Coupled-coherent-states approach for high-order harmonic generation

In this paper, we report a version of the coupled-coherent-states method which is able to accurately compute the high-order harmonic generation (HHG) spectrum of an electron in a laser field in one dimension by the use of trajectory-guided grids of Gaussian wave packets. It is shown that by periodic reprojection of the wave function and dynamically altering the basis set size, the method can account for a wave function which spreads out to cover a large area in phase space while still keeping computational expense low and ensuring the preservation of coherence of the wave function. The HHG spectra obtained show good agreement with those from a time-dependent Schrödinger equation solver. We show also that the part of the wave function which is responsible for HHG moves along a periodic orbit which is far from that of classical motion. Although this paper is a proof of principle and therefore focused on a simple one-dimensional system, future generalizations for the multielectron case are discussed.

Figure 1

The path of the trajectories initially located close to the center of the wave function as they propagate through phase space when not reprojected. The smaller frame shows the path the trajectories take in the region close to the origin. As a result of this, after a short time of approximately 20 a.u., the overlap between the wave-function trajectories and the initial grid points becomes too small for the CCS dynamics to be accurately calculated and the coherent states lose coupling.

Figure 2

Changing number of basis vectors for different values of the basis threshold parameter investigated in a logarithmic fashion. Selected values from a range of $\zeta ={10}^{-1}$ to $\zeta ={10}^{-10}$ are given, namely, $\zeta ={10}^{-1}$ (large dashed line), $\zeta ={10}^{-3}$ (small dashed line), $\zeta ={10}^{-5}$ (dotted line), and $\zeta ={10}^{-8}$ (dot-dashed line), as well as the case where $\zeta =0$ (solid line) and so the number of basis vectors stays constant.

Figure 3

Motion of the wave function over time on a grid of $80\times 20$ coherent states with a separation of $\Delta =1.59$ a.u. and with $\zeta ={10}^{-3}$. Each panel illustrates the entire grid, with white space denoting areas where the coherent states at those grid points have been removed during reprojection. The panels show the motion of the important parts of the wave function after reprojection at $t=20$, 42, 74, and 117 a.u., and clearly illustrates how the external field has an effect on the wave function, drawing a portion of it into an elliptical motion around the core of the wave function. The elliptical motion of this portion of the wave function is illustrated by the dashed line.

Figure 4

(a)–(c) The real, imaginary, and absolute values, respectively, of the ACFs when no reprojection is carried out on the wave function. Here the black dashed line is the CCS result and the red solid line is the result from the TDSE solver.

Figure 5

(a) A comparison between the absolute values of the autocorrelation functions produced by the TDSE solver and the CCS method with $\zeta =0$, showing near total agreement. (b) A comparison between the absolute values of the autocorrelation functions produced by the TDSE solver and the CCS method for the first few values of the basis threshold parameter. It can be seen that the ACF for $\zeta ={10}^{-1}$ shows visible differences compared to the TDSE solver plot, but for values of $\zeta \le {10}^{-3}$, the ACF begins to converge well towards the TDSE result, close enough that values for $\zeta ={10}^{-5}$ agree almost exactly with the TDSE solver plot. Reprojection is carried out at every $\tau =1$ a.u. and the intensity of the external field is set at ${\mathcal{E}}_0=0.1$ with a frequency of ${\omega }_0=0.05$ for both comparisons. In all cases, the wave packet initially starts at $(q,p)=(0,0)$.

Figure 6

Comparison between dipole acceleration graphs for the $\zeta ={\zeta }_{\mathrm{MEC}}$ CCS calculated wave function and the TDSE calculated plot.

Figure 7

Comparisons of HHG spectra generated by the TDSE solver and those generated by the CCS method with various basis threshold parameter values, with (a) $\zeta ={10}^{-1}$, (b) $\zeta ={10}^{-3}$, (c) $\zeta ={\zeta }_{\mathrm{MEC}}$, and (d) $\zeta =0$.