A molecular clock for autoionization decay

Ionization of the H₂ molecule after absorption of a photon with an energy of ∼30 eV proceeds via two paths (see figure 1). The first ('direct') channel corresponds to leaving a vibrationally excited ${{\rm{H}}}_{2}^{+}$ ion in the ground electronic ${\sigma }_{{g}}$ state, starting the 'molecular clock'. The second ('resonant') channel corresponds to populating the Q₁ series of doubly-excited states accesible by one-photon absorption. These states spontaneously decay into the electronic continuum associated to the ground electronic state of ${{\rm{H}}}_{2}^{+}$ by emitting a photoelectron, through the process of AI. The interference of the 'direct' and the 'resonant' ionization channels records their relative phases by mapping them into amplitude modulations of the population of vibrational state in the ${{\rm{H}}}_{2}^{+}$ electronic ground state as a function of the absorbed energy [22, 23, 27, 34].

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Continuum resonances—states that decay by AI—are present in all multielectronic atoms and molecules. In atoms, they lead to the well known Fano resonances [35]. In molecules, the picture can be largely modified by the coupling between electronic and vibrational degrees of freedom [36, 37], resulting even in the dissapearance of the Fano-like profiles in the total photoionization yield. An extreme case is the H₂ molecule. The lifetime of its lowest resonant state, that is also most largely populated by one-photon absorption, is of the order of a few fs (1–3 fs). However, the light mass of the nuclei confers a rapid vibrational dynamics on a comparable time-scale. Consequently, in the H₂ molecule, the ultrafast electronic decay is strongly coupled to the vibrational dynamics of the molecule [27, 38]. A reconstruction procedure that retrieves both phase and amplitude of the 'resonant' wave packet would yield complete information about its corresponding time evolution. Such reconstruction is possible via a time–energy mapping between the AI decay and the vibrational dynamics in the final ${{\rm{H}}}_{2}^{+}$ ion, which is illustrated in figure 1. The decay from the Q₁ band of autoionizing states populates different vibrational levels of the ${{\rm{H}}}_{2}^{+}$ ion at different time-delays between excitation (XUV pump) and autoionization (probe) [38]. The longer the pump-probe delay, the higher is the energy of the populated vibrational states. The time–energy mapping is similar to the one used in the attosecond streak-camera [3, 39] although here the streaking is due to the accelerating motion of the nuclei.

In the following, we demonstrate how the AI decay can be fully reconstructed in time from the vibrationally resolved photoionization spectra of the H₂ molecule, if the 'direct' ionization channel is known or, at least, can be approximated from the experimental data. This reconstruction renders a simple qualitative picture of the time–energy mapping, where each final vibrational state ν provides a temporal attosecond gate for the AI decay. In order to validate the reconstruction procedure, we use the simulated data produced by nearly-exact calculations whose reliability has been amply verified in previous works by comparing them with experimental data [40]. Moreover, the reconstruction procedure presented in this work is general and does not rely on any assumptions or approximations about the underlying dynamics, which are usually performed en route to calculate the dynamics of the 'resonant' states in molecules.

In the basic formalism of one-photon ionization close to a resonant state, we consider a molecule in its ground state $| {{\rm{\Psi }}}_{g}\rangle$ that is excited by a laser field to a manifold of excited states $| {\rm{\Phi }}\rangle$ through the interaction potential V(t). We assume that $| {{\rm{\Psi }}}_{g}\rangle$ is orthogonal to $| {\rm{\Phi }}\rangle$ by construction. Within the first order of perturbation theory, the excited wavefunction $| {\rm{\Phi }}(t)\rangle$ is (atomic units are used throughout)

$$\begin{eqnarray}&&| {\rm{\Phi }}(t)\rangle =-{\imath }{\int }_{-\infty }^{t}{\rm{d}}t^{\prime} \,{{\rm{e}}}^{-{\imath }\hat{H}(t-t^{\prime} )}\,\hat{V}(t^{\prime} )\,| {{\rm{\Psi }}}_{g}\rangle {{\rm{e}}}^{-{\imath }{E}_{g}t^{\prime} }.\end{eqnarray} \tag{ 1 }$$

Consider an observable that projects $| {\rm{\Phi }}(t\to \infty )\rangle$ onto the final state $| \nu \rangle$, which is stationary, non-decaying and has energy $\langle \nu | \hat{H}| \nu \rangle ={E}_{\nu }$. We can further define a state vector $| q\rangle$ that contains all the remaining states of the system, such that $| q\rangle \langle q| +| \nu \rangle \langle \nu | =1$. Thus, we divide the system into a subspace that includes the final stationary state $| \nu \rangle$, and a subspace that includes all the intermediate states $| q\rangle$. These states include resonances that are assumed to irreversibly decay into $| \nu \rangle$ as $t\to \infty$. Hence, the dynamics in each subspace is non-hermitian. The interaction between the two subspaces is governed by $\langle \nu | \hat{H}| q\rangle ={\hat{H}}_{\nu q}$, whose contribution to $| \nu \rangle$ is assumed to vanish as $t\to \infty$, allowing to define $| \nu \rangle$ as the final state after ionization. The outcome of such measurement is

$$\begin{eqnarray}\langle \nu | {\rm{\Phi }}(t)\rangle & = & -{\imath }{\displaystyle \int }_{-\infty }^{t}{\rm{d}}t^{\prime} \,{{\rm{e}}}^{-{\imath }{E}_{g}t^{\prime} }\times ({{\rm{e}}}^{-{\imath }{E}_{\nu }(t-t^{\prime} )}\langle \nu | \hat{V}(t^{\prime} )| {{\rm{\Psi }}}_{g}\rangle \\ & & +{{\rm{e}}}^{-{\imath }{\hat{H}}_{\nu q}(t-t^{\prime} )}\langle q| \hat{V}(t^{\prime} )| {{\rm{\Psi }}}_{g}\rangle ).\end{eqnarray} \tag{ 2 }$$

The approach could be generalized to more complicated cases, although the reconstruction would be more involved. In the case of two distinct intermediate subspaces $| q\rangle$ and $| r\rangle$, it would be enough to separate the $| q\rangle$ and $| r\rangle$ from the $| \nu \rangle$ manifold. Defining the laser field $\hat{V}(t)$ via its Fourier transformation $\hat{V}(t)={\int }_{-\infty }^{\infty }{\rm{d}}{\rm{\Omega }}f({\rm{\Omega }})\,\exp (-{\imath }{\rm{\Omega }}t)\,\hat{D}$, where $f({\rm{\Omega }})$ is the spectral density of the pulse and $\hat{D}$ is the dipole operator and introducing the new time variable $\tau =t-t^{\prime}$ the expression (2)  above becomes

$$\begin{eqnarray}\langle \nu | {\rm{\Phi }}(t)\rangle & = & -{\imath }{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}{\rm{\Omega }}\,f({\rm{\Omega }})\,{{\rm{e}}}^{-{\imath }({\rm{\Omega }}+{E}_{g})t}\,{\displaystyle \int }_{0}^{\infty }{\rm{d}}\tau \,{{\rm{e}}}^{{\imath }({\rm{\Omega }}+{E}_{g})\tau }\\ & & \times ({{\rm{e}}}^{-{\imath }{E}_{\nu }\tau }\langle \nu | \hat{D}| {{\rm{\Psi }}}_{g}\rangle +{{\rm{e}}}^{-{\imath }{\hat{H}}_{\nu q}\tau }\langle q| \hat{D}| {{\rm{\Psi }}}_{g}\rangle )\\ & = & -{\imath }{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}{\rm{\Omega }}\,f({\rm{\Omega }})\,{{\rm{e}}}^{-{\imath }({\rm{\Omega }}+{E}_{g})t}\\ & & \times (\langle \nu | \hat{D}| {{\rm{\Psi }}}_{g}\rangle \delta ({\rm{\Omega }}+{E}_{g}-{E}_{\nu })\\ & & +{\displaystyle \int }_{0}^{\infty }{\rm{d}}\tau \,{{\rm{e}}}^{{\imath }({\rm{\Omega }}+{E}_{g})\tau }\,{{\rm{e}}}^{-{\imath }{\hat{H}}_{\nu q}\tau }\langle q| \hat{D}| {{\rm{\Psi }}}_{g}\rangle ).\end{eqnarray} \tag{ 3 }$$

The resulting amplitude is thus described by a sum of two terms—a 'direct' contribution (first term in the brackets), which is the 1st order interaction with the laser, and a 'resonant' contribution (second term in the brackets), which is mediated by the non-stationary dynamics of the system, e.g. decay of resonances and transient states. These two terms determine the density of probability ${S}_{\nu }(\epsilon )$ of an energy-resolved measurement, e.g., the energy resolved spectra:

$$\begin{eqnarray}{S}_{\nu }(\epsilon ) & = & | { \mathcal F }{ \mathcal T }[\langle \nu | {\rm{\Phi }}(t)\rangle ]{| }^{2}\\ & = & | f(\epsilon -{E}_{g}){| }^{2}\times | {D}_{\nu }(\epsilon )+Q(\epsilon ){| }^{2}\end{eqnarray} \tag{ 4 }$$

with

$$\begin{eqnarray}&&{D}_{\nu }(\epsilon )=\langle \nu | \hat{D}| {{\rm{\Psi }}}_{g}\rangle \,\delta ({E}_{\nu }-\epsilon )\end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray}&&{Q}_{\nu }(\epsilon )={\int }_{0}^{\infty }{\rm{d}}\tau \,{{\rm{e}}}^{{\imath }\epsilon \tau }\,{Q}_{\nu }(\tau ),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{Q}_{\nu }(\tau )={{\rm{e}}}^{-{\imath }{\hat{H}}_{\nu q}\tau }\langle q| \hat{D}| {{\rm{\Psi }}}_{g}\rangle =\langle \nu | {{\rm{e}}}^{-{\imath }\hat{H}\tau }| q\rangle \langle q| \hat{D}| {{\rm{\Psi }}}_{g}\rangle ,\end{eqnarray} \tag{ 7 }$$

where ${ \mathcal F }{ \mathcal T }$ denotes the Fourier transformation.

The full spectrum ${S}_{\nu }(\epsilon )$ is the result of the interference between the 'direct' and the 'resonant' ionization paths. Hence, if both the spectrum $S(\epsilon )$ and the 'direct' ionization paths are measurable or predictable, e.g., from a theoretical calculation, the 'resonant' contribution can be obtained using a procedure similar to heterodyne detection. In the case of single photoionization of H₂, and following equation (4), the full vibrationally resolved photoionization spectrum ${{\rm{S}}}_{\nu }(\epsilon )$ can be expressed as

$$\begin{eqnarray}{S}_{\nu }(\epsilon ) & = & {| {D}_{\nu }(\epsilon )+{Q}_{\nu }(\epsilon )| }^{2}\\ & = & | {D}_{\nu }(\epsilon ){| }^{2}+| {Q}_{\nu }(\epsilon ){| }^{2}+2\,\mathrm{Re}[{D}_{v}(\epsilon )\ Q(\epsilon )],\end{eqnarray} \tag{ 8 }$$

where the corresponding ionization dipole elements are ${D}_{\nu }(\epsilon )$ for the 'direct' and ${Q}_{\nu }(\epsilon )$ for the'resonant' paths, is the final photoelectron energy and ν is the final vibrational state label. Here and in the following, a monochromatic laser pulse is considered, for which $f({\rm{\Omega }})=\delta ({\rm{\Omega }})$ with $\epsilon ={\rm{\Omega }}-{E}_{g}$. Nevertheless, all the results can be straightforwardly generalized to pulses of finite duration by using the spectral density of the pulse $f({\rm{\Omega }})$, for example

$$\begin{eqnarray}&&{\tilde{Q}}_{\nu }(t)={ \mathcal F }{ \mathcal T }[f({\rm{\Omega }}=\epsilon -{E}_{g})\times {Q}_{\nu }(\epsilon )],\end{eqnarray} \tag{ 9 }$$

which is exact for single photon processes. Also, pulses of finite duration will be considered in section 3.3. We further assume that $| {Q}_{\nu }(\epsilon ){| }^{2}/| {D}_{\nu }(\epsilon ){| }^{2}\ll 1$, i.e., that the 'resonant' ionization channel is weaker than the 'direct' channel. This condition is usually satisfied in the case of doubly excited states, and is also satisfied in the case of H₂ ionization considered here—the two-electron excitation upon one-photon absorption is coming purely from the electron correlation terms and therefore are appreciably smaller than the promotion of an electron into the continuum associated to the ground state of the cation. In practice, the reconstruction procedure outlined below works well even when the strength of the 'direct' and 'resonant' channels are comparable. Furthermore, $\mathrm{Im}[{D}_{\nu }(\epsilon )]\approx 0$ is assumed, i.e., that the scattering phase of 'direct' photoionization is negligible. Indeed, the scattering phase is large when multi-electron/multi-channel effects are involved, and small for single-electron ionization, described by the ${D}_{\nu }(\epsilon )$ ionization dipole. Using the assumptions above, the terms in equation (8) can be rearranged to obtain an explicit expression for the 'resonant' ionization path:

$$\begin{eqnarray}&&\mathrm{Re}[{Q}_{\nu }^{(0)}(\epsilon )]=\displaystyle \frac{{S}_{\nu }(\epsilon )-| {D}_{\nu }(\epsilon ){| }^{2}}{2\,\mathrm{Re}[{D}_{\nu }(\epsilon )]}.\end{eqnarray} \tag{ 10 }$$

Although equation (10) allows one to obtain only the real part of the 'resonant' term, it still contains the full information about the decay. This is the case, because in the equation (6) the time integral is performed from 0 to $\infty$ and not from $-\infty$ to $\infty$. Thus, the full time-dependent AI decay ${Q}_{\nu }^{(0)}(t)$ into the final state ν can be recovered by performing a Fourier transformation, as is evident from equation (6). We will demonstrate next that the quantity ${Q}_{\nu }(t)$ that is recovered corresponds to the rate of population transfer from the state Q₁ to state ν.

Since the $| {Q}_{\nu }(\epsilon ){| }^{2}/| {D}_{\nu }(\epsilon ){| }^{2}$ term was neglected in equation (8), the reconstructed 'resonant' contribution ${Q}_{\nu }^{(0)}(\epsilon )$ from equation (10) is only approximate. It can be used to estimate the approximate spectra ${S}^{(0)}=| {D}_{\nu }(\epsilon )+{Q}_{\nu }^{(0)}(\epsilon ){| }^{2}$. By inserting the difference ${\rm{\Delta }}S(\epsilon )=S(\epsilon )-{S}^{(0)}(\epsilon )$ into equation (10), instead of $S(\epsilon )$, a correction to the 'resonant' term ${\rm{\Delta }}{Q}_{\nu }^{(0)}(\epsilon )$ is obtained. It can then be used to estimate a higher order ${Q}_{\nu }^{(1)}(\epsilon )={\rm{\Delta }}{Q}_{\nu }^{(0)}(\epsilon )+{Q}_{\nu }^{(0)}(\epsilon )$. Recursively repeating this procedure, an increasingly more accurate solution ${Q}_{\nu }^{(n)}(\epsilon )$ is obtained. We found accurate results for the 3rd order solution.

The vibrationally resolved photoionization spectra of H₂ are presented in figure 2. The spectra were obtained by solving the full dimensional time-dependent Schrödinger equation (TDSE) [40] and employing a 400 as long XUV pulse with frequency centered around 30 eV and a laser intensity of 10⁹ W cm⁻². For the laser field parameters used here, photoionization takes place within the perturbative regime, where we can define a cross section that is independent of the pulse parameters (length, intensity, etc). The spectra shown in figure 2 are the normalized probabilities to the spectral intensity of the laser pulse as described in [41] to obtain the laser-field independent ionization probability, i.e. and effective vibrationally resolved photoionization cross section. Although an attosecond pulse was used in the calculation, an analogous spectra would be obtained using a time-independent calculation as in previous works asuming monochromatic radiation [27]. In an experiment, such correlated spectra could be obtained by, e.g., measuring the photoelectrons and ${{\rm{H}}}_{2}^{+}$ fragments in coincidence after ionization by an attosecond pulse [42]. However, the present scenario would require to additionally measure the vibrational state of the ${{\rm{H}}}_{2}^{+}$. Alternatively, analogous spectra could be obtained by using a monochromatic laser beam and measuring the photoionization spectra at different laser frequencies, as it was done in [43].

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The methodology employed to obtain a numerical solution of the TDSE for the H₂ molecule is described in detail in [40, 44–46] and will not be presented here. In short, it uses a full expansion of the wavefunction into products of electronic and vibrational Born–Oppenheimer states of H₂. It allows to obtain 'full' spectrum including the effects of vibrational motion and the AI decay. Additionally, the method allows one to selectively truncate the simulation to a given number of desired states or to suppress the coupling between selected eigenstates. It is thus straightforward to remove the Q₁ state manifold such that one obtains the spectrum of a structureless 'direct' photoionization process, i.e. the incoherent background of the 'direct' ionization channel without the 'resonant' contribution. This calculation was used as the 'reference' photoionization channel in the reconstruction. In the abscence of accurate calculation, it could be also approximately extracted from an experimental measurement by extrapolating the data in the energy regions where only direct photoionization is expected.

The shape of vibrationally resolved photoionization spectra can be explained in terms of semi-classical dynamics. It was previously studied in [37] using semi-classical methods for the dissociative ionization channel, with experimental realization in [47]. More recently, it was extended to the photoionization process associated to the bound states of the ion [38]. There, it was shown that the modulations of the photoionization spectra result from the phase accumulated by vibrational wavepacket moving on the Q₁ electronic surface, which is in turn directly related to its classical momentum. Once the vibrational wavepacket is created on the Q₁ electronic surface it starts gaining momenta and therefore preferentially decays into higher ν vibrational states that are characterized by a higher momentum value. Hence, higher ν states are populated at larger delayed times.

We illustrate the reconstruction procedure selecting three vibrational states ν = 0, 6 and 12 that represent low, middle and high energy states from the vibrational progression in the ${{\rm{H}}}_{2}^{+}$ ground state, although it works equally well for any final (bound) vibrational states. The 'full' photoionization spectra for the selected states are shown in figure 3, together with the 'direct' spectra, which were obtained by removing the Q₁ state from the basis of the ab initio calculation. By following the recursive procedure described in the previuos section, we obtain the time-dependent decay rates ${Q}_{\nu }(t)$. The quantity ${{Q}}_{\nu }(t)$, as defined in equation (6) describing the 'resonant' ionization channel, thus corresponds to the rate of population transfer from the Q₁ state to each of the $\nu \leqslant 18$ final states. These rates are plotted as a function of time in figure 4 for the ν = 0, 6 and 12 final vibrational states. The reconstructed AI decay rates reveal that each final vibrational state ν is dominantly populated at a different time ${\tau }_{\nu }$ after ionization. The larger the vibrational number ν, the longer the delayed time. Therefore, each final vibrational state acts as a temporal gate, capturing a 'time-frame' of the decay of the Q₁ doubly excited state. The time resolution achieved between different states ν is just a fraction of a femtosecond.

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The integral of the corresponding decay rates ${{Q}}_{\nu }(t)$ over time yields the total population that decays into each final vibrational state ${P}_{\nu }(t)$. By adding-up these partial populations ${P}_{\nu }(t)$, the total population P(t) of the bound part of the ${{\rm{H}}}_{2}^{+}$ ground state can be retrieved:

$$\begin{eqnarray}&&P(t)=\displaystyle \sum _{\nu }{P}_{\nu }(t)=\displaystyle \sum _{\nu }{\int }^{t}{\rm{d}}\tau \ | {Q}_{\nu }(\tau ){| }^{2}.\end{eqnarray} \tag{ 11 }$$

Note, however, that the integral is performed over the modulus of the decay rates ${{Q}}_{\nu }(t)$. This is justified, since for a given final energy each final vibrational state constitutes an ionization continuum associated to a unique photoelectron state. Therefore, an incoherent integral over the decay time can be taken to extract the total final population.

The accumulated population in the progression of the bound vibrational states (${\sum }_{\nu =0}^{\nu =\nu }{P}_{\nu }(t)$) of the ${{\rm{H}}}_{2}^{+}$ ground state over time is plotted in figure 5(a), including each individual contribution with a different solid color. The increase of population is entirely due to the decay of the lowest Q₁ ${}^{1}{{\rm{\Sigma }}}_{u}^{+}$ resonant state and therefore directly reveals the lifetime of this Q₁ resonance. The 'direct' ionization of the ${{\rm{H}}}_{2}^{+}$ molecule is removed by the reconstruction procedure. The ${{\rm{H}}}_{2}^{+}$ ground state is the only open decay channel for the Q₁ series of ${}^{1}{{\rm{\Sigma }}}_{u}^{+}$ symmetry states, and the lowest state of this series is by far the most dominant autoionization channel.

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We now compare the reconstruction results with the outcome of the ab initio calculation. Since the dependence on the laser pulse is removed during the reconstruction, the reconstructed dynamics corresponds to an instantaneous excitation of the Q₁ state. On the other hand, the ab initio calculation is performed in the time domain, using a finite duration laser pulse. Hence, the finite duration of the laser pulse has to be accounted for in the reconstruction to compare it with the ab initio calculation. This is done by multiplying the reconstructed energy-dependent decay dipoles ${Q}_{\nu }(\epsilon )$ by the spectral representation of the laser pulse $f({\rm{\Omega }})$. In the time domain this corresponds to a convolution with the laser electric field. The reconstructed decay where the finite pulse duration was taken into account is presented in figure 5(b) (full lines).

In order to obtain the ab initio final state populations due to the resonant ionization path alone, additional truncated ab initio calculations were performed. In these calculations, the non-stationary evolution of Q₁ state coupled to the background continua was fully included, but the transition dipole matrix elements between the ground state of H₂ and the background continua associated to the ground state of ${{\rm{H}}}_{2}^{+}$ were set to zero, thus removing the 'direct' ionization channel. In this way, the final vibrational states ν of ${{\rm{H}}}_{2}^{+}$ molecule were populated only by the 'resonant' ionization channel via the Q₁ state, capturing its time-dependent population decay rate.

The population in each final vibrational state ν over time obtained using the truncated ab initio calculation, i.e. removing the contribution of the direct ionization, are plotted in figure 5(b) (dotted lines). Overall, for most of the final vibrational states they show a good agreement with reconstructed populations that account for a finite pulse duration (figure 5(b) full lines). The total population resulting from the extracted rates, equation (11), exhibits an excellent agreement with the results of the ab initio method, thus validating the reconstruction procedure.