First-order correction terms in the weak-field asymptotic theory of tunneling ionization

The weak-field asymptotic theory (WFAT) of tunneling ionization in a static electric field is developed to the next order in field. The first-order corrections to the ionization rate and transverse momentum distribution of the ionized electrons are derived. This extends the region of applicability of the WFAT at the quantitative level toward stronger fields, practically up to the boundary between tunneling and over-the-barrier regimes of ionization. The results apply to any atom or molecule treated in the single-active-electron and frozen-nuclei approximations. The theory is illustrated by calculations for hydrogen and noble-gas atoms.

Figure 1

Ratio of the ionization rate to the field factor (51) as a function of field for hydrogen in the ground state. Solid (black) line: exact results. Dotted (blue) line, WFAT(0): the leading-order WFAT results obtained by omitting the linear in $F$ correction term in Eq. (54). Dashed (red) line, WFAT(1): the WFAT results including the first-order correction in $F$, Eq. (54). Dotted-dashed (green) line, WFAT(2), and dashed-double-dotted (magenta) line, WFAT(10): results from Ref. [25] including terms up to order $F^2$ and $F^{10}$, respectively, in the asymptotic expansion for $\Gamma$.

Figure 2

Ratio of the ionization rate to the field factor (51) as a function of field for hydrogen in states with parabolic quantum numbers $(n_{\xi },n_{\eta },m)$ indicated in the figure. Solid (black) lines: exact results. Dotted (blue) lines, WFAT(0): the leading-order WFAT results. Dashed (red) lines, WFAT(1): the WFAT results including the first-order correction in $F$, Eq. (54).

Figure 3

Transverse momentum distributions for hydrogen in the ground state at two representative values of the field $F$. Solid (black) lines: exact results. Dotted (blue) lines, WFAT(0): the leading-order WFAT results obtained by omitting the linear in $F$ correction term in Eq. (60). Dashed (red) line, WFAT(1): the WFAT results including the first-order correction in $F$, Eq. (60).

Figure 4

Real part of the energy (top panel), ionization rate (middle panel), and its ratio to the structure factor (51) (bottom panel) as functions of field for Ne($2p$). Solid lines: exact results. Dashed (red) line in the top panel: perturbation theory, Eq. (27a). Dotted (blue) lines in the lower two panels, WFAT(0): the leading-order WFAT results obtained by omitting the first-order correction terms in Eq. (53). Dashed (red) lines in the lower two panels, WFAT(1): the WFAT results including the first-order correction, Eq. (53).

Figure 5

Same as in Fig. 4, but for Ar($3p$).

Figure 6

Same as in Fig. 4, but for Kr($4p$).

Figure 7

Same as in Fig. 4, but for Xe($5p$).

Figure 8

Transverse momentum distributions for Ne($2p$) at two representative values of the field $F$. Solid (black) lines: exact results. Dotted (blue) lines, WFAT(0): the leading-order WFAT results obtained by omitting the first-order correction terms in Eq. (58). Dashed (red) lines, WFAT(1): the WFAT results including the first-order correction, Eq. (58).

Figure 9

Same as in Fig. 8, but for Ar($3p$).

Figure 10

Same as in Fig. 8, but for Kr($4p$).

Figure 11

Same as in Fig. 8, but for Xe($5p$).

Figure 12

Transverse momentum distribution $P\left(k_{\perp }\right)$ for Ar($3p$) at $F=0.04$ divided by the Gaussian distribution $P\left(0\right)\exp (-\varkappa k_{\perp }^2/F)$ as a function of the scaled transverse momentum. Solid (black) line: exact results. Dotted (blue) line, WFAT(0): the leading-order WFAT results. Dashed (red) line, WFAT(1): the WFAT results including the first-order correction.