Strong Field Non-Franck-Condon Ionization of H$_2$: A Semi-Classical Analysis

TL;DR

The paper addresses how strong-field ionization of H$_2$ proceeds and how the resulting vibrational distribution in H$_2^+$ departs from Franck-Condon (FC) expectations. It adopts a semi-classical rate framework based on MO-ADK and MO-PPT to compute ionization and Coulomb-explosion channels, explicitly incorporating internuclear distance via Ip$(R)$ and field dependence through $W_H2(R,t)$, with linear and circular polarization treated through $F(t)$ and angular factors. The authors show that circular polarization yields higher ionization efficiency than linear polarization due to greater fluence, and that the vibrational distribution of H$_2^+$ formed during ionization is non-FC, preferentially populating low-lying $v_+$ states because the ionization rate climbs with $R$ (via Ip$(R)$). The MO-PPT model generally agrees with full TDSE results across TI and MPI, whereas MO-ADK performs well mainly in the TI limit; this work thus provides a practically useful link between strong-field ionization rates and the initial vibronic state of the molecular ion, setting the stage for including post-ionization nuclear dynamics and double ionization effects in future work.

Abstract

Single ionization of H$_2$ molecules exposed to strong and short laser pulses is investigated by a semi-classical method. Three laser characteristics are considered: i) The carrier-wave frequency corresponds to wavelengths covering and bridging the two ionization regimes: From tunnel ionization (TI) at 800 nm to multiphoton ionization (MPI) at 266 nm. ii) Values of the peak intensity are chosen within a window to eliminate competing double ionization processes. iii) Particular attention is paid to the polarization of the laser field, which can be linearly or circularly polarized. The results and their interpretation concern two observables, namely the end-of-pulse total ionization probability and vibrational distribution generated in the cation H$_2^+$. The most prominent findings are an increased ionization efficiency in circular polarization and a vibrational distribution of the cation that favors lower-lying levels than those that would be populated in a vertical (Franck-Condon) ionization, leading to non Franck-Condon distributions, both in linear and circular polarizations.

Figures (10)

• Figure 1: Depiction of the ionization paths of the H$_2$ vibrational state $v=0$ (wave function depicted in green) from its initial potential well (brown) into its ionized vibrational states $v_+$ (green) in the H$_2^+$ ground electronic potential (red), and ultimately to its Coulomb explosion potential H$^+$+H$^+$ (blue) with the kinetic energy $E$ (green).
• Figure 2: Panel (a): 800 nm circularly polarized laser electric field components along the $z$-axes ($F_z(t)$, full blue line), and along the $x$-axes ($F_x(t)$, dotted red line) as a function of time. The H$_2$ molecular axis is aligned along $z$-axis. $\theta$ is the angle between the intermolecular axis and the instantaneous field direction. Panel (b): Square modulus of the linear (full blue line) and circular (dashed purple line) fields as a function of time.
• Figure 3: Final populations of H$_2$ ($P_{\mathrm{H}_{2\!}}(t_f)$; dashed blue line), H$_2^+$ ($P_{\mathrm{H}_2^{+\!}}(t_f)$; full green line) and H$^+$ + H$^+$ ($P_{\mathrm{CE}}(t_f)$; Coulomb Explosion, red dotted line), using a 800 nm $(a)$ linearly and $(b)$ circularly polarized field, and the MO-PPT approach for a sin$^2$ pulse envelope of total duration 32 fs corresponding to 12 optical cycles (16 fs FWHM). The vertical purple line in panel (a) indicates the value of the intensity at which the Keldysh parameter is $\gamma=1$.
• Figure 4: Final ionization probability of H$_2$ (in logarithmic scale) using a linearly polarized field of peak intensity $I$ (in logarithmic scale), as given by MO-ADK (dashed red line), MO-PPT (full black line) and TDCI Awasthi_SAE_2008 (grey crosses) methods, with wavelengths of $(a)$ 266 nm over 36 optical cycles, $(b)$ 400 nm over 24 optical cycles and $(c)$ 800 nm over 12 optical cycles. The total pulse duration is therefore fixed at the value 32 fs (16 fs FWHM).
• Figure 5: Final ionization probability of H$_2$ (in logarithmic scale) using a circularly polarized field of peak intensity $I$ (in logarithmic scale), as given by MO-ADK (dashed red line) and MO-PPT (full black line) methods, with wavelengths of $(a)$ 266 nm over 36 optical cycles, $(b)$ 400 nm over 24 optical cycles and $(c)$ 800 nm over 12 optical cycles. The total pulse duration is therefore fixed at the value 32 fs (16 fs FWHM).