Spatial distribution on high-order-harmonic generation of an ${\mathrm{H}}_2{}^{+}$ molecule in intense laser fields

High-order-harmonic generation (HHG) for the ${{\mathrm{H}}_2}^{+}$ molecule in a 3-fs, 800-nm few-cycle Gaussian laser pulse combined with a static field is investigated by solving the one-dimensional electronic and one-dimensional nuclear time-dependent Schrödinger equation within the non-Born-Oppenheimer approximation. The spatial distribution in HHG is demonstrated and the results present the recombination process of the electron with the two nuclei, respectively. The spatial distribution of the HHG spectra shows that there is little possibility of the recombination of the electron with the nuclei around the origin $z=0$ a.u. and equilibrium internuclear positions $z=\pm 1.3$ a.u. This characteristic is irrelevant to laser parameters and is only attributed to the molecular structure. Furthermore, we investigate the time-dependent electron-nuclear wave packet and ionization probability to further explain the underlying physical mechanism.

Figure 1

(a1) and (b1) Electric field (blue solid line) and ionization probability (red dashed line) in a Gaussian laser pulse ($\alpha =0$) and the combination of a Gaussian laser pulse and a static laser field ($\alpha =0.4$), respectively. (a2) and (b2) Dependence of the harmonic order on the ionization time (green circles) and emission time (blue triangles) in a Gaussian laser pulse ($\alpha =0$) (the recombination with the nuclei along both the positive- and negative-$z$ directions is shown) and the combination of a Gaussian laser pulse and a static laser field ($\alpha =0.4$) (only the recombination with the nucleus along the negative-$z$ direction is shown), respectively. The inset in (b2) shows the dependence of the harmonic order on the ionization time (green circles) and emission time (blue triangles) in the region of time from −0.5 to 1.0 optical cycle (o.c.), where only the recombination with the nucleus along the positive-$z$ direction is shown. (a3) and (b3) Electronic probability density $P_z(z,t)$ from −2 to 2 o.c. in a Gaussian laser pulse ($\alpha =0$) and the combination of a Gaussian laser pulse and a static laser field ($\alpha =0.4$), respectively.

Figure 2

(a) Map of the value of $d_A(z,t)$ and (b) spatial distribution of the HHG spectra as a function of the electronic coordinate and harmonic order for the 1D ${\mathrm{H}}_2^{+}$ molecule in the Gaussian laser field ($\alpha =0$). The white dashed lines indicate the positions of the electronic coordinate at the origin $z$ = 0 a.u. and the equilibrium internuclear distance $z=\pm 1.3$ a.u., respectively. The other laser parameters are the same as those in Fig. 1(a1).

Figure 3

Spatial distribution of the HHG spectra as a function of electronic coordinate and harmonic order for different FWHMs (a) $\tau =3\text{fs}$, (b) $\tau =5\text{fs}$, and (c) $\tau =7\text{fs}$ in a Gaussian laser pulse with a static field ($\alpha =0.4$). The other laser parameters are the same as those in Fig. 1(b1). The white dashed lines indicate the positions of the electronic coordinate at the origin $z=0$ a.u. and the equilibrium internuclear distance $z=\pm 1.3$ a.u., respectively.

Figure 4

Contour plot of the time evolution of the nuclear probability distribution $P_R(R,t)$ calculated from the 1D non-Born-Oppenheimer approximation TDSE for the ${\mathrm{H}}_2^{+}$ molecule. The other laser parameters are the same as those in Fig. 1(b1).

Figure 5

Probability density distributions of the coupled electron nuclear wave packet for the time evolution from $t=-1.0$ to 0.75 o.c. with each time interval 0.25 o.c. The other laser parameters are the same as those in Fig. 1(b1).

Figure 6

Combined energy of the Coulomb potential and laser field potential at time (a) $t=0.06$ o.c. and (b) $t=0.75$ o.c. The red dot represents the electron. The other laser parameters are the same as those in Fig. 1(b1).