Coherent suppression and dephasing-induced reentrance of high harmonics in gapped Dirac materials

TL;DR

We address high-harmonic generation in gapped Dirac materials and show that destructively interfering intra- and inter-band currents coherently suppress the HHG spectrum in the small-gap regime, as revealed by Semiconductor Bloch Equation simulations. The authors develop a one-dimensional diabatic analysis to explain the suppression and demonstrate that the effect generalizes to both linear (massive Dirac) and quadratic (bilayer graphene) dispersions. Dephasing can reverse the suppression, yielding re-entrant high harmonics with a phase shift $\phi(\omega)-\pi \approx 1/[ (\omega/\omega_0) \tau_2 ]$ and an amplitude scaling $I(\omega) \sim I^{\text{inter}}(\omega)/(\omega T_2)^2$ for $\omega \gg \omega_0$, offering a practical signature and a route to measure dephasing times. The results indicate broad applicability to graphene-like 2D materials and provide a framework to interpret HHG spectra and to control solid-state HHG via coherence and dephasing.

Abstract

High-harmonic generation in solids by intense laser pulses provides a fascinating platform for studying material properties and ultra-fast electron dynamics, where its coherent character is a central aspect. Using the semiconductor Bloch equations, we uncover a mechanism suppressing the high harmonic spectrum arising from the coherent superposition of intra- vs. inter-band contributions. We provide evidence for the generality of this phenomenon by extensive numerical simulations exploring the parameter space in gapped systems with both linear dispersion, such as for massive Dirac Fermions, and with quadratic dispersion, as e.g. for bilayer graphene. Moreover, we demonstrate that, upon increasing dephasing, destructive interference between intra- and inter-band contributions is lifted. This leads to reentrant behavior of suppressed high harmonics, i.e. a crossover from the characteristic spectral "shoulder" to a slowly decaying signal involving much higher harmonics. We supplement our numerical observations with analytical results for the one-dimensional case.

Figures (9)

• Figure 1: Top: Destructive interference between the inter- and intra-band contributions to the HHG emission causes the total signal to be drastically reduced. Bottom left: Quantifying the degree of reduction of the total HHG signal by $R=\ev{I^{\text{inter}}/I^{\text{intra}}}_{\omega}$, we observe that coherent suppression is most efficient for small multi-photon numbers, $M$, and large strong-field parameters $\zeta$, i.e., in the regime of small gaps and strong driving fields (parameters defined in Eq. (\ref{['eq:parameters']})). Markers refer to example spectra in the top panel and in Fig. \ref{['fig:fig2']}. Bottom right: Schematic illustration of gapped Dirac band structure $\pm\sqrt{v_F^2k^2+\Delta^2}$ (solid purple lines) and diabatic energies $\pm v_F k_x$ (dashed) .
• Figure 2: Total frequency-resolved emission intensity $I(\omega)$ (Eq. (\ref{['eq:IDecomposition']}), shaded blue) compared to intra-band (solid green line) and inter-band (dashed orange line) contributions for different multi-photon numbers, $M$, and strong-field parameters, $\zeta$, defined in Eq. (\ref{['eq:parameters']}). Here, we drive a massive Dirac model, Eq. (\ref{['dirac hamiltonian scaled']}), by the electric field in Eq. (\ref{['driving field']}) with $\sigma=3\pi/\omega_0$. Top row panels show intensities with different $M$ for $\zeta=3.2$, demonstrating coherent suppression (CS) due to the interference term in Eq. (\ref{['eq:IDecomposition']}) (not shown) for small $M$ and inter-band dominance for large $M$. Bottom row panels depict results for various values of $\zeta$ at $M=0.18$, indicating appearance of CS for a wide range of $\zeta$. Markers refer to position in parameter space in Fig. \ref{['fig:fig1']}. For a driving frequency of $\omega_0/2\pi=10THz$, the multi-photon numbers $M\in\{0.18,3.2,7.5\}$ correspond to bandgaps of $\Delta\in\{7.5meV, 130meV, 310meV\}$. At a Fermi velocity of $v_F=5.0d5m/s$, strong-field parameters $\zeta\in\{0.56,3.2,7.5\}$ correspond to peak field strengths of $E\in\{0.015MV/cm, 0.083MV/cm,0.19MV/cm\}$.
• Figure 3: Spectral yields from one-dimensional slices through the Brillouin zone (see Eq. \ref{['eq:kyresolved emission']}) for a massive Dirac model with $\zeta = 7.5$ and $M = 0.18$ (all parameters as in Fig. \ref{['fig:fig2']}f ). (a)-(c) show the decomposition of the total (blue shaded) emission spectrum into intra-band (solid green line) and inter-band (dashed orange line) for three different locations of the cut, $\kappa_y = 0.6$ (a), $\kappa_y = 0$ (b) and $\kappa_y = -1.7$ (c). Panel (d) displays the color-coded and frequency-resolved total emission intensity obtained from several horizontal, one-dimensional slices of the Brillouin zone integral for different $\kappa_y$. (e) shows the $\kappa_y$-dependence of the 5th, 11th, and 17th harmonic. All intensities are normalized to the first harmonic of the $\kappa_y=0$ slice for easier comparison.
• Figure 4: Dephasing-induced HHG -- (a) Emission intensity $I(\omega)$ (Eq. (\ref{['eq:IDecomposition']})) and (b) deviation from $\pi$ of relative phase $\phi$ between inter- and intra-band intensities (Eq. (\ref{['eq:IDecomposition3']})) for different dephasing strengths in the one-dimensional massive Dirac model, Eq. (\ref{['eq:dirac1dhamiltonian']}). Colors correspond to scaled dephasing times ${\tau_2\rightarrow\infty}$ (blue), ${\tau_2=2.0}$ (orange) and ${\tau_2=0.2}$ (green). The total emission (a) coincides with ${I^{\mathrm{inter}}(\omega)/[(\omega/\omega_0) T_2]^2}$ (dotted lines) for ${\omega\gg\omega_0}$. This follows directly from ${\phi\!-\!\pi \approx 1/[(\omega/\omega_0) T_2]}$ (dashed lines in (b)). Parameters used are $\zeta=7.5$,$M=0.18$ and $\sigma=3\pi/\omega_0$ as in Figs. \ref{['fig:fig2']}(f), \ref{['fig:fig4']} and marked by $\blacksquare$ in Fig. \ref{['fig:fig1']}. For a driving frequency of $\omega_0/2\pi=10THz$, the scaled dephasing times $\tau_2\in\{0.2,2\}$ correspond to $T_2\in\{20fs,200fs\}$.
• Figure 5: Effect of dephasing on high harmonic emission intensity $I(\omega)$ for the driven two-dimensional massive Dirac model for dephasing times $\tau_2\rightarrow\infty$ (blue), $\tau_2=20.0$ (purple),$\tau_2=2.0$ (orange), and $\tau_2=0.2$ (green) in units of the laser cycle $2\pi/\omega_0$. Re-entrance of high harmonics beyond the plateau is clearly visible, but less pronounced compared to the one-dimensional model (see Fig. \ref{['fig:fig3']}). Parameters are $\zeta=7.5$, $M=0.18$, and $\sigma=3\pi/\omega_0$ as in Figs. \ref{['fig:fig2']}f), \ref{['fig:fig3']} and marked by $\blacksquare$ in Fig. \ref{['fig:fig1']}). The inset shows the dependence of the 33rd harmonic as a function of dephasing time $\tau_2$, normalized by the 33rd harmonic for $\tau_2=10$. For a driving frequency of $\omega_0/2\pi=10THz$, the scaled dephasing times $\tau_2\in\{0.2,2,20\}$ correspond to $T_2\in\{20fs,200fs,2ps\}$.