Floquet-Bloch states, quasienergy bands, and high-order harmonic generation for single-walled carbon nanotubes under intense laser fields

We study the electronic states and nonlinear motion of $\pi$ electrons in an armchair nanotube driven by monochromatic intense laser fields with polarization parallel to the nanotube axis. The intensity and frequency (photon energy) of the applied laser fields are varied so their effect on the electrons can be understood. In each case, Floquet-Bloch theory is used to calculate the Floquet-Bloch states, quasienergy band, mean energy band, and electron current density. By summing up the current density of all occupied Floquet-Bloch states, the harmonic generation spectrum can be determined. We demonstrate that the deformation of quasienergy band and mean energy band is related to high-order harmonic generation. Only the states deviating from field-free eigenstates may contribute to high-order harmonic generation.

Figure 1

(Color online) Quasienergy band structure $\epsilon (\mu ,k)$ (in eV) of (10, 10) SWNT in the limit where laser field intensity $I_0$ is infinitesimal. The range of the fundamental zone is the same as the photon energy $\hslash \omega$ of the laser field. (a) $\hslash \omega =1.50\mathrm{eV}$; (b) $\hslash \omega =0.94\mathrm{eV}$.

Figure 2

(Color online) Mean energy band structure $⟨⟨E(\mu ,k)⟩⟩$ of (10, 10) SWNT in the limit where laser field intensity $I_0$ is infinitesimal. Different subbands $\mu$ are colored differently in the same manner as Fig. 1. Notice when the intensity vanishes, photon energy $\hslash \omega$ of the laser field has no effect on mean energy band. (a) $\hslash \omega =1.50\mathrm{eV}$; (b) $\hslash \omega =0.94\mathrm{eV}$.

Figure 3

(Color online) Quasienergy band structure $\epsilon (\mu ,k)$ (in eV) of (10, 10) SWNT in the presence of monochromatic laser fields with polarization parallel to the nanotube axis. The applied laser fields have a fixed photon energy $\hslash \omega =1.50\mathrm{eV}$ and different intensities $I_0$: (a) $I_0=1.91\times {10}^{11}\mathrm{W}∕{\mathrm{cm}}^2$; (b) $I_0=1.04\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$; and (c) $I_0=8.46\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 4

(Color online) Mean energy band structure $⟨⟨E(\mu ,k)⟩⟩$ of (10, 10) SWNT in the presence of monochromatic laser fields with polarization parallel to the nanotube axis. The applied laser fields have a fixed photon energy $\hslash \omega =1.50\mathrm{eV}$ and different intensities $I_0$: (a) $I_0=1.91\times {10}^{11}\mathrm{W}∕{\mathrm{cm}}^2$; (b) $I_0=1.04\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$; and (c) $I_0=8.46\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$. Different subbands $\mu$ are colored differently in the same manner as Fig. 1.

Figure 5

(Color online) The third and fourth Fourier components of the electron current density contributed by the occupied Floquet-Bloch states with $\mu =\pm 7$ and $-0.5\le k\le 0.5$ of (10, 10) SWNT in the presence of monochromatic laser fields with polarization parallel to the nanotube axis. The photon energy of the laser field is $1.50\mathrm{eV}$, and the intensity of the laser field is $1.04\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$. From top to down, they are $J_{7,k}\left(3\omega \right)$, $J_{-7,k}\left(3\omega \right)$, $J_{7,k}\left(4\omega \right)$, and $J_{-7,k}\left(4\omega \right)$, respectively. Notice $J_{\pm 7,k}\left(3\omega \right)=J_{\pm 7,-k}\left(3\omega \right)$ and $J_{\pm 7,k}\left(4\omega \right)=-J_{\pm 7,-k}\left(4\omega \right)$.

Figure 6

(Color online) Several Fourier components of the electron current density contributed by occupied Floquet-Bloch states of (10, 10) SWNT in the presence of monochromatic laser fields with polarization parallel to the nanotube axis. The applied laser fields have a fixed photon energy $\hslash \omega =1.50$ eV and different intensities $I_0$: (a) $I_0=1.91\times {10}^{11}\mathrm{W}∕{\mathrm{cm}}^2$; (b) $I_0=1.04\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$; and (c) $I_0=8.46\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$. States with different $\mu$ are colored differently in the same manner as Fig. 1.

Figure 7

(Color online) HHG spectra of (10, 10) SWNT in the presence of monochromatic laser fields with polarization parallel to the nanotube axis. The applied laser fields have a fixed photon energy $\hslash \omega =1.50\mathrm{eV}$ and different field intensities $I_0$: (a) $I_0=1.91\times {10}^{11}\mathrm{W}∕{\mathrm{cm}}^2$; (b) $I_0=1.04\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$; and (c) $I_0=8.46\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 8

(Color online) Quasienergy band structure $\epsilon (\mu ,k)$ (in eV) of (10, 10) SWNT in the presence of monochromatic laser fields with polarization parallel to the nanotube axis. The applied laser fields have a fixed photon energy $\hslash \omega =0.94\mathrm{eV}$ and different intensities $I_0$: (a) $I_0=1.91\times {10}^{11}\mathrm{W}∕{\mathrm{cm}}^2$; (b) $I_0=1.04\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$; and (c) $I_0=8.46\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 9

(Color online) Mean energy band structure $⟨⟨E(\mu ,k)⟩⟩$ of (10, 10) SWNT in the presence of monochromatic laser fields with polarization parallel to the nanotube axis. The applied laser fields have a fixed photon energy $\hslash \omega =0.94\mathrm{eV}$ and different intensities $I_0$: (a) $I_0=1.91\times {10}^{11}\mathrm{W}∕{\mathrm{cm}}^2$; (b) $I_0=1.04\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$; and (c) $I_0=8.46\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$. Different subbands $\mu$ are colored differently in the same manner as Fig. 1.

Figure 10

(Color online) Several Fourier components of the electron current density contributed by occupied Floquet-Bloch states of (10, 10) SWNT in the presence of monochromatic laser fields with polarization parallel to the nanotube axis. The applied laser fields have a fixed photon energy $\hslash \omega =0.94\mathrm{eV}$ and different intensities $I_0$: (a) $I_0=1.91\times {10}^{11}\mathrm{W}∕{\mathrm{cm}}^2$; (b) $I_0=1.04\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$; and (c) $I_0=8.46\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$. States with different $\mu$ are colored differently in the same manner as Fig. 1.

Figure 11

(Color online) HHG spectra of (10, 10) SWNT in the presence of monochromatic laser fields with polarization parallel to the nanotube axis. The applied laser fields have a fixed photon energy $\hslash \omega =0.94\mathrm{eV}$ and different intensities $I_0$: (a) $I_0=1.91\times {10}^{11}\mathrm{W}∕{\mathrm{cm}}^2$; (b) $I_0=1.04\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$; and (c) $I_0=8.46\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$.