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High-order above-threshold dissociation of molecules

Significance Above-threshold ionization of atoms in strong laser fields is extensively studied for its overwhelming importance and universality. However, its counterpart, above-threshold dissociation of molecules in strong laser fields, is hard to be observed, although it has been predicted for decades. In this paper, by measuring the momenta of photoelectron and dissociative fragments coincidently, we successfully obtained distinct nuclear energy peaks of the high-order above-threshold dissociation, which must appear simultaneously with the above-threshold ionization. The coexistence of high-order above-threshold dissociation and high-order above-threshold ionization in molecular dissociative ionization offers a perspective to disentangle the complex electron–nuclear correlation in molecules and to image the molecular orbitals, and so on.

I n 1905, Albert Einstein proposed the hypothesis of light quanta, which postulates that light energy is carried in discrete quantized packets (photons), to explain the law of photoelectric effect. This has been a pivotal step in understanding of the quantized nature of light. Since then, studies of photoionization in strong fields have been greatly boosted and various mechanisms have been understood, e.g., the multiphoton and tunneling ionizations. In 1979, above-threshold ionization (ATI) was observed in the multiphoton ionization of atoms exposed to strong laser fields (1). Multiple photons with energy exceeding the ionization threshold can be simultaneously absorbed, leading to discrete photoelectron spectra on account of the quantized nature of the multiphoton energy absorption. Similarly to ATI, above-threshold dissociation (ATD) (2) was predicted for the multiphoton dissociation of molecules in strong laser fields. In the process of ATD, typically the number of absorbed photons exceeds the minimum required for breaking the molecular bond, appearing as evenly spaced peaks in the kinetic energy release (KER) spectrum of the nuclear fragments.
Despite the fact that the existence of ATD is logical and consistent with fundamental laws of nature, the experimental evidence of the distinct high-order (more than three orders) ATD has never been reported while the theoretical prediction was made more than 20 years ago (2). An early experiment using 532-nm, 100-ps laser pulses showed the first signature of the third-order ATD of H 2 + due to the photon-coupled dipole transition between the 1sσ g and 2pσ u states (3). Since then, significant efforts have been dedicated to enhance the experimental visibility of the high-order ATD of molecules by, e.g., using vibrational cold molecular ions (4) or applying intense few-cycle laser pulses (5). Nevertheless, the observation of the distinct ATD spectra is still an open question.
The essential problem of concealing the distinct high-order ATD lies on the electron-nuclear correlation in molecular dissociative ionization. The photon energy absorbed by a molecule is shared between the electronic and nuclear degrees of freedom, which implies that the ATD could not be revealed if the freed electron is not measured in coincidence with the nuclear fragments (4-7). Previous coincidence measurements of dissociative ionization of molecules in femtosecond UV laser pulses explicitly showed the correlation between electron and nuclear degrees of freedom but no high-order ATD spectrum was observed (8)(9)(10). Here we pursue an alternative route to observe high-order ATD by taking advantage of the joint energy spectrum (JES) (8)(9)(10)(11)(12)(13)(14) of the coincidently measured electron and ion ejected from the same molecule. In particular, the correlated dynamics of the liberated electron and nuclei are bridged by laser-driven inelastic rescattering and can be unveiled by the electron-nuclear JES, which indicates that the electron and nuclei of the molecule as a whole absorb the multiphoton energy. Similarly to the ATI photoelectron spectrum, more than four distinct peaks spaced by the photon energy are observed in the nuclear KER spectrum, which is an unambiguous evidence of high-order ATD. Numerical simulations allow us to trace the periodical emission of the correlated electron-nuclear wave packets in each optical cycle, leading to the coexistence of the discrete ATI and ATD spectra in the dissociative ionization of molecules.

Significance
Above-threshold ionization of atoms in strong laser fields is extensively studied for its overwhelming importance and universality. However, its counterpart, above-threshold dissociation of molecules in strong laser fields, is hard to be observed, although it has been predicted for decades. In this paper, by measuring the momenta of photoelectron and dissociative fragments coincidently, we successfully obtained distinct nuclear energy peaks of the high-order above-threshold dissociation, which must appear simultaneously with the above-threshold ionization. The coexistence of high-order above-threshold dissociation and high-order above-threshold ionization in molecular dissociative ionization offers a perspective to disentangle the complex electron-nuclear correlation in molecules and to image the molecular orbitals, and so on.

Results and Discussion
Hydrogen and its isotopes, as the simplest neutral molecule, have been the prototype for understanding fundamental molecular processes (3,15,16). Here, H 2 molecule is used to demonstrate the ATD of molecules in strong laser fields, that is, H 2 + nZɷ → H + + H + e, hereafter denoted as H 2 (1,0) channel. In this paper, we focus on the electron rescattering-assisted ATD, as illustrated in Fig. 1A. One electron is freed via the tunneling through the laser-dressed Coulomb potential, while, the generated molecular ion H 2 + starts to stretch along the potential surface of the 1sσ g state. The accelerated electron in the laser field may slam back and transfer its energy to H 2 + , launching the nuclear wave packet onto the 2pσ u state, which dissociates afterward into the nuclear fragments with high kinetic energies. Such rescattering events occur periodically in every optical cycle. In the experiment, linearly polarized laser pulses (100 fs, 790 nm) with a peak intensity of 9 × 10 13 W/cm 2 were utilized. The measurements were performed in a reaction microscope setup of the cold target recoil ion momentum spectroscopy (17,18) (Materials and Methods). The proton H + and the correlated electron e from H 2 (1,0) channel were detected in coincidence measurement. The total KER of the nuclei (E N ), that is, the sum of the kinetic energy of the proton (E p ) and the hydrogen atom (E H ), was deduced based on the momentum conservation of the breaking system. Fig. 2A depicts the measured electron-nuclear JES, that is, the yield of H 2 (1,0) channel as a function of the photoelectron energy (E e ) and the nuclear kinetic energy (E N ), using linearly polarized laser fields. Two distinct regions separated at about E N ∼ 2 eV can be recognized, which can be attributed to different dissociation mechanisms. The low-E N region (with E N < 2 eV) originates from one-photon and net two-photon transitions between the 1sσ g and 2pσ u states of H 2 + , which has been widely studied both theoretically and experimentally (19)(20)(21)(22)(23)(24). However, the high-E N region (with E N > 2 eV) is mainly produced via electron recollision, which we will focus on in the following discussion. The rescattering nature of the high-E N pathway is verified by adjusting the polarization of the driving laser from linear to circular. As shown in Fig. 2B, the high-E N pathway is dramatically suppressed in the circularly polarized light (blue curve) compared with the linearly polarized case (red curve) (see SI Text for the electron-nuclear JES in circularly polarized laser fields). The circularly polarized light produces very few events with E N in the range of 2-6 eV which originate mainly from the Coulomb explosion via the charge-resonance enhanced double ionization of H 2 rather than the dissociative single-ionization channel of H 2 (1,0) which we focus on here for the observation of the high-order ATD.
To reveal the underlying mechanism of the rescatteringbridged JES, we numerically simulate the time-dependent Schrödinger equations by including two-electron dynamics and nuclear vibration along the laser polarization axis (Materials and Methods). The calculated JES is shown in Fig. 2D, which shares very similar structures with the experimental results shown in Fig. 2A, especially in the high-E N region. To confirm that the dissociation of H 2 + in the high-E N region is triggered by the electron rescattering rather than the photon resonant coupling between the 1sσ g and 2pσ u states, we omitted the laser-e 2 coupling and diagnosed the wave packet corresponding to H 2 (1,0). This gives almost unchanged high-E N JES but yields a substantially suppressed low-E N JES (see SI Text for details). It clearly indicates the capability of our one-dimensional model to simulate the role of rescattering in producing the high-E N JES, although it cannot deal with the dissociative ionization of H 2 in the circularly polarized laser pulses. As presented in Fig. 2 B and C, both the nuclear and photoelectron spectra of the high-E N region exhibit smooth distributions regardless of the quantized nature of the light energy. However, the JES in Fig. 2A is featured with multiple diagonal energy correlation lines spaced by the photon energy. This indicates that the electron and nuclei absorb photons as a whole, and the ATI and ATD must be measured simultaneously in the dissociative ionization of molecules. Thus, as shown in Fig. 3A, discrete peaks can be reconstructed in the total energy spectrum of all emitted photoelectron and nuclear fragments, that is, E sum = E N + E e . Furthermore, the ATI (ATD) peaks in the photoelectron (nuclear) spectra can be produced by confining the nuclear (photoelectron) energy in a relatively narrow range. For instance, as shown in Fig. 3B, by integrating the electronnuclear JES over E N in the range of 3.8-4.2 eV, discrete ATI peaks appear in the photoelectron spectrum. On the other hand, by integrating the electron-nuclear JES over E e in the range of 0.8-1.2 eV, a photon-energy spaced discrete structure appears in the nuclear spectrum, as shown in Fig. 3C. Thus, multiple photon energy exceeding the dissociation threshold is deposited into the nuclei. As shown in Fig. 3 B and C, the discrete energy of the photoelectron (nuclei) decreases when the nuclear (photoelectron) energy increases owing to their energy sharing of the absorbed photons.
We note that the above-mentioned high-order ATD accessed via the electron recollision differs from the three-photon or the net two-photon ATD explored previously (3,5) in a fundamental manner. In the previously reported works, most energy of the absorbed photons above the ionization threshold is taken by the electron in the ionization step; while the energy gain of the nuclei originates mainly from the succeeding dipole-allowed photon transitions between 1sσ g and 2pσ u states. The ionization step and the dissociation step are thus separated. Our present results also differ from the ATD using the cold ion source HD + (4), where the HD + directly absorbs several photons and then dissociates. In our case, the rescattering of the electron wave packet with the parent H 2 + launches the dissociative wave packet on the 2pσ u state, and the rescattering occurs at every half of the optical  cycle. Thus, the dissociative ionized wave packet can be approximately written as ΨðR, x 1 , x 2 ; tÞ = X j χ j ðR, x 1 ; tÞ ⊗ ϕ j ðx 2 ; tÞ, where j indexes the jth rescattering, R, x 1 , x 2 specify the internuclear distance and the coordinates of the two electrons along the light polarization direction, χ j ðR, x 1 ; tÞ is the wave packet of the dissociative H 2 + , and ϕ j ðx 2 ; tÞ describes the wave packet of the rescattered electron. ΨðR, x 1 , x 2 ; tÞ should satisfy the two-electron exchanging symmetry since the two electrons form the singlet spin state. Note that the neighboring nuclear wave packets χ j ðR, x 1 ; tÞ and χ j+1 ðR, x 1 ; tÞ do not interfere with each other since the corresponding electronic parts ϕ j ðx 2 ; tÞ and ϕ j+1 ðx 2 ; tÞ finally propagate in the opposite direction. The interference of every other rescattering molecular wave packet, that is, the intercycle interference (25,26), induces the joint ATI and ATD. By comparing the highorder ATD with the well-understood nonsequential double ionization of atoms in strong laser fields (27)(28)(29), one may find the correspondence between these two scenarios: The tunneling electron rescatters with its parent ion, and shares its energy with another electron. In molecules, the extra freedom of nuclear repulsion finally inherits the energy stored in H 2 + by inelastic rescattering There is also a certain probability that the inelastic-rescattering excited H 2 + is further ionized by the remaining laser field through the charge-resonance enhanced ionization (30,31) at a critical range of the internuclear distance. Governed by the energy conservation, the slopes of the energy correlation lines in the JES of one electron and two nuclei are expected to be −1 for single ionization and −2 for double ionization if the two electrons have the same energy. The slopes of energy correlation lines in Fig. 2A clearly indicate that the high E N in our measured JES is dominated by the rescattering-induced dissociative single ionization rather than the double-ionization channel, although the two channels may have the overlapping KER distributions.
For the rescattering driven by the oscillating laser field, the electron liberated in the single ionization may revisit the parent ion H 2 + several times before it is eventually released. During this time interval, the ionization-created vibrational wave packet propagates dispersively on the 1sσ g potential energy surface. Different rescattering returns give rise to different KERs of the emitted nuclear fragments. For a given number of absorbed photons, the energy deposition between the electron and correlated nuclei is determined by the internuclear separation at the time of electron recollision. The nuclear wave packet with smaller internuclear separation tends to obtain more energy from the recolliding electron, giving rise to dissociative fragments with higher kinetic energy. The vibrational wave packet is broadened at the time of the third-return recollision, thus resulting in a broader nuclear KER distribution. This accounts for the two bands with E N in the ranges of 2-8 eV and 8-12 eV in the JES (Fig. 2A), which correspond to the two rescattering returns around the internuclear separations of 2.0-3.6 a.u. and 1.7-2.0 a.u., respectively. Our numerical simulations show that the fragments with low KER have larger proportion for a lower laser intensity since the first rescattering does not bring enough energy to excite H 2 + from 1sσ g to 2pσ u at small internuclear distances. The multiple rescattering in the dissociation of molecules has also been discussed in the former works (32)(33)(34). For the electron rescattering-induced dissociation in the current experiment, the kinetic energy band of 2-8 eV is dominated by the third-return recollision and the kinetic energy band of 8-12 eV is attributed to the first-return recollision.

Conclusion
In conclusion, we have presented a conclusive experimental evidence of the high-order ATD featured with the discrete nuclear energy spectrum by means of JES of the coincidently measured electron and nuclei ejected from the same molecule. High-order ATD of H 2 + occurs via the energy sharing during the electronion inelastic rescattering, instead of the direct absorbing photons from the external laser field. Similarly to the well-observed ATI photoelectron spectrum in multiphoton ionization of atoms, periodical emission of the correlated electron and nuclear wave packets in each optical cycle imprints the quantized nature of the light in the joint ATI and ATD spectra of molecules. The phenomenon of electron rescattering-assisted high-order ATD is expected to be general for various molecules (see SI Text for the results of D 2 molecules for instance). However, in the multielectron system, the high energy nuclei depends on the final repulsive electronic state involved in the rescattering-induced excitation of the molecular cation, which might also be influenced by the complex nuclear dynamics when polyatomic molecules are involved. Our results show that the multiparticle correlation in molecules plays key roles in molecular fundamental processes, and the coincidence observation of different fragments in chemical reactions is favor to unveil unambiguously the underlying fascinating dynamics.

Materials and Methods
Experimental Technique. The measurements were performed in an ultrahighvacuum reaction microscope setup of the cold target recoil ion momentum spectroscopy (COLTRIMS) (17,18), where the electrons and ions ejected from a single molecule can be detected in a coincidence measurement by two time-and position-sensitive detectors at the opposite ends of the spectrometer. A beam of linearly polarized femtosecond laser pulses (100 fs, 790 nm, 10 kHz) was down-collimated (2:1) and focused afterward onto a supersonic gas jet of H 2 by a concave silver mirror (f = 75 mm) inside the COLTRIMS. The polarization of the laser pulses can be adjusted to be circular by using a quarter-wave plate. The peak intensity in the laser-molecule interaction zone was estimated to be 9.0 × 10 13 W/cm 2 by tracing the intensitydependent hydrogen's time-of-flight spectrum (35). The maximum kinetic energy of the returning electron is 3.17U p ∼ 16 eV, which is sufficient to excite the ground state 1sσ g to the first excited state 2pσ u of H 2 + but less than the double-ionization threshold. Here, U p is the ponderomotive energy of the electron in the oscillating laser field. The count rate was ∼0.03 ions and 0.06 electrons per laser shot, which ensures a negligible fraction of the false coincidence among the detected electrons and ions. Three-dimensional momenta of the ejected electrons and protons were reconstructed using the measured times-of-flight and positions of the impacts during the offline analysis. The momentum of the undetected neutral H was deduced based on the momentum conservation of the breaking system, whose kinetic energy together with that of H + accounted for the total energy of the nuclei, that is, where μ is the reduced nuclear mass, p R , p 1 , and p 2 are the nuclear relative momentum operator, the first and second electron momentum operators, respectively. AðtÞ = − R t 0 Eðt ' Þdt ' is the laser vector potential with EðtÞ being the laser electric field. The potential VðR, x 1 , x 2 Þ is modified by two R-dependent soft-core parameters (36), and thus the ground-state energy curve of H 2 , the ground state and the first excited state of H 2 + agree with the real potential curves reasonably well, which justifies the simplified one-dimensional model in the energy point of view. The simulation grid extends over [0, 36] a.u., [−1200, 1200] a.u., [−1200, 1200] a.u. along the R, x 1 , and x 2 dimensions and is sampled by 900, 8,000, and 8,000 points, respectively. In simulations, the laser wavelength and intensity are 790 nm and 9 × 10 13 W/cm 2 , respectively, as those used in the experiments. The laser pulse comprises 10 cycles (one-cycle ramp-on, eight-cycle constant amplitude, and one-cycle ramp-off). We propagated the wave function after the laser pulse until the concerned physical results were converged. In boundaries of the simulation box, the absorber cos 1/6 is applied for suppressing the unphysical reflections. Only very few dissociative ionized fragments of H 2 (1,0) are absorbed by the boundaries, which does not change Fig. 2D visibly. The signal of H 2 (1,0) in momentum representation was calculated by Fourier transforming the wave packet entering the following area: R > 10 and from which the JES was built. Note that the potential curves in the dimensional-reduced model are not identical to the ones from the 3D model, and thus the KER in the calculations does not quantitatively agree with the measurement. We noticed that the simulated low E N in the JES (which is not induced by rescattering but by direct absorbing extra photons after the single ionization of H 2 ) does not agree with the experimental measurement; however, the JES related to the rescattering dissociation agrees with the measurement quite well. To verify that the dissociation of H 2 + is induced by the electron rescattering rather than by the direct laser-H 2 + coupling, we artificially neglected the laser-e 2 coupling. By doing this, we obtained the similar JES in the high nuclear energy range for the wave packets that e 2 is final attached to one of the nuclei (see SI Text for details).