Special Issue: Jubilee Issue on Hydrogen: a Fundamental System in All States

This review article focuses on imaging and controlling ultrafast dynamics of the hydrogen molecule and its cation, initiated by ultrashort laser pulses. We discuss the mechanisms underlying these dynamics and theoretical methods to describe them. A broad variety of defining and influencing theoretical and experimental results is presented. We put special emphasis on the required experimental techniques, many of which have been developed in view of imaging the fastest of all nuclear dynamics.

Being the simplest molecule on the planet, H₂, as well as its isotopes, have been the prototype systems for strong-field molecular physics for decades. Photoionization and dissociation of H₂ have been extensively investigated. After single ionization, the electron left in the molecular ion and its microscopic localization around the two dissociating nuclei can be effectively manipulated using intense femtosecond laser fields with broken symmetry. In this paper, we review the recent progress made on the sub-cycle directional control of the dissociative ionization of hydrogen molecules by tailoring the waveform of femtosecond laser fields, including few-cycle pulses and 2-color fields polarized along the same direction (one-dimension) or different directions (two-dimension).

The atomic hydrogen target has played a pivotal role in the development of quantum collision theory. The key complexities of computationally managing the countably infinite discrete states and the uncountably infinite continuum were solved by using atomic hydrogen as the prototype atomic target. In the case of positron or proton scattering the extra complexity of charge exchange was also solved using the atomic hydrogen target. Most recently, molecular hydrogen has been used successfully as a prototype molecule for developing the corresponding scattering theory. We concentrate on the convergent close-coupling computational approach to light projectiles, such as electrons and positrons, and heavy projectiles, such as protons and antiprotons, scattering on atomic and molecular hydrogen.

Calculation of the rotation-vibration spectrum of ${{\rm{H}}}_{3}^{+}$, as well as of its deuterated isotopologues, with near-spectroscopic accuracy requires the development of sophisticated theoretical models, methods, and codes. The present paper reviews the state-of-the-art in these fields. Computation of rovibrational states on a given potential energy surface (PES) has now become standard for triatomic molecules, at least up to intermediate energies, due to developments achieved by the present authors and others. However, highly accurate Born–Oppenheimer energies leading to highly accurate PESs are not accessible even for this two-electron system using conventional electronic structure procedures (e.g. configuration-interaction or coupled-cluster techniques with extrapolation to the complete (atom-centered Gaussian) basis set limit). For this purpose, highly specialized techniques must be used, e.g. those employing explicitly correlated Gaussians and nonlinear parameter optimizations. It has also become evident that a very dense grid of ab initio points is required to obtain reliable representations of the computed points extending from the minimum to the asymptotic limits. Furthermore, adiabatic, relativistic, and quantum electrodynamic correction terms need to be considered to achieve near-spectroscopic accuracy during calculation of the rotation-vibration spectrum of ${{\rm{H}}}_{3}^{+}$. The remaining and most intractable problem is then the treatment of the effects of non-adiabatic coupling on the rovibrational energies, which, in the worst cases, may lead to corrections on the order of several cm⁻¹. A promising way of handling this difficulty is the further development of effective, motion- or even coordinate-dependent, masses and mass surfaces. Finally, the unresolved challenge of how to describe and elucidate the experimental pre-dissociation spectra of ${{\rm{H}}}_{3}^{+}$ and its isotopologues is discussed.

The hydrogen atom is the simplest system of atomic and molecular physics, while a two-qubit system is the simplest of quantum information. Remarkably, they share common symmetry aspects which are described in this paper, based on a correspondence between the four-dimensional unitary group and the six-dimensional rotational group with its non-compact extensions. Both systems involve 15 basic operators. Reductions to Lorentz and Poincare space–time group symmetries of a free particle are also discussed.

Interaction of atoms and molecules with strong electric fields is a fundamental process in many fields of research, particularly in the emerging field of attosecond science. Therefore, understanding the physics underpinning those interactions is of significant interest to the scientific community. One crucial step in this understanding is accurate knowledge of the few-cycle laser field driving the process. Atomic hydrogen (H), the simplest of all atomic species, plays a key role in benchmarking strong-field processes. Its wide-spread use as a testbed for theoretical calculations allows the comparison of approximate theoretical models against nearly-perfect numerical solutions of the three-dimensional time-dependent Schrödinger equation. Until recently, relatively little experimental data in atomic H was available for comparison to these models, and was due mostly due to the difficulty in the construction and use of atomic H sources. Here, we review our most recent experimental results from atomic H interaction with few-cycle laser pulses and how they have been used to calibrate important laser pulse parameters such as peak intensity and the carrier-envelope phase (CEP). Quantitative agreement between experimental data and theoretical predictions for atomic H has been obtained at the 10% uncertainty level, allowing for accurate laser calibration intensity at the 1% level. Using this calibration in atomic H, both accurate CEP data and an intensity calibration standard have been obtained Ar, Kr, and Xe; such gases are in common use for strong-field experiments. This calibration standard can be used by any laboratory using few-cycle pulses in the 10¹⁴ W cm⁻² intensity regime centered at 800 nm wavelength to accurately calibrate their peak laser intensity to within few-percent precision.

Photoionization of triatomic molecular ions has been studied for triangular and linear geometries by bichromatic circularly polarized attosecond UV laser pulses at frequencies ${\omega }_{2}=2{\omega }_{1}$. Simulations are performed on the single electron molecule ${{\rm{H}}}_{3}^{2+}$ by numerically solving time-dependent Schrödinger equations. We measure molecular frame photoelectron momentum distributions (MFPMDs), which show spiral electron vortex patterns as functions of the helicity of the pulse and the molecular geometry. The ionization interference effects arise from multi-pathway ionization, which give rise to the modulation of multi-center photoelectron spectra. We describe these phenomena in MFPMDs based on an ultrafast delta-function ionization model and attosecond perturbation ionization theory. Interference patterns in MFPMDs reflect the helicity and symmetry of the electric fields in the attosecond ionizing pulses.

We investigate l-mixing processes in collisions of excited positronium (Ps) atoms with protons in the range of the principal quantum number n of the Ps atom between 4 and 8. We show first that results of the threshold theory agree very well with convergent close-coupling calculations. We then compare quantum cross sections with results of semiclassical and classical theories. In the semiclassical theory, the Ps atom is treated quantum-mechanically, whereas the relative Ps-p motion is described classically. For l-mixing processes there is a good agreement between quantum and semiclassical results, whereas classical theory strongly underestimates the l-mixing cross sections. The conclusions are important for the interpretation of Classical Trajectory Monte Carlo simulations and the use of their data in the design of experiments.

The ultrafast decay of highly excited electronic states is resolved with a molecular clock technique, using the vibrational motion associated to the ionic bound states as a time-reference. We demonstrate the validity of the method in the context of autoionization of the hydrogen molecule, where nearly exact full dimensional ab-initio calculations are available. The vibrationally resolved photoionization spectrum provides a time–energy mapping of the autoionization process into the bound states that is used to fully reconstruct the decay in time. A resolution of a fraction of the vibrational period is achieved. Since no assumptions are made on the underlying coupled electron–nuclear dynamics, the reconstruction procedure can be applied to describe the general problem of the decay of highly excited states in other molecular targets.

We extended the two-dimensional generalized pseudo-spectral time-propagator to a three-dimensional time-propagator. With this time-propagator, we theoretically investigated the above-threshold ionization of hydrogen atoms in an elliptical field by solving the time-dependent Schrödinger equation. We found that the total ionization probabilities are suppressed about one order, while the excitation probabilities drop about eight orders from a linearly polarized laser to a circularly polarized one with a given laser intensity. The width of the lateral (perpendicular to the laser polarization plane) momentum distribution increases as the ellipticity increases, which is against our intuition that the large peak laser field results in a broad distribution in the lateral direction. Further analysis shows that the narrow lateral momentum distribution for linearly polarized laser fields originates from Coulomb focusing, which is as less or not important for the circularly polarized field. Such a high precision simulation can be used to calibrate the infrared laser intensity to a high precision.

We present a multi-color ladder excitation scheme that exploits Stark-induced adiabatic Raman passage to selectively populate a highly excited vibrational level of a molecule. We suggest that this multi-color coherent ladder excitation provides a practical way of accessing levels near the vibrational dissociation limit as well as the dissociative continuum, which would allow the generation of an entangled pair of fragments with near-zero relative kinetic energy. Specifically, we consider four- and six-photon coherent excitation of molecular hydrogen to high vibrational levels via intermediate vibrational levels, which are pairwise coupled by two-photon resonant interaction. Using a sequence of three partially overlapping, single-mode, nanosecond laser pulses we show that the sixth vibrational level of H₂, which is too weakly coupled to be easily accessed by direct two-photon Raman excitation from the ground vibrational level, can be efficiently populated without leaving any population stranded in the intermediate level. Furthermore, we show that the fourteenth vibrational level of H₂, which is the highest vibrational level in the ground electronic state with a binding energy of 22 meV, can be efficiently and selectively populated using a sequence of four pulses. The present technique offers the unique possibility of preparing entangled quantum states of H atoms without resorting to an ultracold system.

We have carried out velocity map imaging experiments on HOD molecules irradiated by 10 fs long pulses of intense (∼1 PW cm⁻²) laser light (800 nm). We have detected HD⁺ ions as a signature of unimolecular hydrogen migration within the water molecule; ion momentum maps measured at different laser polarizations yield evidence that such hydrogen migration occurs on ultrafast timescales. We have been able to utilize the momentum maps to deduce that (i) the HD⁺ ion that is formed is vibrationally excited, and (ii) that the electronic state of the precursor HOD²⁺ dication has an essentially linear geometrical structure with elongated O–H and O–D bonds. Our results are in agreement with expectations from ab initio quantum chemical computations of potential energy surfaces of the lowest-energy states of HOD, HOD⁺ and HOD²⁺.

Pulsed-field ionization zero-kinetic-energy photoelectron spectra of D₂ have been recorded from the intermediate $\bar{{\rm{H}}}{}^{1}{{\rm{\Sigma }}}_{g}^{+}$ state to determine the positions of bound rovibronic levels of ${{\rm{D}}}_{2}^{+}$ located within 1400 cm⁻¹ of the D⁺ + D(1s) dissociation threshold. The ion-pair character of the $\bar{{\rm{H}}}$ intermediate state resulted in large changes ${\rm{\Delta }}N={N}^{+}-N$ of the rotational quantum number upon photoionization, which enabled the observation of levels of ${{\rm{D}}}_{2}^{+}$ with rotational quantum number ${N}^{+}$ as high as 10. The experimental data cover a range of levels within which the usual hierarchy of timescales of the electronic, vibrational and rotational motions is inverted. The term values of these levels with respect to the X ${}^{1}{{\rm{\Sigma }}}_{g}^{+}(v=0,N=0)$ rovibronic ground state of D₂ and the energy intervals of the ionic states, measured with an accuracy of typically 0.11 cm⁻¹ and 0.02 cm⁻¹, respectively, are compared with positions calculated ab initio at various degrees of approximation, starting from the Born–Oppenheimer approximation and successively including adiabatic, nonadiabatic, relativistic and radiative corrections. The comparison shows that the accuracy of the photoelectron-spectroscopic measurement is sufficient to reveal the effects of the adiabatic, nonadiabatic, relativistic and radiative corrections on the absolute term values. Comparing our calculations, which rely on an approximate evaluation of the nonadiabatic corrections based on effective R-dependent reduced masses, with the theoretical results for ${N}^{+}\leqslant 5$ by Moss (1993 J. Chem. Soc. Faraday Trans. 89 3851) and for ${N}^{+}\leqslant 8$ by Wolniewicz and Orlikowski (1991 Mol. Phys. 74 103) enables the quantification of the errors introduced by our approximative treatment of the nonadiabatic corrections. Improved rotational term values of the $\bar{{\rm{H}}}(v=12)$ level were also derived.

We present the observation of a checkerboard-like interference pattern in transverse momentum distributions measured for near-threshold photoionization of hydrogen atoms in a DC electric field. We analyze the pattern in terms of constructive and destructive interference between electron trajectories that directly leave the vicinity of the ion and indirect trajectories that remain in the vicinity of the ion for one or more orbital periods, and show that the interference pattern can be discussed in terms of ionization time delays between these two classes of trajectories.

We studied the photo double ionization of hydrogen molecules in the threshold region (50 eV) and the complete photo fragmentation of deuterium molecules at maximum cross section (75 eV) with single photons (linearly polarized) from the Advanced Light Source, using the reaction microscope imaging technique. The 3D-momentum vectors of two recoiling ions and up to two electrons were measured in coincidence. We present the kinetic energy sharing between the electrons and ions, the relative electron momenta, the azimuthal and polar angular distributions of the electrons in the body-fixed frame. We also present the dependency of the kinetic energy release in the Coulomb explosion of the two nuclei on the electron emission patterns. We find that the electronic emission in the body-fixed frame is strongly influenced by the orientation of the molecular axis to the polarization vector and the internuclear distance as well as the electronic energy sharing. Traces of a possible breakdown of the Born–Oppenheimer approximation are observed near threshold.

Investigations on the simplest benchmark system ${{\rm{H}}}_{2}^{+}$ can reveal most underlying mechanisms for the intricate dynamics of molecular systems induced by strong laser pulses. However, due to the two-center Coulomb potential and the highly nonlinear nature of the electron dynamics, the accurate computation of the above threshold ionization spectra remains challenging, especially at large internuclear distances and high laser intensities. In the present work, we implement a new Gauss-quadrature approximation (GA) in the framework of finite element discrete variable representation to solve the time-dependent Schrödinger equation of ${{\rm{H}}}_{2}^{+}$ in strong laser fields. By using this GA, one can arrive at a matrix representation of the first derivative operator that keeps its anti-hermiticity. This crucial feature allows a very stable propagation of the wavefunction under the usual Lanczos scheme. Combining with a wavefunction splitting in the asymptotic region, we show that our present numerical method can reliably deal with the electronic dynamics of stretched molecules at large internuclear distances for high laser intensities and long pulse durations. Accurate photoelectron momentum distributions under these conditions are presented and the distinct features due to the two-center potential are discussed.

We consider the hydrogen atom H($2s$) exposed to a short laser pulse with a central frequency ${\omega }_{0}$ ranging from 136 eV to 1.5 keV (${\omega }_{0}\approx 5\mbox{--}55$ au) at the intensity of $3.51\times {10}^{16}$ W cm⁻². We study stimulated Raman scattering transitions to the $1s$ ground state (anti-Stokes) and to the upper ns, np and nd states (with $n\gt 2$) (Stokes). Nondipole (retardation) effects are included up to ${ \mathcal O }(1/c)$ (c is the speed of light in a vacuum). The calculation of the transition probabilities, based on the integration of the time-dependent Schrödinger equation, is confronted with results obtained by applying perturbation theory. The two methods are in very good agreement. We show that retardation effects play an important role for frequencies ${\omega }_{0}$ larger than a few hundred eV and pulse durations of the order of the femtosecond. In this regime, as the frequency ${\omega }_{0}$ increases, the contribution of the ${{\bf{A}}}^{2}$ term dominates over ${\bf{A}}\cdot {\bf{P}}$ (where ${\bf{A}}$ and ${\bf{P}}$ denote the vector potential of the field and the momentum operator, respectively). Results are presented for various values of the pulse frequency and duration.

We report a combined experimental and theoretical study on the low-energy (E₀ = 31.5 eV) electron-impact ionization of molecular hydrogen (H₂). Triple differential cross sections are measured for a range of fixed emission angles of one outgoing electron between ${\theta }_{1}=-70^\circ$ and −130° covering the full 4π solid angle of the second electron. The energy sharing of the outgoing electrons varies from symmetric (${E}_{1}={E}_{2}=8$ eV) to highly asymmetric (E₁ = 1 eV and E₂ = 15 eV). In addition to the binary and recoil lobes, a structure is observed perpendicular to the incoming beam direction which is due to multiple scattering of the projectile inside the molecular potential. The absolutely normalized experimental cross sections are compared with results from the time-dependent close-coupling (TDCC) calculations. Molecular alignment dependent TDCC results demonstrate that these structures are only present if the molecule axis is lying in the scattering plane.

Benchmark reactions involving molecular hydrogen, such as H₂ + D or H₂ + Cl, provide the ideal platforms to investigate the effect of near threshold resonances (NTR) on scattering processes. Due to the small reduced mass of such systems, shape resonances in certain partial waves can provide features at scattering energies up to a few kelvin, achievable in current experiments. We explore the effects of NTRs on elastic and inelastic scattering for various partial waves ℓ (in the case of H₂ + Cl for s-wave and H₂ + D for p-wave scattering) and find that NTRs lead to a different energy scaling of the cross sections as compared to the well known Wigner threshold regime. We give a theoretical analysis based on Jost functions, and we explore the NTR effects for higher partial waves using a simple model that incorporates the key ingredients of coupled-channel scattering problems. The effect of long-range interactions is also discussed, with special attention paid to the appearance of an effective Wigner regime for elastic scattering in certain partial waves.

We discuss a number of aspects regarding the physics of ${{\rm{H}}}_{2}^{+}$ and H₂. This includes low-energy electron scattering processes and the interaction of both weak (perturbative) and strong (ultrafast/intense) electromagnetic radiation with those systems.

Molecular hydrogen is a benchmark system for bound state quantum calculation and tests of quantum electrodynamical effects. While spectroscopic measurements on the stable species have progressively improved over the years, high-resolution studies on the radioactive isotopologues ${{\rm{T}}}_{2}$, $\mathrm{HT}$ and $\mathrm{DT}$ have been limited. Here we present an accurate determination of T₂ $Q(J=0\mbox{--}5)$ transition energies in the fundamental vibrational band of the ground electronic state, by means of high-resolution coherent anti-Stokes Raman spectroscopy. With the present experimental uncertainty of $0.02\,{\mathrm{cm}}^{-1}$, which is a fivefold improvement over previous measurements, agreement with the latest theoretical calculations is demonstrated.

In two recent papers, Blum and Lohmann (2016 Phys. Rev. Lett. 116 033201) and Lohmann et al (2016 Phys. Rev. A 94 032331), the possibility of continuously varying the degree of entanglement between an elastically scattered electron and the valence electron of quasi-one electron targets was discussed. Here we present results for elastic electron scattering from atomic hydrogen in the energy regime of 1−10 eV and the full range of scattering angles $0^\circ -180^\circ$. We confirm previous calculations at very low energies, which predicted that the hydrogen target is not a promising candidate for Bell correlations through electron collisions. This finding remains unchanged in the near-resonance regime of incident electron energies just below 10 eV. In addition to the spin-correlation parameter P, we present the angle-integrated total cross section, as well as the angle-differential cross section at a few representative collision energies.

We study the rescattering dynamics of a one-dimensional hydrogen molecule with the nuclei fixed at various internuclear distances. By analysing the laser-intensity dependence of the yields of the ${{\rm{H}}}_{2}^{+}$ ions, we show that the rescattering process strongly depends on the internuclear distance. The dynamics of the ionic excitation is found to reflect the interplay of the rescattering dynamics and the charge-resonant states of the ${{\rm{H}}}_{2}^{+}$ ion. In addition, we propose an internuclear collisional excitation mechanism for interpreting the enhanced excitation of the ${{\rm{H}}}_{2}^{+}$ ion at large nuclear distances.

${{\rm{H}}}_{2}^{+}$ is an ideal candidate for the detailed study of strong-field coherent control strategies inspired by basic mechanisms referring to some specific photodissociation resonances. Two of them are considered in this work, namely: zero-width resonances (ZWRs) and coalescing pairs of resonances at exceptional points (EPs). An adiabatic transport theory based on the Floquet Hamiltonian formalism is developed within the challenging context of multiphoton dynamics involving nuclear continua. It is shown that a rigorous treatment is only possible for ZWRs, whereas adiabatic transport mediated by EPs is subjected to restrictions. Numerical maps of resonance widths and non-adiabatic couplings in the laser parameter plane help in optimally shaping control pulses. Full time-dependent wavepacket dynamics shows the possibility of selective, robust filtration and vibrational population transfers, within experimentally feasible criteria.

We construct the velocity map images of the proton transfer reaction between helium and molecular hydrogen ion ${{\rm{H}}}_{2}^{+}$. We perform simulations of imaging experiments at one representative total collision energy taking into account the inherent aberrations of the velocity mapping in order to explore the feasibility of direct comparisons between theory and future experiments planned in our laboratory. The asymptotic angular distributions of the fragments in a 3D velocity space is determined from the quantum state-to-state differential reactive cross sections and reaction probabilities which are computed by using the time-independent coupled channel hyperspherical coordinate method. The calculations employ an earlier ab initio potential energy surface computed at the FCI/cc-pVQZ level of theory. The present simulations indicate that the planned experiments would be selective enough to differentiate between product distributions resulting from different initial internal states of the reactants.

We present a new derivation of the proton–electron mass ratio from the hydrogen molecular ion, HD⁺. The derivation entails the adjustment of the mass ratio in highly precise theory so as to reproduce accurately measured ro-vibrational frequencies. This work is motivated by recent improvements of the theory, as well as the more accurate value of the electron mass in the recently published CODATA-14 set of fundamental constants, which justifies using it as input data in the adjustment, rather than the proton mass value as done in previous works. This leads to significantly different sensitivity coefficients and, consequently, a different value and larger uncertainty margin of the proton–electron mass ratio as obtained from HD⁺.

This review article focuses on imaging and controlling ultrafast dynamics of the hydrogen molecule and its cation, initiated by ultrashort laser pulses. We discuss the mechanisms underlying these dynamics and theoretical methods to describe them. A broad variety of defining and influencing theoretical and experimental results is presented. We put special emphasis on the required experimental techniques, many of which have been developed in view of imaging the fastest of all nuclear dynamics.

Experimental and theoretical differential cross sections (DCS) for the electron-impact excitation of molecular hydrogen to the $B{}^{1}{{\rm{\Sigma }}}_{u}^{+}$, $c{}^{3}{{\rm{\Pi }}}_{u}$, $a{}^{3}{{\rm{\Sigma }}}_{g}^{+}$, $C{}^{1}{{\rm{\Pi }}}_{u}$, and the $E(F){}^{1}{{\rm{\Sigma }}}_{g}^{+}$ states are presented at incident energies near to threshold. The experimental DCSs were taken at incident energies of 14, 15, 16 and 17.5 eV and for scattering angles from 10° to 130°. The theoretical DCSs are from the convergent close-coupling method which has recently successfully modeled differential electron scattering from H₂ when compared with available experiment at energies of 17.5 eV and above.