Abstract
Shape resonances in physics and chemistry arise from the spatial confinement of a particle by a potential barrier. In molecular photoionization, these barriers prevent the electron from escaping instantaneously, so that nuclei may move and modify the potential, thereby affecting the ionization process. By using an attosecond two-color interferometric approach in combination with high spectral resolution, we have captured the changes induced by the nuclear motion on the centrifugal barrier that sustains the well-known shape resonance in valence-ionized N2. We show that despite the nuclear motion altering the bond length by only 2%, which leads to tiny changes in the potential barrier, the corresponding change in the ionization time can be as large as 200 attoseconds. This result poses limits to the concept of instantaneous electronic transitions in molecules, which is at the basis of the Franck-Condon principle of molecular spectroscopy.
INTRODUCTION
Shape resonances, due to trapping of particles in potential barriers, are ubiquitous in nature. First discovered by Fermi et al. when studying slow-neutron capture in artificial radioactivity (1, 2), they have since been the focus of countless investigations in physics, chemistry, and biology. They play a crucial role in α-decay of radioactive nuclei (3), molecular fragmentation (4), rotational predissociation (5), electron detachment (6), ultracold collisions (7), low-energy electron scattering (8, 9), and photoionization (10, 11), to name a few. They are also thought to be at the origin of enhanced radiation damage of DNA and other biomolecules (12) and to play an important role in the stability of Bose-Einstein condensates (13). Shape resonances are usually associated to specific spectral features. For instance, they lead to broad peaks in the photoionization spectrum of atoms and molecules (14–17) and to a strong variation in the corresponding photoelectron angular distribution as a function of kinetic energy (18).
The energies at which these resonances are expected to appear and the corresponding trapping times (or conversely, decay lifetimes) are entirely determined by the shape of the barrier seen by the trapped (or ejected) particle, hence the name “shape resonances.” Thus, the analysis of the resonance peaks observed in experimental spectra can be used to infer the actual height and width of the potential barrier seen by the impinging or emitted particle. This can be unambiguously done in atomic systems. However, in molecules, the situation is more complicated, as nuclear motion may cause the potential felt by the electrons to change during a vibrational period, thus affecting the shape of the barrier. This was theoretically predicted back in 1979 by Dehmer, who argued that the anomalous variations of vibrationally resolved photoelectron angular distributions of N2 with photon energy could be the consequence of changes in the shape of the potential barrier with the internuclear distance (10). This prediction has been the subject of debate for years (11). So far, direct time-resolved measurements of the dynamical evolution of shape resonances as the molecule vibrates have not been possible due to the lack of temporal resolution.
In this work, we investigate with attosecond time resolution the changes induced by the vibrational motion on the potential barrier that sustains the
(A) A comb of high-order harmonics (HH), spanning over the photon energy range of 20 to 40 eV, probes the entire shape resonance region in
RESULTS AND DISCUSSION
The RABBIT technique has been widely used to determine accurate photoionization delays in atoms (20, 21) and, following the pioneering work of Haessler et al. (22), has started to be applied to molecular systems as well (23–25). Here, N2 is ionized by a comb of odd high-order harmonics, covering the
In our experiment (see Materials and Methods for details), a near-infrared (NIR) 45-fs laser pulse generated high-order harmonics in argon, corresponding to a train of attosecond pulses in the time domain (26). The harmonics and a weak replica of the NIR pulse (probe) were focused into an effusive gas jet containing N2 molecules. The ejected electrons were detected by a magnetic bottle electron spectrometer with a resolution up to E/ΔE ∼ 80. To use the best possible resolution and hence resolve the vibrational levels in the first two outer valence states of the
We perform simulations to explicitly obtain the vibrationally resolved PES resulting from the interaction of an isolated N2 molecule with an XUV attosecond pulse train and a time-delayed NIR pulse, so that the extracted time delays can be directly compared with experiment. The time-dependent Schrödinger equation is numerically solved including the bound-bound, bound-continuum, and continuum-continuum dipole transition matrix elements between the electronic states. These are computed using the static exchange density functional theory method described in (27). The nuclear motion is taken into account within the Born-Oppenheimer approximation (see the “Theoretical Methods” section in the Supplementary Materials for a detailed explanation), and the laser parameters are chosen to reproduce the experimental conditions.
The measured and calculated PES, obtained with XUV and NIR, exhibit sidebands originating from the interference between two quantum paths: the absorption of a harmonic and an NIR photon and the absorption of the next harmonic and the stimulated emission of an NIR photon (see Fig. 1A). Consequently, the amplitude of the sidebands, ASB, oscillates as a function of τ according to the formula (19)
Figure 2 (A and B) shows the experimental and theoretical data, respectively, corresponding to the difference between the XUV + NIR and XUV-only PES (in color), which oscillates with frequency 2ω0 as a function of τ. Both theory and experiment are in excellent agreement (for details about the theoretical method, see the Supplementary Materials). Figure 2C presents the XUV-only (violet) and XUV + NIR (black) PES obtained by integrating over all delays. Good agreement with the calculated spectra shown in Fig. 2D can be noticed. The difference between theory and experiment in the relative intensities of some of the photoelectron peaks is due to the different position of the shape resonance predicted by theory (see Fig. 3B). In addition, the harmonic comb used in the theoretical calculations was slightly different from the experimental one. Figure 2E presents individual contributions from the X (blue) and A (red) states in the theoretical XUV + NIR PES, which allows us to assign the different features of the experimental spectra. The structures between 14.5 and 16 eV, for example, are due to ionization to the A state by absorption of the 21st harmonic leaving the
Difference between PES obtained with XUV + NIR and XUV only, as a function of delay. Experiment (A) and theory (B). Experimental (C) and theoretical (D) PES for XUV-only (violet) and XUV + NIR (black) photoionization, averaged over all relative delays. (E) Theoretical PES for XUV + NIR photoionization to the X (blue) and A (red) electronic states. By comparing (A) and (E), we can assign the spectral features to different vibrational levels (shaded areas) of the X and A electronic states, as indicated by the vertical blue and red dashed lines, respectively.
(A) Differences in molecular time delays (τmol) between X and A states for v′ = 0. Red circles, experiment; black circles, theory. (B) Partial photoionization cross sections for the X state; open squares, synchrotron-based experiment (30, 31); solid line, theory (this work). The position of the resonance maxima is shifted by almost 6 eV (denoted as ↔) between theory and experiment. This shift is also observed in the relative time delay (A and C). (C) Relative time delay between the vibrational levels v′ = 1 and 0 for the X state. The strong photon energy dependence observed here vanishes completely if one neglects the nuclear motion (see fig. S4). (D) Same as (C), but for the A state.
To determine molecular two-photon ionization time delays τmol, we fitted the measured (Fig. 2A) sideband oscillations to Eq. 1. The same procedure is applied to extract the theoretical ones from the RABBIT spectra computed with a time-dependent numerical approach. Figure 3 (A, C, and D) shows experimental (red circles) and theoretical (black circles) relative time delays for different final states. Since the contribution from the ionizing radiation (τXUV) is the same for all the final states of
The relative molecular time delay for leaving
Figure 3 (C and D) shows the relative molecular time delays, τX(v′ = 1) − τX(v′ = 0) and τA(v′ = 1) − τA(v′ = 0), for the X and A electronic states, respectively. For the A state, the relative delay is very small and practically independent of photon energy, while for the X state, it varies markedly across the shape resonance. Once again, for the reasons described above, the theoretical curve is shifted down in energy with respect to the experimental one by ∼6 eV.
The variation of the molecular time delay differences between the X and A states observed in Fig. 3A is therefore mainly due to the variation of the time delay for the former, which can be attributed to the presence of the shape resonance. The time delay varies with energy because the time spent by the photoelectron in the metastable state before being ejected into the continuum also varies with energy. Since, for the A state, the electron does not have to go through any potential barrier, the corresponding time delay is much smaller than for the X state.
We now analyze the physical meaning of the results presented in Fig. 3 (C and D). In atomic systems, photoionization time delays obtained from RABBIT measurements can often be written as the sum of two contributions, τ1 + τcc. The first term is related to one-photon ionization by the XUV field. For a single or dominant ionization channel containing no sharp structures in the continuum (e.g., narrow Fano resonances), τ1 is given by the derivative of the scattering phase in that particular channel, the so-called Wigner delay (33, 34). The second term, τcc, is the additional time delay due to the continuum-continuum transitions induced by the NIR field (35, 36).
In the vicinity of the shape resonance, one-photon ionization leading to
Modulus (A and C) and phase (B and D) of the dominant terms
Figure 5A shows the absolute square of the product between the initial vibrational wave function, the transition matrix element for the X state (see Fig. 4A) at an electron kinetic energy of 8.2 eV, and the final vibrational wave function (see the Supplementary Materials for details). The initial and final vibrational wave functions correspond, respectively, to the v = 0 level of N2 in the ground electronic state and the v′ = 0 (black) and v′ = 1 (red) levels of
(A) Absolute square of the transition matrix element
Figure 5C shows the photoionization delays resulting from the one-photon dipole transition matrix elements calculated at the two abovementioned internuclear distances as a function of photon energy. The difference in internuclear distance leads to a noticeable shift in the position of the corresponding maxima of photoionization delays, in agreement with the difference in resonance energy discussed above and the positions of the maxima calculated using the Wentzel-Kramers-Brillouin (WKB) approximation, indicated by the black and red dots. In addition, the energy range where the photoionization delays vary substantially is slightly broader for v′ = 1 than for v′ = 0, a direct consequence of the shorter lifetime for v′ = 1. This can also be seen by the horizontal bars, representing the widths of resonance obtained with the WKB approximation. Both effects, due to the shape resonance, contribute to a variation of the time delay difference between the v′ = 1 and v′ = 0 states as indicated by the green curve. Such a simple model predicts the main features of the experimental results in Fig. 3C, particularly the change of sign of the relative photoionization delay at low photon energy and the maximum at approximately the resonance position. It is worth noting that the resonance lifetimes obtained from the WKB model, 139.6 and 163 as for the v′ = 1 and v′ = 0 levels, respectively, are much smaller than the corresponding vibrational period, which is of the order of 16 fs (see the Supplementary Materials for details). As a consequence, the nuclei barely move during the ionization process, thus supporting the above analysis.
CONCLUSION
In summary, we measured vibrationally resolved molecular photoionization time delays between the X and A electronic states in N2 across the
MATERIALS AND METHODS
Experimental methods
The output of a Ti:Sapphire laser system delivering NIR pulses around 800 nm with 5-mJ pulse energy at 1-kHz repetition rate was sent to an actively stabilized Mach-Zehnder–type interferometer. In the “pump” arm, the NIR pulses were focused into a gas cell containing argon atoms to produce a train of attosecond XUV pulses via high-order harmonic generation. A 200-nm-thick aluminum foil was used to filter out the copropagating NIR pulse. The bandwidth of the driving NIR pulse was kept around 50 nm, which ensured the generation of high-order harmonics having full width at half maximum of about 150 meV. This is notably smaller than the energy separation (267 meV) between the lowest vibrational levels (v′ = 1 and v′ = 0) of the X electronic state in
Data analysis
Because of the spectral congestion between ionization by XUV and XUV + NIR radiation to the three different states (X, A, and B) of the
For every sideband, a Fourier transform was performed to make sure that the sideband oscillation did not include frequency components higher than 2ω0. The uncertainty σX (σA) for each measurement of the molecular photoionization time delay τX (τA) was obtained from the fit of the RABBIT oscillation to a cosine function (see Eq. 1). The corresponding uncertainty on the relative time delay, τX − τA, can be expressed as
An identical procedure was used to calculate the uncertainties for the relative time delays between two vibrational levels of the same electronic state.
The final experimental values shown in Fig. 3 were obtained from a weighted average of the data points from several sets of measurements. For N measurements yielding N data points: k1, k2, …, kN with corresponding uncertainties: σ1, σ2, …, σN, the weighted average can be calculated as
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/31/eaba7762/DC1
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
REFERENCES AND NOTES
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