Quantum path interference in the wavelength-dependent below-threshold harmonic generation

We have theoretically investigated the wavelength-dependent below-threshold harmonic (BTH) generation in different laser intensity regimes. Based on the simulations of the time-dependent Schrödinger equation, we find a periodic fluctuation in the wavelength-dependent BTH yields with laser intensities in the magnitude of ${10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. This result shows significant differences from the resonance peak structures in the recent study with intensities about ${10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ [Xiong et al., Phys. Rev. Lett. 112, 233001 (2014)]. By performing both the quantum path analysis and time-frequency transform of the harmonic spectra, we demonstrate that the difference in the wavelength-dependent BTH yields is due to the complicated quantum path distributions of BTHs with the intensities in the magnitude of ${10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. And the periodic fluctuation is further demonstrated to originate from the interference between the two quantum paths that dominate the harmonic generation.

Figure 1

(a) Wavelength dependence of the integrated harmonic yield for the ninth harmonic (H9) with the laser intensity varying from $3\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ to $7\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ (from bottom to top). (b) Same as (a), but for some higher laser intensities that vary from $3\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ to $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ (from bottom to top).

Figure 2

(a) Quantum path distribution for H9 driven by a 810 nm laser field with the intensity varying from $3\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ to $7\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. (b) Same as (a), but for the laser intensity varying from $3\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ to $7\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. (c)–(h) Quantum path distributions for H9 at six specific laser intensities [corresponding to the transverse lines in (a) and (b)]: (c) $4\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, (d) $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, (e) $6\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, (f) $4\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$, (g) $5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$, and (h) $6\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 3

(a) Time-frequency distribution for the harmonics generated in an 810 nm laser field with the intensity of $4\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. (b) Same as (a), but for the intensity of $4\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 4

(a) Variations of the integrated harmonic yield for H11 in a narrow wavelength range around 1003 nm. (b)–(d) Same as (a), but for H13 around 1185 nm, H15 around 1368 nm, and H17 around 1550 nm, respectively. Here, the laser intensity is chosen as $3\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. In each panel, the integrated harmonic has the photon energy close to $I_p$ (ionization threshold of hydrogen atom, 13.6 eV).

Figure 5

Modulation period $\delta \lambda$ as a function of the driving laser wavelength with the laser intensity of $3\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 6

(a),(b) Same as Figs. 4 and 4, respectively, but for a lower driving laser intensity of $6\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$.