High-order Harmonic Generation Spectra Research Articles

Two-Electron Scenario of High-Order Harmonics Generation by Atom in an Intense Infrared Field and Attosecond Pulse

The two-electron scenario is proposed for the high-order harmonic generation (HHG) by an atom interacting with an intense infrared field and attosecond pulse. The two-electron dynamics involved in the proposed scenario is realized for the specific attosecond pulse, which can excite the resonance between the valence and deeper shells. Within the numerical solution of the time-dependent Kohn–Sham equations, we analyze the contribution of the single [Phys. Rev. A 98, 063433 (2018)] and two-electron scenarios of HHG by decreasing the duration of attosecond pulse having carrier frequency detuned from the atomic resonance. Favorable conditions are formulated for the realization of a two-electron scenario, which causes the enhancement of the harmonic yield in the spectral range exceeding the cutoff energy of the HHG spectrum in the infrared field.

Manipulating helical phase of HHG vortex with atomic states

By deploying the strong-field approximation theory, we perform numerical simulation on atomic high-order harmonic generation (HHG) of hydrogen and HHG vortices generated by hydrogen layer. Our results show that the atomic HHG spectra demonstrate peak-shifting and the helical phase of the HHG vortex can be manipulated with initial states of atoms, all ionized by oppositely polarized bicircular ω and 2ω Laguerre–Gaussian fields. The HHG peaks shift to higher frequency by ω if the initial state is switched from ψ100 to ψ211 , and the HHG peaks shift to lower frequency by ω if the initial state is switched from ψ100 to ψ21−1 , the final state is always the ground state ψ100 . The helical phase patterns of the HHG vortices are investigated in connection with the atomic HHG peak-shifting and the angular momentum conservation law. In addition, we observe that the fourth HHG harmonic is strongly dependent on the 2ω field but weakly on the ω field for an initial atomic state of magnetic quantum number −1.

Molecular orbital tomography of HOMO and HOMO-1 of the nonlinear molecule H2O: The study on multi-orbital effects

Molecular orbital tomography (MOT) through high-order harmonic generation (HHG) is a crucial method for probing individual molecular orbitals. Previous studies mainly focused on linear molecules such as N2, and CO, where valence orbitals were imaged by detecting harmonic emission at various alignments. In this study, we investigate the HHG from the water molecules, with the aim of exploring MOT for nonlinear molecules. By employing the time-dependent density functional theory, we obtain HHG spectra for different angles between the laser polarization and molecular symmetry axis. By manipulating the scanning planes (planes perpendicular to the laser propagation direction and contain the field polarization direction), specific orbitals can be selectively imaged. Particularly, a well-reconstructed highest occupied molecular orbital (HOMO) is obtained when the laser scanning plane approximately aligns with the nodal plane of HOMO-1. However, when the laser polarization deviates from the nodal plane, multiple orbitals contribute to the HHG process, and we visualize the orbital distortions resulting from such effects. Furthermore, HOMO-1, can be reconstructed when the laser scanning plane aligns with the nodal plane of HOMO. This research significantly advances the development of MOT and opens up intriguing possibilities for reconstructing orbitals of more complex molecules.

Real-Time Extension of TAO-DFT.

Thermally assisted occupation density functional theory (TAO-DFT) has been an efficient electronic structure method for studying the ground-state properties of large electronic systems with multi-reference character over the past few years. To explore the time-dependent (TD) properties of electronic systems (e.g., subject to an intense laser pulse), in this work, we propose a real-time (RT) extension of TAO-DFT, denoted as RT-TAO-DFT. Moreover, we employ RT-TAO-DFT to study the high-order harmonic generation (HHG) spectra and related TD properties of molecular hydrogen H2 at the equilibrium and stretched geometries, aligned along the polarization of an intense linearly polarized laser pulse. The TD properties obtained with RT-TAO-DFT are compared with those obtained with the widely used time-dependent Kohn-Sham (TDKS) method. In addition, issues related to the possible spin-symmetry breaking effects in the TD properties are discussed.

Orientation-Dependent High-Order Harmonic Generation from Monolayer ZnO

Solid-state high-order harmonic generation (HHG) now is a strong tool for detecting target properties, like band structure, Berry curvature and transition dipole moments (TDMs). However, the physical mechanism of high-order harmonic generation (HHG) in solids has not been fully elucidated. According to previously published works, in addition to the inter-band polarization, intra-band currents, and anomalous currents due to Berry curvature, there is another term which will be called the mixture term (MT). Taking monolayer ZnO as a sample, it is found that the intensity of the mixture term, which has been ignored for a long time in previous works, actually is comparable with other terms. Additionally, we compare the orientation-dependent HHG spectra that originated from different mechanisms. It is found that the inter-band and mixture HHG show similar orientation features. Meanwhile, Berry curvature only produces perpendicularly polarized even harmonics, and intra-band perpendicularly polarized even harmonics show special orientation features which can be explained by the orientation-dependent group velocity. This work will help people understand the mechanisms of solid-HHG better.

Nonadiabatic electron dynamics effects on high-harmonic generation spectrum of [formula omitted]: Minima and oscillatory pattern

We numerically solved the full-dimensional electronic time-dependent Schrödinger equation for H2+ with Born-Oppenheimer approximation under different sin2-shaped and trapezoidal laser pulses at some different wavelengths, with I = 1 × 1013, 3 × 1013 and 6 × 1013 Wcm−2 intensity at 4.73 a.u. and 7.0 a.u. internuclear distances. Some structures such as minima and oscillatory patterns appearing in the high-order harmonic generation (HHG) spectra. We evaluate these structures in the HHG spectrum by considering the electron localization, electron nonadiabatic dynamics, and the Rabi oscillation of the population of the ground and excited electronic states. We have explored that these minima originate from the oscillatory patterns and different phase in the Sgω and Suω spectra due to recombination to the 1σg and 2pσu respectively. These oscillatory patterns in turn are due to the nonadiabatic electronic behavior appearing as a slow oscillation pattern in the ground and first excited electronic states populations. We show that the minima of the HHG spectrum are related to Sgω, Suω, and orbital interference.

Physical origins of complex interference structures in harmonic emission from molecular ions stretched to large internuclear distances

High-order harmonic generation (HHG) from the molecular ions stretched to large internuclear distances is studied numerically and analytically in this work. We focus on the fine structure of the HHG spectrum related to the contribution of short electron trajectory. In our numerically solving the time-dependent Schrodinger equation (TDSE), we use a trajectory-dependent filtering procedure to separate the short-trajectory contribution from other contributions of long trajectory and multiple returns. Our TDSE results reveal that the short-trajectory HHG spectra of molecular ion with larger internuclear distance show some complex interference structures characterized by some remarkable dips, and that the position of the dip is sensitive to the laser parameters. With a developed model arising from strong-field approximation (SFA), we are able to identify the physical origins of the complex interference structures. In this model considered is the charge-resonance effect which induces the strong coupling between the ground state and the first excited state of the molecular ion at large internuclear distance. In this model, the well-known effect of two-center interference occurs in the form of the canonical momentum instead of the momentum related to the instantaneous velocity of the electron in the general SFA. It is shown that some dips in TDSE results arise from two-center interference of the electronic wave between these two atomic cores of the molecule in the ionization process, while others come from that in the recombination process. These ionization and recombination dips alternately appear in the HHG spectra from the formed complex interference structures. The main differences between the interference effects in the ionization process and the recombination process are twofold. Firstly, in the ionization process, the destructive two-center interference strongly suppresses the forming of the continuum wavepacket, while in the recombination process, the recombination of the rescattering electron with other bound eigenstates with small weights can also contribute to HHG bedsides the recombination of the ground state with the first excited state with large weights. As a result, in TDSE results, the ionization dips are deeper and more remarkable than the recombination ones. Secondly, in the recombination process, the Coulomb acceleration remarkably changes the de Broglie wavelength of the rescattering electron and therefore changes the position of the interference-induced dip. While in the ionization process, the Coulomb potential plays a small role in the interference effect. As a result, the interference dips in the ionization process and the recombination process differ from each other.

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.

Calculation of high-order harmonic generation of atoms and molecules by combining time series prediction and neural networks.

High-order harmonic generation (HHG) from the interaction of ultra-intense laser pulses with atoms is an important tabletop short-wave coherent light source. Accurate quantum simulations of it present large computational difficulties due to multi-electron multidimensional effects. In this paper, the time-dependent response of hydrogen atoms is calculated using a time-series prediction scheme, the HHG spectrum is reconstructed very accurately. The accuracy of the forecasting is further improved by using a neural network scheme. This scheme is also applied to the simulation of the harmonic emission on multi-electron systems, and the applicability of the scheme is confirmed by the harmonic calculation of complex systems. This method is expected to simulate the nonlinear dynamic process of multi-electron atoms and molecules irradiated by intense laser pulses quickly and accurately.

High-order harmonic generation in three-dimensional Weyl semimetals with broken time-reversal symmetry

In this paper the nonlinear interaction of a Weyl semimetal (WSM) with a strong driving electromagnetic wave field is investigated. In the scope of the structure-gauge-invariant low-energy nonlinear electrodynamic theory, the polarization-resolved high-order harmonic generation spectra in the WSM with broken time-reversal symmetry are analyzed. The results obtained show that the spectra in the WSM are completely different compared to the two-dimensional graphene case. In particular, at the noncollinear arrangement of the electric and Weyl node momentum separation vectors, anomalous harmonics are generated which are polarized perpendicular to the pump wave electric field. The intensities of anomalous harmonics are quadratically dependent on the momentum space separation of the Weyl nodes. If the right and the left Weyl fermions are merged, we have a four-component trivial massless Dirac fermion and, as a consequence, the anomalous harmonics vanish. In contrast to the anomalous harmonics, the intensities of normal harmonics do not depend on the Weyl nodes' momentum separation vector and the harmonics spectra resemble the picture for a massless three-dimensional Dirac fermion.

Role of symmetry breaking in high-order harmonic generation from Su-Schrieffer-Heeger systems

High-order harmonic generation (HHG) in topological insulators (TIs) is receiving tremendous attention due to the topologically protected surface states. To study the HHG in topological surface, an open-boundary Su-Schrieffer-Heeger model was adopted and we rigorously addressed quantum decoherence via Lindblad equation. We find that role of the symmetry breaking is significant for HHG in TIs. Our results reveal that (i) the edge electrons in TIs radiate pure even-order harmonics via distinguishing the topological bulk and edge contributions; (ii) a short dephasing time makes the even-order harmonic yields lower than that of odd order; (iii) two-channel subcycle interference model and dynamics-symmetry analysis were developed to explain the odd- and even-order characteristic of HHG spectra; (iv) the symmetry breaking selectively enhances emission of certain channel. This work reveals the vital role of the TIs' symmetry in the strong-field HHG, which provides an alternative way to understand and control the HHG from topological surfaces.

Ellipticity dependence of high-order harmonic generation in disordered semiconductors

We analyze the ellipticity dependence of high order harmonic generation (HHG) in disordered semiconductors. We show that a disordered crystal can radiated HHG spectra containing only odd harmonics of the laser frequency for all values of the ellipticity of the laser. Furthermore, we show that the HHG yield of our disordered models decreases monotonically with increasing laser ellipticity as observed in recent experiments. I particular, our numerical calculations, based on a coarse grained model, reproduce many of the qualitative features of the experimental HHG spectra of ZnO and GaAs.

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 beyond mean-field 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 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 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 Mott insulator limit of high electron-electron correlation is approached, the finite-size effect on the electron dynamics becomes negligible.

Multiple collisions in crystal high-order harmonic generation

We theoretically investigate high-order harmonic generation (HHG) in crystals induced by linearly polarized laser fields. We obtain the HHG spectra by solving the semiconductor Bloch equations and analyze the radiation process by different models. Here we propose a multiple collision model, in which the electrons and holes are produced at different times and places. It is found that the multiple collision trajectories can help us comprehensively and better explain the results of the quantum calculation. Moreover, we find that the harmonic suppression occurs due to the overlap of multiple collision trajectories.

Probing subcycle spectral structures and dynamics of high-order harmonic generation in crystals

Subcycle spectral structures and dynamics of high-order harmonic generation (HHG) processes of atoms and molecules driven by intense laser fields on the attosecond time scale have been originally studied theoretically and experimentally. However, the time scale of HHG dynamics in crystals is in the order of sub-femtosecond, and the carrier dynamics of HHG in crystals driven by subcycle laser pulses are largely unexplored. Here we perform a theoretical study of subcycle structures, spectra, and dynamics of HHG of crystals in mid-infrared laser fields subject to excitation by a subcycle laser pulse with a time delay. The HHG spectra as a function of time delay between two laser fields are calculated by using a single-band model for the intra-band carrier dynamics in crystal momentum space and by solving the time-dependent Schrödinger equation in velocity gauge for the treatment of multi-band crystal systems. The results exhibit a complex time-delay-dependent oscillatory pattern, and the enhancement and suppression of the HHG related to subcycle pulse are observed at the given time delay in either single-band or multi-band crystal systems. To understand oscillation structures with respect to the dependence for the subcycle laser fields, the time-frequency characteristics of the HHG as well as the probability density distribution of the radiation are analyzed in detail.

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.

Reply to “Comment on ‘Origin of symmetry-forbidden high-order harmonic generation in the time-dependent Kohn-Sham formulation’”

In reply to the Comment by O. Neufeld et al. [Phys. Rev. A 105, 047101 (2022)], we argue that the conclusions of Phys. Rev. A 103, 043106 (2021) remain valid. We disprove the claim that the unphysical even-order harmonics originate from convergence issues related to reflections at the boundary of the simulation box. By additional calculations, we show that such reflections perturb the high-order harmonic generation spectra by oscillations with periods much smaller than the distance between the harmonics. We also demonstrate that the convergence argument of the Comment, in contrast to our multielectron excitations argument, cannot explain why there are no unphysical even-order harmonics in one-electron systems. Moreover, we show that the argument put forward in the Comment to conclude that the time-dependent Kohn-Sham equations are superior to the time-dependent natural Kohn-Sham equations is not valid.

Macroscopic effects in high-order harmonic generation - a focal-averaging method based on the integral solution of the wave equation.

A macroscopic theory of high-order harmonic generation (HHG) is presented, which applies a focal-averaging method based on the integral solution of the wave equation. The macroscopic high-harmonic yield is the coherent superposition of the single-atom contributions of all atoms of the generating medium, which are positioned at different spatial points of the laser focus and exposed to the space-time-dependent laser pulse. The HHG spectrum obtained in our macroscopic simulations is qualitatively different from the one obtained using the microscopic or single-atom theory of HHG. Coherent intensity focal averaging, the simpler and more approximate of two methods we introduced, gives the spectrum which forms a declining plateau with the same cutoff position as that of the microscopic spectrum. The second, more precise method, which we call coherent spatio-temporal focal averaging, shows that it is possible, changing the macroscopic conditions, to obtain an observable peak in the harmonic spectrum at an energy much lower than the microscopic cutoff energy. Generally, the high-harmonic yield appears to be dominated by the contributions of laser-pulse spatio-temporal regions with lower intensities as well as by interference, so that the high-energy plateau and its sharp cutoff are quenched in the theoretical simulation and, presumably, in the experiment. The height and position of this peak strongly depend on the macroscopic conditions. We confirmed these findings by applying our macroscopic theory to simulate two recent experiments with mid-infrared laser fields, one with a linearly polarized field and the other one with a bicircular field.

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.

Genetic-algorithm retrieval of the molecular alignment distribution with high-order harmonics generated from transiently aligned CO2 molecules

Using genetic algorithm, we propose a method to retrieve the alignment distribution of transiently aligned ${\mathrm{CO}}_{2}$ molecules from the high-order harmonic generation (HHG) spectra. The retrieval method is based on the quantitative rescattering (QRS) model where averaged photorecombination transition dipole can be factored out from the parallel (or perpendicular) harmonic spectra after the propagation of the harmonic fields in the gas medium. We examine how the retrieved alignment distributions are affected by uncertainty in alignment-dependent ionization probability and on multiple orbital contribution to the HHG. We further confirm that alignment distribution is more accurately retrieved by using the minima in the HHG spectra driven by a long-wavelength laser. In addition, we show that earlier experimental data on the ratios between the perpendicular and the parallel HHG components of aligned ${\mathrm{CO}}_{2}$ molecules are in better agreement with the QRS model if the macroscopic propagation and multiple orbital interference are included in the theoretical calculation.