Ultrafast Hole Deformation Revealed by Molecular Attosecond Interferometry

Department of Physics, National University of Defense Technology, Changsha, Hunan, China Innovation Laboratory of Terahertz Biophysics, National Innovation Institute of Defense Technology, Beijing, China Institute of Applied Physics and Computational Mathematics, Beijing, China Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang, Sichuan, China Graduate School of China Academic of Engineering Physics, Beijing, China Shenzhen Key Laboratory of Ultra Intense Laser and Advanced Material Technology, Center for Advanced Material Diagnostic Technology, and College of Engineering Physics, Shenzhen Technology University, Shenzhen, China


Introduction
Since the late 1980s, time-resolved femtosecond techniques have led to tremendous developments in observing and manipulating chemical reactions at a timescale shorter than the period of vibration and rotation, i.e., to form or break chemical bonds in real time [1][2][3][4]. However, if viewed from a faster and a more microscopic scale, a chemical reaction originates from the overlap and departure of electron wave packets (EWPs) [5][6][7][8]. Nowadays, advances in ultrafast science have enabled the detection of these dynamics by pulses or time-resolved gating with attosecond precision [9][10][11].
Coupling among electrons or between nuclei and electrons helps to realize a subfemtosecond-timescale manipulation of electron and hole dynamics [12,13], which is pivotal for controlling the chemical reactions that follow. For instance, a redox reaction is an ultrafast process of gaining and losing electrons [14]. Manipulation of these electrons can migrate a localized charge, thereby forming new chemical bonds between molecules [15].
The natural spatial-temporal scale of electrons is at the atomic and molecular levels, which can be used to selfprobe the dynamics of a hole and EWPs in a controlled fashion [16,17]. In the strong laser fields, EWPs disperse in space after ionization and can recombine with the parent ion to emit high-energy photons: all of these occur consecutively within a laser cycle [18]. Since this process is induced by a highly nonlinear tunneling ionization, instant removal or excitation of an electron can provide a strong mixture of many one-hole configurations: an electron missing in the highest occupied molecular orbital (HOMO) or an electron missing in the inner orbitals [19]. The wave packet of a hole, therefore, exhibits a time-dependent periodic flow with a period T = ℏ/ΔE, where ΔE is the energy spacing between the two states [20,21]. Controlling the periodic flow of EWP offers a unique possibility of steering chemical reactions within the timescale of electrons.
In this article, we demonstrate a new methodology to pump and probe a hole from different molecular alignments, enabling the visualization of the ultrafast hole deformation using molecular attosecond interferometry. Threefold control is achieved here: First, an alignment pulse is used to excite the rotational wave packet, resulting in the field-free alignment of molecules at a later time delay. The rotational wave packet can be traced by monitoring the intensities of harmonics generated by a delayed probing pulse to pinpoint the moment of alignment [23,24], as shown in Figure 1(a) for the ensemble of CO 2 molecules. Second, the attosecondscale EWPs are manipulated by using a two-color laser pulse to form attosecond interferometry. Analogous to Young's spatial two-slit interferometry, we construct a temporal interferometry within one laser cycle by perturbing the dynamics of the EWPs propagating in opposite directions (Figure 1(b)) [25,26]. Third, by varying the polarization between the twocolor laser pulse and the alignment pulse, the hole along different angles in the molecular frame can be created as needed. Since the ionization of molecules is channel-dependent, hole deformation strongly depends on the molecular orbitals involved [21]. For instance, due to the nodes of HOMO, ionization from HOMO will be greatly suppressed; thus, the inner orbitals will become vital at specific angles [27]. The third control, therefore, completes molecular attosecond interferometry by manipulating the amplitude and relative phase between different channels and, in turn, providing hole control, as shown in Figure 1(c). The shape of the hole can be encoded by the recombined EWP [28] and reflect itself on the harmonic and terahertz (THz) emission spectra.
The threefold control involves different timescales from picoseconds to attoseconds, allowing us to observe the interplay between different ionization channels and the resultant complex electron-hole dynamics by comparing with the theoretical model and ab initio calculations. In addition, the yield and modulation of the accompanying THz emission can help to characterize the alignment-dependent ionization probability [29] and relative phase of the two-color laser fields [30], facilitating the identification of different channels during hole formation. The technology established in this work pushes the precise and multidimensional control of electron/hole dynamics into a new level and can be generalized to address the electron correlations in more complex systems. Moreover, varying the alignments and the twocolor phases can modulate the hole wave packet and EWP of molecules, therefore resolving and controlling the nuclear dynamics and the chemical bonds [31]. Figure 2(a) shows the scheme of molecular attosecond interferometry and its general probing mechanism. The experimental setup consists of a Ti:sapphire system (100 fs, 800 nm, 4.4 mJ, and 1 kHz) and vacuum charmers for the generation and collection of high-harmonic and terahertz spectroscopy (HATS) from laser-aligned CO 2 . The second-harmonic (400 nm) pulse is introduced by a 30 μm type-I β barium borate (β-BBO) crystal, and it collinearly propagates with the fundamental pulse to keep the two-color relative phase stable. The phase difference between the two-color pulses is compensated by an α-BBO crystal and can be precisely adjusted by a pair of fused silica wedges. The intensity of the secondharmonic pulse is roughly varied by tilting the pitching angle of the β-BBO. A wire grid polarizer is placed on the generation arm to ensure linear polarization before focusing. The typical intensities of the aligning pulse, the fundamental pulse, and the second-harmonic pulse in the experiment are estimated as 7 × 10 13 W/cm 2 , 2 × 10 14 W/cm 2 , and 2 × 10 12 W/cm 2 , respectively. The alignment angle α ′ , which is the angle between the alignment pulse polarization and the generation pulse, can be continuously varied by a rotatable half-wave plate. A homemade continuous nozzle (200 microns in diameter) is placed 0.2 mm above and 2 mm after the focus of the generating beam with 1 bar backing pressure. The rotational temperature of the molecule ensemble is estimated to be 90 K and the maximum of hcos 2 θi at the alignment moment is about 0.67. Each highharmonic spectroscopy (HHS) accumulates 3000 pulses by an X-ray CCD. The THz waveform is detected by one 1 mm thick (110)-cut ZnTe crystal via electrooptic sampling (EOS).

Ionization Determined by THz Detection.
When a weak second-harmonic pulse is added, an additional phase between the consecutive half-cycles of the Michelson interferometry breaks the time-reversal symmetry of the EWPs and leads to the generation of even harmonics (Figure 2(b)) [25] and the THz wave (Figure 2(c)) [30]. Because they are inherently phase-locked from strong field interactions, the joint measurement of HATS offers an unusual opportunity to observe electron or hole dynamics in multiple dimensions. For instance, the molecular alignment can be monitored by tracing the harmonic yield, which varies with the time delay t D between the aligning pulse and the generation pulse. The transient rotational wave packets of the alignment and antialignment ensembles are sketched in the inset of Figure 1(a).
The following experiments were carried out when t D is 21.3 ps, i.e., the alignment moment around the half-revival of CO 2 rotational wave packet [32].
As demonstrated in our previous work, the measured THz yield can serve as a reference to angular differential 2 Ultrafast Science ionization probability [29]. In Figure 2(d), alignmentdependent THz yields (averaged over eight independent measurements) are depicted under two relative phases ϕ between the two-color laser fields with fixed intensity ratios.
The alignment-dependent THz yields maintain a similar shape under different relative phases, and both peak at 42 ± 2 degrees. This agrees with the previous ion measurements [33,34] but differs from the theoretical predictions [35][36][37]. The arms on the left and right sides of the electron wave packet are constructed by the adjacent half-cycles of the laser. Two symmetric trajectories are generated by using only the fundamental laser pulse (red trajectories), while adding the second-harmonic pulse can break the symmetry (blue trajectories) and introduce the even harmonics and THz emissions. θ is the direction of molecular axis with respect to the generation pulse polarization. (c) Control of hole formation. The calculated HOMO, HOMO-1, and HOMO-2 orbitals are shown with their binding energies, respectively [22]. Ionization of molecules is strongly dependent on the molecular orbitals involved. The angle-resolved hole wave packet in the molecular frame can be reconstructed by EWPs propagating in opposite directions to exhibit the deformation of a hole and therefore, under different azimuth angle θ, form molecular attosecond interferometry (data from H28).

Ultrafast Science
Recent theoretical advances show that dynamic exchange of electrons is indispensable for addressing the differential ionization property of polyatomic molecules [38]. By continuously varying the relative phase between two-color fields, we find that the modulation of THz yields under different alignments is hard to change, as illustrated in the bottom row of Figure 3(a). This result suggests that the alignment angle α ′ and the two-color relative phase ϕ are decoupled in THz generation from CO 2 . The intensity of the THz wave generated from aligned molecules, therefore, can be expressed as a product of the alignment-dependent THz yields and a modulation function of the two-color phase delay, where ϕ op,T is the optimal two-color phase (OP) for the maximal THz yields. Meanwhile, the alignment-dependent THz Alignment-dependent terahertz yields at the optimal two-color phase delay (OP, red dots) and OP+0.5π (blue dots). The scanning angle step of the alignment angle is 2 degrees. 4 Ultrafast Science amplitude E THz ðα′Þ is assumed to be proportional to the alignment-dependent ionization probability Nðα′Þ [29]. It has been demonstrated that the THz yield takes the maximum at ϕ op,T ≈ 0:8π for atoms and small molecules due to the soft collision of the ionized electron with the ionic core [30].
In the present work, electrons can be released from inner orbitals in spite of their higher ionization potentials [38][39][40]. The fact that the optimal phase of the THz yields is independent of the alignment angle also indicates no ionization phase difference exists between different ionization channels, in agreement with the previous study [41].

Molecular Attosecond Interferometry with THz Phase
Reference. In Figure 3(a), two-dimensional (2D) measurements of harmonics from H22 to H32 are compared to the robust phase reference of THz yields. The scanning angle step for these 2D measurements is 5 degrees to save the total measurement time. To highlight the two-color phase delay modulation, each horizontal line is normalized by its maxima under each one specific alignment. It can be seen from Figure 3(a) that the OPs for H22 and H24 are almost the same with different alignments, but for the even harmonics higher than H26, a significant phase jump approaching 5 Ultrafast Science 0.6π can be observed between the parallel and perpendicular alignments, analogous to the results from RABBITT measurement [42]. This can be understood by considering the participation of multiple molecular orbitals [41]. Since no phase variation is observed from the ionization process, the optimal phase for each orbital is determined by the motion of attosecond EWP released from the respective binding orbitals. The optimal phase can be directly mapped to the birth-time of even harmonics [25], reflecting the group delay dispersion of EWPs [43]. Participation of multiple orbitals would modulate the amplitude or phase of the EWP, thus reproducing the angle-dependent OPs. The 2D-measurement opens up avenues for tracing the relative contribution of dominant orbitals by calibrating the intensity modulation of harmonics with the THz phase reference.
The second-harmonic field modifies the phase of the continuum EWP, leading to the modulation of the even harmonics as at each alignment angle jD 2N j 2 ∝ cos 2 ðϕ − ϕ op,2N Þ [25,30], with ϕ op,2N being the OPs for the yields of the even harmonics. The explicit values of ϕ op,2N can be deduced from fitting the experimental data. In Figure 3(b), the experimentally deduced OPs are presented for H22-H32 with the parallel (0 degree), 45-degree, and perpendicular alignments (90 degrees), respectively. It is worth noting that the absence of OP for H26 in Figure 3(a) around the parallel alignment is because the intensity modulation is too weak to define an explicit value of OP, but the regular modulations clearly reappear around the perpendicular alignment. For comparison, we calculate the OPs assuming the individual involvement of the HOMO, HOMO-1, or HOMO-2. The OP for each orbital is determined by ϕ s op,2N = −2ωt 2N r , where the birth time t 2N r is calculated with strong-field approximation (SFA) [25]. The theoretical predictions of HOMO (ϕ s 0 ), HOMO-1 (ϕ s 1 ), and HOMO-2 (ϕ s 2 ) are shown in Figure 3(b). The OPs of HOMO-1 and HOMO-2 are shifted by π from that of the HOMO due to the phase difference of harmonics emitted from different orbitals (see SM for details). With the robust THz-phase reference, we can confirm that the experimentally deduced OPs for H22 and H24 agree well with the SFA predictions from HOMO. While for harmonics from H28 to H32, a strong deviation occurs between the OPs from the experimental results and the predictions of the HOMO at the parallel alignment. For the other two alignments, we also see the disagreements between experimental results and the prediction with only HOMO involved. At the parallel alignment, the phase jump originates from the optimal phase transition between HOMO and HOMO-2 orbitals with comparable contributions, whereas for the perpendicular alignment, the relatively small phase variation in the OPs between the calculations from HOMO and the experimentally deduced results is attributed to the participation of HOMO-1, which is important in preparing the hole during the ionization step.
To further confirm our analysis, we perform ab initio calculations by the time-dependent Hartree-Fock (TDHF) approach [44] for the parallel-aligned CO 2 . The OPs at the parallel alignment for harmonics from H22 to H32 experience a significant phase jump across the 26th har-monics, qualitatively agreeing with the measurements in Figure 3(b). The calculated harmonic spectra with the two-color relative phase ϕ = 0 are shown in Figure 3(c) with the blue and red bars representing the odd-and even-harmonic yields, respectively. By analyzing the individual contribution from each orbital, it is found that the dynamical interference between HOMO and HOMO-2 leads to the spectral minimum and weak modulation around H26 with parallel alignment [41], which is consistent with the experimental results (see SM for details).

Reconstruction of the Hole Deformation.
Based on the molecular attosecond interferometry, the transient hole structure can be reconstructed at the recombination time. The harmonic intensity is proportional to the recombination transition moment between the continuum state jki and the hole state: with jΨ hole ðθ, t 2N r Þi = ∑ i jc i ð2N, θÞje −iφ i ð2N,θÞ jΨ i i being the hole wave packet at the recombination time. The subscript i labels the orbitals, and we only consider the HOMO (jΨ 0 i), HOMO-1 (jΨ 1 i), and HOMO-2 (jΨ 2 i) of CO 2 in reconstruction of the hole. Recently, numerical investigations suggested a certain contribution of HOMO-3 in the high-harmonic generation from aligned CO 2 , and this contribution might become dominant at the cut-off region [45]. In this work, we are dealing with the harmonics in the plateau region; thus, we neglect the contribution of HOMO-3 and only consider the three highest occupied molecular orbitals in reconstructing the hole. To compare with the experimental results directly, the alignment dependence on harmonic yields is considered. The measured even-harmonic yields Mðα′, 2N, t D , ϕÞ could be viewed as the coherent sum among emissions from molecules with different alignments: where A i ðα ′ , 2NÞ is the fitting amplitude of different molecular orbitals and θ is the polar angle of molecular axis. The experimental results are fitted by the coherent summation of the three involved channels with the dipole matrix element F i,k ðα ′ , t D Þ integrated with the alignment distribution. The primed and unprimed notations indicate the laboratory frame and molecular frame, respectively. The relative phase difference among the three channels can be determined directly by analyzing the ionization phase, excursion phase, and recombination phase in harmonic generation [41]. The 2D measurements in Figure 3(a) are fitted using Equation (3) with three involved orbitals. It can be seen in 6 Ultrafast Science Figure 4 that the simulated harmonic yields (Figures 4(b) and 4(e)) are in good agreement with the measurements (Figures 4(a) and 4(d)). The alignment-dependent fitting amplitudes of different orbitals are exhibited in Figures 4(c) and 4(f). The fitting amplitude of the HOMO dominates at all aligned angles for H22 and H24. For H28-H32, the amplitude of HOMO-2 is much greater than the contribution from HOMO for the polar angles less than 30 degrees, resulting in the OP variations from the parallel to the perpendicular alignment.
With the deduced amplitude and phase of each orbital at the recombination time t 2N r , the hole dynamics can be reconstructed by the multi-ionization channels that are inherently coherent. In Figure 5, we show the hole deformation viewed by the EWP with different recolliding momentum k at the parallel and perpendicular alignment angles. Apparently, as the momentum of the recombining electron increases, the hole evolution shows clear transition from the single orbital to the multiple-orbital case at all alignment angles. At the parallel alignment, the reconstructed hole wave packet changes from the HOMO orbital to the HOMO-2dominated orbital, while at the perpendicular alignment, the hole gradually evolves from the HOMO orbital to a combination of HOMO and HOMO-1 orbitals as a function of the recollision time. The absolute values of the hole wave packet are reconstructed by the coefficients |c i | , the phases φ i , and the molecular orbitals via applying the expression of hole wave packet in Equation (2), as shown in Figure 5. (d-f) present the same for H30. For comparison, we present the fitting amplitudes of the three molecular orbitals for H24 (g), H28 (h), and H32 (i). The standard deviation of the fitting amplitudes jA i j is estimated to be less than 5%.

Ultrafast Science
The variation of the hole wave packet reflects the ultrafast dynamics of the coherent molecular ensembles under the strong laser field. Ionization of the ensembles from multi-channels is the first step to create the hole wave packet and the continuum EWP, both of which evolve coherently under the external two-color fields. As the continuum electrons  Ultrafast Science return to their parent ions at different recombining times, the hole dynamics are recorded by this recombination process with a temporal resolution determined by the time spacing between the neighboring two harmonics. For an electron with a smaller momentum, it mainly sees the contribution of the HOMO orbital to the hole wave packet, whereas for an electron with a larger momentum, the inner orbitals can contribute to the formation of the hole. Therefore, the control of a hole wave packet via different alignments of molecular ensembles can be revealed here by the hole dynamics in Figure 5. Strong-field ionization creates the angle-dependent hole wave packet, and the hole dynamics at different alignment angles are determined by the amplitude and the phase of the participated orbital. The hole shape reconstructed from harmonic H22 keeps almost the same for all alignment angles, because the optimal phase for H22 keeps constant in the experimental results; therefore, the reconstructed hole wave packet is mainly contributed by the HOMO orbital. The slight differences in the absolute values of the hole wave packet between different alignments reflect the angular-dependent ionization probabilities. The hole recorded by the higher harmonics exhibits a strong alignment-dependent deformation, which can be directly observed from the reconstructed hole in Figure 5. The shape of the hole gradually changes from a σ-dominant orbital to a π-dominant orbital from the parallel to the perpendicular alignment, reflecting the geometry of the participating orbitals. Meanwhile, the angle-dependent phase variation observed at higher harmonics indicates that the recombination time depends on the alignment angle for electrons with the same recolliding momentum, which is about 400 attoseconds earlier at the perpendicular alignment than the parallel alignment, reflecting the angle dependence of correlation between the EWP and the ionic core. The hole shapes reconstructed by the phase of even harmonics from H28 to H32 exhibit a strong alignment dependence, implying a possible control on the hole by selecting the involved orbitals via different ionization angles.

Discussion
The present research demonstrates how to probe the ultrafast deformation of a molecular hole from threefold control using molecular attosecond interferometry. The impulsive alignment of the molecule is achieved by creating a coherent molecular rotational wave packet by rotational Raman transitions and fixing the molecule axis in the laboratory frame. The opposite two arms of the propagating directions are used to form the interferometry of electrons, and the relative phase can be adjusted by a weak second-harmonic pulse in a resolution of several tens of attoseconds. By jointly measuring the second-harmonic pulse-modulated HATS, harmonic phases and the hole dynamics are precisely extracted with the robust phase reference of the modulated THz yields. Harmonics with alignment-dependent OPs imply alignmentdependent participation of different orbitals. By fitting the amplitudes of these orbitals, it is possible to reconstruct the hole dynamics, as demonstrated in Figure 5. Moreover, we note that the evolution of EWP can also be decoded in the phase variation observed from the molecular attosecond interferometry. These results suggest a precise control over hole dynamics at the attosecond timescale and subangstrom space-scale resolutions. The state-of-the-art ultrafast processes, such as the hole carrier, thermal, and acoustic dynamics in condensed matter, have been investigated by nanoscale pump-probe detection [46,47]. However, we are dealing with a faster process, which occurs within a fraction of one laser cycle from the phase-matched molecules. Although hole dynamics is previously studied from the yields of different orders of harmonics [11], our methodology can provide more precise control over the initial formation and retrieve detailed information on the following dynamics of the hole-electron system, thanks to the HATS technique.
A hole is a left vacant of an absent electron. Previously, the structural deformation of CO 2 molecules was reported in the presence of strong laser fields, and it was interpreted by the coupling between the electronically excited and ground states of ions [48]. Because different molecular states are close to each other in energy spacing, under the exposure of laser pulses, multimolecular states can be excited. These electrons, ejected from not only the HOMO but also the inner orbitals, are released creating a coherent EWP, which is the coherent superposition of electronic eigenstates. The remaining hole, which is a mixture of different one-hole configurations, can exhibit a time-dependent deformation that reflects the coherent nature between different states [11]. Hole deformation can also be controlled by varying the alignment angle of an ensemble of molecules due to the alignment-dependent ionization of different orbitals [21]. In the present work, we experimentally and theoretically advance this concept. The dispersion of the recombined EWP can be precisely tuned by the phase difference between the two-color laser fields, which helps to add a new dimension of detection. Different from the previous research of hole which detects only the intensities and phases of different harmonics and therefore confirms the participation of HOMO and HOMO-1 of N 2 [49], our molecular attosecond interferometry can add an additional phase of the EWP, leading to the reconstruction of hole deformation from more than two orbitals. In other words, our method can be applied to the other polyatomic molecules that exhibit the alignment-dependent ionization channels, for instance, N 2 O 4 [50], C 4 H 6 , C 4 H 10 [51], C 2 HI [11], C 2 H 2 [52,53], CF 3 I [54], and COS [55], in order to understand the channel-dependent dynamics during the strong field process. So far, the present work is limited to study the electron dynamics decoupled with nuclear motion. More fields of study, such as the vibrational dynamics [50,56] and the coupling between electrons and nuclear [7,8,53,57], can benefit from our method when using pulses with shorter pulse duration. Furthermore, we expect that the molecular attosecond interferometry based on the precise control of rotational and electronic wave packets and laser shaping will advance the study of ultrafast electron correlations and multidimensional and multichannel couplings in a range of fields,

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Ultrafast Science such as chemical reactions, molecular orbital tomography, charge migration, and transport.

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

Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this article.

Supplementary Materials
Experimental setup, channel-resolved quantum-orbit analysis for optimal phases of HHG, data analysis by fitting the intensity modulations, and time-dependent Hartree-Fock approach. Figure S1: the detailed experimental setup. Figure  S2: quantum-orbit analysis for the optimal phases of different harmonics under the Strong Field Approximation (SFA). Figure S3: the recombination dipole moments of HOMO, HOMO-1, and HOMO-2 orbitals of CO 2 and the corresponding orbitals. Figure S4: The normalized harmonicorder-dependent coefficients jc i ð2N, θÞj. Figure S5: harmonic spectra calculated by the time-dependent Hartree-Fock approach (TDHF) method. Figure S6: experimentally observed high harmonic spectra with the phase delay ϕ = 0 under the parallel, 45-degree, and perpendicular alignment.