Quantum-path control and isolated-attosecond-pulse generation using H${}_2$${}^{+}$ molecules with moving nuclei in few-cycle laser pulses

We theoretically investigate the high-order harmonic generation (HHG) and isolated-attosecond-pulse generation from a full one-dimensional (1D) model of H${}_2$${}^{+}$ molecule in 3-fs, 800-nm laser pulses by using numerical solutions of the non-Born-Oppenheimer time-dependent Schrödinger equation (TDSE). The numerical results with moving nuclei and the static nuclei are demonstrated. The harmonic spectrum from the 22nd to the cutoff becomes smooth and fewer modulations and an isolated-attosecond pulse with duration 129 as is generated when we consider the nuclear motion. We investigate the emission time of harmonics in terms of the time-frequency analysis, which shows that, with moving nuclei, the long trajectory is suppressed, and the short trajectories is enhanced. We apply the nuclear and electronic probability density and the simulation of classical electron trajectory to illustrate the physical mechanism of HHG.

Figure 1

Harmonic spectrum of the H${}_2$${}^{+}$ molecule generated by a 3-fs, 800-nm laser pulse with the peak intensity $I=4\times {10}^{14}$W/cm${}^2$. (Solid black curve) with static nuclei ($R=2.6$ a.u.); (dotted red curve) with moving nuclei.

Figure 2

The combined Coulomb potential and the laser field for polarization direction of the laser pulse parallel (red dotted line) or antiparallel (green solid line) to the molecular axis, respectively.

Figure 3

The time-frequency distribution of the HHG spectra corresponds to (a) the solid black curve (with static nuclei); (b) the dotted red curve (with moving nuclei) in Fig. 1.

Figure 4

Contour plot of the time evolution of (a) the nuclear probability density P${}_R(R,t)$ and (b) a detailed view of the electronic probability density P${}_z(z,t)$ from $-$1 o.c. to 2 o.c. obtained from the TDSE for the H${}_2$${}^{+}$ molecule with moving nuclei. Selected classical trajectories are also shown in (b) (white solid line). The inset (c) in (a) shows the electric field (blue solid line) and the ionization probability (red dashed line). The parameters are the same as those in Fig. 1.

Figure 5

Harmonic spectrum of H${}_2$${}^{+}$ molecule with moving nuclei generated by a 3-fs, 800-nm laser pulse with different peak intensity. (Solid black curve) $I=3.4\times {10}^{14}$W/cm${}^2$. (Dotted red curve) $I=2.2\times {10}^{14}$W/cm${}^2$. (Dashed green curve) $I=1.6\times {10}^{14}$W/cm${}^2$.

Figure 6

The time-frequency distribution of the HHG spectra corresponds to (a) the solid black curve ($I=3.4\times {10}^{14}$W/cm${}^2$); (b) the dotted red curve ($I=2.2\times {10}^{14}$W/cm${}^2$); (c) the dashed green curve ($I=1.6\times {10}^{14}$W/cm${}^2$) in Figs. 5, 5, and 5 show the contour plot of the time evolution of the nuclear probability density with different peak intensity corresponds to (a), (b), and (c).

Figure 7

The temporal profiles of the attosecond pulses generated by superposing the harmonics (a) from 28th to 59th order and (b) from 35th to 48nd order for the H${}_2$${}^{+}$ with static nuclei. The temporal profiles of the attosecond pulses generated by superposing the harmonics (c) from 30th to 61st order and (d) from 30th to 50th order for the H${}_2$${}^{+}$ with moving nuclei.