Attosecond Delays of High-Harmonic Emissions from Hydrogen Isotopes Measured by XUV Interferometer

High-harmonic spectroscopy can access structural and dynamical information on molecular systems encoded in the amplitude and phase of high-harmonic generation (HHG) signals. However, measurement of the harmonic phase is a daunting task. Here, we present a precise measurement of HHG phase di ﬀ erence between two isotopes of molecular hydrogen using the advanced extreme-ultraviolet (XUV) Gouy phase interferometer. The measured phase di ﬀ erence is about 200 mrad, corresponding to ~ 3 attoseconds ( 1 as = 10 − 18 s ) time delay which is nearly independent of harmonic order. The measurements agree very well with numerical calculations of a four-dimensional time-dependent Schödinger equation. Numerical simulations also reveal the e ﬀ ects of molecular orientation and intramolecular two-center interference on the measured phase di ﬀ erence. This technique opens a new avenue for measuring the phase of harmonic emission for di ﬀ erent atoms and molecules. Together with isomeric or isotopic comparisons, it also enables the observation of subtle e ﬀ ects of molecular structures and nuclear motion on electron dynamics in strong laser ﬁ elds. the two jet positions and the intrinsic phase di ﬀ erence between the two species. By measuring HHG yields for di ﬀ erent gas con ﬁ gurations and jet separations, these two contributions can be disentangled. The XUV radiation from the ﬁ rst jet has negligible e ﬀ ect on the phase of HHG emission in the second jet as was demonstrated in [23, 24] by measuring dependence of HHG yield on separation between the jets containing the same gas.


Introduction
The high-harmonic generation process is a sensitive probe of molecular dynamics and structures [1][2][3][4]. The hydrogen molecule, being the lightest, exhibits the fastest nuclear motion. Also, being the simplest neutral molecule, it allows accurate ab initio quantum mechanical simulations without resorting to severe approximations. Those simulations can be used to understand and validate the experimental results. Moreover, availability of different isotopes adds isotopic comparison as a valuable benchmarking and validation tool. A number of studies with such comparisons have already been performed, including effects of nuclear dynamics on relative HHG yields [5,6] and tunnel ionization rates in H 2 and D 2 [7]. The strong field ionization of H 2 launches a free electron with H + 2 in the 1 sσ g state. The electron wave packet driven by the laser field accelerates away from the parent ion. At the same time, the nuclear wave packet evolves on the potential energy surface of H + 2 (see Figure 1(a)). After a certain time delay, the electron wave packet returns to the parent ion upon the reversal of the driving laser field. Its consequent recombination induces an oscillating dipole, which imprints its amplitude and phase on the emitted HHG photons [8]. This ultrafast HHG process occurs in less than half of a laser oscillation period giving the potential to explore sensitive measurement on molecular electron structure and nuclear dynamics known as high-harmonic spectroscopy [9][10][11][12][13].
The investigation of the nuclear dynamics of hydrogen isotopes was envisioned theoretically [14][15][16] and implemented experimentally [5,6,17] by comparing their high-harmonic yields. However, only two studies explored the isotope effects on the high-harmonic phase [18,19]. The first one estimated the relative phase by measuring harmonic yields in a mixed gas cell with H 2 and D 2 [19]. The second study relied on the determination of group delay by using the reconstruction of attosecond beating by interference of two-photon transitions (RABITT) [18]. However, ensuring that both isotopes are at the same number density in the HHG interaction region is very challenging. Additionally, a sign ambiguity [19] and large experimental uncertainties of these measurements made it difficult to determine the phase difference and absolute delays accurately [18,19]. Neither of those studies actually reproduced the measured phase difference theoretically.
To retrieve the phase information, the high-harmonic interferometers either split the driving laser beam into two paths and focus at different locations within a single gas jet that produce two phase-locked HHG photons [20,21] or use a single driving laser beam in a gas mixture [2,19]. The first method is limited to a single gas species, while the second has low resolution and issues with determining the sign of the relative harmonic phase. Recently, the phase difference between two atomic species was measured by an all-optical attosecond interferometric technique [22], where the delay between two extreme ultraviolet (XUV) pulses is controlled by a two-segment mirror which can provide the temporal resolution of~6 attoseconds. Such an interferometer must maintain subcycle stability of its path difference (displacement between the two mirror segments) for XUV frequencies for the duration of the experiment, while this path difference is being scanned. This is an exceedingly difficult task which severely limits practical utility of the device.
Building an interferometer in the XUV region is quite challenging for two reasons: firstly, it is challenging to control the delay of the XUV pulses precisely between the two arms with subcycle precision; secondly, the highly reflective XUV optics is yet to be developed. Our passively stabilized Gouy phase interferometer [23,24], on the other hand, is an all-optical direct XUV interferometric technique. It does not require calibration of gas pressures to ensure the same number densities. Additionally, it does not require any XUV optics. The technique provides an elegant way to generate two coherent highharmonic pulses without splitting the driving laser and XUV beams. The two mutually coherent XUV pulses are generated by exploiting the inherent properties (Gouy phase) of a single Gaussian focused laser beam. This method has an unprecedented resolution of~300 μrad (~100 zeptoseconds) because unlike other interferometric techniques, it requires optical path length stability on the order of Rayleigh length of the driving infrared laser rather than the XUV wavelength.
We apply the technique to investigate the effect of nuclear dynamics on the electron motion in molecular hydrogen by precise measurement of high-harmonic phase difference (and corresponding HHG phase delays) produced in H 2 and D 2 . Since the ionization potentials of H 2 (I p = 15:43 eV) and D 2 (I p = 15:46 eV) are almost identical [5], the difference in the phase accumulated by electron in the continuum is considered to be negligible. However, due to the nuclear mass difference, the evolution of nuclear wave packet while electron propagates in the continuum and then recombines to the ground state may differ substantially. The harmonic intensity from the heavy isotope was shown to be  Figure 1: Three-step model of HHG with the potential energy surfaces of H 2 and H + 2 , and schematics of the Gouy phase interferometric technique. (a) Tunneling ionization launches an electron wave packet together with a nuclear vibrational wave packet. The ionized electron is driven by the laser electric field, and it returns to the parent ion at a certain recollision time. Simultaneously, the nuclear wave packet evolves on the potential energy surface 1 sσ g of the ground state of H + 2 . Upon electron recombination, a photon is emitted with an amplitude and phase dependent on the correlated electron-nuclear dynamics. (b) Two gas jets are placed sequentially with a separation Δz in a single laser focus. Two coherent high-harmonic pulses are generated simultaneously from the top (H 2 ) and bottom (D 2 ) gas jets. A phase difference between the two pulses includes contributions from the Gouy phase shift of the driving laser field at the two jet positions and the intrinsic phase difference between the two species. By measuring HHG yields for different gas configurations and jet separations, these two contributions can be disentangled. The XUV radiation from the first jet has negligible effect on the phase of HHG emission in the second jet as was demonstrated in [23,24] by measuring dependence of HHG yield on separation between the jets containing the same gas. 2 Ultrafast Science higher compared to the lighter molecule as harmonic emission is sensitive to the nuclear motion [5] though for D 2 , the ionization probability is less than for H 2 [7]. The aim of this work is to measure a small phase difference of HHG signals and to gain an insight into the correlated electron-nuclear dynamics for the two isotopes of molecular hydrogen.

Experimental Methods
The high harmonics are generated by a linearly polarized, 800 nm, 9 fs pulses with a peak intensity of~5 × 10 14 W/ cm 2 . To extract the relative phase difference between two different gases, our advanced Gouy phase interferometer consists of two spatially separated gas jets along the propagation direction in a single laser focus. One of the gas jets (bottom) is fixed in position at the center of laser focus, while the second jet (top) can move along the laser propagation direction before the focus of the laser beam, as shown in Figure 1(b) (see Supplementary Materials for details). The Gouy phase modulates the carrier envelope phase (CEP) of a focused laser pulse which results in a phase shift of XUV pulses generated from the two gas jets given by where q is the harmonic order and Δϕ Gouy is the Gouy phase difference at two gas jet positions (separated by Δz). The diameter of the gas jets is 200 μm, and gas pressure in both jets is kept at 100 Torr even though optimal phase matching happens at a slightly higher pressure. The short interaction region and low gas density help to reduce the macroscopic phase matching effects and to minimize the reabsorption of XUV photons by the second gas jet. The gases are sup-plied from cylinders through individual gas lines. The pressures in each gas jets are controlled by two separate regulated valves and monitored with individual capacitance manometer pressure gauges. The pressures were kept constant throughout the experiments. The gases are switched in the jets by electronically actuated microvalves that allows faster measurement and thus minimizes the errors due to laser fluctuation over time and also compensates for a small difference of intensities (and associated phases) at the two jet positions. The interference fringes of the high harmonics from H23 to H37 generated with H 2 in both jets obtained by varying the separation between the jets are shown in Figure 2(a). To optimize the resolution for all the observable harmonic orders, the isotopic phase difference measurements were performed at jet separations of 0.63 mm and 1.27 mm where the fluctuation of laser intensities is only 0.2% and 0.8%, respectively.
To extract the relative phase difference between the harmonics generated from the two gases, for each of the two jet separations, measurements are performed for two configurations. In the first configuration, D 2 and H 2 are delivered from the top and bottom jets, respectively. The gases are swapped in the second configuration.
The HHG spectra are recorded for two different jet separations and for two gas configurations to account for systematic uncertainties. For each configuration, three HHG spectra are recorded: for both gas jets activated and for individual jet activated. The measured spectra are presented in

Results and Discussions with Theoretical Analysis
The results are presented in Figure 4(a). The left axis corresponds to the phase difference averaged over both separations and both gas configurations. The phase differences measured at gas jet separations of 0.63 mm and 1.27 mm are very similar. This indicates that mixing of gases from the two jets has a negligible effect on our measurements. It should be noted that our method measures both the value and sign of the phase difference, and the two different configurations of gases induce opposite signs. Their absolute values were averaged, and the shown phases correspond to emission from H 2 being delayed relative to D 2 as our measurements indicate (see Supplementary Materials for details). The corresponding phase delays of harmonics from H 2 relative to D 2 are depicted on the right axis of Figure 4(a) and are determined by where ω q is the angular frequency of the q th harmonic order. The harmonics from H 2 are found to be~3 attoseconds delayed in phase with respect to D 2 for all the observable harmonic orders. The experimental measurements are supported by numerical solutions of the four-dimensional time-dependent Schrödinger equation (4D-TDSE) within the single active electron (SAE) approximation [14] (see Supplementary Materials for details). In this model, the electronic and nuclear motions are both confined to a plane containing the molecular vibration and rotation. The simulated results (red crosses) are presented in Figure 4(a), and they agree well with the experimental data (blue stars). Additional simulations showed that Δϕ H 2 −D 2 is relatively insensitive to the laser intensity, so the laser focal-volume averaging was not included.
According to the well-known three-step model, the total HHG phase includes contributions from the ionization, propagation, and recombination processes. Thus, for molecules aligned along different angles θ r towards the laser polarization direction, the dynamics of ionization and recombination should be different, resulting in the θ r -dependent Δϕ H 2 −D 2 .
Numerically, by excluding molecular rotations and restricting the nuclear motion to vibrational degree of freedom, we reduced our 4D-TDSE model to a three-dimensional one. We used that 3D-TDSE model to obtain θ r -dependent Δϕ H 2 −D 2 , presented in Figure 4(b). Only the molecules aligned at small angles (0-30 degrees) from the laser polarisation direction exhibit significant dependence of Δϕ H 2 −D 2 on harmonic order (energy of recolliding electrons) which could be attributed to two-center interference in the recombination step. Due to the different nuclear masses, the molecular bond elongation between the ionization and recombination steps is different for H 2 and D 2 . In the recombination step, the complex twocenter destructive interference occurs at a specific internuclear 23    Ultrafast Science distance R when the recombining electron momentum k meets the condition kR cos ðθ r Þ = π, and the HHG phase undergoes a sudden change at the corresponding frequency ω = k 2 /2 + I p . It is the difference in two-center interference between H 2 and D 2 that is responsible for the angle and energy dependence of the curves in Figure 4(b). More simulations show that the molecular rotation induced by the laser field results in a phase difference not more than 0.03 radian. A more detailed analysis based on the Lewenstein model and the two-center interference as well as the rotation effect estimation can be found in the supplementary information.

Conclusions
In summary, we have demonstrated a novel all-optical Gouy phase interferometric technique for measuring HHG phase difference between two atomic or molecular species. We used this technique to measure the HHG phase difference between H 2 and D 2 to be about 0.2 radian, corresponding to 3 attoseconds phase delay for all the harmonics. We also simulated this phase difference by numerically solving the time-dependent Schrödinger equation. The theoretical results obtained at the highest level of simulations agree quite well with the experiment. The simulations reveal that Δϕ H 2 −D 2 depends on the molecular alignment, the bond stretching, and the two-center interference. The phase difference of harmonics from molecular isotopes can be used as a sensitive probe of ultrafast correlated electron-nuclear dynamics in molecules.

Data Availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.