Frustrated tunnelling ionization during strong-field fragmentation of D₃⁺

In a recent letter, we reported the first observation of D₃⁺ dissociation and ionization in an intense laser pulse [29]. From those measurements we deduced that three-body breakup of D₃⁺ leading to D⁺ + D⁺ + D via reaction channel (2) results predominantly from ionization plus excitation to the repulsive first-excited potential energy surface (PES) of D₃²⁺. This conclusion was based on the fact that dissociation on this PES was kinematically consistent with our measurements. The kinetic energy released in this process is peaked at 10 eV with a high energy tail falling off rapidly above 15 eV as shown in figure 2(a). From the energies involved and the topology of the PES, we could conclude that D₃⁺ breaks up predominantly from an internuclear separation r ∼ 3.5 au, where r is the deuteron–deuteron distance as defined in figure 1(c). Moreover, double ionization to D⁺ + D⁺ + D⁺ via reaction (3) peaks at KER = 25 eV as seen in figure 2(b). By assuming direct Coulomb explosion via the repulsive 3/r (in atomic units) potential, this gives a similar r ∼ 3.2 au for breakup. This r value is roughly consistent with the approximate initial vibrational distribution of D₃⁺(∼2.5–3.0 au) predicted by Anicich and Futrell [59] (see figures 1(b) and (c)) and would thereby indicate only a little stretching of the D₃⁺ ion during dissociative single and double ionization with 7 fs pulses. While this is an informative result, as it seems to rule out charge-resonance enhanced ionization of D₃⁺ or D₃²⁺ at large r (r ∼ 7 au at 1064 nm) as suggested in [39], it is not altogether that surprising as direct ionization has been observed to dominate in other small molecules, particularly for few-cycle laser pulses (e.g. [20, 64]).

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What is unexpected is that for 40 fs pulses one observes an additional high-KER feature centred at 22 eV in the D⁺ + D⁺ + D breakup channel shown in figure 2(c). For 40 fs pulses, since there is more time for the D₃⁺ ion to stretch before being ionized to the repulsive D₃²⁺ potential, one would expect if anything that the KER distribution would be shifted to lower energy than for 7 fs pulses (see, for example, D₂⁺ experiments [65]). Indeed, we do detect a small shift in the lower KER D⁺ + D⁺ + D peak from 10 eV for 7 fs to 8 eV for 40 fs. Remarkably, however, the high-KER peak at 40 fs has almost the same energy as the Coulomb explosion peak (at 23 eV) in the D⁺ + D⁺ + D⁺ channel at 40 fs (figure 2(d)). On reflection, this is the first clue to the origin of the mysterious high-KER feature.

To probe the nature of this high-KER feature further, we considered its ellipticity dependence. This is plotted in figure 3 as the ratio of the integrated number of events above 15 eV to the number of events below 15 eV in the D⁺ + D⁺ + D channel for 40 fs pulses. For each ellipticity value, the laser intensity has been set to match the peak electric field used for linear polarization (with intensity 5 × 10¹⁵ W cm⁻²). For example, for circular polarization twice the laser power was used than for linear polarization. While the number of events for KER < 15 eV remained relatively unchanged with (see inset), for KER > 15 eV the number diminished, resulting in the decrease of the ratio in figure 3 with increasing . As was discussed in the introduction, this decrease is strong evidence for some form of electron recollision process being responsible for the high-KER feature.

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At first, it is reasonable to consider the high-KER as arising from excitation induced by electron recollision. That is, when D₃⁺ is ionized to D₃²⁺ the ejected electron returns to recollide with the parent D₃²⁺ core. Upon colliding, it induces excitation to a highly excited repulsive state of D₃²⁺ that then decays to energetic D⁺ + D⁺ + D fragments. However, we consider this mechanism unlikely on the basis that at the intensity studied (10¹⁶ W cm⁻²) the scattering cross-section for this should be extremely small, for the following reason: it is well documented, for both atoms and molecules (including polyatomics [66, 67]), that electron recollision contributions are only prominent in the non-sequential ionization intensity regime (see e.g. figure 3 of [68]). Generally, when the laser intensity is sufficient to sequentially ionize an atom or molecule, electron recollision contributions to excitation or ionization are heavily out-weighed by the direct tunnel ionization of an electron in the strong laser field. For D₃⁺, an intensity of 10¹⁶ W cm⁻² is in the sequential double ionization regime as evidenced by the large D⁺ + D⁺ + D⁺ yield (cf figure 6(b) that shows that the D⁺ + D⁺ + D⁺ channel dominates over the D⁺ + D⁺ + D channel at intensities above 5 × 10¹⁵ W cm⁻² for 40 fs). Thus, any non-sequential recollision excitation contribution in the single ionization channel would be small and unable to explain the large magnitude of the high-KER D⁺ + D⁺ + D peak observed in figure 2(c).

Rather, previous experimental work for H₂ [23] and D₂ [24], simulations for D₃⁺ [35, 36], as well as evidence that we gather here, would suggest that the mechanism instead involves FTI. What is truly remarkable though, is the large amplitude of the process. At 10¹⁶ W cm⁻², 40 fs, the number of events in the D⁺ + D⁺ + D channel with KER above 15 eV is 29% larger than below 15 eV, and is as much as 13% of the D⁺ + D⁺ + D⁺ channel, showing that this certainly is not a minor process.

We expect that FTI in strong-field fragmentation of D₃⁺ proceeds as follows. When subjected to an intense laser, an electron is first ejected from D₃⁺ to form the intermediate D₃²⁺ complex, followed by the ejection of a second electron, forming the transient D³⁺₃ ion. The similarity of the KER distributions for the D⁺ + D⁺ + D⁺ channel at 7 and 40 fs (peaked at 25 and 23.5 eV, respectively) suggests that both electron ejection steps occur within a relatively short time interval of one another (≲3 fs from a classical estimate of the nuclear motion on the PES). Upon the temporary production of D⁺ + D⁺ + D⁺, the three deuterons mutually repel one another and start moving apart, the so-called 'Coulomb explosion'. At the same time, the two electrons that were born into the continuum still feel the influence of the strong laser field which drives their subsequent motion. Depending on the phase of the linearly polarized laser cycle at which they are released, they will either acquire enough momentum to escape the deuterons' attraction, or fail to gain sufficient momentum and stay in the vicinity of the deuterons. If the latter occurs, the deuteron–electron attraction can lead to an electron being captured to an orbital of the deuteron and remaining bound as an excited D* Rydberg fragment, i.e. overall the D₃⁺ ion undergoes single ionization accompanied by excitation of D₃²⁺, rather than double ionization. As the capture process (recombination) requires the conservation of energy and momentum, the general understanding of FTI is that the electron gradually exchanges its momentum with the laser field as it is gently decelerated over many cycles on the trailing edge of the pulse, before being attracted by the deuteron. In molecules, however, the excess energy and momentum could alternatively be exchanged with one of the other nuclei, if the nuclei are close enough.

It has been proposed that the FTI process can be modelled using classical trajectory calculations [23, 69, 70], as a full quantum-mechanical calculation is theoretically challenging due to the nuclear degrees of freedom and electron dynamics. Indeed, it has recently been demonstrated by Lötstedt et al [35, 36] that the laser-driven dynamics of D₃⁺ are depicted quite well by classical calculations. These classical calculations, however, are based on a Hamiltonian substantially modified with non-trivial momentum-dependent and many-body potentials to mimic essential quantum mechanical properties. Those authors even report signatures of the FTI mechanism in their simulations. Thus, to illustrate the FTI process for D₃⁺, we will also employ a classical model. Since our goal is merely qualitative, we use a simple approach with only the usual pairwise Coulomb interactions rather than the more sophisticated Hamiltonian of [35, 36]. However, we do note that the qualitative behaviour of our simulations is consistent with the more complete treatment in [35, 36].

In our simulations, initially the D⁺ nuclei are located on an x–y grid in the geometry of an equilateral triangle with a nuclei–nuclei spacing of r = 3.2 au and the electrons at the centre. The electrons are assumed to be ejected from the D₃⁺ molecule near the peak of a 40 fs, 790 nm, 10¹⁵ W cm⁻² pulse, with each electron released in consecutive half-cycles of the pulse. In the case of linearly polarized light, the polarization vector is along the x-coordinate (coinciding with one side of the triangle) and the electrons are released with a small lateral momentum (0.1 au), while for circularly polarized light the polarization is in the x–y plane of the molecule. Figure 4 displays example trajectories of the deuterons and electrons for one combination of electron release times, for linearly (panels (a) and (b)) and circularly (panels (c) and (d)) polarized light. For the linearly polarized case, in the plot of the x-coordinate of the particles as a function of time (figure 4(a)), one observes that after the rapid oscillations of the electrons due to the laser influence ceases, electron 1 (e₁) drifts free while electron 2 (e₂) is captured into a periodic orbit of one of the deuterons. The timescale for the capture event is large, more than one hundred femtoseconds, allowing the deuterons to spread apart. This is clearer when viewed in the x–y plane in figure 4(b), where evidently the captured electron follows a large elliptical orbit in what is effectively a Rydberg state of the atom. Thus, instead of both electrons escaping leading to double ionization, D⁺ + D⁺ + D⁺, the final products resulting from FTI are D⁺ + D⁺ + D. We note that, from our simulations, FTI is very sensitive to the timing of the electron release. That is, there is typically only a small window of electron release times near the peak of a laser cycle that leads to electron capture. This can be understood since electrons released at times significantly offset from the peak of a laser cycle generally acquire a large drift momentum in the direction of polarization that drives them far from the nuclei such that they cannot be captured. As a comparison, we show in figures 4(c) and (d) the same calculation but for circular polarization. In this case both electrons are driven far from the deuterons within the first few oscillations of the laser (on the order of 10 fs) and are never captured. This scenario occurs for effectively all electron release times using circular polarization. Hence, FTI is suppressed with circularly—or substantially elliptically—polarized laser pulses.

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If the production mechanism for the high-KER D⁺ + D⁺ + D fragments does involve FTI, then one would expect this channel to bear many of the signatures of the D⁺ + D⁺ + D⁺ channel in addition to an almost-matching KER spectrum, as observed for D₂ [24]. To determine whether our expectation is true, we look at the energy (or momentum) sharing of the KER between the different fragments. A convenient method of doing so for a three-body system is using a Dalitz plot [72], sometimes referred to as a ternary plot. Through the choice of axis scaling, a Dalitz plot maps the energy sharing of the fragments onto different positions within a circle as shown in figure 5(a), the edge of which is confined by momentum conservation. A detailed explanation of Dalitz plots can be found elsewhere [55, 71, 73, 74]. For the purposes of interpretation, an event that maps to the centre of the circle (denoted by the equilateral triangle) indicates that the three fragments share the final energy equally. In contrast, an event that maps to the edge of the circle (denoted by the linear line) indicates the breakup of the fragments along a linear axis. In the case of linear breakup, in the molecular frame one would observe one slow (or stationary) fragment along with two fast fragments (top-left, top-right, and bottom edges of the circle) or one fast fragment along with two slower fragments (top, bottom-left and bottom-right edges of the circle). Positions in between the circle centre and edge indicate some mixture of these configurations as denoted by the sketches.

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The interesting behaviour we are looking for is to compare the measured distributions of the D⁺ + D⁺ + D channel at 40 fs below 15 eV (figure 5(b)) and above 15 eV (figure 5(c)). At first sight the distributions are starkly different reaffirming the stance that the fragmentation mechanisms are different. If we compare the D⁺ + D⁺ + D channel below 15 eV at 40 fs with the 7 fs results (figure 5(d)), these distributions are similar. Indeed the fact that the distributions peak in the lower portion of the plot shows that on average the two D⁺ ions take away higher energy than the D fragment. In a simplistic picture one would expect this since the D⁺ ions feel additional Coulomb repulsion from one another that the neutral D atom does not experience. In contrast, the D⁺ + D⁺ + D⁺ channel at 40 fs in figure 5(e) (and 7 fs not shown) is predominantly peaked near the centre of the circle showing that all fragments share approximately equal energy. The D⁺ ions mutually repel one another, and since they begin as an equilateral triangle in the ground state configuration of D₃⁺, the final energy is about equally apportioned.

The remarkable likeness of the D⁺ + D⁺ + D KER > 15 eV channel in figure 5(c) to the D⁺ + D⁺ + D⁺ channel is compelling evidence in itself for FTI. The capturing of an electron by one D⁺ ion at a later time does not significantly affect the early repulsion of the three D⁺ ions; this is apparent in our classical simulations. Hence, although an electron is reattached to form an excited D* atom, the D⁺ + D⁺ + D channel already bears all the characteristics of the D⁺ + D⁺ + D⁺ channel—that is, similar high-KER and near-equal energy sharing of the fragments.

One question that still remains is why does the FTI mechanism seemingly not appear in the D⁺ + D⁺ + D channel at 7 fs (no distinct high-KER peak in figure 2(a)) despite being so prominent at 40 fs (figure 2(c)). In fact, our classical trajectory calculations would suggest that FTI is more probable at 7 fs as an electron is recaptured for more release phases. The answer to this apparent puzzle is found in figure 6 which shows the measured rates of the various channels for different intensities. At 7 fs the rate of the D⁺ + D⁺ + D⁺ channel is small. Thus, since FTI emanates from this channel it is not surprising that the FTI rate is also small. Coupled with the fact that the D⁺ + D⁺ + D rate (below 15 eV) is very large for 7 fs, the relative FTI contribution is barely visible in figure 2(a). On closer inspection, the inset of figure 2(a), which is an expanded vertical scale, shows that there are counts where FTI is expected to appear. Indeed the fraction of FTI events for 7 fs is ∼20% of the D⁺ + D⁺ + D⁺ events, higher than the 13% observed for 40 fs. The correlation in exhibited behaviour of the high-KER D⁺ + D⁺ + D events and the D⁺ + D⁺ + D⁺ events lends further credence to the conclusion that they are due to FTI as one expects FTI and D⁺ + D⁺ + D⁺ to be closely linked.

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Finally, one may question whether FTI is visible in the dissociation channels. To answer, we first examine two-body breakup, i.e. the D⁺ + D₂ and D + D₂⁺ channels. In these cases, D₃⁺ may be excited to the transient D₃²⁺ which then breaks via the two-body reaction path to D⁺ + D₂⁺. As the D⁺ and D₂⁺ repel apart, either fragment may capture the originally ejected electron, forming excited D* or D*₂. To look for evidence of this process we survey the KER distribution for each of these channels. Unfortunately for 40 fs there are insufficient statistics in our data for these channels to be conclusive. Not to be deterred, the equivalent data for 7 fs, 10¹⁶ W cm⁻² is plotted in figures 7(a)–(c). Noticeably, a small number of high KER fragments between 4 and 12 eV are visible for both D⁺ + D₂ and D + D₂⁺, reflecting the distribution of fragments for the D⁺ + D₂⁺ channel suggesting that FTI is also present in these channels (though a much smaller contribution in comparison with the D⁺ + D⁺ + D breakup at 40 fs (figure 2(c))).

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As before, the true test if the high-KER events are due to FTI is if they disappear using circular polarization. These data, at the same peak electric fields (linear 5 × 10¹⁵ W cm⁻², circular 10¹⁶ W cm⁻²), are shown in figures 7(d)–(f). The KER distributions for linearly and circularly polarized pulses are the same for the D⁺ + D₂⁺ channel, as expected, since no recollision process is involved for this channel. Tellingly, the high-KER peaks in the D⁺ + D₂ and D + D₂⁺ channels are absent for circular polarization, strongly supporting the supposition that they are from FTI.

Since FTI is present for two-body dissociation, it can also be expected for the three-body channel D⁺ + D + D. Despite the poorer statistics for this channel due to the low three-body dissociation rate (see [29, 32]), the KER distribution in figure 8 for linearly polarized 7 fs pulses displays an unmistakable peak at large KER around 10 eV that resembles the D⁺ + D⁺ + D channel (note the peak at lower KER around 2 eV is likely to be from regular dissociation pathways by multiphoton excitation). As indicated by the inset, the high-KER peak for D⁺ + D + D seems to be suppressed using circularly polarized pulses consistent with its origin arising from FTI.

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We speculated earlier on the possibility of electron-recollision excitation being responsible for the surprising high-KER peak in the D⁺ + D⁺ + D channel (instead of FTI) but excluded it on the basis that the intensity used is within the sequential tunnelling ionization regime (rather than the non-sequential recollision regime). The observation of equivalent high-KER peaks in the two- and three-body dissociation channels is further evidence to support this assertion. That is, any explanation based on electron-recollision excitation suggests an initial ionization step to create the recolliding electron wavepacket. Since the dissociation channels do not entail ionization then electron-recollision excitation must be ruled out for dissociation. It should be noted that such an argument is only valid assuming that the initial electron that tunnels out does not recombine upon recollision. If instead the tunnelling electron recombines upon recollision, while exciting a secondary electron to the Rydberg state, then the result could still lead to dissociation (without ionization).

Before summarizing, we reflect on the possible principal quantum number, n, of the Rydberg states produced by FTI. Simulations for He [22] suggest an n distribution peaked around n = 8 and dropping off rapidly for higher n. Experiments on N₂ [25] attempted to directly measure the n distribution and found that states with n > 45 are hardly populated, those with 23 < n < 45 make up over 50% of the population, and those with n < 23 the remainder. The authors also noted that atoms with n ≲ 15 were likely to have decayed in their experiments and may not have been recorded. In our experiments, we did not attempt to measure the n distribution. Nevertheless, it is important to bear in mind that in our spectrometer setup we applied a static electric field of strength 400 V cm⁻¹ for both the 7 and 40 fs experiments—sufficient to field ionize high-lying Rydberg states. From the saddle point field strength relation, $F_{\mathrm {sad}} = \frac {1}{16n^4}$, relating the applied field strength to the lowest n state that can be field-ionized [25], we estimate that only neutral fragments with n ≲ 30 would have survived without being field-ionized by our spectrometer. This number is a little higher, n ≲ 35, if instead one uses the diabatic field ionization formula, $F = \frac {1}{9n^4}$ [25]. Our experiments are, however, fortunate not to suffer from a limitation on the lowest n state that may be detected as the neutral fragments are moving with high energy in the laboratory frame. Thus, even atoms or molecules that have decayed to lower n before reaching the detector will also be recorded.