Modified saddle-point method applied to high-order above-threshold ionization and high-order harmonic generation: Slater-type versus asymptotic ground-state wave function

We present a modified saddle-point method and apply it to the high-order above-threshold ionization (HATI) and high-order harmonic generation (HHG) processes. When we use the Slater-type orbitals to describe the ground-state wave function of the valence electron in a noble gas atom, we cannot apply the ordinary saddle-point method (SPM) because the singular points of the ionization matrix element, as well as the recombination matrix element for HHG, are in the vicinity of the corresponding saddle points. Therefore, we present a modification of the SPM and show that the obtained results are in excellent agreement with the spectra calculated by numerical integration. Furthermore, we compare the results obtained by the modified SPM with those obtained by the ordinary SPM with the asymptotic ground-state wave function. We conclude that both methods work for HATI, but for HHG we show that it is necessary to use the modified SPM because the recombination happens close to the atomic core, where the electron ground state is well described only by the Slater-type orbitals.

Figure 1

Logarithm of the electron probability density $\rho \left(\mathbf{r}\right)=|{\psi }_0{\left(\mathbf{r}\right)|}^2$ (in a.u.) as a function of the distance from the nucleus on the $x$ axis, for the neon atom with $l=1$ and $m=1$, obtained using the STO ground-state wave function (red solid line) and the asymptotic ground-state wave function (black dashed line).

Figure 2

SP solutions for HATI of neon atoms in a linearly polarized laser field with intensity $I=2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and wavelength 1300 nm, for emission in the direction of the laser field. The dependence of the electron kinetic energy $E_{\mathbf{p}}$ (in units of ponderomotive energy $U_p$) on the real parts of the ionization and rescattering times $t_0$ and $t_r$ is shown. Only the backward-scattering solutions with $m=0,1,2$ are presented. The indices $\beta$ and $m$ are labeled near the corresponding ionization times, while the index $\alpha$ is labeled near the corresponding rescattering time. The contribution of the solutions denoted by dashed lines should be discarded after the cutoff.

Figure 3

Saddle-point (black thick lines) and singular-point solutions (red thin lines) in the complex plane of the ionization time for pairs of solutions $(\beta ,m)=(1,0),(2,0),(1,1)$, and (2,1), denoted in Fig. 2. For each SP solution there are four singular-point solutions, corresponding to four STOs in the wave function of the neon atom. The electron kinetic energy changes along each curve from zero to $E_{\mathbf{p}}=15U_p$. The solid and dashed curves correspond to different values of $\alpha =\pm 1$ and approach each other near the cutoff of the corresponding pair.

Figure 4

Logarithm of the differential ionization rate (in a.u.) of neon atoms as a function of electron kinetic energy $E_{\mathbf{p}}$ (in units of ponderomotive energy $U_p$) for the same parameters as in Fig. 2. (a) Spectra obtained by numerical integration and (b) spectra obtained by the SPM. Black dot-dashed lines represent the spectra in which the asymptotic ground-state wave function is used, while the red solid lines represent the spectra in which the STO ground-state wave function is used. (c) Partial contributions of the most dominant SP solutions. Thinner lines represent contributions obtained using the asymptotic ground-state wave function, while thicker lines represent contributions obtained using the STO ground-state wave function.

Figure 5

Saddle-point solutions for HHG of neon atoms in a linearly polarized laser field with intensity $I=2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and wavelength 1800 nm. The dependence of the harmonic photon energy $n\omega$ (in units of the ponderomotive energy $U_p$) on the real parts of the ionization and recombination times $t_0$ and $t_r$ is presented. Only the solutions with $m=0,1,2$ are shown.

Figure 6

Saddle-point (black thick lines) and singular-point solutions (red thin lines) in the complex plane of the ionization time for pairs of solutions $(\beta ,m)=(1,0),(2,0),(1,1)$, and (2,1), denoted in Fig. 5. The results are presented similarly as in Fig. 3 for HATI, but we have shown only singular-point solutions corresponding to the first two STOs of the neon atom.

Figure 7

Logarithm of the harmonic intensity (in a.u.) as a function of the harmonic order $n$ for HHG of neon atoms for the same parameters as in Fig. 5. The spectra are plotted in the same way as for HATI in Fig. 4.

Figure 8

Electron momentum $q_x\left(t\right)=\mathrm{Re}[k_{\mathrm{st},x}+A_x\left(t\right)]$ (in a.u.) as a function of time (in units of period $T$) for the solution $(\alpha ,\beta ,m)=(-1,1,0)$ for two different harmonic orders $n=51$ (black dashed line) and $n=281$ (red solid line) from the moment of ionization $\mathrm{Re}{t}_0$, to the moment of recombination $\mathrm{Re}{t}_r$. The momentum of the electron at the moment of recombination $q_{x,\mathrm{rec}}\equiv q_x\left(\mathrm{Re}{t}_r\right)$ is equal to 1.00 a.u. for $n=51$, while for $n=281$ it is equal to 3.56 a.u.

Figure 9

Logarithm of (a) electron probability density in momentum space $|{\tilde{\psi }}_0{\left(\mathbf{q}\right)|}^2$ (in a.u.) and (b) absolute square of the recombination matrix element $\langle {\psi }_0\left|j\right|\mathbf{k}+\mathbf{A}\left(t\right)\rangle$ (in a.u.), as functions of electron momentum $q_x$ (restricted to the $x$ axis only) for the neon atom with $l=1$ and $m=1$, obtained using the STO or the asymptotic ground-state wave function. In (a) the red solid line corresponds to the case of STO, while the black dashed line corresponds to the case of asymptotic wave function. In (b), the red lines correspond to the case of STO (solid line for the $x$ and dot-dashed line for the $y$ component), while the black lines correspond to the case of asymptotic wave function (dashed line for the $x$ and dotted line for the $y$ component). (c) shows the vector potential $A_x$ (red dashed line) and the electron momentum $q_x$ (blue solid line), for the solution $(\alpha ,\beta ,m)=(-1,1,0)$ at the moment of recombination, as functions of the harmonic order $n$. Additionally, in (a) the electron momenta at the moments of recombination for $n=51$ and $n=281$ are indicated by blue arrows.