High-Harmonic Generation in Mott Insulators

Using Floquet dynamical mean-field theory, we study the high-harmonic generation in the time-periodic steady states of wide-gap Mott insulators under ac driving. In the strong-field regime, the harmonic intensity exhibits multiple plateaus, whose cutoff energies ${\epsilon }_{\mathrm{cut}}=U+m{E}_0$ scale with the Coulomb interaction $U$ and the maximum field strength $E_0$. In this regime, the created doublons and holons are localized because of the strong field and the $m$th plateau originates from the recombination of $m$th nearest-neighbor doublon-holon pairs. In the weak-field regime, there is only a single plateau in the intensity, which originates from the recombination of itinerant doublons and holons. Here, ${\epsilon }_{\mathrm{cut}}={\mathrm{\Delta }}_{\mathrm{gap}}+\alpha E_0$, with ${\mathrm{\Delta }}_{\mathrm{gap}}$ the band gap and $\alpha >1$. We demonstrate that the Mott insulator shows a stronger high-harmonic intensity than a semiconductor model with the same dispersion as the Mott insulator, even if the semiconductor bands are broadened by impurity scattering to mimic the incoherent scattering in the Mott insulator.

Figure 1

(a) HHG spectra in the strong-field regime, (b) HHG spectra in the weak-field regime, and (c) HHG spectra as a function of the field strength ($E_0$) and the harmonic energy ($n\mathrm{\Omega }$). The arrows and the white circle markers show cutoff energies. White lines in panel (c) indicate $n\mathrm{\Omega }=U+m{E}_0$ and the red line is the fit for the weak-field regime. The inset of panel (a) shows the current and electric field during one period for $E_0=4.0$. We use $U=8.0$, $\beta =2.0$, $\mathrm{\Gamma }=0.06$, $\mathrm{\Omega }=0.5$.

Figure 2

(a) Time-averaged local spectral function [$\overline{A}(\omega )$] of the nonequilibrium steady state as a function of $E_0$. (b) Field-strength dependence of the time-averaged doublon density ($d_0=\overline{⟨n_{\uparrow }{n}_{\downarrow }⟩}$) in the NESS. (c) Log-scale plot of the temporal HHG intensity $I_{hh}(\omega ;t_{\text{probe}})$ for $\mathrm{\Omega }=0.5$, $E_0=4.0$. The dashed lines are $\omega =U\pm mE(t)$. Vertical lines indicate $t_{\text{probe}}=0$, $\mathcal{T}/2$, $\mathcal{T}$. Here, $U=8.0$, $\beta =2.0$, $\mathrm{\Gamma }=0.06$, and $W_{\mathrm{bath}}=5$. (d) Field-strength dependence of the cutoff energy in the weak-field regime for various $U$. Dashed lines are linear fits and the arrows at $E_0=0$ indicate the gaps estimated from the local spectral functions.

Figure 3

(a) Schematic picture of the semiconductor model, Eq. (2). (b)–(d) HHG spectra $I_{hh}$ in the plane of $E_0$ and $n\mathrm{\Omega }$. (b) Type 1 semiconductor model. (c) Same model with additional impurity scattering $V_{\mathrm{imp}}=0.55$. (d) Type 2 semiconductor model. The color scale is the same as in Fig. 2 and $\mathrm{\Omega }=0.5$, $U=8$, $\beta =2.0$, $\mathrm{\Gamma }=0.06$.

Figure 4

(a) Momentum dependent spectral function $A(k,\omega )$ of the Mott insulator in equilibrium. The white lines show the peak position predicted by the H1 approximation and the type 1 semiconductor, while the dashed orange lines show the dispersion of the type 2 semiconductor. (b) Comparison between the local spectrum $A_{\mathrm{loc}}(\omega )$ and $A(k,\omega )$ at ${\epsilon }_k=0$ $[A_{k_0}]$ obtained within DMFT, H1, and for the type 1 semiconductor with $V_{\mathrm{imp}}=0.55$ ($\mathrm{S}+\mathrm{I}$).