Correlation effects in high-order harmonic generation from finite systems

Using the Hubbard model we study how the process of high-order harmonic generation (HHG) is modified by electron-electron correlation for both finite and bulk systems. A finite-size enhancement of the HHG signal is found and attributed to electrons backscattering off the lattice edges. Additionally, with the increasing strength of the electron-electron correlation an enhancement of the high-frequency regime of the HHG spectrum is found. This is attributed to the on-site Coulomb repulsion between electrons giving rise to a localized correlation-induced quiver motion of the electrons. The finite-lattice enhancement dominates the HHG spectra from a few harmonic orders until a threshold from which the correlational enhancement dominates. This threshold is determined by the degree of correlation and decreases into the low-frequency regime for increasing electron-electron correlation. This infers that as the electron-electron correlation increases, the finite-size effect on the electron dynamics decreases.

Figure 1

Energies for the six lowest-lying eigenstates for both finite and bulk lattices for a variety of $U$ values. All energies are plotted relative to the ground-state energy of the given system and in units of the laser carrier frequency ${\omega }_L$. Note that degenerate states result in that fewer than six distinct energy levels are visible. See the text for more detail on the degeneracy of the states.

Figure 2

HHG spectra [Eq. (5)] from bulk and finite lattices with (a) $U=0$, (b) $U=0.01t_0$, and (c) $U=0.8t_0$. The spectrum for the finite system with $U=0$ from (a) is indicated in (b) and (c) by a gray curve for direct comparison. See text for laser and system parameters. The two vertical dashed lines indicate the boundaries between the three characteristic regions discussed in the text.

Figure 3

The electron population on the various sites in the finite lattice over time for (a) $U=0$ and (b) $U=0.8t_0$. Plotted on top in red is the driving electric field. Note that the colorbars are different in (a) and (b).

Figure 4

The electric current in both the finite and bulk lattice as a function of time for (a) $U=0$ and (b) $U=0.8t_0$. The green curve shows the applied electromagnetic vector potential $A\left(t\right)$.

Figure 5

HHG spectra generated from the currents between individual sites, see Eq. (12), for bulk and finite lattices and three $U$ values. The spectra of the total current [Eq. (5)] is divided by $N_j^2$ and plotted by full and dashed red curves.

Figure 6

Plots of the electric currents between the different lattice sites for the finite lattice and two different values of $U$. The total current is given in red when divided by $N_j$ [Eq. (6)].