Sub-femtosecond tracing of molecular dynamics during strong-field interaction

TL;DR

This paper presents a novel experimental method for simultaneously measuring nuclear and electronic dynamics in molecules during strong-field interactions with sub-femtosecond resolution, providing new insights into early molecular processes.

Contribution

The authors introduce an experimental technique that captures both nuclear and electronic motions in molecules during strong-field ionization with unprecedented temporal and spatial resolution.

Findings

Direct observation of nuclear wavepacket evolution on H2+

Simultaneous measurement of electronic and nuclear dynamics

Attosecond temporal and picometer spatial resolution achieved

Abstract

We introduce and experimentally demonstrate a method, where the two intrinsic time scales of a molecule, the slow nuclear motion and the fast electronic motion, are simultaneously measured in a photo-electron photo-ion coincidence experiment. In our experiment, elliptically polarized, 750~nm, 4.5~fs laser pulses were focused to an intensity of $9\times10^{14}\mathrm{W/cm}^2$ onto H$_2$. Using coincidence imaging, we directly observe the nuclear wavepacket evolving on the \ssg{} state of H$_2^+$ during its first roundtrip with attosecond temporal and picometer spatial resolution. The demonstrated method should enable insight into the first few femtoseconds of the vibronic dynamics of ionization-induced unimolecular reactions of larger molecules.

Full text

Molecular fragmentation and isomerization processes are of fundamental importance in nature. These nuclear rearrangement processes are initiated and ultimately determined by electronic dynamics that can be influenced by precisely timed distortions of the electronic structure with the electric field of strong, ultrashort laser pulses Kling et al. (2013); Alnaser and Litvinyuk (2017); Xie et al. (2012, 2014). For example, fine variations of the delay between successive ionization events can determine the number of moieties produced during fragmentation of polyatomic molecules Xie et al. (2015). To reveal the dynamics underlying these processes, it is necessary to apply probing techniques that are sensitive to both, the fast electronic dynamics that may take place on attosecond time scales and the slower nuclear motion taking place on the femtosecond time scale. While even the fastest vibrational wavepacket in H${}_{2}^{+}$ has been probed successfully shortly after its creation using a pump-probe scheme with near-single-cycle pulses Ergler et al. (2006); Xu et al. (2015), it is, however, a major obstacle to trace the molecular dynamics during the first few femtoseconds of the interaction of a molecule with a strong laser pulse.

In this Letter, we introduce and experimentally demonstrate a method that is capable of tracing simultaneously electronic and vibrational wavepacket dynamics in a fragmenting molecule on sub-femtosecond times. The method combines the sub-cycle sensitivity inherent to angular streaking that has been applied to studying ionization dynamics in atoms (Maharjan et al., 2005; Eckle et al., 2008a; Pfeiffer et al., 2011; Schöffler et al., 2016; Wustelt et al., 2017) and molecules Wu et al. (2012a, b, 2013); Gong et al. (2014), with the high structural sensitivity of Coulomb explosion imaging Stapelfeldt et al. (1998); Niikura et al. (2003); Légaré et al. (2003, 2005); Erattupuzha et al. (2017) to resolve vibrational motion. Our work is inspired by earlier attempts Staudte (2005) of constructing such a method that has been called The Molecular Clock.

We investigate the applicability of this concept by tracing the nuclear and electronic dynamics in $\text{H}_{2}^{+}$ triggered by the emission of an electron over several femtoseconds. Upon ionization of $\text{H}_{2}^{+}$ various scenarios can take place Ibrahim et al. (2018). The simplest one is that a vibrational wavepacket is created on the $1s\sigma_{g}$ energy level following the Franck-Condon (FC) principle Niikura et al. (2002), see Fig. 1(a). Even for this simplest of all cases it has been shown that the very fast nuclear motion taking place during the ionization event can lead to deviations from the population distribution of vibrational states predicted by the FC principle Urbain et al. (2004). In another scenario, shake-up excitations can lead to the population of vibrational states not only on the $1s\sigma_{g}$ but also on the $2p\sigma_{u}$ level Litvinyuk et al. (2005). In recent photoelectron holography measurements evidence for subcycle population transfer to $2p\sigma_{u}$ was found Haertelt et al. (2016). As the vibrational dynamics proceeds after ionization in the presence of the strong laser field, three- or five-photon resonant excitations, can lead to partial population transfer between the $1s\sigma_{g}$ and $2p\sigma_{u}$ levels Kling et al. (2013); Alnaser and Litvinyuk (2017). We show that we can achieve temporal and spatial resolutions of a few tens of attoseconds and about 1 pm – sufficient to disentangle these different scenarios.

To obtain attosecond temporal resolution we employ angular streaking (Maharjan et al., 2005; Eckle et al., 2008a) illustrated in Fig. 1(b). Angular streaking exploits the direct mapping of electron momentum to the instantaneous laser electric field at the time of ionization. In elliptically polarized light the electric field vector $\textbf{E}(t)$ completes a full rotation within one period of the field, $T=2\pi/\omega$, where $\omega$ is the frequency of the light. This rotation serves as the minute hand of a clock: The ionization phase within a laser cycle $\varphi=\omega t_{i}$ is mapped into the emission angle of the photoelectron via the relation $\textbf{p}_{e}=-\textbf{A}(t_{i})$, valid within the strong-field approximation Faisal (1973); Reiss (1980), where the laser vector potential $\textbf{A}(t)$ is connected to the laser electric field by $\textbf{A}(t)=-\int_{-\infty}^{t}\textbf{E}(t^{\prime})dt^{\prime}$. Thus, measurement of the electron emission angle in the laboratory frame determines the ionization time $t_{i}$ within one cycle. In addition, the magnitude of the emitted electron’s momentum vector $|\textbf{p}_{e}|$ is proportional to the instantaneous field strength and thereby provides a measure about the ionization time within the pulse envelope Pfeiffer et al. (2011).

The hand of the clock on the vibronic time scale is the kinetic energy release (KER) of the protons following double ionization. It has been shown that the kinetic energy released during Coulomb explosion of a molecule into two ionic fragments is a sensitive measure of the distance $R$ where the repulsive Coulombic potential is populated Vager et al. (1989); Stapelfeldt et al. (1998); Chelkowski et al. (1999); Légaré et al. (2005); Katsuki et al. (2006). In H2 the kinetic energy released during Coulomb explosion is $\text{KER}=\frac{1}{2m_{p}}\left(\textbf{p}_{p1}^{2}+\textbf{p}_{p2}^{2}\right)$, where $\textbf{p}_{p1}$ and $\textbf{p}_{p2}$ are the proton momenta and $m_{p}$ is the proton mass. The mapping of internuclear distance $R=\frac{1}{\text{KER}}$ to kinetic energy release is very precise for H2 Niikura et al. (2003); Weber et al. (2004). The first ionization event at time $t_{1}$ initiates a H${}_{2}^{+}$ nuclear wavepacket on the $1s\sigma_{g}$ ground state. The wavepacket propagates on the light-induced potential energy surfaces and is projected onto the Coulombic potential energy curve by the second ionization event at time $t_{2}=t_{1}+\Delta t$, see Fig. 1(a). Measuring the momenta of the two protons provides us with the internuclear distance $R$ at which the second ionization step happened. Hence, if the two ionization events are confined to within the first roundtrip of the nuclear wavepacket on the $1s\sigma_{g}$ potential we can establish a correlation between KER and $\Delta t$.

For the experiment we used reaction microscopy Dörner et al. (2000). We measured the three-dimensional momentum vectors of two protons in coincidence with two electrons emerging from the interaction of H2 molecules with elliptically polarized pulses with a broad spectrum centered around 750 nm (oscillation period $T=2.5$ fs), a full-width at half maximum (FWHM) duration of 4.5 fs in intensity and a peak intensity of $9\text{\times}{10}^{14}\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$. The experimental apparatus consists of a two-stage arrangement to provide an internally cold ultrasonic gas jet of hydrogen, and an interaction chamber with an ultra-high vacuum ($1.3\times 10^{-10}$mbar). Electrons and ions were guided by weak magnetic (12 G) and electric (21 V/cm) fields along the spectrometer axis ($z$-direction) to two position and time sensitive multi-hit detectors. Further details on the reaction microscope can be found in Refs. Xie et al. (2012); Zhang et al. (2014); Xie et al. (2017) and on the optical setup in Ref. Schöffler et al. (2016) as well as in Supplemental Material SMH .

Fig. 1(c) shows the proton momentum distribution in the laser polarization plane for the double ionization pathway. The anisotropy of the proton distribution indicates the alignment of the polarization ellipse in the laboratory frame. In elliptical light, ionization preferentially takes place twice during the optical cycle Eckle et al. (2008b). Hence, sequential double ionization in elliptical light occurs with a delay between the two ionization steps of $\Delta t=nT/2;n=0,1,2,\ldots$. To calibrate the zero of our time-scale we turn to the KER, shown in Fig. 1(d). For instantaneous double ionization, i.e., $\Delta t=0$, the H2 groundstate wavefunction would be projected vertically onto the Coulomb repulsion curve, to yield a KER distribution centered at 19 eV with a FWHM of 2 eV Weber et al. (2004). Thus, based on Fig. 1(d) we can exclude instantaneous double ionization in our experiment.

The observed KER distribution from 10 eV to 18 eV can be related to a timescale, assuming nuclear wavepacket propagation on a given potential energy surface. To this end, we computed the nuclear-stretch motion in between the ionization times $t_{1}$ and $t_{2}=t_{1}+\Delta t$ by solving Newton’s equations on the $1s\sigma_{g}$ energy curve Sharp (1970); see Supplemental Material SMH for details on the simulations. The simulation shows that our experiment covers a range of ionization delays starting from about $\Delta t=0.5T$ ($\textrm{KER}\approx 18$ eV) to roughly $\Delta t=3T$ ($\textrm{KER}\approx 10$ eV), with a maximum of the double ionization probability around $\Delta t=1.5T$ ($\textrm{KER}\approx 13$ eV). Smaller $\Delta t$-values correspond to smaller internuclear distances, $R$, where the ionization potential is greatly increased and double ionization probability is accordingly suppressed, see Fig. 1(a). Double ionization delays of $\Delta t\lesssim 0.5T$ require both the increase of the pulse peak intensity and a pulse duration even shorter than the 4.5-fs pulses used here (Supplemental Material SMH ).

The energy of each photoelectron, shown in Fig. 1(e), labels the light intensity at the instant of ionization. The peak at 40 eV corresponds to an ionization intensity of about $3.8\times 10^{14}\mathrm{W/cm}^{2}$. Therefore, the majority of double ionization events does not occur at the peak of the pulse, in agreement with the prediction of the double ionization delay based on the KER. Sub-cycle sensitivity, finally, is obtained from the relative angle $\alpha=\text{ang}\left(\textbf{p}_{e1},\textbf{p}_{e2}\right)$ between the two photoelectron momenta $\textbf{p}_{e1}$ and $\textbf{p}_{e2}$, shown in Fig. 1(f). By virtue of the relation $\textbf{p}_{e1,e2}=-\textbf{A}(t_{1,2})$ the relative angle $\alpha\in[0^{\circ},180^{\circ}]$ directly measures $\Delta t$ within one laser half-cycle. The minimum at $\alpha\approx 90^{\circ}$ can be attributed to the ellipticity of the pulse. The relative emission angle peaks at $\alpha=180^{\circ}$, thus indicating a preferential double ionization at a delay of half-integer optical cycles, consistent with the $\Delta t=1.5T$ deduced from the maximum of the KER distribution.

However, the estimates based on the isolated observables in Fig. 1(d-f) do not yield a true sub-cycle resolution of the coupled electronic and nuclear dynamics. To obtain this, we will in the following examine the relation between KER, photoelectron emission angle and energy to demonstrate how the sub-cycle dynamics of the nuclear wavepacket can be accessed by correlating these three observables. In Fig. 2(a) we show the correlation between the two main hands of the molecular clock: the relative emission angle $\alpha$ of the photoelectrons and the KER of the protons. A previously indistinguishable maximum in the double ion yield at a KER of 14.5 eV is associated with a relative emission angle of zero degrees, i.e., the parallel emission of the two photoelectrons. The main maximum at a KER of 12.6 eV is slightly lower than in Fig. 1(d) and associated with anti-parallel electron emission. Finally, a third peak at a KER of 11.5 eV and parallel emission is weakly distinguishable.

Thus, by correlating $\alpha$ with KER, we obtain a powerful parametric framework for representing the coupled electron and nuclear dynamics taking place in between the two ionization steps. For example, if the $2p\sigma_{u}$ energy level is populated by shake-up during the first ionization step or also at later times by resonant transitions, the corresponding parametric curves in the $\alpha$-KER frame of reference will look distinctively different from the one corresponding to vibrational motion on the $1s\sigma_{g}$ level; see Supplemental Material SMH for a comparison of curves obtained for different scenarios. Therefore, the ionization-fragmentation dynamics can be studied by comparison of the measured fragmentation yield to a simulated curve for a given scenario, e.g., for motion only on the $1s\sigma_{g}$ energy level.

Further insight into the molecular dynamics within the duration of the pulse can be obtained when the magnitudes of the photoelectron momenta $|\textbf{p}_{e1,e2}|$ are analyzed. $|\textbf{p}_{e1,e2}|$ provide an additional time reference linked to the fast rise time of a few-cycle pulse’s envelope Pfeiffer et al. (2011). Gating on $|\textbf{p}_{e1,e2}|$, thus allows selecting a range of ionization delays $\Delta t$. As a result, it becomes possible to obtain sub-cycle traces of the molecular dynamics for certain ranges of delays $\Delta t$ in the two-electron emission and, moreover, as we will show further below, even attosecond snapshots of the propagating vibrational wavepacket.

For the purpose of the following analysis we show the energies of both photoelectrons in Fig. 2(b). The highest double ionization probability is found for similar momenta of both electrons. Hence, the highest double ionization yield is from events where both electrons are emitted symmetrically around the peak of the pulse envelope. For example, the region A1 indicated in the correlated energy spectrum Fig. 2(b) corresponds to situations where both electrons are emitted within $\Delta t=1.2T-1.7T$. Region A2 at smaller energies corresponds to emissions with longer delay, $\Delta t=2.3T-3.2T$.

The capabilities opened up by the selection of the emission time-window based on $|\textbf{p}_{e1,e2}^{2}/2|$ are demonstrated in Figs. 2(c,d). In these figures we show the measured distribution of the fragmentation yield in the $\alpha$-KER plane for the two regions of electron energies A1 and A2 in Fig. 2(b). Accordingly, the corresponding ranges of the yield distribution in the $\alpha$-KER plane in Figs. 2(c) and (d) show the vibrational wavepacket evolution approximately in the time ranges of $1.2T-1.7T$ and $2.3T-3.2T$ after the first ionization step, respectively. A movie obtained for arbitrary selections of $|\textbf{p}_{e1,e2}|$ is available as Supplemental Material SMH . The measured distributions in Figs. 2(c,d) are compared to the simulated curve obtained for wavepacket motion on the $1s\sigma_{g}$ energy level (described in Supplemental Material SMH ). The measured distributions show no signs of shake-up excitation that would appear for KER$>19$ eV. Likewise, they show negligible yield in the range KER$<10$ eV that corresponds to resonant excitation to the $2p\sigma_{u}$ energy curve. Overall, the measured distributions agree well with the simulated curve for nuclear motion on the $1s\sigma_{g}$ curve, although there are some small deviations visible for the smaller emission delays $\Delta t$, cf. Fig. 2(c).

As $\alpha=0-180^{\circ}$ corresponds to one laser half cycle, by limiting $\alpha$ to small intervals, attosecond snapshots of the KER distribution can be extracted from these traces. By virtue of the Coulomb law, KER$=1/R$ can be converted to internuclear distance $R$. Thus, it becomes possible to obtain snapshots of the absolute value of the vibrational wavepacket, $|\chi(R,\Delta t)|^{2}$, for very short time-intervals around certain values of $\Delta t$.

To obtain $|\chi|^{2}$ we select slices in the $\alpha$-KER distribution for $\alpha=[0^{\circ},20^{\circ}]$ and $\alpha=[160^{\circ},180^{\circ}]$ corresponding to intervals of $\Delta t$ of about 140 as. The yield distributions obtained by these selections are plotted in Fig. 3(a,b) as a function of $R$ together with Gaussian fits. The Gaussian fits agree well with yield distributions obtained by more sophisticated selections (detailed in Supplemental Material SMH ) that also involve gating on $|\textbf{p}_{e1,e2}|$ as described above, see Fig. 3(c). The four distributions in Fig. 3(c) constitute snapshots of $|\chi(R,\Delta t)|^{2}$ with an uncertainty of about $\pm 70$ as around $\Delta t=0.5T,1T,1.5T,2T$ and with a spatial resolution of about 0.02 a.u. (about 1 pm).

The equilibrium internuclear distance of H2 is 1.4 a.u. Fig. 3(c) shows that after half a laser cycle ($\Delta t=1.25$ fs) the H${}_{2}^{+}$ nuclear wavepacket has propagated in distance by about 0.2 a.u. For larger values of $\Delta t$, our measurement reveals how the wavepacket progressively moves to larger internuclear distances and spreads in space. Compared with previous experiments that probed the nuclear motion of H${}_{2}^{+}$ after strong-field laser ionization Niikura et al. (2003), our measurement is, to the best of our knowledge, the first one to reveal not only the position but also the shape of the wavepacket. Additionally, our measurement probes the wavepacket on a finer grid of $\Delta t$ and with significantly higher temporal resolution (i.e., uncertainty in $\Delta t$).

In conclusion, we introduced a new method, that allows tracing molecular dynamics on sub-femtosecond times during strong-field interaction following field-ionization. The method exploits the rotation of the electric field vector of elliptically polarized light as an attosecond temporal reference, and the dependence of the ion fragment energy on the molecular geometry as a clock for nuclear motion. Although demonstrated here for H2, for which the nuclear clock shows a $1/R$ dependence, this is not a pre-requisite of the method; any monotonic dependence of fragment energy on nuclear geometry is suitable. We therefore expect a wide applicability of the presented method and envision that it can also be applied to reasonably fast dissociative few-particle fragmentation channels in polyatomic molecules, as many of them fulfill this requirement. We highlight the high temporal resolution and spatial sensitivity of the method by demonstrating tracing of the vibrational wavepacket evolution in H${}_{2}^{+}$ with a temporal uncertainty of about $\pm 70$ as and a spatial resolution of about 1 pm. The resolution of this technique can be expected to be further improved by exploitation of the carrier-envelope phase of few-cycle laser pulses Schöffler et al. (2016).