Emission times in high-order harmonic generation

We calculate the emission times of the radiation in high-order harmonic generation using the Gabor transform of numerical data obtained from solving the time-dependent Schrödinger equation in one, two, and three dimensions. Both atomic and molecular systems, including nuclear motion, are investigated. Lewenstein model calculations are used to gauge the performance of the Gabor method. The resulting emission times are compared against the classical simple man’s model as well as against the more accurate quantum orbit model based on complex trajectories. The influence of the range of the binding potential (long or short) on the level of agreement is assessed. Our analysis reveals that the short-trajectory harmonics are emitted slightly earlier than predicted by the quantum orbit model. This partially explains recent experimental observations for atoms and molecules. Furthermore, we observe a distinct signature of two-center interference in the emission times for ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$.

Figure 1

Emission times for a hydrogenic atom with $I_p=0.6$ a.u. The dashed (continuous) lines show the emission times from the SM model (QO model). The squares are the emission times obtained from the Gabor analysis of the Lewenstein model results for the harmonic spectrum. (a) Laser intensity $I=1.2\times {10}^{14}$ W/cm${}^2$ (multiphoton regime); (b) laser intensity $I=3\times {10}^{14}$ W/cm${}^2$ (tunneling regime).

Figure 2

Same as Fig. 1, but for an ${\mathrm{H}}_2$ molecule with vibrational motion included in the Lewenstein model.

Figure 3

From left to right: the Gabor analysis for HHG from an atom in 1D, 2D, and 3D. In all cases, the binding potential is long range. In the 3D case, the laser wavelength is 780 nm. The dashed (continuous) lines show the emission times from the SM model (QO model). The laser intensity is $3\times {10}^{14}$ W/cm${}^2$.

Figure 4

Gabor analysis for HHG from a 1D atom with a long-range binding potential. (a) Laser intensity $1.2\times {10}^{14}$ W/cm${}^2$ (multiphoton regime); (b) laser intensity $3\times {10}^{14}$ W/cm${}^2$ (tunneling regime).

Figure 5

Emission times obtained from the Gabor analysis in Fig. 4. The empty black squares are the maxima of the Gabor-transform amplitudes.

Figure 6

Emission times for a 1D atom. The empty black squares (filled red circles) correspond to a long-range (short-range) binding potential. The laser intensity is $3\times {10}^{14}$ W/cm${}^2$. The right panel is a magnified view of the short trajectory from the left panel.

Figure 7

Gabor analysis for various systems. The upper row shows the results for an atom in 1D, and the lower row shows the results for an atom in 2D. Left panels, short-range potential; right panels, long-range potential. The laser intensity is $3\times {10}^{14}$ W/cm${}^2$.

Figure 8

Gabor analysis for HHG from molecules including the vibrational motion. (a) ${\mathrm{H}}_2$; (b) ${\mathrm{D}}_2$. The laser intensity is $1.2\times {10}^{14}$ W/cm${}^2$.

Figure 9

Emission times obtained from the Gabor analysis in Fig. 8. Filled (red) circles, ${\mathrm{H}}_2$. Empty squares (black), ${\mathrm{D}}_2$.

Figure 10

Same as Fig. 8 for the higher laser intensity $3\times {10}^{14}$ W/cm${}^2$. (a) ${\mathrm{H}}_2$; (b) ${\mathrm{D}}_2$.

Figure 11

Emission times obtained from the Gabor analysis in Fig. 10. Filled (red) circles, ${\mathrm{H}}_2$; empty (black) squares, ${\mathrm{D}}_2$.