Role of interband and intraband current in laser interaction with bichromatic quasiperiodic crystals

We study the role of the inter- and intraband current in the laser interaction with the bichromatic quasiperiodic crystals. The interaction dynamics are simulated by solving the time-dependent Schrödinger equation in the $k$ space, and time evolution of the inter- and intraband current is obtained in a gauge-invariant form. We observed that for certain bichromatic potential ratios, the energy band structure of the “valence band” and the “conduction band” facilitates the interband transitions only at the center or at the edge of the Brillouin zone, which leads to a very interesting population transfer mechanism between the bands. The temporal profile of the inter- and intraband current gives a detailed account of the interaction. The higher-order harmonic generation is also studied for these bichromatic optical lattices, and the resultant harmonic yield is commented upon.

Figure 1

The normalized potential denoted by Eq. (1) for 0:1, 1:2, 1:3, and 5:8 are presented in (a). The corresponding band structure showing the first four bands is shown for the ratios 1:2 (c), 1:3 (d), and 5:8 (e). The minimum band gap between the ${\mathrm{VB}}_2$ and ${\mathrm{CB}}_1$ is 9.4 eV for 1:2 (c), 0.85 eV for 1:3 (d), and 0.012 eV for 5:8 ratio (e). The ratio 0:1 is the fairly standard Mathieu-type potential [31], and hence the associated band structure is not presented. The $k$-dependent matrix element of transition from the second band (${\mathrm{VB}}_2$) to the third band (${\mathrm{CB}}_1$) is presented in (b), i.e., $p_k^{23}=\langle {\varphi }_k^2\left|\widehat{p}\right|{\varphi }_k^3\rangle$.

Figure 2

The HHG spectra for the frequency ratios 1:2 (a), 1:3 (b), and 5:8 (c) are presented for $S_{\text{inter}}, S_{\text{intra}}$, and $S_{\text{total}}$. The red arrow represents the minimum band gap between the ${\mathrm{VB}}_2$ and ${\mathrm{CB}}_1$ for respective cases (refer to Fig. 1). The respective inter- and intraband currents are presented for 1:2 (d), 1:3 (e), and 5:8 (f). As mentioned, the inter- and intraband currents in (d)–(f) are plotted on the same axis with different scaling parameters $a$ and $b$.

Figure 3

Temporal evolution of the interband current, along with the ${\mathrm{VB}}_2$ and ${\mathrm{CB}}_1$ band population, is presented for frequency ratio 1:3 (a) and 5:8 (b). The zoomed version of (b) in the range $2.6\tau \le t\le 3.3\tau$ is shown in (c), with $\tau$ being an optical cycle. In (c), the temporal profile of the vector potential (dashed line) is also shown along with the intraband current. In order to represent band population ($\mathcal{P}$) and the interband current on the same scale, an appropriate scale factor $\delta$ is mentioned for interband current, and the quantity $4\mathcal{P}-2$ is plotted, such that $\mathcal{P}=0$ corresponds to $-2$ and $\mathcal{P}=1$ corresponds to $+2$ on the $y$ scale.

Figure 4

Harmonic spectra associated with the interband, intraband, and interference terms [refer to Eqs. (17) and (18)] are presented for the bichromatic ratio 1:3 ($V_0=0.3$ a.u.) (a), 5:8 with $V_0=0.3$ a.u. (b), and $V_0=0.6$ a.u. (c). On the right panel, the respective total HHG spectra are plotted in (d)–(f). The phase difference $\mathrm{\Delta }\phi$ [refer to Eq. (20)] is also shown in the figure.

Figure 5

Harmonic spectra for 1:3 and 5:8 cases are illustrated in (a) and (b), respectively, for different values of the $V_0$ [refer to Eq. (1)]. The energy bands ${\mathrm{VB}}_2, {\mathrm{CB}}_1$, and ${\mathrm{CB}}_2$ for $V_0=0.3$ and 0.6 a.u. cases are illustrated for 1:3 (c) and 5:8 (d) ratios. The temporal evolution of the population of ${\mathrm{CB}}_2$, i.e., ${\mathcal{P}}_{C{B}_2}$, are compared for $V_0=0.3$ and 0.6 a.u. cases for 1:3 (e) and 5:8 (f) ratios.

Figure 6

The variation of the harmonic yield with the potential depth $V_0$ is presented for three different bichromatic ratios in (a). However, the maximum of the band population ${\mathcal{P}}_{C{B}_2}$ dependence on the $V_0$ is shown in (b). In (a), plots above (below), the black dashed line represent the harmonic yield in the energy range 3–10 eV (20–30 eV).

Figure 7

Temporal dependence of $j_{\text{inter}}$ is presented for three different values of $V_0$ for 5:8 ratio case (a). The zoomed version of (a) is illustrated in (b), and the variation of a maximum of $j_{\text{inter}}$ with $V_0$ is shown in (c). It should be noted that for visual appeal, the temporal profiles in (a) for $V_0=0.5$ a.u. and 0.6 a.u. are intentionally shifted by 10 and 20 cycles, respectively, and the arrows in (a) show the region of the temporal domain zoomed in (b).

Figure 8

The time frequency analysis for 5:8 case with $V_0=0.3$ a.u. (a) and 0.6 a.u. (b) is shown. In both the figures, the horizontal dashed gray lines show even harmonics. The temporal profile of the $j_{\text{inter}}$ is also superimposed on these plots.

Figure 9

For 5:8 frequency ratio, the contribution $\mathrm{\Delta }k/\text{BZ}\equiv 30\%$ around $k=0$ is considered, and the HHG spectra for different potential depth is presented in (a). The results presented in (b), (c), and (d) are for the case when $V_0=0.6$ a.u. is used. The harmonic spectra associated with intraband, interband, and interference terms are illustrated in (b). The temporal variation of ${\mathrm{VB}}_1,{\mathrm{VB}}_2, {\mathrm{CB}}_1$, and ${\mathrm{CB}}_2$ are shown in (c). The star on ${\mathrm{VB}}_1$ and ${\mathrm{CB}}_2$ legends in (c) means they are scaled up by the factor 5 and 50, respectively. The inter- and intraband current is presented in (d). The respective scaling factors for inter- and intraband currents are mentioned in the legend of (d).