Two-color harmonic spectroscopy of ultrafast Dirac electron dynamics

Abstract

High-harmonic generation (HHG), the hallmark effect of attosecond science, is a nonperturbative nonlinear process leading to the emission of high-harmonic light from gases and solids. In gases, extreme driving laser pulse intensities can deplete the ground state, suppressing harmonic emission during the trailing edge of the pulse. Here, we report a similar effect, pronounced ultrafast carrier saturation dynamics and harmonic emission suppression during nonperturbative harmonic generation (NPHG) in a gapless Dirac semimetal -- highly oriented pyrolytic graphite (HOPG). Remarkably, HOPG supports NPHG at laser intensities as low as $\sim 10^{10}$ W cm$^{-2}$, facilitated by its vanishing bandgap. Ultrafast carrier saturation strongly modulates the interplay between interband and intraband currents, a key characteristic of NPHG in Dirac materials. Using two-color spectroscopy, we reveal the excitation dynamics of Dirac electron-hole pairs as it affects the emission of harmonics during the presence of the driving laser pulse. The excitation of out-of-equilibrium hot carriers and the concomitant saturation near the Dirac points leads to a marked suppression of interband harmonics and induces measurable temporal shifts. These observations are supported by simulations based on semiconductor Bloch equations. Our finding reveal that field-driven carrier saturation plays a critical role in gapless solid NPHG. We demonstrate the potential of NPHG and HHG as a sensitive, all-optical probe of ultrafast carrier dynamics, offering novel opportunities for ultrafast optoelectronics in Dirac materials.

Figures (5)

• Figure 1: Nonperturbative harmonic generation from HOPG by a two-color pulse.a, Conceptual sketch of nonperturbative harmonic generation in HOPG driven by a strong driving laser pulse with 1980 nm central wavelength and a weak second harmonic pulse. b, Three-step model of interband HHG around the Dirac cone. c, Experimentally measured harmonic spectrum with a laser peak intensity of 72 GW cm$^{-2}$ and $3\%$ second harmonic admixture. d, Intensity scaling of the third and fifth harmonic yield when driven by the fundamental pulses only. The third harmonic exhibits a clear transition from a perturbative regime ($\propto I^{3}$) to a nonperturbative regime ($\propto I^{2.2}$), while the fifth harmonic is purely nonperturbative over the measured range, scaling as $I^{2.5}$ at the highest intensity. e, Dependence of the harmonic intensity on laser ellipticity ($\epsilon = E_x/E_y$), for both $x$- and $y$-polarized components.
• Figure 2: Harmonic intensity as a function of $\omega_0$--$2\omega_0$ delay.a, Experimentally measured harmonic intensities as function of the $\omega_0$--$2\omega_0$ delay for the fourth harmonic (H4) in HOPG with $\omega_0$ peak intensity $72\, \mathrm{GW cm^{-2}}$. b, The same for the sixth harmonic (H6) in ZnO with a laser intensity of $430\,\mathrm{GW cm^{-2}}$. c and d, Corresponding theoretical results simulated based on the semiconductor Bloch equations (Eq. \ref{['SBE3_P']}) for comparison with the experimental data of HOPG and ZnO, respectively. Note that for negative delays, the $2\omega_0$ pulse arrives earlier at the harmonic generation medium than the $\omega_0$ pulse. The peak emission of H4 occurs approximately $17.5 \pm 0.2$ fs before the optimal temporal overlap in the experiment and $\sim$ 17 fs in the simulation.
• Figure 3: Population saturation dynamics around Dirac points.a, Calculated conduction band electron population dynamics and b, interband polarization dynamics in momentum space during the excitation driven solely by the fundamental pulse at an intensity of $72\,\mathrm{GW\,cm^{-2}}$. In the gapless Dirac cone, carrier populations can reach up to 50% at each $k$-point, leading to pronounced saturation effects. c, The total conduction band population dynamics show that the excitation of 50% of the final population occurs well before the peak of the driving field. For comparison, the HHG process in ZnO at a peak intensity of $430\,\mathrm{GW\,cm^{-2}}$ shows that the excitation of 50% of the final population coincides with the peak of the laser pulse. d, Time–frequency analysis of the interband harmonics, obtained via a Gabor transform for 1D (Dirac cone), 2D (graphene), and 3D (HOPG) models, reveals that the maximum intensity of the third harmonic (H3) occurs prior to the peak of the driving field.
• Figure 4: H4 emission time shift as a function of laser intensity.a, Experimentally measured normalized H4 intensity as a function of $\omega_0$--$2\omega_0$ delay at different laser intensities. The relative delay position ($\mu_0$) of the maximum H4 intensity shifts to earlier times as the laser intensity increases. For better visibility, we introduce horizontal offsets. b, Comparison of the maximum H4 intensity delay as a function of $\omega_0$ laser intensity obtained from experimental measurements, SBE calculations, and analytical results based on Eq. \ref{['analy_wt']}. Experimental uncertainties for $\mu_0$ are smaller than the marker size at higher intensities.
• Figure 5: Analysis of Dirac electron population dynamics via emission time shift.a, Electron dynamics associated with fourth harmonic generation. The green shaded area indicates the region in momentum space where electron population contributes to H4 generation as the bandgap energy there corresponds to the H4 photon energy. b, Under excitation by a 2 $\mu$m pulse with a peak intensity of 70 GW cm$^{-2}$, the H4 electron population oscillates as a function of propagation time. The time $t_c$ marks the moment when a quarter of the electron population in that region are excited. c, The absolute value of $t_c$ increases with driving laser intensity, indicating an intensity-dependent excitation process. d, A strong correlation is observed between $t_c$ (in the single-color 2 $\mu$m excitation case) and $\mu_0$ (in the two-color spectroscopy case). The color coding in panel d corresponds to that in panel c.