Role of Excited States In High-order Harmonic Generation

Ultrafast Resonant State Formation by the Coupling of Rydberg and Dark Autoionizing States

Studying the dynamics of dark states is challenging due to their inability to undergo single-photon emission or absorption. This challenge is made even more difficult for dark autoionizing states owing to their ultrashort lifetime of a few femtoseconds. High-order harmonic spectroscopy recently appeared as a novel method to probe the ultrafast dynamics of a single atomic or molecular state. Here, we demonstrate the emergence of a new type of ultrafast resonance state as a manifestation of coupling between Rydberg and a dark autoionizing state dressed by a laser photon. Through high-order harmonic generation, this resonance results in extreme ultraviolet light emission that is more than one order of magnitude stronger than for the off-resonance case. The induced resonance can be leveraged to study the dynamics of a single dark autoionizing state and the transient changes in the dynamics of real states due to their overlap with the virtual laser-dressed states. In addition, the present results allow the generation of coherent ultrafast extreme ultraviolet light for advanced ultrafast science applications.

Control of Electron Wave Packets Close to the Continuum Threshold Using Near-Single-Cycle THz Waveforms

The control of low-energy electrons by carrier-envelope-phase-stable near-single-cycle THz pulses is demonstrated. A femtosecond laser pulse is used to create a temporally localized wave packet through multiphoton absorption at a well defined phase of a synchronized THz field. By recording the photoelectron momentum distributions as a function of the time delay, we observe signatures of various regimes of dynamics, ranging from recollision-free acceleration to coherent electron-ion scattering induced by the THz field. The measurements are confirmed by three-dimensional time-dependent Schrödinger equation simulations. A classical trajectory model allows us to identify scattering phenomena analogous to strong-field photoelectron holography and high-order above-threshold ionization.

Role of Spin-Orbit Coupling in High-Order Harmonic Generation Revealed by Supercycle Rydberg Trajectories

High-harmonic generation is typically thought of as a sub-laser-cycle process, with the electron's excursion in the continuum lasting a fraction of the optical cycle. However, it was recently suggested that long-lived Rydberg states can play a particularly important role in high harmonic generation by atoms driven by the combination of the counterrotating circularly polarized fundamental light field and its second harmonic. Here we report direct experimental evidence of very long and stable Rydberg trajectories contributing to high-harmonic generation in such fields. We track their dynamics inside the laser pulse using the spin-orbit evolution in the ionic core, utilizing the spin-orbit Larmor clock. We confirm their effect on harmonic emission both via microscopic simulations and by showing how this radiation can lead to a well-collimated macroscopic far-field signal. Our observations contrast sharply with the general view that long-lived Rydberg orbits should generate negligible contribution to the macroscopic far-field high harmonic response of the medium.

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.

Ultraviolet supercontinuum generation driven by ionic coherence in a strong laser field

Supercontinuum (SC) light sources hold versatile applications in many fields ranging from imaging microscopic structural dynamics to achieving frequency comb metrology. Although such broadband light sources are readily accessible in the visible and near infrared regime, the ultraviolet (UV) extension of SC spectrum is still challenging. Here, we demonstrate that the joint contribution of strong field ionization and quantum resonance leads to the unexpected UV continuum radiation spanning the 100 nm bandwidth in molecular nitrogen ions. Quantum coherences in a bunch of ionic levels are found to be created by dynamic Stark-assisted multiphoton resonances following tunneling ionization. We show that the dynamical evolution of the coherence-enhanced polarization wave gives rise to laser-assisted continuum emission inside the laser field and free-induction decay after the laser field, which jointly contribute to the SC generation together with fifth harmonics. As proof of principle, we also show the application of the SC radiation in the absorption spectroscopy. This work offers an alternative scheme for constructing exotic SC sources, and opens up the territory of ionic quantum optics in the strong-field regime.

Supercontinuum generation can be utilized for light source development. Here the authors demonstrate ultraviolet supercontinuum generation from ions due to strong field ionization and multiphoton resonance effect.

Time-dependentab initioapproaches for high-harmonic generation spectroscopy

High-harmonic generation (HHG) is a nonlinear physical process used for the production of ultrashort pulses in XUV region, which are then used for investigating ultrafast phenomena in time-resolved spectroscopies. Moreover, HHG signal itself encodes information on electronic structure and dynamics of the target, possibly coupled to the nuclear degrees of freedom. Investigating HHG signal leads to HHG spectroscopy, which is applied to atoms, molecules, solids and recently also to liquids. Analysing the number of generated harmonics, their intensity and shape gives a detailed insight of, e.g., ionisation and recombination channels occurring in the strong-field dynamics. A number of valuable theoretical models has been developed over the years to explain and interpret HHG features, with the three-step model being the most known one. Originally, these models neglect the complexity of the propagating electronic wavefunction, by only using an approximated formulation of ground and continuum states. Many effects unravelled by HHG spectroscopy are instead due to electron correlation effects, quantum interference, and Rydberg-state contributions, which are all properly captured by anab initioelectronic-structure approach. In this review we have collected recent advances in modelling HHG by means ofab initiotime-dependent approaches relying on the propagation of the time-dependent Schrödinger equation (or derived equations) in presence of a very intense electromagnetic field. We limit ourselves to gas-phase atomic and molecular targets, and to solids. We focus on the various levels of theory employed for describing the electronic structure of the target, coupled with strong-field dynamics and ionisation approaches, and on the basis used to represent electronic states. Selected applications and perspectives for future developments are also given.

Time-Dependent Unitary Transformation Method in the Strong-Field-Ionization Regime with the Kramers-Henneberger Picture

Time evolution operators of a strongly ionizing medium are calculated by a time-dependent unitary transformation (TDUT) method. The TDUT method has been employed in a quantum mechanical system composed of discrete states. This method is especially helpful for solving molecular rotational dynamics in quasi-adiabatic regimes because the strict unitary nature of the propagation operator allows us to set the temporal step size to large; a tight limitation on the temporal step size (δt<<1) can be circumvented by the strict unitary nature. On the other hand, in a strongly ionizing system where the Hamiltonian is not Hermitian, the same approach cannot be directly applied because it is demanding to define a set of field-dressed eigenstates. In this study, the TDUT method was applied to the ionizing regime using the Kramers-Henneberger frame, in which the strong-field-dressed discrete eigenstates are given by the field-free discrete eigenstates in a moving frame. Although the present work verifies the method for a one-dimensional atom as a prototype, the method can be applied to three-dimensional atoms, and molecules exposed to strong laser fields.

High-order parametric generation of coherent XUV radiation

Extreme ultraviolet (XUV) radiation finds numerous applications in spectroscopy. When the XUV light is generated via high-order harmonic generation (HHG), it may be produced in the form of attosecond pulses, allowing access to unprecedented ultrafast phenomena. However, the HHG efficiency remains limited. Here we present an observation of a new regime of coherent XUV emission which has a potential to provide higher XUV intensity, vital for applications. We explain the process by high-order parametric generation, involving the combined emission of THz and XUV photons, where the phase matching is very robust against ionization. This introduces a way to use higher-energy driving pulses, thus generating more XUV photons.

Role of exchange and correlation in high-harmonic generation spectra of H2, N2, and CO2: Real-time time-dependent electronic-structure approaches

This study arises from the attempt to answer the following question: how different descriptions of electronic exchange and correlation affect the high-harmonic generation (HHG) spectroscopy of H₂, N₂, and CO₂ molecules? We compare HHG spectra for H₂, N₂, and CO₂ with different ab initio electronic structure methods: real-time time-dependent configuration interaction and real-time time-dependent density functional theory (RT-TDDFT) using truncated basis sets composed of correlated wave functions expanded on Gaussian basis sets. In the framework of RT-TDDFT, we employ Perdew-Burke-Ernzerhof (PBE) and long-range corrected Perdew-Burke-Ernzerhof (LC-ωPBE) functionals. We study HHG spectroscopy by disentangling the effect of electronic exchange and correlation. We first analyze the electronic exchange alone, and in the case of RT-TDDFT with LC-ωPBE, we use ω = 0.3 and ω = 0.4 to tune the percentage of long-range Hartree-Fock exchange and short-range exchange PBE. Then, we added the correlation as described by the PBE functional. All the methods give very similar HHG spectra, and they seem not to be particularly sensitive to the different description of exchange and correlation or to the correct asymptotic behavior of the Coulomb potential. Despite this general trend, some differences are found in the region connecting the cutoff and the background. Here, the harmonics can be resolved with different accuracy depending on the theoretical schemes used. We believe that the investigation of the molecular continuum and its coupling with strong fields merits further theoretical investigations in the near future.

Aurore: A platform for ultrafast sciences

We present the Aurore platform for ultrafast sciences. This platform is based on a unique 20 W, 1 kHz, 26 fs Ti:sapphire laser system designed for reliable operation and high intensity temporal contrast. The specific design ensures the high stability in terms of pulse duration, energy, and beam pointing necessary for extended experimental campaigns. The laser supplies 5 different beamlines, all dedicated to a specific field: attosecond science (Aurore 1), ultrafast phase transitions in solids (Aurore 2 and 3), ultrafast luminescence in solids (Aurore 4), and femtochemistry (Aurore 5). The technical specifications of these five beamlines are described in detail, and examples of the recent results are given.

The quantum-optical nature of high harmonic generation

High harmonic generation (HHG) is an extremely nonlinear effect generating coherent broadband radiation and pulse durations reaching attosecond timescales. Conventional models of HHG that treat the driving and emitted fields classically are usually very successful but inherently cannot capture the quantum-optical nature of the process. Although prior work considered quantum HHG, it remains unknown in what conditions the spectral and statistical properties of the radiation depart considerably from the known phenomenology of HHG. The discovery of such conditions could lead to novel sources of attosecond light having squeezing and entanglement. Here, we present a fully-quantum theory of extreme nonlinear optics, predicting quantum effects that alter both the spectrum and photon statistics of HHG, thus departing from all previous approaches. We predict the emission of shifted frequency combs and identify spectral features arising from the breakdown of the dipole approximation for the emission. Our results show that each frequency component of HHG can be bunched and squeezed and that each emitted photon is a superposition of all frequencies in the spectrum, i.e., each photon is a comb. Our general approach is applicable to a wide range of nonlinear optical processes, paving the way towards novel quantum phenomena in extreme nonlinear optics.

Conventional models of high harmonic generation typically do not provide a full quantum description of all phenomena. Here, the authors develop a fully quantum theory for high harmonic generation and use it to study the emission from a quantum system in a strong field.

Giant non-linear susceptibility of hydrogenic donors in silicon and germanium

Implicit summation is a technique for the conversion of sums over intermediate states in multiphoton absorption and the high-order susceptibility in hydrogen into simple integrals. Here, we derive the equivalent technique for hydrogenic impurities in multi-valley semiconductors. While the absorption has useful applications, it is primarily a loss process; conversely, the non-linear susceptibility is a crucial parameter for active photonic devices. For Si:P, we predict the hyperpolarizability ranges from χ ⁽³⁾/n _3D = 2.9 to 580 × 10^-38 m⁵/V² depending on the frequency, even while avoiding resonance. Using samples of a reasonable density, n _3D, and thickness, L, to produce third-harmonic generation at 9 THz, a frequency that is difficult to produce with existing solid-state sources, we predict that χ ⁽³⁾ should exceed that of bulk InSb and χ ⁽³⁾ L should exceed that of graphene and resonantly enhanced quantum wells.

The Role of Electron Trajectories in XUV-Initiated High-Harmonic Generation

High-harmonic generation spectroscopy is a powerful tool for ultrafast spectroscopy with intrinsic attosecond time resolution. Its major limitation—the fact that a strong infrared driving pulse is governing the entire generation process—is lifted by extreme ultraviolet (XUV)-initiated high-harmonic generation (HHG). Tunneling ionization is replaced by XUV photoionization, which decouples ionization from recollision. Here we probe the intensity dependence of XUV-initiated HHG and observe strong spectral frequency shifts of the high harmonics. We are able to tune the shift by controlling the instantaneous intensity of the infrared field. We directly access the reciprocal intensity parameter associated with the electron trajectories and identify short and long trajectories. Our findings are supported and analyzed by ab initio calculations and a semiclassical trajectory model. The ability to isolate and control long trajectories in XUV-initiated HHG increases the range of the intrinsic attosecond clock for spectroscopic applications.

Role of excited states in molecular alignment-dependent ionization

We introduce an ab initio approach and the modified strong-field approximation to investigate the alignment-dependent ionization of H2+(1π_u) exposed to different few-cycle laser fields. The ab initio calculations are performed by the B-splines one-center method and the Crank-Nicolson method in spherical coordinates. It is shown that the peak ionization probabilities appear around alignment angles 50^° and 40^° at the laser intensities 3×10¹³ W/cm² and 5×10¹³ W/cm², respectively, and the above distinct features come from the resonant excitation of the molecular ion, which is confirmed by calculation including and excluding the state 2σg in the basis expansion. Furthermore, the results obtained by including the state 2σg in the ab initio simulations can be qualitatively reproduced by the modified molecular length gauge strong-field approximation (SFA) taking account of the 1πu and 2σg states simultaneously. Analysis indicates that a part of electron is directly emitted from the 1πu orbital and another portion of electron is released from 2σg orbital and other excited state after the single-photon resonant transition between 1πu and 2σg orbitals.

Nonadiabatic redshifts in high-order harmonic generation from solids

We studied the multi-plateau high-order harmonic generation (HHG) from solids numerically. It is found that the HHG spectra in the second and higher plateaus are redshifted in short laser pulses due to the nonadiabatic effect. The corresponding FWHMs also increase as a function of the harmonic order, suggesting the step-by-step excitation of higher conduction bands in the HHG process. Although the system is symmetric in the coordinate space, even-order harmonics are present. It is due to the fact that the symmetry of electron motions and the population in the higher conduction bands is broken in the k space and time domain based on the indirect step-by-step excitation model. Our numerical results are in good agreement with recent experimental measurements of Ndabashimiye et al. [Nature 534, 520 (2016)].

Time-resolved high harmonic spectroscopy of dynamical symmetry breaking in bi-circular laser fields: the role of Rydberg states

The bi-circular scheme for high harmonic generation, which combines two counter-rotating circular fields with frequency ratio 2:1, has recently permitted to generate high harmonics with essentially circular polarization, opening the way for ultrafast chiral studies. This scheme produces harmonic lines at 3N + 1 and 3N + 2 multiples of the fundamental driving frequency, while the 3N lines are forbidden owing to the three-fold symmetry of the field. It is generally established that the routinely observed signals at these forbidden harmonic lines come from a slight ellipticity in the driving fields, which breaks the three-fold symmetry. We find that this is neither the only nor it is the dominant mechanism responsible. The forbidden lines can be observed even for perfectly circular, long driving pulses. We show that they encode rich information on the sub-cycle electronic dynamics that occur during the generation process. By varying the time delay and relative intensity between the two drivers, we demonstrate that when the second harmonic either precedes or is more intense than the fundamental field, the weak effects of dynamical symmetry breaking caused by finite pulse duration are amplified by electrons trapped in Rydberg orbits (i.e., Freeman resonances), and that the forbidden harmonic lines are a witness of this.

Alignment dependent ultrafast electron-nuclear dynamics in molecular high-order harmonic generation

We investigated the high-order harmonic generation (HHG) process of diatomic molecular ion H₂⁺ in non-Born-Oppenheimer approximations (NBOA). The corresponding three-dimensional time-dependent Schrödinger equation is solved with arbitrary alignment angles. It is found that the nuclear motion can lead to spectral modulation of HHG in both the tunneling and multiphoton ionization regimes. The universal redshifts of the whole spectrum are unique in molecular HHG. The spectral width of HHG increases in NBOA. We calculated possible influences on redshifts of HHG in real experimental conditions and found that redshifts 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. It will be instructive to related experiments.