# High order harmonic generation-based attosecond light sources and applications to quantum phenomena

Eric Constant, S. Nandi, Charles Picot, E. Prost, S. Palakkal, F. Lépine, V.

Loriot

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Eric Constant, S. Nandi, Charles Picot, E. Prost, S. Palakkal, et al. High order harmonic generation-based attosecond light sources and applications to quantum phenomena. APL Photonics, 2025, 10 (1), pp.010907. 10.1063/5.0235171. hal-04947085

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# High order harmonic generation-based attosecond light sources and applications to quantum phenomena

E. Constant; S. Nandi 🌐 ; C. Picot; E. Prost 🌐 ; S. Palakkal 🌐 ; F. Lépine 📄 ; V. Loriot 🌐

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APL Photonics 10, 010907 (2025)

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# High order harmonic generation-based attosecond light sources and applications to quantum phenomena

Cite as: APL Photon. 10, 010907 (2025); doi: 10.1063/5.0235171

Submitted: 26 August 2024 • Accepted: 2 December 2024 •

Published Online: 31 January 2025

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E. Constant, S. Nandi, C. Picot, E. Prost, S. Palakkal, F. Lépine,ª and V. Loriot

## AFFILIATIONS

Universite Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, UMR5306, F-69100 Villeurbanne, France

ª) Author to whom correspondence should be addressed: franck.lepine@univ-lyon1.fr

## ABSTRACT

Attosecond science and technology have emerged as a promising path for improving our understanding of light-matter interaction. High harmonic generation based light sources combine high photon energy, broad spectrum, and short duration with a relatively compact implementation, which makes them flexible tools to study and modify physical and chemical properties of matter by acting directly on electronic degrees of freedom. Because extreme ultraviolet radiation interacts primarily with outer and inner valence electrons, it finds likely applications in atomic, molecular, and material science. Despite their spectacular successes, the development of these sources is still a subject of intense investigations, searching for extreme performances in terms of specifications and reliability. In this perspective article, we present some of the current developments in high harmonic generation sources, as well as related applications in quantum and attosecond dynamics in atoms and molecules.

© 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0235171

## I. INTRODUCTION

High harmonic generation (HHG) represents a highly nonlinear process in which the photons of a fundamental laser beam are converted into photons with much higher energy that typically reaches the extreme ultraviolet (XUV) range (10–100 eV). This phenomenon occurs when an intense laser pulse, reaching an intensity above $10^{13}$ W/cm$^2$ [typically a few tenth of femtoseconds (1 fs = $10^{-15}$ s) pulse with few millijoules] is focused on a gas medium (jet, cell, etc.). Compared to traditional low harmonics generation (second, third, fourth), HHG reaches the non-perturbative regime that is often described in the framework of the so-called three-step model.¹ In this model, the light is considered as a time-dependent electric field that is intense enough that the force exerted on the atoms is able to compensate for the Coulomb force between the electron and the nuclei. Consequently, during this interaction, the atom is tunnel-ionized allowing the electron to escape and to be accelerated by the light electric field. The emitted electron accumulates energy due to this acceleration, and it is brought back to the ionic core after the electric field changes its sign. This electron can

recombine with the nuclei and release its excess energy through the emission of an electromagnetic field in the form of high-energy photons. The re-collision process occurs every half cycle of the fundamental light, and this coherent conversion allows for the production of XUV/x-ray light that is structured in time and space.¹,²

Nearly 40 years after the discovery of HHG in two laboratories in 1987,³,⁴ HHG-based light sources have now found applications in many research fields related to the study of all phases of matter. One of the spectacular advances arose from the generation of attosecond pulses (1 as = $10^{-18}$ s), which permit investigation of electron dynamics on an atomic length scale.⁵ Attosecond pulses allowed us to observe electron motion in atoms and molecules in gas and liquid phases with the perspective of manipulating chemical reactions, giving rise to so-called “attosecond chemistry.”⁶,⁷ It also allows investigating modifications of material properties in the condensed phase, such as ultrafast switches from insulator to conductor, with possible applications in petahertz electronics.⁸ The HHG process itself is based on the sub-optical cycle electron dynamics that are encoded in the resulting harmonic amplitude and phase.⁹⁻¹¹ Beyond such fundamental aspects, HHG technologies are also progressing in
the direction of technological applications such as lithography and nano-structuration of materials. Another exciting development in the XUV ultrafast community is the possibility to observe and control quantum processes on ultrafast timescales to interrogate and manipulate entanglement and coherence down to the attosecond timescale. Interestingly, even if laser technologies and HHG sources made tremendous progress in the past years, these emergent applications are providing new challenges for the current state-of-the-art of the sources, pushing for more control and reliability.

While attosecond pulse generation is becoming routine in laboratories worldwide, experiments are also producing increasingly accurate measurements, which in turn require a detailed knowledge of the XUV pulse properties. A seminal example is the measurement of attosecond delay in photoionization using RABBITT¹² (reconstruction of attosecond beating by interference of two-photon transitions) or streaking techniques.¹³ While RABBITT and streaking were initially thought of as tools to measure the duration of XUV attosecond pulses, with the improvement of attosecond technology, it has been realized that the phase information extracted from such measurements also gives access to the electron scattering phase and photoionization time delays on the angstrom length scale.¹⁴ Consequently, electron scattering time delay measurements required an accurate characterization of the pulse properties allowing us to extract delays down to a few attosecrons. Initially performed in gas phase atomic targets, scattering delay measurements allowed the demonstration of the influence of resonances,¹⁵,¹⁶ angular momenta,¹⁵ or electron correlation.¹⁷ Later applied to molecules,¹⁸,¹⁹ it was possible to perform experiments on small systems and to measure the role played by the nuclear degrees of freedom or the molecular structure.²⁰ When applied to surfaces,²¹ it was possible to measure how the electron path was modified by the band structure. With recent advances, scattering delay has been combined with coincidence spectroscopy²² and complex polarization of the light.²³ It has also been investigated for increasingly complex systems to explore dimensionality effects.²⁴

In general, with the current possibility of generating pulses down to a few tens of attoseconds,²⁵ it becomes compulsory to control HHG sources with a similar accuracy. This also becomes striking when tackling the question of quantum processes on an ultrafast timescale. Nowadays, methods to characterize quantum states have been developed: the study of the interplay between coherence and entanglement is emerging, and the manipulation of complex quantum objects at the ultimate time scale is still in its infancy. In an ideal experiment where an electron quantum state must be prepared and fully characterized, the statistical dispersion of pulse properties, such as phase and amplitude, must be minimized. Here again, it becomes compulsory to control the single shot properties of the pulses.

In this article, we present some of the recent progress and perspectives on the development of HHG sources as well as on recent progress in attosecond scattering phase measurements and the ultrafast quantum aspect of XUV science.

## II. DEVELOPMENT OF ULTRAFAST XUV SOURCES BASED ON HHG

### A. HHG sources of classical light

Soon after its discovery, it was realized that HHG-based sources can provide broadband XUV light pulses, especially since it consists

of a plateau with an almost constant harmonic amplitude that ends up at a cutoff frequency, $v_{cutoff}$, above which no harmonics are emitted. It was also realized that the regular harmonic structure can be a signature of phase locking, and therefore high order harmonics can potentially be used to generate attosecond pulses,²⁶ assuming proper spectral filtering of a large spectrum and regular spectral phase evolution. After years of studying the physics of HHG and questioning the experimental feasibility of attosecond pulse generation,²⁷ techniques for measuring the attosecond temporal profile²⁸⁻³⁰ as well as confinement approaches³¹ to control the XUV duration have been proposed. It was also demonstrated that the resulting attosecond pulses are locked in time with respect to the optical field of the fundamental pulse used for the generation. This allowed using the same fundamental field for harmonic generation and for its characterization. Subsequently, in 2001, it was shown¹²,³² that attosecond pulses can indeed be generated in gases because the harmonics can be properly phase locked.

After these discoveries, the field of so-called “attosecond science”⁵ kept evolving quickly with the rapid evolution of femtosecond laser technologies combined with the improved understanding of the XUV source properties. Nowadays, the repetition rate of HHG-based sources spans over almost ten orders of magnitude. It evolved both toward high repetition rate and low energy sources³³ using Yb fundamental lasers (for instance, compatible with coincidence experiments), and low repetition rate and high energy sources³⁴,³⁵ using Ti:Sa fundamental lasers, which opened new opportunities for non-linear XUV physics.³⁵⁻³⁸ The XUV spectral width has also increased significantly over the years with the advent of new femtosecond laser systems operating in the near infrared (NIR) domain as the cutoff frequency, $h\nu_{cutoff} = I_p + 3.17 U_p$ (here, $I_p$ is the ionization potential of the medium and $U_p$ is the ponderomotive energy of the driving laser), increases rapidly with the fundamental wavelength. On the other hand, it was realized that the XUV emission efficiency can be boosted by using UV femtosecond pulses³⁹,⁴⁰ that lead to a stronger response at the single atom level and are less sensitive to collective effects such as phase matching or ionization perturbed propagation.⁴¹,⁴² Figure 1 illustrates this tunability through several HHG spectra generated from various atomic targets (different $I_p$), different phase matching conditions, and several central wavelengths of the driving pulse ($U_p$[eV] = 6 × I[10¹⁴ W cm⁻²] at $\lambda_0$ = 800 nm and $U_p$[eV] = 1,5 × I[10¹⁴ W cm⁻²] at $\lambda_0$ = 400 nm). Gas-phase media have been mostly employed for the generation of HHGs. Nowadays, solid-state samples⁴³,⁴⁴ can also be used for the same purpose and this approach is also very interesting to study material properties like band structure dynamics. The spatiotemporal structure of the harmonics can become complex by structuring the fundamental beam, such as with angular orbital momentum⁴⁵ or non-uniform polarization.⁴⁶ The optimization of the harmonic sources in terms of XUV flux has also evolved with time, and orders-of-magnitude improvement have regularly been achieved by many laboratories.⁴⁷ Optimizing parameters such as interaction geometry, phase matching, harmonic re-absorption, and shaping of the beam profile are numerous, but nowadays they are well identified.⁴⁰,⁴⁸⁻⁵² They constitute as many control knobs that can be used to optimize the HHG source. The scaling of the parameters⁵³ is also well understood and allows optimizing sources⁴⁸⁻⁵² with very different characteristics. Overall, similar efficiencies have been obtained in many laboratories around the world in a large
<figure><img src="image_10.png" /><figcaption>**FIG. 1.** Tunability of the high harmonic spectrum. By choosing the generation gas [(a) xenon, (b) neon, (c), (d) argon], the fundamental driving pulse frequency [800 nm for (a), (b), (d)] or its second harmonic (c), a plethora of harmonic orders of the fundamental frequency and various envelops selection (dash gray lines) can be generated. Top panels are the focused (a), (c), (d) or the collimated (b) spatial profile and the low panels are the corresponding HHG spectra represented in harmonic order of the 800 nm.</figcaption></figure>

range of experimental configurations, which illustrates the versatility and robustness of HHG sources.

Many key theoretical steps were also achieved in this domain. In brief, it was shown that HHG could be well described with both classical¹ and quantum approaches.² These approaches highlight the role of electrons accelerated in the field during ionization and how the XUV emission depends on the electron trajectories in the strong fundamental field. In the strong field approximation (SFA), the Coulomb attraction of the electron by the nucleus was initially neglected, and the time of XUV emission could be estimated with good accuracy. More precise estimation was mandatory to justify some experimental observations,⁵⁴ and Coulomb attraction can now be included to refine these approaches. It was shown that this attraction influences the return time of the electron and thereby the emission time of the XUV photons.⁵⁵⁻⁵⁷ Multi-electron effects are also currently considered further increasing our capabilities for precise HHG simulations.

Temporal characterization of the harmonic pulses has also evolved greatly.⁵⁸ Initially based on time resolved photoelectron¹²,³²,⁷⁵ or photoion²²⁴ spectroscopy, some all-optical methods are now available. They use an interferometric perturbation of the HHG process⁵⁹⁻⁶¹,²³² or spectral interferometry⁶²,⁶³ to retrieve the spectral phase and characterize the XUV temporal profile. In this paper, we focus on the so-called RABBITT¹² technique that relies on observing energy resolved photoelectron spectra induced by two photon transitions (one XUV and one NIR photon)

as a function of the delay between the pulses. This approach is illustrated in Fig. 2(a) that represents the photoelectron energy levels after absorption of an XUV photon (where the photoelectron energy corresponds to the difference between the photon energy and the ionization potential) or after absorption of one XUV photon and simultaneous absorption or stimulated emission of an NIR photon. This later case leads to additional photoelectron energy peaks, called sidebands. The amplitude of the sidebands is determined by the interference between the two quantum paths that lead to the same final kinetic energy. The quantum character of the electron appears here via the interference between the two-photon transitions. The sidebands oscillate with the XUV-NIR delay and contain the information on the relative phase between the consecutive harmonics.

This approach was initially based on the assumption that the attosecond pulse train was periodic and thereby very long. It has been extended using the so-called FROG-CRAB⁶⁴ approach that allows one to reconstruct the full temporal profile of the attosecond pulse train with finite duration. Figure 2(b) shows a FROG-CRAB measurement (i.e., RABBITT measurement over the entire temporal overlap between the pulses). The photoelectrons are redistributed by the IR dressing pulse during its temporal overlap with the XUV pulse. Oscillations at twice the fundamental frequency are visible in the time-resolved photoelectron spectrum. The oscillation phases of the sidebands are composed of the XUV ionization phase, the dressing continuum-continuum transition phase, and the optical
<figure><img src="image_11.png" /><figcaption>FIG. 2. (a) Energy levels of photoelectrons created by absorption of XUV photons (labeled H11–H17, blue arrows) and IR photons (up or down red arrows). The sideband peaks are created by the absorption of an XUV photon and absorption (up arrow) or stimulated emission (down arrow) of an IR photon. Sidebands are created when the IR and XUV pulse overlap in time. Their amplitudes depend on the IR–XUV delay and the relative dephasing between adjacent harmonics. (b) FROG Crab trace obtained by ionizing argon atoms with harmonics and a phase locked delayed fs IR pulse. Oscillations depend on the relative dephasing between harmonics and the temporal overlap of the XUV and IR. (c) Provides the temporal profile of the attosecond pulse train reconstructed with the RABBITT approach when only the central part of the Frog Crab trace is used.</figcaption></figure>

phase of the harmonics.⁶⁵,⁶⁶ The phase difference between the harmonics can be extracted by removing the other contributions.⁶⁷ The temporal profile of the attosecond pulse train can be reconstructed with the amplitudes and the phases of the harmonics as shown in Fig. 2(c).

To reduce the number of attosecond pulses in the attosec-
ond pulse train, one can reduce the duration of the fundamental
pulse. Post-compression techniques can shorten the pulse dura-
tion close to the single cycle limit.⁶⁸ First, the spectrum is broad-
ened using self-phase modulation in a hollow core fiber⁶⁹ or in
a multi-pass cell,⁷⁰ for instance, and the pulse dispersion is fur-
ther compressed by passive⁷¹,⁷² or active optics.⁷³,⁷⁴ A few-cycle
pulse can produce an isolated attosecond pulse (IAP) that corre-
sponds to a broadband continuous XUV spectrum in phase.⁷⁵,⁷⁶
The IAP generation has rapidly evolved, and several approaches
are currently used to confine the XUV emission. Pioneering exper-
imental work was performed in the group of Hentschel⁷² using
the so-called intensity gating approach, where high-energy cutoff
photons are only emitted around the maximum of the few cycle
pulse. A single sub-100 attosecond IAP pulse could hence be iso-
lated by spectrally isolating the photon energies around 100 eV.⁷⁷
Another solution to confine lower order harmonics emission down
to the IAP level⁷⁸,¹⁰⁷ came with the polarization gating,³¹,⁷⁹,¹⁰⁷
where the polarization state of the few-cycle pulse rapidly evolves
and confines the XUV emission in a time window shorter than
the fundamental pulse. This approach can be upgraded by adding
a weak second harmonic pulse to achieve the so-called double
optical gating⁸⁰,⁸¹ that allows reaching an IAP with a longer fun-
damental pulse duration. Mixing non-commensurate amount of
wavelengths in the fundamental pulse, such as with a waveform syn-

thesizer,⁸²,⁸³ provides a high degree of freedom that can generate high-energy IAP capable of inducing non-linear XUV processes.⁸⁴ Finally, the temporal manipulation of the fundamental wavefront can generate an attosecond pulse train where each attosecond pulse is emitted in a specific direction. An IAP can hence be extracted by spatial selection,⁸⁵ an approach known as the *attosecond lighthouse*.

An important parameter to reach IAP reproducibility consists of locking the carrier envelope phase (CEP) of the driving pulse. The CEP is usually expressed in radians (φ<sub>CEP</sub>) and corresponds to the mismatch between the maximum of the carrier electric field oscillation and the pulse envelope Λ(t) of the electric field, that is, E(t) = Λ(t) × cos(ωt − φ<sub>CEP</sub>), as illustrated in Fig. 3(a). The CEP corresponds to the 0th order of the spectral phase Taylor-expansion and hence, it is the most sensitive term to the environment and dispersion. The CEP can also be expressed in time by comparison with the carrier frequency, that is, E(t) = Λ(t) × cos(ω(t − t<sub>CEP</sub>)), with t<sub>CEP</sub> = φ<sub>CEP</sub>/ω. In this representation, 100 mrad corresponds to a time offset of 42 as at 800 nm, 55 as at 1030 nm, and 210 as at 4 μm. Thus, CEP’s sensitivities are related to attosecond dynamics in the optical regime. Figure 3 illustrates the impact of CEP on the generation of attosecond pulses. The pulses shown in Fig. 3(a) have the same duration but differ in their CEP. Only one clear maximum of the electric field appears for φ<sub>CEP</sub> = 0 and two identical extrema for φ<sub>CEP</sub> = π/2. Since an attosecond pulse can be generated at each maximum of the electric field, essentially one or two attosecond pulses can be generated by a few cycle NIR pulse depending on its CEP. This is schematically illustrated in Fig. 3(b), where the attosecond pulse timing is linked to the phase of the field and its amplitude to a power low of the driving intensity profile. In this figure, the role
<figure><img src="image_12.png" /><figcaption>FIG. 3. Carrier Envelope Phase (CEP) control. Representation of the electric field for a given pulse duration and different CEP values (a). The envelope of the pulse is represented in a gray dashed line. Simulated train of attosecond pulses (b) generated from the pulses shown in (a). (c) Long-term and automatic stabilization/optimization of the CEP adapted from Ref. 135.</figcaption></figure>

of the CEP is clear since the NIR pulse duration is close to the dura-
tion of the carrier field optical cycle. However, such short single- or
few-cycle pulses cannot directly be delivered by the standard laser
technology because of the limited spectral bandwidth of the laser
crystals.⁸⁶

Nowadays, IAPs produced by CEP-stable few-cycle pulses rep-
resent a central tool of attosecond science. For instance, an IAP and
a CEP stable pulse used in a pump-probe configuration can allow
observing electron dynamics through the streaking method. ¹³ In
addition, the electronic, vibrational, and dissociation dynamics can
be observed by measuring the XUV transmission or reflection on an
attosecond timescale. ⁸⁷⁻⁸⁹ The photochemistry of molecules can be
observed by measuring the ion/dication yield of the molecules. ⁹⁰,⁹¹
Nevertheless, the CEP control of intense laser pulses remains a dif-
ficult task. Several methodologies have emerged using active and
passive strategies. ⁹² Stabilizing the CEP of a femtosecond oscillator
consists of stabilizing the frequency comb of the cavity. ⁹³,⁹⁴ Typi-
cally, stabilization based on proportional–integral–derivative (PID)
controllers is used to nullify the various sources of instabilities. A
fast loop can operate up to tens of kHz and allows controlling fast
and low amplitude mechanical or electromagnetic noise. A slow
loop that operates in a few seconds timescale compensates the ther-
mal drifts, which can potentially have a high amplitude but usually
have a lower frequency (mHz). For amplified lasers, ⁹⁵,⁹⁶ the CEP
stabilization may be performed before and after the amplification

steps. The stabilization of the CEP is usually performed by using
an (f – 2f) interferometer.⁹⁷ It consists of measuring the spectral
interference between the fundamental frequency (f) and its sec-
ond harmonic (2f) of an octave-spanning spectrum. Since the CEP
corresponds to the 0th order spectral phase, its second harmonic
corresponds to twice the CEP value of the fundamental. The inter-
ference between both leads to the CEP difference between the pulses
that corresponds to the CEP of the fundamental. Different (f –
2f) configurations are usually used to measure the CEP before and
after the amplification due to the large difference in the repetition
rate. Nowadays, commercially available solutions allow one to sta-
bilize and measure the CEP of amplified pulses up to 100 kHz.⁹⁸
Despite the stabilization of the CEP becoming an accessible tool, it
is still mainly used in specialized laboratories. Solutions based on
software that take care of all the parameters related to the CEP (con-
trol, monitoring, and auto-optimization) are also available⁹⁹ [see
Fig. 3(c)], allowing the benefits of CEP to be accessible in other
research fields.

For femtosecond laser technologies other than Ti:Sa based sys-
tems, non-linear effects provide solutions to passively stabilize the
CEP using the difference frequency generation (DFG)¹⁰⁰⁻¹⁰² pro-
cess. In this process, two pulses with different central frequencies
are generated from the same non-CEP stabilized pulse. If such two
pulses are from the same source and they can maintain the same
relative CEP, DFG can perform the spectral difference between
the two pulses, which, in turn, corresponds to a phase subtrac-
tion between the two spectra, and the CEP is canceled out. Only
thermal drifts of the CEP have to be compensated that is usually
performed by the slow feedback loop. These solutions are currently
under operation for Optical Parametric Chirped Pulse Amplifica-
tion (OPCPA) systems with passive stabilization.¹⁰³ With the rapid
development of post-compression systems down to the few-cycle
regime, the CEP becomes a crucial parameter that plays a sig-
nificant role in the light–matter interaction. Apart from the CEP
performance itself, the possibility to get a CEP stabilization at a
reasonable level with a robust, simple, and safe solution is still an
active quest.

B. Quantum aspects of HHG sources

Besides considering HHG-based sources as classical time-
dependent electromagnetic waves, the quantum properties of these
sources became a topic of increasing interest for the community.
Since its inception via the HHG process, attosecond science has been
in the forefront of probing quantum phenomena. One of the earliest
examples is the description of the HHG process itself in terms of the
time-dependent Schrödinger equation that led to the development of
the strong-field approximation in intense light-matter interaction.²
Later, the HHG process was reinterpreted in terms of the Feyn-
man path-integral approach,¹⁰⁴ leading to the discovery of quantum
orbits for the electron undergoing excursions in the electric field
of the driving laser. This led to a plethora of literature for study-
ing quantum-path interference in HHG processes,¹⁰⁵,¹⁰⁶ especially
in the case of linear molecules,¹⁰⁷ in the early days of attosecond sci-
ence. Nevertheless, the XUV light produced via the HHG process
was considered classical—a Gaussian pulse in the temporal domain
denoting the time-dependent electric field of the electromagnetic
radiation.
In recent times, the quantum optical nature of the HHG process has attracted a lot of attention. Of particular interest is the production of non-classical states of light.<sup>108</sup> It was first shown experimentally in the case of HHG in a xenon gas-medium by counting the photon statistics of the driving NIR field.<sup>109</sup> Later, it was successfully extended toward the HHG process in a semiconductor, namely, ZnO crystals.<sup>110</sup> Eventually, the creation of Schrödinger cat states via macroscopic superposition has become possible in HHG in the gas-medium by incorporating a quantum spectrometer capable of selecting only the NIR photons responsible for the HHG process.<sup>111</sup> All these experimental observations have led to increased exploration of how the HHG process takes place and whether quantum electrodynamical effect,<sup>112</sup> as opposed to the classical electromagnetic field described by the vector potential in the strong-field approximation, can be observed across attosecond timescales.<sup>113</sup>

### III. APPLICATION OF XUV SOURCES

#### A. Attosecond delays in photoionization

One of the capabilities of attosecond science is to unravel the ionization dynamics down to the attosecond and angstrom scales. Ionization is a natural consequence of the interaction of matter with an attosecond pulse. The XUV photons in the attosecond pulse have energies well above the ionization potential of atoms and molecules. After XUV photo-absorption, an electron can be emitted into the continuum. Before being free from the influence of its parent ion at large distances, the electron scatters in the potential of the remaining ion. The outgoing photoelectron wavepacket contains the information about this interaction in its phase. Following the Wigner-Smith-Eisenbud theory, it can semi-classically be linked to a delay in the photoemission process.<sup>114–116</sup> Accessing such phases allows reconstructing the ionization dynamics. Quantum scattering phases cannot be measured directly with a particle detector but are usually reconstructed through an interference with another reference channel. Using the monochromatic synchrotron radiation, the relative phases between the partial waves can be retrieved through their interferences in the photoelectron angular distribution.<sup>117–119</sup> In attosecond science, photoionization delays are measured following the streaking or RABBITT methods as mentioned above. The first method needs an isolated attosecond pulse combined with a moderately strong NIR field that “streaks” the momentum of the electron depending on the ionization time. It has been applied successfully to various systems such as atoms,<sup>14</sup> surfaces,<sup>120</sup> and nanoparticles.<sup>121,122</sup> When considering systems with many electronic states close in energy, it might be preferable to consider the RABBITT method that provides simultaneous spectral and temporal resolution.<sup>123</sup> In contrast to most of the femtosecond time-resolved experiments,<sup>124</sup> the temporal resolution is not determined by the pulse duration but mostly by the interferometric accuracy of the interferometer. On atomic targets, it produces a background-free signal as shown in Figs. 2 and 4, allowing to extract the ionization time delay across a wide range of photon energies unraveling the temporal aspects of the structured ionization continuum.<sup>15,16,125,126</sup> RABBITT experiments are robust and reliable as shown in Fig. 4, where two comparable measurements taken at more than 24 h difference are perfectly overlapping with each other. More recently, similar experiments with molecules are of particular interest with

several measurements in H₂,<sup>127,128</sup> N₂,<sup>18,20,129</sup> CO,<sup>130</sup> NO,<sup>131</sup> O₂,<sup>132</sup> CO₂,<sup>133</sup> H₂O and N₂O,<sup>134</sup> CH₄ and CD₄,<sup>135,136</sup> CF₄,<sup>137,138</sup> C₂H₂,<sup>139</sup> and poly-cyclic aromatic hydrocarbons (PAHs).<sup>24</sup> Here, we restrict the discussion primarily to delay measurements performed with the RABBITT method.

To begin with, it is worth noting the origin of such a time delay in photoemission processes. In Wigner’s original semi-classical formulation,<sup>150</sup> the derivative of the scattering phase with respect to the wave-vector of the scattered particle must be larger than or equal to $-r_s$, where $r_s$ is the scattering length. This follows from the causality principle, which stipulates that the reaction products of a scattering process should not fly apart before they could collide. Quantum mechanically using the S-matrix formulation, the amplitude of the scattered wave in its asymptotic form can be shown to be dependent on the group delay of the wave-packet.<sup>140</sup> In analogy to the classical case, the group delay refers to the energy derivative of the scattering phase of the packet. This can be interpreted as the delay in arrival of the wave-packet to a

<figure><img src="image_13.png" /><figcaption>**FIG. 4.** Day-to-day reproducibility of the attosecond experimental conditions. Two RABBITT measurements have been performed on two consecutive days with the laser and electrodes switched Off and On and re-optimized. (a) High harmonic spectrum after the Sn filter, (b), (c) RABBITT spectrogram, and their corresponding amplitude (d) and phase (e) of oscillation at $2\omega_0$. The two different days are represented in different colors in the panels (a), (d), and (e) and separately in (b) and (c).</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**FIG. 5.** Principle of molecular ionization time measurements. (a) Concept figure: several electronic states of the valence band of a polyatomic molecule can be ionized by a given XUV photon. The ionized electrons scatter into the molecular potential that affects the phase of the photoelectron wavepackets. This phase can be interpreted in terms of ionization time delay following the Wigner theory.¹⁵⁰,¹⁵¹ Such phases can be measured/compared using the RABBITT method that produces photoelectron interferences. (b) Typical RABBITT measurement in a small molecule (adapted from Ref. 175). Here, in C₂H₂, the outer electronic state produces a RABBITT signal at high photoelectron energy that is almost isolated from the other electronic states. At low photoelectron energies, the RABBITT oscillations for each electronic state overlap each other.</figcaption></figure>

detector compared to another packet that does not interact with
the scattering center. Indeed, the Wigner delay is ubiquitous in
different physical phenomena across different timescales, such as
zeptosecond delays in particle-particle collision,¹⁴¹ femtosecond
delays in Goos-Hänchen effect,¹⁴² picosecond delays in quantum
dots,¹⁴³ and nanosecond delays in light-atom scattering,¹⁴⁴ just to
name a few. Photoionization can be considered as a half-scattering
process—following the emission of the electron in the continuum, it
gets scattered in the potential of the residual ion. Due to its interac-
tion with the local potential, the photoelectron wave-packet picks
up a group delay, the information about which can be accessed
by the interferometric technique, such as RABBITT and streaking,
after subtracting the influence of the dressing field.¹⁷,¹⁴⁵,¹⁴⁶ In this
sense, measuring the relative photoionization time-delay between
two different ionization channels provides a measure of the differ-
ence in spatial separation (proportional to the scattering length)
between the scattered particle (photoelectron) and the scattering
center (residual ion) for two different scattering channels. Simi-
lar to photoionization time-delays, tunneling time associated with
electrons traversing a potential barrier has also been measured,
especially using the “attoclock” technique in strong-field ionization
processes.¹⁴⁷⁻¹⁵¹

To determine the time delay of a given electronic state, usu-
ally two measurements are needed: the target ionization channel
and a reference. While in the calculations, the reference is clearly
defined as the electron scattering in the absence of the molecular
potential, experimentally one faces the practical problem of deter-
mining the time “zero.” In the pioneering work of Klunder et al.,¹⁷
the ionization time delay from the 3s² and 3p⁶ states of argon has
been measured and compared within a single RABBITT experi-
ment. As schematically shown in Fig. 5(a), molecules have many
electronic valence states close in energy. Each state can be ionized
and the escaping electron experiences its own scattering in the ionic

potential, and hence gets its own ionization time delay. When the
states are separated enough in the energy, the difference in ion-
ization time delay between the states can be extracted without any
spectral overlap.²⁶,¹⁷⁵ For instance, as shown in C₂H₂ in Fig. 5(b), the
outermost electronic state is separated by several eVs compared to
the inner states, providing a range of photoelectron energy where the
X-state can be measured separately. In the nitrogen molecule, the X-
state (ionization from the 3σg orbital with 15.6 eV ionization energy)
experiences a shape resonance at around 30 eV photon energy. The
escaping electron can be trapped into the centrifugal potential bar-
rier, leading to a delay in photoionization. In contrast, the A-state
(ionization from the 1πu orbital with 17 eV ionization energy) of N₂
has a flat continuum at the same photon energy and can be used as
Refs. 20 and 165. The comparison between such states leads to the
measurement of the electron trapping time in the shape resonance,
which has been studied experimentally using different wavelengths
of the dressing field in the RABBITT technique. In addition for res-
onances, for molecules with D∞h symmetry, the time delays can be
affected by Young-type two-center interferences.¹⁵² While this has
recently been explored in the case of krypton dimers,¹⁵³ for multi-
electronic linear molecules probing, such effects require removing
core-electrons. With the availability of attosecond x-ray pulses from
free-electron lasers, these types of experiments can indeed be fore-
seen.¹⁵⁴ For time-delay experiments, an atom with a flat continuum
at the photon energy of interest can be used as a reference and mixed
with the target molecular or atomic gas. By measuring the elec-
tron and the ion in coincidence,²²,¹⁶³,¹⁶⁶ the photoelectrons can be
sorted as a function of their corresponding ionic parent. The samples
are studied under the exact same experimental conditions, allowing
direct comparison. For small molecular systems with a dissociative
channel, such as in H₂,¹⁶³ it allows working in the molecular frame
to observe the coupling between the electron and ion dynamics. For
isotopologues, the signature of the nuclear dynamics on the ioniza-
tion time delay can be extracted.¹⁷¹ In diatomic molecules such as CO or NO as well as in polyatomic molecules such as $CF_4$,¹⁷⁴ the asymmetry of the molecular potential can lead to scattering time that strongly depends on both emission angles in the molecular frame¹⁶⁶ and resonances.⁷⁷,¹⁵³ For molecular clusters, the ionization time delay can vary as a function of the number of molecules in the cluster. This follows the general intuition that the ionization time delay increases with the size of the scattering center.⁵² To perform experiments using a single photoelectron spectrometer, a specific strategy involving gas mixtures can also be employed.²⁴ The technique consists of measuring the two samples (molecule and atomic reference) separately and performs a third measurement with a mixture of both. The measurement using gas mixture corresponds to a linear combination of the isolated measurements. Any possible inherent time drift is efficiently canceled by using an optimization algorithm. It has been successfully applied to $C_2H_2$¹⁷⁵ and PAH molecules²⁴ using argon as a reference. As a perspective, another approach would be to perform a first measurement in a molecular gas and refocus the outgoing XUV and NIR beams into a reference atomic gas. This can be performed using multifocus beamlines,¹⁵⁵ which are normally used to correct temporal jitter between the pulses.¹⁵⁶

## B. Quantum applications

While the HHG process itself can act as a self-probe toward various quantum effects in the attosecond domain, the XUV pulses thus produced can also be used to unlock different physical phenomena pertinent to quantum mechanics. Very early on, it was realized that the attosecond pulses generated via the HHG process are phase-locked due to the unique driving pulse. It means that the XUV pulses have a high degree of coherence, both spatially and temporally.¹⁵⁷ Thus, a major motivation for the attosecond science community has always been to characterize and control various photon-induced processes at ultrafast timescales using the coherent nature of these pulses. Pertinent examples include observing charge migration in complex molecules,¹¹² coherent diffractive imaging,¹³⁸ and multi-band high-harmonic coherence spectroscopy.¹⁵⁹

Recently, another aspect of the attosecond light-matter interaction has gained momentum, namely, to observe directly the effect of entanglement and/or coherence in the light-induced processes, especially in the case of photoionization. As it is well-known, entanglement is a purely quantum phenomenon without any classical counter-part—when the composite wavefunction of a many-body system cannot be factorized in terms of the wavefunction of its constituents, they are considered to be entangled with each other.¹⁶⁰ For photoionization, the emitted photoelectron and the residual ion are fully entangled since the measurement of the kinetic energy of the former provides accurate information about the quantum state of latter.¹⁶¹ This provides a unique opportunity to study the manifestation of the entanglement between two massive particles, namely, the photoelectron and the photoion, as was shown in the pioneering experiments conducted for diatomic molecules using narrowband soft x-rays from synchrotron radiation sources.¹⁵² In this connection, the coherent nature of the attosecond XUV pulses provides an additional knob to control and possibly characterize the entanglement. The question is why one

would care about the entanglement when one of the main goals of the attosecond science is to generate, control, and characterize coherence?¹⁶²

In its simplest form, the quantum coherence can be generated by coherently exciting and/or ionizing two closely spaced levels with a light pulse having a bandwidth larger than the level spacing versus.¹⁶³ As can be understood, in this situation, by measuring the kinetic energies of photoelectrons, one cannot determine the state of the residual ion, since the signal can be generated from any of the coherently excited levels. Thus, coherence implies complete lack of entanglement and vice versa.¹⁶² Following photoionization, the emitted electron and the ion are inherently entangled—it makes the generation of coherence as envisioned by attosecond XUV-pulse-induced ionization processes in isolated systems a challenging task. This is especially true for molecules in the gas phase, where the non-adiabatic coupling between electrons and nuclei can lead to decoherence hindering the observation of quantum coherence across longer timescales.¹⁶⁴ As pointed out elsewhere, this can be considered as the entanglement between the electronic and the nuclear degrees of freedom close to the singularities in potential energy surfaces of the molecules (conical intersections). While the large bandwidth of the attosecond pulses can result in a coherent superposition of several electronic states and thus, pure electronic coherence in the photo-ionized molecule, observing such coherence in the case of larger systems across longer timescales remains a daunting task in itself because of this entanglement. Indeed, as evidenced in recent experimental studies, the observation of quantum coherence is limited to small polyatomic systems with typical coherence times below 50 femtoseconds.¹⁶⁵ In other words, understanding better the origin of quantum entanglement in attosecond photoionization processes holds the key toward observing coherence.¹⁶²

One of the earliest examples where the manifestation of entanglement may have been observed includes photoionization of hydrogen molecules by a broadband attosecond XUV-pulse.¹⁶⁶ If $H_2^+$ is produced in a gerade ionic state, the emitted electron is necessarily in an ungerade state and vice versa, thus forming a photoelectron and ion bipartite state. Instead, electron localization was observed, which requires coherent superposition of ionic gerade and ungerade states, mediated by a XUV$_{pump}$-NIR$_{probe}$ scheme. Another contemporary example includes studying the build-up of electronic coherence in multiply ionized krypton atoms in a strong optical field.¹⁶⁷ The emergence of electron-ion entanglement was hypothesized to be a limiting factor for the electronic coherence observed in the attosecond transient absorption signal.¹⁶⁸ The effect of decoherence in attosecond interferometric methods, most notably in the case of RABBITT, was studied for atomic targets, such as neon.¹⁶⁹ It was shown that because of several experimental uncertainties, such as instrumental resolution and temporal jitter, the retrieved phase of the wavefunction describing the emitted electron wavepacket could be the same for coherent and incoherent processes. Hence, the wave-packet must be described in terms of the density matrices of the electron wave-packet whose off-diagonal terms give access to the coherence. This was later elaborated for autoionizing Fano-resonances in argon atoms, where decoherence was found to be the direct result of entanglement between radial and angular degrees of freedom of the emitted photoelectron.¹⁷⁰ For homo-nuclear diatomic molecules, femtosecond ultraviolet pulses
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>FIG. 6. (a) Avoided crossing observed in XUV-induced ionization of Helium atoms undergoing Rabi-oscillations. Note the blue shift of the avoided crossing compared to the atomic transition (denoted by the vertical dashed line).<sup>175</sup> Experimental photoelectron spectra (b) without and (c) with quantum entanglement between the photoelectron and XUV-dressed He<sup>+</sup> ion. In the first case, the spectra follow Einstein's photoelectric equation (dashed line), whereas in the latter case, it shows a clear avoided crossing as predicted by the Rabi model for a dressed-atom.<sup>177</sup></figcaption></figure>

have been used to prepare Bell-like states by initiating disso-
ciation and probing with multiphoton ionization using intense
NIR pulses.<sup>171</sup>

In all these cases, the manifestation of the entanglement was
observed in various pump-probe schemes using attosecond XUV

pulses, but robust methods to generate and control entanglement
over ultrafast timescales did not exist. A scheme to perform Bell-
type tests for atomic photoionization at attosecond timescales has
been hypothesized.¹⁷² While the chirp of ionizing XUV pulses was
perceived as a control-knob to tune the degree of entanglement, it
has not been verified experimentally. ¹⁷³ Recently, these challenges
were tackled in both attosecond and femtosecond photoionization
of model systems. In a pioneering example, a combination of two
XUV attosecond pulse trains along with an NIR pulse was used to
study time-resolved photoionization of H₂ molecules.¹⁷⁴ A nested
Mach–Zehnder interferometer, with one interferometer responsi-
ble for the delay between the two XUV pulses integrated inside
a larger interferometer responsible for time-delayed NIR pulses,
allowed to perform a Ramsey-like experiment for retrieving the
ionic coherence at different XUV–XUV delays. The entanglement
between the emitted electron and the H₂⁺ ions can be controlled
efficiently by changing the delay between the two XUV-pulses, since
the absorption of a second XUV-photon may lead to the creation of
a different vibrational state at any given temporal instant, leading
to the loss of vibrational coherence induced by the first XUV-
photon. The loss of coherence was observed as a change in the
Ramsey-type fringes associated with various vibrational levels of
the H₂⁺-ion.

While attosecond XUV-pulses produced via HHG processes
can provide a means to observe and control the manifestation of
entanglement in ultrafast photoionization, coherent XUV-pulses
from free-electron lasers (FELs), especially from seeded FELs, can
also be used to probe similar processes. This offers new perspectives
for future tabletop HHG-based light sources. In the first example of
coherent light-matter interaction, intense XUV-pulses with dura-
tion close to 50 femtoseconds from FERMI were used to drive Rabi-
oscillations in 1s → 4p bound-bound transition (energy: 23.742 eV)
in neutral helium atoms.175 They were probed “in situ” via photoion-
ization of the helium atom undergoing the oscillations. Quantum
interferences between resonant one-photon and non-resonant two-
photon pathways were found to result in a blue shift of the observed
avoided crossing in the photoelectron spectrogram [denoted by the
vertical dashed line in Fig. 6(a)]. The avoided crossing provided a
clear experimental proof of the generation of the “atom + photon”
dressed-states at XUV wavelengths.176 Afterward, two-photon ion-
ization of helium atoms were studied at photon energies near the
1s → 2p transition (energy: 40.817 eV) in He⁺ ion.177 This is higher
than the ionization energy of the neutral helium atom but signif-
icantly lower than the single-photon double ionization of helium.
As a result, following the absorption of one XUV-photon, an elec-
tron is emitted in the continuum. Because the XUV-pulses from
the FEL can be intense enough, reaching up to a value of more
than 10¹³ W/cm² in effective intensities, the absorption of a second
XUV-photon from the same pulse can take place. In this situation,
the remaining electron in the He⁺ ion undergoes Rabi-oscillation
under the presence of the electromagnetic field. The question is
whether the emitted photoelectron already in the continuum can
be aware of the oscillation in the residual ion. It was first addressed
theoretically by Grobe and Eberly, who proposed that it could be
possible via transfer of coherence mediated by electron–electron
correlation.178 Previous experiments performed in calcium atoms
did not yield a clear signature of the proposed coherence trans-
fer due to the excitation of several closely spaced electronic levels
by a broadband femtosecond pulse.¹⁷⁹ However, recent experiments using XUV pulses from a seeded FEL demonstrate clearly that such an observation is possible only if there is entanglement between the emitted electron and the Rabi-cycling atom.¹²¹ As shown in Fig. 6(b), in the absence of entanglement, the photoelectron spectrum is a single photo-line, whose kinetic energy increases with an increase in photon energy, as predicted by Einstein's photoelectric equation (the black dashed line). However, in the presence of entanglement, the emitted electron becomes aware of the Rabi oscillations in the XUV-dressed He⁺ ion, and consequently, the photoelectron spectra develop a level anti-crossing [see Fig. 6(c)], reminiscent of the Rabi dynamics [see Fig. 6(a)]. By changing the intensity of the FEL pulse, it was observed that the manifestation of the entanglement could be controlled.¹²¹ For the electron and the dressed ion to be fully entangled, it requires the electronic population to undergo at least one complete Rabi cycling, though once developed, the entanglement can persist during the entire duration (around 70 fs) of the XUV-FEL pulse. These experiments clearly show that it is not only possible to generate entanglement in ultrafast photoionization processes but also one can control its manifestation by various properties of the driving ultrashort XUV pulses.

One of the challenges is to characterize the entanglement generated in these ultrafast processes. It is worth noting that the degree of entanglement is usually quantified in terms of the entropy of entanglement, which is a notoriously difficult quantity to measure, with a first experimental realization reported in 2015 only,¹⁸⁰ as it requires access to both the diagonal and off-diagonal elements of the density matrix of the bipartite system. In typical photoelectron spectroscopy, collecting the electrons provides access to the diagonal elements only. For accessing the off-diagonal elements, a quantum state tomography method has recently been put forward to retrieve the off-diagonal elements of the density matrix.¹⁸¹ In here, a broadband XUV attosecond pulse is used in combination with a time-delayed bi-chromatic NIR field. The time-dependent electron spectrograms are obtained at different relative frequencies of the two components of the bichromatic field. Depending on the relative difference, one can have access to the different off-diagonal elements of the density matrices.¹⁸² It has led to a complete characterization of the quantum state of the photoelectron. Given its practicability, this method can possibly be extended to complex molecular systems as well.

# IV. PERSPECTIVES

## A. HHG source

Perspectives on attosecond pulse generation via HHG in gases include going toward the increased control of attosecond light sources to make them more reliable and better controlled. For isolated attosecond pulse generation, sculpting of the fundamental light cycle is now possible by separate amplification and recombination of CEP-stable multicolor pulses in a so-called light wave synthesizer.¹¹⁹ Despite the complexity of the process, this allows for very fine control of the harmonic emission in terms of bandwidth, duration, or even control of the XUV chirp using metallic filters,¹⁸³ chirped mirrors,¹⁸⁴ or gas media dispersion.¹⁸⁵ As the global efficiency of the process is roughly constant after optimization and scaling factors are known,⁵³ increasing the photon flux calls for

fundamental lasers with higher power. The development of fs Yb laser allows a significant power increase as compared to fs Ti:Sa laser. Long pulse duration has been a limitation for generating short attosecond pulse trains or isolated attosecond pulses using mid-IR pulses from Yb laser, but new approaches are being developed to achieve high throughput post-compression.¹⁸⁶ Now, Yb laser can deliver short pulses with high repetition rates and high power, desirable for attosecond pulse generation and applications. The low energy per pulse at high repetition rate implies strong focusing with short focal lengths for HHG and therefore strongly diverging sources. This implies difficulties in collimating or refocusing the XUV sources, and work on controlling the XUV beam spatial characteristics is under progress. Dynamic studies with attosecond pulses can be performed with an XUV pulse combined with another XUV pulse,¹⁸⁷⁻¹⁸⁹ but more often with a short, delayed femtosecond pulse. In the latter case, the second intense pulse can potentially perturb the medium under study, and significant efforts are devoted toward the development of non-linear physics induced by XUV that holds the promise to perform studies where both the pump and probe are perturbative. For many years, high-energy pulses were used to induce non-linear XUV transitions, which called for laser systems with low repetition rates. Recently, significant advances have been made in this domain as it was shown that high repetition rate sources could be used if the effective harmonic source has a small size. This source size can be demagnified by imaging, and XUV-intensities large enough for multi-photon ionization can be obtained.³⁸ Many directions can be opened in this field provided harmonic beams can be focused properly.

Improving XUV-focusing is also an active research domain that can bring significant improvement for the applications of these sources. The spatial characteristics of high-order harmonic beams are complex and unique in many aspects. Often called laser-like beams, the harmonic beams are co-propagating with the fundamental beam and can exhibit many spatial profiles like Gaussian, flattop, annular, and so on. It is known that the spatial characteristics of harmonic beams have a significant impact on the duration of the XUV pulses, especially when refocused. For instance, it is a key element for the generation of IAP.¹⁹⁰ Thus, many studies have been conducted on harmonic spatial properties with an ever-increasing diversity of tools. Harmonics generated in gases are also known to have a good spatial coherence. This was first studied via mutual coherence between two separated XUV sources,¹⁷⁷ and the observation of fringes occurring from mutual coherence implies that both beams are spatially coherent. Several studies were then performed to study this spatial coherence.¹⁹¹ This good spatial coherence of the harmonic beam is a key for coherent diffractive imaging that can have a spatial resolution in the 10-nm range. It was predicted earlier¹⁹² that high-order harmonics generated in gases have spatial characteristics that evolve with the harmonic order. This is visible in Fig. 1(b), where the harmonics generated in neon are observed far from the focus and exhibit spatial profiles evolving with the harmonic order. In turn, this implies that, when refocused, the XUV focus position changes longitudinally with the harmonic order. With the advent of precise techniques for XUV beam characterization (SWORD,¹⁹³,¹⁹⁴ XUV Hartmann,¹⁹⁵ HASO,¹⁹⁶ or all optical approaches¹⁹⁷), the understanding that the spatial properties have a major impact on the attosecond temporal profiles¹⁹⁸ and the potential development of industrial activities, major interest in
the spatial control of harmonic beams is growing. It was observed with a spatially resolved XUV spectrometer that each plateau harmonic beam often exhibits two zones with different divergences:¹⁹⁹ a central zone with low divergence and an outer zone with larger divergence. The low divergence zone is attributed to harmonics emitted via the so-called short quantum path emission, where the phase of the harmonics is little influenced by the laser.²⁰⁰ The outer zone with larger divergence arises from harmonics that are emitted via the so-called long quantum path where the XUV phase is strongly influenced by the laser. This separation in short and long paths has a classical analogy in terms of electron trajectories, and the phase of the radiation is a direct signature of the quantum aspect of the HHG process. Both zones are merging in the cutoff where the divergence is an average of the two former. This spatial separation in the far field allows a selection of the short quantum path emission simply by clipping the XUV beam or by controlling the phase-matching condition that can be different for both paths.²⁰¹ Usually, experiments are mostly performed after selection of the short quantum path. This has a direct impact on the XUV temporal profile, as it is known that attosecond pulses emitted via the short and long quantum paths are shifted in time. Therefore, selecting one quantum path makes the XUV temporal profile less congested. The SWORD approach developed in 2009 for XUV harmonics¹⁹³ allows measuring the wavefront curvature and intensity profile of each harmonic in a plane. It was observed that harmonics generated in a jet with a Gaussian beam exhibit a wavefront curvature that changes with the harmonic order.¹⁹⁴ Using phase and intensity profiles and by back-propagating the light, the apparent XUV source can be identified for each harmonic. It was observed that the source position changes longitudinally with the order as well as the source size. This observation was confirmed in several laboratories with SWORD or alternative techniques.²⁰²⁻²⁰⁷ Controlling the separation between harmonic sources is not an easy task, but it is mainly imposed by their wavefront curvature, which depends on many controllable parameters (laser intensity, position of the generating medium, wavelength of the fundamental laser, etc.). One of them is the spatial evolution of the fundamental beam intensity profile in the generating medium. Indeed, the harmonic wavefront curvature depends on the intensity profile of the fundamental. A Gaussian fundamental beam imposes a given curvature but it was shown that harmonics can be generated with spatially shaped fundamental beams.²⁰⁸⁻²¹¹ Use of temporally and spatially shaped beams will change the wavefront curvature and its dependency with the harmonic order. A recent study²¹² showed that using fundamental beams with a flattop intensity profile is indeed a way to reduce the sensitivity on the harmonic order and thereby reduce the XUV chromatic aberrations. With an ideal flattop shaped profile and a thin generating medium, all harmonics would have the same wavefront curvature and would refocus at the same position. Furthermore, it was shown that flattop spatial shaping is a way to increase the brilliance of individual harmonics²¹¹ and it is likely that this control of the XUV chromatic aberration by spatial shaping of the fundamental beam will be improved with alternative approaches in the future. This control will be of utmost importance when attosecond control is applied to complex systems, as it is important that all harmonics be refocused in the same way to control the systems efficiently.

Developing new strategies to obtain tunable XUV sources is also a timely objective for the community, and many approaches have been developed.²¹³⁻²¹⁷ Figure 7 illustrates the tunability of the XUV comb that can be obtained by chirping the fundamental pulse when a short NIR pulse is used for HHG. Approaches such as frequency mixing or tailoring of the fundamental spectrum yield good results, and the choice of one or the other approach depends on the wavelength range and bandwidth to use.

Progresses on attosecond quantum light sources are also ongoing. In parallel to the experimental development, exciting theoretical development concerning the HHG process driven by quantum light has also taken place.²¹⁸ Among other things, it led to the appearance of many-body effects in the HHG process—how tailoring the HHG process can be used to tailor those many-body effects.²¹⁹ Another emerging effect is the entanglement between the emitted radiation and the emitting media under the phase-matching condition.²²⁰ This is particularly important for time-resolved spectroscopy of ultrafast light-induced processes since the production of entangled XUV-photon pairs holds the key toward a spectral resolution beyond what is determined by the Fourier-limit. A plausible way to observe squeezed light in the HHG process has also been hypothesized.²¹⁸,²²¹

## B. Quantum effects

While the entanglement between photoelectron and photoion in light-matter interaction is a well-known phenomenon, the advent of attosecond technology has given it a new perspective in terms of how to utilize it and how to circumvent it when necessary. The tabletop HHG-based sources providing highly coherent broad-band pulses provide a unique opportunity to study these quantum mechanical events with unprecedented temporal resolution. While so far the applications have been limited to only model atomic (He, Ar, Ne, etc.) or diatomic ($H_2, N_2, O_2$, etc.) systems, the emergence of new-type of attosecond sources with improved intensity and stability can allow us to reach a major goal of attosecond science—to make or break a chemical bond by means of controlling the pure electronic coherence in photochemical reactions, thereby impacting profoundly our understanding and capacities of control of fundamental properties of light-matter interaction in ultrafast timescales.¹⁶² Apart from this, two equally important aspects of coherence and entanglement can be envisioned in the upcoming future, namely, bypassing the Fourier-limit in time-resolved spectroscopy using entangled XUV-photons and the generation and control of entanglement at macroscopic length-scales over ultrafast time-scales. The usefulness of quantum light at NIR wavelength from single photon sources in time-resolved spectroscopy is already shown to provide spectacular insight about ultrafast photochemical processes such as photosynthesis.²²² Several theoretical propositions are put forward to observe the light-induced non-adiabatic dynamics near conical intersections in complex molecules using entangled photons enabling beyond the Fourier-limit resolution.²²³ The HHG-process driven quantum light can prove to be a stepping-stone toward this direction at shorter wavelength. Similarly, for massless particles such as photons, entanglement over hundreds of kilometers was shown to exist. However, for massive particles, such as atoms or its constituents, generating entanglement over macro-
<figure><img src="image_18.png" /><figcaption>FIG. 7. Fine tuning of harmonic spectra obtained by controlling the chirp of the input fundamental pulse. This map is a concatenation of several harmonic spectra generated in an argon cell after a chirp is added to the fundamental pulse. The chirp is controlled by adding calibrated thicknesses of fused silica on the beam path. Two specific spectra are shown to illustrate tunability. The short driving pulses have a minimum duration of 8.2 fs and allow complete tunability between harmonics around 35 eV.</figcaption></figure>

scopic length-scales is rather complicated since it is difficult to isolate the particles from the surrounding environment. In this sense, using femto- or attosecond XUV-pulses, one could generate entanglement between massive particles within ultrashort timescales, before they get a chance to interact with the environment. This was evident in the experiment described in Ref. 177, where within the XUV-pulse duration, the wave-packet of the emitted free-electron was estimated to expand up to a mesoscopic distance of 200 nm. The availability of coherent ultrashort x-ray pulses from FELs²²⁴ can lead to ionization of core-level electrons, allowing the study of the entanglement between photo- and Auger electrons.²²⁵ Having higher kinetic energy of the electrons can allow them to move by a macroscopic length within a few tens of femtoseconds, enabling the observation of entanglement between massive particles separated by macroscopic distances. The remarkable advances in both tabletop HHG-based XUV-sources as well as FELs providing ultrashort light pulses ranging from XUV to x-ray unveil plethora of opportunities in generating, controlling, and characterizing entanglement as well as coherence in ultrafast timescales.

With these sources, spatial resolution is usually limited by the wavelength of the emitted light. To circumvent this limit, an active domain of research concentrates on the use of nano-tips.²²⁶⁻²²⁸ They can confine spatially the electron emission much below the light wavelength. When the emission is triggered by ultrashort laser, the electron emission can be confined temporally. These sources have therefore many interesting properties that can also open new opportunities in the quantum domain.

### C. Attosecond delays

As the number of atoms increases, many electronic states appear closer in energy in the valence band, which pushes the ionization-time-delay concept toward a more general description. The electronic states do not have the same symmetry and can a priori behave independently. This statement contrasts with the intuition from scattering theory where, larger is the extent of the

scattering center, larger is the scattering time, and hence the ionization delay. For many-atom systems, the valence band can almost be represented by a continuum of states. This spectral region can be seen as an internal electron reservoir. During the excursion of the photoelectrons, they probe the residual species and imprint a detailed picture of the ionic potential. The ionization time delay may follow the size of the molecular potential, and the electronic states may present a common global-trend. This statement is consistent with the experiment with water clusters where the bigger the cluster, the higher the ionization time delay.²² However, this general rule does not always hold, especially for molecules with reduced-dimensionality.²⁴

As recently demonstrated using 2D planar molecules, the quadrupole term of the multipole expansion of the molecular potential significantly affects the electron scattering and carries a signature of the ionization dynamics in general.²⁴ Because the hole is smeared over the molecular plan, the outgoing electron interacts with a less attractive potential compared to its 3D molecular counterpart with similar chemical composition that corresponds to a repulsive term and leads to a less-delayed electron in the continuum, that is, a negative contribution in the ionization time delay. Similarly, for 1D molecules, different terms of the multipolar expansion will have a strong influence on the resulting ionization time delay, as shown for $C_2H_2$.¹³⁹ The molecular symmetry is imprinted into the photoionization-time-delay metrology. The RABBITT method is well suited to measure the ionization time delay of an isolated electronic state with a comb of XUV harmonics where the signal is free of any spectral congestion.¹²³ However, the density of electronic states increases with the size of the molecule. To circumvent the spectral congestion problem, one solution is to isolate a few harmonics from the comb using metallic filters to reduce the number of channels.¹³⁴ Alternatively, the arbitrary spectral shaping of the XUV comb is possible using a time-compensated monochromator.²²⁹ In this case, the electronic states can be compared at a given XUV photon energy. It may allow accessing the inner valence states. To reduce even more the spectral congestion, an approach is to dress the molecule with a photon that corresponds
to the energy between two consecutive XUV harmonics, thereby overlapping the sidebands with the main harmonic peaks.²³,²³¹ This solution is efficient because the consecutive oscillations are becoming almost in phase enhancing the detected signal. This is in line with the actual development of high repetition rate laser (Yb-based or OPCPA) where the central photon energy is reduced, implying a low difference of energy between two consecutive harmonics. Up until now, most of the molecules under study are composed by H, C, N, O, and F atoms. One of the natural evolution is to explore molecules composed with other atoms. As already demonstrated using strong field (streaking¹³), the ionization of an iodine atom can be isolated allowing to investigate the influence of its environment.²³² Ionization from core shell in the x-ray regime, implemented recently with the streaking version¹⁵⁴ is also promising.

## V. CONCLUSION

With the rapid development of attosecond technology, a plethora of applications has emerged, ranging from pure optics to atomic physics, chemistry, or material science. State-of-the-art installations worldwide have allowed pushing new investigations, giving access to the inner working of natural or technological processes. With the new exciting phenomena that have been discovered with HHG sources, arise the questions of the current limit of these sources. In this review, we have discussed some aspects of potential further developments, ranging from spatial and temporal to spectral control. Other aspects such as polarization states and angular momenta of light in general are also a fascinating path to explore and pioneering work have already demonstrated its importance in the context of chirality or magnetism. Pushing the control of the HHG sources up to the quantum precision will also open new exciting research area. We discussed two topics that are currently the subject of intense research activities: the measurement of attosecond ionization delays and the characterization of quantum aspect in light-matter interaction (entanglement and coherence). These examples are chosen as seminal cases that will strongly benefit from highly performing sources, and progress in those areas will rely on such performances. Attosecond physics is at the heart of the ultimate control of light-matter interaction and a bright future is expected in the field that will affect other topics with societal impact, from photovoltaics, to new materials or medical applications.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**E. Constant**: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **S. Nandi**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

**C. Picot**: Data curation (equal); Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – review & editing (equal). **E. Prost**: Data curation (equal); Formal analysis (equal); Writing – review & editing (equal). **S. Palakkal**: Data curation (equal); Formal analysis (equal); Writing – review & editing (equal). **F. Lépine**: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **V. Loriot**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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