Electro-nuclear dynamics of single and double ionization of H$_2$ in ultrafast intense laser pulses

TL;DR

This work addresses the challenge of modeling single and double ionization of H2 in ultrafast laser fields by coupling semi-analytical PPT ionization rates with a quantum-mechanical, vibrationally resolved description of H2+ dynamics. The authors implement a time-dependent four-wavepacket framework that tracks H2 and H2+ on relevant electronic surfaces, including inter-state coupling and laser-driven transitions, and extract ionization probabilities and proton kinetic energy release spectra. Key findings show that vibrational motion is crucial for accurately predicting double ionization and KER spectra, especially at longer wavelengths, where nuclear dynamics occur on timescales comparable to the pulse duration; at shorter wavelengths, vibrational effects may be modest for ultrashort pulses but become prominent as intensity grows and multiple ionization channels open. The approach offers a computationally efficient alternative to full electronic-nuclear simulations and is well-suited for inclusion in PIC codes and other practical modeling frameworks in laser-plasma interactions, providing physically realistic guidance across a broad range of frequencies, intensities, and pulse shapes.

Abstract

We present an efficient method for modeling the single and double ionization dynamics of the H$_2$ molecule in ultrashort intense laser fields. This method is based on a semi-analytical approach to calculate the time-dependent single and double molecular ionization rates and on a numerical approach to describe the vibrational motion that takes place in the intermediate molecular ion H$_2^+$. This model allows for the prediction of the single and double ionization probabilities of the H$_2$ molecule to be made over a wide range of frequencies and laser intensities with limited computational time, while providing a realistic estimate of the energy of the products of the dissociative ionization and of the Coulomb explosion of the H$_2$ molecule. The effect of vibrational dynamics on ionization yields and proton kinetic energy release spectra is demonstrated and, in the case of the latter, discussed in terms of basic strong-field molecular fragmentation mechanisms.

Figures (6)

• Figure 1: Representation of the first and second ionization paths of the H2 molecule, initially in its electronic and vibrational ground states. The lowest energy potential curve, shown in black, corresponds to the $X^1\Sigma_g^+$ electronic state of H2. The two red potential curves, solid and dashed, correspond to the two lowest energy electronic states, $X^2\Sigma_g^+$ (1s$\sigma_g$) and $A^2\Sigma_u^+$ (2p$\sigma_u$), of the H2+ molecular ion. Finally, the highest energy potential curve, shown in blue, corresponds to the Coulomb repulsion of the doubly ionized H+ + H+ system. The initial $v=0$ wave function of H2 is shown in green. Single ionization generates a vibrational wave packet, shown in purple, on the ground electronic curve of the H2+ molecular ion. This wave packet evolves while coupled to the first dissociative excited state of H2+, giving rise to a second wave packet, shown in orange. The nuclear dynamics of these two wave packets in the 1s$\sigma_g$ and 2p$\sigma_u$ electronic states is accompanied by a possible second ionization, which ultimately leads to the Coulomb explosion process, schematically represented here by the three pink wave packets. $W_{\ch{H2}}(R,t)$ is the instantaneous single ionization rate of H2 at internuclear distance $R$ and time $t$. Similarly, $W_g(R,t)$ and $W_u(R,t)$ are the instantaneous ionizations rates of H2+ in the lowest 1s$\sigma_g$ and highest 2p$\sigma_u$ electronic states. Finally, $V_{ug}(R,t)$ is the radiative coupling between the 1s$\sigma_g$ and 2p$\sigma_u$ states at distance $R$ and time $t$.
• Figure 2: Coulomb explosion spectra and ionization dynamics for one-optical-cycle pulses at 266 nm and 800 nm with a peak intensity of $I=10^{15}$ W/cm$^2$. The first row (panels a and c) corresponds to 266 nm, while the second is for 800 nm. The dashed curves with circles are obtained by freezing the nuclear dynamics, while the solid curves take the nuclear dynamics into account. The left side of the figure (panels a and b) shows the Coulomb explosion spectra $P_{\mathrm{c}}(E)$. The right side of the figure (panels c and d) shows the time evolution of the H2 populations in black, H2+ in red and H+ + H+ in green on a logarithmic scale (left axis). The inset in panel d shows the time evolution of the average internuclear distance $\langle R \rangle$ (in atomic units) of the H2+ 1s$\sigma_g$ wave packet in the case where the nuclear dynamics is frozen (purple dashed line with circles) and in the case where it is not (purple solid line). The dotted black lines in panels c and d show the time dependence of the normalized electric field $E(t)/E_0$ on a linear scale (right axis).
• Figure 3: Final populations in H2 (black lines), in H2+ (red lines), and in the Coulomb explosion channel H+ + H+ (green lines) as a function of the peak intensity (log scale between $5 \times 10^{13}$ W/cm$^2$ and $5 \times 10^{15}$ W/cm$^2$) using a 800 nm linearly polarized field with a $\sin^2$ pulse envelope of total duration of 32 fs, corresponding to 12 optical cycles. The dashed lines with circles show the results obtained by freezing the vibrational dynamics, while the solid lines take it into account. The inset shows the time evolution of the average internuclear distance $\langle R \rangle$ (in atomic units) of H2+ between $t=0$ and $t=0.7\,t_f$ for the peak intensity $6.3 \times 10^{14}$ W/cm$^2$ in the case where the nuclear dynamics is frozen (lower decreasing purple line) and in the case where it is not (upper oscillating orange line).
• Figure 4: Single (panel a) and double (panel b) ionization probabilities in a log-log scale as a function of the peak intensity between $4 \times 10^{13}$ W/cm$^2$ and $2 \times 10^{15}$ W/cm$^2$ (panel a) and between $1.5 \times 10^{14}$ W/cm$^2$ and $4 \times 10^{15}$ W/cm$^2$ (panel b) using a 800 nm linearly polarized field with a $\sin^2$ pulse envelope of total duration of 32 fs, corresponding to 12 optical cycles. The black dashed lines with circles show the results obtained by freezing the vibrational dynamics, while the solid red lines take it into account. The blue crosses were extracted from Awasthi2008 who used an $R-$fixed TDCI approach with the same laser parameters.
• Figure 5: Proton Kinetic Energy Release (KER) spectra calculated at 266 nm for a peak intensity of $2 \times 10^{14}$ W/cm$^2$ (panels a and a'), $10^{15}$ W/cm$^2$ (panel b), $1.2 \times 10^{15}$ W/cm$^2$ (panel c) and $1.5 \times 10^{15}$ W/cm$^2$ (panel d). Panel a' is a simple zoom of panel a between 4 and 9 eV. The electric field envelope is characterized by a sin$^2$ shape, and the total pulse duration is 32 fs, corresponding to 36 optical cycles. The total KER spectra $P(E)$ are shown with blue solid lines, and the red dashed lines correspond to the KER spectra $P_{\mathrm{c}}(E)$ associated only with the Coulomb explosion channel. The total probabilities of photodissociation and Coulomb explosion are 1.2 % and 0 % in panel (a), 8.4 % and 3.6 % in panel (b), 10.4 % and 27.6 % in panel (c), and 8.8 % and 43.8 % in panel (d), respectively. The acronyms BS, ATD, SCE and DCE stand for Bond Softening, Above Threshold Dissociation, Sequential Coulomb Explosion and Direct Coulomb Explosion respectively.