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Three-dimensional (3D) analysis on the growth process of high-order harmonic generation (HHG) is achieved by integrating in situ measurement of a 3D phase-matching profile with interaction length control. Incorporating the phase-matching profile, calculation of high-order harmonic wave generation, and propagation reproduces the measured harmonics growth curve as well as distinctive features of the far-field beam profiles. The method resolves contributions of HHG from different positions of the driving beam affected by different sources of phase mismatch, and thus provides illuminating insight into the HHG process.

The Mollow triplet is a fundamental signature of quantum optics, and has been observed in numerous quantum systems. Although it arises in the 'strong driving' regime of the quantized field, where the atoms undergo coherent oscillations, it can be typically analyzed within the rotating wave approximation. Here we report the first observation of high-order effects in the Mollow triplet structure due to strong driving. In experiments, we explore the regime beyond the rotating wave approximation using concatenated continuous driving that has less stringent requirements on the driving field power. We are then able to reveal additional transition frequencies, shifts in energy levels, and corrections to the transition amplitudes. In particular, we find that these amplitudes are more sensitive to high-order effects than the frequency shifts, and that they still require an accurate determination in order to achieve high-fidelity quantum control. The experimental results are validated by the Floquet theory, which enables the precise numerical simulation of the evolution and further provides an analytical form for an effective Hamiltonian that approximately predicts the spin dynamics beyond the rotating wave approximation.

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., 14 pages, 6 figures, RevTeX

High-order harmonic generation (HHG) is expected to produce high-quality coherent ultrashort light sources. We propose a scheme similar to laser mode locking to selectively enhance and compress HHG in crystals through nonuniform illuminations and phase matching. The selective enhancement of harmonics reflects attosecond-resolved electron dynamics. And these harmonic characteristics originate from the interference of harmonics from different lattices. It paves the way for the practical application of solid HHG with high conversion efficiencies and the detection of electron-hole dynamics with ultrahigh spatial-temporal resolution.

We explore the possibility of bicircular high-order harmonic spectroscopy to probe the laser-induced dynamics of molecules in a non-Born-Oppenheimer treatment. The numerical solutions of the time-dependent Schr\"odinger equation for aligned ${\mathrm{H}}_{2}$ and its isotopologs in $\ensuremath{\omega}\text{\ensuremath{-}}2\ensuremath{\omega}$ bicircular fields show that the intensity ratio between ${\mathrm{D}}_{2}$ and ${\mathrm{H}}_{2}$ for harmonic orders $3q$ is lower than that for orders $3q\ifmmode\pm\else\textpm\fi{}1\phantom{\rule{4pt}{0ex}}(q\ensuremath{\in}\mathbb{N})$. Based on the strong-field approximation, we demonstrate that the interplay of vibrational wave-packet motion and dynamical-symmetry breaking leads to the different ratio. In general, the vibrational motion causes the ratio between isotopologs to increase with $q$ for both harmonic orders $3q$ and $3q\ifmmode\pm\else\textpm\fi{}1$. On the other hand, the emission of orders $3q$ is possible only because of the alignment-induced breaking of the dynamical symmetry. The faster nuclear motion in ${\mathrm{H}}_{2}$ enhances the symmetry breaking, resulting in the lower ${\mathrm{D}}_{2}$/${\mathrm{H}}_{2}$ ratio for the orders $3q$. Therefore, the harmonic orders $3q$ give access to the attosecond probing of dynamical symmetries in molecules.

We examine giant resonant enhancement of high-order harmonics generation (HHG) in the Mn atom and its singly charged ${\mathrm{Mn}}^{+}$ ion. Our theoretical model combines single active electron tunneling and propagation with correlation-enhanced recombination. Previously, this approach demonstrated its useability for the Xe atom [Phys. Rev. A 100, 013404 (2019)]. Spin-polarized Hartree-Fock and random-phase approximation with exchange calculations are carried out to evaluate the correlation enhancement factor in Mn and ${\mathrm{Mn}}^{+}$, which is then compared with other reported calculations and the experiment.

The semiconductor Bloch equations (SBEs) are routinely used for simulations of strong-field laser-matter interactions in condensed matter. In systems without inversion or time-reversal symmetries, the Berry connections and transition dipole phases (TDPs) must be included in the SBEs, which in turn requires the construction of a smooth and periodic structure gauge for the Bloch states. Here, we illustrate a general approach for such a structure-gauge construction for topologically trivial systems. Furthermore, we investigate the SBEs in the length and velocity gauges and discuss their respective advantages and shortcomings for the high-harmonic generation (HHG) process. We find that in cases where we require dephasing or separation of the currents into interband and intraband contributions, the length-gauge SBEs are computationally more efficient. In calculations without dephasing and where only the total current is needed, the velocity-gauge SBEs are structure-gauge independent and are computationally more efficient. We employ two systems as numerical examples to highlight our findings: a one-dimensional model of ZnO and the two-dimensional monolayer hexagonal boron nitride (hBN). The omittance of Berry connections or TDPs in the SBEs for hBN results in nonphysical HHG spectra. The structure- and laser-gauge considerations in the current work are not restricted to the HHG process and are applicable to all strong-field matter simulations with SBEs.

We study high-order harmonic generation (HHG) from oriented isotope molecules beyond the Born-Oppenheimer approximation numerically and analytically. We compare the symmetric cases of ${\mathrm{H}}_{2}$ and its isotopic variant ${\mathrm{T}}_{2}$ to the asymmetric ones of ${\mathrm{HeH}}^{+}$ and ${\mathrm{HeT}}^{+}$. Different from symmetric molecules, the laser-induced stretching for asymmetric molecules depends strongly on the molecular orientation. In addition, the structures of the HHG spectra for oriented ${\mathrm{H}}_{2}$ versus ${\mathrm{T}}_{2}$ differ remarkably from each other, and one can trace the stretching of these symmetric isotope molecules directly from the spectra. However, the spectral structures for oriented ${\mathrm{HeH}}^{+}$ versus ${\mathrm{HeT}}^{+}$ are basically the same. In this situation, we propose a procedure for probing the stretching of these asymmetric isotope molecules using odd-even HHG, with a high time-space resolution.

We propose a simple quasi-phase-matching (QPM) scheme in high-order harmonic generation (HHG) based on the approximately linear dependence of the phase mismatch on harmonic order. With this scheme, essentially all the harmonic orders are phase matched simultaneously, with harmonic order $q$ experiencing a $q\mathrm{th}$-order QPM. We validate this proposal by simulations using a semiclassical numerical model of HHG.

Spontaneous parametric down conversion (SPDC) has been one of the foremost tools in quantum optics for over five decades. Over that time it has been used to demonstrate some of the curious features that arise from quantum mechanics. Despite the success of SPDC, its higher-order analogues have never been observed, even though it has been suggested that they generate far more unique and exotic states than SPDC. An example of this is the emergence of non-Gaussian states without the need for post-selection. Here we calculate the expected rate of emission for nth-order SPDC with and without external stimulation (seeding). Focusing primarily on third-order parametric down-conversion (TOPDC), we estimate the photon detection rates in a rutile crystal, for both the unseeded and seeded regimes., 11 pages, 6 figures

Laser-induced high harmonic generation (HHG) in solids strongly depends on the band structure of the target. In this work, we studied the band structure of bulk wurtzite ZnO under different strains. The changes of band structure induced by shear, uniaxial, biaxial, and isotropic strains are compared. It is found that the band gap can be narrowed or widen by modulating the external conditions, which offers an efficient way to engineer the wide-gap crystals to produce HHG. The strain effects on HHG are investigated by numerically solving the semiconductor Bloch equations. Our calculations show that the HHG yield depends sensitively on the change of band structure caused by stress- and temperature-induced strains. In addition, we found that the emission times of HHG and the ratio between inter and intraband contributions to the total solid HHG spectra can be controlled by the magnitude of strain imposed on the solid targets.

High harmonic generation has been considered for many years to be a coherent source of upconverted laser radiation. We propose a method for the generation of essentially incoherent high harmonic generation with a controlled degree of temporal coherence, obeying random-walk statistics. This is achieved by a unique combination of single-atom and macroscopic effects involving preparation of the medium in an excited state with coordinate-dependent population inversion and using an intense off-resonant and phase-mismatched pumping.

We propose to correct the ionization factor in the quantitative rescattering (QRS) theory for a higher precision calculation of macroscopic high harmonic generation (HHG) in a gaseous medium. Compared to the first-order strong-field approximation (SFA-1) and the Ammosov-Delone-Krainov (ADK) ionization models, the Perelomov-Popov-Terent'ev (PPT) model achieves the best agreement with the macroscopic HHG spectra of Ar based on the solution of the time-dependent Schr\"odinger equation (TDSE). The investigation of the ionization probability dependence on laser intensity indicates that the PPT model is consistent with the TDSE, both in the tunneling and multiphoton ionization regimes, and the SFA-1 or the ADK mostly works in the tunneling regime. This explains the difference in the total macroscopic harmonic spectra, mainly due to off-axis harmonic emissions generated by small laser intensities. The examples for Ne and Xe by using three ionization models in the QRS theory also show the significance of ionization correction by the PPT model to precisely simulate the macroscopic HHG spectra. This study will help further increase the accuracy of simulating measured HHG spectra quantitatively under a variety of macroscopic conditions.

International audience; We demonstrate experimentally the control of the polarization (ellipticity and polarization axis) of femtosecond high-order harmonics. The method relies on a two-color (omega - 2 omega) configuration, where both omega and 2 omega generating fields have linear polarizations with a variable crossing angle. We correlate our measurements with the conservation rules of energy and linear momentum accounting for harmonic generation in a two-color field. We evidence that the generation process, and especially the number of omega and 2 omega photons absorbed for generating a given harmonic, strongly depends on the crossing angle. Finally, relying on a simple formalism, we derive analytical formulae for calculating both polarization axis and ellipticity of the separate harmonics. The model corroborates our results and represents a base for future investigations.

We theoretically investigate the influence of dynamic core-electron polarization (DCeP) of CO molecules on high-order harmonic generation (HHG) by solving the time-dependent Schr\"odinger equation (TDSE) within the single-active-electron (SAE) approximation. The effect of DCeP is shown to depend strongly on the molecular orientation angle $\ensuremath{\theta}$. Particularly, compared to the calculations without DCeP, the inclusion of this effect gives rise to an enhancement of harmonic intensity at $\ensuremath{\theta}={0}^{\ensuremath{\circ}}$ when the electric field aligns along the O-C direction and to a suppression at $\ensuremath{\theta}={180}^{\ensuremath{\circ}}$ when the field heads in the opposite direction. Meanwhile, when the electric field is perpendicular to the molecular axis, the effect is almost insignificant. The phenomenon is thought to be linked to the ionization process. However, this picture is not completed yet. By solving the TDSE within the SAE approximation and conducting a classical simulation, we are able to obtain the ionization probability as well as the ionization rate and prove that HHG, in fact, receives a major contribution from electrons ionized at only a certain time interval, rather than throughout the whole pulse propagation. Including DCeP, the variation of the ionization rate in this interval highly correlates to that of the HHG intensity. To better demonstrate the origin of this manifestation, we also show the alternation DCeP makes on the effective potential that corresponds to the observed change in the ionization rate and consequently the HHG intensity. Our results confirm previous studies' observations and, more importantly, provide the missing physical explanation. With the role of DCeP now better understood for the entire range of the orientation angle, this effect can be handled more conveniently for calculating the HHG of other targets.

Molecular orientation has wide applications in physics and chemistry. However, the degree of orientation achieved in experiments is difficult to evaluate, especially when three-dimensional (3D) orientation is involved. We show numerically and analytically that odd-even high-harmonic generation (HHG) from asymmetric molecules is sensitive to both the degree of orientation and the molecular structure. Based on the observations, a simple procedure is proposed to probe the degree of orientation with odd-even HHG, without the need of solving the rotational equation. This procedure can be applied to 3D-oriented top molecules such as polar or nonpolar planar ${\mathrm{H}}_{3}{}^{2+}$ with high resolution.

We propose an intuitive method, called the time-dependent population imaging (TDPI) method, to map the dynamical processes of high-order-harmonic generation (HHG) in solids by solving the time-dependent Schr\"odinger equation. It is shown that the real-time dynamical characteristics of HHG in solids, such as the instantaneous photon energies of emitted harmonics, can be read directly from the energy-resolved population oscillations of electrons in the TDPI picture. Meanwhile, the short and long trajectories of solid HHG are illustrated clearly from the TDPI picture. By using TDPI, we investigate the effects of the carrier-envelope phase in few-cycle pulses and demonstrate intuitively the HHG dynamics driven by two-color fields. Our results show that TDPI provides a powerful tool to study the ultrafast dynamics in strong fields for various laser-solid configurations and to gain insight into HHG processes in solids.

Citation: Tros, J., Ren, X., Makhija, V., Mondal, S., Kumarappan, V., & Trallero-Herrero, C. A. (2017). N2 HOMO-1 orbital cross section revealed through high-order-harmonic generation. Physical Review A - Atomic, Molecular, and Optical Physics, 95(3). doi:10.1103/PhysRevA.95.033419

Citation: Jin, C., & Lin, C. D. (2016). Spatially coherent high-order harmonics generated at optimal high gas pressure with high-intensity one- or two-color laser pulses. Physical Review A, 94(4), 6. doi:10.1103/PhysRevA.94.043804

We develop an approach describing nonlinear-optical processes in the strong-field domain characterized by the nonperturbative field-with-matter interaction. The polarization of an isolated atom in the external field calculated via the numerical solution of the time-dependent Schr\"odinger equation agrees with our analytical findings. For the practically important case of one strong laser field and several weaker fields, we derive and analytically solve propagation equations describing high-order (HO) wave mixing, HO parametric amplification, and HO stimulated scattering. These processes provide a way of efficient coherent xuv generation. Some properties of HO processes are new in nonlinear optics: essentially complex values of the coefficients in the propagation equations, the superexponential (hyperbolic) growing solutions, etc. Finally, we suggest conditions for the practical realization of these processes and discuss published numerical and experimental results where such processes could have been observed.

We employ a rotating frame of reference to elucidate high-harmonic generation of circularly polarized light by bicircular driving fields. In particular, we show how the experimentally observed circular components of the high-harmonic spectrum can be directly related to the corresponding quantities in the rotating frame. Supported by numerical simulations of the time-dependent Schr\"{o}dinger equation, we deduce an optimal strategy for maximizing the cutoff in the high-harmonic plateau while keeping the two circular components of the emitted light spectrally distinct. Moreover, we show how the rotating-frame picture can be more generally employed for elliptical drivers. Finally, we point out how circular and elliptical driving fields show a near-duality to static electric and static magnetic fields in a rotating-frame description. This demonstrates how high-harmonic generation of circularly polarized light under static electromagnetic fields can be emulated in practice even at static field strengths beyond current experimental capabilities., Comment: 10 pages, 4 figures

We compare harmonic spectra from hydrogen based on the numerical solution of the time-dependent Schrödinger equation and three approximate models: (i) the strong field approximation (SFA), (ii) the Coulomb-Volkov modified strong field approximation (CVA), and (iii) the strong field approximation with the stationary phase approximation applied to the momentum integrals (SPSFA). At laser intensities in the range of (1-3)×1014W/cm2 we find good agreement when comparing the SFA and CVA with exact results. In general the CVA displays an overall better agreement with ab initio results, which reflects the role of the Coulomb field in the ionization as well as in the recombination process. Furthermore, it is found that the widely used SPSFA breaks down for low-order harmonic generation; i.e., the approximation turns out to be accurate only in the outer part of the harmonic plateau region as well as in the cutoff region. We trace this deficiency to the singularity of the SPSFA associated with short trajectories, i.e., short return times. When removing these, we obtain a version of the SPSFA which works rather well for the entire harmonic spectrum. Fil: Pugliese, Silvina Norma. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina Fil: Simonsen, A. S.. University Of Bergen; Noruega Fil: Førre, M.. University Of Bergen; Noruega Fil: Hansen, J. P.. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. University Of Bergen; Noruega. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina

We theoretically investigate the nonadiabatic spectral redshift of high-order harmonics with the help of a VUV pulse. It is found that the nonadiabatic spectral redshift of high-order harmonics can be observed when a weak VUV pulse is properly added in the falling part of the fundamental laser due to the nonadiabatic response of the dipole to rapid change of laser intensity. Further time-frequency analysis shows that the high-order harmonics are mainly generated in the falling part of the fundamental pulse. This is because the VUV pulse enhances the ionization in the falling part of the fundamental pulse by the 1s-2p transition of $\text{H}{e}^{+}$. In addition, this scheme is also used to observe the nonadiabatic spectral blueshift of high-order harmonics by changing the time delay between the fundamental laser and the VUV pulse.

We have performed the quantum trajectory analysis for high-order-harmonic generation (HHG) with different driving laser wavelengths. By defining the ratio of HHG yields of the $N$th and first rescattering events (${Y}_{N}/{Y}_{1}$), we quantitatively evaluate the HHG contributions from multiple rescatterings. The results show that the HHG yield ratio increases gradually with the increase of the laser wavelength, which demonstrates that high-order rescatterings provide ascendent contributions to HHG at longer wavelength. By calculating the classical electron trajectories, we find significant differences exist in the electron behaviors between the first and high-order rescatterings. Further investigations have demonstrated that the increasing HHG yield ratio is mainly attributed to the relatively smaller contributions from the short path of the first electron rescattering at longer laser wavelength.

We point out the possibility of probing the nuclear vibration by using the high-order harmonic generation (HHG) with ultrashort laser pulses. By analyzing the HHG spectra obtained from the numerical solution of the time-dependent Schr\"odinger equation, we find that the intensity of the emitted harmonics is strongly determined not only by the molecular configuration but also by the initial velocity of nuclei. The calculations show that the intensity of HHG is enhanced nearly the time when the internuclear separation takes the equilibrium value and the nuclei are moving closer together. In contrast, with the same initial internuclear separation but with the opposite direction of nuclear velocity, the intensity of emitted light is reduced noticeably. We use that evidence as a tool to probe the nuclear vibration.

This work was partially supported by the Chemical Sciences, Geosciences and Biosciences Division of the Office of Basic Energy Sciences, Office of Sciences, US Department of Energy. We also acknowledge the partial support of St. Petersburg State University (Grant No. 11.38.654.2013).

High-order frequency mixing experiments using the radiation of a high-power Ti:sapphire laser and its second harmonic are described and discussed. Linearly and circularly polarized light fields with comparable intensities have been used. For the theoretical description a three-dimensional quantum-mechanical calculation with a \ensuremath{\delta}-function potential has been applied, showing quite good agreement with the experiments.

Guoli Wang,1,2 Cheng Jin,2 Anh-Thu Le,2 and C. D. Lin2 1Key Laboratory of Atomic and Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China 2J. R. Macdonald Laboratory, Physics Department, Kansas State University, Manhattan, Kansas 66506-2604, USA (Received 29 February 2012; published 10 July 2012)

We theoretically investigate high-order harmonic generation in argon gas driven by two orthogonally polarized laser pulses of single color, with the consideration of the macroscopic propagation effect including fundamental and harmonic fields. Both the three-dimensional propagation simulation and a simple analytical model based on a single-atom level demonstrate that the harmonic intensity of any harmonic order varies periodically when the delay between two pulses is scanned. This significant intensity modulation provides an experimentally observable contrast, from which the electron excursion time of a short quantum path associated with a certain harmonic order can be directly read. The result shows that the attosecond time-resolved electron motion can be mapped onto the macroscopic harmonic intensity modulation.

The mechanism of the formation of doorway quasistates (transient species) prior to the start of ionization of the two-electron molecular H${}_{2}$ system subjected to an eight-cycle ultrashort intense laser pulse is investigated by solving exactly the one-dimensional (1D) electronic time-dependent Schr\"odinger equation, including nuclear motion semiclassically. Space of the electron density of the two-electron 1D H${}_{2}$ system around the nuclei is partitioned into four physicochemically significant partitions, including homolytic $({e}_{1}{\mathrm{H}}_{\ensuremath{\alpha}}^{+}\ensuremath{-}{\mathrm{H}}_{\ensuremath{\beta}}^{+}{e}_{2})\ensuremath{\sim}({e}_{2}{\mathrm{H}}_{\ensuremath{\alpha}}^{+}\ensuremath{-}{\mathrm{H}}_{\ensuremath{\beta}}^{+}{e}_{1})$ and ionic $({\mathrm{H}}_{\ensuremath{\alpha}}^{+}\ensuremath{-}{\mathrm{H}}_{\ensuremath{\beta}}^{\ensuremath{-}})\ensuremath{\sim}({\mathrm{H}}_{\ensuremath{\alpha}}^{\ensuremath{-}}\ensuremath{-}{\mathrm{H}}_{\ensuremath{\beta}}^{+})$ transient species. The underlying mechanisms responsible for the formation and evolution of the homolytic and ionic transient species are explored based on probing variation of the two-electron norms of these states and their corresponding time-dependent high-order harmonic generation spectra.

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We report the experimental observation of generation of four and six strongly correlated and anticorrelated laser fields via atomic spin coherence in triple- and quadruple-\ensuremath{\Lambda}-type ${}^{85}$Rb atomic systems. The anticorrelated Stokes and anti-Stokes fields, simultaneously produced through scattering of the applied laser fields off the atomic spin coherence established in advance by the coupling and probe fields, are also highly anticorrelated, respectively, with the corresponding scattering fields. This method can, in principle, be used to create correlated and anticorrelated narrowband multiple fields of any desired order with long correlation times. It paves the way to further study of the generation of multifield quantum correlations and anticorrelations of arbitrarily high order.

A striking spectral-modulation effect has been observed by means of a proper postselection procedure under conditions where the spatial shift of the wave trains greatly exceeds the coherence length of the neutron beams traversing an interferometer. It is shown that Schr\"odinger-cat-like states are created by the superposition of two coherent states generated in the interferometer. These entangled states exhibit, under certain circumstances, characteristic squeezing phenomena, indicating a highly nonclassical behavior. Analogies with optical experiments are discussed.

We report the observation of harmonics up to the 119th in helium, generated using a 1.053-\ensuremath{\mu}m, 1-ps chirped-pulse-amplification laser at intensities up to 3\ifmmode\times\else\texttimes\fi{}${10}^{14}$ W/${\mathrm{cm}}^{2}$, and the measurement of their far-field spatial distribution. Complex spatial distributions are found for harmonics in the plateau region and near the cutoff; their angular distribution narrows to approximately that predicted by lowest-order perturbation theory.

High-order harmonic generation (HHG) by hydrogenlike atoms or ions in the field of counterpropagating laser beams of standing-wave configuration, with linear polarizations and relativistic intensities, is studied. The relativistic quantum theory of HHG in such field configurations (homogeneous), at which the impeding factor of relativistic magnetic drift of superstrong laser fields can be eliminated, is presented.

We performed a series of simulations of resonant high-order harmonic generation (HHG) by means of a multiconfigurational time-dependent Hartree-Fock (MCTDHF) approach for three-dimensional fullerenelike systems. The results proved the theory of resonant recombination proposed in this article and showed the ways of resonant HHG optimization. The results of MCTDHF calculation of the HHG for C{sub 60} were in good qualitative agreement with reported experimental data.