High-order Harmonic Generation In Solids Research Articles

Solid high-order harmonic generation: Review of the semiconductor Bloch equations

Solid high-order harmonic generation: Review of the semiconductor Bloch equations

High-Order Harmonic Generation in Solids: The Role of Intraband Transitions in Extreme Nonlinear Optics.

High-order harmonic generation (HHG) in gases is frequently used nowadays to produce attosecond pulses and coherent radiation in the visible-to-soft X-ray spectral range. HHG in solids is a natural extension of the idea of HHG in gases, and its first observation about ten years ago opened the door to investigations on attosecond electron dynamics in solids and the development of solid-state attosecond light sources. The common process in both types of HHG is nonlinear photocarrier generation, and thus, transitions between different bands (interband transitions) are always important for HHG. As well, in the case of solids, the transitions within a band (intraband transitions) also need to be considered, because efficient carrier acceleration is possible due to them. This Perspective focuses on experimental findings that show how intraband transitions can be controlled because such an understanding will be essential in the development of unique optoelectronics that can operate at petahertz frequencies.

Laser-field-strength dependence of solid high-order harmonic generation from doped systems

Laser-field-strength dependence of solid high-order harmonic generation from doped systems

Robust retrieval method of crystal transition dipole moments by high-order harmonic spectrum

According to the three-step model for solid high-order harmonic generation, there is a one-to-one correspondence between the emitted photon energy and the band gap where the electron-hole pair is annihilated. In the tunneling excitation regime, as the electron-hole pair is mostly created in the vicinity of the minimum band gap, the conversion efficiency of the high-energy photon should be approximately proportional to the square of the transition dipole moment at the $\mathbf{k}$ point where the high-harmonic photon is emitted. Based on this picture we propose that a high-order harmonic spectrum could be a strong tool to reconstruct the shape of $\mathbf{k}$-dependent transition dipole moments with the band dispersion and the laser field known. Two real systems, e.g., MgO and ZnO, are taken as samples to verify our idea. The reconstructed shape of the transition dipole moments shows small variation as the laser parameters, such as intensity, wavelength, and pulse duration, are tuned in wide ranges, which proves this scheme is robust.

Trajectory-controlled high-order harmonic generation in ZnO crystals.

We experimentally and theoretically study high-order harmonic generation in zinc oxide crystals irradiated by mid-infrared lasers. The trajectories are mapped to the far field spatial distribution of harmonics. The divergence angles of on-axis and off-axis parts exhibit different dependences on the order of the harmonics. This observation can be theoretically reproduced by the coherent interference between the short and long trajectories with dephasing time longer than 0.5 optical cycle. Further, the relative contribution of the short and long trajectories is demonstrated to be accurately controlled by a one-color or two-color laser on the attosecond time scale. This work provides a reliable method to determine the electron dephasing time and demonstrates a versatile control of trajectory interference in the solid high-order harmonic generation.

Reconstructing the Semiconductor Band Structure by Deep Learning

High-order harmonic generation (HHG), the nonlinear upconversion of coherent radiation resulting from the interaction of a strong and short laser pulse with atoms, molecules and solids, represents one of the most prominent examples of laser–matter interaction. In solid HHG, the characteristics of the generated coherent radiation are dominated by the band structure of the material, which configures one of the key properties of semiconductors and dielectrics. Here, we combine an all-optical method and deep learning to reconstruct the band structure of semiconductors. Our method builds up an artificial neural network based on the sensitivity of the HHG spectrum to the carrier-envelope phase (CEP) of a few-cycle pulse. We analyze the accuracy of the band structure reconstruction depending on the predicted parameters and propose a prelearning method to solve the problem of the low accuracy of some parameters. Once the network is trained with the mapping between the CEP-dependent HHG and the band structure, we can directly predict it from experimental HHG spectra. Our scheme provides an innovative way to study the structural properties of new materials.

Size-controlled quantum dots reveal the impact of intraband transitions on high-order harmonic generation in solids

Since the discovery of high-order harmonic generation (HHG) in solids, much effort has been devoted to understanding its generation mechanism and both interband and intraband transitions are known to be essential. However, intraband transitions are affected by the electronic structure of a solid, and how they contribute to nonlinear carrier generation and HHG remains an open question. Here, we use mid-infrared laser pulses to study HHG in CdSe and CdS quantum dots (QDs), where quantum confinement can be used to control the intraband transitions. We find that both the HHG intensity per excited volume and the generated carrier density increase when the average QD size is increased from about 2 nm to 3 nm. We show that the reduction of the subband gap energy in larger QDs enhances intraband transitions, and this in turn increases the rate of photocarrier injection by coupling with interband transitions, resulting in enhanced HHG.

Polarization-resolved analysis to solid high-order harmonic generation

We propose a quantitative polarization-resolved quantum trajectory approach to analyse the solid high-order harmonic generation (HHG). By using this approach, we can distinguish the ionization channels of different polarization components in HHG. We take the HHG in hBN under two counter-rotating circularly polarized fields as an example. We find that the right and left circular polarized high harmonics are contributed by the electron trajectories with the ionization channel around the high symmetry points K and K′ respectively. Furthermore, the intensity of the right and left circular polarization harmonics is determined by the interference between these electron trajectories. Our work offers a quantitatively polarization-resolved analysis to understand the underlying mechanism in HHG in the view of electron reciprocal space trajectory.

Non-classical high harmonic generation in graphene driven by linearly-polarized laser pulses.

Recent studies in high-order harmonic generation (HHG) in solid targets reveal new scenarios of extraordinary rich electronic dynamics, in comparison to the atomic and molecular cases. For the later, the main aspects of the process can be described semiclassically in terms of electrons that recombine when the trajectories revisit the parent ion. HHG in solids has been described by an analogous mechanism, in this case involving electron-hole pair recombinations. However, it has been recently reported that a substantial part of the HHG emission corresponds to situations where the electron and hole trajectories do not overlap in space. According to the present knowledge, HHG from this imperfect recollisions reflects the quantum nature of the process, arising in systems with large Berry curvatures or for elliptically polarized driving fields. In this work, we demonstrate that imperfect recollisions are also relevant in the more general case. We show the signature of such recollisions in the HHG spectrum from monolayer graphene -a system with null Berry curvature- irradiated by linearly polarized driving fields. Our calculations also reveal that imperfect multiple-order recollisions contribute to the harmonic emission when electron-hole excursion times exceed one cycle of the driving field. We believe that our work adds a substantial contribution to the full understanding of the sub-femtosecond dynamics of HHG in solid systems.

High-order harmonic generation from the interference of intra-cycle trajectories in the k-space.

Considering the crystal momenta of the entire k-space, we demonstrate that constructive intra-cycle interference of electrons enhances the high-order harmonic generation (HHG) of a GaN crystal from dominant interband Bloch oscillations. This results in a higher plateau of the HHG spectrum at a driven yield strength below the Bloch field strength. This phenomenon is confirmed in both the two-band and three-band models. Using two-color laser fields, the constructive or destructive interference of interband Bloch oscillations can be tuned. Our findings reveal the essential impact of intra-cycle interference in the full k-space on the HHG in solids.

Resonance-enhanced solid high-order harmonic generation by a chirped laser pulse

The resonant high-order harmonic generation (HHG) from solid in the periodic potential is investigated by a chirped laser pulse. We show the dependence of the intensity of the resonance peak on the chirp parameters β, and it is found that the increase of the resonant harmonic intensity with β is significantly higher than that of the nonresonance ones. The frequency of the resonance peaks remains at 4.2 eV, which equals to the band gap energy between the valance band (VB) and the conduction band (CB1). By adjusting the chirp parameters, the electron will be injected into the conduction band which results in the band gap emission. Through the time-dependent population imaging (TDPI) and the time–frequency (TF) investigation, we demonstrate that the resonance peak originates from the transition of electrons from the VB toward CB1.

Control of high-order harmonic generation in solids by orthogonally polarized laser fields

We theoretically investigate the high-order harmonic generation (HHG) from ZnO crystals driven by orthogonally polarized two-color (OTC) laser fields. By solving the semiconductor Bloch equation (SBE) in two dimensions, it is found that the HHG yields show a high sensitivity to the relative phase between the two fields. By changing the laser intensity, the yields of HHG show two maxima in the high-order region. To reveal the physical origin of these phenomena, we analyze the behavior of the electron-hole pair in the real and reciprocal spaces by the recollision model, respectively. It could qualitatively interpret the results by SBE simulations. Our analysis demonstrates that the OTC laser fields could control the electron-hole dynamics and HHG processes efficiently.

Interference between harmonics of different crystal momentum channels in solid high-order harmonic generation

We theoretically investigate high-order harmonic generation (HHG) in one-dimensional model solids with linearly polarized laser pulses. By identifying crystal-momentum-resolved contributions, we show that the reason behind the odd-order harmonics-dominated harmonic spectrum originates from the interferences between harmonics generated by different crystal momentum channels. Symmetry analysis show that constructive interferences between odd-order harmonics and destructive interferences between even-order harmonics are determined by the inversion symmetry of the band dispersion and the time translation symmetry of the laser field. We also propose and numerically demonstrate a method to select the short and long trajectories for HHG in solids. It is shown that the classical trajectories predicted by quasiclassical model are associated with the quantum orbits that originate from a single crystal momentum channel. However, the classical trajectories predicted by the recollision model are associated with all of the crystal momentum channels in the entire first Brillouin zone, and the interferences between harmonics generated by different crystal momentum channels play a key role in the formation of recollision quantum orbits. Moreover, we find that the effect of dephasing time is the damping of the current rather than changing the harmonic phases. The present results may contribute to disentangling the underlying mechanism of HHG in solids.

Quantum decoherence in high-order harmonic generation from solids

For the high-order harmonic generation (HHG) from solids, the dephasing process induced by many-body interactions has been discussed extensively in the studies of using the semiconductor Bloch equations (SBEs). However, the role of dephasing in solid HHG is always ignored in the simulations of using a time-dependent Schr\"odinger equation under the independent electron approximation. To solve this problem, we introduce the imaginary potential to phenomenologically depict the dephasing process in the solid HHG. Compared with the results of experiment and SBEs, the validity of this approach has been verified by the laser intensity- and wavelength-dependent HHG spectra. Diffusion of the quantum wave packet controls the time-frequency characteristic in solid HHG. To obtain semiclassical trajectories whose predictions are consistent with the quantum simulations, we propose an open-trajectory model by relaxing the zero displacement condition in the tunneling and recollision steps. In addition, the quantum decoherence adjusts the chirp of the emission time profile via modulating the coherent overlap between recombined wave packets, which further paves a way to generate an attosecond pulse from solids and probe the dephasing time via the high-harmonic spectroscopy.

Enhanced extreme ultraviolet high-harmonic generation from chromium-doped magnesium oxide

High-order harmonic generation (HHG) from crystals is emerging as a new ultrashort source of coherent extreme ultraviolet (XUV) light. Doping the crystal structure can offer a new way to control the source properties. Here, we present a study of HHG enhancement in the XUV spectral region from an ionic crystal, using dopant-induced vacancy defects, driven by a laser centered at a wavelength of 1.55 μm. Our numerical simulations based on solutions of the semiconductor Bloch equations and density-functional theory are supported by our experimental observations and demonstrate an increase in the XUV high harmonic yield from doped bulk magnesium oxide (MgO) compared to undoped MgO, even at a low defect concentration. The anisotropy of the harmonic emission as a function of the laser polarization shows that the pristine crystal's symmetry is preserved. Our study paves the way toward the control of HHG in solids with complex defects caused by transition-metal doping.

Effects of quantum interferences among crystal-momentum-resolved electrons in solid high-order harmonic generation

We theoretically demonstrate the effects of quantum interferences among valence electrons with various initial crystal momenta in the high-order harmonic generation (HHG) in solids. By investigating crystal-momentum-dependent harmonics from solids in linearly polarized laser fields, some unique radiation characteristics of the observed overall harmonics are attributed to the consequences of interferences among valence electrons with different crystal momenta. It is shown that the electron pairs with opposite crystal momenta result in the destructive interferences of even-order harmonics for a crystal with the inversion symmetry, which eventually leads to the only odd orders in observed overall harmonic spectra for a semiconductor material because of the complete pairing of all valence electrons. Additionally, each of the harmonic plateaus in the overall multiple-plateau harmonic spectrum is identified to be contributed by the valence electrons within different crystal momentum zones. We also find that the solid-phase HHG in the below-band-gap region is substantially suppressed due to the collective responses of multiple valence electrons. This work sheds light on the essential impacts of interferences among crystal-momentum-resolved electrons on the HHG in solids and is helpful to explore ultrafast coherent processes in condensed matter.

Exploration of the high-order harmonic generation from periodic potentials by Bohmian trajectories

The recombination processes of the electrons in solid are illustrated by solving the time-dependent Schrödinger equation. The results show that the Bohmian trajectories and the time evolution of the electronic probability density agrees very well, which demonstrates that we can use the Bohmian trajectories to investigate the recombination processes of the electrons in solid. We select the region where the probability density of the electron reached the strongest and the weakest as the initial position of the calculated Bohmian trajectories, one can see that the Bohmian trajectories have similar structures. In addition, our results show that the emission time of the solid high-order harmonic generation (HHG) spectra from the time-frequency distribution agrees well with the time that the Bohmian trajectories change direction. By regulating the phase of the electric field, the electrons can move farther, which will result in the broad cutoff of the HHG. We have also demonstrated that the similar structure of the Bohmian trajectories of the solid with a defect for different initial positions disappears due to the broken periodic structure, which further illustrates that the motion processes of the electrons in solid depend on the structure of the solid.

Spectral red-shift of solid high-order harmonic generation combining a fundamental field and a terahertz field

High-order harmonic generation (HHG) from a solid by combining a fundamental field and a terahertz field is investigated by solving the time-dependent Schrödinger equation. The numerical results illustrate that the red-shift from odd harmonics is more obvious in the combined field than the fundamental field. We have also investigated solid HHG with different relative phases. We find that the HHG spectra exhibit a larger red-shift from odd harmonics for the relative phase φ = 0 and the red-shift from odd harmonics is gradually weakened by increasing the relative phase. For the relative phase φ = π, the red-shift of solid HHG disappears and only odd harmonic order can be observed. The time-dependent population imaging picture is used to illustrate the physical mechanism of the red-shift in solid HHG.

Two-color-driven enhanced high-order harmonic generation in solids

We theoretically investigate the emission of high-harmonic (HH) radiation in model crystals by bichromatic few-cycle driving pulses that are composed as the phase-coherent superposition of a mid-infrared fundamental pulse and its second harmonic. Adjusting the model-crystal parameters to reproduce the lowest band gap of MgO, we examine the extent to which distinct domains of the HH spectrum can be controlled and enhanced by tuning the temporal profile of the bichromatic driving laser electric field. We change the driving-pulse shape by varying its fundamental-versus-second-harmonic pulse amplitude ratio and delay, while keeping the energy of the driving laser pulse fixed. For suitable amplitude ratios and delays, we find an up to fivefold enhancement of the spectral HH yield and significant shifts of the HH cutoff frequency.

Intense-laser-driven electron dynamics and high-order harmonic generation in solids including topological effects

A theory for laser-driven electron dynamics and high-harmonic generation in bulk solids with two lattice sites per unit cell of arbitrary dimension is formulated. In tight-binding approximation, such solids can be described by $2\times 2$ Bloch-Hamiltonians. Our theory is able to fully capture topological effects in high-harmonic generation by such systems because no simplifications beyond tight-binding, dipole approximation, and negligible depletion of the valence band are made. An explicit, analytical expression for the electron velocity is given. Exemplarily, the theory is applied to the Su-Schrieffer-Heeger chain and the Haldane model in strong laser fields.