Suppression of strong-field ionization by optimal pulse shaping: Application to hydrogen and the hydrogen molecular ion

We investigate the ability of quantum optimal control theory to shape pulses suppressing strong-field ionization of a hydrogen atom and a ${\mathrm{H}}_2{}^{+}$ molecule. We show that considerable suppression of the ionization yield can be achieved for both H and ${\mathrm{H}}_2{}^{+}$ with optimal pulse shaping for a fixed fluence and pulse length. The mechanisms responsible for ionization suppression and the shape of the optimized pulse are different for infrared and ultraviolet laser fields. In the low-frequency regime the optimized pulse reduces the ionization yield by suppressing the highest peaks of the laser field. For the higher laser frequencies considered the ionization yield of H can be decreased by exciting low-lying resonances.

Figure 1

(a) Total bound-state occupation, (b) ground-state occupation, and (c) ionization yield (arbitrary units) of an H atom at the end of the laser pulse given by Eq. (4) as functions of the pulse amplitude $F_0$ and the frequency $\omega$. The pulse length is $T=206.71$ a.u. (5 fs), and the number of cycles ranges from 7 to 16 for $\omega =0.2$ a.u. and $\omega =0.5$ a.u., respectively.

Figure 2

(a) Total bound-state [dashed (red) curve] and ground-state [solid (blue) curve] occupations in H at the end of the laser pulse as a function of the pulse frequency. (b) Ionization yield as a function of pulse frequency. The field strength is $F_0=0.15$ a.u., and the pulse length is as in Fig. 1. Dashed vertical lines indicate the frequencies studied in Fig. 3.

Figure 3

Occupations of some of the lowest states [(a), (c), (e), and (g)], and total bound-state occupations along with continuum populations [(b), (d), (f), and (h)] in H during a laser pulse with $F_0=0.15$ a.u. for four laser frequencies: (a, b) 0.35 a.u., (c, d) 0.4 a.u., (e, f) 0.45 a.u., and (g, h) 0.5 a.u. The pulse length is as in Figs. 1 and 2.

Figure 4

(a) Initial laser pulse with $F_0=0.15$ a.u. and $\omega =0.4$ a.u. [dashed (blue) curve] and optimized pulse [solid (red) curve] with a fixed fluence equal to 1.16 a.u. (b) Fourier spectra of the pulses. (c) Occupations of some of the bound states during the optimized pulse. (d) Total bound-state occupation and the population in the continuum during the optimized pulse.

Figure 5

(a) Initial laser pulse with $F_0=0.1$ a.u. and $\omega =0.0459$ a.u. [dashed (blue) curve] and optimized pulse [solid (red) curve]. The pulse fluence is fixed to 2.07 a.u. (b) Fourier spectra of the pulses. (c) Time dependence of the ground-state occupation in an H atom for the initial pulse [dashed (blue) curve] and the same dependence for the optimized pulse [solid (red) curve]. (d) Continuum occupation during the initial [dashed (blue) curve] and the optimized [solid (red) curve] laser pulses.

Figure 6

Intensity dependence of the ionization yield of an ${\mathrm{H}}_2{}^{+}$ molecule for the initial [dashed (blue) curve] and optimized [solid (red) curve] laser pulses. The internuclear axis is parallel to the polarization direction. The frequency of all initial pulses is $\omega =0.114$ a.u. The fluence and the pulse duration are fixed to 3.1 and 206.71 a.u. (5 fs), respectively. The maximum frequency allowed in the optimization is (a) ${\omega }_{\max }=0.23$ a.u. and (b) ${\omega }_{\max }=0.114$ a.u.

Figure 7

Initial laser pulse [dashed (blue) curve] for $\omega =0.114$ a.u. and two optimized pulses [dash-dotted (green) curve and solid (red) curve] that suppress the ionization of the ${\mathrm{H}}_2{}^{+}$ molecule at the intensity of $2.25\times {10}^{15}$ ${\mathrm{W}/\mathrm{cm}}^2$. Dash-dotted (green) and solid (red) curves correspond to ${\omega }_{\max }=0.23$ a.u. and ${\omega }_{\max }=0.114$ a.u., respectively.