal tool because their short wavelengths ensure a negligible distortion of the molecular potential. In this work, we propose the use of two twin extreme UV pulses to create a molecular interferometer from direct and sequential two-photon ionization processes that leave the molecule in the same final state. We theoretically demonstrate that such a scheme allows for a complete identification of both electronic and nuclear phases in the wave packet generated by the pump pulse. We also show that although total ionization yields reveal entangled electronic and nuclear dynamics in the bound states, doubly differential yields (differential in both electronic and nuclear energies) exhibit in addition the dynamics of autoionization, i.e., of electron correlation in the ionization continuum. Visualization of such dynamics is possible by varying the time delay between the pump and the probe pulses.
been used as a benchmark in many IR/IR (3, 25, 26) and XUV/IR pump–probe experiments (27–29). In this paper, we present a complete analysis of a realistic numerical experiment in which H2 is ionized by using an XUV pump/XUV probe scheme with identical pulses and subfemtosecond temporal resolution. We show that, from such a scheme, amplitudes and phases associated with the different electronic and nuclear components of the molecular wave packet (WP) generated by the pump pulse can be determined, so that the coupled electron and nuclear dynamics can be disentangled. Visualization of such dynamics is possible by varying the time delay between the pump and the probe pulses. The ab initio theoretical approach used here, which includes all electronic and vibrational (dissociative) degrees of freedom (only rotation is neglected), has been described elsewhere (Materials and Methods and refs. 30, 31). Its predictive character has been demonstrated by the excellent agreement with experimental measurements in different contexts (32–34), particularly XUV-IR pump–probe experiments involving isolated (27) and trains (29) of attosecond pulses. To make the problem tractable while keeping high accuracy, we consider the case of linearly polarized light and molecules parallel to the polarization direction. This situation can be realized experimentally by using current techniques, for instance, by measuring the momentum of all ejected protons and selecting only those whose momenta correspond to a parallel orientation of the H2 molecule. For this molecular orientation, according to the dipole selection rule, only Σg ↔ Σu transitions are allowed by absorption of a single photon. In any case, our conclusions regarding the method to extract phases and amplitudes of the generated molecular wave packets will remain valid for other molecular orientations, hence for random orientation. The purpose of using two identical pulses is to create a molecular interferometer that contains information about different temporal and frequency quantum paths leading to the same final state. This idea has been theoretically explored in atoms (35–37) to learn about electron dynamics and also experimentally explored in diatomic molecules to extract information on the much slower nuclear dynamics (38–41) (by using much longer pulses, from hundreds of fs up to ps). The challenge we address and solve in this paper is to resolve coupled electron and nuclear dynamics in excited electronic states of the H2 molecule. The XUV pump/XUV probe pulse sequence that we use in this work is described in Fig. 1. The field intensity (1012 W·cm−2) is chosen so that distortion of the molecular potential by the external field is negligible [precluding strong-field effects such as bond softening or hardening (42)] and is experimentally accessible in HHG (18) and FEL (43). We use two identical carrier envelope phase (CEP) stabilized pulses of 2-fs duration and centered at a photon energy of 12.2 eV, which is in resonance with the vertical one-photon transition from the ground state (X1 Σ+g ) to the first excited state (B1 Σ+u ) of H2. τ is the time delay between the pulses. The first pulse not only excites the molecule but can also efficiently ionize it after absorption of a second photon from the B1 Σ+u state (a particular case of resonance enhanced multiphoton ionization). One can distinguish two different ionization channels: direct (Fig. 1A), when the two photons come from the same pulse or from different pulses that overlap in time, and sequential (Fig. 1C), when one photon comes from the pump and the other one comes from the probe. In the sequential case, the electronic plus nuclear WP created by the pump pulse evolves freely before the second photon is absorbed (Fig. 1B). In either case, ionization can leave the residual H+2 ion in dissociative and nondissociative channels and can be accompanied by autoionization of the Q11 Σ+g doubly excited states (DESs) embedded in the electronic continuum. In addition, the probe pulse generates replicas of the WPs launched by the pump pulse (Fig. 1A), thus leading to additional interferences. 3974 | www.pnas.org/cgi/doi/10.1073/pnas.1316762111
A
B
C
D
Fig. 1. Schematic representation of the XUV pump/XUV probe scheme, with two pulses of 2 fs of duration with a time delay of 6 fs among them. Each panel shows the relevant potential energy curves of the H2 molecule: the ground state (X1 Σ+g ), the two lowest excited states 1 Σ+u (B and B′), the lowest Q1 1 Σ+g doubly excited states, and the two lowest ionization thresholds 1sσ g and 2pσ u. The blue shaded areas represent the squares of the WP generated by the pump pulse at the times indicated in the figure: t = 2 (A), 4 (B), and 8 (C) fs. The red shaded areas represent the WPs generated by the probe pulse t = 8 fs. The time delay between the pump and the probe pulses is τ = 6 fs (D). The orange arrows represent direct two-photon ionization by the pump and probe pulses, and the green arrow represents the absorption of a single photon from the probe pulse after free evolution of the WP generated by absorption of a single photon from the pump pulse (sequential two-photon ionization process). The WP created in the sequential process is represented by the green shaded area. The black curves superimposed on the blue and red WPs in C represent the WP resulting from the interference between the red and the blue WPs.
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 t = τ, differential in both the electron energy ee and the nuclear energy eN , for a final total energy Ef = ee + eN , can be approximately written as 2 X iΔEf 0 τ iΔEfm τ + Pf ðτÞ = af 1 + e bfm e [1] ; m where af is the complex amplitude corresponding to direct twophoton ionization (Fig. 1A); bfm 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); ΔEf 0 = ðEf − E0 Þ; and ΔEfm = ðEf − Em Þ. Note that the formula includes a coherent sum over all m states contributing to the intermediate WP and that the energy differences ΔEf 0 and ΔEfm involve total energies, i.e., electronic plus nuclear energies, for the final state (Ef ), the ground state (E0 ), and all m intermediate states (Em ) of the Palacios et al.
The observed beatings can be easily understood by assuming that the af and bfm 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 to
A
~=A α β~ = B +
[4a] X
Cmm′ cos½ðEm′ − Em Þτ
[4b]
m;m′>m
B ~γ =
X
F m cos½ðEm − E0 Þτ;
[4c]
C
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).
molecule. We also note that, in general, m runs over vibronic states associated with various electronic states. To better visualize the phases involved in the process, we decompose Eq. 1 as Pf = αf + βf + γ f ;
[2]
where 2 αf = 2af 1 + cos Ef − E0 τ βf =
X 2 X p bfm + cos½ðEm′ − Em Þτ 2Re bfm bfm′ m
γf =
X
[3a] [3b]
m;m′>m
2Re apf bfm cos Ef − Em τ + cos½ðEm − E0 Þτ : [3c]
m
αf is thus associated with the direct ionization process and its 2 time-delayed replica [i.e., af ð1+eiΔEf 0 τ Þ ], βf is associated with 2 P the sequential process (i.e., m bfm eiΔEfm τ ), and γ f is the cross ~ = α ~ + β~ + ~γ , is obtained by term. The total ionization yield, P integrating Pf over all electronic continuum energies ee and all vibrational and dissociative energies eN associated with that elec~ and ~γ terms to ~, β, tronic continuum. The contribution of the α 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 (β~ term), whereas the fast ones (∼0.35 fs period) come from the cross term (~γ ). The contribution from the direct process (~ α term) is nearly independent of time delay. Palacios et al.
where A, B, Cmm′ , and F m are constants. It is therefore clear that the phases Em − Em′ , 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 B1 Σ+u 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 B1 Σ+u state. The fast beating is controlled by the phases ðEm − E0 Þ, which involve the energies of the ground and the B1 Σ+u vibronic states. Therefore, it results from the coherent superposition of the direct and sequential two-photon ionization processes. The above equations also ~ barely depends on time delay: the ðEf − E0 Þ phase explain why α is practically washed out by integration over ee and eN . 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 ðEm − E0 Þ 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 ðEm − E0 Þ beatings (∼300–400 as) and smaller than the period of the nuclear component of the WP in the B1 Σ+u 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: B1 Σ+u and B′1 Σ+u (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 PNAS | March 18, 2014 | vol. 111 | no. 11 | 3975
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m
A
B
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.
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 B1 Σ+u state. Additional features are observed at around 14 eV. They reflect the motion of another nuclear WP component in the B′1 Σ+u 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. 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 E0 − Em and Em − Em′ beatings in the molecular WP. The Em − Em′ 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 Em − Em′ 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 3976 | www.pnas.org/cgi/doi/10.1073/pnas.1316762111
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. 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 B1 Σ+u 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: αf , βf , and γ f . 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 αf 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 1 + cos½ðee + eN − E0 Þτ, 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 (Ef = ee + eN = E0 + 2ω − DI, where DI = 18.15 eV is the dissociative ionization energy of H2). As shown above, these oscillations, as well as those related to the ðEf − Em Þ 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).
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 B1 Σ+u electronic state (in black) and the vibrational states associated with the B′1 Σ+u one (in red). Palacios et al.
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 βf term (Fig. 5, fourth row), and therefore, it is due to the sequential ionization process. After 4 fs, the nuclear WP component in the B1 Σ+u state has reached the region of internuclear distances (R > 2:5 a.u.) where absorption of the probe photon can lead to ionization above the 2pσ u threshold. Due to the pure dissociative character of the 2pσ u potential energy curve, the contribution from the 2pσ u 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 2pσ u 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 2pσ u appears. When the intermediate nuclear WP component turns back and moves toward smaller internuclear distances, the motion of the 2pσ u peak is reversed. This motion repeats with a periodicity of 28–30 fs, i.e., that of the nuclear WP motion in the B1 Σ+u state. All remaining features in the βf 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
Conclusion and Perspectives In conclusion, we have shown how an attosecond XUV pump/ XUV probe scheme can be used to obtain complete information about electronic and nuclear dynamics in molecules, without introducing significant distortions associated with the pump and probe pulses. This is in contrast with previously proposed pump–probe schemes in which the relatively strong interaction of the probe field with the system is responsible for the observed dynamics. Our analysis is based on an accurate solution of the time-dependent Schrödinger equation for the hydrogen molecule and on the use of a simple model that allows one to extract the relevant dynamical information as in a real experiment. The key point in this scheme is the use of two identical XUV pulses, which build up a molecular interferometer based on direct and sequential two-photon ionization processes leading to the same final state. The use of identical pulses also simplifies its experimental implementation. Total (dissociative) ionization yields provide information about the amplitudes and phases (energy beatings) that build up the molecular wave packet (electronic plus nuclear) created by the pump pulse but not about the wave packet created by the probe pulse. Information about the latter wave packet can only be retrieved from doubly differential yields. For this, one needs to measure all ejected charged particles in coincidence. The resulting 2D spectrum provides the necessary energy resolution to analyze all of the final states populated by the probe pulse, the only limiting factor being the energy resolution of the detectors. Time resolution comes from recording these 2D spectra as a function of time delay.
Palacios et al.
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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 αf , βf , and γ f contributions defined in Eq. 3.
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 1 Σ+g 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, τ ≤ 2 fs, only the lowest Q1 1 Σ+g 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 B1 Σ+u 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 B1 Σ+u state and reaches again the region of internuclear distances at which transition to the DESs is possible. Fig. 5 shows that the γ f term takes positive and negative values, but these are significantly smaller than those of the αf and βf terms. Furthermore, as for the total yield, the observed pattern oscillates very rapidly with time delay [due to the large value of the phases ðEm − E0 Þ in Eq. 3c] and always around a mean value of zero. Therefore, the physical information contained in γ f 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 2pσ u 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 onephoton 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.
The investigated dynamics result from the coherent superposition of vibronic states associated with the various electronic states populated by the XUV pump pulse. Although this coherent superposition evolves in time, thus generating a coupled electronic plus nuclear wave packet, population of the different states does not vary with time. To see significant variations in the populations of the intermediate electronic states, one could dress the XUV pump pulse with a weak IR pulse of similar duration that resonantly couples two electronic states. Then the use of another XUV pulse to probe the system, in combination with the interpretation scheme as that described in the present work, should also allow one to reveal the variations of the intermediate-state population with time. Finally, although the present work has been restricted to the H2 molecule, a similar scheme can be applied to more complex molecules, e.g., to investigate and control the dynamics through conical intersections (50). Materials and Methods
electronic continuum states associated with the 1sσ g and 2pσ u ionization channels, and the lowest Q1 and Q2 series of doubly excited states. The electronic part of the vibronic wave function is calculated in a box of 60 a.u., and the nuclear part is calculated in a box of 12 a.u. The sizes of these boxes are large enough to ensure that there are no significant reflections of electronic and nuclear WPs in the box boundaries for propagation times smaller than τ + T, where T is the duration of the XUV pulses and τ is the time delay (from maximum to maximum). Pump and probe pulses are identical with central frequency of 12.2 eV (or 14 eV), duration T = 2 fs, and cosine square temporal envelope. The intensity is 1012 W·cm−2. Apart from truncation of the basis of vibronic states at a high enough energy (∼65 eV above the H2 ground state), the only approximations made in our calculations are (i) the dipole approximation in describing the laser–molecule interaction and (ii) the neglect of nonadiabatic couplings and molecular rotations. In previous work, this method has been shown to yield integral and differential crosssections for dissociative ionization of H2 and D2 in good agreement with various experimental measurements (e.g., refs. 27, 29, 33, 34).
The theoretical method has been explained in detail in previous works (30, 31). It includes all electronic and vibrational (dissociative) degrees of freedom. Only the rotational motion is neglected. Briefly, we solve the timedependent Schrödinger equation by expanding the time-dependent wave function in a basis of fully correlated H2 vibronic stationary states of Σ+g and Σ+u symmetries built upon the bound electronic states, the nonresonant
ACKNOWLEDGMENTS. We acknowledge Dr. J. Feist for fruitful discussions. This work was accomplished with an allocation of computer time from Mare Nostrum Barcelona Supercomputing Center and was partially supported by the Ministerio de Ciencia e Innovación Projects FIS2010-15127 and CSD 200700010, the European Research Area-Chemistry Project PIM2010EEC-00751, the European Grants MC-ITN CORINF and MC-RG ATTOTREND 268284, and the Advanced Grant of the European Research Council XCHEM 290853.
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Palacios et al.
Edited by Philip H. Bucksbaum, Stanford University, Menlo Park, CA, and approved January 30, 2014 (received for review September 5, 2013)
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attosecond molecular dynamics free electron lasers generation XUV pump-probe spectroscopy
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dvances in optical technology in the last 2 decades have enabled a variety of applications in which ultrashort laser pulses can be used to probe, manipulate, and eventually control electron and fast nuclear dynamics in molecules. A successful approach, inspired by femtochemistry (1), is to use a combination of ultrashort pump and probe pulses, in which the dynamics are launched by the pump pulse and the imaging of these dynamics is obtained by scanning the time delay between both pulses. By controlling optical coherence, early experimental works made use of intense infrared (IR) pulses with durations of the order of femtoseconds (fs) to steer electronic motion in atoms and molecules and to probe it, in a subfemtosecond time scale, by analyzing either the high harmonic emission, the photoelectron spectrum resulting from rescattered electrons, or the kinetic energy of the generated ions (2–4). More recent experimental approaches have developed pump–probe schemes in which attosecond extreme UV (XUV) pulses have been used to ionize and induce some dynamics in the target, and phase-locked IR probe pulses have been used to trace it, both in atoms (5–7) and molecules (8, 9). A step forward in pump–probe schemes has been the advent of new detection and analysis strategies, such as interferometric methods to reconstruct electron wave packets in atoms (10) or the attosecond streak camera (7, 11, 12, 13) to retrieve phases and time-varying intensity envelopes of attosecond XUV pulses from their imprint in photoelectron spectra (14). The optical characterization of attosecond pulses is thus directly linked to the understanding of photoionization (13, 15, 16). In all of the above schemes, however, the dynamics launched by the pump pulse are usually hidden by that induced by the IR probe field, which is strong enough to significantly distort the molecular potential. A possible way out to explore field-free molecular dynamics is the use of “weak” XUV pulses for both www.pnas.org/cgi/doi/10.1073/pnas.1316762111
the pump and the probe because their short wavelengths ensure that the ponderomotive energy, i.e., the energy acquired by an electron due to its interaction with the external field, is negligible in comparison with the electron–electron and electron–nucleus interactions. Very recent experimental work (17) has succeeded in implementing such a pump–probe scheme and in providing ionization yields for Xe (17) and H2 (18) with a time resolution of the order of 1 fs. The key point in these experiments is the use of XUV pulses generated from high harmonic generation (HHG), which are short enough (∼1 fs) to ensure high temporal resolution. Still, XUV pulses generated in this way are so weak that measurements are generally affected by rather low statistics, which can eventually blur the electron dynamics. An alternative to HHG is the use of XUV pulses generated by free electron lasers (FEL). Here intensities are substantially higher (although still low enough to avoid distortions of the molecular potential), but pulse durations are usually too high to achieve the necessary temporal resolution for electron dynamics. Significant efforts are progressively reducing the duration of FEL pulses, and in fact, proof-of-principle experiments have already been carried out with time resolutions of tens of fs (19, 20). In either HHG or FEL approaches, pump–probe schemes can strongly benefit from existing multicoincidence (21) and velocity map imaging (22) detection methods, so efforts along this direction are also in progress (19, 20, 23, 24). It is thus timely to guide these experimental efforts with the help of accurate theoretical methods that provide realistic information on different experimental observables and on possible ways to extract the molecular dynamics that are sought. Nearly exact theoretical methods able to describe the coupled electron and nuclear dynamics in molecules are limited, even nowadays, to the hydrogen molecule H2. Because of this, H2 (and D2) has Significance Current attosecond technologies designed to study fast electron and nuclear motion in molecules make use, at some stage, of infrared laser pulses that strongly perturb the molecular potential, thus modifying the dynamics inherent to the system. To access the actual dynamics of the molecule, gentler laser sources must be used. Very recently, extreme UV pulses have been successfully combined to provide a picture of the intrinsic electron dynamics in atoms with subfemtosecond time resolution. Here we show that by using two identical extreme UV pulses, one can also obtain a complete description of the coupled electronic and nuclear dynamics in molecules. Visualization of such dynamics is possible by varying the wavelength and/or the time delay between the two pulses. Author contributions: A.P. and F.M. designed research; A.P., A.G.-C., and F.M. performed research; A.P., A.G.-C., and F.M. contributed new reagents/analytic tools; A.P., A.G.-C., and F.M. analyzed data; and A.P. and F.M. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. 1
To whom correspondence should be addressed. E-mail: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1316762111/-/DCSupplemental.
PNAS | March 18, 2014 | vol. 111 | no. 11 | 3973–3978
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Understanding the coupled electronic and nuclear dynamics in molecules by using pump–probe schemes requires not only the use of short enough laser pulses but also wavelengths and intensities that do not modify the intrinsic behavior of the system. In this respect, extreme UV pulses of few-femtosecond and attosecond durations have been recognized as the ideal tool because their short wavelengths ensure a negligible distortion of the molecular potential. In this work, we propose the use of two twin extreme UV pulses to create a molecular interferometer from direct and sequential two-photon ionization processes that leave the molecule in the same final state. We theoretically demonstrate that such a scheme allows for a complete identification of both electronic and nuclear phases in the wave packet generated by the pump pulse. We also show that although total ionization yields reveal entangled electronic and nuclear dynamics in the bound states, doubly differential yields (differential in both electronic and nuclear energies) exhibit in addition the dynamics of autoionization, i.e., of electron correlation in the ionization continuum. Visualization of such dynamics is possible by varying the time delay between the pump and the probe pulses.
been used as a benchmark in many IR/IR (3, 25, 26) and XUV/IR pump–probe experiments (27–29). In this paper, we present a complete analysis of a realistic numerical experiment in which H2 is ionized by using an XUV pump/XUV probe scheme with identical pulses and subfemtosecond temporal resolution. We show that, from such a scheme, amplitudes and phases associated with the different electronic and nuclear components of the molecular wave packet (WP) generated by the pump pulse can be determined, so that the coupled electron and nuclear dynamics can be disentangled. Visualization of such dynamics is possible by varying the time delay between the pump and the probe pulses. The ab initio theoretical approach used here, which includes all electronic and vibrational (dissociative) degrees of freedom (only rotation is neglected), has been described elsewhere (Materials and Methods and refs. 30, 31). Its predictive character has been demonstrated by the excellent agreement with experimental measurements in different contexts (32–34), particularly XUV-IR pump–probe experiments involving isolated (27) and trains (29) of attosecond pulses. To make the problem tractable while keeping high accuracy, we consider the case of linearly polarized light and molecules parallel to the polarization direction. This situation can be realized experimentally by using current techniques, for instance, by measuring the momentum of all ejected protons and selecting only those whose momenta correspond to a parallel orientation of the H2 molecule. For this molecular orientation, according to the dipole selection rule, only Σg ↔ Σu transitions are allowed by absorption of a single photon. In any case, our conclusions regarding the method to extract phases and amplitudes of the generated molecular wave packets will remain valid for other molecular orientations, hence for random orientation. The purpose of using two identical pulses is to create a molecular interferometer that contains information about different temporal and frequency quantum paths leading to the same final state. This idea has been theoretically explored in atoms (35–37) to learn about electron dynamics and also experimentally explored in diatomic molecules to extract information on the much slower nuclear dynamics (38–41) (by using much longer pulses, from hundreds of fs up to ps). The challenge we address and solve in this paper is to resolve coupled electron and nuclear dynamics in excited electronic states of the H2 molecule. The XUV pump/XUV probe pulse sequence that we use in this work is described in Fig. 1. The field intensity (1012 W·cm−2) is chosen so that distortion of the molecular potential by the external field is negligible [precluding strong-field effects such as bond softening or hardening (42)] and is experimentally accessible in HHG (18) and FEL (43). We use two identical carrier envelope phase (CEP) stabilized pulses of 2-fs duration and centered at a photon energy of 12.2 eV, which is in resonance with the vertical one-photon transition from the ground state (X1 Σ+g ) to the first excited state (B1 Σ+u ) of H2. τ is the time delay between the pulses. The first pulse not only excites the molecule but can also efficiently ionize it after absorption of a second photon from the B1 Σ+u state (a particular case of resonance enhanced multiphoton ionization). One can distinguish two different ionization channels: direct (Fig. 1A), when the two photons come from the same pulse or from different pulses that overlap in time, and sequential (Fig. 1C), when one photon comes from the pump and the other one comes from the probe. In the sequential case, the electronic plus nuclear WP created by the pump pulse evolves freely before the second photon is absorbed (Fig. 1B). In either case, ionization can leave the residual H+2 ion in dissociative and nondissociative channels and can be accompanied by autoionization of the Q11 Σ+g doubly excited states (DESs) embedded in the electronic continuum. In addition, the probe pulse generates replicas of the WPs launched by the pump pulse (Fig. 1A), thus leading to additional interferences. 3974 | www.pnas.org/cgi/doi/10.1073/pnas.1316762111
A
B
C
D
Fig. 1. Schematic representation of the XUV pump/XUV probe scheme, with two pulses of 2 fs of duration with a time delay of 6 fs among them. Each panel shows the relevant potential energy curves of the H2 molecule: the ground state (X1 Σ+g ), the two lowest excited states 1 Σ+u (B and B′), the lowest Q1 1 Σ+g doubly excited states, and the two lowest ionization thresholds 1sσ g and 2pσ u. The blue shaded areas represent the squares of the WP generated by the pump pulse at the times indicated in the figure: t = 2 (A), 4 (B), and 8 (C) fs. The red shaded areas represent the WPs generated by the probe pulse t = 8 fs. The time delay between the pump and the probe pulses is τ = 6 fs (D). The orange arrows represent direct two-photon ionization by the pump and probe pulses, and the green arrow represents the absorption of a single photon from the probe pulse after free evolution of the WP generated by absorption of a single photon from the pump pulse (sequential two-photon ionization process). The WP created in the sequential process is represented by the green shaded area. The black curves superimposed on the blue and red WPs in C represent the WP resulting from the interference between the red and the blue WPs.
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 t = τ, differential in both the electron energy ee and the nuclear energy eN , for a final total energy Ef = ee + eN , can be approximately written as 2 X iΔEf 0 τ iΔEfm τ + Pf ðτÞ = af 1 + e bfm e [1] ; m where af is the complex amplitude corresponding to direct twophoton ionization (Fig. 1A); bfm 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); ΔEf 0 = ðEf − E0 Þ; and ΔEfm = ðEf − Em Þ. Note that the formula includes a coherent sum over all m states contributing to the intermediate WP and that the energy differences ΔEf 0 and ΔEfm involve total energies, i.e., electronic plus nuclear energies, for the final state (Ef ), the ground state (E0 ), and all m intermediate states (Em ) of the Palacios et al.
The observed beatings can be easily understood by assuming that the af and bfm 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 to
A
~=A α β~ = B +
[4a] X
Cmm′ cos½ðEm′ − Em Þτ
[4b]
m;m′>m
B ~γ =
X
F m cos½ðEm − E0 Þτ;
[4c]
C
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).
molecule. We also note that, in general, m runs over vibronic states associated with various electronic states. To better visualize the phases involved in the process, we decompose Eq. 1 as Pf = αf + βf + γ f ;
[2]
where 2 αf = 2af 1 + cos Ef − E0 τ βf =
X 2 X p bfm + cos½ðEm′ − Em Þτ 2Re bfm bfm′ m
γf =
X
[3a] [3b]
m;m′>m
2Re apf bfm cos Ef − Em τ + cos½ðEm − E0 Þτ : [3c]
m
αf is thus associated with the direct ionization process and its 2 time-delayed replica [i.e., af ð1+eiΔEf 0 τ Þ ], βf is associated with 2 P the sequential process (i.e., m bfm eiΔEfm τ ), and γ f is the cross ~ = α ~ + β~ + ~γ , is obtained by term. The total ionization yield, P integrating Pf over all electronic continuum energies ee and all vibrational and dissociative energies eN associated with that elec~ and ~γ terms to ~, β, tronic continuum. The contribution of the α 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 (β~ term), whereas the fast ones (∼0.35 fs period) come from the cross term (~γ ). The contribution from the direct process (~ α term) is nearly independent of time delay. Palacios et al.
where A, B, Cmm′ , and F m are constants. It is therefore clear that the phases Em − Em′ , 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 B1 Σ+u 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 B1 Σ+u state. The fast beating is controlled by the phases ðEm − E0 Þ, which involve the energies of the ground and the B1 Σ+u vibronic states. Therefore, it results from the coherent superposition of the direct and sequential two-photon ionization processes. The above equations also ~ barely depends on time delay: the ðEf − E0 Þ phase explain why α is practically washed out by integration over ee and eN . 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 ðEm − E0 Þ 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 ðEm − E0 Þ beatings (∼300–400 as) and smaller than the period of the nuclear component of the WP in the B1 Σ+u 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: B1 Σ+u and B′1 Σ+u (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 PNAS | March 18, 2014 | vol. 111 | no. 11 | 3975
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m
A
B
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.
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 B1 Σ+u state. Additional features are observed at around 14 eV. They reflect the motion of another nuclear WP component in the B′1 Σ+u 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. 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 E0 − Em and Em − Em′ beatings in the molecular WP. The Em − Em′ 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 Em − Em′ 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 3976 | www.pnas.org/cgi/doi/10.1073/pnas.1316762111
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. 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 B1 Σ+u 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: αf , βf , and γ f . 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 αf 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 1 + cos½ðee + eN − E0 Þτ, 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 (Ef = ee + eN = E0 + 2ω − DI, where DI = 18.15 eV is the dissociative ionization energy of H2). As shown above, these oscillations, as well as those related to the ðEf − Em Þ 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).
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 B1 Σ+u electronic state (in black) and the vibrational states associated with the B′1 Σ+u one (in red). Palacios et al.
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 βf term (Fig. 5, fourth row), and therefore, it is due to the sequential ionization process. After 4 fs, the nuclear WP component in the B1 Σ+u state has reached the region of internuclear distances (R > 2:5 a.u.) where absorption of the probe photon can lead to ionization above the 2pσ u threshold. Due to the pure dissociative character of the 2pσ u potential energy curve, the contribution from the 2pσ u 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 2pσ u 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 2pσ u appears. When the intermediate nuclear WP component turns back and moves toward smaller internuclear distances, the motion of the 2pσ u peak is reversed. This motion repeats with a periodicity of 28–30 fs, i.e., that of the nuclear WP motion in the B1 Σ+u state. All remaining features in the βf 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
Conclusion and Perspectives In conclusion, we have shown how an attosecond XUV pump/ XUV probe scheme can be used to obtain complete information about electronic and nuclear dynamics in molecules, without introducing significant distortions associated with the pump and probe pulses. This is in contrast with previously proposed pump–probe schemes in which the relatively strong interaction of the probe field with the system is responsible for the observed dynamics. Our analysis is based on an accurate solution of the time-dependent Schrödinger equation for the hydrogen molecule and on the use of a simple model that allows one to extract the relevant dynamical information as in a real experiment. The key point in this scheme is the use of two identical XUV pulses, which build up a molecular interferometer based on direct and sequential two-photon ionization processes leading to the same final state. The use of identical pulses also simplifies its experimental implementation. Total (dissociative) ionization yields provide information about the amplitudes and phases (energy beatings) that build up the molecular wave packet (electronic plus nuclear) created by the pump pulse but not about the wave packet created by the probe pulse. Information about the latter wave packet can only be retrieved from doubly differential yields. For this, one needs to measure all ejected charged particles in coincidence. The resulting 2D spectrum provides the necessary energy resolution to analyze all of the final states populated by the probe pulse, the only limiting factor being the energy resolution of the detectors. Time resolution comes from recording these 2D spectra as a function of time delay.
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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 αf , βf , and γ f contributions defined in Eq. 3.
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 1 Σ+g 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, τ ≤ 2 fs, only the lowest Q1 1 Σ+g 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 B1 Σ+u 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 B1 Σ+u state and reaches again the region of internuclear distances at which transition to the DESs is possible. Fig. 5 shows that the γ f term takes positive and negative values, but these are significantly smaller than those of the αf and βf terms. Furthermore, as for the total yield, the observed pattern oscillates very rapidly with time delay [due to the large value of the phases ðEm − E0 Þ in Eq. 3c] and always around a mean value of zero. Therefore, the physical information contained in γ f 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 2pσ u 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 onephoton 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.
The investigated dynamics result from the coherent superposition of vibronic states associated with the various electronic states populated by the XUV pump pulse. Although this coherent superposition evolves in time, thus generating a coupled electronic plus nuclear wave packet, population of the different states does not vary with time. To see significant variations in the populations of the intermediate electronic states, one could dress the XUV pump pulse with a weak IR pulse of similar duration that resonantly couples two electronic states. Then the use of another XUV pulse to probe the system, in combination with the interpretation scheme as that described in the present work, should also allow one to reveal the variations of the intermediate-state population with time. Finally, although the present work has been restricted to the H2 molecule, a similar scheme can be applied to more complex molecules, e.g., to investigate and control the dynamics through conical intersections (50). Materials and Methods
electronic continuum states associated with the 1sσ g and 2pσ u ionization channels, and the lowest Q1 and Q2 series of doubly excited states. The electronic part of the vibronic wave function is calculated in a box of 60 a.u., and the nuclear part is calculated in a box of 12 a.u. The sizes of these boxes are large enough to ensure that there are no significant reflections of electronic and nuclear WPs in the box boundaries for propagation times smaller than τ + T, where T is the duration of the XUV pulses and τ is the time delay (from maximum to maximum). Pump and probe pulses are identical with central frequency of 12.2 eV (or 14 eV), duration T = 2 fs, and cosine square temporal envelope. The intensity is 1012 W·cm−2. Apart from truncation of the basis of vibronic states at a high enough energy (∼65 eV above the H2 ground state), the only approximations made in our calculations are (i) the dipole approximation in describing the laser–molecule interaction and (ii) the neglect of nonadiabatic couplings and molecular rotations. In previous work, this method has been shown to yield integral and differential crosssections for dissociative ionization of H2 and D2 in good agreement with various experimental measurements (e.g., refs. 27, 29, 33, 34).
The theoretical method has been explained in detail in previous works (30, 31). It includes all electronic and vibrational (dissociative) degrees of freedom. Only the rotational motion is neglected. Briefly, we solve the timedependent Schrödinger equation by expanding the time-dependent wave function in a basis of fully correlated H2 vibronic stationary states of Σ+g and Σ+u symmetries built upon the bound electronic states, the nonresonant
ACKNOWLEDGMENTS. We acknowledge Dr. J. Feist for fruitful discussions. This work was accomplished with an allocation of computer time from Mare Nostrum Barcelona Supercomputing Center and was partially supported by the Ministerio de Ciencia e Innovación Projects FIS2010-15127 and CSD 200700010, the European Research Area-Chemistry Project PIM2010EEC-00751, the European Grants MC-ITN CORINF and MC-RG ATTOTREND 268284, and the Advanced Grant of the European Research Council XCHEM 290853.
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