High-order-harmonic generation of vibrating ${{\mathrm{H}}_2}^{+}$ and ${{\mathrm{D}}_2}^{+}$

We study the isotope effect on high-order-harmonic generation (HHG) of vibrating ${{\mathrm{H}}_2}^{+}$ and ${{\mathrm{D}}_2}^{+}$ molecular ions aligned parallel to the polarization of the laser field. The time-dependent Schrödinger equation is solved accurately and efficiently by the time-dependent generalized pseudospectral and Fourier grid methods in three spatial coordinates, one of them being the internuclear separation and the other two describing the electronic motion. The laser pulses have a carrier wavelength of 800 nm and duration of 10 or 16 optical cycles. The peak intensities used in the calculations are $2\times {10}^{14}$ and $3\times {10}^{14}\text{W}/{\mathrm{cm}}^2$. The effect of nuclear vibration is visible in both ${{\mathrm{H}}_2}^{+}$ and ${{\mathrm{D}}_2}^{+}$ but more pronounced in the lighter ${{\mathrm{H}}_2}^{+}$ molecule. Striking differences from the fixed nuclei case are a total disappearance of the traditional plateau in the HHG spectrum at the higher intensity and significant redshift of the harmonic peaks in the central part of the spectrum. These phenomena are explained based on the analysis of the dynamics of the nuclear vibrational wave packet.

Figure 1

HHG spectra of ${{\mathrm{H}}_2}^{+}$ (left) and ${{\mathrm{D}}_2}^{+}$ (right) initially in the ground state ($v=0$) for a peak intensity of the laser field of $2\times {10}^{14}\text{W}/{\mathrm{cm}}^2$ (top panels) and $3\times {10}^{14}\text{W}/{\mathrm{cm}}^2$ (bottom panels). Pulse duration is 10 optical cycles. The red dashed line in all panels shows the spectrum for the nuclei fixed at $R=2$ a.u.

Figure 2

HHG spectra of ${{\mathrm{H}}_2}^{+}$ (left) and ${{\mathrm{D}}_2}^{+}$ (right) initially in the ground state ($v=0$) for a peak intensity of the laser field of $2\times {10}^{14}\text{W}/{\mathrm{cm}}^2$ (top panels) and $3\times {10}^{14}\text{W}/{\mathrm{cm}}^2$ (bottom panels). Pulse duration is 16 optical cycles. The red dashed line in all panels shows the spectrum for the nuclei fixed at $R=2$ a.u.

Figure 3

Time-dependent nuclear density distribution of ${{\mathrm{H}}_2}^{+}$ (left) and ${{\mathrm{D}}_2}^{+}$ (right) initially in the ground state ($v=0$) for a peak intensity of the laser field of $2\times {10}^{14}\text{W}/{\mathrm{cm}}^2$ (top panels) and $3\times {10}^{14}\text{W}/{\mathrm{cm}}^2$ (bottom panels). Pulse duration is 10 optical cycles.

Figure 4

Time-dependent nuclear density distribution of ${{\mathrm{H}}_2}^{+}$ (left) and ${{\mathrm{D}}_2}^{+}$ (right) initially in the ground state ($v=0$) for a peak intensity of the laser field of $2\times {10}^{14}\text{W}/{\mathrm{cm}}^2$ (top panels) and $3\times {10}^{14}\text{W}/{\mathrm{cm}}^2$ (bottom panels). Pulse duration is 16 optical cycles.

Figure 5

Time-frequency dipole acceleration SST spectrum of ${{\mathrm{H}}_2}^{+}$ initially in the vibrational states $v=0$ for a peak intensity of the laser field of $3\times {10}^{14}\text{W}/{\mathrm{cm}}^2$ and a pulse duration of 10 optical cycles. The horizontal yellow line shows vertical ionization potential at $R=2$ a.u.

Figure 6

HHG spectra of ${{\mathrm{H}}_2}^{+}$ (solid blue line) and ${{\mathrm{D}}_2}^{+}$ (dashed green line) initially in the ground state ($v=0$) for a peak intensity of the laser field of $3\times {10}^{14}\text{W}/{\mathrm{cm}}^2$ a and pulse duration of 16 optical cycles. Black vertical lines indicate odd harmonic orders.