Edge-state and bulklike laser-induced correlation effects in high-harmonic generation from a linear chain

We explore the significance of laser-induced correlation effects for high-order harmonic generation (HHG) in a linear chain model of a generic band gap material for both single-IR-pulse and VUV-pump–IR-probe HHG schemes. We examine various pump-pulse excitations, and categorize signatures of laser-induced correlation effects into two classes related to edge-state and bulklike effects. The relative importance of the edge-state ef- fects decreases with increasing system size, while the bulklike effects remain. We identify regimes where these effects may alter the harmonic yield by an order of magnitude, but also regions where they may be neglected. We characterize the underlying laser-induced dynamics of the correlational mechanisms, using both an electronic population description in a static field-free basis and a comparison to a band-structure picture with laser-induced time-dependent energy shifts. From this we provide a guideline on when to account for laser-induced electron correlations and give an estimate of the timescale in which the laser-induced electron-electron interactions can cause correlational changes in the harmonic spectra. We find this timescale to be a few tens of femtoseconds.

Figure 1

Part of the band structure resulting from plotting the norm squared of the Fourier transformed occupied and unoccupied KS orbitals for the finite-size system ($N=80$), as a function of their corresponding energies and the Fourier variable $k$. The occupied valence bands are labeled as $\text{VB}i$ with $i=1,2$ and the unoccupied conduction bands are labeled as $\text{CB}i$ with $i=1,2,3,4,5...$. Due to the surrounding vacuum in the simulation box, the free-space (FS) dispersion of $k^2/2$ is also visible.

Figure 2

(a) HHG spectra for a finite-size system ($N=80$) when applying an $n_d=15$-cycle IR driving pulse with ${\omega }_d=0.023$ and $F_{0,d}=0.00552$ with independent electrons and correlated electrons. (b) Electron population in CB4 as a function of time in units of the driving field duration $T_d=2\pi /{\omega }_d$.

Figure 3

(a) Zoom-in of VB2 and CB1 from Fig. 1. The ESs are indicated below the CB by the horizontal line at $\approx -0.18$. The free-space (FS) dispersion is visible. Arrows denote central photon energies, the smallest red arrows depicting the $11{\omega }_d$ multiphoton transition, the solid blue arrow 1 illustrating the preexcitation pulse when resonant with the ES, and arrows 2 denoting preexcitation pulses with frequencies corresponding to energies close to the BG frequency which allows for the blue dash-dotted transition to the lowest energy state in the CB or the blue dashed transition from the VB to the ES at another $k$. (b) Real-space norm squared of the ES KS orbital below the first CB and of the lowest energy state within the first CB for an $N=80$ system. (c) Pulse sequence with the VUV preexcitation and the IR driving laser pulse in units of the period of the IR laser, $T_d$, and with each pulse normalized to the peak field strength of the IR pulse. The pulses drive the transitions in (a); see text for parameters.

Figure 4

HHG spectra (a) for a finite-size system ($N=80$) with independent electrons, (b) for a bulk system with independent electrons, (c) for a finite-size system including laser-induced correlations, and (d) for a bulk system including laser-induced correlations. The spectra are generated applying an IR field ($n_d=15$-cycle IR pulse with ${\omega }_d=0.023$ and $F_{0,d}=0.00552$) to a system that has been preexcited with an ES preexcitation frequency (ES-Pre.), a system preexcited with a BG preexcitation frequency (BG-Pre.) and a reference system without preexcitation (Ref.). The preexcited systems were prepared at a time $\tau =26T_d$ earlier than the center of the IR pulse using an $n_p=153$-cycle VUV pulse with ${\omega }_{P,\mathrm{ES}}=0.235$ or ${\omega }_{P,\mathrm{BG}}=0.239$, $F_{0,p}=0.0005$, and resonant with the ES or BG [see Figs. 3 and 3]. The signals have been scaled by $N^{-2}$ to compare the systems on an equal footing.

Figure 5

Projections of the electronic wave packet onto the initial band structure [using Eq. (14)] during the simulation of the preexcited finite-size system ($N=80$) with parameters as in Fig. 4.

Figure 6

Static band-structure resolved electron populations during the simulation of the preexcited finite-size systems ($N=80$) as given from the projections of Fig. 5 and Eq. (15). In (a) the population in all CBs higher than CB1 is shown. Parameters are defined as in Fig. 4 and the notations VB1, VB2, and CB1 are defined in Fig. 1.

Figure 7

(a) HHG spectra of a finite ($N=80$) system as a function of time delay between preexcitation and driving pulse applying parameters as defined in Fig. 4. The signals have been scaled by $N^{-2}$.

Figure 8

HHG spectra for a finite-size system ($N=80$), with independent electrons and including laser-induced correlations for the ES preexcitation (a, c) and for the BG preexcitation (b, d) with $\tau =0$ and $4T_d$. The parameters are identical to Fig. 7, the spectra of which are cross sections at the given $\tau$. The signal has been scaled by $N^{-2}$.