Effect of nuclear mass on carrier- envelope-phase-controlled electron localization in disassociating molecules

We explore the effect of nuclear mass on the laser-driven electron localization process. We dissociate a mixed H2/D2 target with intense, carrier-envelope-phase (CEP) stable 6 fs laser pulses and detect the products in a reaction microscope. We observe a very strong CEP-dependent asymmetry in proton/deuteron emission for low dissociation energy channels. This asymmetry is stronger for H2 than for D2. We also observe a large CEP offset between the asymmetry spectra for H2 and D2. Our theoretical simulations, based on a one-dimensional two-channel model, agree very well with the asymmetry spectra, but fail to account properly for the phase difference between the two isotopes. Coherent control of molecular fragmentation with few-cycle laser pulses of well-defined carrierenvelope phase (CEP) has become an active research topic in ultrafast science due to its potential application for control of chemical reactions. In particular, molecular hydrogen (H2) and its heavy isotope deuterium (D2) have attracted a lot of attentions [1-7], since these simple molecules can help us understand how more complicated larger molecules interact with strong laser field. Even for simple molecules, complete ab initio treatment is still a daunting task and approximate models continue to be commonly used. It is therefore important to explore validity and limitations of those approximate models by comparing their predictions with experimental observations. Any such quantitative comparison would usually require a precise knowledge of laser parameters, and must also take into account temporal and spatial variations of those parameters over the interaction region. Studying the isotope dependence allows one to keep all the parameters constant while focusing on a singular effect of nuclear mass. Here we report the first such experimental comparative study of CEP-dependent dissociation in H2 and D2. The CEP control of dissociative ionization of D2 with 5 fs 1×10 W/cm laser pulses was first demonstrated by Kling et al [1]. That study reported a significant (~20%) CEP-dependent asymmetry for a channel with high kinetic energy release (KER). Their result was attributed to the recollision excitation from the bound ground 1sσg (gerade) state directly to the repulsive excited 2pσu (ungerade) state of D2, shortly after the first ionization of D2. However, no significant asymmetry was observed by Kling and co-workers in the dominant low-KER region, which corresponds to radiative excitation/deexcitation channels. In another experiment with H2, Kremer et al [2] reported a pronounced asymmetry modulation of ~15% for low-KER channels with significantly higher laser intensity (4×10 W/cm). In our own experiments on H2 with even more intense pulses (6×10 W/cm), we observed the highest CEP-dependent asymmetry reported to date, 40%, as well as modulation of up to 5% in dissociation yield [3]. More recently, studies of CEP-dependent dissociation using an H2 ion beam have also been reported [8, 9]. These experiments used 4.5 fs [8] and 5 fs [9] pulses of 4×10 W/cm peak intensity and also measured significant asymmetry, up to 30% for [8], in the low-KER channels. None of those experimental studies makes a comparison between the two isotopes of hydrogen. H2 and D2 remain the only targets for which full dimensionality ab initio theoretical modelling of laser-induced dissociation is currently feasible, however these are only applicable within the Born-Oppenheimer approximation [10] and for sufficiently low intensities avoiding ionization [9]. In general, low-KER ions are produced via radiative excitation/deexcitation by absorbing/emitting a certain number of photons from the driving laser field at large inter-nuclear separations. Figure 1 shows the evolution on the molecular potential and the major dissociation channels, which were identified some time ago using multi-cycle laser pulses. Low-KER dissociation occurs through bond softening (BS, net one-photon process [11]) and above-threshold dissociation (ATD, net two-photon process [12]). Some later experimental [13] and theoretical [14] work indicated that a net three-photon process (TPD) might also be relevant to hydrogen dissociation. According to the general theory of CEP effects [15], CEP-dependent asymmetries can be interpreted in terms of interfering quantum pathways with different (by odd number mostly Δn = 1) number of absorbed photons. The relevant dissociation pathways are shown in figure 1(a). To observe the interference, the KER spectra for the interfering channels must overlap, so that broadband (few-cycle) pulses are required for CEPcontrolled asymmetry. Optimal asymmetry modulations will be achieved when the two interfering pathways have similar transition amplitudes. That condition can only be satisfied at intensities of a few ×10 W/cm, an order of magnitude higher than in the original experiment by Kling and coworkers [1]. Nuclear mass also plays an important role in the observation of CEP-dependent asymmetries. Since we start with a neutral molecule in its ground state, the ionization creates a dynamic nuclear wavepacket propagating on the ground state potential energy surface of molecular ion. That molecular ion will resonantly absorb (and sometimes also emit) photons from the field later during the laser pulse, as the nuclear wavepacket continues its propagation towards dissociation along various pathways. The timing and probability of those resonant radiative transitions will sensitively depend on wavepacket velocity, which in turn depends on nuclear mass. For example, the nuclei in H2 move faster and will reach the coupling region sooner than the heavier D2 nuclei, so they experience higher intensities in the falling edge of the laser pulse. It is therefore expected that CEP control of electron localization will also be very sensitive to nuclear mass. A pioneering theoretical work [16] has discussed such mass effect in dissociation of isotopes of hydrogen ions, while no supporting experimental evidence has been reported so far. The dissociation of various isotopes of hydrogen molecule has been studied experimentally in previous works [1,2,17], but a direct comparison of those experiments is not possible due to the influence of different laser parameters (intensity, pulse duration, CEP and focusing geometry). Here, we present an experiment performed with a H2/D2 mixed gas target interacting with a CEP-stabilized few-cycle laser pulses. By properly selecting the laser intensity, a strong CEP-dependent asymmetry for both H2 and D2 is observed. In a mixed gas jet, the H2 and D2 molecules will experience exactly the same laser parameters. Hence, the measured CEP dependence for H2 and D2 is suitable for quantitative study of the role of nuclear mass in the CEP-controlled molecular fragmentation which is free of systematic errors. We focus our attention on low-KER (< 4 eV) fragments: at the intensities used in our experiment, more energetic fragments are dominated by double ionization, which obviously cannot display any asymmetry. Our experimental apparatus is shown in Figure 1 (b), and its detailed description can be found in our previous paper [3]. The pulse duration of the few-cycle laser is measured to be ~6 fs (FWHM of intensity profile). The CEP of the pulse is stabilized by using a feedback loop based on f-2f interferometer (Menlo Systems), and is manipulated by rotating a piece of fused silica plate around Brewster’s angle. The laser beam is focused with a 75 mm focal length spherical mirror onto a supersonic mixed-gas jet, which contains 50% H2 and 50% D2. A very small amount (<1%) of HD molecules is also observed in the experiment, but its influence on the measurement could be neglected. The width of the mixed-gas jet is localized along the laser propagation direction to <100 μm by an adjustable slit, which is much smaller than the Rayleigh length (~700 μm) of the laser beam. This helps us minimize the influence of the Gouy phase effect and focal volume averaging effect, yielding higher asymmetry modulation. Before entering the REMI chamber, the laser beam is truncated by an adjustable iris to tune the laser intensity in the interaction region. A quarter-wave-plate is employed to switch the polarization of the laser between linear and circular without affecting the pulse duration. Circularly polarized pulses are used to ionize a neon jet, and the laser peak intensity is calibrated in situ by measuring the momentum distribution of Ne ions [18, 19]. The electric field of the linearly polarized pulse is parallel to the time-of-flight axis of REMI and perpendicular to both laser beam and gas jet propagation directions. The H and D fragments are steered by a static electric field, which is high enough to avoid overlapping of the time-of-flight spectra of H and D, towards a timeand position-sensitive detector, where their 3-dimensional momentum vectors are determined. The experimental KER spectra as well as KERand CEP-dependent asymmetry are shown in Figure 2 (a, b, e, f). As the strongest coupling occurs for molecules parallel to the laser polarization, H/D emitted into a small angle ( ) around the laser polarization are selected for analysis. For those ions emitted towards the detector (up), < 30° is selected, while for ions emitted away from the detector (down), > 150° is selected. The KERand CEP-dependent asymmetry parameter is defined as , , , (1), where and are the ion yield in the up and down directions respectively, is the CEP averaged total ion yield in both directions. As predicted in ref [16], can be expressed as a sinusoidal function of CEP with a phase offset, when only interference of neighbouring (Δn = 1) photon channels is important. However, the bandwidth of 6 fs pulse is broad enough to support interference between n and n+2 photon channels, which will lead to a π-periodic modulation of as a function of CEP. We observed such a modulation at the 5% level previously [3]. Consequently, the asymmetry parameter defined in a traditional way as / will not be a simple sine function of CEP. To correct for this, we use CEP averaged instead of so that the asymmetry parameter can be described by , · sin (3), where and are the KER-resolved amplitude and phase shift parameter of the asymmetry modulation respectively. The measured CEP-dependent asymmetry for H2 and D2 is then fitted by equation (3) to retrieve the A and B parameters, which are presented in Figure 3 (b) and (c). We compare our experimental results with theoretical calculations based on a one-dimensional twochannel approximation to the time-dependent Schrödinger equation (TDSE). Figure 2 shows the results of these calculations. We model the first ionization step, which creates the molecular ion, by projecting the neutral ground-state nuclear wave packet onto the 1sσg state of the molecular ion (Franck-Condon approximation). In the Born-Oppenheimer approximation, the wave function of the molecular ion can be expressed as , , , , , , , where R is the internuclear separation, x denotes the electronic coordinate, , and , are the two lowest electronic states with opposite parities, 1sσg and 2pσu, with potential energy curves for these states given by Vg(R) and Vu(R). The corresponding nuclear wave functions are denoted , and , . We now solve the following two-channel TDSE equation [2, 4], with initial wave packets created at each local maximum of the few-cycle pulse, and their relative probabilities are weighted according to the ADK [20] ionization rate: , , , , , , (2), where , is the reduced mass, , is the dipole coupling. In our simulation, the spatial and temporal steps are ∆R = 0.04 a.u. and ∆t = 1 a.u. The spatial grid covers the range R 0 to 500 a. u.. In this model, we assume that the molecular axis is parallel to the laser, and ignore molecular rotations. Focal volume averaging and Gouy phase variation are also ignored in the calculation. Since the thickness of the molecular beam in the experiment is much smaller than the Rayleigh range of the laser focus, those effects should not significantly affect the results. As seen from Figure 2, the asymmetry results for H2 and D2 are qualitatively similar, with two distinct energy regions characterized by different modulation depths with opposite tilting of the stripes in the asymmetry spectra. For H2, the two energy regions are KER < 1.3 eV and KER > 1.3 eV, which are assigned as corresponding to BS/ATD interference and ATD/TPD interference [3], respectively. For D2, the two energy regions are KER < 1.8 eV and KER > 1.8 eV, with the boundary shifted by 0.5 eV to higher energy. That energy shift is reproduced by the model calculation of this work. We note that similar energy shifts in hydrogen molecular ion and its heavier isotopes have also been predicted theoretically in [16]. In figure 2 (e, f), a KER window is selected for each energy region, within which the asymmetry curves are calculated and presented. For addressing the observed 0.5 eV shift, the KER windows for D2 (see figure 2 (f)) are shifted towards higher energy by 0.5 eV comparing to those for H2 (see figure 2 (e)). The asymmetry modulation depth for H2 in the ATD/TPD region could be as high as 50%, while for D2 a somewhat lower 40% modulation is observed. The calculated asymmetry (figure 2 (c, d)) agrees very well with experimental results for both molecules. Comparison of the experimental data for H and D reveals three main differences between the two isotopes. (i) The total dissociation yield is much higher (almost twice for BS and even more for ATD and TPD) for the light isotope (figure 3(a)). That is likely due to the fact that in H2 the nuclear wavepacket reaches the strong-coupling region faster than in D2 and experiences higher intensities and higher radiative transition rates. (ii) The asymmetry modulation amplitude is somewhat larger for the light isotope, particularly for KER > 2 eV corresponding to ATD/TPD interference (figure 3(b)). That is due to higher relative population of the TPD pathway in H2, which also can be explained by its faster expansion into the radiative coupling region. (iii) The CEP-dependent asymmetry does not maximize at the same CEP for the two isotopes – there is a substantial phase shift in both experimental and theoretical asymmetry plots (figure 3(c, d)). This phase shift is not unexpected – the phase will accumulate at different rates as different isotopes propagate along different pathways. The phase difference between the two interfering pathways will determine the relative phase of the resulting CEP dependence. KER dependence of the phase (manifested as a tilt in the 2D asymmetry plots) for the same two-channel interference has similar origin. It is noteworthy that the theory fails to predict the experimentally observed phase shift. The phase shift of ~1 rad is measured for KER between 2 and 4 eV (figure 3(c)), while the theory predicts a much larger 3 rad phase difference between H2 and D2 (figure 3(d)). Obviously, our simple one-dimensional model fails to properly account for the phase. The discrepancy could be due to neglect of rotations (much different for the two isotopes) and non-Born-Oppenheim terms (more important for H2) or due to the omitted interference of the nuclear wavepacket generated in different optical cycles. Regardless of the reasons, a note of caution is due here. Looking at the good agreement between theory and experiment for each of the two isotopes taken separately (figure 2), one might be tempted to use that comparison for an absolute CEP calibration. However, side-by-side comparison of the two isotopes immediately reveals that the theoretical model does not reproduce the phase correctly and therefore cannot be used for such absolute CEP measurement. More generally, one has to be particularly careful when trying to extract absolute CEP information from ad hoc simplified models. To conclude, we compare the CEP dependence of the dissociation of H2 and its heavy isotope D2 by intense CEP-stabilized few-cycle laser field under identical experimental conditions. We report the strongest CEP-dependent asymmetry measured to date for either isotope – 50% for H2 and 40% for D2. Quantitative comparison of the two isotopes reveals significant mass effects, which can be generally explained by faster motion of lighter nuclei in H2. While the overall agreement between the experiment and theoretical predictions is very good, our one-dimensional two-channel BornOppenheimer model fails to reproduce the phase difference between the two isotopes. A more realistic model will be required to account for that measurement.