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The time-dependent Schrodinger equation (TDSE) is usually treated in real space in the textbook. However, it makes the numerical simulations of strong-field processes difficult due to the wide dispersion and fast oscillation of the electron wave packets under the interaction of intense laser fields. Here we demonstrate that the TDSE can be efficiently solved in the momentum space. The high-order harmonic generation and above-threshold ionization spectra obtained by numerical solutions of TDSE in momentum space agree well with previous studies in real space, but significantly reducing the computation cost.

High harmonic generation in solids provides us compact and coherent UV sources. Our simulations illustrate abnormal harmonic yield dependence on the intensity of driving or pre-excitation pulses. A multielectron interference model is proposed to reveal the mechanism of the extreme nonlinear optical phenomena. We find that the $k$-resolved intraband harmonic demonstrates different symmetry, phase, and amplitude, which can be observed by using a resonant pre-excitation pulse. The position of the interference minimum obtained by solving the semiconductor Bloch equations agrees well with that obtained by our $k$-resolved semiclassical model. This interference model can help us probe the $k$-resloved band structure and optimize the ultrafast electron-hole dynamics for solid harmonic processes.

Various interference effects are known to exist in the process of high harmonic generation (HHG) both at the single atom and macroscopic levels. In particular, the quantum path difference between the long and short trajectories of electron excursion causes the HHG yield to experience interference-based temporal and spectral modulations. In solids, due to additional phenomena such as multi-band superposition and crystal symmetry dependency, the HHG mechanism appears to be more complicated than in gaseous atoms in identifying accompanying interference phenomena. Here, we first report experimental data showing intensity-dependent spectral modulation and broadening of high harmonics observed from bulk sapphire. Then, by adopting theoretical simulation, the extraordinary observation is interpreted as a result of the quantum path interference between the long and short electron/hole trajectories. Specifically, the long trajectory undergoes an intensity-dependent redshift, which coherently combines with the short trajectory to exhibit spectral splitting in an anomalous way of inverse proportion to the driving laser intensity. This quantum interference may be extended to higher harmonics with increasing the laser intensity, underpinning the potential for precise control of the phase matching and modulation even in the extreme ultraviolet and soft X-ray regime. Further, this approach may act as a novel tool for probing arbitrary crystals so as to adjust the electron dynamics of higher harmonics for attosecond spectroscopy.

We studied the high-order harmonic generation (HHG) from 2D solid materials in circularly and bichromatic circularly polarized laser fields numerically by simulating the dynamics of single-active-electron processes in 2D periodic potentials. Contrary to the absence of HHG in the atomic case, circular HHGs below the bandgap with different helicities are produced from intraband transitions in solids with $C_4$ symmetry driven by circularly polarized lasers. Harmonics above the bandgap are elliptically polarized due to the interband transitions. High-order elliptically polarized harmonics can be generated efficiently by both co-rotating and counter-rotating bicircular mid-infrared lasers. The cutoff energy, ellipticity, phase, and intensity of the harmonics can be tuned by the control of the relative phase difference between the 1$\omega$ and 2$\omega$ fields in bicircularly polarized lasers, which can be utilized to image the structure of solids.

We study numerically the Bloch electron wavepacket dynamics in periodic potentials to simulate laser-solid interactions. We introduce a new perspective in the coordinate space combined with the motion of the Bloch electron wavepackets moving at group and phase velocities under the laser fields. This model interprets the origins of the two contributions (intra- and interband transitions) of the high-order harmonic generation (HHG) by investigating the local and global behavior of the wavepackets. It also elucidates the underlying physical picture of the HHG intensity enhancement by means of carrier-envelope phase (CEP), chirp and inhomogeneous fields. It provides a deep insight into the emission of high-order harmonics from solids. This model is instructive for experimental measurements and provides a new avenue to distinguish mechanisms of the HHG from solids in diffrent laser fields.

We studied the multi-plateau high-order harmonic generation (HHG) from solids numerically. It is found that the HHG spectrum in the second or higher plateau is redshifted in short laser pulses due to the nonadiabatic effect. The corresponding FWHMs also increase, suggesting the step-by-step excitation process of higher conduction bands in the HHG process. Even order harmonics in the higher plateaus of HHG are present due to the break of symmetry in the k space. It may stimulate related experimental work.

We investigated the high-order harmonic generation (HHG) process of diatomic molecular ion $\mathrm{H}_2^+$ in non-Born-Oppenheimer approximations. The corresponding three-dimensional time-dependent Schr\"odinger equation is solved with arbitrary alignment angles. It is found that the nuclear motion can lead to spectral modulation of HHG. Redshifts are unique in molecular HHG which decrease with the increase of alignment angles of the molecules and are sensitive to the initial vibrational states. It can be used to extract the ultrafast electron-nuclear dynamics and image molecular structure.

We study numerically the Bloch electron wave-packet dynamics in periodic potentials to simulate laser-solid interactions. We introduce a quasi-classical model in the \emph{k} space combined with the energy band structure to understand the high-order harmonic generation (HHG) process occurring in a subcycle timescale. This model interprets the multiple plateau structure in HHG spectra well and the linear dependence of cutoff energies on the amplitude of vector potential of the laser fields. It also predicts the emission time of HHG, which agrees well with the results by solving the time-dependent Schr\"{o}dinger equation (TDSE). It provides a scheme to reconstruct the energy dispersion relations in Brillouin zone and to control the trajectories of HHG by varying the shape of laser pulses. This model is instructive for experimental measurements.

We theoretically study high-order harmonic generation (HHG) from solid-phase systems in spatially inhomogeneous strong laser fields originated by resonant plasmons within a metallic nanostructure. The intensity of the second plateau in HHG may be enhanced by two-three orders and be comparable with the intensity of the first plateau. This is due to bigger transition probabilities to higher conduction bands. It provides us a practical way to increase the yields of HHG with laser intensity below the damage threshold. It presents a promising way to triple the range of HHG spectra in experimental measurements. It also allows to generate intense isolated attosecond pulse from solids driven by few-cycle laser fields.

We investigate the high harmonic generation (HHG) from solids by simulating the dynamics of a single active electron in periodic potentials. The corresponding time-dependent Schr\"odinger equations (TDSE) are solved numerically by using B-spline basis sets in coordinate space. The energy band structure and wave vectors can be directly retrived from the eigenfunctions. The harmonic spectra obtained agree well with the results simulated by TDSE in $k$ space using Bloch states and show a two-plateau structure. Both of the cutoff energies of the two plateaus in the harmonic spectrum scale linearly with the field strength. We also study HHG driven by intense few-cycle laser pulses and find that the cutoff energy of the harmonic spectrum is as sensitive to the changes of the carrier envelope phase, as to HHG from gas samples, which suggests recollision pictures in HHG as found by recent experiments (Nature {\bf 522}, 462 (2015)).

Crank-Nicolson (C-N) Method combined with $ B $-spline basis set can be used in both time-dependent and time-independent calculations of the photoionization cross sections of atoms and molecules. For time-independent systems, it is found that C-N method in imaginary-time propagation (ITP) can converge directly to not only the ground state, but also excited and continuum states by controlling the time step size. The C-N method can also be directly applied in time-dependent calculations. Both time-dependent and time-independent calculations agree very well with previous results. This method can also be extended to two-electron systems directly.