Space-time contours to treat intense field-dressed molecular states. II. Applications

This second article in the two back-to-back articles presents a numerical application to support and strengthen two theoretical findings extensively discussed in the previous article (article I). In I, we found that introducing the space-time contours enables to distinguish between $N$⁠, the number of nuclear Schrödinger equations to be solved, and $L$⁠, the number of field-free states that become populated by the external field (in the ordinary, perturbative approaches this distinction is not apparent). In the numerical study we show, employing the electronic transition probability matrix $P(s,t)$ [which closely is related to the transformation matrix $ω(s,t)$—see Eqs. (21) and (25) in I], that the $N=L$ case is rare and in most cases we have $N<L$⁠. Since the perturbative approach can be shown to follow when $N=L$ (see Sec. III C in I) the numerical study implies that in most cases the perturbative approach is not reliable. The second issue that is studied is related to the diabatization process. It is shown, numerically, that the $N<L$ case, in general, does not lead to field-dressed diabatic potentials which are single valued. However, if $N$ is chosen to be identical to the number of field-free states that yield field-free single-valued diabatic potentials in a given spatial region then the corresponding $N$ field-dressed states also yield single-valued (field-dressed) diabatic potentials. This result is independent of $L$⁠. The numerical study is carried out for an eigenvalue problem based on the Mathieu equation.