Dynamic tunnelling ionization of H2 in intense fields

Intense-field ionization of the hydrogen molecular ion by linearly-polarized light is modelled by direct solution of the fixed-nuclei time-dependent Schrödinger equation and compared with recent experiments. Parallel transitions are calculated using algorithms which exploit massively parallel computers. We identify and calculate dynamic tunnelling ionization resonances that depend on laser wavelength and intensity, and molecular bond length. Results for λ ∼ 1064 nm are consistent with static tunnelling ionization. At shorter wavelengths λ ∼ 790 nm large dynamic corrections are observed. The results agree very well with recent experimental measurements of the ion spectra. Our results reproduce the single peak resonance and provide accurate ionization rate estimates at high intensities. At lower intensities our results confirm a double peak in the ionization rate as the bond length varies. The mechanism of high-intensity ionization by infrared and optical wavelength light is often considered a static tunnelling process. The simplicity of this model is hugely appealing because of the ease of calculation. The ionization rates are effectively independent of wavelength, and to some extent the internal structure of the molecule can be ignored [1]. A rough criterion for validity of this model is given by the Keldysh parameter, γk ≡ √ |Ei|/2Up, where the internal binding energy is (|Ei|) and the external laser-driven kinetic energy is (Up). When the conditions are such that γk ≪ 1, the ionization process is dominated by static tunnelling in which the shape of the potential strongly (exponentially) affects the ionization rate. At certain critical distances between the nuclei, discovered by Codling and co-workers [2], the ionization rate can rise sharply producing a sequence of fast fragments ions at sharply-defined energies. Predictions for ion yields and energies based on classical arguments [1] agree very well with experiments even for large diatomic molecules such as I2. The presence of critical distances would be evident in polyatomic molecules and is also seen in small rare-gas clusters [3]. The tunnelling process is generally relatively fast compared to the vibrational motion of the molecule, so the fixed-nuclei approximation is reasonable. However the tunnelling time may be longer than the optical period of the laser. Under these conditions the process is more accurately termed a dynamic tunnelling process. In this paper we provide evidence of just such effects for Ti:Sapphire light λ ∼ 790 nm at intensities I ∼ 10 W cm. Our theoretical results in this wavelength region do not agree with cycleaveraged static field models, however the results do agree well with the features observed in experimental studies. In a molecule with few electrons the ionization process can be studied quantally with few approximations. For one-electron models, static-field ionization resonances in the potential Letter to the Editor 2 wells [4, 5, 6, 7] occur at distances far from the equilibrium internuclear separation and tend to produce low-energy ions. Experiments have confirmed the existence of enhanced multiphoton ionization in the hydrogen molecular ion at infrared wavelengths [9, 10] but at intensities such that dynamic effects of the field cannot be neglected [11]. The wellestablished Fourier-Floquet analysis [6, 12] is not particularly suitable for the study of long-wavelength excitations as the number of frequency components required is very large. Moreover, this approach supposes continuous wave conditions such that the state decays exponentially from an isolated resonance state with a lifetime longer than the optical cycle or natural orbital period. Conversely, long-wavelength pulses can be described by quasistatic fields under the conditions such that γk ≪ 1. However, simple tunnelling formulae assume exponential decay from a single isolated resonance connected adiabatically to the field-free state. This neglects nonadiabatic transitions within the well [7, 13] and rescattering of the continuum electron [14, 15]. Given these difficulties, the direct solution of the Schrödinger equation has distinct advantages. It is suited for all intensities and electronic states and all wavelengths and pulse shapes. In particular it is capable of describing pulse-shape effects and nonadiabatic transitions. Thus it is highly appropriate for realistic modelling of experiments at infrared wavelengths such that γk ∼ 1. Tunable Ti:Sapphire light λ ∼ 780 − 800 nm interacting with atomic hydrogen for example, achieves the pure tunnelling regime, γk ≈ 0.1, only for intensities I > 1 × 10W cm, while for λ = 800 nm with I ∼ 3 × 10W cm [9], γk ∼ 0.7. Under these latter conditions the ionization rate is well-defined, however a static tunnelling model is unlikely to give a correct estimate of the resonance positions and rates; we show that this is indeed the case. In fact, the dynamic effects displace the critical distances, change the ionization rates, and create electron excitation resonances. In the present work we solve the electronic dynamics exactly by a direct solution of the time-dependent Schrödinger equation (TDSE). This does not include broadening due to the finite focal volume and the corrections due to nuclear motion. Using atomic units, the ground state of the molecular ion is characterized by a bond length Re = 2.0 and rotational constant Be = 1.36 × 10. If the laser pulse duration is relatively short (∼ 20 fs compared with rotational timescale for this molecule (Trot ∼ 1/Be ∼ 200 fs), then the laser-molecule interaction can be regarded as sudden in comparison to the rotation of the system. Neglect of rotation effects is therefore reasonable. In spite of these simplifications the results are very promising and in remarkably good agreement with experiment for the ion energy spectrum. The indications are that appropriate refinements of the model would improve agreement, but this remains a goal for future work. Making these approximations the TDSE, in atomic units, reads