Tout

Anthony Dit

Wu Meiyi

Xuan Je Remercie Aussi

In this thesis, I studied theoretically atoms and molecules interacting with a short, low-frequency and intense laser pulse, in the typical regime of high-order harmonic generation (HHG). We use HHG as a self-probe process to examine electronic and nuclear dynamics on the attosecond scale with Ångström resolution, focusing on the spectral phase. By using simple models with reduced dimensionality, we are able to solve extensively the time-dependent Schrödinger equation, either numerically or with the so-called Strong Field Approximation (SFA). Our models give us valuable physical insights on the underlying dynamical processes and intuitive explanations while keeping a predictive propensity. Equipped with efficient tools developed specifically to analyze our numerical results, we first investigate the ionization dynamics through a shape resonance in a model molecule such as N 2 2 2 . Secondly, we take another look at two-center interferences, and uncover a very interesting behavior which is linked to the dressing of the electronic ground-state by the laser field. It is indeed confirmed by additional developments of molecular SFA. We predict that this behavior can be observed experimentally using quantum path interferences. Finally, we examine the effect of nuclear vibration in diatomic molecules by coupling consistently electronic and nuclear motions. Our results show that with short pulses, nuclear motion in the neutral molecule can be triggered by impulsive stimulated Raman scattering. Thus, we invalidate an uncorrelated theory, so-called Lochfraß, which focuses on the dependence of the ionization yield with internuclear distance as an explanation. Lastly, we question the extension within SFA of the notion of ionization potential in molecules.

Dans cette thèse, j'ai étudié théoriquement l'interaction d'atomes et de molécules avec des impulsions laser brèves, intenses et de fréquences basses. En insistant sur la phase spectrale, nous utilisons la génération d'harmoniques d'ordres élevés comme processus auto-sonde pour étudier les dynamiques attoseconde. Nous résolvons l'équation de Schrödinger avec des modèles simples à dimensionnalité réduite, numériquement ou en utilisant une théorie semi-analytique nommée SFA (Strong-Field Approximation, approximation du champ fort), nous permettant ainsi d'obtenir des informations approfondies sur les processus physiques mis en jeu, à travers des explications intuitives, tout en gardant une propension prédictive. Avec des outils développés spécifiquement pour analyser nos résultats numériques, nous étudions d'abord la dynamique d'ionisation dans une molécule modèle telle que N 2 . Puis, en réexaminant les interférences à deux centres, nous mettons au jour un comportement très intéressant, lié à l'habillage de l'état fondamental par le laser, et confirmé par des développements analytiques d'une adaptation du SFA aux molécules. Nous prédisons la possibilité d'observer ce phénomène expérimentalement par l'intermédiaire des interférences de chemins quantiques. Enfin, nous étudions les effets de la vibration des noyaux dans les molécules diatomiques en couplant le mouvement des électrons avec celui des noyaux. Nous montrons que pour de telles impulsions laser, l'excitation vibrationnelle de la molécule neutre peut être induite par effet Raman. Nous invalidons alors une théorie non corrélée, nommée Lochfraß, qui base son interprétation sur la dépendance du rendement d'ionisation avec la distance internucléaire. Enfin, nous proposons de prolonger au SFA la notion de potentiel d'ionisation dans les molécules. At the beginning of the 20th century, physics and chemistry have experienced a turning point through the emergence of a new theory that disrupted contemporary science: Quantum Mechanics. Since then, tremendous efforts have been made to develop this fascinating theory and verify its capability to describe nature on the infinitesimal scale. Originated from a necessary change of paradigm, it was the starting point of new philosophical concepts about the universe. Technological innovations have followed this fundamental breakthrough, and have rapidly become powerful tools to probe matter even further. In the meantime, these innovations have been used for industrial applications in everyday life.

Among them, the discovery of the laser was of major importance. First pioneered by the invention of the maser (microwave amplification by stimulated emission of radiation) in the 50s, the laser (light amplification by stimulated emission of radiation) has been developed in the 60s, based on the principle of stimulated emission theoretically described by Einstein in 1917. Basically, optical or electrical pumping is carried out between atomic energy levels. Then the populated excited states radiate photons of frequency corresponding to the energy gap between the atomic levels. Radiation is amplified in a cavity, making the laser an intense source of coherent light at fixed frequency, opening great perspectives on non-linear optics and the study of radiation-matter interaction.

Ever since, laser properties have been intensely increased, with a constant obsession for reaching higher and higher intensities in order to test further fundamental properties of matter. The evolution of the laser intensities being attained is summarized in Fig. 1.1 along with the corresponding physical regimes that could be reached for these intensities. As one could suspect, it was not linear in time, but marked with punctual discoveries such as the revolutionary Chirped Pulse Amplification (CPA) technique [1]. The CPA breakthrough has led to the construction of very intense lasers until the most so far: HERCULES [2]. In the meantime, it made possible the development of table-top intense laser sources, leading to the discovery of High-order Harmonic Generation (HHG) in the late 80s simultaneously by an American [3] and a French [4] teams.

The discovery required the focusing of intense femtosecond pulses on a gaseous medium. Once again, it was enabled by the CPA, which offers the capability of generating laser pulses of short duration and high intensity. These developments were motivated by the will to probe matter dynamically on very short scales. At that time, Zewail et al realized femtochemistry experiments with a laser described as "the world fastest camera", and explored molecular dynamics on the femtosecond (10 -15 s) scale such as the breaking of chemical bonds. Almost ten years after, the highly nonlinear phenomenon of HHG has become very important to the scientific community, revealing the capability of generating sub-femtosecond pulses in the ultra-violet (UV) or extreme UV (XUV) range [5,6]. Improvements, summarized in Fig. 1.2, have made accessible the generation of pulses of few tenths of attoseconds (10 -18 s) [START_REF] Antoine | Attosecond Pulse Trains Using High-Order Harmonics[END_REF][START_REF] Antoine | Generation of attosecond pulses in macroscopic media[END_REF][START_REF] Christov | Generation and propagation of attosecond x-ray pulses in gaseous media[END_REF][START_REF] Paul | Observation of a Train of Attosecond Pulses from High Harmonic Generation[END_REF][START_REF] Mairesse | Attosecond Synchronization of High-Harmonic Soft X-rays[END_REF][START_REF] Dudovich | Measuring and controlling the birth of attosecond XUV pulses[END_REF][START_REF] Gaarde | Macroscopic aspects of attosecond pulse generation[END_REF], opening a new field of research: attosecond science [START_REF] Scrinzi | Attosecond physics[END_REF][START_REF] Corkum | Attosecond science[END_REF][START_REF] Krausz | Attosecond physics[END_REF].

Nowadays, HHG offers multiple applications in spectroscopy. First, the generated XUV attosecond pulses can serve in ultra-fast pump-probe experiments to detect molecular dynamics through photoemission. For example, they have been used to probe photoionization in gaseous media such as helium [START_REF] Larsson | Two-colour time-resolved spectroscopy of helium using high-order harmonics[END_REF] and argon [START_REF] Klünder | Probing Single-Photon Ionization on the Attosecond Time Scale[END_REF], to reveal resonance features in nitrogen molecule [START_REF] Haessler | Phase-resolved attosecond near-threshold photoionization of molecular nitrogen[END_REF][START_REF] Caillat | Attosecond Resolved Electron Release in Two-Color Near-Threshold Photoionization of N_{2}[END_REF] and potentially spin-forbidden transitions in SO 2 [START_REF] Lévêque | Direct observation of spin-forbidden transitions through the use of suitably polarized light[END_REF]. These pulses can also be used on condensed matter [START_REF] Cavalieri | Attosecond spectroscopy in condensed matter[END_REF], resolve Auger decay [START_REF] Drescher | Time-resolved atomic inner-shell spectroscopy[END_REF] and tunnel ionization [START_REF] Uiberacker | Attosecond real-time observation of electron tunnelling in atoms[END_REF] in real-time, and allow the reconstruction of the instantaneous incident laser field with the streaking technique [START_REF] Constant | Methods for the measurement of the duration of high-harmonic pulses[END_REF][START_REF] Itatani | Attosecond Streak Camera[END_REF].

Secondly, HHG can also be employed as a self-probing tool to monitor attosec- ond dynamics in the constituents of the generating medium [START_REF] Haessler | Self-probing of molecules with high harmonic generation[END_REF]. Indeed, the underlying structure and dynamical processes are encoded in the generated radiation which is coherent per se, and can thus be partially or completely reconstructed from the collected light. For instance, it was used so to probe attosecond nuclear dynamics in molecular hydrogen [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF][START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF][START_REF] Baker | Dynamic Two-Center Interference in High-Order Harmonic Generation from Molecules with Attosecond Nuclear Motion[END_REF], image electronic wave-packets [START_REF] Haessler | Attosecond imaging of molecular electronic wavepackets[END_REF][START_REF] Zhao | Positioning of Bound Electron Wave Packets in Molecules Revealed by High-Harmonic Spectroscopy[END_REF], or to implement tomography of molecular orbitals [START_REF] Itatani | Tomographic imaging of molecular orbitals[END_REF][START_REF] Vozzi | Generalized molecular orbital tomography[END_REF][START_REF] Salières | Imaging orbitals with attosecond and Ångström resolutions: toward attochemistry?[END_REF].

The goal of this PhD was to investigate HHG theoretically with the self-probing approach in mind. What a theoretician is left with is the so-called Time-Dependent Schrödinger Equation (TDSE) which rules the quantum world. In its most general form, it reads: where Ĥ(t) is the time-dependent Hamiltonian and |Ψ(t) refer to the time-dependent wave-function of the system. The latter encodes all the properties and all the dynamics of the system one needs to know to predict its evolution, given the initial conditions. However, the resolution of the TDSE is a tremendous task. Indeed, it can only be solved analytically for very simple systems involving only one electron, such as the hydrogen atom. Otherwise, its analytical resolution is impossible. As soon as more than one particle is involved, numerical methods are invoked to solve the TDSE. Thanks to the swift progress in computer science and the increasing computational power, the TDSE can be solved ab initio with programs that implement smart algorithms for small systems but greater than one particle. Yet, the rapidly growing complexity of chemical and, more critically, biological systems prevents to treat them exactly via the TDSE. One must consequently use other numerical methods based on appropriate approximations on the system. The first task of a quantum theoretician is thus to find the best way to solve the TDSE that conciliate proper approximations and decent computational demand. Then, the second work is to extract meaningful physical information from the computations. The knowledge of the exact wave-function |Ψ(t) of a large system is sufficient to describe its evolution but does not give easily proper insights into the underlying processes by itself. Appropriate analysis of the TDSE outcomes is required. For this reason also, simple models are often more powerful to gain physical understanding and predictive results than intensive all-in-one resolution of the exact TDSE. Furthermore, their low computational cost enables extensive computations unlike the latter, which is very useful to approach multiple aspects of a problem. Therefore, we planned to simulate HHG in atoms and diatomic molecules with very simple models, but still sufficient to reproduce experimental features and have predictive capabilities.

The thesis is organized as follows. In Chapter 2, we first introduce HHG through the background of laser-matter interaction, and expose its properties via a simple semi-classical model. Then, we present in Chapter 3 our theoretical models and the resolution methods we used to simulate HHG in atoms and diatomic molecules. Intermediately, we dedicated Chapter 4 to the analysis tools that we used and developed to extract physical meaning from the resolution of the TDSE. Finally, we present our most interesting results in Chapter 5 and their discussion. We will first speak about the Wigner distribution within HHG. Then we demonstrate the spectral phase signature of a shape-resonance in nitrogen molecule. Furthermore, we will revisit two-center interferences and emphasize amazing findings on the spectral phase. Finally, we will explore consistently the effect of vibration in diatomic molecules in HHG, leading to decisive consideration on the definition of the ionization potential.In the entire document, unless otherwise specified, all the equations are expressed in atomic units as defined in Appendix A.1.

Chapter 2

High-order Harmonic Generation CHAPTER 2. HIGH-ORDER HARMONIC GENERATION

Introduction

This chapter is intended to draw a brief overview of laser-matter interaction, from intensities small enough to treat the laser as a perturbation to intensities high enough to place the system in the strong-field regime. Namely, in strong-field physics, the strength of the laser field is comparable to the one of the electron binding field, but still electrons are accelerated to energies allowing non-relativistic propagation. The experimental study of such a regime has led to the discovery of HHG in the late 80s. At the beginning of the 90s, a semi-classical picture of the phenomenon which brought great physical understanding has been successfully proposed in accordance with experiments.

Laser-matter interaction 2.2.1 Perturbative regime

Interaction between light and matter can be successfully described with the Perturbation Theory (PT) as long as the intensity of radiation is sufficiently small. Under this assumption, solutions can be expanded as power series around the perturbation up to a desired order [38].

In Quantum Mechanics, the unperturbed system is fully described by its Hamiltonian H 0 , whose eigenvectors |ϕ n and eigenvalues E n , i.e. solutions of:

Ĥ0 |ϕ n = E n |ϕ n , (2.2.1)
are the eigenstates and the corresponding energies, respectively. Being in the initial state |ϕ i , the system interacts at t = 0 with a radiation which is described as a timedependent perturbation V (t). We assume that this perturbation is small compared to Ĥ0 so that we can write V (t) = λ Ŵ (t), where λ 1 and Ŵ (t) corresponds to an observable of the same order of magnitude as for Ĥ0 (i.e. as the energies E n ). The perturbed system is ruled by the TDSE (in a.u.):

i d dt |Ψ(t) = Ĥ0 + λ Ŵ (t) |Ψ(t) . ( 2

.2.2)

Writing the wave-function in the eigenstate basis {|ϕ n }, i.e. |Ψ(t) = ∑ n c n (t)|ϕ n , with c n (t) = ϕ n |Ψ(t) , the TDSE reads, after projection on a given eigenstate ϕ n :

i d dt c n (t) = E n c n (t) + λ ∑ k ϕ n Ŵ (t) ϕ k c k (t).
(

.2.3)

If λ = 0, the straightforward solutions are c n (t) = b n exp(-iE n t): state n accumulates phase -E n t and b n is time-independent. When λ is not zero but still very small compared to 1, one can expect the solutions to be of the form c n (t) = b n (t) exp(-iE n t), where now b n (t) is a function of time. Hence, the problem is equivalent to searching for solutions b n (t) of:

i d dt b n (t) = λ ∑ k ϕ n Ŵ (t) ϕ k b k (t)e i(E n -E k )t .
(2.2.4)

The PT allows one to know the probability p n (t) = |c n (t)| 2 = |b n (t)| 2 to populate state |ϕ n , through the interaction with the perturbative light, as a function of time. It relies on the expansion of the coefficients in power series in λ : (q-1) k

(t)e i(E n -E k )t , for q > 0.

(2.2.7)

Initially, the system starts in state |ϕ i . Hence:

b (0) n (t = 0) = δ ni , (2.2.8) b (q)
n (t = 0) = 0, for q > 0.

(2.2.9)

Given Eq. (2.2.6), b

n (t) = δ ni , and Eq. (2.2.7) for q = 1 leads to: b

n (t) = -i t 0 dt 1 ϕ n Ŵ (t 1 ) ϕ i e i(E n -E i )t 1 . (1)
(2.2.10)

Injecting this expression in Eq. (2.2.7) for q = 2, we have: b

n (t) = -∑ k t 0 dt 2 ϕ n Ŵ (t 2 ) ϕ k e i(E n -E k )t 2 t 2 0 dt 1 ϕ k Ŵ (t 1 ) ϕ i e i(E k -E i )t 1 . (2. (2)

2.11)

The intermediate quantity λ is used to visualize the expansion in power series, and must be replaced consistently. Usually, one recasts λ = 1 in order to rewrite Ŵ (t) = V (t). However, we will see that the interaction potential V (t) is proportional to the electric field of the laser, which amplitude E L is small compared to 1. Hence, we set here λ = E L to explicitly show the dependency on it. Finally, the zeroth order probability to populate |ϕ n =i is null, while the first order probability is:

p (1)
n (t) = E 2 L |b (1)
n (t)| 2 ,
(2.2.12)

and expresses single photons transitions. The second order probability:

p (2) n (t) = E 4 L |b (2)
n (t)| 2 , (2.2.13)
however describes two-photon processes (such as Raman scattering.) One may need to derive the PT up to qth order to describe processes involving a number q of photons, for example to describe Multi-Photon Ionization (MPI) which is the absorption of a number of photons up to the ionization threshold of the system. Nevertheless, the expansion in power series of the PT (Eq. (2.2.5)) is a divergent series. Thus, in practice, one needs extra care when dealing with high orders of the PT.

Above-threshold ionization

At high intensities, typically from 1 × 10 13 to 1 × 10 14 W.cm -2 for low-frequency lasers such as Ti:sapphire (wavelength of 800 nm), the requirement on which the PT is based may not be ensured. The system can absorb a higher number of photons than needed to be ionized, releasing electrons at higher kinetic energies. The resulting photo-electron spectrum shows peaks separated by the incident photons energy. This phenomenon, called Above-Threshold Ionization (ATI) has been discovered by Agostini et al in 1979 [START_REF] Agostini | Free-Free Transitions Following Six-Photon Ionization of Xenon Atoms[END_REF]. First observed with nanosecond pulses, it has been then studied with few-cycles pulses [START_REF] Miloševi | Above-threshold ionization by few-cycle pulses[END_REF]. As the high intensity regime does not allow the use of PT, the phenomenon could be studied through the numerical resolution of the TDSE [START_REF] Bardsley | Multiphoton ionisation from a short-range potential by shortpulse lasers[END_REF] or strong field analytical methods.

Tunnel regime

At even higher intensities, i.e. typically greater than 1 × 10 14 W.cm -2 , a new ionization process competitive to MPI is observed. At such intensities, the atomic or molecular potential is strongly affected by the electric field of the laser. Indeed, consider a hydrogen-like atom, for the sake of simplicity, subjected to a laser pulse. The Coulomb interaction between the nucleus and the electron results in a potential V (r) = -1/ r , where r is the vector distance from the nucleus to the electron. The electron interacts with the electric field E(t) of the incident laser pulse: the interaction potential r • E(t) (the electron charge being -1 in a.u.) adds up with the Coulomb potential. At peak intensity 1 × 10 14 W.cm -2 , the electric field magnitude is E L 0.053 a.u., hence, at a distance r = 1/ E L 4.3 a.u. the interaction potential is on the same order of magnitude as the Coulomb potential, which strongly distorts the latter. The resulting potential exhibits a barrier that can be crossed by the electron via the tunnel effect, as depicted in Fig. 2.1 along the polarization direction of E(t). To summarize the different regimes, we can introduce an adiabacity parameter γ, also known as the Keldysh parameter, named after his work [START_REF] Keldysh | Ionization in the field of a strong electromagnetic wave[END_REF]. Depicting the barrier in its simplest form as in Fig. 2.2, one can consider that the electron being in the ground state at energy -I p (where I p is the ionization potential) needs to cross a barrier of length L b = I p /E L . The average velocity of the tunneling electron under the "classically forbidden barrier" is v t = 2I p /2, given by the virial theorem [START_REF] Dewitt | Calculating the Keldysh adiabaticity parameter for atomic, diatomic, and polyatomic molecules[END_REF]. Therefore the duration τ of tunneling and the corresponding frequency ω t are:

Keldysh parameter

τ = L b v t = 2I p E 2 L , ω t = 1 τ = E 2 L 2I p . ( 2

.2.14)

Every half laser cycle, the barrier appears along the laser polarization on the left or right side of the potential depending on the sign of the electric field. Hence, in order to observe tunneling, ω L /ω t 1 is required, leading to the definition of the where:

U p = E 2 L 4ω 2 L (2.2.16)
is the ponderomotive energy, associated with the free oscillation of an electron in the laser field. Consequently, with γ, we can define the three regimes that we presented:

• The tunnel regime is attained for γ 1. (Yet, too small values of γ could lead to barrier suppression, i.e. the ground state is above the barrier. Thence, the system would be strongly ionized within half a period, that we want to avoid.)

• MPI is dominant when γ > 1.

Note that the limit γ = 1 is just an indication on whether tunnel ionization or MPI is dominant. In fact, both regimes overlap and can be observed at the same time in experiments. Moreover, the Keldysh parameter has been derived within a very simple model. Hence, the relevance of the limit γ = 1 may vary between species mainly due to the shape of orbitals involved in tunneling.

The generation of high-order harmonics 2.3.1 Discovery

High-order Harmonic Generation (HHG) has been discovered simultaneously by an American [3] and a French [4] teams at the end of the 80s. In the first group, rare gases (He, Ne, Ar, Kr and Xe) have been irradiated with 350 fs pulses of KrF * laser (248 nm) with focused intensity in the range of ∼ 10 15 -10 16 W.cm -2 . In the second one, they used 30 ps pulses of Nd:YAG laser (1064 nm) with focused intensity of 3 × 10 13 W.cm -2 on Ar, Kr and Xe. In both cases, odd harmonics of the incident laser frequency have been observed up to 17th order in Ne with KrF * laser and up to 33th order in Ar with Nd:YAG laser.

At this point, we propose to examine, by calculating the Keldysh parameter in both experiments, in which regime the atoms have been placed. Results are compiled in Table 2.1. For the all the experiments, the Keldysh parameter is close to one and the system was under the conditions of tunnel ionization.

Characteristics

High-order harmonic spectra have interesting recurrent features. For instance, we present in Fig. 2.3 data obtained by Ferray et al [4] in argon (details of the laser parameters have been presented in the previous section). First of all, harmonics collected are odd multiples of the laser frequency. Secondly, the spectrum has a particular shape. We observe at first a swift decrease of the harmonic intensity. Then, in the so-called plateau region, harmonics have a roughly constant intensity, until a cutoff beyond which the intensity finally rapidly decreases. [3] with a KrF * laser (248 nm) for intensities of 1× × ×10 1 1 15 5 5 and 1× × ×10 1 1 16 6 6 W.cm ---2 2 2 , and Ferray et al [4] with a Nd:YAG laser (1064 nm) at intensity of 3× × ×10 1 1 13 3 3 W.cm ---2 2 2 .

Semi-classical picture 2.4.1 Three-step model

We acknowledged the fact that most of the first discoveries of HHG have been realized in the experimental conditions required to place the atoms in the tunnel regime. Based on these considerations, a semi-classical model has been proposed to explain the phenomenon [START_REF] Kulander | Dynamics of Short-Pulse Excitation, Ionization and Harmonic Conversion[END_REF][START_REF] Corkum | Plasma perspective on strong field multiphoton ionization[END_REF][START_REF] Schafer | Above threshold ionization beyond the high harmonic cutoff[END_REF] and successfully validated by later studies built on the resolution of the TDSE. Called the three-step model, it is extensively invoked to picture HHG using three simply understandable steps. First, we explained previously that when the laser field amplitude is high for a given half laser period, an electron is first pulled off from the atom or molecule by tunnel ionization. In the continuum, it is then accelerated by the oscillating laser field which changes sign and drives it back into the vicinity of the parent ion during the following half-cycle. It finally may recombine into its initial state and radiates its accumulated kinetic energy. The collected emitted photons constitute the harmonic spectrum. three-step model with a 1D atomic scheme. Approximate time-ranges during which each step happens are presented along the oscillating electric field to give an idea on typical timings for which each process is involved.

In this model, while tunnel ionization and radiative recombination are purely quantum phenomenon, the electron motion in the continuum can be considered as the one of a classical free particle subjected to an oscillating electric field. Its motion is thus characterized by trajectories.

Classical trajectories

In this section, we analyze the classical electron trajectories by solving the Newton equation for a free particle of charge -1 and mass 1 (in atomic units) in an oscillating electric field [START_REF] Corkum | Plasma perspective on strong field multiphoton ionization[END_REF][START_REF] Paulus | Rescattering effects in above-threshold ionization: a classical model[END_REF]. We consider a linearly polarized electric field along the x-axis of the form:

E(t) = E L cos(ω L t), (2.4.1)
where E L is the amplitude and ω L the frequency of the laser field. Driven only by the electric field, the motion of the electron is thus restricted to the x-axis. In atomic units, the Newton equation reads: We assume that tunnel ionization releases an electron in the continuum at time t i , at x = 0. Namely, we neglect the length of the barrier, which is of the order of few a.u. when the field magnitude is high. Its velocity is exactly zero at t i . Hence, the velocity ). In color scale, we display trajectories of electrons that recollide with the nucleus (i.e. for which x(t i ,t r ) = 0), the color shade traducing their kinetic energy E k (t i ,t r ) = v 2 (t i ,t r )/2 in units of U p . The dashed black line displays the recolliding electron trajectory for which the maximum kinetic energy is attained, i.e. 3.17U p . Solid black lines are for trajectories of electrons that never recollide with the nucleus.

d 2 x dt 2 = -E L cos(ω L t).
v and position x of the electron at time t are:

v(t i ,t) = E L ω L sin(ω L t i ) -sin(ω L t) , (2.4.3)
x(t i ,t) = E L ω 2 L sin(ω L t i )(ω L tω L t i ) + cos(ω L t)cos(ω L t i ) .

(2.4.4)

We take an interest in the couples of times (t i ,t r ) for which the position of the electron is zero, i.e. the electron is freed at ionization time t i and rescattered onto the nucleus at recollision time t r . We scan the values of t i and find numerically the corresponding t r . In Fig. 2. 5 we drew the electron trajectory x(t i ,t), t ∈ [t i ,t r ] for each couple (t i ,t r ).

We reported with a color scale the kinetic energy of the electron at recombination time t r , that is:

E k (t i ,t r ) = 1 2 v 2 (t i ,t r ).
(2.4.5)

We also show some trajectories of electrons that never recollide with the core as being freed too early in the rise of the laser cycle. We found numerically that the maximum kinetic energy possibly gained by recolliding electrons is 3.17U p . Thus, this energy corresponds to the cutoff observed in the spectrum, i.e. the maximum photon energy possible is [START_REF] Huillier | High-order Harmonic-generation cutoff[END_REF]:

Ω cut = I p + 3.17U p .
(2.4.6)

Notwithstanding this limit, we still observe exponentially decreasing harmonic emission beyond this value. This is due to the quantum nature of electrons. As a matter of fact, the ground-state, with which freed electrons recombine, is spatially spread with exponential tails. Hence, the electronic wave-packet can recombine at different

SEMI-CLASSICAL PICTURE

Eletron kinetic energy (U units) p 3.17 3.17U p . We displayed time regions corresponding to the short and the long trajectories for ionization and recollision, and the further electron returns (2-5). We also report the electric field amplitude (red lines). For the first return (1), we easily observe that two trajectories correspond to the same kinetic energy at recollision time (see black circles).

positions than zero, where its kinetic energy can be greater than the limit at x = 0, being even more accelerated.

Interestingly, we see that, on both sides of the trajectory of maximum kinetic energy, two possible trajectories lead to the same kinetic energy at recollision time. The ones that have the shortest duration t i -t r , namely for which electrons are freed lately in the decrease of the laser field and recollide early after, are called short trajectories. Conversely, the one that have the longest duration t i -t r , corresponding to the earliest freed and latest recolliding electrons, are called long trajectories. Most interestingly, we find that the spatial extension of the short trajectories is bounded by [START_REF] Corkum | Plasma perspective on strong field multiphoton ionization[END_REF]:

x α = E L ω 2 L , (2.4.7)
whereas the spatial extension of the long trajectories is bounded by 2x α . Moreover, long trajectory electrons will always go above x α . Furthermore, we find that for the long trajectories only, greater values of t r fulfill the recollision requirement x(t i ,t r ) = 0. That is to say, denoting all the solutions t (n) r , n = 1, 2, ..., electrons that return for the first time at t (1) r can be pushed away by the laser field and brought back again to the nucleus at a second time t (2) r , and so on. However, this is not the case for short trajectory electrons which never recollide with the core after the first return. Figure 2.6 displays ionization and recollision times for short and long trajectories as well as for further returns up to the 5th occurrence, for 3 laser cycles. We observe that each return is also split into two class of trajectories leading to the same recollision kinetic energy, and as for the first return, the shortest trajectory electrons will not be able to further recollide while the longest can lead to subsequent recollision.

//potential Potential potential(potType); potential.setParams(a_,R,V0_,U0,L1,L2,npwr,al,be,ga,de); potential.setCharges(Z,Z1,Z2); cout << potential.whichType() << endl;

Simulation Methods

Introduction

In the previous chapter, we have presented the basis of HHG through a semi-classical model where the electron motion in the continuum driven by the laser is treated classically, solving the Newton equation for a point charge. However, since HHG occurs at the atomic scale, where electrons and nuclei must be treated with a quantum approach, one cannot have a completely valid description through a classical model. One must solve the TDSE to obtain exact dynamics of a quantum system. In the most general case, the TDSE reads: where Ĥ(t) and |Ψ(t) are respectively the time-dependent Hamiltonian and wavefunction of the system. In this chapter, we present the theoretical approaches and models that we use to solve the TDSE for an atom or molecule in a strong laser field. First, we developed a numerical resolution in 1D with fixed nuclei for which electronic and nuclear motion are uncoupled, i.e. within the Born-Oppenheimer Approximation (BOA). We also used the so-called Strong-Field Approximation (SFA), a semi-analytical approach which gives more direct physical insight. Finally, to study vibrational effects in HHG, we developed 1D×1D simulations beyond the BOA, coupling electronic and nuclear motion exactly.

One dimensional ab initio computations

Assumptions

In this section we present the theoretical framework of our 1D ab initio computations 1 1 These computations are usually referred as "TDSE computations", not to be mistaken with other computations such as the strong-field approximation (see Sec. 3.3.1) which also solves the TDSE but with a semi analytical approach and additional approximations.

for an atom or molecule subjected to the influence of a low-frequency strong laser field.

1D model We consider a laser field which is linearly polarized along the coordinate x. Consequently, the electrons under its influence move along its polarization, and the transverse directions accounts only for the spreading of the electronic wave-packets, which can be neglected. Thus, we limit the description of our system in 1D along x.

Single-active electron HHG involves the temporary ionization of one electron typically from the valence shell of an atom, or the Highest-Occupied Molecular Orbital (HOMO) of a molecule. As a result, we treat the motion of a Single-Active Electron (SAE) and neglect the effect of the other electrons. Hence, there is no electron correlation in this model. (In some molecules, though, for which several occupied orbitals can be close together and contribute to HHG, the study must be carried out beyond the SAE.)

Non-relativistic electron The maximum energy of the electron can be evaluated by the maximum kinetic energy attained after ionization by the laser field. It is approximately 10U p in a semi-classical picture [START_REF] Paulus | Rescattering effects in above-threshold ionization: a classical model[END_REF]. The resulting electron velocity is v max = 20U p . In the usual working conditions, such as Ti:sapphire IR laser of 800 nm wavelength (frequency ω L = 0.057 a.u.) and peak intensity I L ∼ 5 × 10 14 W.cm -2 , U p ∼ 1 a.u. Thus v max ∼ 4.5 a.u., which is much lower than the velocity of light c = 137 a.u. Hence, neglecting relativistic effects to study HHG is valid.

ONE DIMENSIONAL AB INITIO COMPUTATIONS

Classical laser field In such laser intensities the number of photons is very high. We thus consider that the electro-magnetic field is not quantized and will work with its classical form.

Dipole approximation

The laser wavelength is large compared to the de Broglie wavelength of the electron. Namely, at a given time the spacial variations of the electric field on the atomic scale can be neglected. We assume that the field is seen as a Plane Wave (PW) on the atomic scale, and denote k the wave vector. Formally, the dipole approximation means that we use the zeroth order approximation of the term

e ik•r = 1 + ik • r -(k • r) 2 /2 + ...,
that is e ik•r 1. Thus, the interaction of the electron with the laser field is, in the length gauge:

Vint (t) = xE(t). (3.2.1)
Here the interaction with the magnetic field of the pulse has been neglected because it is very weak compared to the interaction with the electric field. This consideration validates the study of the electron only in 1D, along the laser polarization.

Born-Oppenheimer approximation

The proton mass (1836.15 a.u.) is much greater than the electron mass. Consequently, the motion of the electron is much faster than the motion of the nuclei. Hence, the electronic and nuclear dynamics can be uncoupled: the electron "sees" frozen nuclei. This is the basis of the so-called BOA.

No direct interaction between the laser field and the nuclei For the same reasons, the effect of the laser field on the position of the nuclei takes place at a much larger time scales than for the electrons. As a result, we consider that the charged nuclei are not subjected to the laser field.

Hamiltonian and time-dependent Schrödinger equation

Within these assumptions, the Hamiltonian of the system reads:

Ĥ(t) = Ĥ0 + xE(t), (3.2.2)
where: Ĥ0 = -1 2

∂ 2 ∂ x 2 + V (3.2.3)
is the Hamiltonian of the atom or molecule. The first term is the kinetic energy operator and the second is the potential interaction between the electron and the rest of the system, the ion. The TDSE in the x-coordinate writes:

i ∂ ∂t Ψ(x,t) = - 1 2 ∂ 2 ∂ x 2 +V (x) + xE(t) Ψ(x,t). (3.2.4)
The expressions used for V (x) are presented in the following section.

Potentials

As we explained it, the potential V (x) in Eq. (3.2.2) is the mean interaction potential between the single-active electron and the other particles of the system. Its form is adjusted depending on which physical property of the system we need to reproduce. We present here three different cases we studied.

Atom In the atomic case, the interaction potential between the electron and the singly-charged ion is the Coulomb interaction -1/x. This potential presents a singularity at x = 0. In 1D, the electron motion is restricted to the x-axis. Driven by the laser field, is "forced" to cross x = 0, while in 3D the electronic wave-packet can avoid this pole. To get rid of numerical problems arising with this pole, we use a regularized Coulomb potential, also-called soft-Coulomb potential [START_REF] Eberly | High-order harmonic production in multiphoton ionization[END_REF][START_REF] Su | Model atom for multiphoton physics[END_REF]:

V at (x) = -1 √ x 2 + a 2 . (3.2.5)
The regularization parameter a is adjusted in order to reproduce the ionization potential of the considered atom.

Diatomic molecule

To model a homo-nuclear diatomic molecule we use a doublewell potential made of two soft-Coulomb potential shifted by the internuclear distance R. The origin of the x-axis is chosen midway between the two atomic centers, and the positive charge is equally distributed on the two centers such that:

V dia (x) = - 1/2 x - R 2 2 + a 2 - 1/2 x + R 2 2 + a 2 . (3.2.6)
Here again, a is adjusted in order to reproduce the ionization potential of the considered molecule.

Model of shape-type resonance A shape resonance in a molecule is a highly excited state which is coupled with the continuum due to the particular form of the potential [START_REF] Klaiman | On Resonance[END_REF]. A way to model it is to structure the continuum with barriers above the ionization threshold. The effect of the barriers is the introduction of one or several pseudo-bound states, which are coupled with the continuum. They have an inherent energy width corresponding to their life-time. If we want to specifically focus our study on the shape resonance we do not need to reproduce the different centers of the molecule (which could introduce other effects polluting our observations, as we will see later). Hence our potential is based on an atomic soft-Coulomb for which we symmetrically add two super-Gaussian barriers B ± (x):

V B (x) = -1 √ x 2 + a 2 + V 0 + 1 √ x 2 + a 2 B + (x) + B -(x) , (3.2.7)
with:

B ± (x) = exp ln 1 2 2x ± (L 1 + L 2 ) L 2 2n . (3.2.8)
Here, L 1 is the distance between the two barriers, L 2 the width and V 0 the height of the barrier, and n the rank of the super-Gaussian function. Figure 3.1 is a representation of the three potentials we presented (the parameters used are specified in the figure caption). The choice of the parameters, specified in the caption, has no physical justification. It is based on visual convenience only. In order to specifically describe some physical aspects of atomic or molecular systems with these potentials, the adjustment of the parameters must be based on physical quantities. First, we explained that the parameter a is adjusted to reproduce the ionization potential of the atom or the molecule. Secondly, the characteristics of the resonance states must reproduce the ones of "real" physical systems. Therefore, we need to know what are the eigenstates and energies of the electron in these potentials. x (a.u.) x (a.u.)

x (a.u.)

-8 -4 0 4 8 -8 -4 0 4 8 a b c L 2 L 1 R V 0

Search of eigenvalues and eigenvectors

To find the eigenvalues and eigenvectors of the Hamiltonian Ĥ0 , one has to solve the time-independent Schrödinger equation (TISE):

Ĥ0 |ϕ E = E|ϕ E . (3.2.9)
The eigenvalue E is the energy of the eigenstate |ϕ E . This equation can be solved with different methods. The most straightforward way is to diagonalize the Hamiltonian Ĥ0 . In this way, all the energies and eigenstate of the system are obtained. Yet, in most of the cases, it can be computationally costly compared to other approaches which targets specific energies and eigenstates, such as the inverse iteration. We will first present the inverse iteration and then discuss the diagonalization of the Hamiltonian which can be useful to compute the density of states and observe resonances.

Inverse iteration

For the atomic or molecular potential given by Eqs. (3.2.5) and (3.2.6) the ionization potential I p is deduced from the energy E 0 of the Electronic Ground State (EGS):

I p = -E 0 . (3.2.10)
If one only need to know the value of E 0 and the wave-function of the EGS, one can use the inverse iteration method. Knowing an estimate of E 0 , this method allows one to find iteratively the correct value of the energy and the corresponding eigenstate.

The system:

( Ĥ0 -ε)|ξ (k+1) = |ξ (k) N (k) , (3.2.11)
where

N (k) = ξ (k) |ξ (k) , is inverted n times, i.e. for k ∈ [[0, n]].
Let us introduce the meaning of each terms through the description of the iterative process. We start with a guess ε of the eigenvalue E 0 and a trial function |ξ (0) which norm is N (0) . We invert the system and obtain a new function ξ (1) , which is then normalized by N (1) and served as a new guess. The process is repeated and converges in few iterations to the eigenstate |ϕ 0 . To understand in which extent this iterating process converges to the desired eigenstate, we express the guess ξ (0) and the new function ξ (1) in the basis of the eigenstates {|ϕ p } of the system:

|ξ (0) = ∑ p c (0) p |ϕ p ,
p = c (0) p E p -ε , ∀p (3.2.14)
As a result, if ε is close enough to E 0 , the dominant term in Eq. (3.2.13) corresponds to the eigenstate |ϕ 0 . The normalization step ensures that the process does not explode. The next step, the function |ξ (2) will be even more closer to |ϕ 0 . The convergence criterion is usually a check on the norm of |ξ (k+1) -|ξ (k) . In our computations, we stop the iterative process if this norm is lower than 10 -10 . After the convergence, a correction ∆E to the energy can be evaluated as :

∆E = 1 (N (n) ) 2 . (3.2.15) Indeed, ξ (n) |ξ (n) = ∑ p |c (n) p | 2 , (3.2.16)
and similarly to Eq. (3.2.14), we have:

c (n) p = c (n-1) p E p -ε , ∀p.
(3.2.17)

Since our iterative process has converged, and due to the normalization at step n -1:

c (n) 0 ∼ 1 E 0 -ε (3.2.18) c (n) p ∼ 0, ∀p ≥ 1, (3.2.19)
and thus:

1

(N (n) ) 2 = 1 ξ (n) |ξ (n) = E 0 -ε = ∆E. (3.2.20)
We can repeat a few times the complete inverse iteration, each time with the corrected energy ε = ε + ∆E, and using an additional convergence criterion we finally obtain the correct value of the energy E 0 .

This method can also be applied to the excited states as long as their separation in energy is sufficiently large. It becomes tricky for Rydberg states and completely useless for the continuum states. Imaginary-time propagation Such as the inverse iteration, the Imaginary-Time Propagation (ITP) enables one to find some eigenvalues and eigenvectors of Ĥ0 [52]. Though the procedure is very different than the inverse iteration, the demonstration of its convergence is similar. It relies on the fact that one already has a robust method to propagate the TDSE, whereas the inversion of the Hamiltonian would need further computational efforts. The idea is to translate time in the imaginary axis: t → -iτ.

The TDSE is transformed into a diffusion equation:

∂ ∂ τ |Ψ(τ) = -Ĥ0 |Ψ(τ) , (3.2.21)
which is formally equivalent to:

|Ψ(τ) = e -Ĥ0 τ |Ψ(τ = 0) . (3.2.22)
Such as before, the wave-function at imaginary time τ = 0 can be expressed in the basis of the eigenstates {|ϕ p } of the system:

|Ψ(τ = 0) = ∑ p c (0) p |ϕ p , (3.2

.23)

leading to:

|Ψ(τ) = ∑ p e -E p τ |ϕ p . ( 3

.2.24)

When τ approaches infinity, the dominant term of the sum is the ground-state, having the lowest (negative) energy. As a result, the wave-function converges to the eigenvector |ϕ 0 . The propagation is initiated from a trial wave-function |Ψ(τ = 0) , and each time-step ∆τ the new wave-function is normalized so that the solution does not explode. In this way, the ITP leads only to the ground-state. However, one can compute a number of excited states by propagating simultaneously the same number of trial wave-functions and constraining them at each time step to be orthogonal to each other. This can be achieved by diagonalization of the overlap matrix or using Gram-Schmidt orthogonalization.

The propagation is stopped if the energy has converged with a sufficient precision. Here, we use a criterion of 10 -5 in the relative energy error between two time-steps. It is much greater than the convergence criterion used in the inverse iteration in order to reduce computational cost, because we used the ITP for the 1D×1D correlated model that we will present in Sec. 3.4.2. Nevertheless, we checked the sufficiency of this criterion.

... ...

Ψ(x) Ψ(x ) i Ψ(x ) 0 x i x 1 x 2 x 3 x 4 x Ψ(x ) N x

Nx

Δx Fig. 3.3 Scheme of the discretization of the wave-function Ψ(x) on a grid.

Diagonalization of the Hamiltonian

To obtain all the eigenstates of the system and their energies, one can diagonalize the Hamiltonian (3.2.2) on a grid. Let's write x i the discrete positions of the grid, for i ∈ [[1 :

N x ]],
where N x is the number of points in the grid. The grid length is 2L x such as -L x ≤ x i ≤ L x , x 1 = -L x and x N x = L x . The electronic wave function |Ψ is expressed as a vector of its value at each position of the grid:

|Ψ = Ψ(x 1 ), Ψ(x 2 ), • • • , Ψ(x N x ) . (3.2.25)
As sketched in Fig. 3.3, this can be viewed as the decomposition of |Ψ in a basis of Dirac delta-functions such as:

Ψ(x) = N x ∑ i=1 Ψ(x i )δ (x -x i ). (3.2.26)
The expression of the kinetic energy operator on the grid is approximated by Euler's method:

- 1 2 ∂ 2 Ψ ∂ x 2 -Ψ(x i+1 ) + 2Ψ(x i ) -Ψ(x i-1 ) 2∆x 2 + O(∆x 2 ), (3.2

.27)

where ∆x is the step between two consecutive points on the grid. The expression of the potential is simply V (x i ). In the basis of the positions, the N x × N x matrix of the Hamiltonian is thus tridiagonal:

H 0 =           -1 ∆x 2 +V (x 1 ) -1 2∆x 2 (0) -1 2∆x 2 -1 ∆x 2 +V (x 2 ) -1 2∆x 2 -1 2∆x 2 -1 ∆x 2 +V (x 3 ) . . . . . . . . . -1 2∆x 2 (0) -1 2∆x 2 -1 ∆x 2 +V (x N x )           (3.2.28)
and is diagonalized using the QL algorithm [START_REF] Press | Numerical recipes, the art of scientific computing[END_REF].

The benefit of the diagonalization method is that it ends up with all the energies of the bonded states, and also the positive energies of continuum states. However, the number of state is finite, defined by the size of the grid N x , while the continuum must display states continuously at every energy. Nonetheless, the diagonalization can be used to compute the Density Of States (DOS) in the continuum. This approach is very useful to observe structures and resonances in the continuum.

Density of states

The DOS is defined as the number of states per unit of energy. Namely, it corresponds to:

ρ(E) = ∆n ∆E . (3.2.29)
As explained before, diagonalizing the Hamiltonian H 0 results in numerous positive energies, corresponding to the "continuum" states. In our symmetric potentials, the degeneracy of the continuum, with even and odd states at the same energy, leads us to compute the DOS for even and odd states separately. We apply this method in the case of the potential with barriers V B as expressed by Eq. (3.2.7) in order to monitor the resonances introduced in the continuum. Preliminary results are reported in Fig. 3.4a. The parameters used are (in a.u.) a = 0.8975, L 1 = 1.25, L 2 = 2.395, V 0 = 0.8 and n = 4 in order to reproduce the shape resonance in N 2 observed in ionization channel X [START_REF] Davenport | Ultraviolet Photoionization Cross Sections for N 2 and CO[END_REF][START_REF] Dehmer | Shape-Resonance-Enhanced Nuclear-Motion Effects in Molecular Photoionization[END_REF][START_REF] Lucchese | Studies of differential and total photoionization cross sections of molecular nitrogen[END_REF][START_REF] Plummer | Partial photoionization cross sections of N 2 and CO using synchrotron radiation[END_REF] (and its ionization potential I p = 15.58 eV [START_REF] Watanabe | Ionization potentials of some molecules[END_REF]). It displays a high dependence of the DOS in the energy which hides the resonances.

This dependency is estimated to be about 1/ √ E. Indeed, solving the Schrödinger equation for an electron on a grid, of finite dimension by nature, can be viewed as an equivalent problem of an electron in a 1D box. As it allows analytical calculations, we will first derive the expression of the DOS in a potential that is zero inside the box and infinite outside. We recall that the energies of the electron are:

E = n 2 π 2 2L 2 , n ∈ N, (3.2.30)
where L is the size of the box 2 2 In fact, the correct expression is: . This expression shows that the energy depends on the state number n, and that these states are not equally distributed. As the size of the box L increases, ρ(E) as given by Eq. (3.2.29) can be cast into its continuous limit:

E = 2 ∆x 2 sin
ρ(E) = dn dE . (3.2.31)
Indeed, for sufficient large boxes the separation between the states is squeezed and the energy distribution becomes quasi-continuous. From Eq. (3.2.30) we obtain n = L √ 2E/π, and thence:

ρ(E) = L π √ 2E . (3.2.32)
We thus normalized our even and odd DOS with 1/ √ E. However, this energy dependency is based on the infinite box approximation. Here, the size of the box is limited for computational reasons. The normalization with the energy dependency is not satisfactory, as shown in Fig. 3.4b. An efficient way to get rid of the energy dependency is to subtract the odd DOS to the even DOS. The DOS difference will thus exhibit the even and odd resonances in the continuum, as we see Fig. 3.4c. They are easy to identify: the even (odd) resonances are negative (positive) peaks. The Full-Width at Half Maximum (FWHM) of the peaks can give an estimation of the energy width δ E r of the resonant states. Thence the life-time of these pseudo-bound states is τ r = 1/δ E r .

Laser field

The electric field of the laser pulse is of the form:

E(t) = E L f (t) sin(ω L t), (3.2.33)
where E L is the amplitude, ω L the frequency and f (t) the envelope. Experimental laser pulses have usually a Gaussian spatio-temporal shape. We use two forms for the temporal envelope f (t). The first is a square-sine envelope which tends to reproduce the Gaussian shape:

f (t) =    sin 2 ω L 2N c t if t ∈ [0 : N c T L ] 0 elsewhere, (3.2

.34)

where N c is the number of oscillations (or number of cycles) under the envelope and T L = 2π/ω L the period of the oscillations. The advantage of a square-sine envelope compared to a Gaussian one is that is it is exactly zero at the beginning and the end. Indeed, mathematically, a Gaussian function is never zero. Hence, one has to propagate over a long duration to ensure that it is negligible and that the atomic Hamiltonian really drives the dynamics. As a matter of fact, abrupt starts of the pulse lead to numerical problems such as intense excitation along the whole frequency range, which is not physically acceptable in our case 3 3 This effect is to be related to the excitation of a physical system with a Dirac delta impulse. All the frequencies are present in the input signal, hence all the normal modes of the system are excited. This phenomenon is used for example in the field of acoustics to measure the impulse response of a room, office or concert hall to monitor reverberation time and resonating frequencies.

. The shortcomings of such pulses is, first, that the envelope distorts the oscillations, introducing a small frequency chirp, and, second, that the intensity of each cycles seen by the electron is not the same. Since the dynamics is driven by the instantaneous laser field, the resulting spectrum is an average on the different instantaneous intensities.

The second envelope used overcome these drawbacks. It has a trapezoidal shape: plugged with a linear ramp, it has then a flat top, and is finally unplugged with a linear ramp:

f (t) =                t N r T L if 0 ≤ t < N r T L 1 if N r T L ≤ t < (N c -N r )T L N c N r - t N r T L if (N c -N r )T L ≤ t < N c T L 0 elsewhere, (3.2.35)
where N r is the number of cycles under the ramps. In the following, we will use the nomenclature N r -(N c -N r ) -N r to characterize a trapezoidal pulse (e.g. a pulse of one-cycle ramp and eight-cycles flat top will be denoted 1 -8 -1). Numerically, the HHG can be studied only on the flat top where the artifacts of Gaussian or square-sine pulses are absent. Figure 3.5 shows two laser pulses with both presented envelopes.

Numerical propagation of the TDSE

The TDSE given by Eq. (3.2.4) is solved iteratively on a temporal and spatial grid. Formally, given a time-step ∆t, the evolution of the wave-function |Ψ from time t to time t + ∆t can be written according to the TDSE:

|Ψ(t + ∆t) = U(t,t + ∆t)|Ψ(t) , (3.2

.36)

where we introduced the propagator:

Û(t,t + ∆t) = exp -i t+∆t t Ĥ(τ)dτ . (3.2.37)
If the ∆t is sufficiently small the propagator can be approximated by:

Û(t,t + ∆t) = exp -i Ĥ(t + ∆t 2 )∆t . (3.2.38)
We used the Crank-Nicolson algorithm [START_REF] Crank | A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type[END_REF] to solve this evolution. This algorithm is stable and unitary and the error is in ∆t 3 [START_REF] Press | Numerical recipes, the art of scientific computing[END_REF]. The basic idea is to write the propagator as:

Û(t,t + ∆t) = exp -i Ĥ(t + ∆t 2 ) ∆t 2 exp +i Ĥ(t + ∆t 2 ) ∆t 2 , (3.2.39)
and expand it to the first-order in ∆t into:

Û(t,t + ∆t) 1 -i Ĥ(t + ∆t 2 ) ∆t 2 1 + i Ĥ(t + ∆t 2 ) ∆t 2 (3.2.40)
The time-propagation equation (3.2.36) thus reads:

1 + i Ĥ(t + ∆t 2 ) ∆t 2 |Ψ(t + ∆t) = 1 -i Ĥ(t + ∆t 2 ) ∆t 2 |Ψ(t) . ( 3

.2.41)

Given an initial wave-function |Ψ(t 0 ) , one has to invert the system to get the wavefunction at the next step. Iteratively, one can compute the wave-function at each time

t n = t 0 + n∆t, n ∈ [[0 : N t -1]],
where N t is the number of points in the time grid. Usually, the initial wave-function is taken as the EGS:

|Ψ(t 0 ) = |ϕ 0 .
Since the interaction potential xE(t) is diagonal and the time-independent Hamiltonian Ĥ0 is tridiagonal as explained previously, the time-dependent Hamiltonian H(t) is tridiagonal. Hence, the inversion of the system (3.2.41) is carried out with a simple tridiagonal-matrix inverter.

Absorbing boundaries

High energy freed electrons that cannot recombine with the nucleus would do so if they are reflected on the edges of the spatial box, and thereby would subsequently emit photons of higher energy beyond the cutoff. To avoid this numerical artifact, we place absorbing conditions at the edges of the box. Two simple solutions are available: first, we can multiply the wave-function at each time-step by an absorbing function A(x) of the form:

A(x) =          cos 1/8 |x| -L x + L abs 2L abs π if |x| > L x -L abs 1 if |x| ≤ L x -L abs 0 elsewhere. (3.2.42)
Secondly, one can use a Complex Absorbing Potential (CAP) [START_REF] Sahoo | The complex absorbing potential method (CAP) to study the Stark effect in hydrogen and lithium[END_REF][START_REF] Muga | Complex absorbing potentials[END_REF], which is a potential V CAP of the form:

V CAP = -iC CAP L abs -L x + |x| N CAP if |x| > L x -L abs 0 elsewhere. (3.2.43)
The CAP is added to the potential of the system V (x) and the propagation of the TDSE is carried out with the resulting potential V (x) +V CAP (x), absorbing the wavefunction in the desired region. In the first case, L abs is adjusted knowing that the efficiency of the absorber depends on the time-step dt in such a way that it is, for a given dt, equivalent to an absorber in cos κ/8 with a time-step dt/κ. In the second case, C CAP , N CAP and L abs are adjusted to obtained the optimal efficiency. Note that each method cannot absorb perfectly electrons (residual reflection is always present) and that the efficiency depends on the electron energy. Thence, we must adjust the absorber to operate for the energy range from the cutoff to the highest energetic electrons. With that in mind, we can show that both methods are (with their own set of parameters) equivalent 4 4 We are aware of a more optimal absorber proposed by Scrinzi [START_REF] Scrinzi | Infinite-range exterior complex scaling as a perfect absorber in time-dependent problems[END_REF] but we did not employ it here since it requires involved computations. .

Dipole and spectrum

One is now able to propagate the wave-function with the TDSE and compute the wave-function each time t. Classically, the power radiated by an accelerated charge in proportional to the second derivative of its displacement, that is its acceleration a(t). However, in Quantum Mechanics, there is no acceleration operator. Thus, the electron acceleration is usually casted as the second derivative of the mean value of the position operator x:

a(t) = d 2 x dt 2 , (3.2.44)
where:

x (t) = Ψ(t) | x| Ψ(t) . (3.2.45)
The Ehrenfest theorem is then invoked to write [START_REF] Burnett | Calculation of the background emitted during highharmonic generation[END_REF]:

a(t) = d 2 x dt 2 = d 2 x dt 2 = dV dx + E(t) = Ψ(t) dV dx + E(t) Ψ(t) . (3.2.46)
In our TDSE computations, we thus call "dipole" the expectation value of the acceleration: d(t) = a(t). The amplitude and phase of the high-order harmonics are computed from the Fourier Transform (FT) of the dipole d(t):

D(ω) = TF[d](ω) = d(t)e -iωt dt = a(t)e -iωt dt. (3.2.47)
The intensity of the HHG spectrum is proportional to the square modulus of D(ω), while the phase of the harmonics is encoded in the phase of D(ω). The algorithm of choice to compute the FT is the Fast Fourier Transform (FFT) which scales in O(N t ln N t ) instead of O(N 2 t ) for the direct integration of Eq. (3.2.47) [START_REF] Press | Numerical recipes, the art of scientific computing[END_REF].

The strong-field approximation

Atomic SFA

The well-known SFA is an analytical theory of HHG developed by Lewenstein et al in 1994 [START_REF] Lewenstein | Theory of high-harmonic generation by low-frequency laser fields[END_REF]. It brings a clear physical picture of HHG in atoms which corroborates the three-step model. In addition, its numerical efficiency and ability to reproduce experiments have motivated extensive use of this theory ever since. Further improvements have also been proposed to supplement missing features or to adapt the theory to other systems such as molecules, for instance.

In this section, we recall here the basics of SFA in its general 3D formulation. However, the only difference is that we will use the velocity form of the dipole instead of the length form used in [START_REF] Lewenstein | Theory of high-harmonic generation by low-frequency laser fields[END_REF]. We will explain later our motivations. The TDSE reads:

i ∂ ∂t Ψ(r,t) = - 1 2 ∇ 2 +V (r) -r • E(t) Ψ(r,t), (3.3.1)
where r is the electron coordinate and E(t) is polarized along the x-coordinate. Initially, the system is in the ground-state |ϕ 0 . To solve the TDSE, the SFA relies on the three following assumptions:

1. The working intensities are large enough so that the system is in the tunneling regime, i.e. the Keldysh parameter γ < 1. It is then valid to neglect the possible involvement of excited states or dynamical resonances. The only contribution of the bound states of the system is the one of the ground-state |ϕ 0 .

2. The intensities are however small enough to avoid over barrier ionization. Hence, the fraction of ionized electron is small compared to the remaining bound part so that the depletion of the ground state is neglected.

3. The electrons in the continuum are considered as free particles undergoing the effect of the laser field, i.e. the effect of the potential V (r) is neglected for the continuum states. As a consequence, the freed electrons are described with Plane-Waves (PW) denoted |k from their kinetic momentum k, i.e. r|k = e ik•r . This is the so-called Plane-Wave Approximation (PWA).

Within these assumptions, the wave function of the system writes:

|Ψ(t) = e iI p t |ϕ 0 + d 3 k b(k,t)|k . (3.3.2)
By replacing this expression in the TDSE given by Eq. (3.3.1) and projecting it on a PW |k , one obtains an expression of the PW coefficients:

b(k,t) = i t 0 dt d ion (k -A(t) + A(t ),t ) × exp -i t t dτ k -A(t) + A(τ) 2 2 + I p . (3.3.3)
Here, we introduced the "reduced" vector potential A which is defined such that 5 5 Note that the exact definition of the vector potential is A such that:

E(t) = - 1 c ∂ A ∂t ,
where c is the speed of light. However, in all the equations the quantity A/c is encountered. Hence, we use the reduced vector potential A = A/c as defined above.

:

E(t) = - ∂ A ∂t , (3.3.4)
and the ionization dipole matrix element:

d ion (k) = k xE(t ) ϕ 0 . (3.3.5)
We have seen previously that the power spectrum is given by the Fourier transform of d(t) (see Eq. (3.2.47)). In TDSE computations, we use the acceleration form of the dipole, i.e. d(t) = a(t). In SFA calculations as performed by Lewenstein et al [START_REF] Lewenstein | Theory of high-harmonic generation by low-frequency laser fields[END_REF] they used the dipole in the length form. However, based on macroscopic considerations, Baggesen and Madsen [START_REF] Baggesen | On the dipole, velocity and acceleration forms in high-order harmonic generation from a single atom or molecule[END_REF] recently demonstrated that the generated harmonic field is proportional to the dipole velocity. We then derive the SFA with the velocity form of the dipole, as in Refs. [START_REF] Śpiewanowski | High-order-harmonic generation from field-distorted orbitals[END_REF][START_REF] Śpiewanowski | Field-induced orbital distortion in high-order-harmonic generation from aligned and oriented molecules within adiabatic strong-field approximation[END_REF]. We will see in the upcoming section that it is crucial for the derivation of SFA in diatomic molecules. Nonetheless, one can show that these three approaches are equivalent. Indeed, after integration by parts and arguing that the acceleration and the velocity of the electron vanish at ±∞, we can write: where v(t) = Ψ(t) |-i ∇| Ψ(t) • e x is the velocity and x(t) = Ψ(t) |r| Ψ(t) • e x the length dipoles along the laser polarization. After some manipulations, using Eqs. (3.3.2) and (3.3.3) and introducing a canonical momentum p = k -A(t), we find 6 6 We mention here that, using the dipole velocity form, the continuumcontinuum transitions k |-i ∇| k are treated exactly, with no need of additional assumption unlike in the Lewenstein model [START_REF] Lewenstein | Theory of high-harmonic generation by low-frequency laser fields[END_REF].

D(ω) = a(t)e -iωt
: is the semi-classical action, and:

v(t) = i t 0 dt d 3 p d rec (p + A(t))e -iS(
d rec (k) = ϕ 0 |-i ∇| k • e x , (3.3.10)
the recombination dipole matrix element along the laser polarization. The notation "c.c." stands for the complex conjugate of the preceding expression. The expression of v(t) has a clear physical interpretation. Subjected to the laser field, the electron is freed at ionization time t with the probability amplitude d ion (p + A(t ),t ). It then accumulates phase in the continuum described by the semi-classical action, and finally recombines at recollision time t with the probability amplitude d rec (p+A(t)). Hence, this equation encodes the three-step picture. Finally, the power spectrum is

|D(ω)| 2 = |TF[v(t)](ω)| 2
, where: The computation of D(ω) as given by Eq. (3.3.11) involves a triple integration which is numerically costly. One can however note that the term exp[-iS(p,t,t , ω)] is swiftly oscillating. Thus, the triple integration can be reduced to the evaluation of the integrand at the stationary solutions of S, i.e. for which: This procedure is the so-called Saddle-Point Approximation (SPA). It leads to the search of solutions of the following equations: while the stationary ionization and recollision times and t at and t at are numerically found by steepest-descent method. Interestingly, we find two valid couples of solutions denoted (t s at ,t s at ) and (t l at ,t l at ), which correspond to short and long travel times in the continuum. Hence, the SFA along with the SPA reveals short and long quantum paths of the freed electron, usually called trajectories such as in the classical picture. The total dipole (Eq. (3.3.11)) reduces to: For the sake of clarity, no distinction will be made for either the short (s) or the long (l) trajectories in the following notations. As a consequence, the sum over the short and long trajectories in the expression of D at will not be explicitly included, such that we can write: In this expression, C is the saddle-point prefactor and reads [START_REF] Chipperfield | Tracking individual electron trajectories in a high harmonic spectrum[END_REF]: comes from the evaluation of Eq. (3.3.11) at t at and t at .

D(ω) = -ω dt t 0 dt d 3 p d rec (p + A(t))d ion (p + A(t ),t )e -iS(
∂ S ∂ p = 0, ∂ S ∂t = 0,
t t dτ[p + A(τ)] = 0 , (3.3
D at (ω) = -ω ∑ j=s,l C(t j
D at (ω) = -ω C(
C(t at ,t at ) = 2π ε + i(t at -t at ) 3/2

Molecular SFA

In this section, we recall the equations of molecular SFA as they have been developed in Refs. [START_REF] Chiril | Strong-field approximation for harmonic generation in diatomic molecules[END_REF][START_REF] Faria | High-order harmonic generation in diatomic molecules: A quantum-orbit analysis of the interference patterns[END_REF], with the only difference that we used the velocity form of the dipole. The electronic ground state ϕ 0 (r) in the two-center molecular potential V (r) can be approximated by a Linear Combiation of Atomic Orbitals (LCAO). As it is symmetric (for example, see Fig. 3.2b in the case of a 1D double soft-Coulomb potential), it becomes the sum of two symmetric atomic orbitals φ s centered on each nucleus:

ϕ 0 (r) = 1 2 1 + w(R) φ s r - R 2 + φ s r + R 2 , (3.3.22)
where w(R) is the overlap between the two φ s orbitals. We temporarily note:

N = 1 2 1 + w(R)
.

(3.3.23)

THE STRONG-FIELD APPROXIMATION

The recombination dipole matrix element writes:

d rec (k) = ϕ 0 |-i ∇| k • e x = d 3 rϕ 0 (r)(-i ∇)e i k•r • e x = 1 2 1 + w(R) d 3 r φ s r - R 2 + φ s r + R 2 ke i k•r • e x = k • e x 2 1 + w(R) d 3 rφ s (r)e i k•r e i k• R 2 + d 3 rφ s (r)e i k•r e -i k• R 2 = k • e x 2 1 + w(R) φs (k) cos k • R 2 , (3.3.24)
where φs is the FT of φ s . Note that for an antisymmetric molecular state one should replace "cos" by "sin" and replace w(R) by -w(R).

The expression that has been derived and used in the first formulation of the molecular SFA by Chirilȃ et al [START_REF] Chiril | Strong-field approximation for harmonic generation in diatomic molecules[END_REF] arose from the length form of the dipole. Namely, it consists in evaluating d L rec (k) = ϕ 0 |r| k • e x . Similar manipulations than before lead to:

d L rec (k) = i 2 1 + w(R) ∂ φs ∂ k cos k • R 2 -φs (k) R 2 sin k • R 2 • e x , (3.3.25)
which is analogous to Eq. (3.3.24) but with an additional term with linear dependence in R. Unphysical results have been attributed to this term [START_REF] Chiril | Strong-field approximation for harmonic generation in diatomic molecules[END_REF][START_REF] Chiril | Assessing different forms of the strong-field approximation for harmonic generation in molecules[END_REF]. Thus, in [START_REF] Faria | High-order harmonic generation in diatomic molecules: A quantum-orbit analysis of the interference patterns[END_REF] it has been dropped out, without clear physical justification, in order to use only:

d L rec (k) i 2 1 + w(R) ∂ φs ∂ k • e x cos k • R 2 (3.3.26)
This procedure is somehow equivalent to deriving the expression of the recombination dipole matrix element in the velocity form, as we did above 7 7 We mention that in [START_REF] Lein | Interference effects in high-order harmonic generation with molecules[END_REF] the acceleration form of the dipole is used.

. Chirilȃ et al [START_REF] Chiril | Assessing different forms of the strong-field approximation for harmonic generation in molecules[END_REF] have investigated the effect of the length, velocity and acceleration form of the recombination dipole matrix element and shown that the velocity form gives results in better agreement with TDSE computations. As a conclusion, these considerations together with the study of Baggesen and Madsen [START_REF] Baggesen | On the dipole, velocity and acceleration forms in high-order harmonic generation from a single atom or molecule[END_REF] lead us to use the velocity dipole. Hence we use:

d rec (k) = k • e x 2 1 + w(R) φs (k) cos k • R 2 .
[recalling Eq. (3.

3.24)]

Given the fact that Eq. (3.3.5) is obtained before any assumption about the form of the dipole, the ionization dipole matrix element is however the same using either the length, velocity or acceleration dipole:

d ion (k) = -i 2 1 + w(R) ∂ φs ∂ k cos k • R 2 -φs (k) R 2 sin k • R 2 • E(t). (3.3.27)
The basic idea of the molecular SFA is to work directly with the exponential forms of cos(k • R/2) and sin(k • R/2):

-R x /2 0 R x /2 x at 1 2 at 11 12 21 22 ϕ s (x-R/2) ϕ s (x) ϕ s (x+R/2) V (x)+xE(t)
cos k • R 2 = e ik• R 2 + e -ik• R 2 2 , (3.3.28)
sin k • R 2 = e ik• R 2 -e -ik• R 2 2i . ( 3
d rec (k) = 2R(k) cos k • R 2 = R(k) e ik• R 2 + e -ik• R 2 , (3.3.30)
d ion (k,t ) = I 1 (k,t )e ik• R 2 + I 2 (k,t )e -ik• R 2 , (3.3.31)
where:

R(k) = k • e x 1 2 1 + w(R) φs (k) , (3.3.32)
I α (k,t ) = - 1 2 1 + w(R) i ∂ φs ∂ k + (-1) α φs (k) R 2 • E(t ), α ∈ {1, 2}. (3.3.33)
We can now find an expression of the velocity form of the SFA dipole, Eq. (3.3.11). The product of the transition dipole matrix elements reads: where: This leads to the following saddle-point equations [START_REF] Faria | High-order harmonic generation in diatomic molecules: A quantum-orbit analysis of the interference patterns[END_REF]:

d rec (p + A(t))d ion (p + A(t ),t ) = 2 ∑ α=1 2 ∑ β =1 R(p + A(t))I α (p + A(t ),t )e -iΦ
Φ αβ (p,t,t ) = (-1) α [p + A(t )] • R 2 -(-1) β [p + A(t)] • R 2 , α, β ∈ {1,
t t dτ[p + A(τ)] + (-1) α R 2 -(-1) β R 2 = 0 , (3.3.38) [p + A(t)] 2 2 + I p -ω + (-1) β E(t) • R 2 = 0 , (3.3.39) [p + A(t )] 2 2 + I p + (-1) α E(t ) • R 2 = 0 , (3.3.40)
for which we find four groups of stationary solutions (p αβ ,t αβ ,t αβ ). They correspond to four classes of trajectories, represented in Fig. 3.6. They express the ionization from one center and the recollision to either the same or the other one. Like in the atomic case, each class contains a short and a long path. Within the saddle-point approximation, the total dipole reduces to:

D(ω) = -ω 2 ∑ α=1 2 ∑ β =1 C αβ (t αβ ,t αβ )R(p αβ + A(t αβ ))I α (p αβ + A(t αβ ),t αβ )
× e -iS αβ (p αβ ,t αβ ,t αβ ,ω) + c.c. with C αβ (t αβ ,t αβ ) the resulting saddle-point prefactors.

Beyond the Born-Oppenheimer approximation

Vibration in molecules

In a molecule, the electrons are shared in "bonds" (i.e. orbitals), which are responsible for the cohesion of the nuclei together. Let us consider a diatomic molecule at equilibrium, subjected to an intense laser field. The effect of the laser pulse on the nuclei is twofold:

• Subsequently to the ionization of an electron, the remaining molecular ion is no longer at the equilibrium since the cohesion of the molecule is no longer ensured by the same number of electrons. The system needs to relax to the equilibrium internuclear distance of the ion (which is very often, but not always, larger than the equilibrium internuclear distance of the neutral molecule). Hence, vibrational motion in the ion is induced by the removal of an electron. In the usual quantum chemistry picture, the ionization is expressed as the projection of the nuclear wave-packet of the neutral molecule to the potential energy surface of the ion. Excited vibrational levels of the ion are populated following the Frank-Condon (FC) factors, and consequently the nuclear wave-packet starts to oscillate in the ion. Namely, vibrational motion has been induced in the ion by the laser pulse through ionization.

• The laser pulse can also excite vibrational levels of the neutral molecule through multi-photon excitation to electronic excited states and desexcitation to higher vibrational levels of the EGS. The most probable process is a two-photon Raman-like vibrational excitation, that we will call Impulsive Stimulated Raman Scattering (ISRS) as in Ref. [START_REF] Wagner | Monitoring molecular dynamics using coherent electrons from high harmonic generation[END_REF].

In light molecules (e.g. H 2 or D 2 ) the vibrational time-scales are of the order of the duration of a single cycle of HHG (the three steps followed by the electron), i.e. the laser period. Indeed, for example, in the case of H 2 the separation between the fundamental and the first excited vibrational state in the ion is ω + 01 = 0.010 a.u., corresponding to a period of oscillations T + 01 = 15 fs. Let us recall that the period of an IR laser such as Ti:sapphire is T L = 2.6 fs. Therefore, for a given HHG process the electron recollision will happen with a ion which has significantly evolved since the ionization. Theses considerations led Lein et al [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF] to predict the possibility of measuring vibrational motion through HHG recollision, being later proved experimentally by Baker et al [START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF].

Still in the example of H 2 , the separation between the fundamental and the first excited vibrational state in the neutral molecule is ω 01 = 0.019 a.u., corresponding to a period of T 01 = 8.0 fs. Hence, vibrational excitation of the neutral molecule can be induced and probed by HHG, provided that the laser bandwidth is large enough. Such experiments have been carried out by Wagner et al [START_REF] Wagner | Monitoring molecular dynamics using coherent electrons from high harmonic generation[END_REF] for SF 6 molecules.

Correlated model

In regard of these considerations, we developed a 1D×1D model for HHG in diatomic molecules beyond the BOA, which couples exactly electronic and nuclear dynamics. It uses all the approximations listed in Sec. 3.2.1 but the BOA. The coordinates are the internuclear distance R and the position x of the electron along the internuclear axis. The Hamiltonian of the system in these coordinates reads:

H 0 = - 1 2µ ∂ 2 ∂ R 2 - 1 2 ∂ 2 ∂ x 2 +V NN (R) +V Ne (x, R) , (3.4.1)
where µ is the reduced mass of the nuclei, and V NN and V Ne are the nucleus-nucleus and nuclei-electron effective interaction potential, respectively. The former is the potential energy curve of the ion and the latter is a double soft-Coulomb potential as presented in Sec. 3.2.3 except that here the regularization parameter a is a function of the internuclear distance R, and is adjusted such as the energy of the ground-state corresponds to the energy difference between V NN (R) and the potential energy curve of the neutral molecule.

The system is interacting with an electric field polarized along the internuclear axis. The TDSE reads:

i ∂ ∂t Ψ(x, R,t) = H 0 + xE(t) Ψ(x, R,t) , (3.4.2)
and is solved by expressing the wave function of the system Ψ(x, R,t) in the vibrational eigenstates basis {|χ + v } of the ion:

Ψ(x, R,t) = N v ∑ v =0 Φ v (x,t)χ + v (R) , (3.4.3)
where Φ v (x,t) are electronic wave functions and encode the dynamics of the system. The infinite basis {|χ + v } is practically truncated to a number N v + 1, and the convergence of the simulations with N v is verified. Denoting ε v the energy of the vibrational eigenstate |χ + v of the ion, the TDSE becomes a set of N v coupled equations acting on the electronic wave-functions:

i ∂ ∂t Φ v (x,t) = - 1 2 ∂ 2 ∂ x 2 + ε v + xE(t) ≡ H 1 (t) Φ v (x,t) + N v ∑ w =0 χ + v VNe χ + w ≡ V Φ w (x,t) . (3.4.4)
The last term exhibits the coupling between the electronic and the nuclear dynamics. The TDSE expressed in the truncated basis {|χ

+ v } v ∈[[0:N v ]
] and in the x-grid can also be written in the matrix form as:

i ∂ ∂t |Ψ(t) = (H 1 (t) + V)|Ψ(t) , (3.4.5)
where:

• |Ψ(t) is a (N v + 1) × N x vector:

|Ψ(t) = |Φ 0 (t) , |Φ 1 (t) , • • • , |Φ v (t) , • • • , |Φ N v (t) , (3.4.6)
with:

|Φ v (t) = Φ v (x 1 ,t), Φ v (x 2 ,t), • • • , Φ v (x N x ,t) , (3.4.7)
• H 1 (t) is a block-diagonal matrix:

H 1 (t) =              H 1 (t) + ε 0 N x H 1 (t) + ε 1 N x (0) . . . H 1 (t) + ε v N x (0)
. . .

H 1 (t) + ε N v N x              (3.4.8)
with N x the identity matrix of dimension N x , and each block H 1 (t) a N x × N x tridiagonal matrix:

H 1 (t) =           -1 ∆x 2 + x 1 E(t) -1 2∆x 2 (0) -1 2∆x 2 -1 ∆x 2 + x 2 E(t) -1 2∆x 2 -1 2∆x 2 -1 ∆x 2 + x 3 E(t) . . . . . . . . . -1 2∆x 2 (0) -1 2∆x 2 -1 ∆x 2 + x N x E(t)           (3.4.9)
• and where V is a

(N v + 1)N x × (N v + 1)N x dense matrix: V =             χ + 0 VNe χ + 0 χ + 0 VNe χ + 1 • • • χ + 0 VNe χ + w • • • χ + 0 VNe χ + N v χ + 1 VNe χ + 0 χ + 1 VNe χ + 1 • • • χ + 1 VNe χ + w • • • χ + 1 VNe χ + N v . . . . . . . . . . . . . . . χ + v VNe χ + 0 χ + v VNe χ + 1 • • • χ + v VNe χ + w • • • χ + v VNe χ + N v . . . . . . . . . . . . . . . χ + N v VNe χ + 0 χ + N v VNe χ + 1 • • • χ + N v VNe χ + w • • • χ + N v VNe χ + N v             (3.4.10) each block χ + v VNe χ + w being a N x × N x matrix.
The evolution of the system from t to t + ∆t following the TDSE as given by Eq. (3.4.5) is equivalent to:

|Ψ(t + ∆t) = e -i[H 1 (t+ ∆t 2 )+V]∆t |Ψ(t) , (3.4.11)
and can be approached, using the split-operator method [START_REF] Feit | Solution of the Schrödinger equation by a spectral method[END_REF], by: |Ψ(t + ∆t) e -iV ∆t 2 e -iH 1 (t+ ∆t 2 )∆t e -iV ∆t 2 |Ψ(t) .

(3.4.12)

Within this procedure, the time-independent Hamiltonian V is diagonalized once and applied to the system in the diagonal basis, while the time-dependent Hamiltonian H 1 (t) is applied using the Crank-Nicolson algorithm as in Sec. 3.2.6 for each tridiagonal block H 1 (t). The error of the split-operator method is in O(∆t 3 ) per step.

The study of the vibration in the neutral molecule is carried out through the vibronic states expressed in the nuclear basis of the ion. Namely, the vibronic wavefunction corresponding to the neutral vibrational state v is: where ϕ v,v (x) is the electronic part of state v along the ionic vibrational state v .

ψ v (x, R) = N v ∑ v ϕ v,v (x)χ + v (R) .
Given the complex expression of the Hamiltonian H 0 in the basis of the vibrational eigenstates of the ion which complicates both its diagonalization and its inversion, and given the fact that we already have a robust time-propagation of the TDSE with the split-operator method, the ITP seems to be the best choice to find the vibronic eigenstates and their energies [52].

Introduction

We are now equipped with efficient methods allowing extensive computations to simulate HHG. However, the simulations based on the numerical resolution of the TDSE are, to simplify, like a black box in which we input the forms of the interaction potentials (atomic or molecular, and laser) and get a spectrum in output. Notwithstanding that they constitute "exact" references for models and computations based on other approaches such as the semi-analytical SFA, in turn, physical understanding of the underlying quantum processes is not direct. As a consequence, we may need additional "tools" to analyze the raw TDSE results, extract fundamental information and shed light on implicitly included physical processes. The aim of this chapter is to present the collection of analysis tools that we used and developed to understand TDSE results.

Time-frequency analysis 4.2.1 Introduction

Generally speaking, the time-frequency analysis of any signal denotes the methods used to represent the frequencies of the signal as a function of time. The most usual representation is a 3D diagram displaying the intensity of the frequencies varying with time. It is commonly called spectrogram, especially in the field of acoustics. In most of the cases, a spectrogram is displayed as a 2D color map with time in abscissas and frequency in ordinates, and the color shade for the intensity. In the case of HHG, time-frequency analysis is performed on the dipole in order to determine the emission times of each harmonic from the numerical resolution of the TDSE. The existing methods of time-frequency analysis are multiple [START_REF] Cohen | Time-frequency Analysis[END_REF]. In this section, we will present four methods. The first one is the Short-time Fourier transform (STFT), usually known as Gabor transform in strong field physics [START_REF] Antoine | Time profile of harmonics generated by a single atom in a strong electromagnetic field[END_REF][START_REF] Chelkowski | High-harmonic generation from a coherent superposition of electronic states: Controlling interference patterns via short and long quantum orbits[END_REF][START_REF] Chiril Ȃ | Emission times in high-order harmonic generation[END_REF], and the second one is the wavelet transform. Both are commonly used to perform timefrequency analysis of the HHG dipole. As they rely on the Fourier-transform, they are both endowed with a limited resolution. Moreover, the main problem with these approaches is the arbitrary choice of the parameters. For a given signal, an infinite number of spectrograms can be generated. Hence, they can only be used to obtain qualitative information about the harmonic emission times. Besides, while the wavelet transform has been proposed as a more suitable method than the STFT we believe that it not adapted as it has inherent misleading artifacts. We will thus present and discuss these two methods.

Thirdly, we have developed a method which is quantitative by nature. Based on the notion of Instantaneous Frequency (IF), it determines an unequivocal frequency for each time. Consequently, it can be used to extract exact harmonic emission times [START_REF] Risoud | Quantitative extraction of the emission times of high-order harmonics via the determination of instantaneous frequencies[END_REF]. The last method we will introduce is the Wigner-Ville transform which computes a quasi-distribution of the intensity of the signal, and reveals interesting interference features [START_REF] Kim | Wigner time-frequency distribution of high-order harmonics[END_REF]. For completeness' sake, we inform the reader of a comprehensive study of the available time-frequency methods used for laser-driven quantum processes proposed by Sheu et al [START_REF] Sheu | Exploring laser-driven quantum phenomena from a time-frequency analysis perspective: a comprehensive study[END_REF].

Short-time Fourier transform (Gabor transform)

The STFT is a Fourier transform with a moving window function. The signal is multiplied by a window function which is almost zero everywhere except in short time-range. The window function is moved along the time axis and scans the whole duration of the signal. For each window position, the Fourier transform reveals the frequencies of the signal in the time width of the window function. The mathematical expression of the STFT of a signal d is:

STFT w [d](ω,t) = d(τ)w(τ -t)e -iωτ dτ, (4.2.1)
where w is the window function. It is commonly chosen as a Gaussian function:

w(t) = 1 δ √ π e - t 2 2δ 2 , (4.2.2)
where the parameter δ is adjusted to have the desired window width.

Wavelet transforms

Mathematically, the STFT is the scalar product in L 2 space of d with:

F ω,t (τ) = w(t -τ)e -iωτ . (4.2.3)
In other words, it is a decomposition of the signal d on a basis of functions {F ω,t }. Each coefficient of the decomposition is directly mapped into a point (ω,t) in the time-frequency plane. These functions F ω,t are usually called Gabor atoms.

The wavelet transform relies on the same decomposition principle and is alsocalled wavelet decomposition. We will use the Continuous Wavelet Transform (CWT):

CWT W [d](ω,t) = d(τ)W ω,t (τ)dτ, (4.2.4)
where the basis of functions W ω,t is a family of wavelets:

W ω,t (τ) = 1 κ(ω) W τ -t κ(ω) , (4.2.5)
deduced from a generating function W (also-called mother function). The function κ depends on ω. Any kind of function W is theoretically valid to generate wavelets. However, we need to associate to each wavelet its own frequency to be able to map the decomposition into the (ω,t)-plane. Hence, W must have a clear single frequency component in terms of Fourier analysis. This is the case of the Morlet function, commonly employed for the analysis of HHG [START_REF] Antoine | Time profile of harmonics generated by a single atom in a strong electromagnetic field[END_REF][START_REF] Daniele | Wavelet Analysis of the Spectrum Emitted by a One-Dimensional Atom Driven by a Strong Laser Pulse[END_REF][START_REF] Chen | Time-frequency analysis of molecular high-harmonic generation spectrum by means of wavelet transform and Wigner distribution techniques[END_REF][START_REF] Chelkowski | High-order-harmonic generation from coherent electron wave packets in atoms and molecules as a tool for monitoring attosecond electrons[END_REF][START_REF] Telnov | Effect of nuclear vibration on high-order-harmonic generation of aligned H + 2 molecules[END_REF][START_REF] Jun | Probing dynamic interference in high-order harmonic generation from long-range molecular ion: Bohmian trajectory investigation[END_REF][START_REF] Miller | High-harmonic spectroscopy of laser-driven nonadiabatic electron dynamics in the hydrogen molecular ion[END_REF]:

W Mo (τ) = 1 δ √ π e - τ 2 2δ 2 -iω 0 τ . (4.2.6)
This function is nothing but the generator of the family {F ω 0 ,t } for the STFT for which κ(ω) = 1. The key point of the CWT is that here κ(ω) = ω 0 /ω such that the number of oscillations below the Gaussian function remains the same for all the wavelets. Additionally, this number can be chosen with the adjustment of the Gaussian parameter δ . Not surprisingly, we mention that these wavelets are also-called Gabor wavelet.

For the sake of completeness, we also investigated the Mexican-hat wavelets, generated by:

W MH (τ) = 2π -1/4 √ 3σ 1 - t 2 σ 2 e - τ 2 2σ 2 .
(4.2.7)

For this family, κ(ω) = √ 2/σ ω. Contrarily to the Morlet wavelet, the number of oscillations is fixed by the definition of the Mexican hat (see Fig. 4.1). As we explained it, a depends on the frequency ω such as the width of the wavelet is adapted for each frequency. As a result, contrarily to the STFT, the tiling of the time-frequency plane is not regular, as sketched in Fig. 4.2.

To speed up the computations of the CWT, we use the FFT. Indeed, the CWT is the convolution product of the signal d with the wavelet W ω,t . Given the efficiency of the FFT, it is much faster to compute for each ω the convolution product through the equality:

CWT

W [d](ω,t) = FT -1 FT[d] × FT[W ω,t ] , (4.2.8)
where FT stands for the Fourier transform and FT -1 for the inverse Fourier transform.

Comparison

In this section, we compare the STFT with the wavelet transforms. known element, however the following study is general and simple parameters have been chosen as a proof of principle. The laser pulse has a trapezoidal 1-8-1 shape and the laser intensity is I L = 3.5 × 10 14 W.cm -2 (E L = 0.1 a.u.).

STFT Figure 4.3 displays the spectrogram of the dipole computed with the STFT on the second and the third laser cycle along with the dipole and the harmonic spectrum. We used a 1-2-1 trapezoidal pulse, for which harmonic generation occurs during the second laser cycle. Indeed, the first half-cycle in the ramp is too weak to allow tunnel ionization, while the electrons freed in the second half-cycle of the ramp recombine with the ion during the first half-cycle of the flat-top. Hence, results will be presented only for the flat-top of the pulse (i.e. cycles 1 and 2).

The STFT is a wealth of information about the underlying quantum processes in HHG. First of all, we identify intense bell-shaped distributions, which correspond to the emission times of the harmonics, i.e. the electron recollision times. These bellshaped distributions are called attobursts. For the first laser cycle, we see that one harmonic is emitted at two different times, which correspond to the short and long trajectories. Every half laser cycle, the same process is repeated, that is to say the bell-shaped distributions are identical. The four attobursts are denoted by Roman numbers I -IV . These results corroborate the semi-classical computations based on the three-step model presented in Sec. 2.4.2.

Additionally, under the second one, we identify a small contribution which corresponds to the second return of the electron. Namely, as we explained in Sec. 2.4.2, the electron that did not recombine when it was brought back to the ion for the first time propagates further and then is driven back a second time to the ion, where it can recombine again and emit harmonic radiation. As this process repeats itself, in the third half-cycle, second return and third return are added together, and so on for the next cycles. Their separation is difficult with STFT. However, one should note that this is an artifact of the 1D model which tends to over-represent the multiple returns. Actually, there are two reasons that further returns are mostly inexistent in experimental spectra. The first is quantum: in a real 3D world, the electronic wave packet spreads rapidly in the transverse direction, thus the later the recombination, the lower the contribution. The second is macroscopic: the generation is carried out by a focused laser beam in a gaseous medium. The spatial shape of the beam is usually Gaussian which introduces phase-mismatch between harmonics generated at different locations in the medium [START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF]. Note that flat-top pulses are being experimentally developed [START_REF] Boutu | High-order-harmonic generation in gas with a flat-top laser beam[END_REF][START_REF] Dubrouil | Controlling high harmonics generation by spatial shaping of high-energy femtosecond beam[END_REF][START_REF] Valentin | Top-hat beam output of a single-mode microstructured optical fiber: Impact of core index depression[END_REF][START_REF] Kong | Flat-top mode from a 50 µm-core Yb-doped leakage channel fiber[END_REF] and theoretically studied [START_REF] Hernández-García | Resolving multiple rescatterings in high-order-harmonic generation[END_REF]. In the latter reference, flat-top shape appears to strongly reduce the phase mismatch and higher-order returns are resolved with high amplitude even in 3D. Hence, phase mismatch seems to be the major reason of not observing further returns experimentally.

Finally, the continuous emission below harmonic 13 (corresponding to the ionization potential of this "atom") is attributed to multiphoton absorption in the perturbative regime.

CWT A comparison of the Morlet and Mexican-hat CWT is presented in Fig. 4.4. The Morlet wavelet which is used has the same width than the Gabor window at harmonic 45, i.e. the parameters of the Morlet generating function are ω 0 = 45ω L and δ = 7 a.u.

First of all, we discard the Mexican-hat CWT, which appears to have a too short window duration to describe the frequencies with an acceptable resolution. Mexicanhat CWT is not as useful for HHG as for image compression [START_REF] Pearlman | Wavelet Image Compression[END_REF].

In the light of the STFT in Fig. 4.3, one can observe the adaptive tiling of the timefrequency plane. The Morlet CWT exhibits the same features than the STFT but with significant deformations. First, the low frequencies are not well resolved in time and hence start to interfere, forming odd harmonics. Secondly, the high frequencies are arbitrarily stretched so that the cutoff seems to be enhanced. Thus, extra care must be used while interpreting wavelet transforms. For instance, it seems that the study of cutoff extension by Carrera et al [START_REF] Carrera | Extension of high-order harmonic generation cutoff via coherent control of intense few-cycle chirped laser pulses[END_REF] by means of wavelet transforms is hazardous.

We conclude that STFT is more adapted to express quantum processes in HHG in the time-frequency plane than wavelet transforms. Wavelet transforms should be used in specific cases where low frequencies are present for long durations and high frequencies for very brief instants. This was the case, for instance, for the breakthrough gravitational wave chirp observed recently at the Laser Interferometer Gravitational-Wave Observatory in United States, signature of a binary black hole merger [START_REF] Abbott | Observation of Gravitational Waves from a Binary Black Hole Merger[END_REF].

Shortcomings of the methods

As mentioned at the beginning of the section, there are two major drawbacks of these methods. The first arises from the Fourier decomposition of the signal. The basis functions are localized in time and frequency with a certain width that cannot be lower than the so-called Gabor limit given by the Fourier uncertainty principle:

∆t∆ω ≥ 1 2 , (4.2.9)
where ∆t (∆ω) is the time width (frequency width) of the basis function. For the Gaussian window (or equivalently the Morlet wavelet), the equality in Eq. (4.2.9) is achieved. The consequence of the Gabor limit is that a wide window allows a good frequency resolution at the expense of temporal resolution, and conversely a narrow window allows a good temporal resolution at the expense of frequency resolution. No one can achieve an arbitrary small resolution in both directions.

The second drawback is linked to the first one. An infinite number of timefrequency representations can be achieved between a transformation with an infinitely wide window (which is actually the spectrum) and a transformation with a Dirac window function (which is the signal itself). With no further justification, the width of the window is arbitrarily adjusted for each signal to achieve a convenient visualization. A problem appears when it is used to extract supposedly quantitative times and frequencies. Indeed, in Ref. [START_REF] Chiril Ȃ | Emission times in high-order harmonic generation[END_REF], the harmonic emission times have been extracted using the position of the maximum amplitude of the STFT. However, as shown in Table 4.1, this method depends strongly on the window width (through the parameter δ in Eq. (4.2.2)) especially close to the cutoff. In our case, the variations of the extracted timings are up to 49 as for the short trajectories and 98 as for the long trajectories, which is considerable.

Several times, people have argued us that HHG is endowed with a limited resolution by nature, as a quantum process, ruled by the Heisenberg uncertainty principle. As a result, the broad distributions observed in the STFT or CWT are the consequence of the quantum character of HHG.

We objected with two arguments. First, one must not forget that the dipole is the mean value of the acceleration. The position and momentum of the electronic wave-packet are averaged over their inherent width by the computation of the dipole. Secondly, if this limited resolution had a physical nature, it would not depend arbitrarily neither on the nature of the time-frequency analysis nor on the shape of the window or basis function.

Theses considerations led us to develop a new method to circumvent the Gabor limit. It combines filtering and IF and allows one to compute exactly the time that is associated to each generated frequency.

Instantaneous Frequency

The preliminary step is the filtering of the dipole. As exposed in Sec. 4.2.4, two contributions are observed in the dipole:

• The unstructured part below the harmonic 13,

• The bell-shaped distributions above, that correspond to the high-order harmonic emission.

The former is filtered out with the introduction of a low-cut filter. The latter, restricted to the first half-cycle, can be identified as a unique frequency emitted at each time.

Setting aside the broad resolution imposed by the Gabor limit, it can be viewed as the representative curve of a function, mathematically speaking. It is thus possible to use the notion of IF. The term frequency defines the number of repeated events of a cyclic process in a unit of time. It is unambiguously defined in the stationary case. However, in the non-stationary case the definition of frequency is unequivocal, and its instantaneous character seems paradoxical. These considerations have raised controversies about its definition, physical meaning, measurement and application [START_REF] Mandel | Interpretation of Instantaneous Frequencies[END_REF][START_REF] Gupta | Definition of instantaneous frequency and frequency measurability[END_REF]. For a deeper insight on the notion of IF and its interpretation, we refer the reader to comprehensive review proposed by Boashash [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal[END_REF].

The filtered dipole is shown in Fig. 4.5 for the first laser cycles. We identify four different regions that we call attobursts, corresponding to the periodic HHG process as discussed before. We clearly see that the first attoburst is a chirped sinusoidal function, composed of only one time-evolving frequency. The second attobursts exhibits beatings resulting from the sum of the main attoburst and the contribution of the second return. As explained previously, the attobursts III and IV are the superposition of multiple returns of greater order. More precisely, the attoburst I is an "Intrinsic Mode Function" as introduced by Huang et al [START_REF] Huang | The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis[END_REF]: [START_REF] Cohen | Time-frequency Analysis[END_REF][START_REF] Huang | The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis[END_REF]. Indeed, it fulfills the "narrow band" requirement, i.e. the number of zero crossings are equal to the number of extrema [START_REF] Gupta | Definition of instantaneous frequency and frequency measurability[END_REF][START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal[END_REF][START_REF] Schwartz | Communication Systems and Techniques[END_REF]].

• It is "monocomponent"
• The upper and lower envelopes of the signal are symmetric with respect to zero.

We are thus in the conditions required to compute the IF for the first attoburst.

The definition of the IF arises with the transformation the real-valued dipole d(t) into an analytical complex function z d (t). The creation of an analytical function from a signal is not unique. However, Gabor showed that it can be uniquely constructed by the mean of a Hilbert tranform H d [START_REF] Gabor | Theory of communication[END_REF]: where:

z d (t) = d(t) + iH d (t) = a d (t)e iΦ d (t) ,
H d (t) = 1 π P d(t ) t -t dt , (4.2

.11)

with P the Cauchy principal value. As for the CWT, the Hilbert transform is a convolution product and is computed with the FFT algorithm. The modulus and phase of z d (t) are thus:

a d (t) = d 2 (t) + H 2 d (t) , Φ d (t) = arctan H d (t) d(t) . ( 4

.2.12)

As pointed out by Huang et al [START_REF] Huang | The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis[END_REF], the IF can be derived from the analytical dipole z d (t) through its Fourier transform:

FT[z d ](ω) = z d (t)e -iωt dt = a d (t)e i(Φ d (t)-ωt) dt . (4.2.13)
Using the stationary-phase method, the maximum contribution to FT[z d ](ω) is given by the frequency ω that satisfies:

d dt (Φ d (t) -ωt) = 0 . ( 4

.2.14)

The frequency verifying this equation is the IF:

ω IF (t) = dΦ d (t) dt . ( 4

.2.15)

The IF has by definition a single value for each time. It is thus relevant only for monocomponent signals. However, Eq. (4.2.10) and consequently Eq. (4.2.15) can be computed for any signal. As shown previously, the attoburst II is bicomponent, i.e. the sum of two monocomponent signals. The corresponding analytical dipole is the sum of two chirped complex amplitudes oscillating at two distinct frequencies ω 1 (t) and ω 2 (t). For simplicity reasons, we assume that the corresponding amplitudes a 1 and a 2 are constant in time:

z d (t) = a 1 e iω 1 (t)t + a 2 e iω 2 (t)t = a(t)e iΦ d (t) , (4.2

.16)

After straightforward manipulations, one can show that the derivative of the phase is [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal[END_REF]:

dΦ d (t) dt = Ω 1 + Ω 2 2 + Ω 1 -Ω 2 2 a 2 1 -a 2 2 a 2 1 + a 2 2 + 2a 1 a 2 cos [(ω 1 -ω 2 )t] , (4.2.17)
with

Ω i = ωi t + ω i , i ∈ [[1, 2]]. It oscillates at the beating frequency ω B (t) = ω 1 (t) - ω 2 (t)
, consequence of the interferences between the two components. 4.2 for harmonics 37 and 55, and compared with the harmonic emission times derived from SFA computations performed as in Sec. 3.3.1 and classical times obtained from the resolution of the Newton equation. As expected, classical times give terrible results, while SFA times are much better approaching the exact numerical times. Still, discrepancies up to 50 as are observed, which are of the same order as the typical timings we want to achieve in attoscience, for instance few tens of attosecond in the measure of delays in photoionization [START_REF] Klünder | Probing Single-Photon Ionization on the Attosecond Time Scale[END_REF][START_REF] Schultze | Delay in Photoemission[END_REF][START_REF] Shafir | Resolving the time when an electron exits a tunnelling barrier[END_REF]. These results are linked with the ones of Ref. [START_REF] Smirnova | Direct XUV Probing of Attosecond Electron Recollision[END_REF]. 4.1, we show the emission times for harmonic 37 in the plateau region and harmonic 55 close to the cutoff, for short (S) and long (L) trajectories. In the case of classical calculations which underestimates the cutoff, the emission times of harmonic 55 are not defined.

IF

We can use these oscillations to extract the timings of the second return. Since the HHG process is periodic, the main component is known from the first attoburst, i.e. ω 1 (t) is known in Eq. (4.2.17). By extracting the beating frequency ω B (t) one can then deduce ω 2 (t) = ω 1 (t)ω B (t), that is the IF of the second return, displayed in dashed lines in Fig. 4.6.

Figure 4.7 summaries the comparisons of the IF with SFA and classical computations for the first and the second return. We clearly see that the reconstructed recollision times of the second return are in qualitative agreement with either SFA or classical recollision times (which here are close together), particularly for the latest recollision times (t/T L > 2) because of the appearance of the third attoburst. Further extraction on the attoburst III of the recollision times of the third return is impossible because the three interfering chirped frequencies are difficult to separate by filtering. Moreover, these interferences cause several poles in the Hilbert transform (we can see the diverging behavior of the IF in Fig. 4.6), which would lead to important computational errors.

To summarize, we have designed a robust time-frequency method to extract quantitative and unequivocal harmonic emission times. It also allows us to extract qualitatively the timings of the second return.

Wigner-Ville distribution

Subsequently to its work on the IF, Ville proposed the so-called Wigner-Ville Distribution (WVD) [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal[END_REF]. For the dipole d, it is given by:

WVD[d](ω,t) = d t + τ 2 d t - τ 2
e -iωτ dτ. and:

WVD[d](ω,t)dt = |TF[d](ω)| 2 , (4.2.20)
which is the harmonic spectrum. While it is nicely interpretable for simple signals, it exhibits non-local interference cross-terms that may be hard to understand for more complex signals. Their physical meaning has often been questioned. However, they encode highly valuable information. To be able to understand the following results, we remind that we have chosen laser-related parameters. We recall that ω L is the laser frequency and T L = 2π/ω L the laser period.

these cross-terms patterns are found exactly in between two "real" spectral densities. Indeed, in Fig. 4.8 we represented the WVD of a compass signal, which is a timefrequency analogous of the compass state 8 8 named after the four cardinal points that the WVD displays.

defined by Zurek in Ref. [START_REF] Zurek | Sub-Planck structure in phase space and its relevance for quantum decoherence[END_REF]. We chose this very example for its pertinence to understand the WVD. The signal, noted C(t), is the sum of four Gabor atoms, i.e. four frequencies localized at different times:

C(t) = 4 ∑ j=1 e - (t-t j ) 2 2δ 2 +iω j t , (4.2

.21)

where δ = 12 a.u., and parameters t j and ω j are presented in Table 4.3. We see the interferences patterns in diagonal between two Gabor atoms. We also see at the center of the figure a chessboard pattern which corresponds to a coherent superposition of the interferences between the top and the bottom atoms and between the left and the right ones. Interference structures are smaller than the Gabor limit. Indeed, for two atoms distant by T , the separation between positive and negative values of the fringe pattern is ∆ω = 2π/T [START_REF] Zurek | Sub-Planck structure in phase space and its relevance for quantum decoherence[END_REF]. For the compass state, the distance between west and east atoms is T = 2T L leading to a fringe pattern in ∆ω = π/T L , and the distance between north and south atoms is Ω = 12ω L leading to a fringe pattern ∆t = 2π/Ω = T L /12. hence, the tile dimension A of the chessboard pattern is :

A = ∆t∆ω = π 12 < 1 2 , (4.2.22)
lower than the Gabor limit. Figure 4.9 presents the WVD of the same dipole than in the previous section. It reveals strong interference patterns, which quickly saturates on scale A = 4π 2 /T Ω, where T and Ω are the temporal and frequency extensions, respectively, containing the signal [START_REF] Zurek | Sub-Planck structure in phase space and its relevance for quantum decoherence[END_REF]. Though we believe that they encode valuable physical information, we are aware of several methods that have developed to remove these interferences: the smoothed-pseudo WVD (which is endowed with limited resolution such as the STFT) or reassignment methods [START_REF] Sheu | Exploring laser-driven quantum phenomena from a time-frequency analysis perspective: a comprehensive study[END_REF], and the optimal-kernel time frequency representation [START_REF] Jones | An adaptive optimal-kernel time-frequency representation[END_REF]. However, all these methods suffer from the choice of the parameters which is arbitrary by nature.

We observe similar bell-shaped structures except that they are twice as many as in the spectrograms (see Fig. 4. 3 or Fig. 4.4a). However, these structures are not equivalent. Following a half-cycle periodicity, we observe an alternating of "physical" attobursts, marked with the same numbers than before, and "virtual" ones denoted by v X-Y where X and Y are the nearest physical attobursts. The virtual bursts are the interference patterns of two consecutive attobursts. They encode destructive (constructive) interferences for even (odd) harmonics with negative (positive) values.

The intense interferences bellow harmonic 33 result from the intense low frequency part of the dipole. They can be removed, again by using a low-cut filter.

Here, we provide a zoom on attobursts I, v I,II and II of the WVD for harmonics 33 to 69 in Fig. 4.10. Interestingly, attoburst II exhibits harmonic structure. Indeed, it is the sum of the physical attoburst II and the virtual interferences between bursts I and III. Hence, information about past and future processes are encoded in attoburst II. As bursts I and III are the consequence of electrons ionized and propagating in the same direction, their interferences display both positive even and odd harmonics. They display a slight blue shift, as already observed by Kim et al [START_REF] Kim | Wigner time-frequency distribution of high-order harmonics[END_REF] by means of WVD. Following Eq. (4.2.20), the sum of the WVD over the laser cycles is the HHG spectrum. Consequently, we understand that the summation will add the physical bursts with the virtual ones: even harmonics will be killed while odd harmonics will be revealed.

Trajectory separation

Introduction

As discussed before, the process of HHG is characterized by a periodic release of electrons in the continuum. Two possible trajectories lead to the same kinetic energy when the electron recollides to the ionic core, with consequent emission of photons of same frequency. By definition, these short and long trajectories are related to different timings. However, the HHG spectrum is the coherent sum of both trajectory contributions, over all the generation cycles. Hence, if one wants to study dynamic effects on the HHG spectrum, it is useful to separate the short and long trajectory in order to examine individually their contributions.

Case of SFA

As explained in Sec. 3.3, the additional use of the saddle-point approximation on the SFA dipole leads to pairs of solutions that correspond to the short and long trajectories. Hence, the separation and the study of each contribution is straightforward.

Windowed analysis

In light of the previous section (Sec. 4.2), one can claim that direct separation can be achieved by windowing the dipole to select temporally either the short (t/T L < 1.425) or the long (1.425 < t/T L < 1.63) trajectory contributions (see . This method has been used, for example, by Śpiewanowski et al [START_REF] Śpiewanowski | Field-induced orbital distortion in high-order-harmonic generation from aligned and oriented molecules within adiabatic strong-field approximation[END_REF] to study two-center interferences from field-distorted orbitals. Yet, this procedure is similar to the STFT which performs a windowed FT. As a result, it ends up with nonphysically broadened features. As we want to study precisely small signatures in HHG spectra, time-window separation is not appropriate.

Absorbing conditions in TDSE propagation

It is known that the dynamics of the short and long trajectories differs. Hence, we can use their spatial properties to separate them and sort out their distinct contribution to the HHG phase. Classical computations (see Sec. 2.4.2) showed us that the excursion of the electrons in the continuum is characterized by a distance x α = E L /ω 2 [START_REF] Corkum | Plasma perspective on strong field multiphoton ionization[END_REF]. The short trajectories never go above x α while the long trajectories always cross this limit. Therefore, we first solve the TDSE with a numerical absorber as in Sec. 3.2.7, but placed at x α . It absorbs the electrons which go further away, corresponding to the long trajectories. As a consequence, the surviving part of the wave function is the one of the short trajectories. Thence, to recover the long trajectories, we compute the wave function in the same conditions without the numerical absorber at the position x α and subtract to it the wave function computed with the absorber. The remaining part thus corresponds to the long trajectories. to the dipole after being separated with an absorber. The absorber must be short enough to resolve spatially the short and long trajectory electrons at the expense of its efficiency. Hence, reflection of a small part of electrons beyond the cutoff is observed and is the cause of a non-physical cutoff enhancement. It turns out that this artifact is useful for our studies, as discussed later on.

Macroscopic propagation of the radiation in a gaseous medium

Experimentally, high-order harmonics are not generated from a single atom or molecule, but by focusing a laser beam onto a gaseous medium 9 9 Nowadays, HHG from liquids [START_REF] Kurz | High-order-harmonic generation from dense water microdroplets[END_REF][START_REF] Dichiara | An investigation of harmonic generation in liquid media with a mid-infrared laser[END_REF], bulk materials [START_REF] Ghimire | Observation of high-order harmonic generation in a bulk crystal[END_REF], nanoparticles [START_REF] Ganeev | High-order harmonic generation in nanoparticle-containing laser-produced plasmas[END_REF][START_REF] Ganeev | High-order harmonic generation in Ag nanoparticle-containing plasma[END_REF][START_REF] Shaaran | High-order-harmonic generation by enhanced plasmonic near-fields in metal nanoparticles[END_REF] or solids [START_REF] Der Linde | Generation of high-order harmonics from solid surfaces by intense femtosecond laser pulses[END_REF][START_REF] Dichiara | Scaling of High-Order Harmonic Generation in the Long Wavelength Limit of a Strong Laser Field[END_REF][START_REF] Vampa | Linking high harmonics from gases and solids[END_REF] is trendy. However, this is not the topic of this thesis.

. Hence the collective response of the atoms or molecules constituting the medium is observed. The spatiotemporal shape and the focusing of the beam, together with the highly non-linear response of each atom or molecule, consequently affect the phase-matching between harmonics and the final build-up of the spectrum.

Interestingly, as the phase of short and long trajectory contributions differs, the focusing of the laser beam allows in some configurations the separation of these trajectory contributions. Therefore, we will use this method to study each contribution. Moreover, and very importantly, this study allows us to verify at the same time whether our single-body conclusions can be experimentally observed within the macroscopic response.

We will use the model, first developed by L'Huillier et al [START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF][START_REF] Huillier | Propagation effects in high-order harmonic generation in rare gases[END_REF][START_REF] Huillier | Theoretical aspects of intense field harmonic generation[END_REF], which solves the Maxwell Wave Equation (MWE) in the Fourier domain. We follow the derivation proposed by Gaarde et al [START_REF] Gaarde | Macroscopic aspects of attosecond pulse generation[END_REF] in the light of the work of Geissler et al [START_REF] Geissler | Light propagation in field-ionizing media: extreme nonlinear optics[END_REF]. We would like to mention that similar work has been carried out by Puthumpally-Joseph et al [START_REF] Puthumpally-Joseph | Quantum Interferences in the Dynamics of Atoms and Molecules in Electromagnetic Fields[END_REF], applied for example to study dipole-induced electromagnetic transparency [START_REF] Puthumpally-Joseph | Dipole-Induced Electromagnetic Transparency[END_REF][START_REF] Puthumpally-Joseph | Theoretical analysis of dipole-induced electromagnetic transparency[END_REF].

Maxwell wave equation

We consider a laser beam propagating in the z direction, and having a revolution symmetry. It is thus pertinent to treat the problem in the cylindric coordinates (r, θ , z) as presented in Fig. 4.12, and we can drop the dependence in θ . The total electric field propagating in the medium is E(r, z,t) = E(r, z,t) + E h (r, z,t), conveniently separated into the sum of the laser field E and the harmonic radiated field E h . The MWE writes:

∇ 2 r + ∂ 2 ∂ z 2 - 1 c 2 ∂ 2 ∂t 2 E(r, z,t) = 1 ε 0 c 2 ∂ 2 ∂t 2 P(r, z,t), (4.3.1)
where c is the speed of light, ε 0 the vacuum permittivity 10 10 In atomic units, c = 137 and ε 0 = 1/4π, see Appendix A.1.

, and P(r, z,t) is the timedependent polarization of the medium. Let us note v g the group velocity of the laser pulse. The easiest way to solve Eq. (4.3.1) is to translate the spatio-temporal coordinate system in the moving frame at the velocity v g :

(r, z,t) -→ r, z = z,t = t - z v g , (4.3.2)
which implies:

∂ 2 ∂ z 2 = ∂ 2 ∂ z 2 - 2 v g ∂ 2 ∂ z ∂t + 1 v 2 g ∂ 2 ∂t 2 , (4.3.3) ∂ 2 ∂t 2 = ∂ 2 ∂t 2 . (4.3.4)
Hence, the MWE in the moving frame reads:

∇ 2 r - 2 v g ∂ 2 ∂ z ∂t + 1 v 2 g - 1 c 2 ∂ 2 ∂t 2 + S S S ∂ 2 ∂ z 2 E(r, z ,t ) = 1 ε 0 c 2 ∂ 2 ∂t 2 P(r, z ,t ). (4.3.5)
The Slowly Evolving Wave Approximation (SEWA) [START_REF] Brabec | Nonlinear Optical Pulse Propagation in the Single-Cycle Regime[END_REF] enables us to neglect the term in ∂ 2 z . The SEWA imposes that the amplitude and phase of the electric field

TRAJECTORY SEPARATION

do not change significantly over a distance comparable to the wavelength λ L = ω L /c of the pulse, i.e. ∂ 2 z E λ L E [START_REF] Geissler | Light propagation in field-ionizing media: extreme nonlinear optics[END_REF]. The SEWA must not be mistaken with the Slowly Varying Envelope Approximation (SVEA) which only requires that the amplitude of the pulse does not change significantly over λ L . In return, the duration of the pulse can be as short as one cycle in the SEWA contrarily to the SVEA [START_REF] Brabec | Nonlinear Optical Pulse Propagation in the Single-Cycle Regime[END_REF].

From now on, we drop the primes (z ≡ z and t ≡ t) for the sake of clarity. The MWE is solved in the Fourier domain [START_REF] Gaarde | Macroscopic aspects of attosecond pulse generation[END_REF][START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF][START_REF] Huillier | Propagation effects in high-order harmonic generation in rare gases[END_REF][START_REF] Huillier | Theoretical aspects of intense field harmonic generation[END_REF][START_REF] Geissler | Light propagation in field-ionizing media: extreme nonlinear optics[END_REF]:

∇ 2 r + 2iω v g ∂ ∂ z + ω 2 1 c 2 - 1 v 2 g Ē(r, z, ω) = - ω 2 ε 0 c 2 P(r, z, ω), (4.3.6)
where Ē(r, z, ω) and P(r, z, ω) are the FT of the electric field and the polarization, respectively.

Polarization of the medium The polarization regroups three terms of different physical origins. The first one is linear in E and represents the absorption and dispersion of the medium. The second one manifests the effect of the ionized electrons on the propagation and is related to plasmas, while the third one corresponds to the generation of high-order harmonics and is linked to the HHG dipole. Both are nonlinear. Consequently, the polarization writes:

P(r, z, ω) = ε 0 χ(ω) Ē(r, z, ω) + Pion (r, z, ω) + Pdip (r, z, ω), (4.3.7)
where the electric susceptibility χ is a complex number, connected to the complex refractive index n following:

n(ω) = 1 + χ(ω) = η(ω) + iκ(ω). (4.3.8)
Here, the real part η expresses the dispersion of the medium. The imaginary part κ is the extinction coefficient and manifests itself as the absorption of the medium. Thus, the MWE reads:

∇ 2 r + 2iω v g ∂ ∂ z +ω 2 n 2 (ω) c 2 - 1 v 2 g Ē(r, z, ω) = - ω 2 ε 0 c 2 Pion (r, z, ω) + Pdip (r, z, ω) . (4.3.9)
The ionization term of the source is computed with the introduction of a timedependent current density J such as:

∂ ∂t P ion (r, z,t) = J(r, z,t), (4.3.10)
where J(r, z,t) = J p (r, z,t) + J abs (r, z,t) according to Rae et al [START_REF] Rae | Detailed simulations of plasma-induced spectral blueshifting[END_REF]. Here J p is the plasma oscillation term defined by:

∂ ∂t J p (r, z,t) = N e (t)E(r, z,t), (4.3.11)
and:

J abs (r, z,t) = Γ(t)N e (t)I p E(r, z,t) E(r, z,t) 2 , (4.3.12)
is due to the energy loss from the laser due to ionization. In these expressions, Γ is the ionization rate, I p the ionization potential and N e the number of free electrons defined by:

N e (t) = N(z) 1 -exp - t 0 Γ(τ)dτ , (4.3.13)
where N(z) is the atom density of the gas jet. Regarding the ionization rate, on can use the Ammosov-Delone-Krainov (ADK) theory [START_REF] Ammosov | Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field[END_REF] for atoms, or the molecular-ADK (MO-ADK) [START_REF] Tong | Theory of molecular tunneling ionization[END_REF] as performed by Jin et al [START_REF] Jin | Medium propagation effects in high-order harmonic generation of Ar and N 2[END_REF][START_REF] Jin | Analysis of effects of macroscopic propagation and multiple molecular orbitals on the minimum in high-order harmonic generation of aligned CO 2[END_REF]. Gaarde et al [START_REF] Gaarde | Macroscopic aspects of attosecond pulse generation[END_REF] used conveniently normalized ADK rates relative to TDSE computations. However, we use directly the TDSE rates. We perform atomic or molecular simulations as in Sec. 3.2 and compute at each time t the probability of the electron to be bound as the sum of the probabilities of the electronic wave-function |Ψ(t) to be in the bound states |ϕ n of the potential. The ionization rate is therefore:

Γ(t) = d dt 1 -∑ n | ϕ n |Ψ(t) | 2 . (4.3.14)
While J abs is small in the usual conditions of HHG, J p can be important especially in highly ionized media. It expresses the oscillatory motion of the free electrons induced by the laser field and gives rise to variation of the refractive index in space and time. It is the cause of self-defocusing and phase modulation of the beam. As a consequence, it introduces supplementary phase mismatch during the generation and propagation of the harmonics. Using the FT properties, we have simply:

- ω 2 ε 0 c 2 Pion (r, z, ω) = 1 ε 0 c 2 FT ∂ ∂t J(r, z,t) (ω). (4.3.15)
The dipole source term is directly proportional to the HHG dipole d. In [START_REF] Gaarde | Macroscopic aspects of attosecond pulse generation[END_REF], it is computed within the SFA. Hence, to consistently take into account the ionization in the medium they set the proportionality coefficient equal to the number of remaining (non-ionized) atoms or molecules, i.e. N(z) -N e (t). Here, we compute the dipole with our 1D ab initio model following Sec. 3.2. Hence, the ionization is already consistently included 11 11 This is also the case if one uses an improved version of SFA which takes into account the depletion of the ground state [START_REF] Antoine | Theory of high-order harmonic generation by an elliptically polarized laser field[END_REF].

into d, and the proportionality coefficient is simply N(z). The radiated dipole depends on the local intensity seen by the atoms or molecule. It is thus a function of the spatial coordinates (r, z). We consider a linearly polarized electric field along the direction e x . The polarization of the radiated light is the same than the laser polarization. As a result: In the Fourier domain, we have then:

P dip (r, z,t) = N(z) d(r, z,t)e x .
Pdip (r, z, ω) = N(z) D(r, z, ω)e x , (4.3.17)
where D is the FT of the dipole, as defined by Eq. (3.2.47) in Sec. 3.2.8.

Practically, for each position z in the gas jet, we calculate the amplitude and phase of the harmonics by computing a HHG spectrum D(r, z, ω) for each electric field value at each position r of the beam.

Weak harmonic field and other approximations

We are going to take in consideration the following additional approximations. The first stands that the generated harmonic field E h is weak compared with the driving laser field E and thus does not affect it through the nonlinear source terms. Namely, E is not sensitive to any change in the harmonic field. Moreover, transparency of the plasma to the high-order harmonics is assumed such that the harmonic field is not subjected to the ionization term of the source. This allows one to separate Eq. (4.3.6) into two equations:

∇ 2 r + 2iω v g ∂ ∂ z + ω 2 n 2 (ω) c 2 - 1 v 2 g Ē(r, z, ω) = 1 ε 0 c 2 FT ∂ ∂t J(r, z,t) (ω), (4.3.18) ∇ 2 r + 2iω v g ∂ ∂ z + ω 2 n 2 h (ω) c 2 - 1 v 2 g Ēh (r, z, ω) = - ω 2 ε 0 c 2 N(z) D(r, z, ω), (4.3.19)
where we have projected on e x since all the vectors are zero in the other directions. Furthermore, since the gas density is low (the pressure of typical experimental gas jets is between 10 -100 Torr, which corresponds to 1.9 × 10 -7 -10 -6 atoms per (a.u.) 3 ), we can propose supplemental approximations. First, the dispersion of the laser field and the generated harmonics in the medium is neglected, i.e. η(ω) = 1 and η h (ω) = 1. As a result:

v g = c η + ω ∂ η ∂ ω = c. (4.3.20)
Secondly, as the laser intensity is very high, the number of absorbed photons of frequency ω L is a small fraction of the total. Hence, we neglect the absorption of the laser field, i.e. κ = 0. Nonetheless, as mentioned before, the harmonic field is weak, thus the absorption of generated photons by the medium cannot be neglected. Consequently, n h (ω) = 1 + iκ(ω). Note that most of generated frequencies are higher than the ionization potential of the constituent of the medium. Thence, we are in the photoionization regime: the absorption of one harmonic photon takes off an electron. However, we neglected this source of free electron in the expression of Pion again because the harmonic field is weak compared to the laser field. The fraction of photoelectrons is much lower than the tunnel-ionized electrons. For all frequencies κ(ω) 1 and thereby n 2 h (ω) 1 + 2iκ(ω). Finally, we need to solve:

∇ 2 r + 2iω c ∂ ∂ z Ē(r, z, ω) = 1 ε 0 c 2 FT ∂ ∂t J(r, z,t) (ω), (4.3.21)
∇ 2 r + 2iω c ∂ ∂ z + iω c α(ω) Ēh (r, z, ω) = - ω 2 ε 0 c 2 N(z) D(r, z, ω), (4.3.22)
where α is the absorption coefficient, defined by:

α(ω) = 2κ(ω) ω c . (4.3.23)
The values of the absorption coefficient are computed consistently with the TDSE by irradiating our 1D model with photons of frequency ω.

Numerical resolution

The propagation of Eqs. (4.3.21) and (4.3.22) along z on a grid

{z n } n∈[[0,N z -1]]
, is carried out using the Crank-Nicolson scheme with a source term. We defined N z the number of points and ∆z the grid step. Formally speaking, if we consider the following differential equation:

∂ u ∂ z = Au (4.3.24)
we show that the Crank-Nicolson scheme on the grid is exactly half of the sum of a fully explicit scheme and a fully implicit one [START_REF] Press | Numerical recipes, the art of scientific computing[END_REF]:

u n+1 -u n ∆z = Au n , (fully explicit) u n+1 -u n ∆z = Au n+1 , (fully implicit) u n+1 -u n ∆z = A 2 (u n+1 + u n ). (4.3.25)
Indeed, this expression is equivalent to:

1 - 1 2 A∆z u n+1 = 1 + 1 2 A∆z u n , (4.3.26)
which is the Crank-Nicolson shceme presented in Sec. 3.2.6. Now considering the non-homogeneous differential equation: In our case, the source term g depends non-linearly on f and is known only at n. Hence, we treat it fully explicitly, leading to the scheme:

∂ u ∂ z = Au + g.
1 - 1 2 A∆z u n+1 = 1 + 1 2
A∆z u n + g n ∆z.

∂ ∂ z Ē(r, z, ω) = ic 2ω ∇ 2 r A Ē(r, z, ω) - i 2ε 0 ωc FT ∂ ∂t J(r, z,t) (ω) g , (4.3.29) ∂ ∂ z Ēh (r, z, ω) = ic 2ω ∇ 2 r - α(ω) 2 A h Ēh (r, z, ω) + iω 2ε 0 c N(z) D(r, z, ω) g h . ( 4

.3.30)

As the system has a revolution symmetry (no dependence in θ ), the expression of the radial Laplace operator ∇ 2 r in cylindric coordinates is:

∇ 2 r = 1 r ∂ ∂ r + ∂ 2 ∂ r 2 . ( 4

.3.31)

The matrices A and A h expressed in the r-grid are thus tridiagonal, similarly to the Hamiltonian in Sec. 3.2.4. Hence, inversion of the matrices (1 -A∆z/2) and (1 -A h ∆z/2) is performed with the same algorithm as before.

Far-field profile The propagation of the MWEs is carried out along the gas jet. At the end of the propagation, z = z e , we obtain the near-field profile. Experimentally, the harmonics are detected in the far-field. We obtain the profile of the harmonic field at a point (r , z ) of the far-field using a Hankel transformation [START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF][START_REF] Jin | Medium propagation effects in high-order harmonic generation of Ar and N 2[END_REF][START_REF] Jin | Analysis of effects of macroscopic propagation and multiple molecular orbitals on the minimum in high-order harmonic generation of aligned CO 2[END_REF][START_REF] Salières | Coherence Control of High-Order Harmonics[END_REF][START_REF] Gaarde | Spatiotemporal separation of high harmonic radiation into two quantum path components[END_REF]:

Ēh (r , z , ω) = -i ω c Ēh (r, z e , ω) z -z e J 0 ωrr c(z -z e ) exp iω(r 2 + r 2 ) 2c(z -z e ) r dr, (4.3.32)
where J 0 is the zeroth order Bessel function. The propagation of the harmonic field in the vacuum from the near-field to the far-field enhances the separation between short and long trajectories.

Gaussian laser beam Eqs. (4.3.29) and (4.3.30) are solved for a real laser field as in [START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF]. The laser beam has a Gaussian shape focused at z = 0. The total expression of the electric field of the laser pulse is:

e z z r b w(z) w 0 w 0 √2 Fig. 4.13 Scheme of the Gaussian beam. E(r, z,t) = E G (r, z) f (t)sin(ω L t), (4.3.33)
where f (t) is the temporal envelope as defined in Sec. 3.2.5, and:

E G (r, z) = E L b b + 2iz exp - kr 2 b + 2iz , (4.3.34)
is the spatial Gaussian envelope, with k = ω/c the wave vector and b the confocal parameter [START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF][START_REF] Huillier | Propagation effects in high-order harmonic generation in rare gases[END_REF][START_REF] Huillier | Theoretical aspects of intense field harmonic generation[END_REF]. To calculate the intensity and phase of such a beam, we introduce the Rayleigh length z r = b/2, the beam waist:

w(z) = w 0 1 + z 2 z 2 r , (4.3.35)
where w 0 = 2z r /k is the minimal beam waist reached at z = 0, and the radius of curvature:

R(z) = z 1 + z 2 r z 2 .
(4.3.36) Figure 4.13 explicits the introduced quantities. After some manipulations, one can show that [START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF][START_REF] Huillier | Theoretical aspects of intense field harmonic generation[END_REF]:

E G (r, z) = E L w 0 w(z) exp - r 2 w 2 (z) exp iζ (z) -i kr 2 2R(z) , (4.3.37)
where ζ (z) = arctan(z r /z) is the so-called Gouy phase. The resulting laser intensity is:

I(r, z) = I L w 2 0 w 2 (z) exp - 2r 2 w 2 (z) , (4.3.38)
with I L = E 2 L the maximum peak intensity reached at the center of the beam.

Gas jet

We model the atom density of the gas jet with a truncated Lorentzian function [START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF]:

N(z) =        N 0 1 + 8 z -z c L jet if - L jet 2 < z -z c < L jet 2 0 elsewhere, (4.3.39)
such as L jet is its width and N 0 the peak density at the center z c of the jet. The introduction of the parameter z c allows us to change the position of the gas jet relative to the confocal point of the laser beam. This is crucial to separate short and long trajectories. Indeed, positioning the gas jet after the focus leads to short trajectories constructed on-axis and long trajectories almost killed, whereas positioning it before the focus leads to short trajectory contributions on-axis and long trajectory contributions constructed on a ring around the propagation axis [START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF][START_REF] Salières | Coherence Control of High-Order Harmonics[END_REF], as we can see in Fig. 4.14. It displays the far-field profiles for harmonics 13, 23 and 33 for a gasjet before and after the focus. We also propagated separately the short and the long trajectory contributions of each harmonic in order to identify the nature of each part of the far-field profile. We clearly see that in both cases, short trajectories are constructed on-axis, while long ones are constructed on a ring around the axis when the jet is before the focus, and almost destructed when the jet is after the focus. We see that the higher the harmonic order, the less efficient the trajectory separation. Indeed, the closer to the cutoff, the smaller phase-difference between trajectories.

Quantum path interferences

Introduction

While the dependence of harmonic intensity with the laser intensity has been first theoretically investigated by Lewenstein et al [START_REF] Lewenstein | Theory of high-harmonic generation by low-frequency laser fields[END_REF][START_REF] Lewenstein | Phase of the atomic polarization in high-order harmonic generation[END_REF] within SFA, it has been carried out experimentally for the first time by Zaïr et al [START_REF] Zaïr | Quantum Path Interferences in High-Order Harmonic Generation[END_REF]. The term Quantum Path Interferences (QPI) have been consequently proposed to denote the practice of scanning the harmonic intensity as a function of the laser intensity to observe interference patterns encoded by the interfering short and long quantum paths of HHG. Theoretical and experimental studies followed this work [START_REF] Auguste | Theoretical and experimental analysis of quantum path interferences in high-order harmonic generation[END_REF][START_REF] Holler | Ionization effects on spectral signatures of quantum-path interference in high-harmonic generation[END_REF][START_REF] Schapper | Spatial fingerprint of quantum path interferences in high order harmonic generation[END_REF], and most importantly it has been shown than molecular internal dynamics can be probed using QPI [START_REF] Zaïr | Molecular internal dynamics studied by quantum path interferences in high order harmonic generation[END_REF]. We will thus use QPI as an analysis tool to monitor dynamic properties within HHG.

Numerical QPI

Numerically, a number of HHG spectra are computed for a range of laser intensities, typically between the intensity for which the tunneling regime is just attained and the one corresponding to ionization above the barrier. Then each harmonic intensity can be plotted as a function of the laser intensity. Formally, following Eq. (3.2.47), we compute:

D I L (ω) = TF[d I L ](ω) = d I L (t)e -iωt dt, (4.4.1)
where the dipole d I L (t) is computed at laser intensity I L . We then map it into an intensity dependent function for each harmonic q of frequency ω q : D I L (ω q ) ↔ D q (I L ). As an illustration, Fig. 4.15 displays a typical QPI obtained with the atomic TDSE computations as presented in Sec. 3.2, for harmonic 27 (i.e. it displays |D 27 (I L )| 2 as a function of I L ). The regularization parameter a of the soft-Coulomb potential is set equal to 1 (see Eq. (3.2.5)). The laser frequency is ω L = 0.057 a.u. (Ti:sapphire). We chose a 2-cycle square-sine envelope (see Eq. (3.2.34)) for which only one attoburst is observed, in order to avoid multiple returns. Indeed, during the first laser halfcycle, the intensity is too weak and we are not in the tunneling regime. The electron is thus freed during the second half-cycle and then recombines with the ion during the third and the beginning of the fourth one. The electron released during the third half-cycle cannot recombine because the fourth half-cycle is too weak, and the fifth one is inexistent. For this reason, we do not observe further returns of the electron.

From the lowest laser intensity to 1.25×10 14 W.cm -2 , we observe an exponential growth of the harmonic intensity. Harmonic 27 is in the cutoff until I L = 1.25 × 10 14 W.cm -2 . Above this intensity, harmonic 27 lies in the plateau region. We observe clear interference patterns as the intensity grows. They manifest the presence of the short and the long trajectory contributions, and are not polluted by returns of greater order through the choice of the laser pulse. Above about 4.35 × 10 14 W.cm -2 , the overall harmonic intensity decreases, because ionization becomes very strong since we are in the barrier suppression regime.

Fourier transform and I-α analysis of the QPI

As we discussed it before, we observe in the QPI the beatings of the interfering short and long trajectories. They express the intensity dependence of the phase of the harmonics. To monitor it, one can examine the FT of the QPI signal for each harmonic: we observe the components of the signal as a function of the conjugate variable of the intensity which is usually called α. Hence, we know at which "frequency" the harmonic phase evolves at the function of intensity. Furthermore, as well as timefrequency analysis, one can perform windowed FT of QPI to resolve the evolution of α with intensity. This will call this method I-α transform and denote it T I-α . It reads: [START_REF] Krausz | Attosecond physics[END_REF] presents the I-α transform together with the FT of the 27th harmonic QPI. We observe two branches, one with α 1 corresponding to the short trajectories, and one with α 20 corresponding to the long trajectories. These branches get tighter and merge at I L 1.25 × 10 14 W.cm -2 , i.e. at the cutoff. Unlike the observations of Balcou et al [START_REF] Balcou | Quantum-path analysis and phase matching of highorder harmonic generation and high-order frequency mixing processes in strong laser fields[END_REF] at high intensities, the branch of the long trajectories gets closer to the one of the short trajectories. The origin of this shift is not well understood, but could be attributed either to the form of the laser pulse we have chosen or also to a blue-shift which occurs when ionization is high.

T I-α [D q ](I L , α) = D q (I L )w(I -I L )e -iαI dI,
In Refs. [START_REF] Gaarde | Spatiotemporal separation of high harmonic radiation into two quantum path components[END_REF][START_REF] Balcou | Quantum-path analysis and phase matching of highorder harmonic generation and high-order frequency mixing processes in strong laser fields[END_REF][START_REF] Gaarde | Quantum path distributions for high-order harmonics in rare gas atoms[END_REF], only the phase of the harmonic QPI is transformed. This allowed them to get rid off the strongly varying amplitude in the cutoff and resolve the component α beyond the cutoff, which displays a unique value attributed to a unique trajectory. However, we believe that the notion of electron trajectory beyond the cutoff is irrelevant. Indeed, the exponential decrease of the cutoff corresponds to the exponential tail of the fundamental state allowing recombination at different values than x = 0. The electron can cross the ionic core and be slightly more accelerated; finally, it may recombine at x > 0 with greater velocity values, and consequently emit photons of greater frequencies. This phenomenon has no connection with the attribution of a unique beyond-the-cutoff trajectory. Moreover, as a drawback of analyzing only harmonic phase, all the components are represented with the same amplitude. Hence the I-α analysis is corrupted by many different contributions to the phase that do not appear if we analyze the whole dipole.

Analytical model of QPI

We propose here a model to study analytically the QPI. Based on previous works [START_REF] Gaarde | Spatiotemporal separation of high harmonic radiation into two quantum path components[END_REF][START_REF] Schapper | Spatial fingerprint of quantum path interferences in high order harmonic generation[END_REF][START_REF] Balcou | Quantum-path analysis and phase matching of highorder harmonic generation and high-order frequency mixing processes in strong laser fields[END_REF][START_REF] Gaarde | Quantum path distributions for high-order harmonics in rare gas atoms[END_REF][START_REF] Hostetter | Semiclassical approaches to below-threshold harmonics[END_REF][START_REF] Kretschmar | Spatial contributions of electron trajectories to high-order-harmonic radiation originating from a semi-infinite gas cell[END_REF] and on our observations provided in the previous section, a given harmonic has two phase components depending on laser intensity I L , which correspond to the short and long trajectories. For a given harmonic q of frequency ω q , we can write:

D q (I L ) = ∑ j=s,l A j q (I L )e -iα j q (I L )I L , (4.4.4)
where A j q is the amplitude and α j q the conjugate intensity of the short ( j = s) and long ( j = l) trajectories. We model the two branches with: Here, I cut (ω q ) denotes the intensity for which harmonic q is at the cutoff, α cut is the constant, conjugated to the intensity, at and beyond the cutoff, "shared" by the two trajectories, and ∆I is introduced for a normalization purpose (chosen equal to 1 × 10 1 4 W.cm -2 ). Finally, α j plat (ω q ) is the asymptotic conjugated intensities of the plateau region:

α j q (I L ) =    α cut if I L < I cut (ω q ) α j plat (ω q ) + α cut -α j plat (ω q ) e -I L +I cut (ω q ) ∆I if I L ≥ I cut (ω q ).
α s plat (ω) = α s 0 + δ s ω ω L , (4.4.6) α l plat (ω) = α l 0 -δ l ω ω L , (4.4.7)
where δ s (δ l ) is a small parameter expressing the growth (decrease) of the asymptotic conjugated intensity α s 0 (α l 0 ) in the plateau for the short (long) trajectories. See Fig. 4.17 for an example of the shape of the conjugated intensity for three harmonics. The amplitude is assumed to be exponential in cutoff and constant in the plateau: The last analysis tool we used is the Wigner distribution, which currently finds applications in numerous branches of quantum physics. Developed by Wigner to appraise quantum corrections in statistical thermodynamics [START_REF] Wigner | On the quantum correction for thermodynamic equilibrium[END_REF], it allows the formulation of non-relativistic quantum mechanics in phase space. It obeys the quantum Liouville equation [START_REF] Bauke | Visualizing quantum mechanics in phase space[END_REF][START_REF] Donoso | Quantum Tunneling Using Entangled Classical Trajectories[END_REF][START_REF] Kull | Position-momentum correlations in electron-ion scattering in strong laser fields[END_REF], derived from the Schrödinger equation which operates on the wave-function. Hence, it is often related to classical mechanics as formulated in phase space by Hamiltonian mechanics, for which the classical probability distribution satisfies the Liouville's theorem. For this reason, the Wigner distribution is viewed as a probability distribution in phase space. However, it has the undesired property to result in negative values, and thus is often called a quasiprobability distribution.

A j q (I L ) =    A plat e - I L -I cut (ω q ) ∆I 2 if I L < I cut (ω q ) A plat if I L ≥ I cut (ω q ) ,

Definition

The Wigner distribution operates on the density matrix ρ = |Ψ Ψ| of a quantum system. It translates the wave-function of the system into phase space. We will compute it as a function of time, using the time-dependent density matrix ρ(t) = |Ψ(t) Ψ(t)|. In 1D, it is defined as:

W (x, p,t) = 1 π ρ x - y 2 , x + y 2
,t e ipy dy. The wave function |Ψ(t) being a pure state, it becomes:

W (x, p,t) = 1 π Ψ * x - y 2 ,t Ψ x + y 2
,t e ipy dy. We can therefore draw the parallel between this expression and the WVD as presented in Sec. 4.2.7 (see Eq. (4.2.18)). The Wigner distribution has the same interference properties as the WVD (see the compass signal in Fig. 4.8, analogous of the compass state for which the Wigner distribution has been computed in [START_REF] Zurek | Sub-Planck structure in phase space and its relevance for quantum decoherence[END_REF]). Hence, as difficult as it is to extract, the interferences encode valuable information about the dynamic of the system. All the conclusions provided in Sec. 4.2.7 hold here. All we need is to make the analogy between the Gabor limit and the time-energy uncertainty principle (recalled explicitly with h): Note that the two equations are simply identical if we identify ω = E/h. Finally, like the WVD, the Wigner distribution has the following projection properties: A complete list of the properties of the Wigner distribution is available in Ref. [START_REF] Bauke | Visualizing quantum mechanics in phase space[END_REF].

∆t∆ω ≥ 1 2 ≡ ∆t∆E ≥ h 2 .
W (x, p)d p = |Ψ(x)| 2 ,

SFA case

We propose here to derive the expression of the Wigner distribution in the SFA, starting from the SFA wave function given by Eq. (3.3.2). From Eq. (3.3.2), we write in 1D:

W (x, p,t) = 1 π e -iI p t ϕ 0 (x + y) + dk b * (k,t)e -ik(x+y) (4.5.6)
× e iI p t ϕ 0 (xy) + dk b(k,t)e ik(x-y) e 2ipy dy. It can thus be written as the sum of four terms:

W (x, p,t) = W 0 (x, p) +W 1 (x, p,t) +W 2 (x, p,t) +W free (x, p,t), (4.5.8)
where:

W 0 (x, p) = 1
π dy e 2ipy ϕ 0 (x + y)ϕ 0 (xy),

W free (x, p,t) = 1 π dy e 2ipy dk dk b * (k,t)b(k ,t)e i[(k-k )x-(k+k )y] . (4.5.12)
First, W 1 (x, p) is the Wigner function of the fundamental state (and does not depend on time). Secondly, W free (x, p,t) is the Wigner function of the free electron in the continuum. Finally, W a (x, p,t) and W b (x, p,t) are cross terms, and we can show they are complex conjugate, i.e. W * a (x, p,t) = W b (x, p,t). The expression of W free (x, p,t) given by Eq. (4.5.12) can be simplified:

W free (x, p,t) = 1 π dk dk b * (k,t)b(k ,t)e i(k-k )x dy e i[2p-(k+k )]y] δ (k+k -2p) (4.5.13) = 1 π dk b * (k,t)b(-k + 2p,t)e 2i(k-p)x , (4.5.14)
and with a change of variable k → p + k,

W free (x, p,t) = 1 π dk b * (p + k,t)b(p -k,t)e 2ikx , (4.5.15)
which is, interestingly, the inverse Wigner transform translating the momentum dependent continuum coefficients b in phase space. We can show numerically that the cross terms W a (x, p,t) and W b (x, p,t) are negligible relative to W 0 (x, p) and W free (x, p,t). Furthermore, the Wigner function of the ground state brings only information about the static system, while W free (x, p,t) encodes the electron dynamics in the continuum.

We will thus study only the last term. Identifying its expression as a convolution product (see Eq. (4.5.14)), its computation is performed using FFT in order to speed up the calculations.

Negative probability

The Wigner distribution exhibits negative and positive values. As a result, it is often given by the meaning of a quasiprobability distribution. The interpretation of the mind-blowing concept of negative probability has raised a lot of research and discussions in physics [START_REF] Bartlett | Negative probability[END_REF] (see also Ref. [START_REF] Burgin | Interpretations of negative probabilities[END_REF] and references therein) such as in finance [START_REF] Haug | Why so negative to negative probabilities?[END_REF]. Yet, as done in Refs. [START_REF] Burgin | Interpretations of negative probabilities[END_REF][START_REF] Haug | Why so negative to negative probabilities?[END_REF], we would like to quote Dirac's 1947 paper [START_REF] Dirac | Bakerian Lecture. The Physical Interpretation of Quantum Mechanics[END_REF]:

Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative sum of money, since the equations which express the important properties of energies and probabilities can still be used when they are negative. Thus, negative energies and probabilities should be considered simply as things which do not appear in experimental results.

The parallel with negative money made by Dirac provides intuitive understanding of negative probabilities. Indeed, in our monetary world governed by the laws of economics and finance, there is nothing more common than the use of negative money, and this notion is well accepted by all human beings. Actually, just as the total money in the world is positive, the Wigner function can never be only negative, and the positive values are always more numerous than the negative ones. This is, indeed, a result of the projection properties (Eqs. (4.5.4) and (4.5.5)). Hence, negative probabilities can only exist in presence of positive ones. States with negative probabilities can be viewed as local companions of states with positive probability, but at the end, none of them can be observed experimentally.

An extension of the Wigner distribution which is known to have only positive values and is in this way rigorously a probability distribution in phase space is the Husimi distribution [START_REF] Husimi | Some Formal Properties of the Density Matrix[END_REF][START_REF] Harriman | Some properties of the Husimi function[END_REF][START_REF] Mundarain | Husimi's Q(α) function and quantum interference in phase space[END_REF]. However, it is somehow a windowed Wigner transform, and as for the STFT, introduces a width on the computed quantities.

Wigner distribution as a wave-function

Recent work of Bondar et al [START_REF] Bondar | Wigner function's negativity demystified[END_REF][START_REF] Bondar | Wigner phase-space distribution as a wave function[END_REF] claimed to solve the ambiguity of negative probability in Wigner distribution. They showed that the Wigner distribution is actually a probability amplitude like a wave-function. Their demonstration is based on the Operational Dynamic Modeling (ODM) [START_REF] Bondar | Operational Dynamic Modeling Transcending Quantum and Classical Mechanics[END_REF], which is a general continuation of Koopman-von Neumann (KvN) classical mechanics into phase-space, and by defining appropriate operators and their commutation relations is rigorously linked to the Schrödinger equation. At the end, the Wigner distribution is solution of any intermediate ODM representation between classical KvN and the Schrödinger equation, hence plays the same role as a wave-function.

Signature of Coherence

In fact, as mentioned previously, the alternating between positive and negative values in some regions of the Wigner distribution is an expression of coherence between quantum states, and a signature of their interferences. They express the relevance of phase in quantum mechanics. Regarded as the convolution product of the offdiagonal terms of the density matrix, the Wigner distribution exhibits the coherence between states. The Wigner distribution of completely independent states (i.e. represented by a diagonal density matrix) do not present interference terms, thus has no negative value. Let us illustrate this statement with a simple example. Let us start with a system in a superposition of two orthogonal states |0 and |1 , described by the wave-function:

|ψ = a 0 |0 + a 1 |1 , (4.5.16)
where a 0 and a 1 are the respective amplitudes. In the position representation, the wave-function reads: ψ(x) = a 0 ϕ 0 (x) + a 1 ϕ 1 (x).

(4.5.17)

The density matrix ρ = |ψ ψ| in the basis {|0 , |1 } is simply: and the Wigner distribution (after trivial manipulations) writes:

ρ = |a 0 | 2 a * 0 a 1 a 0 a * 1 |a 1 | 2 .
W (x, p) = |a 0 | 2 W 0 (x, p) + |a 1 | 2 W 1 (x, p) + 2 π Re a 0 a * 1 ϕ * 1 x - y 2 ϕ 0 x + y 2 . (4.5.19)
The two first terms are composed of the Wigner distributions W 0 (x, p) and W 1 (x, p) of states |0 and |1 respectively, and the last term encodes interferences between the two states. Decoherence operates through the interaction of the system with the environment 12 12 We will not develop the theory of decoherence here. For a comprehensive reading about this topic, please refer to Ref. [START_REF] Zurek | Decoherence and the transition from quantum to classical -REVISITED[END_REF].

, and rapidly reduces the density matrix to:

ρ = |a 0 | 2 0 0 |a 1 | 2 , (4.5.20)
for which the Wigner distribution reads: The dissipation of coherence between the states is expressed by the vanishing of the off-diagonal terms of the density matrix and consequently the interference patterns in the Wigner distribution. Interestingly, the interference patterns have structures on the sub-Planck scale, as demonstrated by Zurek [START_REF] Zurek | Sub-Planck structure in phase space and its relevance for quantum decoherence[END_REF]. They are analogous to the structures observed in the Wigner-Ville transform below the Gabor limit, as discussed previously. Since they are inferior to the universal physical Planck constant h below which nothing can be experimentally observable, together with their potential negativity as discussed previously, these structures are often stated unphysical. Reconciliation between the Wigner distribution and our physically observable world can be achieved by convolving with a phase-space Gaussian (i.e. averaging the Wigner distribution over a physically observable Gaussian state of dimension larger than h). This leads to the Husimi representation [START_REF] Husimi | Some Formal Properties of the Density Matrix[END_REF][START_REF] Harriman | Some properties of the Husimi function[END_REF][START_REF] Mundarain | Husimi's Q(α) function and quantum interference in phase space[END_REF] which, because of its construction, no-longer encodes coherence.

W (x, p) = |a 0 | 2 W 0 (x, p) + |a 1 | 2 W 1 (x, p).
It has been showed that this sub-Planck scale (which is not universal but depends on the system) has great importance for decoherence [START_REF] Zurek | Sub-Planck structure in phase space and its relevance for quantum decoherence[END_REF]. Indeed a displacement of the system by that scale is sufficient to cause decoherence, unlike what has been stated before, that is to say that a displacement of at least h is needed. Therefore, even infinitely tiny, this scale has physical relevance. Interferences encode phase information that may not be observable but are essential in quantum mechanics. These considerations raised a lot of philosophical and theoretical work about information nature of the quantum, and led to the emergence of physics of information [START_REF] Zurek | Decoherence and the transition from quantum to classical -REVISITED[END_REF][START_REF] Landauer | Information is Physical[END_REF][START_REF] Toyabe | Experimental demonstration of information-toenergy conversion and validation of the generalized Jarzynski equality[END_REF][START_REF] Guillemant | Characterizing the transition from classical to quantum as an irreversible loss of physical information[END_REF]. Information may be described in other dimensions perpendicular to space-time, as needed for String Theory or Loop Quantum Gravity to unify quantum mechanics and Einstein's general relativity.

Chapter 5

Results

CHAPTER 5. RESULTS

Introduction

In this chapter, I shall present the different studies I carried out during my PhD. The order followed is not chronological but corresponds to our will that the reader gain a progressive insight on HHG and on the underlying quantum processes as the studies grow in complexity. First, we will present the results that we obtained by examination of the Wigner distribution in the framework of HHG (Sec. 5.2). Despite the apparent complex nature of the Wigner distribution and the fundamental questions it raises about the quantum, it provides a practical illustration of HHG in phase space. More precisely, it brings a clear picture of the dressing of the bound electronic state by the laser field. The two following studies, which investigate HHG in presence of a shape resonance such as in nitrogen molecule (Sec. 5.3) and interference effects in diatomic molecules (Sec. 5.4), will both question the role of dressing effects in HHG. Finally, by consistently adding an additional degree of freedom, vibrating diatomic molecules are considered in Sec. 5.6 in order to examine effects of the vibration in strong-field processes, in particular for light molecules.

Wigner distribution

Introduction

Besides the fundamental interrogations raised by the Wigner distribution, we will use it here in a more practical way, to study HHG and to question whether it can bring valuable insight on the underlying quantum processes. We will use it as a posttreatment tool to examine features of the wave-function in phase space.

The Wigner distribution has already been used to study strong-field processes such as tunneling [START_REF] Watson | Quantum signatures in the stabilization dynamics[END_REF][START_REF] Czirják | The Wigner function for tunneling in a uniform static electric field[END_REF][START_REF] De Bohan | Dynamique de l'interaction laser-atome: moment canonique et approximation du champ fort[END_REF][START_REF] Baumann | Wigner representation of ionization and scattering in strong laser fields[END_REF], dressing effects [START_REF] Watson | Quantum signatures in the stabilization dynamics[END_REF] and electron scattering [START_REF] Kull | Position-momentum correlations in electron-ion scattering in strong laser fields[END_REF][START_REF] Baumann | Wigner representation of ionization and scattering in strong laser fields[END_REF]. The Wigner flow [START_REF] Steuernagel | Wigner Flow Reveals Topological Order in Quantum Phase Space Dynamics[END_REF] has also been studied to monitor topological order in quantum processes.

More than an ad hoc theoretical tool, recent experimental achievements on manipulating quantum entanglement and monitoring decoherence with atoms and photons in cavities [START_REF] Kurtsiefer | Measurement of the Wigner function of an ensemble of helium atoms[END_REF][START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF][START_REF] Bertet | Direct Measurement of the Wigner Function of a One-Photon Fock State in a Cavity[END_REF][START_REF] Deléglise | Reconstruction of non-classical cavity field states with snapshots of their decoherence[END_REF] have allowed the reconstruction of the Wigner function [START_REF] Kurtsiefer | Measurement of the Wigner function of an ensemble of helium atoms[END_REF][START_REF] Deléglise | Reconstruction of non-classical cavity field states with snapshots of their decoherence[END_REF][START_REF] Smithey | Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum[END_REF]. In these experiments, positive and negative value were obtained, but not with natural observations. Indirect ways were needed to reconstruct the Wigner function. Thence, the experimental reconstruction of the Wigner function has been successfully used to monitor decoherence [START_REF] Deléglise | Reconstruction of non-classical cavity field states with snapshots of their decoherence[END_REF]. Nevertheless, its reconstruction has not been managed in strong-filed physics at the moment.

Wigner transform of the TDSE wave-function

First of all, we examine the Wigner distribution of the wave-function obtained from TDSE computations. We performed 1D atomic simulations for an electron in the ground state of the soft-Coulomb potential V at presented in Sec. 3.2.3, where the regularization parameter a is set to unity. The "atom", whose ionization potential is I p = 0.67 a.u., is perturbed by an intense IR laser field of frequency ω L = 0.057 (i.e. a wavelength of 800 nm as for an experimental Ti:sapphire laser). The intensity is I L = 3.5 × 10 14 W.cm -2 (E L = 0.1 a.u.), such as the Keldysh parameter equates 0.66. Therefore, we place our system in the tunnel regime, in which HHG is analyzed below. The x-grid size is 2L x = 200 a.u., and an absorber of length L abs = 20 a.u. is placed at the edges (such as it begins at |x| = 80 a.u.).

To provide a representation of HHG in phase space, we computed the Wigner It occurs continuously from panels d-h, and in the meantime the freed electronic wave-packet is accelerated and driven away. The fastest electronic part which reaches the edges is absorbed (arrow 2, panels j-n). Meanwhile, the electric field reverses sign, and slower electrons are brought back to the ion. Then, they go further away in the opposite direction (arrow 3, panels l-q), whereas tunnel ionization occurs also towards negative x (arrow 4, panels l-q). The newly freed electrons and the older part which has crossed the ion, then cannot be completely brought back because the electric field is not intense enough, and they slowly decelerate while being absorbed if they reach they negative edge (panels q-t). We also observe a late electronic part from the first tunnel ionization process that recollides during this period (arrow 5, panels q-s). Only a small part of the newly ionized electrons seems to come back to the ion (arrow 6, panels s-t) which is not sufficient to complete the full harmonic generation process.

The computation of the Wigner function for more cycles in a longer pulse would have increased its complexity due to the presence of interfering contributions in many regions of the phase space.

Monitoring tunnel ionization This representation gives a picture of the tunnel ionization step. In Ref. [START_REF] Czirják | The Wigner function for tunneling in a uniform static electric field[END_REF], from the observation of similar Wigner distribution and comparison with classical trajectories, it has been stated that tunneling electrons are released with a velocity greater than zero. Here, it seems to be also the case. This observation is in physical contradiction with the usual picture of tunneling process, for which it is known that the electrons are freed with a velocity equal to zero just after the barrier. However, these conclusions may be misled by interference artifacts and additional studies should be carried out to corroborate or invalidate them. Monitoring recollision However, the picture is not that simple for recombination due to rapidly interfering contributions and the spreading of the electronic wave-packet over time (the slowest electron are delayed from the fastest ones, resulting in the slanted rectilinear distribution observed in Fig. 5.2l-q). A smarter way to monitor recollision via the Wigner distribution is to watch it along p over time at x = 0, i.e. to plot W (0, p,t). This will give an idea of the momentum distribution of the electronic wave-packet at recollision times, i.e. harmonic emission times. Yet, the remaining population in the ground-state |ϕ 0 , which is maximum at x = 0 and much greater than the population of electrons in the continuum, covers all the representation and prevents us to examine the interesting contributions. Hence, we project the wave-function on the fundamental state and remove its contribution, such as remains only the rest of the wave-function. Namely, we compute:

W (p,t) = 1 π Ψ * - y 2 ,t Ψ y 2 ,t e ipy dy, (5.2.1)
where:

Ψ(x,t) = Ψ(x,t) -ϕ 0 |Ψ(t) ϕ 0 (x). (5.2.2)
Such computations are reported in Fig. 5.3. First, we periodically observe "blobs" every half-cycles (see the structure between t = 0.5 and t = 1 laser cycle). These contributions are maximum when the electric field magnitude is maximum, and disappear when the electric field is zero. As a matter of fact, they are the signature of the dressing of the ground-state by the laser field. The projection of the wave-function on the undressed fundamental state |ϕ 0 is not perfect when the electric field is high because of the dressing. The "blobs" are almost identical to the Wigner distribution of the first exited electronic states |ϕ 1 . Further projection on it and its removal reveal the presence of additional contributions, from higher excited states. Attempts to remove completely this bound-state has failed. Successive projection on the excited bound-states is not complete, which is why we tried to compute the dressed state with Floquet theory as performed in Ref. [START_REF] Wassaf | Théorie de l'ionisation des atomes en champ laser intense : étude des structures observées aux hautes énergies[END_REF]. We do not detail this theory here but refer the reader to Refs. [START_REF] Bai | Floquet Eigenstate, Phase Space Distribution and Entropy in the Atomic Stabilization Dynamics[END_REF][START_REF] Bai | Determination of Floquet states and quasi-energy by correlation function[END_REF][START_REF] Burke | R-matrix-Floquet theory of multiphoton processes[END_REF][START_REF] Chu | Threshold shift and above-threshold multiphoton ionization of atomic hydrogen in intense laser fields[END_REF][START_REF] Chu | Intense Field Multiphoton Ionization via Complex Dressed States: Application to the H Atom[END_REF][START_REF] Maquet | Stark ionization in dc and ac fields: An L 2 complex-coordinate approach[END_REF][START_REF] Millack | Hyper-Raman Lines Produced During High Harmonic Generation[END_REF][START_REF] Millack | Floquet analysis of wavefunctions of atoms interacting with a short, strong laser pulse[END_REF][START_REF] Sambe | Steady states and quasienergies of a quantum-mechanical system in an oscillating field[END_REF][START_REF] Shirley | Solution of the Schrödinger equation with a Hamiltonian periodic in time[END_REF]. However, we found a state which is adiabatically plugged into the complete TDSE wave-function, including continuum electrons. Hence, we cannot remove it to isolate only the continuum part of the Wigner function.

The other contributions of the Wigner distribution, the wave-like structure with lot of interferences, is thus attributed to the continuum part of the wave-function. Interestingly, ignoring the rapidly alternating interferences that are signatures of the coherence on a sub-Planck scale as explained above, we observe dominating contributions that we marked with a red arrow. They correspond to the trajectories of firstly returning electrons, with different momenta (i.e. with different velocities) and follow the periodicity of the laser field. By a continuation of these line-shaped trajectories, one can identify them following their path in the interferences, below the dominant trajectory (see the black arrows). Indeed, they correspond to further returning electrons with lower velocities, and thus lower energies, in accordance with classical computations (cf. Sec. 2.4.2) and with HHG spectrograms (cf. Sec. 4.2).

Wigner within SFA and saddle-point solutions

We computed the Wigner distribution of SFA wave-function, as explained in Sec. 4.5.3, in order to first examine whether SFA reproduces our TDSE observations, and second, if this condition is fulfilled, to allow comparison with the saddle-point solutions. The search of the stationary points in the action indeed enabled to find a stationary momentum p at solution of Eq. (3.3.14). The representation of p at as a function of recollision time t would allow to identify the leading contributions in the Wigner distribution.

Figure 5.4 presents the Wigner distribution at x = 0 of the free electronic part given by Eq. (4.5.15), together with the values of the stationary momentum p at . We chose an intensity of 5 × 10 14 W.cm -2 (E L = 0.12 a.u.). The laser has no envelope since SFA computations are not sensitive to the branching of the pulse, i.e. :

E(t) = E L sin(ω L t). (5.2.3)
The Wigner distribution is very similar to the one from the TDSE wave-function after removing the fundamental state. Interestingly, whereas excited bound-states are not explicitly included in SFA computations, the "blobs" corresponding to the dressing of the ground-state are still observed with an equivalent shape. Hence, though it cannot be described here via the temporary population of excited bound-states, the dressing is accurately included in SFA because of the coupling with the continuum.

Most interestingly, the stationary momentum p at , computed for the first electronic return and the further returns, allows us to identify the electron trajectories in the Wigner distribution. As expected, it is in exact concordance with the dominant paths that we may have identified within the bunch of interferences.

In fine, the Wigner distribution enabled us to have a great picture of tunnel ionization, questioning the value of the electron velocity after tunneling, such as in Ref. [START_REF] Czirják | The Wigner function for tunneling in a uniform static electric field[END_REF]. Secondly, the distribution at x = 0 as a function of time allowed us to identify trajectories of electrons that are driven back to the ion. In the meantime, it showed the signature of the dressing of the ground-state by the laser field in both TDSE and SFA.

Introduction

The photoionization of molecular nitrogen (N 2 ) has been comprehensively studied, both experimentally [START_REF] Plummer | Partial photoionization cross sections of N 2 and CO using synchrotron radiation[END_REF][START_REF] Lofthus | The spectrum of molecular nitrogen[END_REF][START_REF] Raoult | Ab initio approach to the multichannel quantum defect calculation of the electronic autoionisation in the Hopfield series of N 2[END_REF][START_REF] Dehmer | = 0 and 1 near threshold[END_REF][START_REF] Zubek | Near-threshold photoionisation studies of N 2[END_REF][START_REF] Cacelli | Gaussian-type-orbital basis sets for the calculation of continuum properties in molecules: The differential photoionization cross section of molecular nitrogen[END_REF] and theoretically [START_REF] Davenport | Ultraviolet Photoionization Cross Sections for N 2 and CO[END_REF][START_REF] Dehmer | Shape-Resonance-Enhanced Nuclear-Motion Effects in Molecular Photoionization[END_REF][START_REF] Lucchese | Studies of differential and total photoionization cross sections of molecular nitrogen[END_REF][START_REF] Dehmer | Shape Resonances in K-Shell Photoionization of Diatomic Molecules[END_REF]. In photoionization spectra, a large shape resonance from ionization channel X (i.e. ionization from the HOMO, labeled X 2 Σ + g ) has been predicted theoretically [START_REF] Davenport | Ultraviolet Photoionization Cross Sections for N 2 and CO[END_REF][START_REF] Dehmer | Shape-Resonance-Enhanced Nuclear-Motion Effects in Molecular Photoionization[END_REF][START_REF] Lucchese | Studies of differential and total photoionization cross sections of molecular nitrogen[END_REF] and confirmed experimentally [START_REF] Plummer | Partial photoionization cross sections of N 2 and CO using synchrotron radiation[END_REF].

More recently, the shape resonance in N 2 has been observed and angularly resolved with HHG [START_REF] Bertrand | Revealing the Cooper minimum of N 2 by Molecular Frame High-Harmonic Spectroscopy[END_REF][START_REF] Ren | Measuring the angle-dependent photoionization cross section of nitrogen using highharmonic generation[END_REF][START_REF] Jin | Intensity dependence of multiple orbital contributions and shape resonance in high-order harmonic generation of aligned N 2 molecules[END_REF]. As part of collaborative work with the team of Pascal Salières, CEA Saclay, we developed a model to study HHG in N 2 with emphasis in the phase signature of the shape resonance. Indeed, Antoine Camper, during his PhD, developed techniques for angularly resolved high-harmonic phase spectroscopy called MAMMOTH (for Mixed Approaches for the MeasureMent Of the Total Harmonic phases) and CHASSEUR (for Combined Harmonic Attosecond Spectroscopy by two-Source EUv interferometry and RABBIT) [START_REF] Camper | Spectroscopie de phase multi-dimensionnelle de l'émission attoseconde moléculaire[END_REF], the latter being a combination of TSOIN (for Two-Source Optical INterferometry) [START_REF] Smirnova | High harmonic interferometry of multi-electron dynamics in molecules[END_REF][START_REF] Zhou | Molecular Recollision Interferometry in High Harmonic Generation[END_REF][START_REF] Camper | High-harmonic phase spectroscopy using a binary diffractive optical element[END_REF] and RABBIT (for Reconstruction of Attosecond Beating by Interference of Two-photon Transition) [START_REF] Véniard | Phase dependence of (N + 1)-color (N > 1) IR-UV photoionization of atoms with higher harmonics[END_REF][START_REF] Muller | Reconstruction of attosecond harmonic beating by interference of two-photon transitions[END_REF]. These techniques enable both spectrally and angularly resolved phase cartography of molecular species and have been applied to N 2 as a proof of principle.

In the experiments, a particular phase signature has been attributed to the shape resonance, that we could confirm theoretically with a simple model.

TDSE computations

We performed 1D TDSE simulations on a very simple model of N 2 with shape resonance. We used the potential V B as expressed by Eq. (3.2.7). To clarify the form of this potential, note that we only want to appraise the effect of shape resonance in the harmonic phase and avoid any competing effect. As it is known that two-center interferences in diatomic molecules lead to structural phase-jumps in the spectrum (and that we will closely investigate in the next section), we model N 2 with only one center. The shape resonance is introduced with tailored barriers above ionization threshold, creating a pseudo-bound state at positive energy with an inherent bandwidth ∆E (i.e. a lifetime τ ∼ 1/∆E).

As presented in Sec. 3.2.4, where we computed the DOS, we used (in a.u.) a = 0.8975, L 1 = 1.25, L 2 = 2.395, V 0 = 0.8 and n = 4, so that the first pseudo-bound state is observed at about 15 eV above ionization threshold ad has a bandwidth ∆E 10 eV (see Fig. 3.4), according to the quantum chemistry calculations performed by Lucchese et al [START_REF] Lucchese | Studies of differential and total photoionization cross sections of molecular nitrogen[END_REF].

In Fig. 5. 5 we display the HHG spectrum obtained with a 1-8-1 trapezoidal laser pulse of frequency ω L = 0.0456 (wavelength 1 µm) and peak intensity I L = 2.86 × 10 14 W.cm -2 (E L = 0.09 a.u.). We observe the typical shape of a harmonic spectrum. The cutoff is expected at harmonic 79, as we can see it in the figure. However, a strong enhancement of the harmonic yield is found in a large region around harmonic 25. This harmonic order corresponds to a photon energy of 25ω L = 1.14 a.u. = 31 eV, i.e. about 15 eV above ionization threshold (I p = 15.58 eV for N 2 ). Thence, this is the signature of the shape resonance. We chose a longer wavelength than the Ti:sapphire laser in order to have a greater harmonic separation between the ionization threshold (here at 12.5ω L ) and the resonance.

We can provide a simple explanation of the harmonic enhancement. Indeed, elec- trons that recollide with the ion with the energy of 15 eV are trapped in the pseudobound state, which decays slowly over time, allowing the recombination with the ground-state at different times over a long duration, strongly improving the overall recombination yield. These results are in accordance with the work of Tudorovskaya et al [START_REF] Tudorovskaya | High-order harmonic generation in the presence of a resonance[END_REF] where shape resonances have been studied with similarly tailored potentials.

Spectral phase and transition dipole matrix element

We can access the harmonic phase by computing the phase of the FT of the dipole. As we explained it, the phase evolves swiftly along the harmonic order, due to the electron dynamics imprinted in the harmonic emission. Indeed, harmonic emission times correspond to the electron recollision times and follow the attochirp. The effects of the shape resonance on the harmonic phase are expected to be small compared to the contribution of the attochirp. Hence, to appraise them, we need to remove the attochirp contribution. As it depends only on the ionization potential and the laser field, we remove from the harmonic phase obtained in a model of N 2 the harmonic phase of a reference atom of identical ionization potential obtained with the same laser pulse. We model our reference atom with the soft-Coulomb potential V at , with parameter a = 1.20865 to have exactly the same ionization potential. The value of a is significantly different than the one used for the potential V B (a = 0.8975) because the high and large barriers close to the nucleus affect importantly the position of the ground state relative to the same soft-Coulomb potential without the barriers.

To identify the phase signature of the shape resonance, we compared our results with the phase of the recombination dipole matrix element ϕ 0 |x| χ E -as it encodes the harmonic emission -following the work of Le et al [START_REF] Le | Extraction of the species-dependent dipole amplitude and phase from high-order harmonic spectra in rare-gas atoms[END_REF] on rare gases Ar, Xe and Ne. The continuum state χ E is the exact scattering wave at energy E, computed with the shooting method which allows one to obtain continuum states at arbitrary large distances without restriction due to a spatial box and the underlying bounding artifacts. Briefly, it relies in the reformulation of the TISE as follows:

Ĥ0 |χ E = E|χ E [recalling Eq. (3.2.9)] ⇔ - 1 2
∂ 2 ∂ x 2 +V (x) χ E (x) = Eχ E (x) (5.3.1)
⇔

χ E (x) = -2 E -V (x) χ E (x), (5.3.2)
which is solved as a differential equation with "initial" conditions, using the 4th order Runge-Kutta algorithm (RK4) (see Appendix A.3). The scattering waves are real valued [START_REF] Gaillac | Attosecond photoemission dynamics encoded in real-valued continuum wave functions[END_REF] and orthonormal in energy in the sense of the Strömgren's normalization procedure [START_REF] Seaton | The Determination of Phases of Wave Functions[END_REF] (see Appendix A.4). To obtain the even scattering wave at a given energy E = k 2 /2, where k is its momentum, we use the "initial" conditions χ E (x = 0) = 1 and χ E (x = 0) = 0. Conversely, for the odd scattering wave, we use χ E (x = 0) = 0 and χ E (x = 0) = 1. As the ground-state ϕ 0 is even, the involved scattering wave is odd. The phase of the recombination dipole matrix element ϕ 0 |x| k is deduced from the asymptotic phase-shift η(E) (for x → ∞) between a given scattering wave χ E and a reference scattering wave χ ref E . It can be computed with the help of the Wronskian functional W[ f , g] = f gg f as [START_REF] Joachain | Quantum Collision Theory[END_REF][START_REF] Messiah | Quantum Mechanics -Two Volumes Bound as One[END_REF]:

η(E) = -arctan W[χ E , χ ref E ] W[χ E , χ ort E ] , (5.3.3)
where a complementary wave function χ ort E , orthogonal to χ ref E , is needed [START_REF] Gaillac | Attosecond photoemission dynamics encoded in real-valued continuum wave functions[END_REF] (see Appendix A.5). Intuitively, the reference χ ref E is chosen as the odd scattering wave at energy E associated with the same soft-Coulomb potential without the barriers (i.e. V at with a = 0.8975). Its orthogonal counterpart χ ort E is implicitly the even scattering wave at energy E in the reference potential. Figure 5.6a displays the potential V B and the reference with their corresponding scattering waves at 0.84 eV to provide an illustration of the phase-shift η(E), and Fig. 5.6b shows the transition dipole matrix element along with the phase-shift η(E), and the previously computed DOS as a comparison. The phase-shift η(E) features a large phase-jump of almost π, characteristic of a resonant state of large energy width.

In order to follow continuously the phase variations along the spectrum, we use a 2-cycle sin 2 pulse, for which only one emission burst is observed. The harmonic phase difference obtained at peak intensity I L = 2.86 × 10 14 W.cm -2 is presented in Fig. 5.7 along with the phase-shift η(E). Despite the spiky structures, we observe a very good agreement of the overall harmonic phase with η(E) between harmonics H 19 and H 85 . Below H 19 it is difficult to extract meaningful phase for two main reasons. First, the harmonic yield is strongly reduced, as we can see in Fig. 5.5. We attribute it to the difficulty of the electron to recombine in presence of the barriers below the resonant state. Secondly, harmonics below the ionization threshold are mainly emitted by multiphoton processes, hence phase-normalization with the reference is inappropriate in this region. The same applies beyond H 85 , after the cutoff, where the harmonic intensity is strongly reduced. The spikes are attributed to the interferences between short and long trajectories. We see here the need to separate short and long trajectory contributions, allowing, in the meantime, to question whether the shape resonance phase-jump behave identically for both trajectories.

Trajectory separation

First of all, we used absorbing conditions at x α = E L /ω 2 and compared in Fig. 5.8a the phase difference between N 2 and the reference for the short and the long trajectories with the phase of the transition dipole matrix element as computed previously. We used the same laser parameters as in the previous section. Notwithstanding the residual oscillations due to the imperfect separation, the agreement is very good for the long trajectories. For the short ones, the phase-jump is also very close to the phase of the transition dipole matrix element, yet with consequent deviations below H 19 that we can attribute, for the moment, to the same reasons as exposed previously.

Secondly, we performed a macroscopic propagation of the harmonic radiation in a gaseous medium solving the MWE. The gas jet peak pressure is 10 Torr and its temperature is 75K, such as the peak atomic density is 1.9 × 10 -7 at/a.u. 3 . Its length is set to L jet = 1 mm. The confocal parameter of the laser beam is b = 5. To separate short and long trajectory contributions, we placed the center of the gaz jet 1.5 mm before the focus (i.e. z c = -1.5 mm). Once again, the same laser parameters as before are used. However, the above mentioned difficulties to extract long trajectories, especially for high harmonic orders, lead us to use a 1-8-1 trapezoidal pulse to avoid deformation of the attochirp. Consequently, we were only able to extract the phase at the odd harmonic orders for a few harmonics. These results are shown in Fig. 5.8b. Once again, the phase-jump is well-reproduced.

W.cm ---2 2 2 .
The last results are of great importance. We can conclude that the phase signature of the shape resonance survives the propagation in the gaseous medium, and can thus be observed experimentally. Moreover, the fact that it behaves identically for short and long trajectories validates the Quantitative Rescattering Theory (QRS) [START_REF] Le | Theory of high-order harmonic generation from molecules by intense laser pulses[END_REF][START_REF] Le | Quantitative rescattering theory for high-order harmonic generation from molecules[END_REF][START_REF] Le | Quantitative rescattering theory of high-order harmonic generation for polyatomic molecules[END_REF] in this case, as used in [START_REF] Le | Extraction of the species-dependent dipole amplitude and phase from high-order harmonic spectra in rare-gas atoms[END_REF] to explain the extraction of spectral phase for rare gases Ar, Xe, and Ne. The key argument of this theory is that the dipole can be factorized in two terms: the electronic wave-packet, encoding the dynamics, and the transition dipole matrix element encoding the structure of the system. The former is common to all species while the latter is specific to each one. Therefore, one can compute exactly the spectrum for a known atomic reference, for example with the resolution of the TDSE for which the electronic dynamical part is exact, and then obtain the HHG spectrum for more complex species by computing only their transition dipole matrix elements or using experimental photoionization cross sections.

Intensity dependence of the shape resonance

In order to determine whether the use of QRS is suitable for shape resonances, we studied the spectral dependence of the resonance with laser intensity, in amplitude and phase, in order to find eventual dressing effects. Figure 5.9 compiles the harmonic intensity and phase of short and long trajectories around the resonance for laser intensities 1 -5 × 10 14 W.cm -2 , obtained with a 2-cycle sin 2 pulse. The trajectories have been separated with absorbing conditions.

Regarding the intensity of the harmonic spectra (Fig. 5.9a-b), we observe residual oscillations. They are attributed to presence of trajectories that are not perfectly separated, which small contributions lead to constructive and destructive interferences adding to the resonant peak. Hence, taking the maximum to find the position of the resonance may be misleading -the small shifts that we would observe are attributed to persistent interferences.

Here, the inflection point of the phase-jump allows a better evaluation of the position of the resonance, which does not seem to move with the intensity for the short and long trajectories (cf. Fig. 5.9c-d). However, for the short trajectories, we observe that the phase-jump is not well reproduced for the low harmonic part, i.e. between H19 and H25. To be sure that this is not an artifact of the separation method, using absorbers, we also studied the phase-jump as a function of laser intensity after propagation of the radiation in a gaseous medium. We extracted the spectral phase for the short and long trajectories, presented in Fig. 5.10. Again, for the long trajectories, the phase-jump does coincide with the phase of the transition dipole matrix element, while the shape for the short trajectories is similar to the one extracted with absorbing conditions. As a consequence, the phase-jump for the short trajectories is not distorted by the method of trajectory separation but by physical effects that depend on the intensity of the laser.

To clarify this question, we used the R-box method [START_REF] Maier | A spherical-box approach to resonances[END_REF]. This method of choice to study resonances relies on the computational artifact that the number and the density of continuum states described by the diagonalization of the Hamiltonian on a grid -of finite dimension by nature -is directly related to the number of points in the grid and its size. The larger the grid size, the denser the distribution of the continuum states. However, resonant states are independent on the grid size as long as it is sufficiently large to describe their bounded part. When representing the computed continuum energies while increasing the size of the grid, we observe avoided cross- ings at resonance energies. Thence, we can extract energy and width of resonances from these avoided crossings. The advantage of this method compared to the DOS is that it does not rely on the symmetry of states. Hence, it can be used for asymmetrical potentials, such as here, allowing to find resonance energies and widths in presence of an electric field.

As an illustration, Fig. 5.11 displays continuum energies as a function of grid size for the unperturbed potential V B and for two laser intensities (3 and 5 × 10 14 W.cm -2 . For the unperturbed potential, we observe the bound states, Rydberg states and the shape resonance at the expected energy. When the laser intensity is not zero, the bound states are coupled with the continuum, acquire a finite lifetime and behave like resonances. We observe that the ground state does not move significantly, while other bound states and Rydberg states are dramatically shifted above ionization potential: the laser induces resonances in the continuum. In case b, it seems that the shape resonance is split in two. In fact, this is the effect of a light induced resonance, found around the energy of shape resonance.

Table 5.1 compiles the energy of the ground state and the resonance as a function of laser intensity in the range 0-5×10 14 W.cm -2 , obtained at the inflection points of the continuum energy lines. Aside accidental values, due to the presence of induced resonances, the energy of the shape resonance does not vary dramatically with the laser intensity. The mean value for all intensities is 0.5793 a.u., i.e. 0.086 a.u. (1.5%) greater than the value at I L = 0, indicating that the resonance slightly shifts towards higher energies with laser intensity. This shift is of the order of the magnitude of the ground-state Stark shift, -0.076 a.u., but in opposite direction. In total, at high electric field values, electrons trapped in the resonant state recombine with the ground state emitting a photon of higher energy, up to 0.162 a.u. (∼ 3.5ω L ) at I L = 5 × 10 14 W.cm -2 . Note that these values are much smaller than U p , which characterizes the change of ionization potential in the presence of a strong laser field.

To study dressing effects in HHG, on either the ground-state or the resonance, one must take into account the magnitude of the electric field at recombination time, i.e. when the harmonic is emitted, which can be different from recollision time. Indeed, the electron, which recollides with the ion at a given time, can be trapped into the resonant state, and thus, can recombine later, along the decay of this pseudo bound-state characterized by its life-time. In Fig. 5. [START_REF] Dudovich | Measuring and controlling the birth of attosecond XUV pulses[END_REF] we depicted these consider- ations, where the decay of the resonance is sketched by vanishing purple shades. Hence, the harmonic intensity at the resonance is a coherent sum of harmonic emissions over different times. And consequently, these emissions occur for different electric field values, as we can see in Fig. 5.12. As we used a 2-cycle sin 2 laser pulse, electrons of energy E res recollide in presence of a high electric field magnitude for the short trajectories, in presence of very low magnitude for the long trajectories.

In this way, short trajectories must be more sensitive to dressing effects than long trajectories. This may explain that the amplitude peak in the short trajectory contributions (cf. in Fig. 5.9a), underneath residual oscillations, tends to shift to higher energies, while in average, no such effect seems to be observed in the long trajectory contributions. Whether these dressing effects or the presence of induced resonances, or both together, are the cause of additional phase that distort the phase-jump for the short trajectories, cannot be answered simply with this study and should be inquired closely with other methods.

To summarize, dressing effects on the shape resonance are of the order of the Stark-shift of the ground state. The final shifts are small compared to the energy width of the resonance, even at high laser intensities, but still correspond to shifts up to few harmonic orders in HHG spectra. Furthermore, the shape of the phase-jump for the short trajectory contributions is rather sensitive to the intensity of the laser. Hence, the dressing effects cannot be neglected, and therefore, when working at high intensities, the QRS fails for systems presenting shape-resonances. However, it can be used, with care, for small intensities. In the next section, we will see that the QRS is not appropriate for all laser intensities in the case of two-center interferences. Purple shades scheme the recombination at different time following the decay of the resonance.

Introduction

We explained that each harmonic of the spectrum is mainly emitted by two possible trajectories of the recolliding electron: the short and the long ones. Both lead to the same kinetic energy but with different timings when the electron recollides with the parent ion. The long trajectories correspond to electrons emitted before and recolliding after the short ones. We saw that, treated in a quantum theory, they interfere and shape the total spectrum with constructive and destructive patterns. In the case of diatomic molecules, one may envision that electrons ionized from one center and recombining to an other will form additional classes of trajectories. Therefore, the structure of the system is encoded through additional interferences in the spectrum, that are reminiscent to Young's two-slit phenomenon.

In a standard SFA approach, the structure of the molecule was formerly introduced via the transition matrix elements, whereas the electron dynamics remains atomic [START_REF] Zhou | Alignment dependence of high-order harmonic generation from N 2 and O 2 molecules in intense laser fields[END_REF][START_REF] Zhou | Role of molecular orbital symmetry on the alignment dependence of high-order harmonic generation with molecules[END_REF][START_REF] Kamta | Three-dimensional time-profile analysis of high-order harmonic generation in molecules: Nuclear interferences in H + 2[END_REF]. For diatomic molecules, destructive interferences are observed in the harmonic spectrum, which correspond to the zeros of the recombination dipole matrix element. This results in a discontinuous phase-jump of exactly ±π.

However, this behavior does not compare well with experimental data [START_REF] Wagner | Extracting the phase of high-order harmonic emission from a molecule using transient alignment in mixed samples[END_REF][START_REF] Boutu | Coherent control of attosecond emission from aligned molecules[END_REF], nor with ab initio computations such as the exact resolution of the TDSE [START_REF] Lein | Interference effects in high-order harmonic generation with molecules[END_REF][START_REF] Lein | Role of the Intramolecular Phase in High-Harmonic Generation[END_REF][START_REF] Van Der Zwan | Two-center interference and ellipticity in high-order harmonic generation from H + 2[END_REF]. The phase-jumps at the destructive interferences are found far from being sharp: they are smooth, less than π and cover several harmonics.

More recently, a method to account for the effect of the molecular structure on the electron dynamics in SFA computations, called "molecular SFA", has been proposed by Chirilȃ and Lein [START_REF] Chiril | Strong-field approximation for harmonic generation in diatomic molecules[END_REF] and closely investigated by Faria [START_REF] Faria | High-order harmonic generation in diatomic molecules: A quantum-orbit analysis of the interference patterns[END_REF]. It relies on the separation of the dipole into four terms. Each one exhibits an additional phase incorporated to the electronic action. The SPA is carried out for each term and leads to four groups of possible electron trajectories (with their subset of short and long paths). They correspond to the ionization from one center and the recollision to either the same or the other one. To our knowledge, the shape of the phase-jumps in such calculations has not been studied or addressed, except in Ref. [START_REF] Etches | Two-center minima in harmonic spectra from aligned polar molecules[END_REF] where similar computations have been performed in the case of CO 2 . Interestingly, the phase-jumps are smoothed, which is what we will confirm with our simulations.

Earlier attempts to explain such a smooth behavior based either on the full integration of the SFA [START_REF] Chiril | Explanation for the smoothness of the phase in molecular high-order harmonic generation[END_REF] or on describing the continuum with Coulomb corrections [START_REF] Ciappina | Influence of Coulomb continuum wave functions in the description of high-order harmonic generation with H + 2[END_REF] have been proposed. These explanations had to be incomplete, since molecular SFA uses the saddle-point approximation and plane-waves to describe the continuum and still exhibit smoothed phase-jumps. Other attempts based on the computation of SFA with field distorted orbitals display smoothed phase-jumps [START_REF] Śpiewanowski | High-order-harmonic generation from field-distorted orbitals[END_REF][START_REF] Śpiewanowski | Field-induced orbital distortion in high-order-harmonic generation from aligned and oriented molecules within adiabatic strong-field approximation[END_REF].

We propose here a complete examination of the phase of two-center interferences in homo-nuclear diatomic molecules. Beyond the smoothing only, we observe and explain a peculiar behavior which has not been characterized before. We demonstrate that the value of the electric field at the recombination plays a crucial role in the shape of the phase-jump through dressing effects.

TDSE computations

We first performed 1D-TDSE computations on diatomic molecules with potential V dia (see Eq. (3.2.6)) for different internuclear distances R. For each value R the regularization parameter a is adjusted such as the desired ionization potential is kept constant. We have chosen the one of H 2 : I p = 0.567 a.u = 15.43 eV [START_REF] Watanabe | Ionization potentials of some molecules[END_REF].

By the means of spatial separation with absorbers we investigated the two-center interference signature in both short and long trajectory contributions to HHG spectrum. The results are presented in Fig. 5.13. We have reported intensity and phase of short and long trajectory contributions for several internuclear distances between 1.4 and 1.7 a.u., generated with a 2-cycle sin 2 laser pulse at frequency ω L = 0.057 a.u. (Ti:sapphire) and peak intensity I L = 2.55 × 10 14 W.cm -2 . Due to the normalization with a reference atom, for the same reasons presented in the previous section to study phase signature of shape resonance, the reported phase is the phase difference between the molecular and the atomic computations, thus removing the quadratic contribution and allowing the observation of the two-center interference phase.

We observe a strong reduction of harmonic intensity in the case of the short trajectories (Fig. 5.13a), and well pronounced minima for the long trajectories especially for particular values of R (Fig. 5.13c). This is the signature of the destructive interference induced by the two centers of the molecule. Its position corresponds to a zero in the recombination dipole matrix element, that we computed using again the shooting method. We present it in Fig. 5.14a for different internuclear distances R, and report the corresponding harmonic order for which the zero is observed at a given internuclear distance in Fig. 5.14b. The positions coincide with the minima observed in the spectra, that we have extracted for the long trajectories from for the damped minima. These positions can be analytically obtained from the zeros of the recombination dipole matrix element d rec given by Eq. (3.3.24), which is derived within SFA by the use of plane-waves as continuum waves (see Sec. 3.3 and Ref. [START_REF] Lein | Interference effects in high-order harmonic generation with molecules[END_REF]). The zeros of d rec are encountered at particular momentum values:

k q = (2q + 1)π R , q ∈ Z.
(5.4.1)

We focus on the first value k 0 as it is only one contained in our harmonic spectra generated by such short-wavelength laser, as confirmed by Fig. 5.13. For symmetry reasons (the wave function is even), this recombination dipole matrix element is a real number. Thus, constrained to the real axis, its evolution around k 0 from a positive to a negative real part implies a discontinuity of its phase, with a jump of ±π. The corresponding harmonic frequency is:

Ω (PW) 0 = I p + k 2 0 2 = I p + π 2 2R 2 .
(5.4.2)

However, the analytical value Ω (PW) 0 is not in agreement with our TDSE results, as seen in Fig. 5.14b (e.g. it predicts a minimum at harmonic 46 while it is observed at harmonic 41 for R = 1.55 a.u.). It does not correspond to the zeros of the exact recombination dipole matrix element. As a matter of fact, while the continuum is described by plane-waves in SFA, our TDSE and shooting computations consistently include the exact Coulomb waves. To overcome this limitation, it is usually claimed that the effective energy of the Coulomb waves is increased by I p [START_REF] Lein | Interference effects in high-order harmonic generation with molecules[END_REF], leading to the approximate expression:

Ω0 π 2 2R 2 .
(5.4.3)

Still, this approximation is not perfect as we can see in Fig. 5.14b (e.g. it would predict a minimum at harmonic 36 for R = 1.55 a.u.).The correction by I p is overestimated. Furthermore, we find that a constant correction, independent of R, is not correct as curves are not parallel with the exact zeros values. As already observed in Refs. [START_REF] Lein | Interference effects in high-order harmonic generation with molecules[END_REF][START_REF] Lein | Role of the Intramolecular Phase in High-Harmonic Generation[END_REF][START_REF] Van Der Zwan | Two-center interference and ellipticity in high-order harmonic generation from H + 2[END_REF], and what we confirmed in Figs. 5.13b and 5.13d, the interference phase-jumps are smoothed, covering several harmonics, and with a magnitude lower than π. Here, in addition, they exhibit more surprising features. Whereas the phase-jump is smoothed for all values of R for short trajectories and is positive (Fig. 5.13b), the situation is completely different for the long trajectories. It is smoothed for small R and negative, then is steep at a critical internuclear distance R c = 1.55 a.u. for which the behavior reverses into a finally smoothed jump for R > R c , being here positive as for the short trajectories (Fig. 5.13d). Moreover, the depth of the minimum in the amplitude is directly related to the shape of the phase-jump: the steeper the phase-jump, the deeper the minimum.

One should note that the spatial separation of short and long trajectories artificially enhances the cutoff over a few orders. This is due to a reflection of a small part of the fastest electrons by the absorber. This artifact is an asset for this study since it allows us to observe the totality of the smoothed phase-jumps even after the predicted cutoff. The remaining oscillations in the low harmonics are also attributed to reflections of the slowest electrons by the absorber.

We performed time-frequency analysis of the harmonic dipole using Gabor transforms to evaluate, for different values of R, the emission times of the harmonics for which the destructive interference occurs. In Fig. 5.15a-b we chose the particular case of R = R c . We found that the recollision time for the long trajectories corresponds to an almost zero electric field (that is E rec 0.015 a.u.). Moreover, greater (lower) values of R lead to recollision with a negative (positive) electric field for the long trajectories while it is always positive for the short trajectories.

To further examine the relation between the value of the electric field at the recombination and the shape of the phase-jump, we tune the peak laser intensity I L from 2 to 4×10 14 W.cm -2 at fixed R. We chose R = 1.425 a.u. to model H 2 molecule. Results are shown in Fig. 5.16. As expected, the position of the destructive interference does not change with I L and we observe a behavior for the phase-jumps which is similar to the study in R. It is smoothed for all intensities for the short trajectories, while an equivalent smoothing and inversion behavior is present for the long trajectories. We found the critical intensity I c = 3.24 × 10 14 W.cm -2 for which we observe the inversion with a discontinuous ±π phase-jump (see black-dashed curve in Fig.

5.16b).

Using Gabor analysis once again, reported in Fig. 5.15c-d, this change of behavior is directly related to the change of sign of the instantaneous electric field at recombination time, which is almost zero at Ω 0 for an intensity I c . To get rid of the limited Gabor resolution and get the exact recombination time, we used the instantaneous frequency and obtained the corresponding value of the electric field E rec = 0.011 a.u. (≡ I rec = 4.24 × 10 12 W.cm -2 , that is ∼ 1.3% of I c ).

Recombination with a dressed state

In the observation of the results above, we have shown that we could link the smoothness and inversion of direction in the two-center ∼ π phase-jump with the value of the instantaneous laser field at harmonic emission time, corresponding to the time when the electron recollides with the parent ion and looses its kinetic energy by a radiative recombination in the ground-state. To examine whether these effects originate from the dressing of the molecular ground-state, we derive an expression of the recombination dipole matrix element with a dressed molecular ground-state, being expressed as a LCAO [START_REF] Chiril | Strong-field approximation for harmonic generation in diatomic molecules[END_REF][START_REF] Lein | Interference effects in high-order harmonic generation with molecules[END_REF][START_REF] Etches | Two-center minima in harmonic spectra from aligned polar molecules[END_REF][START_REF] Odžak | Interference effects in high-order harmonic generation by homonuclear diatomic molecules[END_REF][START_REF] Odžak | Molecular high-order harmonic generation: analysis of a destructive interference condition[END_REF]. This is to be related to the work of Śpiewanowski et al [START_REF] Śpiewanowski | High-order-harmonic generation from field-distorted orbitals[END_REF][START_REF] Śpiewanowski | Field-induced orbital distortion in high-order-harmonic generation from aligned and oriented molecules within adiabatic strong-field approximation[END_REF], where they included a priori dressed orbitals in SFA, computed with quantum chemistry methods. A complete derivation of the dressing within SFA has also been discussed by Smirnova et al [START_REF] Smirnova | Anatomy of strong field ionization II: to dress or not to dress?[END_REF].

Decomposition in LCAO We want to write the fundamental wave-function ϕ 0 as a symmetric state σ g (field-free molecular ground-state), sum of two symmetric atomic orbitals φ s centered on ±R/2, and the first excited state ϕ 1 as an antisymmetric state σ u (field-free first excited state), difference of the two symmetric orbitals which served to construct the σ g state [START_REF] Etches | Two-center minima in harmonic spectra from aligned polar molecules[END_REF][START_REF] Odžak | Interference effects in high-order harmonic generation by homonuclear diatomic molecules[END_REF][START_REF] Odžak | Molecular high-order harmonic generation: analysis of a destructive interference condition[END_REF]. This decomposition in LCAO is fully valid for large internuclear distances R, when the overlap w between the two atomic orbitals is close to zero. We will show that it is still a good approximation for shorter distances. We write wave functions of states σ g and σ u as the following LCAO:

ϕ σ g (x) = 1 2(1 + w) φ s x + R 2 + φ s x - R 2 , (5.4.4)
ϕ σ u (x) = 1 2(1 -w) φ s x + R 2 -φ s x - R 2 .
(5.4.5)

We want ϕ 0 = ϕ σ g and ϕ 1 = ϕ σ u . Hence we obtain:

φ s x ± R 2 = 1 √ 2 √ 1 + w ϕ 0 (x) ± √ 1 -w ϕ 1 (x) .
(5.4.6)

We can therefore compute φ s iteratively, starting with w = 0 and imposing φ s symmetric (i.e. even). The process converges to a symmetric wave-function which leads to the best possible decomposition. Figure 5.17 displays the exact molecular states and their decomposition in LCAO for H 2 (R = 1.425 a.u.). The final overlap is w = 0.943. Projections of the LCAO states on the exact states are:

ϕ σ g |ϕ 0 = 0.9897, (5.4.7)
ϕ σ u |ϕ 1 = 0.9856.

(5.4.8)

Hence, this decomposition is still a good approximation even for short internuclear distances, as is the case for H 2 .

Dressed ground-state We treat the laser field as a perturbation in order to derive an expression of the dressed fundamental wave-function within the PT. As the laser frequency is low compared to the dynamics of the system, we can derive the dressed state, denoted |Φ 0 , using the time-independent PT for a given value E t = E(t) of the electric field, at a "frozen" time t. At first order, the dressed wave-function reads [38]: The molecule symmetry imposes that for all even states (n = 2q), ϕ 2q |x| ϕ 0 = 0. Moreover, the couplings ϕ 2q+1 |x| ϕ 0 for q > 0 become negligible compared to ϕ 1 |x| ϕ 0 . Hence, we simply write:

|Φ 0 (E t ) = |ϕ 0 + E t ∑ n>0 ϕ n |x| ϕ 0 E 0 -E n |ϕ n .
|Φ 0 (E t ) |ϕ 0 + c t |ϕ 1 , (5.4.10)
where:

c t = E t ϕ 1 |x| ϕ 0 E 0 -E 1 .
(5.4.11)

Using the decomposition in LCAO (Eqs. (5.4.4) and (5.4.5)), we find after simple manipulations:

c t = E t R 2(E 0 -E 1 ) √ 1 -w 2 = -αE t ,
(5.4.12)

where we find α = 7.3. We also performed R-box simulations to compute the dressed ground-state energy, and obtain by inverse iteration the dressed wave-function for different electric field values. We then projected the dressed wave-function on the first excited state |ϕ 1 , presented in Fig. 5. [START_REF] Larsson | Two-colour time-resolved spectroscopy of helium using high-order harmonics[END_REF] and get a numerical expression of c t . We confirm that c t is proportional to E t , with a factor α = 4.6. It is of the same order as the factor obtained before (α = 7.3). The discrepancies are attributed to the two approximations we made to obtain analytical expression (5.4.12). First of all, the expression of the dressed state, expressed via time-independent PT, is truncated to the first excited state. Secondly, the ground-state and first excited states are expressed as a LCAO, that is not perfectly exact.

Recombination dipole matrix element

The velocity form of dressed recombination dipole matrix element at time t is (with similar manipulation than in Sec. 3.3.2):

drec(k,t) = Φ 0 (E t ) |-i ∇| k = 2k φs (k) 1 2(1 + w) cos k R 2 + i αE t 2(1 -w) sin k R 2 . (5.4.13)
Interestingly, this expression exhibits an imaginary part in sin(kR/2), directly proportional to the electric field at recombination time t. To understand its effect, we represented the dressed recombination dipole matrix element in the complex plane in Fig. 5.19a. As k is growing, instead of being restricted to the real axis (purple dashed arrow), the transition dipole matrix element describes an ellipsis on the complex plane, which width is directly proportional to the instantaneous amplitude of the laser field. As a consequence, the phase exhibits a discontinuous ±π jump when the electric field is zero, while it smoothly increases (decreases) when the electric field is positive (negative), as shown in Fig. 5.19b. The direction of the phase-jump is thus dictated by the sign of the instantaneous laser field. For the sake of simplicity, we write:

drec(k,t) ∝ cos k R 2 + iα E t sin k R 2 , (5.4.14)
where:

α = 2(1 + w) 2(1 -w) α (5.4.15)
With the overlap at R = 1.425 (w = 0.943) we have α = 5.8α. Mathematical study of the complex number given by Eq. (5.4.14) is provided in Appendix A. 6 Its phase is:

θ (k,t) = arctan α E t tan k R 2 .
(5.4.16)

For a non-zero electric field E t , we find that:

• The modulus of drec exhibits a damped minimum at the same positions k q = (2q + 1)π/R as derived previously in the undressed case,

• The value of the minimum is |α E t |,

• The phase jumps smoothly around k q , the direction being dictated by the sign of E t ,

• The slope of the phase-jump at k q is exactly (α E t ) -1 .

These findings can be linked with the conclusions in the work of Han et al [START_REF] Han | Internuclear-distance dependence of the role of excited states in high-orderharmonic generation of H + 2[END_REF] where they investigated the extinction of the two center interference with the increasing contribution of the first exited state while R is growing.

Although using an ad hoc dressed recombination dipole matrix element gives nice physical interpretation of the behavior of the phase-jump, quantitative agreement with TDSE computations is not met for several reasons. The first lies in the fact that the expression of the dressed state is approximate, as we mentioned before. Taking into account other excited states or greater PT orders may displace the condition of inversion in the long trajectories form zero electric field to a non-zero value closer to TDSE results. Secondly, the values of momenta k depend on time, hence electric field value evolves along harmonic order. The consequence is that drec(k,t) should either be expressed as a function of momentum k or of time t only, the two variables not being independent. For that reason, drec(k,t) does not actually describe ellipsis in the complex plane but more complex paths. However, a path crossing the origin of the plane still exists, and paths above and below this one are present. Hence, the conclusions on the smoothing and change of direction of the phase-jump remain. Finally, the last reason and what we believe to be the main one, is that we need to artificially shift the interference to fit the position found in TDSE (as we did in Fig. 5.19b) because, as explained previously, the recombination dipole matrix element is derived in the PWA. As a consequence, the whole dynamical process is shifted regarding to R: conditions on momenta k (and times t) are completely changed.

At this time we can conclude that the smoothing of the phase-jump does not originate from Coulombic correction, as it was claimed in Ref. [START_REF] Ciappina | Influence of Coulomb continuum wave functions in the description of high-order harmonic generation with H + 2[END_REF]. Besides, since the recollision dipole matrix element depends on the instantaneous electric field, and thereby is different for each harmonic, the factorization proposed by QRS cannot be validated in this case.

Molecular SFA

Now that we have the TDSE results as a non-equivocal reference, we propose now to study two-center interference phase-jump using molecular SFA as developed in Sec. 3.3.2. We would like to remind the reader that this theory we will use has been developed in Refs. [START_REF] Chiril | Strong-field approximation for harmonic generation in diatomic molecules[END_REF][START_REF] Faria | High-order harmonic generation in diatomic molecules: A quantum-orbit analysis of the interference patterns[END_REF]. The only difference is that we take the velocity form of the dipole, which is in fine equivalent to take the truncated length form of the recombination dipole matrix element as Carla Figueira de Morisson Faria did with no justification [START_REF] Faria | High-order harmonic generation in diatomic molecules: A quantum-orbit analysis of the interference patterns[END_REF]. We present here the specific case of aligned molecules along the laser polarization (which is equivalent to restricting the study to a 1D case). Same results are obtained when molecules form an angle θ with the laser polarization, except that interference conditions involve R cos(θ ) instead of R. For these computations, we chose a Gaussian function as atomic orbital φ s in the decomposition in LCAO, in order to derive fully analytical expressions:

φ s (x) = α 0 π e -α 0 x 2 2
(5.4.17)

where α 0 ∼I p [START_REF] Lewenstein | Theory of high-harmonic generation by low-frequency laser fields[END_REF]. The overlap between the two atomic orbitals centered at ±R/2 is:

w(R) = 1 2 α 0 π e -α 0 R 2 4 .
(5.4.18)

The FT φs (with our conventions) and its first and second derivatives are:

φs (k) = e - k 2 2α 0 (5.4.19) ∂ φs ∂ k = - k α 0 e - k 2 2α 0 (5.4.20) ∂ 2 φs ∂ k 2 = k 2 α 2 0 - 1 α 0 e - k 2 2α 0 (5.4.21)
We report in Fig. 5.20 the phase of the short and long trajectories as a function of the laser intensity. Again this phase is relative to the phase of atomic SFA presented in

I c = 3.71 W.cm ---2 2 2 .
Sec. 3.3.1, following [START_REF] Lewenstein | Theory of high-harmonic generation by low-frequency laser fields[END_REF]. The separation of the short and long trajectories in SFA is straightforward, being inherent to the search of saddle-point solutions. Consequently, no residual oscillation is encountered. Nevertheless, the cutoff prevents the computation of harmonics above it which limits the observation of the entire phase-jumps for the lowest intensities. We observe that the behavior for the phase is very similar to the TDSE simulations. The major difference is the position of the destructive interference due to the PW approximation in SFA as mentioned before. Therefore, the intensity for which the phase is discontinuous and jumps to ±π is shifted to I c = 3.71 W.cm -2 . Hence, the destructive interference is observed exactly at the expression of the frequency given by Eq. ((5.4.2)), which is shifted from the TDSE case.

SFA computations allowed us to obtain the precise value of the electric field at recombination time for the long trajectories at the inversion intensity I c . It is equal to E rec = 0.0034 a.u., that is comparable to the one observed in TDSE computations. The differences are attributed to the difference between TDSE and SFA trajectory times that we observed previously using IF in Sec. 4.2.6 (see Fig. 4.7 and Ref. [START_REF] Risoud | Quantitative extraction of the emission times of high-order harmonics via the determination of instantaneous frequencies[END_REF]). The intensity for which the laser field is zero at recombination time is I z = 3.65 × 10 14 W.cm -2 , ∼ 0.06 lower than I c . This intensity does not lead to an exact ±π discontinuous jump.

We have seen with TDSE computations that the phase behavior depends strongly on the type of trajectory (short vs long) and on the value of the electric field at recombination time. Above, we have proposed a very intuitive explanation using ad hoc dressed recombination dipole matrix element, which gives qualitative agreement with TDSE. Performing equivalent simulations using molecular SFA, we found identical behavior than for TDSE computations. Thanks to the fact that SFA theory is an analytical formulation of the problem, we are able to get further physical insight with additional development of molecular SFA. We will inquiry whether dressings effects could be derived analytically from molecular SFA. The idea is to express the deviations of the molecular solutions regarding to the atomic ones by mean of Taylor expansions.

Short trajectories

Long

Taylor expansions of molecular SFA

Ionization and recollision times As already explained, we normalized our numerical results with the atomic ones in order to remove the quadratic harmonic phase. Moreover, internuclear distances are small compared to the travel path of the freed electrons in the continuum. Thus, the times for molecular trajectories are close to the ones of an atomic trajectory. These considerations conducted us to perform Tay-lor expansions on the molecular SFA saddle-point equations (Eqs. around the atomic ionization and recombination times t at and t at . We recall that we deal with the specific case of aligned molecules along the laser polarization. The absence of perpendicular contribution in the momentum p allows us to derive equations which reveal straightforward physical meaning. We will evaluate the differences introduced in the general 3D case later on. We first derive an expression of the time deviations: Guided by numerical evaluations (see Since ω L 1 we neglect the term ω 2 L A(t at ) • R 2 ∆t αβ and get:

∆t αβ = t αβ -t at , (5.4
∆t (1) αβ = (-1) β R/2
2(ω -I p ) = (-1) β δ .

αβ = i (-1) α R/2 2I p = i (-1) α δ .
δ = R/2 2(ω -I p )
, and δ = R/2 2I p .

( verifying the assumption we made above.

In addition, in order to improve comparison with the "exact" molecular SFA we obtained expressions for the time deviations up to the second order:

∆t (2) αβ = (-1) β R/2 2(ω -I p ) -(-1) β E(t at )δ /2 , (5.4.33) ∆t (2)
αβ = i (-1) α R/2 2I p + (-1) α E(t at )δ /2 .
(5.4.34)

The additional terms in the denominator can be viewed as corrections to the velocity of the outgoing and returning electrons, driven by the electric field. As expected, the second-order values are in better agreement with the numerical values. However, the first-order analytical values lies between the numerical values of the short and long trajectories with small discrepancies (∼ 0.02 a.u.). And still, the differences in the numerical time deviations between the short and the long trajectories are small. Hence, the approximate expressions given by Eqs. (5.4.28) and (5.4.29) will be used in the following developments.

The time deviations expressed in Eqs. (5.4.28) and (5.4.29) have a simple physical picture of the HHG process in a molecule. First, the ionization times are shifted by a pure imaginary delay ∆t αβ which can be interpreted as a correction to the tunneling time with respect to the atomic one. If the electron is ionized from center α = 1 we find that Im(∆t 1β ) < 0 which means that the time spent "below the barrier" is smaller. This is consistent with a smaller barrier to cross as we can see in Fig. 3.6. The opposite situation is observed for an electron ionized from center α = 2 for which Im(∆t 2β ) > 0 where a larger barrier is to cross. Secondly, the recollision times are shifted by a real delay ∆t αβ with respect to the atomic ones. For an electron recombining with center β = 1, we find that ∆t α1 < 0 which means that the electron recollides earlier compared to the atomic one. Indeed, as pictured in Fig. 3.6, the electron travels a shorter path. On the contrary, an electron recolliding with center β = 2 for which ∆t α1 < 0 travels a longer path. These observations are consistent with the formal expression of these time delays which can be regarded as the ratio between the additional distance to cross ±R/2 and the velocity of the electron.

Action We now develop S αβ at the first-order around the atomic times as we did previously: where:

S αβ (p αβ ,
∂ R ∂ k (k) = 1 2(1 + w(R)) φs (k) + k ∂ φs ∂ k , (5.4.42) K α (k,t ) = 1 2(1 + w(R)) × i ω 2 L A(t ) ∂ φs ∂ k -E 2 (t ) ∂ 2 φs ∂ k 2 + (-1) α ω 2 L A(t ) φs (k) R 2 -E 2 (t ) ∂ φs ∂ k R 2 . (5.4.43)
Prefactor C is of second-order in the deviations around the saddle points. Hence, as a first approximation we can consider that C αβ (t αβ ,t αβ ) = C(t at ,t at ). The exact effect of this approximation will be discussed later. Finally, after regrouping all the terms, the total dipole (3.3.41) can be refolded into: D(ω) = -ωC(t at ,t at ) drec (p at + A(t at ),t at ) dion (p at + A(t at ),t at )e -iS at × e -iE(t at ) R 2 δ e -E(t at ) R 2 δ + c.c. . (5.4.44)

We have derived an expression which is akin to the atomic dipole given by Eq. (3.3.11).

It is evaluated at atomic times t at and t at and involves atomic action S at . However, it exhibits modified transition dipole matrix elements dion and drec . The expression drec is simply:

drec (k,t) = 2R(k) cos k R 2 + 2iE(t)δ ∂ R ∂ k sin k R 2 ,
(5.4.45)

while the expression of dion is:

dion (k,t ) = -i 2 1 + w(R) × ∂ φs ∂ k E(t ) -ω 2 L A(t ) φs (k) R 2 δ + E 2 (t ) ∂ φs ∂ k R 2 δ cos k R 2 -φs (k) R 2 E(t ) + ω 2 L A(t ) ∂ φs ∂ k δ -E 2 (t ) ∂ 2 φs ∂ k 2 δ sin k R 2 . (5.4.46)
The dipole also displays an additional phase term exp[-iE(t at )Rδ /2] and a damping term exp[-E(t at )Rδ /2]. Now, the effect of each term will be closely examined. As it contains an imaginary part 2E(t)δ ∂ k R sin(kR/2) which is of first-order in δ and in quadrature with respect to the imaginary part, it is not anymore restricted to the real axis and it describes a path in the complex plane when varying the harmonic order. This imaginary part depends on the instantaneous electric field at the instant of recombination. The higher the electric field magnitude at recombination, the farther the path from the origin, and consequently, the smoother the phase-jump. As noticed previously, the path described is quite different than an ellipsis because of the dependence of the factors in momentum k (i.e. in harmonic order), as we can see in Fig. 5.21a, which displays the values of drec in the complex plane for short and long trajectories along harmonic order, for different laser intensities.

The electric field at recombination is always positive for short trajectories. We thus always observe a positive phase-jump as shown in Fig. 5.21b. For the lowest

TWO-CENTER INTERFERENCES IN MOLECULES

intensities the phase-jump is not complete due to the low cutoff value. However, for long trajectories, the recombination happens across two laser half cycles: the electric field is positive, crosses zero and is then negative, as shown previously. As we can see in Fig. 5.21a, for the particular intensity I c = I z (black dashed line), the path followed by drec crosses the origin. It corresponds to zero electric field at recombination time and thus to zero imaginary part. Therefore, we retrieve the ideal case for which the phase is discontinuous and jumps to ±π (see Fig. 5.21c). For the other intensities, the laser field is not zero and the phase evolves slowly, varying from zero to π (-π) for a positive (negative) electric field at recombination.

We would like to mention, for the sake of completeness, that the dependency with k of factors R and E(t)δ ∂ k R slightly shifts the harmonic frequency of the minimum in | drec | regarding to Ω 0 (see Appendix A.6). This explains the deviations observed previously between the minima extracted from the TDSE computations for the long trajectories and the exact non-modified recombination dipole matrix element obtained by shooting method, in Fig. 5.14. It would have been found exactly at Ω 0 if they were independent on k. However, the shift is a fraction of harmonic order and is not visible in the figures.

We fully understood the behavior of the recombination dipole matrix element which encodes the phase of the destructive interference. Its modifications from the ideal case are driven by the dynamics of the electron in the molecule. They lead to phase-jumps whose shape depends on the electric field at recombination time, characteristic of strong dressing effects. Since electrons having short or long trajectories recombine with the molecule in different time windows, the corresponding phasejumps behave completely differently.

Inversion condition

In this section we will examine in details the conditions required to observe inversion in long trajectory phase-jump, either as a function of internuclear distance R (cf. For long trajectory contributions, the electric field is zero at a single harmonic frequency denoted Ω z . Thence, harmonics below the cutoff and above Ω z are emitted in presence of a positive electric field, and harmonics below Ω z in presence of a negative electric field.

The value of Ω z can be obtained within atomic classical calculations framework, that we developed in Sec. 2.4.2. We thus solved the Newton equation for an electron freed at ionization time t and brought back to its initial position x = 0 at recollision time t under the influence of the laser field. The recollision time for which the laser field is zero is t z = 3π/2ω L (see Eq. (2.4.1)), leading to the corresponding ionization time:

t z 1 3π 2 -2 3π ω L . ( 5

.4.47)

In this way, the harmonic frequency for which the electric field is zero reads:

Ω z = I p + E 2 L 2ω 2 L [sin(ωt z ) + 1] 2 ≈ I p + 2.98U p , ( 5

.4.48)

We recall that the ponderomotive energy (U p = E 2 L /4ω 2 L = I L /4ω 2 L ) depends on laser intensity I L . Therefore, Ω z is a function of I L , while Ω 0 is a function of R, according to Eq. (5.4.2).

Thus, modifying R at a fixed laser intensity, we move the position of Ω 0 in the spectrum. Conversely, we can shift Ω z by scanning the laser intensity at a fixed internuclear distance. Then, can put our system in the three cases that we previously discussed:

• Ω 0 < Ω z , the phase-jump for long trajectories is smoothed and have an opposite direction than the one of short trajectories;

• Ω 0 = Ω z , the phase-jump for long trajectories is discontinuous and equal to ±π;

• Ω 0 > Ω z , the phase-jump for long trajectories is smoothed and have the same direction as the one of short trajectories.

In most of the situations, the first case is attained in usual working conditions. In the second case, equality of Ω 0 (Eq. (5.4.49)

In the last case, except in our TDSE computations where spacial separation of trajectories artificially enhanced the cutoff, the entire phase-jump is not observed because Ω z is close to cutoff frequency Ω cut = I p + 3.17U p (cf. Eq. (2.4.6)), as we can see in Figs. 5.20 and 5.21. Unfortunately, one cannot expand the region between these two frequencies by changing the laser intensity since both depend linearly on it. The analysis of drec leads to associate the discontinuous phase to the recombination in presence of a zero electric field. However, both for TDSE and molecular SFA computations, we observed that the sharpest phase-jump is obtained when the electric field at recombination time is not exactly zero. In the following section we will analyze in details the remaining terms in D(ω) (Eq. (5.4.44)) and explain the origin of this shift.

Extra phase factors to HHG spectrum

Ionization dipole matrix element The modified ionization dipole matrix element dion is a pure imaginary number; the extra term does not affect the shape of the phasejump. To justify this affirmation further, we examined two other versions of molecular SFA. The first only includes the exponential form of the recombination dipole matrix element in the action, and the second includes only the exponential form of the ionization dipole matrix element. Interestingly, the first reproduces almost identically the full molecular SFA, while the second matches almost perfectly the non-modified version of SFA. Consequently, this justifies that the relevant quantity for the study of the interference is the recombination dipole matrix element, as it has already been argued [START_REF] Chiril | Strong-field approximation for harmonic generation in diatomic molecules[END_REF][START_REF] Faria | High-order harmonic generation in diatomic molecules: A quantum-orbit analysis of the interference patterns[END_REF].

Additional exponential terms

The terms exp(-iE(t at )Rδ /2) and exp(-E(t at )Rδ /2) in D(ω) are responsible for a global phase shift and damping and then do not encode any dynamics into the two-center destructive interference.

Exact prefactors

We have discussed so far the case of C αβ (t αβ ,t αβ ) = C(t at ,t at ) as a first approximation for the Taylor expansion of molecular SFA. However, the deviations of C αβ (t αβ ,t αβ ) from C(t at ,t at ) are not entirely negligible and give rise to additional shifts in the harmonic phase. Since the analytical first-order expansion of C αβ (t αβ ,t αβ ) is cumbersome and leads to extremely tedious expressions, we chose to identify its role numerically.

As noticed previously, the sharp ±π-jump is observed at a small but non-zero electric field. Indeed, the exact C αβ (t αβ ,t αβ ) shifts the condition with an extra phase factor, which is independent on the electric field. As a matter of fact, Fig. 5. [START_REF] Lévêque | Direct observation of spin-forbidden transitions through the use of suitably polarized light[END_REF] shows the value of the minimum in HHG intensity for the long trajectory contributions for Harmonic intensity (arb. units) In both cases, we observe a clear ditch, the bottom of which corresponds to the minimal value reached, thus to the sharpest phase-jump. In the first case, it coincides with a zero electric field at the recombination, as expected. In the second case, it is slightly shifted. However, the ditches are almost parallel, indicating that the additional shift introduced by C αβ (t αβ ,t αβ ) is independent on the electric field. The prediction of the minimal values given by the analytical formula of Eq. (5.4.49) is also reported in Fig. 5.22. While it is close to both calculations, we can attribute the slight discrepancies to the well known differences between quantum and classical computations. The latter allows to derive analytical formulae but depicts only qualitatively HHG spectra [START_REF] Risoud | Quantitative extraction of the emission times of high-order harmonics via the determination of instantaneous frequencies[END_REF].

Slope of the phase-jump

The values of the slope of the phase-jump at Ω 0 can be numerically extracted at the inflexion point of the curve. We denote it γ. Figure 5.23 displays the inverse of the slope of the phase-jump (1/γ) for the short and long trajectories as a function of laser intensity. It compares the results for drec with the ones obtained in the harmonic spectra from molecular SFA, i.e. with the correct use of C αβ (t αβ ,t αβ ) and TDSE. An approximate expression of the slope is obtained by assuming that the dependency of the factors R and E(t)δ ∂ k R with k does not move the inflexion point (see

Appendix A.6): γ app = R(k) E(t)δ ∂ R ∂ k . ( 5

1/γ

Laser intensity (×10 14 W.cm -2 ) Short traj.

Long traj. 1/γ app (thick solid lines) with the slopes for the modified recombination dipole matrix element drec (dot-dashed lines) and the one obtained in the spectrum of molecular SFA (dashed lines). We also report the inverse of the slope extracted from our TDSE computations, horizontally shifted so that the zero in the long trajectories coincides with the one of the molecular SFA. The TDSE values for the short trajectories are not displayed in this frame (they lie above 0.32).

We have:

R(k) = k φs (k) 2(1 + w(R))
, and

∂ R ∂ k = 1 2(1 + w(R)) φs (k) + k ∂ φs ∂ k .
(5.4.51)

Given Eqs. (5.4.19) and (5.4.20), we find that:

γ app = 1
E(t)δ 1 k - k α 0 . ( 5

.4.52)

The values of 1/γ app are also reported in Fig. 5.23. Whereas they depict the expected behavior, they are still very different, especially for the short trajectories. This shows that the dependence of R and E(t)δ ∂ k R with k strongly affects the shape of the phase-jump (as demonstrated in Appendix A.6). The inverse of the slope crosses zero in the long trajectories, at an intensity for which the phase-jump is the sharpest. We see a shift between drec and the molecular spectrum of 0.06 × 10 14 W.cm -2 , confirming the shift found previously.

We also present the slope of the phase-jump extracted from our TDSE computations. As we explained it, the turning intensity is not the same as the one of molecular SFA. Thus, we shifted horizontally the values so that the zero in the long trajectories coincides with the one obtained in molecular SFA. For long trajectories, the agreement between TDSE and molecular SFA is good. However, since residual oscillations are strongly present in the phase-jump of short trajectories, the extraction of the slope leads to very distant values which cannot be displayed in Fig. 5

Deviations in the 3D case

We now deal with the general 3D case where H 2 molecules are aligned with an angle θ between the laser polarization and the molecular axis, as depicted in Fig. As explained previously, analytical expressions for time deviations ∆t αβ and ∆t αβ are cumbersome in the 3D case, because of components of momentum p along both e x and e y . Hence, we numerically identified the deviations introduced on ∆t αβ and ∆t αβ by the angle θ . We solved the molecular saddle-point equations (Eqs. leading to six equations on p x , p y , t and t .

5.24.

E(t)

In Table 5. 4 we compare the values of ∆t αβ and ∆t αβ for trajectories 12 and 21 (for which necessarily p y = 0) for different angles θ with the values in the 1D case where we used the corresponding internuclear distance R = R cos(θ ), for harmonic 51. Trajectories of electrons ionized from one center and returning to the same one (i.e. trajectories 11 and 22) do not differ than the ones of 1D computations with R, since they require p y = 0. Therefore, we do not present them in Table 5.4.

The differences are very small even for angles close to π/2 (the limit for which two-center interferences are no longer observed). Hence, all the conclusions we have drawn in the case of aligned molecules along the laser polarization are valid in the general case where θ = 0.

Introduction

In the previous section, we have completely characterized and explained the phasejump of two-center interference in HHG spectra. Due to strong dressing effects, related to the amplitude of the electric field at recombination time, its comportment is different for harmonics emitted by electrons having short and long trajectories as they recollide during disjoint time windows. We explained that the inversion of the phasejump in the long trajectory contributions happens very close to the cutoff. Hence, in most of the cases, where destructive interference is observed in the middle of the plateau, short trajectories lead to a positive smoothed jump while long trajectories to a negative smooth jump, practically symmetrically. Therefore, along harmonic order, phase-difference between short and long trajectory contributions evolves continuously from nearly zero to almost 2π, passing through π around the harmonic order of the destructive interference due to the quasi-symmetry of the phase-jumps.

For this reason, we believe that this phase signature can be observed experimentally with QPI. Indeed, the consequence is that interference patterns of QPI, encoding phase difference between short and long trajectories, must be slowly dephased and then rephased along harmonic order. By comparison with an atomic reference of same ionization potential, or with unaligned molecules, QPI patterns must be found in phase with low harmonic orders, out of phase around destructive interference, and finally in phase at high harmonic orders again.

We propose here to verify this intuitive hypothesis, first with an analytical model of QPI with the use of appropriate additional phase-jumps, and then with unequivocal TDSE computations.

Analytical model

We presented in Sec. 4.4.4 an analytical model of QPI. It mimics the dependency of harmonic phase with laser intensity, including contributions of short and long trajectories. To take into account additional phase introduced by two-center interferences, we simply add to Eq. (4.4.4) supplementary phase, function of harmonic order q and laser intensity I L , θ j q (I L ), where j = s or j = l for either short or long trajectories:

D q (I L ) = ∑ j=s,l
A j q (I L )e -iα j q (I L )I L +iθ j q (I L ) ,

(5.5.1)

with A j q (I L ) and α j q (I L ) as defined previously. The form of θ j q (I L ) must be carefully chosen in order to reproduce the behavior we observed in two-center interference phase-jump.

As we use here an analytical model to depict qualitatively HHG through its QPI, we start with the simplest form of the two-center interference phase derived from the dressed recombination dipole matrix element in Sec. 5.4.3. That is to say, we start with:

θ (k,t) = arctan α E(t) tan k R 2 .
[recalling Eq. (5.

4.16)]

We need to transform it into a function of harmonic order q and laser intensity I L . First, we know that harmonic order q is related to momentum k with the relation: Hence:

qω L = I p + k 2 2 .
θ (k,t) ≡ θ (q, I L ,t) = arctan α E(I L ,t) tan 2(qω L -I p ) R 2 ,
(5.5.3)

where we explicitly added the dependency in I L in the electric field. We recall that in fact, variables q and t are not independent. We thus need to express t as a function of q, what will lead us to separate it into two functions, one for short and one for long trajectories, neglecting further electron returns. We use the classical times, computed as in Sec. 2.4.2, as qualitative description is sufficient here. We found two simple analytical fits (with similarities to Ref. [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF]): that are displayed in Fig. 5.25, so that we finally have: and for an atomic reference where θ j q (I L ) = 0. We clearly observe the behavior we expected: interference patterns are almost in phase for H 35 , then the pattern of H 2 smoothly shifts until H 51-55 where it is out of phase with the reference, and finally shifts until H 75 where it is again in phase.

θ j q (I L ) = arctan α E I L ,t j (q, I L ) tan 2(qω L -I p ) R 2 , j = s, l.

TDSE computations

To support these observations, we performed TDSE computations of QPI with the same H 2 molecule and reference atom as used in the previous TDSE simulations in Sec. 5.4.2. The results are presented in Fig. 5.27. We recall that within our TDSE computations, the destructive interference is observed at harmonic 49. Note that it Harmonic Amplitude (arb. units)

Laser intensity (×10 14 W.cm -2 ) is accompanied with a strong depletion of harmonic intensity, fact that we did not take into account in the analytical model of QPI. Besides the fact that oscillations are consequently less visible, we still clearly observe the behavior that we described previously.

The need of a reference

In the previous computations, we identified a 2π smooth jump by comparison of molecular data with atomic ones. We will show here that on cannot extract the twocenter interference phase as a function of harmonic order qfrom molecular QPI without a reference. Whereas for the analytical case it seems possible to identify a phaseshift by looking at the first oscillation after the cutoff, it is not that simple for TDSE computations, as we can see in Fig. 5.27.

An idea is to perform FT of all our harmonic QPI to retrieve two-center interference phase as a function of q. The FT of the analytical formulation of QPI (Eq. (5.5.1)) reads:

FT[D q ](α) = ∑ j=s,l A j
q (I L )e -iα j q (I L )I L +iθ j q (I L ) e iαI L dI L .

(5.5.7)

We first identify that the intensity dependent amplitude A j q (I L ) will make difficult the extraction of meaningful phase. Hence, we can analyze the QPI phase only, as done in Refs. [START_REF] Gaarde | Spatiotemporal separation of high harmonic radiation into two quantum path components[END_REF][START_REF] Balcou | Quantum-path analysis and phase matching of highorder harmonic generation and high-order frequency mixing processes in strong laser fields[END_REF][START_REF] Gaarde | Quantum path distributions for high-order harmonics in rare gas atoms[END_REF], leading to the FT:

FT[arg(D q )](α) = ∑ j=s,l e i[αI L -α j q (I L )I L +θ j q (I L )] dI L .

(5.5.8)

By restricting the FT to the plateau region, we can neglect the dependency of α j q (I L ) on I L , and have only the asymptotic branches α j plat (ω q ) (see Eq. (4.4.5)).

CHAPTER 5. RESULTS

The resulting FT is:

FT plat [arg(D q )](α) = ∑ j=s,l

I max I cut (ω q )
e i[αI L -α j plat (ω q )I L +θ j q (I L )] dI L ,

(5.5.9)

where we explicitly denoted by I max the maximum intensity reached in our QPI. Using stationary-phase method, the maximum contribution is given by the intensity I L that satisfies:

∂ ∂ I L [αI L -α j plat (ω q )I L + θ j q (I L )] = 0 (5.5.10) ⇔ α -α j plat (ω q ) + S S S ∂ θ j q ∂ I L = 0.
(5.5.11)

We can make the additional approximation that θ j q does not depend strongly on I L , which is true for short trajectories but not quite exact for long trajectories, especially in the region close to the cutoff. However, beyond intensity I z (ω q ) close to the cutoff intensity I cut (ω q ), the laser field is always negative and, as for θ s q , θ l q does not depend strongly on I L . Thence, we can neglect its first derivative in Eq. (5.5.11), leading to the condition: α α j plat (ω q ), (5.5.12)

which allows to retrieve the branches α j plat (ω q ) by finding the maxima of the FT. Consequently, by taking the FT at a given maximum (corresponding to trajectory j = s or l), we obtain the mean value over intensity of the two-center interference phase:

FT plat [arg(D q )](α j plat (ω q ))

I max I cut (ω q )
e iθ j q (I L ) dI L .

(5.5.13)

To sum up, one could only extract intensity averaged two-center interference phase, and only if the following conditions are verified:

• α j q (I L ) does not depend on laser intensity in the plateau region,

• θ j q (I L ) does not depend strongly on laser intensity,

• interval [I cut (ω q ); I max ] is large enough to have a sufficient resolution to find precisely the maxima in the FT.

Unfortunately, the first condition is not verified in practice. The second is not true close to the cutoff for long trajectories, as we explained it. And the third cannot be achieved for arbitrary high harmonic order, because I cut (ω q ) grows with q while I max cannot be larger than the intensity leading to above-the-barrier ionization, since too strong ionization completely kills harmonic signal (and hence QPI).

For this application, QPI cannot be used as a self-referenced tool to probe twocenter interference phase. It must be used in parallel with an atomic reference to confront the interference patterns.

Introduction

Over the last decades, the ability to generate short lasers pulses has raised a great interest in creating and measuring vibrational wave-packets in excited electronic states or in the EGS of molecules [START_REF] Weiner | Femtosecond Pulse Sequences Used for Optical Manipulation of Molecular Motion[END_REF][START_REF] Wittmann | fs-Pulse Synthesis Using Phase Modulation by Impulsively Excited Molecular Vibrations[END_REF][START_REF] Weinacht | Coherent learning control of vibrational motion in room temperature molecular gases[END_REF][START_REF] Bartels | Nonresonant Control of Multimode Molecular Wave Packets at Room Temperature[END_REF]. As explained previously, the possibility of measuring nuclear motion on the attosecond time-scale through HHG has been first predicted by Lein [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF]. The experimental proof has then been provided by Baker et al [START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF] in the case of H 2 . Moreover, the formation of vibrational wave-packets in the EGS of SF 6 , probed with HHG, has been carried out by Wagner et al [START_REF] Wagner | Monitoring molecular dynamics using coherent electrons from high harmonic generation[END_REF]. In this experiment, the vibrational excitation is the consequence of ISRS, i.e. of two-photon excitation induced by a short and intense low-frequency laser pulse. Consequently, several models to describe HHG beyond the BOA in order to consistently take into account the nuclear motion have been developed (see Ref. [START_REF] Nguyen | Probing nuclear vibration using high-order harmonic generation[END_REF] and references therein).

We have seen that tunnel ionization plays an important role in HHG as the first step of the process. In the case of diatomic molecules, the definition of ionization potential is equivocal. Generally, it is claimed that the ionization potential depends on internuclear distance R [START_REF] Tong | Theory of molecular tunneling ionization[END_REF][START_REF] Saenz | Behavior of molecular hydrogen exposed to strong dc, ac, or low-frequency laser fields. II. Comparison of ab initio and Ammosov-Delone-Krainov rates[END_REF][START_REF] Urbain | Intense-Laser-Field Ionization of Molecular Hydrogen in the Tunneling Regime and Its Effect on the Vibrational Excitation of H + 2[END_REF][START_REF] Kjeldsen | Strong-field ionization of diatomic molecules and companion atoms: Strong-field approximation and tunneling theory including nuclear motion[END_REF][START_REF] Miloševi | Strong-field approximation for ionization of a diatomic molecule by a strong laser field[END_REF][START_REF] Goll | Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses[END_REF]. That is to say, the ionization potential is taken as the energy difference between the ionic and the neutral potential energy curves, at a given R.

A surprising theory based on these considerations called Lochfraß has been developed to explain the formation of vibrational wave-packets in the EGS of diatomic molecules and discussed for the specific case of H 2 [START_REF] Goll | Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses[END_REF]. This uncorrelated model acts only on nuclear wave-functions and states that the R-dependent ionization rate causes an anisotropic depletion of the vibrational eigenstate in the EGS. This would be the origin of vibrational excitation in the EGS that is different from ISRS.

Using our 1D×1D correlated model, presented in Sec. 3.4.2, we will study vibrational effects on HHG in light molecules. First of all, we will ensure the consistency of the model and compare our theoretical results with the theoretical predictions of Lein [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF] and the experiment carried out by Baker et al [START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF]. We will then investigate vibrational excitation in the EGS in the conditions where Lochfraß is predicted, and compare with these simulations to address the limitations of such uncorrelated model. Last but not least, we will study the relevance of R-dependent ionization potential, based on the analysis provided by Camille Lévêque during his PhD in order to explain experiments in SO 2 [START_REF] Lévêque | Pump-probe spectroscopy of vibronic dynamics using high-order harmonic generation: general theory and applications to SO2[END_REF].

Monitoring the correlated wave function in H 2

Parameters of the system

We simulated H 2 molecule subjected to a strong laser field with the 1D×1D correlated model presented in Sec. 3.4.2. As we explained it, the nucleus-nucleus potential V NN (R) is the BO potential energy curve of the ion H + 2 computed ab initio [START_REF]Etude de l'ion moléculaire H + 2 . Rapport de stage de Maîtrise de Chimie Physique[END_REF] (cf. Fig. 5.28), and the nuclei-electron interaction potential V Ne (x, R) is a double soft-Coulomb potential as presented in Sec. 3.2.3 (i.e. V Ne (x, R) = V dia (x, R) as given by Eq. (3.2.6)). The regularization parameter a is adjusted for each internuclear distance R such as the energy of the electron in the ground-state of this potential corresponds to the energy difference between V NN (R) and the potential energy curve of the neutral molecule taken in Ref. [START_REF] Kołos | New Born-Oppenheimer potential energy curve and vibrational energies for the electronic ground state of the hydrogen molecule[END_REF] (cf. Fig. 5.28). We used a 2-cycle sin 2 laser pulse at frequency ω L = 0.057 a.u. (Ti:sapphire) and peak intensity I L = 2.24 × 10 14 W.cm -2 (E L = 0.08 a.u.).

Movie of the evolving wave-function

We computed the 1D×1D vibronic wave-packet as a function of time along the laser pulse. In Fig. 5. [START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF] we present 24 snapshots of the wave-function at different times. The times are reported in Fig. 5. [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF] and marked with letters a-x. The wave-function is plotted along with the total interaction potential V NN (R) +V Ne (x, R) + xE(t). We presented the particular contour line corresponding to the ground-state energy at E 0 at equilibrium distance R = 1.425. The full movie is available at the following URL: http://phd.ekqnp.me/H2

The observation of the evolution of the wave-function is a wealth of information. First of all, during the second laser half-cycle, the electric field is strong enough to significantly lower the molecular potential. Thus, when the temporarily formed barrier is small, the electronic part of the wave-function starts tunneling out towards positive x (g-m). Then, the electric field reverses, and the ionized part is brought back to the core during the third laser half-cycle (n-v). In the meantime, electrons are released towards negative x by tunnel effect, similarly to (g-m). However, the corresponding wave-function presents oscillating blob-patterns. Indeed, they are the results of interference between the newly tunneling electrons and the previously released electrons that have been returned and pass through the core, going further away towards negative x.

Last but not least, we see that the electronic part which is brought back to the core (n-v), returns with increasing values of R. This is essentially the signature of nuclear motion of the ion subsequently to tunnel ionization.

Evolution of the mean internuclear distance

We propose here to examine closely the previous observation on the increasing internuclear distance of the ionized part of the vibronic wave-function Ψ(x, R,t). The idea is to integrate this wave-function on the electronic coordinate x over the ionized part on either the positive or negative region. To improve the selection of only the ionized part, we will analyze the wave-function:

Ψ(x, R,t) = Ψ(x, R,t) -ψ 0 |Ψ(t) ψ 0 (x, R), (5.6.1)
which is the wave-function Ψ(x, R,t) from which we removed the vibronic boundstate ψ 0 (x, R), as defined by Eq. (3.4.13), by projection. Introducing a distance x b for which |x| < x b corresponds to the bound part of the wave-function, we define the integrals:

C + (R,t) = +∞ x b | Ψ(x, R,t)| 2 , (5.6.2) C -(R,t) = x b -∞ | Ψ(x, R,t)| 2 .
(5.6.3)

These integrals represent the probability amplitude at a given internuclear distance R as a function of time t. with the electric field. As expected, we confirm that electrons are released to the right side at the maximum magnitude of the electric field during the second half-cycle, and to the left side at the maximum magnitude of the third half-cycle. Then, the position R grows with time, as observed previously.

Interpretation We can give a simple explanation of the phenomenon at stake within the BOA. Following ionization, the nuclear wave packet is promoted to the ionic potential energy curve and starts to move towards greater internuclear distances. When the electron recollides few tens of attoseconds after, it "sees" an ion with greater mean internuclear distance (see the picture provided in Fig. 5.32). As electrons are continuously recolliding during time, they can probe the evolution of the mean internuclear distance on the attosecond time-scale.

From C + and C -we can compute the mean internuclear distance for the positive and negative regions as follows:

R ± (t) = RC ± (R,t)dR C ± (R,t)dR .
(5.6.4)

We verified that the mean internuclear does not depend on x b . The analysis of Ψ(x, R,t) instead of Ψ(x, R,t) ensures that we can choose a small value of x b . Here, we retained x b = 5 a.u. We compare the results with the mean internuclear distance computed in the BOA by propagation of the neutral vibrational ground-state in the ionic potential (i.e. step 2 b in Fig. 5.32, as in [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF]). For this purpose, we need to solve the purely nuclear TDSE: acting on the nuclear wave function ξ (R,t), where µ is the reduced mass of the system. At initial time t = 0, the vibrational wave-function is the neutral vibrational ground-state, i.e. : ξ (R,t = 0) = χ 0 (R), because it presents strong oscillations due to the already mentioned presence of returning electrons adding to the newly ionized ones.

i ∂ ∂t ξ (R,t) = H N ξ (R,t) = - 1 2µ ∂ 2 ∂ R 2 +V NN (R) ξ (R,
Our BOA results are in agreement with the predictions of Lein [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF] and experiments of Baker et al [START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF], as both used the BOA to predict and explain their results. However, important discrepancies are observed between our correlated model and the BOA. First, nuclear motion in the correlated model seems not as fast as in the BOA. Secondly, we see that electrons are not ionized at the equilibrium mean internuclear distance but at about 0.09 a.u. greater, which is reminiscent to the work of Urbain et al [START_REF] Urbain | Intense-Laser-Field Ionization of Molecular Hydrogen in the Tunneling Regime and Its Effect on the Vibrational Excitation of H + 2[END_REF]. Indeed, as we can see in Fig. 5.30g-m, ionization is not aligned with the mean internuclear distance (which is about the maximum of the vibronic wavepacket) because the barrier is thiner at greater R. Hence, the correlation between electronic and nuclear motion plays a paramount role.

Effect on two-center interferences phase Because the mean internuclear distance is increasing with time after tunnel ionization, electrons then "see" a wider ion when they recombine. This consequently affects the signature of two-center interferences in the spectrum. Indeed, the destructive interference is observed at a lower harmonic order, due to the relation given by Eq. (5.4.2). Lein relied mainly on this effect to predict the possibility of probing vibrational dynamics on the attosecond scale [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF], what has been experimentally demonstrated by Baker et al [START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF][START_REF] Baker | Dynamic Two-Center Interference in High-Order Harmonic Generation from Molecules with Attosecond Nuclear Motion[END_REF], as already explained. Hence, we expect that the two-center interference phase-jump is also strongly affected by nuclear dynamics. We performed TDSE simulations with our correlated model and we extracted the phase of short and long trajectories, relative to an atomic reference of equal ionization potential, by the mean absorbing conditions as in Sec. 5.4.2. We used five different nuclei masses, m = 1 (H 2 ), m = 2 (D 2 ), m = 3 (T 2 ) and also m = 5 and m = 10 (in a.u.), in order to gradually decrease vibrational motion. Our results are presented in Fig. 5.34.

As we can see, the position of the destructive interference moves as expected, but the effect of nuclear motion is certainly more dramatical on the behavior of the phase-jump. The inversion of the phase jump is observed in the short trajectory contributions for H 2 and D 2 , and in both short and long trajectories for T 2 and the model molecule with m = 5. At high mass, m = 10, the nuclear motion is very slow and we observe once again the behavior of the uncorrelated case.

Three reasons are invoked to explain these results. First, the smaller the mass, the wider the neutral vibrational ground-state that is promoted to the ionic potential energy curve. Consequently, we observe an average of phase contributions over a large range of internuclear distance. Since we have shown that the value of the internuclear distance and the laser intensity are critical on the shape of the phase jump, this averaging must strongly modify the shape of the phase jump at particular intensities. Secondly, additional phase is accumulated during the propagation of the vibrational wave-packet and contributes to the total harmonic phase. Thirdly, as the evolution of the mean internuclear distance depends on time, it consequently depends on harmonic order because harmonics are emitted at different times. Hence, electrons are not equivalently recombining with the ion, which causes supplemental deviations from the case of a static molecule.

We thus would like to draw the experimentalist attention to the fact that the fea-tures we proposed to probe with QPI may be hard to reveal on H 2 molecule. One must use molecules with greater nuclei mass, such as CO 2 which behaves like a twocenter molecule due to the symmetry of the HOMO, and would not display much of a nuclear dynamics within a laser period.

Lochfraß vs impulsive stimulated Raman scattering

We have a robust method to describe HHG in diatomic molecules taking into account exactly the correlation between electronic and nuclear motion. Now, we place our system in the conditions for which Lochfraß has been predicted [START_REF] Goll | Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses[END_REF] in order to examine the relevance of this theory. Namely, we study the excitation of vibrational levels v in the EGS of the molecule in presence of a laser field. We used again a Ti:sapphire laser (ω L = 0.057 a.u.). As explained in Sec. 3.4.2, the study of the vibrational levels v in the EGS is carried out by projection of the correlated wavefunction on the vibronic states Ψ v (x, R) as expressed by Eq. (3.4.13).

Before the interaction with the laser pulse, the molecule is in the fundamental vibrational state v = 0 of the EGS. The initial wave function is thus the vibronic state Ψ 0 (x, R). The probability to be in the vibrational state v = 1 of the EGS is given by the square modulus of the projection of the wave function Ψ(x, R,t) (Eq. (3.4.3)) on the vibronic state Ψ 1 (x, R):

p (C) 1 (t) = dx dR Ψ * 1 (x, R)Ψ(x, R,t) 2 = dx dR ∑ v ϕ * 1,v (x)χ + v (R) ∑ w Φ w (x,t)χ + w (R) 2 = ∑ v ∑ w dx ϕ * 1,v (x)Φ w (x,t) dR χ + v (R)χ + w (R) δ v w 2 p (C) 1 (t) = ∑ v dx ϕ * 1,v (x)Φ v (x, t) 2 .
(5.6.8)

Here the superscript (C) stands for "correlated model".

We compare this probability with Lochfraß simulations performed as in [START_REF] Goll | Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses[END_REF]. The idea of this theory is to describe the vibrational motion of a diatomic molecule in presence of a laser field within the BOA, and introduce the ionization process as an imaginary potential equal to the anisotropic ionization rate Γ, which is responsible for the loss of population in the neutral molecule. The ionization rate depends on the internuclear distance R and the instantaneous magnitude of the electric field. Thus, the TDSE within these assumptions reads [START_REF] Goll | Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses[END_REF]:

i ∂ ∂t ξ (R,t) = - 1 2µ ∂ 2 ∂ R 2 +V NN (R) + E 0 (R) -i Γ(R,t) 2 ξ (R,t), (5.6.9)
and acts on the nuclear wave function ξ (R,t). We recall that V NN (R) + E 0 (R) corresponds to the potential energy curve of the neutral molecule. Before the interaction with the laser, the molecule is in its vibrational ground-state, i.e. ξ (R, 0) = χ 0 (R).

The ionization rate Γ(R,t) has different strengths along R, and is directly linked to the R-dependent ionization potential I p (R), defined as the energy difference between the ionic and the neutral potential energy curves. Hence, this is the cause of an anisotropic depletion of the nuclear ground-state, as depicted in Fig. 5.35. At a given time t, the vibrational wave-function ξ (R,t) is no longer an eigenstate of the EGS but a wave-packet: a vibrational motion is induced. The ionization rate, introduced as an imaginary potential, consequently drives the vibrational excitation in the EGS of the molecule. In Ref. [START_REF] Goll | Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses[END_REF], few points of the ionization rate are computed ab initio [START_REF] Saenz | Enhanced ionization of molecular hydrogen in very strong fields[END_REF], and the rest is interpolated and extrapolated using the Molecular-ADK (MO-ADK) rates [START_REF] Tong | Theory of molecular tunneling ionization[END_REF][START_REF] Saenz | Behavior of molecular hydrogen exposed to strong dc, ac, or low-frequency laser fields. II. Comparison of ab initio and Ammosov-Delone-Krainov rates[END_REF], which is an extension of the Ammosov-Delone-Krainov (ADK) theory for tunnel ionization [START_REF] Ammosov | Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field[END_REF] in the molecular case, taking into account a R-dependent ionization potential. Here, we used an approach which is consistent with our fully correlated model in order to enable the comparison. We computed the ionization rate from the exact resolution of the electronic TDSE, within the BOA:

i ∂ ∂t φ (x,t; R) = - 1 2 ∂ 2 ∂ x 2 +V Ne (x; R) + xE(t) φ (x,t; R) , (5.6.10)
where φ (x,t; R) are electronic wave-functions, R being a parameter. We used a constant electric field E L for each internuclear distance R. In this way, we computed Γ(R, E L ) for a range of amplitudes E L , which is then mapped into Γ(R,t) in the Lochfraß computations by taking the instantaneous electric field magnitude at each time t, i.e. :

Γ(R,t) = Γ(R, |E(t)|).
(5.6.11)

In this way, the exact ionization rate of our molecular system is plugged into our Lochfraß computations.

In the Lochfraß framework, the probability of populating vibrational state v = 1 is simply given by: p ( )

1 (t) = dR ξ * (R,t)χ 1 (R) 2 , (5.6.12)
where the superscript (ß) denotes Lochfraß simulations.

Vibrational excitation via Lochfraß relies on the fact that the ionization rate depends strongly on R. Therefore, in a system for which the ionization rate does not depend significantly on R, vibrational excitation should not be observed by solving

Internuclear distance R (a.u.) Eq. (5.6.9). In this way, we either studied H 2 and an artificial H 2 -like molecule where the neutral and ionic potential energy curves are parallel such as the ionization potential is constant all along R. It is set equal to the mean ionization potential of H 2 in the vibrational ground state, that is:

I p = dR I p (R)|χ 0 (R)| 2 = 0.5995 a.u.
(5.6.13)

The neutral potential is taken as the one of H 2 and we call this molecule G 2 . A scheme of these two systems is presented aside in Fig. 5.36. In Fig. 5. [START_REF] Zhao | Tailoring a 67 attosecond pulse through advantageous phase-mismatch[END_REF] we present the ionization rates for several laser intensities as a function of R. In H 2 , the ionization rate depends strongly on R. On the contrary, in G 2 the ionization rate is almost constant along R. Variations in the region of interest R ∈ [0.71 : 2.39] defined by the expansion of the neutral vibrational ground-state χ 0 are around 15% in G 2 while they are about 1100% in H 2 .

H 2 I = f(R) P I = const. P H 2 + G 2 G 2 +
We compare in Fig. 5.38 the probability p 1 for our correlated and Lochfraß simulations in H 2 and G 2 at a laser intensity of 3×10 14 W.cm -2 . Apart from the fact that v = 1 is less populated in G 2 than in H 2 , the overall temporal profiles of the probabilities for the correlated computations in H 2 and G 2 are very similar. Lochfraß does not reproduce either the temporal profile nor, and more importantly, the value at the end of the laser pulse of the probabilities p 1 computed with the fully correlated model. Moreover, while vibrational excitation is still strong in the case of G 2 , it dra- matically falls with Lochfraß (by a factor 2100), as expected, since the ionization rate is almost constant along R.

In Lochfraß process, very short laser pulses play an essential role [START_REF] Goll | Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses[END_REF], for it is claimed that in long pulses the effect of the R-dependence is washed out by nuclear motion. Yet, this requirement is also essential in ISRS, for which pulses of broad energy distribution are needed to effectively populate an excited state. In Fig. 5. 38 we observe that p 1 is greater for short pulses than for long ones, which is in accordance with ISRS. Moreover, the fact that v = 1 is still strongly populated in G 2 with a comparable temporal profile than in H 2 demonstrates that the origin of the vibrational excitation is different than Lochfraß. These considerations lead us to investigate whether the vibrational excitation is the consequence of ISRS.

As ISRS is a perturbative phenomenon, we derived the second order perturbation theory (2PT) in H 2 and G 2 , selecting the appropriate states. We assumed that the dominant contribution arises from the coupling between the EGS and the first excited electronic state (FEES) by the laser field. The first photon transition occurs from the vibrational ground-state | f , 0 of the EGS to the vibrational states |e, k , k ∈ N, of the FEES. The second photon transition happens from these vibrational states |e, k to first excited vibrational state | f , 1 of the EGS. The direct one-photon transition between | f , 0 and | f , 1 is forbidden by symmetry (selection rule). We start with the second order probability derived in Sec. 2.2.1, i.e. Eq. (2.2.13) combined with Eq.

(2.2.11):

p (2PT) n (t) = λ 4 ∑ k t 0 dt 2 ϕ n Ŵ (t 2 ) ϕ k e i(E n -E k )t 2 × t 2 0 dt 1 ϕ k Ŵ (t 1 ) ϕ i e i(E k -E i )t 1 2 , (5.6.14)
where the time-dependent perturbation is λ Ŵ (t) = x E(t) = x E L f (t) sin(ω L t) (see Eq. (3.2.33)), i.e. λ = E L and Ŵ (t) = x f (t) sin(ω L t). The states in consideration are: (5.6.17)

|ϕ i = | f , 0 ,
Hence, the probability for the system to be in | f , 1 as a function of time is given by:

p (2PT ) 1 (t) = E 4 L ∑ k f , 1 | x| e, k e, k | x| f , 0 t 0 dt 2 e i(E f ,1 -E e,k )t 2 f (t 2 ) sin(ω L t 2 ) × t 2 0 dt 1 e i(E e,k -E f ,0)t 1 f (t 1 ) sin(ω L t 1 ) 2 . (5.6.18)
The transition dipole matrix elements f , i | x| e, k (i = 0, 1) are computed in the BOA as:

f , i | x| e, k = dR χ f i (R)d f ,e (R)χ e k (R) , (5.6.19)
where χ f i (R) and χ e k (R) are the vibrational states of the EGS and FEES respectively, and:

d f ,e (R) = dx ϕ BO f (x, R)x ϕ BO e (x, R) , (5.6.20)
with ϕ BO f (x, R) and ϕ BO e (x, R) the fundamental and the first excited electronic states of the potential V Ne (x, R) respectively.

We also studied ISRS with a Λ-system, with analogous states of the involved molecular states. Two close states |0 and |1 , corresponding to v = 0 and v = 1 of the EGS, are coupled by the laser field to a third state |2 which is high in energy. The direct transition between |0 and |1 is forbidden. A scheme of the system is provided aside in Fig. 5.40. The energies E 0 and E 1 are set equal to the energies of the vibrational levels computed in the correlated model. The energy E 2 is adjusted a posteriori in order to reproduce the probability amplitude in H 2 . The corresponding TDSE reads:

E 0 E 1 E 2 V(t) V(t)
i    ȧ0 ȧ1 ȧ2    =    E 0 0 V 0 E 1 V V V E 2       a 0 a 1 a 2    , (5.6.21)
were a i are the time-dependent coefficients along the states |i . The time-dependent coupling V (t) is directly proportional to the electric field. The proportionality coefficient is estimated by integrating the transition dipole matrix elements over the vibrational states of the FEES. We solved the Λ-system with the RK4 algorithm (see Appendix A.3). The probability to find the system in state |i is simply:

p (Λ) i (t) = |a i (t)| 2 .
(5.6.22)

The results of these simulations are presented in Fig. 5.39. The overall temporal profiles for the 2PT and the Λ-system reproduce very accurately the correlated computations. This shows that, at intensity 3×10 14 W.cm -2 , the vibrational excitation is due to ISRS. However, while p 1 is 6.8 times weaker in H 2 than in G 2 for the correlated computations (see Fig. 5.38), it is 6.3 greater in the 2PT.

To explain these deviations, we examined the transition dipole matrix elements f , i | x| e, k for all the vibrational states k of the FEES. In the case of G 2 , we found that, fortuitously, one particular state |e, k p is strongly coupled to both | f , 0 and | f , 1 . Therefore, the evaluation of the probability p 1 within the 2PT is overestimated. This kind of state does not exist, however, in H 2 . We computed p 1 for G 2 by discarding only state k p , and we retrieved more acceptable amplitudes, i.e. 2.5 times weaker than in the case of H 2 , in better agreement with the predictions of the fully correlated model. The existence of this state in G 2 is closely linked to the particular symmetry of the system, and shows that the evaluation of the transition dipole matrix elements within the BOA is inaccurate. This state may not be present in the vibronic description of the molecule. Furthermore, it may be washed out by vibronic couplings during the interaction with the laser field. To this extent, we have shown once more the importance of the correlation between electronic and nuclear motion to describe strong-field processes in molecules.

We then studied p 1 at the end of the pulse as a function of laser intensity I L , to reach intensities used in [START_REF] Goll | Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses[END_REF] and [START_REF] Ergler | Quantum-Phase Resolved Mapping of Ground-State Vibrational D_{2} Wave Packets via Selective Depletion in Intense Laser Pulses[END_REF] and even higher ones. The results are presented in Fig. 5.41. As expected, the probability for the 2PT and the Λ-system depends on the square of the laser intensity. The probability for G 2 depends also on I 2 L for the correlated computations. In the case of H 2 , the slope is slightly greater than 2. This is attributed to a non-negligible contribution of a 4-photon transition, due to a high coupling between the EGS and the excited electronic states. The probability for the correlated simulations in G 2 is ∼10 times weaker than in H 2 . It is overestimated for the 2PT in G 2 , as we discussed it previously, while it seems to be underestimated in H 2 . In the case of Lochfraß, as already explained, it is expected to be almost canceled. Indeed, it is 2 orders of magnitude weaker in G 2 than in H 2 . These comparison leads us to the conclusion that even in high intensity regimes, the vibrational excitation observed is the consequence of ISRS.

As mentioned previously, the experiment of Ergler et al [START_REF] Ergler | Quantum-Phase Resolved Mapping of Ground-State Vibrational D_{2} Wave Packets via Selective Depletion in Intense Laser Pulses[END_REF] in D 2 has successfully created and detected vibrational wave-packets in the EGS of D 2 . In this pumpprobe experiment, vibrational motion was initiated and measured with the same IR laser pulse. From a Kerr-lens mode locked Ti:sapphire laser (795 nm wavelength) delivering 25 fs pulses, they tailored very breve pulses of 7 fs duration (FWHM) and peak intensity between 3 -5 × 10 14 W.cm -2 by the mean of post compression in a gas-filled hollow fiber. Using a Mach-Zehnder-type interferometer, they split the laser beam in two pulses separated by a time delay which could be varied from 0 to 3 ps. The first pulse, the pump, was used to excite the vibrational levels of the molecule, and the second, the probe, to ionize the molecule after a delay. The ions D + 2 and also the dissociated fragments D + are then detected, and their yield can be reported as a function of the time delay.

Following the vibrational excitation of the molecule by the pump, a vibrational wave-packet is created and starts to oscillate in the EGS of the molecule. In accordance with the R-dependence of the ionization rate, the ion yield may vary periodically with the position of the vibrational wave-packet. The periodicity of these variations must thus coincide with the oscillation period of the wave-packet. This was confirmed by their experimental observations. Therefore, they were able to reconstruct the oscillations of the vibrational wave-packet in the EGS of D 2 , with a period of T = 11.101 [START_REF] Faria | High-order harmonic generation in diatomic molecules: A quantum-orbit analysis of the interference patterns[END_REF] fs, the fastest nuclear motion in a molecule observed so far in the time domain. Based upon the very fact that the ionization rate depends on the internuclear distance R, this experiment was explained by Lochfraß theory. As we established that Lochfraß does not reproduce our fully correlated simulations, we now show furthermore that our model is able to explain the experiment of Ergler et. al. Indeed, we performed simulations in D 2 in the same experimental conditions. The laser peak intensity is 4 × 10 14 W.cm -2 and the wavelength is 795 nm. We computed the mean value of R and we reproduced perfectly the experimental data, as reported in Figure 5. [START_REF] Keldysh | Ionization in the field of a strong electromagnetic wave[END_REF] which shows the experimental data that we extracted from Ref. [START_REF] Ergler | Quantum-Phase Resolved Mapping of Ground-State Vibrational D_{2} Wave Packets via Selective Depletion in Intense Laser Pulses[END_REF] together with our simulations. As a consequence, we can state that in this experiment the formation of vibrational wave-packets in the EGS of D 2 is due to two-photon excitation induced by the short and intense low-frequency laser pulse.

We have once again demonstrated the importance of the correlation between electronic and nuclear motion to describe HHG. The exact resolution of a 1D×1D TDSE coupling nuclear and electronic dynamics served as an unbiased benchmark to compare with Lochfraß simulations, an uncorrelated model. By performing an exact derivation of the 2PT in the molecule with the appropriate states, and computations on a simple Λ-system where the energies and coupling are estimated with the help of the 1D×1D correlated system, we reproduced very faithfully the correlated simulations. Hence, vibrational excitation in the EGS is the consequence of ISRS, as already observed experimentally in SF 6 [START_REF] Wagner | Monitoring molecular dynamics using coherent electrons from high harmonic generation[END_REF]. Consequently, as our model reproduces very well the observations of the experiment carried out by Ergler et al, the vibrational excitation they observed in D 2 was caused by ISRS. This study shows that Lochfraß can only be seen as a qualitative description of the vibrational excitation, which was fortuitously able to reproduce the experiment. Since the actual cause of vibrational excitation in the EGS is the ISRS, Lochfraß cannot be assimilated as new phenomenon itself, contrary to what was claimed [START_REF] Goll | Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses[END_REF].

Relevant definition of the ionization potential

Ionization potential in diatomic molecules The description of tunnel ionization by intense low-frequency laser fields for diatomic molecules commonly makes use of an ionization potential which depends on the internuclear distance R. Indeed, in Refs. [START_REF] Tong | Theory of molecular tunneling ionization[END_REF][START_REF] Saenz | Behavior of molecular hydrogen exposed to strong dc, ac, or low-frequency laser fields. II. Comparison of ab initio and Ammosov-Delone-Krainov rates[END_REF], the Ammosov-Delone-Krainov (ADK) theory for tunnel ionization [START_REF] Ammosov | Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field[END_REF] has been extended to the molecular case, in particular for diatomic molecules, where a R-dependent ionization potential is defined as the energy difference between the potential energy curves of the neutral molecule and the ion. This new theory, called Molecular-ADK (MO-ADK), has been widely used to describe the ionization step within SFA [START_REF] Kjeldsen | Strong-field ionization of diatomic molecules and companion atoms: Strong-field approximation and tunneling theory including nuclear motion[END_REF][START_REF] Miloševi | Strong-field approximation for ionization of a diatomic molecule by a strong laser field[END_REF]. To some extent, the MO-ADK reproduces more accurately ab initio and experimental data than the ADK only This statement arises from the description of ionization within the BOA: when the electron is freed, the vibrational wave-packet is promoted onto the potential energy curve of the ion (see Fig. 5.32). In this picture, energy conservation law imposes that the energy required for the molecule to be ionized is equal to the energy needed to put the neutral vibrational wave-packet in the ionic potential energy curve. For this reason, the vertical energy difference between the neutral and ionic curves at equilibrium internuclear distance R eq = χ 0 |R| χ 0 is considered, which is equivalent to the definition of a R-dependent ionization potential I p (R), taken at R eq as in Ref. [START_REF] Chiril | Influence of nuclear vibration on harmonic generation in molecules[END_REF] 13 13 To be exact, they used here the energy difference between the vibrational ground-sate energy and the ionic potential energy curve . More precisely, to take into account the distribution in R of the vibrational groundstate χ 0 , or to describe ionization from vibrationally excited molecules, integration over R, weighted by the probability density of the wave-packet, is needed.

Notwithstanding good results provided by the use of I p (R) as defined previously [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF][START_REF] Chiril | Influence of nuclear vibration on harmonic generation in molecules[END_REF], Camille Lévêque showed during his PhD that the experimental results of Hans Jakob Wörner and coworkers on SO 2 could not be explained using this approach [START_REF] Lévêque | Pump-probe spectroscopy of vibronic dynamics using high-order harmonic generation: general theory and applications to SO2[END_REF]. In this experiment, vibrational motion in the EGS of SO 2 was initiated and probed using HHG. Surprisingly, no signature of the oscillations of the neutral vibrational wave-packet was observed, in disagreement with the predictions that make use of I p (R). These contradictions necessarily led to reconsider the theory, and more precisely, the definition of the ionization potential in molecules.

Because of the quantum nature of the nuclei, a set of vibrational eigenstates exists in the neutral and ionic electronic states. This energy quantization is omitted in the R-dependent approach. Treating the ionization potential as a continuous function of R is equivalent to allowing the vibrational wave-packet to make transitions to a continuous distribution of states. This approach can only be valid if the ionic potential energy surface is dissociative. If this is not the case, vibrational states of the ion are quantized: the promotion of the neutral vibrational wave-packet is exactly the projection on the ionic vibrational states. In fact, this is the well-known Franck-Condon principle that applies. Hence, ionization must not be described with an integration over R, weighted by the density of the vibrational wave-packet and using I p (R), but with the summation over the all the populated vibrational states, weighted by the Franck-Condon factors, defining a v-dependent ionization potential as the energy difference between the neutral an ionic vibrational states. Using this vibrational approach, Camille Lévêque was able to reproduce the experiment on SO 2 .

Therefore, we propose here to investigate closely the two approaches in order to corroborate the predictions settled by Camille Lévêque and inquiry whether we could rigorously discard one method to the benefit of the other. Being based on the BOA, both are approximate descriptions of tunnel ionization in diatomic molecules. However, comparison with our 1D×1D model, where ionization is consistently included and takes into account exactly the electron-nuclei correlation, will allow us to inquiry their relevance. This examination will be carried out through HHG. In the harmonic spectra, two features are strongly dependent on the ionization potential:

• the cutoff position,

• the phase difference between short and long trajectory contributions.

The latter manifests itself as interference patterns in the spectrum. They are easily observable in spectra generated by very short pulses of few cycles, as harmonics cannot be formed by repeated processed during successive laser cycles. As explained before, we have continuous spectra with all photon energies, shaped with the interference of short and long trajectories. Hence, we will use again 2-cycle sin 2 pulses with frequency of Ti:sapphire laser.

Computations In order to avoid the signature of two-center interferences in the spectra, which constitutes supplemental contributions that we are not interested in and that might contaminate our examination for being sensitive to nuclear dynamics, we used a single soft-Coulomb potential, i.e. V Ne (x, R) = V dia (x, R) as given by Eq. (3.2.5). We solve the purely electronic TDSE (in the BOA) given by Eq. (5.6.10), using different values of I p by adjusting the regularization parameter a of the potential. We then sum the spectra obtained according to the method under consideration. For the correlated model, a is a function of R which is again computed so that the electronic ground-state energy consistently reproduces energy difference between the BO potential energy curves.

In the R-approach, the FT of the dipole is computed as:

D (R) (ω) = FT dR |χ 0 (R)| 2 d R (t) = dR |χ 0 (R)| 2 D R (ω), (5.6.23)
where |χ 0 is the neutral vibrational ground state and d R (t) and D R (ω) are the dipole and its FT obtained for an ionization potential I p (R), as defined previously, at internuclear distance R. In the v-approach, the FT of the dipole is computed as:

D (v) (ω) = FT ∑ v χ + v |χ 0 2 d v (t) = ∑ v χ + v |χ 0 2 D v (t), (5.6.24)
where d v (t) and D v (ω) are the dipole and its FT obtained with the ionization potential being the energy difference between the neutral ground state |χ 0 and the ionic eigenstate |χ + v . Note that the sum is weighted by the square of the Franck-Condon factors χ + v |χ 0 because they intervene once in the ionization step and once in the recombination step.

Important deviations form conventional Franck-Condon distributions have been attributed to the rapid variation of the ionization rate with R [START_REF] Urbain | Intense-Laser-Field Ionization of Molecular Hydrogen in the Tunneling Regime and Its Effect on the Vibrational Excitation of H + 2[END_REF]. Here, the correct ionization rates are implicitly included in the dipoles d R (t) and d v (t) as computed by the exact resolution of the TDSE.

Toy model First, we used a test model designed to enhance the discrepancies between the R-approach and the v-approach. The neutral and ionic potential energy surfaces are represented in Fig. 5. [START_REF] Dewitt | Calculating the Keldysh adiabaticity parameter for atomic, diatomic, and polyatomic molecules[END_REF] for different situations. The neutral potential energy surface is harmonic with a small depth:

V neutral (R) = A 2 2 (R -R 0 ) 2 -B , (5.6.25)
with A = 0.6, B = 2 and R 0 = 3 a.u., and the ionic one is given by:

V NN (R) = - 3σ 2 (R -R 0 -R shift ) 2 + σ 2 +C(σ ) , (5.6.26)
where σ is varied between 0.05 and 0.8 to modulate the deepness of the well while adjusting C(σ ) to keep the ionic ground state at the same energy for each value of σ , so that the same I p is involved in the v-approach. Parameter R shift is varied from 0 to 0.3 a.u. and allows us to modify the equilibrium position of the ion relative to the neutral one. Let us start with the molecule at equilibrium, i.e. with the vibrational groundstate of the molecule. With such potentials, for low values of σ we see that, in the R-approach, the ionization potential depends strongly on R and that at equilibrium distance R eq its value is very small compared to the first I p (between the neutral and ionic ground-states) in the v-approach. Odd ionic states do not contribute to the v-approach when the neutral and ionic potentials are aligned (R shift = 0), since the Frank-Condon are zero for symmetry reason. However, when R shift = 0 they start to contribute.

In Figs. 5. [START_REF] Kulander | Dynamics of Short-Pulse Excitation, Ionization and Harmonic Conversion[END_REF] and 5. [START_REF] Corkum | Plasma perspective on strong field multiphoton ionization[END_REF] we present HHG simulations in 16 situations, by moving either R shift and σ as follows:

• R shift = 0, 0.1, 0.2 and 0.3 a.u.,

• σ = 0.15, 0.35, 0.55 and 0.75.

The laser intensity used is I L = 2.86 × 10 14 W.cm -2 (E L = 0.09 a.u.). In all the cases, the v-approach reproduces almost identically the correlated results. The Rapproach gives terrible results when σ is small, and better results when σ is high, i.e. when the energy levels of the ionic vibrational states are closer. This proves that the relevant definition of the ionization potential is the one based on the energy difference between vibrational levels. If the ionic states are tightened, we are in a situation close to a continuum of states and thereby the R-approach gives satisfactory results. Hydrogen molecule and isotopes To corroborate these conclusions, we performed equivalent simulations in "real" 1D molecules, such as H 2 , D 2 and T 2 , reported in Fig. 5.46. In these three cases, the greater the nuclei mass, the smaller the energy separation between the vibrational states of the ion, as shown in Table 5.5. Both R-approach and v-approach reproduce very accurately the correlated results. Differences between the R-approach and the v-approach, in favor of the latter, are still observed but are very small. They tend to reduce as nuclei mass increases. This confirms that R-approach is still very good in systems where ionic vibrational levels are very close. 4]]. We present the correlated simulations (green lines) and the spectrum from the R-approach (red dashed lines) and from the v-approach (black lines).

v = 1 v = 2 v = 3 v = 4
We also computed spectra starting from excited vibrational states v ∈ [ [1 : 4]], presented in Fig. 5.47. Again, though both approaches give very close results, the v-approach is in better agreement with the correlated simulations. Residual shift between the minima of interference patterns, also observed in Fig. 5.46 may be attributed to the contribution of excited vibrational states of the neutral molecule, populated during the HHG process by the laser pulse via ISRS, as described in the previous section.

To conclude, care must be taken when modeling tunnel ionization in molecules. The usual picture is to define a R-dependent ionization potential as the energy difference between the BO potential energy curves. However, we have shown that the rigorous definition is a vibrational ionization potential, taken as the energy difference between the involved vibrational states in the molecule and the ion. Although the R-approach gives good results in the case of H 2 and its isotopes, where the energy separation of vibrational states is very small, the v-approach is always the best one. Moreover, the predictions of the R-approach can be completely wrong, such as in SO 2 for which the v-approach has shown good agreement with the experiment [START_REF] Lévêque | Pump-probe spectroscopy of vibronic dynamics using high-order harmonic generation: general theory and applications to SO2[END_REF].

To finalize this study, we should first investigate R-approach and v-approach in the case of a dissociative ion. We expect that both methods would give exactly the same results. Secondly, we should examine both, in the case of vibrational excited molecules. By exciting vibrational levels of the neutral molecule, we create a vibrational wave-packet which evolves through time. The R-approach would integrate over this wave-packet for each ionization time while the v-approach will involve additional contribution of the excited vibrational states for which accumulated phase should be consistently taken into account. We insist on the fact that the dynamics of the vibrational wave-packet, or the accumulated phase, should be evaluated at ionization times t i . To do so, one must work in the Fourier domain and associate to each harmonic frequency ω an ionization time t i (ω) (which can be, for example, evaluated with SFA computations using the SPA). ---------------::::::-::::::::::::::::::-::////:/+//+o+osssssosooooooosssossyyyssyyyssyss ddhhdhhhhdddhyysssyhhyyyysysyhhhdhhhhyyssssssyssssoo++++++o++////+///:::::::::::::--------.-.----........ .----------::/:::::::-:-:::///://:++///:/+/++++++ooo+ooooss+ssssssssssssyyy hysyyhyyydhhyyyyssyysssyyshyyyyyyyyyysooooooosso+oooo/-:+sy+/////:::::::----------..........-.-..````........-----:::::::/:::::-:-::::::++////::////+++++/++ooooo/ososssssssosososyh sysssoossyyyysoyosooooyhhhyyyyyssosoo+ooooooo+o+++shy/.:++///::::---------.........``````....`````````````..... .------------::::/--::::://///:::://///++++++ooossssssssssoooo+//+ssy ssoooo+oooooossosooooosydhysssssooooo++o++++++oo+oyso+++++::::--------......````````````````` ``` ``````..............----::-:::::::::::::-:::///:++++ooosssssooooooo+/. .ooo oooooooo+++oosssooooooosyyssssosooo++++///////++o++////+/::--------......`````` `` ```````..`..........--------::::::::::::::////+++++ossssoo++oooo+o/`` .+oo +++oooo+o++oooooooooooosssssssoooo+++/////////////////:::-------....```````` ``````......`......-------::::::::::::.:///++++/++oooo++++sossyso++/ooss +++ooooooooossssosssooooooosssoso++///::::::/://::////::------........```` ` ```.`.--..```.....---::::-:::::::::::-://:://++++oooo++ossssysssssossss +++o+oo+ooossssss+ssooosoooo+oo+++//::-::::://:/:+/+//+/-------.....`````` ` `````.-.`.`......--:///:::::::/:/:::://///++++ooooo++ssssssssyysso/oy ++++oooossssyyyyssosoosooooo++++///:::--:---:::::/::::::.------....`````` ..`````.--------::::::://::/:///////+++//+++++ooosssossosssssyyyyy oooo++oosssyyysssosoooosoo+++++///:::------------::::-::::::--....`````` ` ` ```````.-/::--:/::::::::::::///////++++/++o+ooo++osoossosssyyyyyys ss+soooooosyyysssss+oooooo+++++///:::-------------:::::--:--....```````` ` ` `` ```````..---::--:::::::::-::///////++o+sooo+oo+o+osooosssysysysyyyh ssosyssoosoooooooooo+/+++o+++///////::-:---------.--------... ` ` `````.:+-::/::--:////++o+s++oso+osoo+//:+++o+o+/://++++o++++o+oo++///++/+oosssyyhhyd oosossssyyoyysso+yysoooooooyhmdyhysshhyyhmddhhsoss+++++/+///+/////+/:---///o//:.::---.---:------/:---:+//+oo+/:////++ooo+/+/+o+///+/:+/:+/:-+++++///++++//::+++o+++++++++oosyyhhshdm sssysyyyyyssssoosyyyysooo+ssss++o+ooyyyyshhhhhyysssssss+soooooo++++ooy+/:/:://:::////:::::::/:://///++//:::.-/+//++++s+/+/++///+/+ss/+++o/:./////////++/+//:/++ooo++o+ssosso+ossydmd ysyyyyyyyyyhysyyyyssyyyysosssoo+////oooo+++ooooooooo+++oo+++soo+++/+++///////:::/:-://-:::://///:://////+///o+///++++++/+//+//+++/++//////://::::::///+o+/ooo++++++ooyyysyysyyyoohdh ysyyyyyyyyysysysssssyyyyyyssoso+++++++/+++/+++++ooo++ooo+/++++++////+++////::/:::/+///://://///+/////+++///++///////++++++o++///////://////////::///++++o+ooooo+++osyhyssyysyys+:ydd yyyysyyyssysysyyyyysyyyyyhyooo+:/:////:///+/////++s++///+++o+++////++++////////::///////+++//////////+///////////////////++///++++++////+//+/++++///+/++++oo+++/::oshhyhyhhyyhysoydy ddddhhhyyhhhdhdhyyssyhhhshyoysysysoooo++/++o+o++/////////++++/++++/++++++////+///+//:::://///+//+/////////// We investigated HHG by the means of simple unidimensional models of atoms and molecules. Either with the exact numerical computations or by the application of the so-called SFA, we solved the TDSE for these models. The low computational cost required by the limited dimensionality of the systems and the consideration of only one active electron (SAE) allowed us to perform extensive simulations. In addition, we could incorporate "by hands" the desired features and get rid off unwanted ones, allowing deep investigation of each independently.

The development of analysis tools has been necessary to extract physical meaning, either from the dipole or directly from the wave-function. First, the study of the dynamics of harmonic generation from the dipole required the use of time-frequency analysis. Afterwards, harmonic emission times can be obtained. They correspond to the times when the electron recombines in its ground-state, and are equal in most cases to the recollision times. The exception is found when a pseudo bound-state in the continuum, i.e. a resonance, traps the electron, which can thus recombine further later according to the decay of the resonant state.

Among the large panel of available methods to compute the spectrogram of the dipole, we demonstrated that the STFT, or commonly called Gabor transform in strong-field physics, is more relevant than the wavelet transforms though they have experienced a renewed interest recently. In the STFT, however, such as in every spectrogram analysis, the results are obtained within a certain width -the Gabor limitinherent to the Fourier transform. We thus elaborated a new time-frequency analysis adapted to the dipole, based on the notion of instantaneous frequency as computed by the means of Hilbert transforms, which provides quantitative harmonic emission times for energies above ionization threshold. Its particularities allowed us to extract emission times for the first and second electron returns. Unfortunately, this method cannot be applied for greater returns.

The Wigner-Ville distribution has also been explored. This method has the subtlety not to be limited by the Gabor limit but is swiftly saturated by patterns of alternating negative and positive values that complicates the analysis. However, these patterns encode the interferences between successive processes and could make one able to extract valuable phase information. In this thesis, this method is also exposed in order to provide the reader with an understanding about the Wigner distribution of the wave-function. Indeed, we examined what information the Wigner distribution could provide as a direct analysis of the wave-function. Despites rapidly saturating interference patterns, as we explained it, we could picture tunnel ionization and identify electrons trajectories directly in the wave-function. This have been achieved in TDSE computations and verified through SFA and the additional use of saddle-point solutions. Moreover, the Wigner distribution displayed a nice picture of the dressing of the ground-state by the laser field. In this way, we could disclose the importance of dressing effects, that have been closely examined afterwards.

Thanks to classical computations, time-frequency analysis of the TDSE dipole, and corroborated by the saddle-point solutions from SFA, we know that harmonic generation is the consequence of two possible electron trajectories in the continuum. As their timing are significantly different relative to the time-scale given by a half laser cycle, the generation of a given harmonic may be different, for example, because it occurs with different conditions of dressing of the ground-state. To examine closely distinct effects on either short and long trajectories, we thus needed robust methods to separate their contributions in TDSE.

Since the temporal windowing of the dipole can possibly enlarge the results due to the width imposed by the Gabor limit, we first chose to act directly on the wavefunction. Classical computations allowed us to state that short and long trajectories are bounded in distinct regions, i.e. that there is a limit that short trajectories cannot go beyond and long trajectories always will. By absorbing the part that goes beyond, we remove the long trajectories from the wave-function and keep only the contribution of the short trajectories. To get only the long ones, we simply subtract the remaining part to the full wave-function propagated without absorbing conditions. The drawbacks of this method are the imperfect efficiency of absorbers which leads to residual participation of undesired trajectories and artificial cutoff enhancement. The second method relied on macroscopic considerations. Experimentally, harmonics are usually generated by focusing a Gaussian laser beam in a gas jet. The laser field and the generated radiation propagate, following Maxwell's equations. Interestingly, the propagation of the harmonic radiation is highly sensitive to the phase, and as it is different for short and long trajectories, one can spatially kill the long trajectories or split the two contributions. In the meantime, this method can be used to validate macroscopically our single-body conclusions.

We first studied the effect of shape resonance in HHG, with a very simple model for which a pseudo-bound state is introduced in the continuum with potential barriers above the ionization threshold. Their shape has been adjusted to fit experimental data and quantum chemistry computations for the shape resonance in N 2 . Nevertheless, this model is versatile and can be used to simulate any shape resonance. We put the emphasis of our study on the spectral phase. Only one center has been considered in order to avoid two-center interferences which would have added unwanted phase contributions to the total harmonic phase. In the case of N 2 , it is however very pertinent because two-center interferences have never been observed experimentally for this molecule [START_REF] Śpiewanowski | High-order-harmonic generation from field-distorted orbitals[END_REF]. According to theoretical and experimental data, we found that the shape resonance enhances the harmonic yield and conveys a smooth π-jump in the phase. Moreover, this phase jump seems to be identical for short and long trajectories and survives after propagation in the gaseous medium. Finally, it appears that the position of the resonant state and the phase jump are not very sensitive to the intensity of the laser. Discrepancies observed for short trajectories are attributed to numerous induced resonances which arose from the coupling between Rydberg states and the continuum by the laser field. This validates the use of the QRS theory to describe HHG with atoms and molecules presenting shape resonances.

However, our second study completely invalidated QRS. As a matter of fact, we took another look at the already intensively studied two-center interferences in HHG, with once again particular attention on the spectral phase. We found that dressing effects play a very important role in the shape of the phase-jump corresponding to destructive interferences. We confirmed that the phase-jump is smooth, as already observed, instead of being discontinuous and exactly π -what would predict the static recombination dipole matrix element. Furthermore, we observed that the phase-jump behaves differently for short and long trajectories and have a very peculiar comportment for the long trajectories. We could link this behavior to the instantaneous value of the laser field. We first studied recombination with a dressed ground-state, similarly to the recent work of Śpiewanowski et al [START_REF] Śpiewanowski | High-order-harmonic generation from field-distorted orbitals[END_REF][START_REF] Śpiewanowski | Field-induced orbital distortion in high-order-harmonic generation from aligned and oriented molecules within adiabatic strong-field approximation[END_REF], where ab inito field-dressed molecular orbitals are used within SFA. A very simple formulation of the dressing with the perturbation theory in a LCAO approach allowed us to derive analytically an expression of the dressed recombination dipole matrix element which reproduces qualitatively the behavior of the phase-jump. By manipulating molecular SFA equations with the use of Taylor expansions around atomic quantities, we were able to retrieve a modified expression of the recombination dipole matrix element, very close to the dressed one, which characterizes and explains completely the phase-jump. Finally, we predicted that our conclusions can be verified experimentally by comparison of QPI for aligned molecules with QPI of a reference with the same ionization potential. Whether the reference can be the unaligned molecule or needs to be an atom shall be investigated.

Finally, we studied nuclear vibration in diatomic molecules, with a 1D×1D model that correlates electronic and nuclear motion and consistently solves the TDSE. First, the direct study of the 1D×1D wave-function demonstrated that nuclear motion is initiated in the ion after tunnel ionization: the nuclei start to spread apart from each others and when the electron is brought back by the laser, it "sees" a larger molecular ion, in agreement with former predictions [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF] and experiments [START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF][START_REF] Baker | Dynamic Two-Center Interference in High-Order Harmonic Generation from Molecules with Attosecond Nuclear Motion[END_REF]. We then used this model to examine the effect of nuclear motion in the two-center interference phase-jump. Dramatical consequences are observed in H 2 and its isotopes, that completely change the behavior predicted for static molecules. However, molecules with high nuclear masses are not affected by nuclear motion, and what we predicted could still be observed in heavy molecules such as O 2 or CO 2 .

Our model for studying correlated electronic and nuclear dynamics also enabled us to reassess the theory called Lochfraß, and invalidate its physical basis. This model is a diverted way to explain vibrational excitation in the EGS with an uncorrelated model, which is, in fact, only Raman-like excitation. Our model could reproduce experimental data of a pump-probe observation of vibrational excitation in the EGS, that was previously explained by Lochfraß.

Based on these considerations, we also reviewed the definition of the ionization potential in molecules, and addressed the relevance of a R-dependent ionization potential. By the use of a toy model, we established that the correct approach is to consider an ionization potential that is defined as the energy difference between the vibrational states of the molecule and the ion. We then demonstrated, with the specific case of H 2 , that if the energy gap between vibrational levels is small, both R-dependent and vibrational approaches give very good results. Nonetheless, the vibrational approach still gave slightly better results. This study should be terminated with the examination of molecules prepared in a superposition of vibrational levels, and should inquiry how nuclear dynamics in the neutral molecule affect both approaches.

Since its discovery in the late 80s, HHG has aroused strong interest in the scientific community. It decisively opened the attosecond world to scientists through multipurpose applications in ultra-fast spectroscopy. Intense works have been carried out either theoretically or experimentally to understand HHG, control the generation of attosecond pulses and probe matter on the attosecond scale. Basis of HHG are now quite well understood, and thanks to the rapidly improving experimental facilities and computational powers, the community turns towards larger and more complex systems, such as liquids [START_REF] Dichiara | An investigation of harmonic generation in liquid media with a mid-infrared laser[END_REF] and water microdroplets [START_REF] Kurz | High-order-harmonic generation from dense water microdroplets[END_REF], crystals [START_REF] Ghimire | Observation of high-order harmonic generation in a bulk crystal[END_REF] and solids [START_REF] Der Linde | Generation of high-order harmonics from solid surfaces by intense femtosecond laser pulses[END_REF][START_REF] Dichiara | Scaling of High-Order Harmonic Generation in the Long Wavelength Limit of a Strong Laser Field[END_REF][START_REF] Vampa | Linking high harmonics from gases and solids[END_REF], and even nanoparticles [START_REF] Ganeev | High-order harmonic generation in nanoparticle-containing laser-produced plasmas[END_REF][START_REF] Ganeev | High-order harmonic generation in Ag nanoparticle-containing plasma[END_REF][START_REF] Shaaran | High-order-harmonic generation by enhanced plasmonic near-fields in metal nanoparticles[END_REF].

However, we have demonstrated that even with very simple models, and even by following the footsteps of widely examined problems, a lot of valuable information and physical understanding can be brought out from the shadows. In this way, we believe that a lot of crucial work still needs to be done in the fundamental basis of HHG.

Ultra-fast physics has a bright future, towards zeptosecond (10 -21 s) spectroscopy [START_REF] Hernández-García | Zeptosecond High Harmonic keV X-Ray Waveforms Driven by Midinfrared Laser Pulses[END_REF] or even yoctosecond (10 -24 s) [START_REF] Ipp | Yoctosecond Photon Pulses from Quark-Gluon Plasmas[END_REF]. However, one may remind that quantum mechanics is ruled by the time-energy uncertainty principle. Its consequences are already important in attoscience, for instance when defining delays in photoionization [START_REF] Caillat | Attosecond Resolved Electron Release in Two-Color Near-Threshold Photoionization of N_{2}[END_REF][START_REF] Gaillac | Attosecond photoemission dynamics encoded in real-valued continuum wave functions[END_REF][START_REF] Maquet | Attosecond delays in photoionization: time and quantum mechanics[END_REF][START_REF] Caillat | Photoionization dynamics: Transition and scattering delays[END_REF]. Indeed, these delays are defined from the maximum of an elec-tronic wave-packet that has a broad energy distribution, and become equivocal when the wave-packets present non-Gaussian structures. Hence, the physics of zeptosecond and yoctosecond may suffer from these limitations.

To go beyond Heisenberg and time-energy uncertainty principles, as we have seen through the study of the Wigner distribution, and to completely move aside physics of equations that may not be solved other than with numerical methods, scientists may turn towards physics of information, that have settled attractive presages on the informational nature of the universe.

A.1 Atomic units

Atomic units (abbreviated a.u.) are designed for atomic and molecular physics computations, to simplify the notations and get rid of redundant atomic constants that make analytical expressions cumbersome. It also allows to express numerical quantities in atomic relevant values instead of very small values in SI units (International System of units) that may be less intuitive. The following fundamental constants are set to unity:

• the electron mass (m e ),

• the elementary charge (e),

• the reduced Planck's constant (h = h/2π),

• and the Coulomb constant (1/4πε 0 ). This is often summarized in a dimensionless equation:

m e = e = h = 1 4πε 0 = 1. (A.1.1)
We gathered in Table A.1 the expressions of these fundamental constants and other physical constants encountered in our models, in atomic units. Their value in SI units is specified. Note also that 1 a.u. of time is 24.1888 as. Within these notations, no distinction is made between frequency and energy E, since a particle of frequency ω has an energy of E = hω = ω in atomic units. Note that rigorously, ω is the angular velocity, linked to frequency f by the relation ω = 2π f . However, it is usually called "frequency".

A.2 Energy of an electron in a discretized box

We demonstrate here the expression of the energy of an electron in a box of length L which is discretized with a step ∆x. The wave-function is represented as: This method is a fourth-order scheme, i.e. the error on each step is on the order of O(∆t 5 ) and the total accumulated error is on the order of O(∆t 4 ).

Ψ(x) = N x ∑ j=1 Ψ(x j )δ (x -x j
We recall that in Sec. 5.6.3, we solved a matrix equation operating on vector functions:

i    ȧ0 ȧ1 ȧ2    =    E 0 0 V 0 E 1 V V V E 2       a 0 a 1 a 2    ,
[recalling Eq. (5.6.21)] and in Sec. 5.3.3, the equation under consideration operated on scalar functions:

χ E (x) = -2 E -V (x) χ E (x), [recalling Eq. (5.3.2)]
where space variable x was considered as the time variable t we used above. Here, the equation involves the second derivative of χ E . Hence, its treatment requires to solve the system:

ξ (x) = -2 E -V (x) χ E (x) χ E (x) = ξ (x).
(A.3.7)

A.4 Strömgren's normalization procedure

We explain here the Strömgren procedure to normalize, on the energy scale, real scattering wave-functions obtained numerically in 1D, following Ref. [START_REF] Seaton | The Determination of Phases of Wave Functions[END_REF]. The considered scattering wave-function is obtained by searching the solution of the TISE: derivative:

χ E (x) = -2 E -V (x) χ E (x), [ recalling
χ (a) E = 1 √ π dk -1 2 dx sin(θ ) + k -1 2 d sin(θ ) dθ dθ dx = 1 √ π dk -1 2 dx sin(θ ) + k 1 2 d sin(θ ) dθ , (A.4.10) χ (a) E = 1 √ π   d 2 k -1 2 dx 2 sin(θ ) + H H H H H H H dk -1 2 dx k cos(θ ) + H H H H H H dk 1 2 dx cos(θ ) -k 3 2 sin(θ )   = 1 √ π d 2 k -1 2 dx 2 sin(θ ) -k 3 2 sin(θ ) = d 2 k -1 2 dx 2 k 1 2 -k 2 χ (a) E (x). (A.4.11)
We end up with a reformulation of the TISE as an equation on k:

d 2 k -1 2 dx 2 k 1 2 -k 2 + 2 E -V (x) = 0. (A.4.12)
We can solve approximately this equation by introducing A = 2 E -V (x) and κ = A 1 2 . We then have: from which one build: In the asymptotic region, k is slowly varying and thus one can state k k.

dκ -1 2 dx = dA -1 4 dx = - 1 4 dA dx A -5 4 , (A.4.13) d 2 κ -1 2 dx 2 = d 2 A -1 4 dx 2 = - 1 4
d 2 A dx 2 A -5 4 + 5
k2 = A + d 2 κ -1 2 dx 2 κ 1 2 = A - 1 4A d 2 A dx 2 + 5
Even states Starting with a cosine rather than a sine in the expression of the asymptotic wave-function, and following the same procedure in order to normalize even scattering wave-functions, does not change the final expressions of C and k.

A.5 Scattering-wave phase-shift

We develop here the expression of the phase-shift based on the Wronskian functional W:

η(E) = -arctan W[χ E , χ ref E ] W[χ E , χ ort E ]
,

[recalling Eq. (5.

3.3)]

in the specific case of 1D scattering wave-functions, in order to prove that it actually gives the desired phase-shift. Let us consider the even scattering wave in the asymptotic limit x → +∞, for which k = √ 2E:

χ E (x) = 1 √ πk cos(kx -η(E)), (A.5.1)
where η(E) is the unknown phase-shift relative to a reference scattering wave:

χ ref E (x) = 1 √ πk cos(kx). (A.5.2)
The orthonormal reference scattering wave is simply: We directly see that the determination of coefficients A and B allow the determination of η(E) as:

χ ort E (x) =
η(E) = arctan B A . (A.5.7)
The Wronskian functionals in Eq. (5.3.3) give:

W[χ E , χ ref E ] = χ E χ ref E -χ ref E χ E = (Aχ ref E + Bχ ort E )χ ref E -χ ref E (Aχ ref E + Bχ ort E ) = (-kAχ ort E + kBχ ref E )χ ref E + kχ ort E (Aχ ref E + Bχ ort E ) = kB[(χ ref E ) 2 + (χ ort E ) 2 ] = B π , (A.5.8) W[χ E , χ ort E ] = - A π . (similarly)
(A.5.9)

Finally, B A = - W[χ E , χ ref E ] W[χ E , χ ort E ] , (A.5.10)
and thus we retrieve the expression for the phase-shift η(E) given by Eq. (5.

3.3).

This treatment still works with Coulomb-like functions as long as the asymptotic behaviors are similar.

A.6 Complex dressed recombination dipole matrix element

We propose here to study the complex function:

z(x) = cos(x) + iα(x) sin(x),

S.1 Introduction

La Physique et la Chimie ont connu un essor immense depuis la formulation de la Mécanique Quantique au début du vingtième siècle. De façon remarquable, la capacité de cette nouvelle théorie à décrire l'infiniment petit, conformément à l'expérience, n'a encore jamais été mise en défaut. Intimement liée à un changement radical de paradigme, elle soulève toutefois de nombreux paradoxes alimentant les réflexions sur la nature de l'Univers. De multiples innovations technologiques ont alors émergé, permettant ainsi de sonder les propriétés fondamentales de la matière encore plus loin, tout en aboutissant à des applications industrielles et de la vie de tous les jours. Parmi elles, l'invention du laser est de loin l'une des plus remarquables. Basée sur le principe d'émission stimulée décrit par Einstein en 1917, elle est la conséquence immédiate de la découverte du maser (acronyme anglais de microwave amplification by stimulated emission of radiation, amplification microonde par émission stimulée de rayonnement) à la fin des années 50. Le laser (light amplification by stimulated emission of radiation, amplification de lumière par émission stimulée de rayonnement) la généralise au début des années 60 en rendant accessible la génération de rayonnement monochromatique à différentes longueurs d'onde en fonction du milieu amplificateur utilisé. Son principe peut être résumé de façon simple : un pompage électrique ou optique entre niveaux d'énergies atomiques permet une inversion de population. Les états excités se dépeuplent ensuite de façon radiative, émettant ainsi des photons de fréquence fixe correspondant à la séparation en énergie des niveaux mis en jeu. Le rayonnement est alors amplifié de façon cohérente dans une cavité, donnant lieu à une source intense de lumière cohérente. La laser a donc ouvert de grandes perspectives pour l'étude de l'optique non linéaire ou de l'interaction rayonnement-matière.

Depuis lors, les technologies de laser n'ont cessé d'évoluer, avec pour obsession d'atteindre des intensités toujours plus grandes, afin d'explorer les constituants fondamentaux de l'Univers dans des régimes extrêmes. Comme nous pouvons le voir sur la figure S.1, l'évolution de l'intensité lumineuse atteinte par les lasers n'a pas été linéaire mais marquée par des découvertes ponctuelles comme la Chirped Pulse Amplification (CPA) [1] qui consiste à amplifier une impulsion laser préalablement étalée spectralement, permettant ainsi de protéger le matériel optique de dommages irréversibles et d'atteindre de très grandes intensités après recompression finale de l'impulsion. Cette technique a non seulement permis de développer des lasers extrêmement intenses comme HERCULES [2] mais aussi, et sans doute de façon plus profitable pour la communauté scientifique, de rendre accessibles des sources intenses table-top, c'est à dire sur des installations de tailles raisonnables tenant littéralement sur une (grande) table de laboratoire.

Simultanément, de nombreuses avancées rendant possible la génération d'impulsions laser de plus en plus brèves ont vu le jour, aboutissant à des expériences de dynamique moléculaire à l'échelle de la femtoseconde (1 fs = 10 -15 s) réalisées avec un laser décrit comme « l'appareil photo le plus rapide du monde » par Zewail dans les années 80. Ces techniques ont alors permis la découverte simultanée par une équipe américaine [3] et par une équipe française [4] à la fin des années 80 d'un phénomène hautement non-linéaire, la Génération d'Harmoniques d'Ordres Élevés (GHOE, ou HHG pour High-order Harmonic Generation en anglais). La GHOE deviendra dix ans après un sujet de recherche très important dans la communauté scientifique grâce à la première démonstration expérimentale d'impulsions, générées par GHOE, de quelques centaines d'attosecondes (1 as = 10 -18 s) de l'ultra-violet (UV) à l'extrême UV (XUV) [5,6]. Elles offrent alors la possibilité de sonder la matière à des échelles de temps caractéristiques du mouvement électronique dans les molécules, ouvrant ainsi un pan entier de recherche: l'attoscience [START_REF] Scrinzi | Attosecond physics[END_REF][START_REF] Corkum | Attosecond science[END_REF][START_REF] Krausz | Attosecond physics[END_REF]. Les améliorations qui s'ensuivirent ont permis de générer des impulsions de quelques dizaines d'attosecondes [START_REF] Antoine | Attosecond Pulse Trains Using High-Order Harmonics[END_REF][START_REF] Antoine | Generation of attosecond pulses in macroscopic media[END_REF][START_REF] Christov | Generation and propagation of attosecond x-ray pulses in gaseous media[END_REF][START_REF] Paul | Observation of a Train of Attosecond Pulses from High Harmonic Generation[END_REF][START_REF] Mairesse | Attosecond Synchronization of High-Harmonic Soft X-rays[END_REF][START_REF] Dudovich | Measuring and controlling the birth of attosecond XUV pulses[END_REF][START_REF] Gaarde | Macroscopic aspects of attosecond pulse generation[END_REF], comme présenté en figure S.2.

La GHOE offre de multiples applications en spectroscopie ultra-rapide. Elle peut être utilisée dans un premier temps de façon indirecte, en utilisant les impulsions XUV préalablement générées comme pompes ou sondes de processus moléculaires détectés ensuite par photoémission. Par exemple, ces impulsions ont été utilisées pour sonder la photoionisation dans un milieu gazeux comme l'hélium [START_REF] Larsson | Two-colour time-resolved spectroscopy of helium using high-order harmonics[END_REF] et l'argon [START_REF] Klünder | Probing Single-Photon Ionization on the Attosecond Time Scale[END_REF], pour révéler des résonances dans le diazote [START_REF] Haessler | Phase-resolved attosecond near-threshold photoionization of molecular nitrogen[END_REF][START_REF] Caillat | Attosecond Resolved Electron Release in Two-Color Near-Threshold Photoionization of N_{2}[END_REF] et de potentielles transitions interdites de spin dans SO 2 [START_REF] Lévêque | Direct observation of spin-forbidden transitions through the use of suitably polarized light[END_REF]. Elles ont également été employées sur de la matière condensée [START_REF] Cavalieri | Attosecond spectroscopy in condensed matter[END_REF], pour sonder le déclin Auger [START_REF] Drescher | Time-resolved atomic inner-shell spectroscopy[END_REF] et l'ionisation tunnel [START_REF] Uiberacker | Attosecond real-time observation of electron tunnelling in atoms[END_REF] en temps réel, et permettent également la reconstruction du champ électrique laser instantané grâce à la technique dite du streaking (caméra à balayage de fente attoseconde) [START_REF] Constant | Methods for the measurement of the duration of high-harmonic pulses[END_REF][START_REF] Itatani | Attosecond Streak Camera[END_REF].

Dans un second temps, et de façon très intéressante, la GHOE peut aussi servir d'outil auto-sonde pour explorer directement la dynamique attoseconde dans le milieu générateur [START_REF] Haessler | Self-probing of molecules with high harmonic generation[END_REF]. En effet, la structure orbitalaire du système générateur et les processus dynamiques sous-jacents à la GHOE sont encodés dans le rayonnement, qui, grâce à sa nature cohérente, permet leur reconstruction complète ou partielle. Par exemple, la GHOE a été utilisée pour sonder la vibration de molécules telles que H 2 ou son isotope D 2 à l'échelle de l'attoseconde [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF][START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF][START_REF] Baker | Dynamic Two-Center Interference in High-Order Harmonic Generation from Molecules with Attosecond Nuclear Motion[END_REF], pour obtenir des images de ce résumé, les équations sont exprimées en unités atomiques (u.a.) pour lesquelles la masse m et la charge e de l'électron, ainsi que la constante de Planck réduite h et la constante de Coulomb 1/4πε 0 sont fixées à 1.

S.2 La génération d'harmoniques d'ordres élevés

La GHOE a été découverte à la fin des années 80 par irradiation de gaz rares avec des impulsions monochromatiques intenses obtenues par un laser KrF * (longueur d'onde de 248 nm) [3] ou un laser Nd:YAG (1064 nm) [4], à des intensités entre 10 13 et 10 15 W.cm -2 . Dans ce régime, le gaz émet alors un rayonnement qui peut être collecté. Il s'agit d'un rayonnement harmonique de la fréquence fondamentale du laser qui possède des propriétés intéressantes. D'une part, les harmoniques sont uniquement des multiples impairs de la fréquence laser et atteignent des ordres élevés. D'autre part, le spectre a une forme bien particulière (cf. figure S.3) : après une rapide décroissance des premières harmoniques, on observe un long plateau où l'intensité des harmoniques est presque constante jusqu'à une coupure (le cutoff en anglais) au delà de laquelle l'intensité harmonique chute très rapidement.

Un modèle simple, semi-classique, pour expliquer ce phénomène a été proposé par Corkum [START_REF] Corkum | Plasma perspective on strong field multiphoton ionization[END_REF]. Ce modèle en trois étapes est basé sur le constat qu'à ces régimes de fréquence et d'intensité laser, le potentiel atomique vu par un électron est fortement perturbé, à tel point que lorsque la valeur du champ laser instantané est très élevée, l'électron "voit" un potentiel résultant présentant une barrière qu'il peut traverser par effet tunnel (figure S.4(1)). Dans un second temps, l'électron libre est accéléré dans le continuum sous l'effet du champ laser oscillant. Il est d'abord éloigné puis, lorsque le champ change de signe, ramené sur l'ion (figure S.4(2)). À cet instant de recollision, l'électron a une probabilité non nulle de se recombiner dans l'état fondamental en libérant l'énergie cinétique accumulée sous forme de rayonnement lumineux. L'ensemble des photons émis constitue ainsi le spectre de GHOE. Si l'électron ne se recombine pas, il continue sa course et, un demi-cycle laser plus tard, peut être ramené à nouveau sur l'ion, pouvant émettre encore une fois de la lumière, et ainsi de suite. Nous appelons cela les retours multiples. Ce modèle décrit avec succès la GHOE et donne des résultats en accord qualitatif avec l'expérience et les résolutions ab initio de l'ESDT. Il considère que l'électron libre se comporte comme une particule classique chargée dans un champ électrique oscillant, et permet ainsi de calculer ses trajectoires dans le continuum en résolvant les équations de Newton. Considérons x la position de l'électron le long de la polarisation du laser d'amplitude E L et de fréquence ω L telles que E(t) = E L cos(ω L t). Trois points majeurs découlent de l'étude de ces trajectoires :

• Il existe deux trajectoires possibles conduisant à la même énergie cinétique de l'électron à l'instant de recollision, correspondant donc à la même énergie du photon émis. La première correspond à un électron libéré avant et revenant sur l'ion après la seconde. On peut alors les classer selon leur durée: la trajectoire longue et la trajectoire courte.

• Il existe une énergie cinétique maximale atteinte par l'électron lui permettant de revenir sur l'ion ; les électrons plus énergétiques ne peuvent revenir. Cette énergie correspond donc à l'énergie maximale observable sur un spectre: il s'agit de la coupure. Elle est donnée par E c,max = 3.17U p , où U p = E 2 L /4ω 2 L est l'énergie pondéromotrice de l'électron dans le champ laser. La fréquence correspondant à la coupure est donc Ω cut = I p + E c,max = I p + 3.17U p .

• Les trajectoires courtes et longues sont également spatialement séparées par une distance caractéristique x α = E L /ω 2 L . Les trajectoires courtes sont bornées par x α , tandis que les trajectoires longues dépassent nécessairement cette valeur.

Nous verrons comment cette dernière propriété nous a permis de séparer les trajectoires courtes des longues. Nous allons maintenant présenter succinctement les méthodes théoriques utilisées pour résoudre l'ESDT et simuler la GHOE pour des atomes ou des molécules diatomiques.

S.3 Méthodes de simulation

S.3.1 Calculs ab initio à une dimension

Dans un premier temps nous proposons de résoudre numériquement et de façon exacte une simplification adaptée de l'ESDT avec les approximations suivantes :

• Nous ne considérons qu'un seul électron actif, subissant l'effet d'un potentiel effectif dû au(x) noyau(x) et aux autres électrons.

• L'électron est non-relativiste dans les régimes d'intensités et de fréquences laser considérés.

• Le champ électrique du laser est classique car, mettant un jeu un nombre très important de photons, sa quantification est négligeable.

• On se place dans l'approximation dipolaire, ce qui revient à négliger les variations spatiales du champ à l'échelle atomique, car la longueur d'onde du laser est grande devant la longueur d'onde de De Broglie de l'électron. Le potentiel d'interaction avec le laser s'écrit donc, comme présenté précédemment: Vint (t) = xE(t).

• Dans le cas de molécules, nous considérons dans un premier temps l'Approximation de Born-Oppenheimer (ABO). Les noyaux étant beaucoup plus massifs que l'électron, ils sont considérés comme fixes : leur mouvement n'est pas corrélé à ceux de l'électron.

• Pour les mêmes raisons, on néglige l'interaction directe entre le champ laser et les noyaux.

Dans ce cadre, l'ESDT s'écrit : Des conditions absorbantes sont ajoutées aux limites de la boite de simulation afin d'éviter la réflexion des électrons les plus énergétiques qui ne reviennent normalement pas sur l'ion. On utilisera indistinctement une fonction absorbante qui décroit en cos 1/8 aux extrémités et que l'on multiplie à la fonction d'onde, ou un potentiel complexe absorbant [START_REF] Sahoo | The complex absorbing potential method (CAP) to study the Stark effect in hydrogen and lithium[END_REF][START_REF] Muga | Complex absorbing potentials[END_REF].

i ∂ ∂t Ψ(x,t) = - 1 2 ∂ 2 ∂ x 2 +V (x) + xE(t) Ψ(x,
La connaissance de la fonction d'onde à chaque instant nous permet de calculer le dipôle comme étant la valeur moyenne de l'accélération, en invoquant le théorème d'Ehrenfest :

a(t) = d 2 x dt 2 = d 2 x dt 2 = dV dx + E(t) = Ψ(t) dV dx + E(t) Ψ(t) , (S.3.3)
et dont la Transformée de Fourier (TF) donne le spectre harmonique : où les éléments de matrice dipolaire d'ionisation et de recombinaison sont : Dans le cas des molécules diatomiques, nous utilisons le SFA moléculaire développé dans les références [START_REF] Chiril | Strong-field approximation for harmonic generation in diatomic molecules[END_REF][START_REF] Faria | High-order harmonic generation in diatomic molecules: A quantum-orbit analysis of the interference patterns[END_REF], avec encore une fois pour différence l'utilisation de la forme vitesse du dipôle. Le calcul de l'élément de matrice dipolaire de recombinaison est conduit en approchant l'orbitale moléculaire (symétrique) par une combinaison linéaire d'orbitales atomiques (LCAO pour Linear Combination of Atomic Orbitals), dont nous donnons l'expression finale : Similairement au cas atomique, l'étude de la phase stationnaire abouti à résoudre les équations saddle-point suivantes : Pour cela, nous avons développé un modèle 1D×1D qui couple le mouvement des électrons et des noyaux ensemble. Nous ne nous limitons alors plus à l'ABO. En posant R la distance internucléaire et x la position de l'électron le long de l'axe internucléaire, l'Hamiltonien du système est : est résolue en exprimant la fonction d'onde totale du système Ψ(x, R,t) dans la base {|χ + v } des états vibrationnels de l'ion : où ε v est l'énergie de l'état |χ + v . Cette équation est résolue en utilisant la méthode du split-operator, dont les détails techniques sont écartés de ce résumé.

D(ω) = TF[a](ω) = a(t)
d ion (k) = k xE(t ) ϕ 0 , (S.3.7) d rec (k) = ϕ 0 |-i ∇| k • e x , (
d rec (k) = k • e x 2 1 + w(R) φs (k) cos k • R 2
d ion (k) = -i 2 1 + w(R) ∂ φs ∂ k cos k • R 2 -φs (k) R 2 sin k • R 2 • E(t)
H 0 = - 1 2µ ∂ 2 ∂ R 2 - 1 2
∂ 2 ∂ x 2 +V NN (
Ψ(x, R,t) = N v -1 ∑ v =0 Φ v (x,t)χ + v (R

S.4 Outils d'analyse S.4.1 Analyse temps-fréquence

Comme nous l'avons expliqué, des outils d'analyse sont nécessaires pour comprendre les résultats obtenus à partir de nos simulations et extraire des informations physiques intéressantes. Les premiers outils que nous avons utilisés permettent une analyse temps-fréquence. Nous avons d'abord examiné précisément la TF fenêtrée (ou communément appelée transformée de Gabor) en comparaison avec les transformées en ondelettes devenues populaires dans le domaine de la GHOE [START_REF] Antoine | Time profile of harmonics generated by a single atom in a strong electromagnetic field[END_REF][START_REF] Daniele | Wavelet Analysis of the Spectrum Emitted by a One-Dimensional Atom Driven by a Strong Laser Pulse[END_REF][START_REF] Chen | Time-frequency analysis of molecular high-harmonic generation spectrum by means of wavelet transform and Wigner distribution techniques[END_REF][START_REF] Chelkowski | High-order-harmonic generation from coherent electron wave packets in atoms and molecules as a tool for monitoring attosecond electrons[END_REF][START_REF] Telnov | Effect of nuclear vibration on high-order-harmonic generation of aligned H + 2 molecules[END_REF][START_REF] Jun | Probing dynamic interference in high-order harmonic generation from long-range molecular ion: Bohmian trajectory investigation[END_REF][START_REF] Miller | High-harmonic spectroscopy of laser-driven nonadiabatic electron dynamics in the hydrogen molecular ion[END_REF]. Notre conclusion est que la transformée de Gabor est bien plus adaptée à la GHOE que les transformées en ondelettes, ces dernières introduisant des déformations qui ne conviennent pas aux processus physiques étudiés.

Le constat que toutes ces méthodes d'analyse temps-fréquence conduisent à des spectrogrammes dont la résolution est limitée par la limite de Les distributions présentent cependant des structures d'interférences qui alternent rapidement entre valeur positives et valeurs négatives, ces dernières rendant équivoques la notion de probabilité de cette distribution. Nous montrons cependant que ces interférences encodent des informations de phase très importantes et peuvent être extraites pour comprendre plus finement la façon cohérente dont les harmoniques s'additionnent.

S.4.2 Séparation des trajectoires courtes et longues

Comme les électrons ayant une trajectoire courte se recombinent avant les électrons ayant une trajectoire longue, les caractéristiques dynamiques encodées dans la GHOE sont susceptibles d'avoir des signatures différentes. Pour examiner cela, nous devons trouver un moyen de séparer les contributions issues des trajectoires courtes de celles des longues. Dans le cas du SFA, cette séparation est immédiate avec la recherche des solutions saddle-point. En revanche, elle est plus difficile à effectuer à partir des calculs ab initio. Nous avons étudié trois possibilités. La première se base sur l'analyse temps-fréquence et consiste à séparer temporellement les trajectoires courtes des longues à l'aide d'une fenêtre, comme par exemple dans la référence [START_REF] Śpiewanowski | Field-induced orbital distortion in high-order-harmonic generation from aligned and oriented molecules within adiabatic strong-field approximation[END_REF]. Nous montrons qu'elle n'est pas adaptée car elle introduit une largeur inhérente à la méthode, nous empêchant d'étudier des contributions et des caractéristiques fines avec précision.

La seconde consiste à utiliser le fait que les trajectoires sont spatialement distinctes, comme nous l'avons expliqué plus haut. Ainsi, en plaçant un absorbeur à la position x α , nous pouvons absorber les trajectoires longues (qui nécessairement iront plus loin que x α ). On obtient donc la partie de la fonction d'onde correspondant à l'état lié et les trajectoires courtes. Les trajectoires longues sont alors retrouvées en soustrayant cette partie de fonction d'onde à la fonction d'onde totale, propagée en l'absence d'absorbeur à x α .

Enfin, la dernière méthode repose sur le processus physique de propagation du rayonnement dans le milieu gazeux. La forme gaussienne du faisceau laser focalisé sur le gaz fait que les harmoniques générées dans le milieu s'additionnent de façon particulière. L'accord de phase entre les harmoniques est différent entre celles issues des trajectoires longues et celles issues des trajectoires courtes. De plus, cet accord dépend de la largeur et de la position du jet de gaz par rapport au point focal du faisceau. En modifiant ces paramètres, il est possible de séparer trajectoires longues, qui se retrouvent sur un anneau hors-axe, et trajectoires courtes qui se retrouve dans l'axe du faisceau laser. En outre, cette méthode nous permet de vérifier si les prédictions que nous effectuons dans le cas d'un atome ou d'une molécule isolée sont toujours valables dans le cas d'une réponse collective d'un milieu gazeux. Nous ne rentrons pas ici dans les détails de simulation, et renvoyons le lecteur aux références [START_REF] Gaarde | Macroscopic aspects of attosecond pulse generation[END_REF][START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF][START_REF] Huillier | Propagation effects in high-order harmonic generation in rare gases[END_REF][START_REF] Huillier | Theoretical aspects of intense field harmonic generation[END_REF][START_REF] Geissler | Light propagation in field-ionizing media: extreme nonlinear optics[END_REF]. En bref, nous résolvons les équations de Maxwell dans le domaine de Fourier en incluant rigoureusement la polarisation du milieu comme proportionnelle au dipôle, ce dernier étant calculé pour différentes valeurs d'intensité laser en amont.

S.4.3 Interférences de chemins quantiques

Les interférences de chemins quantiques, nommées QPI en anglais (pour Quantum Path Interferences) constituent un outil intéressant pour l'analyse de processus attoseconde induisant des différences de phase entre trajectoires courtes et trajectoires longues [START_REF] Zaïr | Quantum Path Interferences in High-Order Harmonic Generation[END_REF][START_REF] Auguste | Theoretical and experimental analysis of quantum path interferences in high-order harmonic generation[END_REF][START_REF] Holler | Ionization effects on spectral signatures of quantum-path interference in high-harmonic generation[END_REF][START_REF] Schapper | Spatial fingerprint of quantum path interferences in high order harmonic generation[END_REF][START_REF] Zaïr | Molecular internal dynamics studied by quantum path interferences in high order harmonic generation[END_REF]. En effet, elles consistent simplement à calculer ou mesurer l'amplitude d'une harmonique donnée en fonction de l'intensité du champ laser. La différence de phase entre trajectoires courtes et trajectoires longues évoluant, pour une harmonique donnée, en fonction de l'intensité laser, on observe des structures d'interférences alternant entre destruction et construction cohérente de lumière. Des changements dans cette différence de phase seront donc directement encodés dans ces structures.

S.4.4 Distribution de Wigner

La distribution de Wigner est similaire à la transformée de Wigner-Ville mais traite directement de la fonction d'onde [START_REF] Wigner | On the quantum correction for thermodynamic equilibrium[END_REF]. Elle permet son étude dans l'espace des phases [START_REF] Bauke | Visualizing quantum mechanics in phase space[END_REF][START_REF] Donoso | Quantum Tunneling Using Entangled Classical Trajectories[END_REF][START_REF] Kull | Position-momentum correlations in electron-ion scattering in strong laser fields[END_REF]. De même que la transformée de Wigner-Ville, elle possède des valeurs négatives qui rendent équivoque le sens de distribution de probabilité [START_REF] Bartlett | Negative probability[END_REF][START_REF] Bondar | Wigner function's negativity demystified[END_REF]. Donnant une image claire de la cohérence entre les états quantiques et des processus de décohérence, elle a connu un regain d'intérêt dans la communauté des atomes et photons piégés en cavités [START_REF] Kurtsiefer | Measurement of the Wigner function of an ensemble of helium atoms[END_REF][START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF][START_REF] Bertet | Direct Measurement of the Wigner Function of a One-Photon Fock State in a Cavity[END_REF][START_REF] Deléglise | Reconstruction of non-classical cavity field states with snapshots of their decoherence[END_REF] et possède des propriétés de projection intéressantes [START_REF] Bauke | Visualizing quantum mechanics in phase space[END_REF]. Un moyen de faire disparaître les interférences de la distribution de Wigner est d'utiliser la distribution de Husimi [START_REF] Husimi | Some Formal Properties of the Density Matrix[END_REF][START_REF] Harriman | Some properties of the Husimi function[END_REF][START_REF] Mundarain | Husimi's Q(α) function and quantum interference in phase space[END_REF]. Nous ne l'étudierons cependant pas, car nous montrerons que les informations de phases de nature quantique sont primordiales.

S.5 Résultats

S.5.1 Distribution de Wigner

L'intérêt tout particulier donné à la distribution de Wigner est qu'elle permet une visualisation claire de la cohérence d'un système quantique. Les structures d'interférences sont l'expression directe d'une information de phase traduisant la cohérence du système. Dans des systèmes complexes hautement corrélés, elles saturent rapidement l'espace de phases avec des structures inférieures à l'échelle de Planck, c'est à dire en dessous de la limite imposée par le principe d'incertitude de Heisenberg ∆t∆E ≥ h/2 [START_REF] Zurek | Sub-Planck structure in phase space and its relevance for quantum decoherence[END_REF]. Bien que non observable physiquement, cette échelle joue un rôle primordial dans la décohérence [START_REF] Zurek | Sub-Planck structure in phase space and its relevance for quantum decoherence[END_REF]. En mettant de côté les considérations fondamentales de mécanique quantique soulevées par la distribution de Wigner [143-145, 147, 148, 153-155], nous avons envisagé d'utiliser cette dernière comme un outil pour comprendre les processus quantiques liés à la GHOE en étudiant directement la fonction d'onde du système. Certains travaux théoriques ont déjà été effectués dans l'étude des processus en champ fort comme l'ionisation tunnel [START_REF] Watson | Quantum signatures in the stabilization dynamics[END_REF][START_REF] Czirják | The Wigner function for tunneling in a uniform static electric field[END_REF][START_REF] De Bohan | Dynamique de l'interaction laser-atome: moment canonique et approximation du champ fort[END_REF][START_REF] Baumann | Wigner representation of ionization and scattering in strong laser fields[END_REF], des effets d'habillage des états par le laser [START_REF] Watson | Quantum signatures in the stabilization dynamics[END_REF] et la diffusion d'électrons sur un potentiel [START_REF] Kull | Position-momentum correlations in electron-ion scattering in strong laser fields[END_REF][START_REF] Baumann | Wigner representation of ionization and scattering in strong laser fields[END_REF]. Le flux de Wigner a également été étudié et a permis de suivre un ordre topologique dans des processus quantiques [START_REF] Steuernagel | Wigner Flow Reveals Topological Order in Quantum Phase Space Dynamics[END_REF]. Cependant, aucune expérience n'a permis de reconstituer la fonction de Wigner pour des atomes ou molécules en champ fort.

L'étude de la distribution de Wigner issue de calculs ab initio nous permet de donner une image claire de l'ionisation tunnel, confirmant ainsi les travaux antérieurs, et de la recombinaison. De plus, elle permet de visualiser l'habillage de l'état fondamental par le laser. Malgré les interférences, on identifie des trajectoires électroniques. La comparaison avec la fonction de Wigner issue de calculs SFA à laquelle nous superposons les solutions saddle-point nous permet de confirmer ces trajectoires et d'identifier les retours multiples, comme présenté en figure S.6. Dans cette figure, nous représentons la fonction de Wigner SFA en x = 0 afin de visualiser la quantité de mouvement (donc la vitesse) de l'électron lorsqu'il se recombine avec l'ion, en fonction du temps.

S.5.2 Résonance de forme dans N 2 2 2

En collaboration avec les expérimentateurs du LIDyL (IRAMIS, CEA, Saclay), nous avons cherché à modéliser la GHOE dans le diazote, en mettant l'accent sur une résonance de forme large observée dans les spectres de photoionisation à 15 eV au dessus du seuil d'ionisation (I p = 15.58 eV) [START_REF] Davenport | Ultraviolet Photoionization Cross Sections for N 2 and CO[END_REF][START_REF] Dehmer | Shape-Resonance-Enhanced Nuclear-Motion Effects in Molecular Photoionization[END_REF][START_REF] Lucchese | Studies of differential and total photoionization cross sections of molecular nitrogen[END_REF][START_REF] Plummer | Partial photoionization cross sections of N 2 and CO using synchrotron radiation[END_REF][START_REF] Lofthus | The spectrum of molecular nitrogen[END_REF][START_REF] Raoult | Ab initio approach to the multichannel quantum defect calculation of the electronic autoionisation in the Hopfield series of N 2[END_REF][START_REF] Dehmer | = 0 and 1 near threshold[END_REF][START_REF] Zubek | Near-threshold photoionisation studies of N 2[END_REF][START_REF] Cacelli | Gaussian-type-orbital basis sets for the calculation of continuum properties in molecules: The differential photoionization cross section of molecular nitrogen[END_REF][START_REF] Dehmer | Shape Resonances in K-Shell Photoionization of Diatomic Molecules[END_REF].

Pour ce faire, nous utilisons un potentiel effectif V (x) avec des barrières au dessus du seuil d'ionisation :

V (x) = -1 √ x 2 + a 2 + V 0 + 1 √
x 2 + a 2 B + (x) + B -(x) , le comportement particulier observé sur le saut de phase, dans le cas des trajectoires courtes.

Cette étude permet donc de confirmer les travaux effectués par Le et al [START_REF] Le | Extraction of the species-dependent dipole amplitude and phase from high-order harmonic spectra in rare-gas atoms[END_REF] dans les gaz rares et de valider -dans ce cas précis -l'utilisation de la théorie QRS (Quantum Rescattering Theory) qui stipule que le spectre de GHOE peut être factorisé en une partie purement dynamique concernant le paquet d'onde électronique dans le continuum et une autre partie correspondant à l'élément de matrice de transition dipolaire [START_REF] Le | Theory of high-order harmonic generation from molecules by intense laser pulses[END_REF][START_REF] Le | Quantitative rescattering theory for high-order harmonic generation from molecules[END_REF][START_REF] Le | Quantitative rescattering theory of high-order harmonic generation for polyatomic molecules[END_REF].

S.5.3 Interférences à deux centres dans les molécules

Dans le cas d'une molécule diatomique dans la formulation de la théorie SFA moléculaire, nous avons expliqué que quatre classes de trajectoires électroniques sont possibles pour générer une seule harmonique. Cette génération étant cohérente, des interférences supplémentaires sont attendues. En effet, aux interférences entre trajectoires courtes et longues s'ajoutent des interférences entre les deux centres atomiques de la molécule. Nous nous intéressons plus particulièrement aux interférences destructives induites par les deux centres et proposons d'aborder ce sujet largement étudié [START_REF] Śpiewanowski | High-order-harmonic generation from field-distorted orbitals[END_REF][START_REF] Śpiewanowski | Field-induced orbital distortion in high-order-harmonic generation from aligned and oriented molecules within adiabatic strong-field approximation[END_REF][START_REF] Chiril | Strong-field approximation for harmonic generation in diatomic molecules[END_REF][START_REF] Faria | High-order harmonic generation in diatomic molecules: A quantum-orbit analysis of the interference patterns[END_REF][START_REF] Lein | Interference effects in high-order harmonic generation with molecules[END_REF][START_REF] Zhou | Molecular Recollision Interferometry in High Harmonic Generation[END_REF][START_REF] Zhou | Alignment dependence of high-order harmonic generation from N 2 and O 2 molecules in intense laser fields[END_REF][START_REF] Zhou | Role of molecular orbital symmetry on the alignment dependence of high-order harmonic generation with molecules[END_REF][START_REF] Kamta | Three-dimensional time-profile analysis of high-order harmonic generation in molecules: Nuclear interferences in H + 2[END_REF][START_REF] Wagner | Extracting the phase of high-order harmonic emission from a molecule using transient alignment in mixed samples[END_REF][START_REF] Boutu | Coherent control of attosecond emission from aligned molecules[END_REF][START_REF] Lein | Role of the Intramolecular Phase in High-Harmonic Generation[END_REF][START_REF] Van Der Zwan | Two-center interference and ellipticity in high-order harmonic generation from H + 2[END_REF][START_REF] Etches | Two-center minima in harmonic spectra from aligned polar molecules[END_REF][START_REF] Chiril | Explanation for the smoothness of the phase in molecular high-order harmonic generation[END_REF][START_REF] Ciappina | Influence of Coulomb continuum wave functions in the description of high-order harmonic generation with H + 2[END_REF] en insistant particulièrement sur la phase des harmoniques.

En effet, l'interférence destructive est attendue à des valeurs précises de la quantité de mouvement k ou de la distance internucléaire R qui annulent l'élément de matrice de recombinaison (équation (S.3.14)) : k q = (2q + 1)π R , q ∈ Z. Nous avons effectué des simulations ab initio avec un potentiel à deux centres : La phase obtenue est lissée et saute lentement de (presque) π en couvrant plusieurs harmoniques. Ces résultats confirment les observations précédentes [START_REF] Chiril | Explanation for the smoothness of the phase in molecular high-order harmonic generation[END_REF][START_REF] Ciappina | Influence of Coulomb continuum wave functions in the description of high-order harmonic generation with H + 2[END_REF]. Cependant, contrairement à toute attente, le saut de phase a un comportement très particulier pour les trajectoires longues. Tout d'abord, il a un sens opposé à celui des trajectoires courtes. De plus, en variant l'intensité laser (ou la distance internucléaire R), on voit évoluer le saut de phase d'un sens à l'autre, en passant par un cas où l'on retrouve un saut de π discontinu (cf. figure S.8).

V (x) = - -1/2 x - R 2 2 + a 2 - -1/2 x + R 2
Des analyses temps-fréquence nous ont permis de relier ce comportement à la valeur instantanée du champ électrique à la recombinaison. Nous avons donc cherché à vérifier si ce comportement est le résultat d'un habillage de l'état fondamental par le champ laser. Dans un premier temps, nous calculons l'élément de matrice dipolaire de recombinaison en présence d'un état fondamental habillé. L'habillage est traduit, en suivant la théorie des perturbations, par une population dans le premier état excité, proportionnelle au champ électrique instantané E(t). Une décomposition LCAO nous permet alors d'écrire l'élément de matrice de recombinaison habillé comme :

drec(k,t) ∝ cos k R 2 + iαE(t) sin k R 2 ,
(S.5.6) lequel présente alors une partie imaginaire en sin(kR/2), proportionnelle au champ électrique E(t) à l'instant de recollision. Cette forme explique le comportement du saut de phase, lissé par la partie imaginaire et dont le sens est dicté par le signe de E(t). Or, les trajectoires courtes se recombinent avec un champ toujours de même signe, tandis que les longues sur une fenêtre temporelle où le champ s'annule et change de signe. On reproduit qualitativement le comportement observé, et le relie directement à l'habillage de l'état fondamental par le champ laser. Les résultats ab initio sont confirmés par les calculs du SFA moléculaire. L'idée est alors d'essayer de retrouver la signature dynamique de l'habillage dans de cette théorie analytique. Guidés par des résultats numériques, nous avons effectué des développements limités au premier ordre des grandeurs moléculaires autour des grandeurs atomiques, en partant des équations saddle-point moléculaires (équations dont la forme est très similaire à l'élément de matrice de recombinaison habillé (cf. équation (S.5.6)). On montre donc que l'incorporation des termes de phases de l'élément de matrice de recombinaison dans l'action totale du système permet d'y imprimer la signature de l'habillage de l'état fondamental et d'expliquer entièrement le comportement du saut de phase. Une analyse fine des termes de phase supplémentaires, qui apparaissent lors des développements limités, permet d'identifier un léger décalage de la condition d'interférence destructive ainsi que de la condition d'inversion du saut de phase dans les trajectoires longues. Le saut discontinu de ±π apparait à une valeur du champ électrique très proche mais pas égale à zéro, pour compenser un terme de phase additionnel, en accord avec les observations des calculs ab initio où les timings ont pu être extraits précisément avec la méthode de la fréquence instantanée.

Cette étude invalide clairement la théorie QRS. L'élément de matrice dipolaire de recombinaison ne peut être simplement factorisé car il est modifié de façon différente le long du spectre et pour les trajectoires courtes et les longues, de par des effets d'habillage de l'état fondamental par le champ laser.

S.5.4 Phase des interférences à deux centres sondée par les QPI

À l'aide d'un modèle analytique basé sur des travaux antérieurs [START_REF] Gaarde | Spatiotemporal separation of high harmonic radiation into two quantum path components[END_REF][START_REF] Schapper | Spatial fingerprint of quantum path interferences in high order harmonic generation[END_REF][START_REF] Balcou | Quantum-path analysis and phase matching of highorder harmonic generation and high-order frequency mixing processes in strong laser fields[END_REF][START_REF] Gaarde | Quantum path distributions for high-order harmonics in rare gas atoms[END_REF][START_REF] Hostetter | Semiclassical approaches to below-threshold harmonics[END_REF][START_REF] Kretschmar | Spatial contributions of electron trajectories to high-order-harmonic radiation originating from a semi-infinite gas cell[END_REF], nous pouvons montrer que le comportement différent du saut de phase observé dans les trajectoires courtes et les longues s'imprime dans l'analyse QPI. En effet, dans la majeure partie des cas, le saut de phase pour les deux trajectoires a un sens opposé. Or, du fait de son lissage, son évolution est progressive le long des ordres harmoniques, ce qui implique que la différence de phase entre les deux trajectoires évolue lentement de 0 à 2π, en passant donc par π à peu près à l'harmonique de l'interférence destructive. Ainsi, par rapport à une référence atomique, les QPI se déphasent progressivement jusqu'à l'interférence destructive, puis se rephasent. Les calculs ab initio de QPI, pour H 2 et une référence atomique, corroborent ces prédictions, comme montré en figure S.9, donnant ainsi la perspective de leur observation et donc de leur confirmation expérimentale.

S.5.5 Vibration dans les molécules diatomiques

La possibilité de créer et de mesurer des paquets d'onde vibrationnels dans l'état électronique fondamental ou les états excités de molécules a été rendue possible grâce aux développements rapide des technologies laser [START_REF] Weiner | Femtosecond Pulse Sequences Used for Optical Manipulation of Molecular Motion[END_REF][START_REF] Wittmann | fs-Pulse Synthesis Using Phase Modulation by Impulsively Excited Molecular Vibrations[END_REF][START_REF] Weinacht | Coherent learning control of vibrational motion in room temperature molecular gases[END_REF][START_REF] Bartels | Nonresonant Control of Multimode Molecular Wave Packets at Room Temperature[END_REF]. En parallèle, la possibilité de mesurer le mouvement des noyaux de l'ion moléculaire avec la GHOE a été prédite par Lein [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF] puis vérifiée expérimentalement par Baker et al [START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF] dans le cas de molécules de dihydrogène. De plus, Wagner et ses collaborateurs ont montré expérimentalement la possibilité de créer un paquet d'onde vibrationnel dans l'état électronique fondamental de SF 6 et de le sonder par la GHOE [START_REF] Wagner | Monitoring molecular dynamics using coherent electrons from high harmonic generation[END_REF]. Enfin, une théorie purement nucléaire appelée Lochfraß a été développée en prenant en compte un potentiel d'ionisation dépendant de la distance internucléaire R pour tenter d'expliquer la formation de paquets d'onde vibrationnels dans l'état électronique fondamental de molécules diatomiques [START_REF] Goll | Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses[END_REF]. Cette théorie stipule que, à l'étape d'ionisation tunnel, l'état vibrationnel fondamental est rogné de façon anisotrope par le taux d'ionisation dépendant fortement de R. Enfin, la création et la détection de paquets d'onde vibrationnels dans l'état électronique fondamental de D 2 a été conduite expérimentalement par Ergler et al [START_REF] Ergler | Quantum-Phase Resolved Mapping of Ground-State Vibrational D_{2} Wave Packets via Selective Depletion in Intense Laser Pulses[END_REF] et expliquée par la théorie Lochfraß.

Ces considérations nous ont amené à développer un modèle couplant le mouvement des noyaux avec celui de l'électron. Dans un premier temps, nous validons notre modèle en vérifiant sa capacité à reproduire les résultats de Lein et de Baker et al [START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF][START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF].

Puis, nous nous intéressons au saut de phase des interférences à deux centres dans H 2 avec vibration. Le mouvement rapide des noyaux légers pour une telle molécule modifie complètement le comportement du saut de phase décrit dans la partie précédente. En étudiant des isotopes de H 2 et en augmentant (artificiellement) la masse des noyaux jusqu'à m = 10, on montre que l'on retrouve le cas où R était fixé. En conclusion, s'il est vraisemblablement possible de prouver expérimentalement nos prédictions sur le saut de phase, il faut cependant utiliser une molécule "lourde", telle que CO 2 qui présente un comportement à deux centres de par la forme de sa HOMO.

Dans un second temps, nous nous plaçons dans les conditions où le Lochfraß est prédit [START_REF] Goll | Formation of Ground-State Vibrational Wave Packets in Intense Ultrashort Laser Pulses[END_REF]. On utilise H 2 et une molécule fictive, que l'on nomme G 2 , pour laquelle la courbe d'énergie potentielle du neutre est parallèle à la courbe d'énergie potentielle de l'ion. Dans une telle molécule, le potentiel d'ionisation est constant le long de R, et par conséquent le taux d'ionisation est quasi constant. Ainsi, le Lochfraß prédirait une excitation vibrationnelle quasi inexistante dans G 2 . Nous comparons alors ces résultats avec ceux de notre modèle exact corrélé. On observe, avec notre modèle, une importante excitation vibrationnelle dans l'état électronique fondamental de la molécule. Ces résultats nous amènent donc à envisager une autre origine à cette excitation vibrationnelle.

Par comparaison avec un système Λ, où trois niveaux sont mis en jeux (correspondant aux deux premiers niveaux vibrationnels de l'état électronique fondamental et d'un niveau excité) couplés par le laser (cf. figure S.10) , on reproduit les résultats de notre modèle corrélé. Ainsi, cette excitation vibrationnelle aurait pour origine un effet Raman dans la largeur du photon infrarouge de l'impulsion laser très brève. De plus, l'évolution de la probabilité de l'état v = 1 avec l'intensité laser I L montre une dépendance en I 2 L , tandis que le Lochfraß a une dépendance très non-linéaire, confirmant l'effet Raman (cf. figure S.11). Enfin, nous montrons que l'on peut expliquer les observations expérimentales de Ergler et al [START_REF] Ergler | Quantum-Phase Resolved Mapping of Ground-State Vibrational D_{2} Wave Packets via Selective Depletion in Intense Laser Pulses[END_REF] avec notre modèle corrélé.

E 0 E 1 E 2 V(t) V(t)
À la lumière de ces simulations, nous envisageons de proposer une définition Ainsi, il dépend de la distance internucléaire R. Nous pensons que cette définition est inappropriée, et qu'il faut en fin de compte définir un potentiel d'ionisation comme la différence d'énergie entre les niveaux vibrationnels du neutre et de l'ion. En utilisant des potentiels non physiques mais façonnés spécifiquement pour accentuer la différence entre l'approche R-dépendante et l'approche vibrationnelle, et par comparaison avec notre modèle corrélé où l'ionisation est prise en compte implicitement et de façon exacte, on montre que l'approche vibrationnelle donne des résultats beaucoup plus appropriés. Enfin, en effectuant les mêmes comparaisons avec une molécule réelle (H 2 et ses isotopes) on montre que l'approche vibrationnelle donne toujours des meilleurs résultats mais que l'approche R-dépendante donne également de très bons résultats. On conclut en affirmant que l'approche vibrationnelle est la plus pertinente mais que l'approche R-dépendante est appropriée lorsque les niveaux vibrationnels considérés sont très proches, et on pense que cette méthode serait exacte dans le cas d'une courbe d'énergie potentielle dissociative pour l'ion. L'étude doit cependant être poursuivie en étudiant l'effet dynamique d'une superposition d'états vibrationnels comme état de départ, c'est-à-dire en ayant préalablement excité vibrationnellement la molécule.

S.6 Conclusion

Nous avons étudié théoriquement atomes et molécules diatomiques soumis à une impulsion laser intense, brève et de fréquence faible (domaine infra-rouge) dans les conditions de la GHOE. En utilisant des modèles simples à dimensionnalité réduite et en considérant un seul électron actif, nous résolvons de façon extensive l'ESDT, soit de façon ab initio à l'aide de méthodes numériques adaptées, soit avec le modèle semi-analytique SFA. Nous avons également construit un modèle prenant en compte la vibration des molécules diatomiques.

Nous avons ensuite développé et utilisé de nombreux outils d'analyse pour étudier les résultats bruts obtenus par les méthodes ab initio et les comparer le cas échéant aux calculs SFA.

Équipés de ces programmes, plusieurs études ont été conduites. Nous avons, tout d'abord, examiné les informations que la distribution de Wigner peut donner dans le cas de la GHOE. Puis, nous avons étudié l'action d'une résonance de forme et des interférences à deux centres sur la phase spectrale, en montrant que dans le premier cas l'habillage de l'état fondamental (et de l'état résonant) par le laser est négligeable, tandis que dans le deuxième cas il apparaît comme crucial pour expliquer un comportement très particulier, dépendant du champ électrique instantané à la recombinaison. Enfin, nous avons étudié l'effet de la vibration sur les spectres de GHOE, nous permettant d'examiner la pertinence de la théorie du Lochfraß et de proposer une autre définition du potentiel d'ionisation moléculaire.

Nos études ont permis de montrer que nous avons encore beaucoup à apprendre de la GHOE et des problèmes en champs intenses. Bien qu'à l'ordre du jour la GHOE est conduite sur des systèmes plus importants comme les liquides [START_REF] Dichiara | An investigation of harmonic generation in liquid media with a mid-infrared laser[END_REF] et les micro-gouttes d'eau [START_REF] Kurz | High-order-harmonic generation from dense water microdroplets[END_REF], les cristaux [START_REF] Ghimire | Observation of high-order harmonic generation in a bulk crystal[END_REF] et les solides [START_REF] Der Linde | Generation of high-order harmonics from solid surfaces by intense femtosecond laser pulses[END_REF][START_REF] Dichiara | Scaling of High-Order Harmonic Generation in the Long Wavelength Limit of a Strong Laser Field[END_REF][START_REF] Vampa | Linking high harmonics from gases and solids[END_REF], et même les nanoparticules [START_REF] Ganeev | High-order harmonic generation in nanoparticle-containing laser-produced plasmas[END_REF][START_REF] Ganeev | High-order harmonic generation in Ag nanoparticle-containing plasma[END_REF][START_REF] Shaaran | High-order-harmonic generation by enhanced plasmonic near-fields in metal nanoparticles[END_REF], son étude fondamentale à l'échelle de l'atome ou de la molécule unique continue de dévoiler des connaissances importantes pour la compréhension de ce phénomène.

Fig. 1 . 1

 11 Fig. 1.1 Maximal laser intensities reached over years and the corresponding physical regimes. In particular, the intensity regime for which High-order Harmonic Generation (HHG) operates is presented. This figure has been inspired by Mourou et al [17].

Fig. 1 . 2

 12 Fig. 1.2 Laser pulse duration over years. Data have been taken from Ref. [15], completed with the last point corresponding to the shortest pulse generated so far: 67 as [37].

  b n (t) = b

n

  (t) + λ b

( 1 )

 1 n (t) + λ 2 b

( 2 )

 2 n (t) + ..., which are substituted in Eq. (2.2.4). Identifying the terms of same order in λ on each side leads to: (t) ϕ k b

Fig. 2 . 1

 21 Fig. 2.1 1D scheme of tunnel ionization. We display the interaction potential xE(t) (red line), the resulting distorted atomic potential sum of the Coulomb potential V (x) and xE(t) (blue line), and the electronic wave-packet (black line and blue shade) constituted of the localized ground state and a small part which has crossed the barrier by tunnel effect. The values below the blue line are hatched in order to emphasize that this part could not be physically crossed by a classical particle.

  Fig. 2.2 Scheme of the barrier according to Keldysh [42].

Fig. 2 . 3

 23 Fig. 2.3 First experimental observation of HHG in Ar by Ferray et al [4]. Data have been extracted from this reference. We marked the cutoff position with an arrow at harmonic 27. Color-shades symbolize experimental error bars and are used as a visual guide for the different regions of the spectrum.

Figure 2 .Fig. 2 . 4

 224 Fig. 2.4 Scheme of the three-step model for an atom. The upper part displays the amplitude of the electric field (thick red lines) over one and quarter period, with approximate time ranges during which each step occurs. The bottom part illustrates in 1D the deformed atomic potential, result of the sum of the Coulomb potential V (x) and the interaction potential xE(t), at the three different steps: (1) tunnel ionization, (2) propagation, (3) recombination.

αFig. 2 . 5

 25 Fig. 2.5 Representation of classical electron trajectories. We display the position x in units of x α (see Eq. (2.4.7)) along with the laser field (dark red line) of amplitude E L = 0.1 a.u. (intensity I L = 3.5 × 10 14 W.cm ---2 2 2). In color scale, we display trajectories of electrons that recollide with the nucleus (i.e. for which x(t i ,t r ) = 0), the color shade traducing their kinetic energy E k (t i ,t r ) = v 2 (t i ,t r )/2 in units of U p . The dashed black

Fig. 2 . 6

 26 Fig. 2.6 Classical ionization (blue lines) and recollision (purple lines) times for 3 laser cycles. The ordinates displays the electron kinetic energy at recollision time in units of U p . A red line marks the maximum kinetic energy attained, i.e. 3.17U p . We dis-

  for (i=0; i<Nx; i++) { V0[i] = potential.at(x[i]); dV0_dx[i] = potential.d_dx_at(x[i]); } //laser Laser<double> laser(w,E0,Nc,laserType); laser.setEnv(Ncenv); laser.setEnvPwr(Nenvpwr); cout << laser.whichType() << endl; double Tpulse = Nc*2*PI/w; for (j=0; j<Nt; j++) env[j] = pow(sin(PI*j*dt/Tpulse),2); //fondamental state in the potential bool recalc = optionParser.isSet("-fond"); E = fondam(suffixP,Nx,dx,xmax,a_,R,V0,freal,recalc); for (i=0; i<Nx; i++) { f[i] = freal[i]; fcore[i] = freal[i]; } //timers LoopTimer loopTimerCN("crank-nicolson"); LoopTimer loopTimerQPI("qpi"); //COMPUTATION OF THE DIPOLE WITH CRANK-NICOLSON ALGORITHM #pragma omp parallel for schedule(dynamic) firstprivate(i,j,h,laser) for (m=0; m<Nqpi; m++) { if (qpi) { loopTimerQPI.affPercentTime(progress,Nqpi); for (i=0; i<Nx; i++) f[i] = freal[i]; E0 = sqrt((I1+m*dI)/350); laser.setE0(E0); } for (j=jinit; j<Nt; j++) { if (!qpi) loopTimerCN.affPercentTime(j,Nt); //set the tridiagonal system for (i=0; i<Nx; i++) { field = laser.at(j*dt); atemp = I*(1./dx2+V0[i]+Vcap[i]+x[i]*field)*dt/2.; b[i] = c[i] = -I*(1./(2.*dx2))*dt/2.; a[i] = 1.+atemp; aa[i] = 1.-atemp; bb = -b[i]; cc = -c[i]; } //calculate the second member u[0] = aa[0]*f[0]+cc*f[1]; for (i=1; i<Nx-1; i++) u[i] = bb*f[i-1]+aa[i]*f[i]+cc*f[i+1]; u[Nx-1] = bb*f[Nx-2]+aa[Nx-1]*f[Nx-1]; //invert the tridiagonal system tridiag(a,b,c,u,f); //calculate the dipole and absorb the ionized part of the wave function d[j] = 0; for (i=0; i<Nx; i++) { d[j] += real(conj(f[i])*(dV0_dx[i]+laser.at(j*dt))*f[i])*dx; f[i] *= VecAbsorber[i]; } } Chapter 3

Fig. 3 . 1

 31 Fig. 3.1 Representation of the 1D potentials exposed in this section. The atomic soft-Coulomb of Eq. (3.2.5) is displayed with a = 1 (a). For the molecular double soft-Coulomb expressed in Eq. (3.2.6) we used a = 0.5 and a deliberately large internuclear distance R = 4 a.u. in order to separate the two wells (b). The potential with barriers given by Eq. (3.2.7) is plotted with (in a.u.) a = 1, L 1 = 5, L 2 = 3, V 0 = 0.2 and n = 4.

Figure 3 . 2

 32 displays the five first bound states in the atomic and molecular potentials given by Eq. (3.2.5) and Eq. (3.2.6) respectively with the same parameters as in Fig.3.1.

Fig. 3 . 2

 32 Fig. 3.2 Five first bound states plotted at their corresponding energies for the atomic (a) and molecular (b) potentials. Same parameters as in Fig. 3.1 are used.

Fig. 3 . 4

 34 Fig. 3.4 Even and odd DOS for the potential with barriers (Eq. (3.2.7)) (a). Normalized even and odd DOS by the law given in Eq. (3.2.32) (b). Difference between the even and the odd DOS, where even (odd) resonances are emphasized with blue (green) shades(c).

Fig. 3 . 5 1 - 8 - 1 "

 35181 Fig. 3.5 Electric field of the laser pulse for a 10 cycles square-sine envelope (black) and a "1-8-1" trapezoidal envelope (red). The respective envelopes are reported in dashed lines.

Fig. 3 . 6

 36 Fig. 3.6 One dimensional scheme of the molecule and the four possible classes of trajectories α → β , with α, β ∈ {1, 2}. We present the case of aligned molecules along the laser polarization (R x = R • e x ). We pictured the ionization towards the positive xcoordinate, i.e. with a negative laser field E(t). We shown in black solid line the resulting potential, sum of xE(t) and the double-well potential V (x) = V dia (x), and the atomic orbitals ϕ s centered on each nucleus used for the computation of the transition dipole matrix elements.

Fig. 4 . 1

 41 Fig. 4.1 Representation of two Morlet wavelets (blue lines) and two Mexican-hat wavelets (black lines) at different times and frequency.

Fig. 4 . 2

 42 Fig. 4.2 Scheme of the tiling of the time-frequency plane in the case of the STFT with fixed window width (a) and for the CWT with frequency-adaptive window width (b).

Figure 4 . 1

 41 shows two Morlet wavelets for different frequency ω.

Fig. 4 . 3

 43 Fig. 4.3 Representation of the STFT of the dipole in the time-frequency plane. The dipole is computed by solving the 1D TDSE for an atom subjected to a 1-2-1 trapezoidal laser pulse of intensity I L = 3.5 × 10 14 W.cm ---2 2 2 . The Gaussian window has a FWHM of ∼283 as (parameter δ = 7 a.u.). It shows the intensity (in log scale) as a function of the harmonic order and time. We also reported the dipole as a function of time (bottom) and the harmonic spectrum (left).

Fig. 4 . 4 3 .Table 4 . 1

 44341 Fig. 4.4 Representation of the CWT of the dipole in the time-frequency plane, with Morlet wavelets (a) and Mexican-hat wavelets (b), in the same conditions as in Fig. 4.3.

Fig. 4 . 5

 45 Fig. 4.5 Low-cut filtered dipole for the first laser cycles. Four attobursts are identified with Roman numbers I-IV.

Fig. 4 . 6

 46 Fig. 4.6 Instantaneous frequency (solid black lines) of the low-cut filtered dipole over its spectrogram computed with the STFT. We also report the extracted second return (dashed black lines).

Figure 4 . 6

 46 Figure 4.6 displays the IF computed for the whole filtered dipole over its STFT. It is well defined for first attoburst and lies in the width of the STFT bell-shaped distributions. Concerning the second attoburst, the IF oscillates due to the bicomponent nature of the dipole. Precise values are given in Table4.2 for harmonics 37 and 55, and compared with the harmonic emission times derived from SFA computations performed as in Sec. 3.3.1 and classical times obtained from the resolution of the Newton equation. As expected, classical times give terrible results, while SFA times are much better approaching the exact numerical times. Still, discrepancies up to 50 as are observed, which are of the same order as the typical timings we want to achieve in attoscience, for instance few tens of attosecond in the measure of delays in photoionization[START_REF] Klünder | Probing Single-Photon Ionization on the Attosecond Time Scale[END_REF][START_REF] Schultze | Delay in Photoemission[END_REF][START_REF] Shafir | Resolving the time when an electron exits a tunnelling barrier[END_REF]. These results are linked with the ones of Ref.[START_REF] Smirnova | Direct XUV Probing of Attosecond Electron Recollision[END_REF].

Fig. 4 . 7

 47 Fig. 4.7 Comparison of the IF for the first return and the second extracted return with SFA and classical recollision times.

( 4 . 2 . 18 )

 4218 It computes the spectral density of d as a function of time. It is a reversible translation of the signal in the time-frequency domain and do not suffer from limited resolution like the STFT and the CWT. Furthermore, the WVD have interesting projection properties: WVD[d](ω,t)dω = |d(t)| 2 ,

Fig. 4 .

 4 Fig. 4.8 WVD of the compass signal as defined by Eq. (4.2.21). The temporal signal and its spectrum are reported. The WVD amplitude is sketched in yellow and red for positive valued and in blue for negative values.

Fig. 4 . 9

 49 Fig. 4.9 Wigner-Ville distribution of the dipole in the same conditions than in the previous section. This spectral density can have positive (red) and negative (blue) values. The dashed line is a visual guide between the short and the long trajectories of the first attoburst.

Fig. 4 .

 4 Fig. 4.10 Zoom on attobursts I, v I,II and II of the WVD provided in Fig. 4.9 for harmonics 33 to 69. The dashed line is a visual guide between the short and the long trajectories of attoburst I. The solid line marks the position of harmonic 57 in burst II.

Fig. 4 .

 4 Fig. 4.11 Spectrogram of the short (a) and the long (b) trajectory parts of the dipole, separated using a cos 1/8 -absorber of length L abs = 25 a.u. placed at x α (the number of points per laser cycle is 8192). As in Sec. 4.2, the atomic potential parameter is a = 1. We used a 1-2-1 trapezoidal pulse of amplitude E L = 0.1 a.u. (peak intensity I L = 3.5 × 10 14 W.cm ---2 2 2 ) and frequency ω L = 0.057 a.u. such that x α = 30.8 a.u.

Fig. 4 . 6 ,

 46 Fig. 4.7 or Fig. 4.10 for example)

Figure 4 .

 4 Figure 4.11 displays the STFT of the short and the long trajectory contributions to the dipole after being separated with an absorber. The absorber must be short enough to resolve spatially the short and long trajectory electrons at the expense of its efficiency. Hence, reflection of a small part of electrons beyond the cutoff is observed and is the cause of a non-physical cutoff enhancement. It turns out that this artifact is useful for our studies, as discussed later on.

  Fig. 4.12 Scheme of the cylindric coordinates.

( 4 . 3 . 22 )

 4322 can be rewritten as:

Fig. 4 .

 4 Fig. 4.14 Far-field profile (black lines) of harmonics 13, 23 and 33 for a gas jet at 1.5 mm before the beam focus (a-c) and for a gas jet (N 0 = 1.9 × 10 -6 atoms per (a.u.)3 3 3 , L jet = 1 mm) at 0.5 mm after the focus (d-f). We also display with green and blue lines the farfield profile of the short and long trajectory contributions, respectively, separated with the absorber technique. Green and blue shades are added under the black-lines as a visual guide. The width of the gas jet is 1 mm and the confocal parameter of the beam is b = 5. The temporal envelope of the beam has a sin 2 2 2 form of 2 cycles duration with peak intensity I L = 3.5 × 10 14 W.cm ---2 2 2 and frequency ω L = 0.057 a.u.

27 Fig. 4 .

 274 Fig. 4.15 Example of QPI obtained numerically with atomic TDSE computations, for harmonic 27 (i.e. |D 27 (I L )| 2 , see Eq. (4.4.2)). The regularization parameter a of the soft-Coulomb potential is set equal to 1. The laser pulse has a 2-cycle square-sine envelope and a frequency ω L = 0.057 a.u.

( 4 . 4 . 3 )Fig. 4 .

 4434 Fig. 4.16 I-α analysis of the QPI of harmonic 27 as presented in Fig. 4.15, together with the QPI and its Fourier transform. The abscissa is the conjugate intensity called α.

Figure 4 .

 4 Figure 4.[START_REF] Krausz | Attosecond physics[END_REF] presents the I-α transform together with the FT of the 27th harmonic QPI. We observe two branches, one with α 1 corresponding to the short trajectories, and one with α 20 corresponding to the long trajectories. These branches get tighter and merge at I L 1.25 × 10 14 W.cm -2 , i.e. at the cutoff. Unlike the observations of Balcou et al[START_REF] Balcou | Quantum-path analysis and phase matching of highorder harmonic generation and high-order frequency mixing processes in strong laser fields[END_REF] at high intensities, the branch of the long trajectories gets closer to the one of the short trajectories. The origin of this shift is not well understood, but could be attributed either to the form of the laser pulse we have chosen or also to a blue-shift which occurs when ionization is high.

Fig. 4 .

 4 Fig. 4.17 Representation of the analytical conjugate intensities α j q (I L ) as given by Eq. (4.4.5) for harmonics 23, 45 and 67. Here α s 0 = 1, α l 0 = 25 and α cut = 7.5 (×10 -14 cm 2 .W -1 ).

W

  (x, p)dx = | Ψ(p)| 2 .

( 4 .

 4 5.5)

( 4 . 5 . 9 )

 459 W a (x, p,t) = 1 π dy e 2ipy ϕ 0 (x + y) dk b(k,t)e ik(x-y) , (4.5.10) W b (x, p,t) = 1 π dy e 2ipy ϕ 0 (xy) dk b * (k,t)e -ik(x+y) , (4.5.11)

Fig. 5 . 1

 51 Fig. 5.1 Times and corresponding electric field field for which snapshots (a-t) of the Wigner distribution have been taken and displayed in Fig. 5.2.

- 2 -Fig. 5 . 2

 252 Fig. 5.2 Snapshots at different times (a-t, see Fig. 5.1) of Wigner function for 1D TDSE simulations of an atom subjected to a 2-cycle sin 2 2 2 laser pulse. The arrows 1-6 are used for the discussion.

1Fig. 5 . 3

 53 Fig. 5.3 Wigner distribution at x = 0 of the TDSE wave-function after having removed the fundamental state (a) for a 1-2-1 trapezoidal laser pulse of intensity I L = 5 × 10 14 W.cm ---2 2 2 (b). The red arrow indicates the dominant continuum contribution corresponding to electrons recolliding for the first time. The black arrows indicate greater order contributions issued from further returning electrons.

Fig. 5 . 4

 54 Fig. 5.4 Free electronic part of the SFA Wigner function at x = 0 as a function of time (color shades) along with the real values of the stationary momentum p at (black circles), for a laser intensity I L = 5 × 10 14 W.cm ---2 2 2 .

Fig. 5 .

 5 Fig. 5.5 HHG spectrum of our model of molecular nitrogen featuring shape resonance.We propagated the 1D TDSE for the potential V B with the same parameters than in Fig.3.4, subjected to a 1-8-1 trapezoidal laser pulse of frequency ω L = 0.0456 (wavelength of 1 µ µ µm) and peak intensity I L = 2.86×10 14 W.cm ---2 2 2 . The dashed line marks the position of harmonic 25, having the maximum intensity in the spectrum.

Fig. 5 . 6

 56 Fig. 5.6 Potential with barriers V B (x) (solid blue lines) and its reference (dashed blue lines) along with the ground-state (brown lines), the scattering wave χ E (x) associated with V B (x) (solid red lines) and the reference scattering wave χ ref E (x) (dashed red lines) (a). Amplitude of the transition dipole matrix element (red lines) and phase-shift η(E) (green lines) (b). As a comparison, we provide the DOS (black dashed lines) from Fig. 3.4, normalized.

Fig. 5 . 7

 57 Fig. 5.7 Harmonic phase difference between the potential with barriers V B (x) and the reference (black lines) compared with the transition dipole matrix element phase (i.e. the phase-shift η(E)) (green lines). The harmonics were generated using a 2-cycle sin 2 2 2 pulse of frequency ω L = 0.0456 and peak intensity I L = 2.86 × 10 14 W.cm ---2 2 2 .

Fig. 5 . 8

 58 Fig. 5.8 Comparison of the phase-shift η(E) (green lines) with the harmonic phase difference between the potential with barriers V B (x) and the reference, for the short (red) and long (blue) trajectory contributions, separated with an absorber (a) and with macroscopic propagation in the gaseous medium (b). The harmonics were generated using a 2-cycle sin 2 2 2 pulse of frequency ω L = 0.0456 and peak intensity I L = 2.86 × 10 14

Fig. 5 . 9

 59 Fig. 5.9 Intensity for short (a) and long (b) trajectory contributions and their respective phase (c-d) around the resonance as a function of laser intensity. We compare with the amplitude (a-b) and phase (c-d) of the transition dipole matrix element (black lines).

Fig. 5 .

 5 Fig. 5.10 Harmonic phase of short (a) and long (b) trajectory contributions after propagation in a gaseous medium, as a function of laser intensity. We compare with phase of the transition dipole matrix element (black lines).

Fig. 5 .

 5 Fig. 5.11 R-box computations for potential V B (x) without electric field (a), and in presence of an electric field of intensity I L = 3 × 10 14 W.cm ---2 2 2 (b) and I L = 5 × 10 14 W.cm ---2 2 2 (c).Color shades represent the width of the resonance, while red lines are guides for the position of the resonance energies, taken at the inflection points of the Hamiltonian energies (black lines).

Fig. 5 .

 5 Fig. 5.12 Representation of ionization (blue) and recombination times (purple) along with the 2-cycle sin 2 2 2 laser pulse (red). The energy of the resonance E res is reported.

Fig. 5 .

 5 Fig. 5.13 TDSE computations of high-order harmonic spectra for a 2-cycle sin 2 2 2 laser pulse of peak intensity I L = 2.55 × 10 14 W.cm ---2 2 2 for molecules of different internuclear distances R (from 1.4 to 1.7 a.u.). We report harmonic intensity and phase for short (a-b) and long (c-d) trajectory contributions.

Fig. 5 .Fig. 5 .

 55 Fig. 5.14 Amplitude of exact recombination dipole matrix element computed with the shooting method for different internuclear distances (a). In (b) we report the exact harmonic order (ex., black lines) corresponding to the zeros obtained from panel (a) (see zoom inset) along with the analytical formulae Ω (PW) 0 (green) and Ω0 (blue) given by Eqs. (5.4.2) and (5.4.3) respectively, and compare them with the position of the minima extracted from the TDSE computations for the long trajectories (red circles) in Fig. 5.13c.

Fig. 5 .

 5 Fig. 5.15 Gabor transform of the dipole for the critical internuclear distance R c = 1.55 a.u. (a) along with the amplitude of the laser field of peak intensity is I L = 2.55 × 10 14 W.cm ---2 2 2 (b). Same in (c) and (d) but for H 2 2 2 (R = 1.425 a.u.) at critical intensity I c = 3.24 × 10 14 . We marked with dashed-lines the corresponding time of the minimum observed in the long trajectories.

Fig. 5 .

 5 Fig. 5.16 Harmonic phase of H 2 2 2 (relative to the atomic reference) for short (a) and long (b) trajectories as a function of laser intensity. The dashed line corresponds to intensity I c = 3.24 × 10 14 W.cm ---2 2 2 .

  Fig. 5.17 Decomposition in LCAO of molecular ground-state and first excited state of H 2 2 2 . We present the exact ground-state (thick blue lines) and first excited state (thick green lines) and the reconstructed ground-state (thin blue lines) and first excited state

6 Fig. 5 .

 65 Fig. 5.18 Projection of the dressed wave-function, computed with R-box method and inverse iteration, on the first excited state |ϕ 1 , as a function of electric field magnitude (green dots). With a linear fit (black lines) we obtained the slope a = 4.6.

Fig. 5 .

 5 Fig. 5.19 Not-to-scale illustration of the dressed recombination dipole matrix element in the complex plane (a). Phase of the dressed recombination dipole matrix element given by Eq. (5.4.13) (b) for different electric field amplitudes E t reported inset (in a.u.).

Fig. 5 .

 5 Fig. 5.20 Molecular SFA computations of the harmonic phase of H 2 2 2 normalized by the atomic SFA for a 2-cycle sin 2 2 2 laser pulse with peak intensities in the range 3 -5 × 10 14 W.cm ---2 2 2 for short (a) and long trajectories (b). The dashed line corresponds to intensity

( 5 . 4 . 29 )In Eqs. ( 5 . 4 . 28 ) and ( 5 . 4 . 29 )

 542954285429 we used:

Fig. 5 . 5 . 4 . 6

 5546 Fig. 5.21 Modified recombination dipole matrix element drec (Eq. (5.4.45)) in the com- plex plane for short and long trajectories (a). The paths are presented for different laser intensities between 3 -5 × 10 14 W.cm ---2 2 2 and for the specific intensity of I c = I z (black dashed line). They correspond to the evolution of drec with the harmonic order H (see black arrow). The black solid line represents the cutoff positions (they do not merge because in SFA we need to impose a restriction on short trajectories whose saddle-points diverge close to the cutoff, leading to stop the computation few orders before the cutoff). Note that the position of the destructive interference corresponds approximately to the intersection between the curves and the imaginary axis (i.e. zero real part). The phase of drec is reported for short (b) and long (c) trajectories. For long trajectories, we clearly see the inversion of the behavior from a smoothed positive jump to a smoothed negative jump, through a discontinuous π jump at intensity I c

Fig. 5 . 13 )

 513 or as a function of laser intensity I L (cf. Fig.5.[START_REF] Krausz | Attosecond physics[END_REF] or Fig.5.20).

( 5 . 4 . 2 )

 542 ) and Ω z (Eq. (5.4.48)) can be obtained for infinity of couples (I L , R), approximately verifying the equation:

Fig. 5 .

 5 Fig. 5.22 Value of the minimum in the intensity of long trajectory contribution to HHG spectrum as a function of laser intensity I L and internuclear distance R. We present in (a) the first-order expanded molecular SFA with C αβ (t αβ ,t αβ ) = C(t at ,t at ) (Eq. (5.4.44)) and in (b) the exact computation of the molecular SFA (Eq. (3.3.41)). The position of the minimal value, is marked with a black dot-dashed line for case (a) and a black dashed line for case (b). We also compare these curves with the classical prediction (Eq. (5.4.49)) in black solid line.

.4. 50 )

 50

Fig. 5 .

 5 Fig. 5.23 Inverse of the phase-jump slope γ as a function of laser intensity, for short (blue) and long trajectories (red). We compare the approximate analytical values of

  Fig. 5.24 3D scheme of H 2 2 2 molecule aligned with an angle θ with the laser polarization. We work in the plane formed by the laser polarization and the molecular axis, identified by {e x , e y }.

( 3 . 3 .R

 33 38)-(3.3.40)) in the general 3D case with: E(t) = E(t)e x , = R cos(θ )e x + R sin(θ )e y , (5.4.55) p = p x e x + p y e y ,

Fig. 5 .

 5 Fig. 5.25 Analytical fits of recollision times (black lines and circles), for short (green lines, Eq. (5.5.4)) and long (red lines, Eq. (5.5.5)) trajectories.

( 5 . 5 . 4 )

 554 t s (q, I L ) = 12.3 arcsin qω L -I p 1.585U p (I L ) -1 + 86.5, t l (q, I L ) = 10.5 arccos qω L -I p 1.585U p (I L ) -1 + 104.8,

( 5 . 5 . 6 )Figure 5 .

 5565 Figure 5.26 displays D q (I L ) for 11 harmonics between the 35th and the 75th, as a function of laser intensity, for a modeled H 2 molecule (i.e. R = 1.425 in Eq. (5.5.6))and for an atomic reference where θ j q (I L ) = 0. We clearly observe the behavior we expected: interference patterns are almost in phase for H 35 , then the pattern of H 2 smoothly shifts until H 51-55 where it is out of phase with the reference, and finally shifts until H 75 where it is again in phase.

Fig. 5 .

 5 Fig. 5.26 Analytical QPI for several harmonics between the 35th and the 75th, for the modeled H 2 2 2 molecule with R = 1.425 a.u. (blue lines) and the atomic reference (black lines).

Fig. 5 .

 5 Fig. 5.27 TDSE QPI for several harmonics between the 13th and the 79th, for H 2 2 2 (blue lines) and the atomic reference (black lines).

Fig. 5 .Fig. 5 .

 55 Fig. 5.28 Potential energy curves of H 2 2 2 (i.e. V NN (R), light blue lines) and H + 2 2 2 (dark blue lines) and the energy difference (black dashed lines) corresponding to the ground-state energy E 0 (R) of V Ne (x, R) as a function of R.

xFig. 5 .Fig. 5 .

 55 Fig. 5.30 Snapshots at different times (a-x, see Fig. 5.29) of the 1D×1D probability density (white contour lines) for H 2 2 2 subjected to a 2-cycle sin 2 2 2 laser pulse. The color map presents the total potential energy surface V NN (R) + V Ne (x, R) + xE(t) from -1 (blue) to 1 (red) a.u., along with contour lines every 0.2 a.u. (black lines). The pink contour line displays the value of I p for the total potential.

Figure 5 . 2 EFig. 5 .

 525 Fig. 5.32 Three-step model in the case of H 2 with nuclear motion within the BOA. We present the potential energy curves of H 2 2 2 (light blue lines) and H + 2 2 2 (dark blue lines) and the nuclear (blue shades) and electronic (red shades) wave-packets. This scheme depicts the promotion of the neutral vibrational ground-state in the ionic potential subsequent to electronic tunnel ionization (1) and its evolution during electronic propagation (2) until recombination (3).

H 2 D 2 Fig. 5 .

 225 Fig. 5.33 Mean internuclear distance of H 2 2 2 (blue) and D 2 2 2 (green) computed from the correlated model ( R ± (t), light color) and within the BOA ( R BOA (t -t i ), where t i = 0.66 a.u., dark color). The dashed lines present the BOA computations shifted to match R ± at the beginning.

( 5 . 6 . 6 )

 566 computed by inverse iteration on Hamiltonian H N The TDSE is solved using the Crank-Nicolson algorithm again. The mean internuclear distance is computed as: R BOA (t) = R|ξ (R,t)| 2 dR.

( 5 . 6 . 7 )Figure 5 .

 5675 Figure5.[START_REF] Zhao | Positioning of Bound Electron Wave Packets in Molecules Revealed by High-Harmonic Spectroscopy[END_REF] presents the values of R + (t) and R BOA (t) for H 2 (µ = m p /2, where m p is the proton mass, cf. Appendix A.1) and D 2 (µ = m p ). It does not display R -(t) because it presents strong oscillations due to the already mentioned presence of returning electrons adding to the newly ionized ones.Our BOA results are in agreement with the predictions of Lein[START_REF] Lein | Attosecond Probing of Vibrational Dynamics with High-Harmonic Generation[END_REF] and experiments of Baker et al[START_REF] Baker | Probing Proton Dynamics in Molecules on an Attosecond Time Scale[END_REF], as both used the BOA to predict and explain their results. However, important discrepancies are observed between our correlated model and the BOA. First, nuclear motion in the correlated model seems not as fast as in the BOA. Secondly, we see that electrons are not ionized at the equilibrium mean internuclear distance but at about 0.09 a.u. greater, which is reminiscent to the work of Urbain et al[START_REF] Urbain | Intense-Laser-Field Ionization of Molecular Hydrogen in the Tunneling Regime and Its Effect on the Vibrational Excitation of H + 2[END_REF]. Indeed, as we can see in Fig.5.30g-m, ionization is not aligned with the mean internuclear distance (which is about the maximum of the vibronic wavepacket) because the barrier is thiner at greater R. Hence, the correlation between electronic and nuclear motion plays a paramount role.

10 Fig. 5 .

 105 Fig. 5.34 Correlated computations of two-center interference phase in HHG for different isotopes of H 2 2 2 as a function of the laser intensity, for the short (top) and long (bottom) trajectory contributions. The corresponding masses (in a.u.) are reported on the figure.

Fig. 5 .

 5 Fig. 5.35 Scheme of the Lochfraß process in H 2 2 2 . We present the R-dependent ionization potential I p (R) = -E 0 (R) (a) and, as in Fig. 5.28, the potential energy curves of H 2 2 2 (light blue lines) and H + 2 2 2 (dark blue lines) (b). We drew the vibrational groundstate whose red part is "absorbed" due to the anisotropic ionization rate Γ(R,t) and promoted to the ion. The residual vibrational wave-packet (blue part) is no longer an eigenstate and thus a vibrational motion is initiated in the EGS of H 2 2 2 .

Fig. 5 .

 5 Fig. 5.37 Neutral vibrational ground-state χ 0 (a) and ionization rates (in log scale) of H 2 2 2 (b) and G 2 2 2 (c) as a function of R for different laser intensities. The dashed lines are visual guides for the region spanned by the vibrational ground-state.

Fig. 5 2 .

 52 Fig. 5.36 Representation of the ionic (solid) and neutral (dashed) potential energy curves for H 2 2 2 and a model molecule called G 2 2 2 . In the latter the potential energy curves are set parallel such as the ionization potential is constant along R.

Fig. 5 .

 5 Fig. 5.38 Probability p 1 as a function of time for different laser pulses and different computations: the correlated TDSE simulations (Eq. (5.6.8)) in the case of H 2 2 2 (a) and G 2 2 2 (b) and the Lochfraß simulations (Eq. (5.6.12)) for H 2 2 2 (c) and G 2 2 2 (d). From the thinest to the thickest curve, we show results for lasers pulses with a sin 2 2 2 envelope of total duration of 3.8, 5.7 and 9.4 fs (≡ 4 cycles, 6 cycles and 10 cycles). We chose a laser intensity I L = 3 × 10 14 W.cm ---2 2 2 . All the probabilities have been multiplied by a factor shown inset to compare with case (a).

( 5 . 6 . 15 )( 5 . 6 . 16 )

 56155616 |ϕ k = |e, k , k ∈ N, |ϕ n = | f , 1 .

Fig. 5 .

 5 Fig. 5.40 Scheme of the Λ-system.

Fig. 5 .

 5 Fig. 5.39 Same as in Fig. 5.38, but for the 2PT in the case of H 2 2 2 (a) and G 2 2 2 (b) and for the Λ-system (c). Again the probabilities have been multiplied by a factor shown inset to compare with the correlated case of H 2 2 2 presented in Fig. 5.38a.

Fig. 5 .

 5 Fig. 5.41 Probability p 1 at the end of the laser pulse as a function of laser intensity for different computations (see inset legend). The laser pulse duration is 3.8 fs. (4 cycles)

Fig. 5 .

 5 Fig. 5.42 Mean value of R as a function of the reduced time τ/T . Experimental data extracted from Ergler et al [234] (black dots with error bars) are reported together with our correlated simulations in D 2 (black line). Experimentally, τ is the delay between the pump and the probe in [234] and T the period of the oscillations.

75 Fig. 5 .

 755 Fig. 5.43 Neutral (dashed lines) and ionic (Eq. (5.6.26), solid lines) potential energy surfaces, in different configurations: R shift =0 and 0.3 a.u. and σ = 0.15, 0.35, 0.55 and 0.75. We also present the energy levels of the neutral ground-state and the first ionic vibrational states.

Fig. 5 .

 5 Fig. 5.45 Same as in Fig. 5.44 but for a shifted ion by 0.2 and 0.3 a.u. (see the values displayed in the figure).

Fig. 5 . 2 - 9 . 2 - 9 . 2 - 9 .

 5292929 Fig. 5.46 HHG spectra for H 2 , D 2 and T 2 . Such as in Figs. 5.44 and Fig. 5.45, we present |D (R) (ω)| 2 (red dashed lines), |D (v) (ω)| 2 (black lines) and the spectra obtained with the fully correlated model (green line).

Fig. 5 .

 5 Fig. 5.47 HHG spectra for H 2 starting with excited vibrational states v ∈ [[1 : 4]]. We

  .....``` . `````.------:--:://::::/:///++/++++oo+ooooooooo+++++ooosyyssyyys/syyh sooosos+sooosoooooo+++++/+/++/////::::::---------.-..............````` ` ` ` `. `..`..-:--:------://+/////++ooo+o+++++o++o++ooo++/+///++osssyyoossyhyy soossssyysyyosssso++++++++/+///////--:/::::--------...............```` ` ` ` `.` `````...----.---::/++ooso+/++/+++++/++++++++++o+////+++++oosssy+osyyyo+y yssossssssssssosoooooo+o/-/+++++++////////::::::------...........```````` `.```..--..-.-----::///+y+syso+++++o++++++//++so+++++++/+++++/+oooosossyooyhhh sssosysssssysso:+o+ooooo+:+ssssoooo+++++/+++////::::::----..-..---.......`````````

Fig. A. 1

 1 Fig. A.1 Scheme of the construction of k ( j) n , j = 1, 2, 3, 4 for the RK4 algorithm.

  with positive energy E. Our potential dies at large distances, i.e. We consider the asymptotic region |x| ∈ [x a , +∞[ for which E -V (x) > 0. When normalized on the energy scale, the asymptotic solution χ

(A. 5 . 3 )(A. 5 . 4 )B

 5354 One can write χ E as a linear combination in the orthonormal basis set {χ ref E , χ ort E }:χ E (x) = Aχ ref E + Bχ ort E .Yet, cos(kxη(E)) = cos(kx) cos(η(E)) + sin(kx) sin(η(E)), and thus:A = cos(η(E)), = sin(η(E)).

(A. 6 . 1 )

 61 where |α(x)| < 1 for all x. The modulus and phase are, respectively:|z(x)| = cos 2 (x) + α 2 (x) sin 2 (x), (A.6.2) θ (x) = arg(z(x)) = arctan(α(x) tan(x)).

Fig. S. 1

 1 Fig. S.1 Intensités lasers maximales atteintes au court du temps et régimes physiques correspondants. Nous présentons en particulier le régime de la Génération d'Harmoniques d'Ordres Élevés (GHOE). Cette figure a largement été inspirée par Mourou et al[START_REF] Mourou | Optics in the relativistic regime[END_REF].

Fig. S. 3

 3 Fig. S.3 Première observation expérimentale d'un spectre de GHOE dans l'argon par Ferray et al [4]. Les données ont été extraites de cette référence. La position de la coupure a été marquée par une flèche à l'harmonique 27. Les régions en couleur correspondent aux barres d'erreur expérimentale et servent de guide visuel.

Fig. S. 4

 4 Fig. S.4 Représentation schématique du modèle en trois étapes. On présente le potentiel atomique déformé (courbe bleue), résultat de la somme du potentiel coulombien V (x) et du potentiel d'interaction avec le laser xE(t) (courbe rouge), où E(t) est le champ électrique instantané du laser. Le paquet d'onde électronique est esquissé en bleu. Les trois étapes sont (1) l'ionisation tunnel, (2) la propagation dans le continuum sous l'effet du champ et (3) la recombinaison radiative avec l'état fondamental.

(S. 3 . 15 ) 2 ,ϕFig. S. 5

 31525 Fig. S.5 Schéma à une dimension de la molécule diatomique et des quatre trajectoires électroniques possibles α → β , avec α, β ∈ {1, 2}. Le cas de molécules alignées le long de la polarisation du laser est présenté (R x = R • e x ). Les courbes noires montrent la somme de xE(t) (pointillés) et du potentiel moléculaire V (x) à deux puits. Les orbitales atomiques ϕ s centrées sur chaque centre sont présentées en rouge et bleu, et comparées au cas atomique (vert).

  Fig. S.6 Distribution de Wigner calculé avec SFA en x = 0 en fonction du temps (carte de couleurs) avec les solutions saddle-point (cercles noirs).

Fig. S. 7

 7 Fig. S.7 Phase spectrale pour les trajectoires courtes (courbes rouges) et longues (courbes bleues) comparées à la phase de l'élément de matrice de transition (courbes vertes), séparées à l'aide d'un absorbeur (a) et par propagation macroscopique dans le milieu générateur (b). Les spectres sont générés avec une impulsion laser de 2 cycles en sin 2 2 2 avec une fréquence ω L = 0.0456 et une intensité pic I L = 2.8 × 10 14 W.cm ---2 2 2 .

(S. 5 . 3 ) 2 0 2 = I p + π 2 2R 2 (S. 5 . 4 )Fig. S. 8

 53222548 Fig. S.8 Phase des harmoniques dans H 2 2 2 (par rapport à une référence atomique) pour les trajectoires courtes (a) et longues (b) en fonction de l'intensité laser. La ligne pointillée correspond à l'intensité I c = 3.24 × 10 14 W.cm ---2 2 2 pour laquelle le saut discontinu de π est observé dans les trajectoires longues.

2 + a 2 ,(S. 5 . 5 )

 2255 et séparé les contributions des trajectoires courtes des longues à l'aide d'un absorbeur. On simule le dihydrogène (R = 1.425) en ajustant le paramètre de régularisation a pour obtenir le potentiel d'ionisation I p = 0.567 u.a. La phase des harmoniques est à nouveau calculée par rapport à une référence atomique afin d'enlever la contribution quadratique de la GHOE.

(S. 3 .(S. 5 . 7 ) 8 ) 2 +

 35782 20)-(S.3.22)). On obtient tout d'abord une expression analytique des différences de temps d'ionisation et de recombinaison entre les trajectoires atomiques et les trajectoires moléculaires :∆t αβ = t αβt at = (-1) β R/2 2(ω -I p ) = (-1) β δ . ∆t αβ = t αβt at = i (-1) α R/2 2I p = i (-1) α δ . (S.5.Ces expressions peuvent s'interpréter très facilement. La première montre que l'électron qui revient sur un centre ou un autre doit parcourir une distance supplémentaire ±R/2 par rapport à l'atome, comme on le voit simplement sur la figure S.5, à une vitesse 2(ω -I p ) (directement reliée à l'énergie du photon émis par conservation de l'énergie). La seconde expression, concernant l'ionisation, est imaginaire pure. Elle montre ainsi que le "temps tunnel" nécessaire à l'électron pour traverser la barrière est modifié par rapport à l'atome de référence, avec pour distance caractéristique ±R/2 à nouveau. Finalement, après quelques manipulations, nous pouvons contracter la quadruple somme du dipôle moléculaire (équation (S.3.23)) et l'exprimer en fonction de temps atomiques, en exaltant un élément de matrice de recombinaison modifié : drec (k,t) = 2R(k) cos k R

Fig. S. 9

 9 Fig. S.9 Calculs ab initio de QPI pour plusieurs harmoniques pour H 2 (courbes bleues) et une référence atomique (courbes noires).

Fig. S. 10

 10 Fig. S.10 Schéma du système Λ. On présente les énergies E i et le couplage avec le laser V (t).

2.3. THE GENERATION OF HIGH-ORDER HARMONICS

	Keldysh parameter:
	γ =	2ω L ω t	=	I p 2U p	,

(2.2.15)

CHAPTER 2. HIGH-ORDER HARMONIC GENERATION

	Atom	I p (eV)		γ
			McPherson et al	Ferray et al
			1 × 10 15	1 × 10 16	3 × 10 13
	He	24.58	1.46	0.46	-
	Ne	21.57	1.37	0.43	-
	Ar	15.76	1.17	0.37	1.48
	Kr	14.00	1.10	0.34	1.38
	Xe	12.13	1.03	0.32	1.30

Table 2 .1 Keldysh parameters in the conditions of the first HHG experiments carried out by McPherson et al

 2

3.8) 3.3. THE STRONG-FIELD APPROXIMATION

	where:	S(p,t,t ) =	t	t	dτ	[p + A(τ)] 2 2	+ I p ,

p,t,t ) d ion (p + A(t ),t ) + c.c., (3.(3.3.9)

  p,t,t ,ω) + c.c.,

	(3.3.11)
	by introducing the canonical action:

S(p,t,t , ω) = S(p,t,t ) + ωt . (3.3.12)

.14)

	[p + A(t)] 2 2	+ I p -ω = 0 ,

(3.3.15)

	[p + A(t )] 2 2	+ I p = 0 .

(

3.3.16)

  We will denote by (p at ,t at ,t at ) the solutions of

Eqs. (3.3.14)-(3.3.16) to

  emphasize that they are related to atomic derivations as compared to molecular derivations we are going to present in the next section. We also define the atomic action S at = S(p at ,t at ,t at , ω). For clarity purposes, we dropped the dependence of t at , t at and p at on ω. The expression of the stationary momentum p at is directly obtained from

Eq. (3.3.14):

	p at (t at ,t at ) = -	t at t at t at -t at A(τ)dτ	,	(3.3.17)

(3.3.19)

  t at ,t at )d rec (p at + A(t at ))d ion (p at + A(t at ),t at )e -iS at + c.c. .

3.3.20)

  In this expression, the first term arises from the evaluation of

	1 det(S )| at	.

(

Eq. (3.3.11) at

										p at and ac-
	counts for the spreading of the electronic wave-packet in the continuum. The second
	term, for which:
	det(S )| at =	∂ 2 S ∂t 2	at	∂ 2 S ∂t 2	at	-	∂ 2 S ∂t∂t at	2	,	(3.3.21)

.3.29) Replacing them into Eqs. (3.3.24) and (3.3.27) leads to:

2}. (3.3.35) 3.4. BEYOND THE BORN-OPPENHEIMER APPROXIMATION

	Thus, the total dipole reads:
	D(ω) = -ω	2 ∑ α=1	2 ∑ β =1	dt	0	t	dt dp R(p + A(t))I α (p + A(t ),t )e -iS αβ (p,t,t ,ω)
	+ c.c. ,					(3.3.36)
	as a sum of four terms that can be evaluated independently by searching the stationary
	points of the modified actions:
			S αβ (p,t,t , ω) = S(p,t,t , ω) + Φ αβ (p,t,t ) .

(

3.3.37)

Sec. 3.2. The

  All the simulations have been performed by solving the TDSE for a atom following parameter a of the soft-Coulomb potential as given by

Eq. (3.2.5) is

  set equal to 1. The ionization potential is I p = 0.67 a.u. = 18.23 eV. It does not correspond to any

Table 4 .2 Harmonic emission times in attosecond (as) obtained with the IF and com- pared with the ones of SFA and classical computations. As in Table

 4

				SFA		Classical
		S	L	S	L	S	L
	H 37	740 1568	772 1576 812 1548
	H 55 1000 1355 1050 1328	-	-

Table 4 .3 Parameters of the compass signal given by Eq. (4.2.21). As they are arbitrary,

 4

17 Decomposition in LCAO of molecular ground-state and first excited state of H 2 2 2 . We present the exact ground-state (thick blue lines) and first excited state (thick green lines) and the reconstructed ground-state (thin blue lines) and first excited state (thin green lines). The obtained atomic orbital

φ s is drawn in dashed black lines at R/2 and -R/2.

Table 5 .2 Values of the numerical stationary momenta |p αβ | as compared to the atomic ones |p at | for six harmonics (column denoted by the H symbol) of the plateau of the spectrum generated at a laser intensity of

 5

	trajectories

5 × 10 14 W.cm -2 . We see that the |p αβ | are really close to the |p αβ |, confirming the assumption used to obtain Eqs. (5.4.28) and (5.4.29).

Table 5

 5

.2) we first assume that p αβ = p at and obtain the first-order expansion of Eq. (3.3.39) as:

[p at + A(t at )] 2 2 + I pω -[p at + A(t at )]E(t at )∆t αβ + (-1) β E(t at ) + ω 2 L A(t at )∆t αβ

R 2 = 0 . (5.4.26) Using the atomic saddle-point (Eq. (3.3.15)) and Eqs. (5.4.24) and (5.4.25) it reduces to: -2(ω -I p ) E(t at )∆t αβ + (-1) β E(t at ) R 2 + (-1) β ω 2 A(t at ) R 2 ∆t αβ = 0. (5.4.27)

Table 5 .3 Values of the numerical time deviations ∆t αβ and ∆t αβ for the same harmon- ics at the same laser intensity as in Table 5.2, compared with the analytical ones given by Eqs. (5.4.28) and (5.4.29). We present the numerical values for the short (S) and the long (L) trajectories. We do not present the imaginary part of

 5

	.4.30)

p at × (t att at ) + t at t at A(τ)dτ + (p αβp at ) × (t att at ) + p at × (∆t ∆t αβ being bounded by 3 × 10 -3 a.u., not the real part of ∆t αβ which is bounded by 5 × 10 -3 a.u.

The time devia- tions are small compared to the imaginary part of the ionization times (7.83

-15.8 a.u.) and the real part of the recollision times (32.9 -107 a.u.). Using Eqs. (3.3.14)-(3.3.16), Eqs. (5.4.28) and (5.4.29) and since t at = t at , we indeed obtain that: ∆p (1) αβ = p αβp at = 0 , (5.4.32)

Table 5 .

 5 3 displays the numerical values of the ionization and recollision time deviations ∆t αβ and ∆t αβ for the short and long trajectories together with the values of the first-order (Eqs.

	(5.4.28) and (5.4.29)) and second-order expressions (Eqs. (5.4.33) and (5.4.34)).

  t αβ ,t αβ , ω) = S αβ (p at ,t at ,t at , ω) + S αβ (p αβ ,t αβ ,t αβ , ω) = S at + Φ αβ (p at ,t at ,t at ) + E(t at ) I α (p αβ +A(t αβ ),t αβ ) = I α (p at +A(t at ),t at )-i(-1) α K α (p at +A(t at ),t at )δ , (5.4.41)

		R 2	δ -i E(t at )	R 2	δ . (5.4.39)
	Prefactors and dipoles Finally, we perform first-order expansions around the atomic
	times for prefactors R and I α (Eqs. (3.3.33) and (3.3.32)):
	R(p αβ + A(t αβ )) = R(p at + A(t at )) -(-1) β ∂ R ∂ k	(p at + A(t at ))E(t at )δ ,	(5.4.40)
	and:
								∂ S αβ ∂t at	∆t αβ +	∂ S αβ ∂t at	∆t αβ . (5.4.35)
		∂ S αβ ∂t at	=	∂ S ∂t at	+	∂ Φ αβ ∂t at	,	(5.4.36)
	which becomes using Eq. (3.3.15):
		∂ S αβ ∂t at	= (-1) β E(t at )	R 2	.	(5.4.37)

The partial derivative of the modified action (3.3.37) with respect to t at t at writes: Similarly, together Eq. (3.3.16) the partial derivative of the modified action over t at t at reads: ∂ S αβ ∂t at = -(-1) α E(t at ) R 2 , (5.4.38) 5.4. TWO-CENTER INTERFERENCES IN MOLECULES and Eq. (5.4.35) becomes:

.23.

				Re(∆t αβ )			Im(∆t αβ )
			3D		1D		3D		1D
	αβ	θ	S	L	S	L	S	L	S	L
	12	π/6	0.277	0.289	0.278	0.289	-0.590	-0.596	-0.591	-0.596
		π/3	0.171	0.177	0.173	0.177	-0.358	-0.362	-0.361	-0.362
		0.48π	0.0201	0.0238	0.0228	0.0229	-0.0435	-0.0459	-0.0467	-0.0467
	21	π/6	-0.290	-0.279	-0.290	-0.279	0.567	0.562	0.566	0.562
		π/3	-0.179	-0.172	-0.177	-0.173	0.353	0.350	0.351	0.350
		0.48π	-0.0257	-0.0220	-0.0229	-0.0228	0.0497	0.0473	0.0465	0.0465

Table 5 .4 Time deviations ∆t αβ and ∆t αβ (in a.u.) for the 3D molecular SFA computa- tions on H 2 2 2 for different molecular angles θ , compared with the equivalent 1D compu- tations for which we replaced

 5 R by R = R cos(θ ), for trajectories 12 and 21.

Table 5 .5 Energies (×10 -2 a.u.) of the ten first vibrational states of ions H

 5

	+ 2 2 2 , D + 2 2 2 and T + 2 2 2 .

Table A .1 Expressions of fundamental atomic constants and derived constants in atomic units, along with their values in SI units.

 A

2 Durée des impulsions laser au court du temps. Les données ont été extraites de la référence [15] et complétées par l'impulsion la plus courte générée jusqu'alors : 67 as [37]. du

  paquet d'ondes électronique[START_REF] Haessler | Attosecond imaging of molecular electronic wavepackets[END_REF][START_REF] Zhao | Positioning of Bound Electron Wave Packets in Molecules Revealed by High-Harmonic Spectroscopy[END_REF] ou pour réaliser la tomographie d'orbitales moléculaires[START_REF] Itatani | Tomographic imaging of molecular orbitals[END_REF][START_REF] Vozzi | Generalized molecular orbital tomography[END_REF][START_REF] Salières | Imaging orbitals with attosecond and Ångström resolutions: toward attochemistry?[END_REF].Le but de ce doctorat est d'examiner de façon théorique la GHOE dans son approche auto-sonde. Pour cela, on doit résoudre l'équation de Schrödinger dépendante du temps (ESDT), qui s'exprime dans sa forme la plus générale par : Pour des systèmes plus complexes, il est alors nécessaire d'utiliser des méthodes numériques. Grâce aux progrès rapides en informatique et à la croissance continuelle des capacités de calcul, une approche ab initio, c'est à dire qui cherche à résoudre exactement l'ESDT pour un système considéré, peut être envisagée. Cependant la complexité des systèmes chimiques et biologiques augmente très vite et rend impossible cette résolution en un temps limité et un coût raisonnable. Il devient alors nécessaire de trouver des approximations pertinentes et des algorithmes appropriés afin de réduire considérablement le temps de calcul.Le second défi qui se présente est d'arriver à extraire de |Ψ(t) des informations physiques significatives. Ceci nécessite une analyse fine de cette fonction d'onde. Ainsi, les modèles simples sont souvent beaucoup plus féconds que les calculs intensifs, et le fait qu'ils ne requièrent que peu de temps de calcul permet de les utiliser de façon extensive et d'aborder de multiples facettes d'un même problème. Nous proposons alors de simuler la GHOE dans les atomes et les molécules diatomiques en utilisant des modèles très simples mais suffisamment robustes pour reproduire certains aspects expérimentaux et avoir une capacité prédictive.La thèse complète, rédigée en anglais, est articulée de la manière suivante: après un court chapitre introduisant la GHOE et ses caractéristiques, nous présentons les méthodes de simulation puis les outils d'analyse utilisés, et enfin nos résultats. Ce résumé suit ce plan, sans cependant s'attarder sur le formalisme et les aspects techniques -nous renverrons autant que possible aux références appropriées. Dans l'ensemble 5

	Intensité des harmoniques (unités arb.)	10 -1 10 -2 10 -3 10 -4 10 -5 10 -6 10 -7 10 -8 1	7	9	11 13	1 fs 10 fs 100 fs 1 ps 10 ps 100 ps 100 as 10 as 15 17 19 21 23 25 27 29 31 33 35 1970 1980 1990 2000 2010 Durée des impulsions GHOE Premières impulsions attosecondes	?
					Fig. S.ih Ordre harmonique	d dt	|Ψ(t) = Ĥ(t)|Ψ(t) .

(S.1.1)

Ici, |Ψ(t) est la fonction d'onde, contenant toutes les propriétés et toute la dynamique du système, et Ĥ(t) l'Hamiltonien qui prédit son évolution à partir de conditions initiales données. Cependant, la résolution de l'ESDT n'est possible analytiquement que pour des systèmes très simples, comme l'atome d'hydrogène par exemple qui ne met en jeu qu'un électron et qu'un noyau.

  Hamiltonien sur une grille. Chaque méthode a des points forts et des points faibles. Son utilisation sera adaptée au système et aux résultats souhaités. Les deux premières, par exemple, permettent d'obtenir rapidement les premiers états liés du système, tandis que la seconde, plus gourmande en temps de calcul, permet d'obtenir les états du continuum et de calculer une densité d'états, ce qui est pertinent pour décrire des résonances dans le continuum.Le champ laser interagissant avec le système est une impulsion de durée brève :

	E(t) = E L f (t) sin(ω L t),	(S.3.2)
	où f (t) est son enveloppe, définie soit de forme trapézoïdale, soit en sin 2 . La première
	permet d'avoir des oscillations identiques dans le plateau du trapèze et ainsi d'exciter
	le système avec une longueur d'onde identique au cours du temps. La seconde nous
	permet de générer des impulsions très brèves (deux cycles sous l'enveloppe, cor-
	respondant à environ un cycle à mi-hauteur) pour lesquelles un seul demi-cycle de
	génération est observé, empêchant les retours multiples.

t). (S.3.1)

Nous étudierons différent potentiels V (x), dont la forme est ajustée en fonction des propriétés physiques que l'on souhaite étudier à travers la GHOE. La recherche des valeurs propres (énergies) et vecteurs propres (états) du système non perturbé (E(t) = 0) est nécessaire pour établir les conditions initiales et étudier les caractéristiques propres à chaque potentiel. Pour cela, nous utiliserons l'itération inverse, la propagation en temps imaginaire [52] ou la diagonalisation directe de l'La fonction d'onde est propagée et connue à chaque instant en résolvant l'ESDT à l'aide de l'algorithme de Crank-Nicolson

[START_REF] Crank | A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type[END_REF]

.

  e -iωt dt.La théorie SFA, pour Strong-Field Approximation, approximation du champ fort, est une approche semi-analytique de la GHOE et permet donc d'obtenir un connaissance approfondie de certains processus. Dans le cas d'un atome, nous nous conformons aux travaux de Lewenstein et al[START_REF] Lewenstein | Theory of high-harmonic generation by low-frequency laser fields[END_REF]. Nous ne rappelons donc pas les hypothèses et présentons très brièvement les équations qui en résultent. La seule différence est que nous utilisons ici la forme vitesse du dipôle comme dans les références[START_REF] Śpiewanowski | High-order-harmonic generation from field-distorted orbitals[END_REF][START_REF] Śpiewanowski | Field-induced orbital distortion in high-order-harmonic generation from aligned and oriented molecules within adiabatic strong-field approximation[END_REF]

	(S.3.4)
	L'intensité des harmoniques est proportionnelle au module au carrée de D(ω) et leur
	phase est celle de D(ω).
	S.3.2 La théorie SFA

,

Baggesen et

Madsen ayant récemment démontré la pertinence de ce choix

[START_REF] Baggesen | On the dipole, velocity and acceleration forms in high-order harmonic generation from a single atom or molecule[END_REF]

. Ainsi, nous calculons le dipôle:

D(ω) = iω v(t)e -iωt dt, (S.3.5) où v(t) = Ψ(t) |-i∇| Ψ(t) • e x ,

ce qui est rigoureusement équivalent à l'expression (S.3.4) après intégration par partie. Ici, nous avons considéré un champ électrique polarisé le long de e x , à savoir E(t) = E(t)e x . En définissant le potentiel vecteur A(t) tel que E(t) = -∂ A/∂t, la théorie SFA permet alors d'aboutir à : D(ω) = -ω dt t 0 dt d 3 p d rec (p + A(t))d ion (p + A(t ),t )e -iS(p,t,t ,ω) + c.c., (S.3.6)

  Il convient donc de chercher les solutions qui annulent les dérivées partielles de S, ce qui conduit aux équations saddle-point (du point de selle) :Leur solutions sont notées (p at ,t at ,t at ) pour insister sur le caractère atomique de ces calculs. Cela permet de réduire la triple intégration à :D at (ω) = -ω C(t at ,t at )d rec (p at + A(t at ))d ion (p at + A(t at ),t at )e -iS at + c.c. , (S.3.13)où l'on a noté S at = S(p at ,t at ,t at , ω). Le terme C(t at ,t at ) provient de l'évaluation des intégrales aux solutions saddle-point et n'est pas explicité ici. Il traduit entre autres l'étalement du paquet d'onde électronique dans le continuum.En réalité, pour chaque fréquence ω on trouve deux triplets de solutions (p at ,t at ,t at ), correspondant aux trajectoires courtes et aux trajectoires longues. Le dipôle final est donc une somme sur ces deux contributions. Cependant, aucune distinction formelle entre ces deux trajectoires n'est faite dans l'expression (S.3.13) et la somme a été enlevée volontairement dans un soucis de clarté.

	t
	dτ[p + A(τ)] = 0 ,				(S.3.10)
	t
	[p + A(t)] 2 2	+ I p -ω = 0 ,				(S.3.11)
	[p + A(t )] 2 2	+ I p = 0 .				(S.3.12)
								S.3.8)
	respectivement, et :
		S(p,t,t , ω) =	t	t	dτ	[p + A(τ)] 2 2	+ I p + ωt,	(S.3.9)

l'action canonique. Le calcul complet du dipôle donné par l'équation (S.3.6) est coûteux car il met en jeu une triple intégration. Cependant, le terme de phase que constitue l'action canonique oscille rapidement. En utilisant l'approximation de la phase stationnaire, il est possible de simplifier cette expression. En effet, en résultat de l'intégrale, seuls les termes où l'action est stationnaire contribuent, les autres s'annihilant entre eux.

3.22) pour

  lesquelles quatre classes de solutions (p αβ ,t αβ ,t αβ ) sont obtenues. Elles correspondent à quatre classes de trajectoires, traduisant l'ionisation d'un centre atomique donné et la recombinaison sur le même centre ou sur l'autre, et inversement, comme schématisé par la figure S.5. Finalement, le dipôle total est évalué comme suit : αβ (t αβ ,t αβ )R(p αβ + A(t αβ ))I α (p αβ + A(t αβ ),t αβ )× e -iS αβ (p αβ ,t αβ ,t αβ ,ω) + c.c. 'équilibre et les deux atomes commencent à bouger (généralement à s'écarter, la liaison chimique étant affaiblie par le retrait d'un électron) et à osciller autour de l'équilibre : les niveaux vibrationnels excités de l'ion ont été peuplés. D'autre part, les niveaux vibrationnels de la molécule neutre peuvent être peuplés par excitation multi-photon de type Raman. Ces effets doivent donc être pris en compte de façon cohérente.

		t	t	dτ[p + A(τ)] + (-1) α R 2	-(-1) β R 2	= 0 ,	(S.3.20)
		[p + A(t)] 2 2	+ I p -ω + (-1) β E(t) •	R 2	= 0 ,	(S.3.21)
		[p + A(t )] 2 2	+ I p + (-1) α E(t ) •	R 2	= 0 ,
		2		2
	D(ω) = -ω	∑	∑
		α=1	β =1
							(S.3.23)

(S.C S.3.3 Au delà de l'approximation de Born-Oppenheimer L'effet du champ laser intense sur la vibration d'une molécule diatomique peut jouer un rôle important. En effet, d'une part, suite à l'ionisation d'un électron, l'ion n'est pas à l

  R) +V Ne (x, R).Ici µ est la masse réduite des noyaux, et V NN et V Ne sont les potentiels d'interaction effectifs noyau-noyau et électron-noyaux, respectivement. L'ESDT :

	(S.3.24)

i ∂ ∂t Ψ(x, R,t) = H 0 + xE(t) Ψ(x, R,t), (S.3.25)

  Ici, Φ v (x,t) sont les fonctions d'ondes électroniques et encodent toute la dynamique du système. Avec cet ansatz, l'ESDT devient :

	i	∂ ∂t	Φ v (x,t) = -	1 2	w =0 ∂ N v -1 ∑	χ + v VNe χ + w Φ w (x,t) ,

). (S.3.26) 2 ∂ x 2 + ε v + xE(t) Φ v (x,t) + (S.3.27)

  Gabor (principe d'incertitude de Fourier) : nous a conduit à développer un outil d'analyse qui s'affranchit de ces limitations. Il est basé sur le concept, relativement controversé[START_REF] Mandel | Interpretation of Instantaneous Frequencies[END_REF][START_REF] Gupta | Definition of instantaneous frequency and frequency measurability[END_REF], de fréquence instantanée. En effet, on peut montrer qu'après un filtrage adéquat des basses fréquences, le dipôle rempli toutes les conditions nécessaires à l'utilisation de la fréquence instantanée définie avec une transformée de Hilbert[START_REF] Cohen | Time-frequency Analysis[END_REF][START_REF] Gupta | Definition of instantaneous frequency and frequency measurability[END_REF][START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal[END_REF][START_REF] Huang | The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis[END_REF][START_REF] Schwartz | Communication Systems and Techniques[END_REF]. À partir du dipôle d(t) filtré on peut alors construire de façon unique un "dipôle analytique" z d (t) à l'aide de sa transformée de Hilbert H d :z d (t) = d(t) + iH d (t) = a d (t)e iΦ d (t) .La fréquence instantanée est définie comme la dérivée de la phase Φ d (t) de cette fonction complexe. Cette méthode nous permet d'extraire les temps d'émission des harmoniques du plateau, quantitativement pour le premier retour et qualitativement pour le second retour[START_REF] Risoud | Quantitative extraction of the emission times of high-order harmonics via the determination of instantaneous frequencies[END_REF]. Une comparaison avec les calculs SFA montrent des différences allant jusqu'à la cinquantaine d'attosecondes, typiquement de l'ordre des timings que l'on souhaite atteindre en attoscience, comme par exemple dans la mesure des délais de photoionisation[START_REF] Klünder | Probing Single-Photon Ionization on the Attosecond Time Scale[END_REF][START_REF] Schultze | Delay in Photoemission[END_REF][START_REF] Shafir | Resolving the time when an electron exits a tunnelling barrier[END_REF].Le calcul de la Distribution de Wigner-Ville (DWV) est une autre méthode d'analyse temps fréquence qui permet de s'affranchir de la limite de Gabor. Elle est donnée par :

						(S.4.2)
	DWV[d](ω,t) = d t +	τ 2	d t -	τ 2	e -iωτ dτ.
	∆t∆ω ≥	1 2	,

(S.4.1) (S.4.3)

  . Dans le cas d'une fonction d'onde électronique Ψ dépendant d'une variable x, elle s'écrit :

	W (x, p,t) =	1 π	Ψ * x -	y 2	,t Ψ x +	y 2	,t e ipy dy,	(S.4.4)

11 Probabilité, à la fin de l'impulsion laser, de peupler le premier état vibra- tionnel excité du neutre en fonction de l'intensité laser. L'impulsion est en sin 2 2 2 et dure 4 cycles.

	1
		modèle corrélé H 2
		modèle corrélé G 2
		Lochfraß H 2
	Probabilité (log scale)	Lochfraß G 2 Λ-system
		1	2	3	4	5	6 7 8
		Intensité laser (×10 14 W.cm -2 , log scale)

Fig. S.adéquate du potentiel d'ionisation. Habituellement, le potentiel d'ionisation d'une molécule diatomique est défini comme la différence d'énergie entre la courbe d'énergie potentielle du neutre et la courbe d'énergie potentielle de l'ion [126, 225, 227, 228].

Theoretical study of attosecond dynamics in atoms and molecules using high-order harmonic generation as a self-probe

		© 2016 François Risoud
		All rights reserved
		Printed in Paris, 2016
	Laboratoire de Chimie Physique -Matière et Rayonnement
		UMR 7614, CNRS -UPMC
		11 rue Pierre et Marie Curie
		75005 PARIS
		France
	This thesis was typeset with	using the book class
	and a custom model designed by Benjamin Varin
		www.maison-solide.fr
	All the pictures were created by François Risoud, except
	chapter 1, 3 & 6: Benjamin Varin
	chapter 2: experimental spectrum, Lou Barreau

Remerciements (S.5.2)

Ici, on définit L 1 la distance entre les deux barrières, L 2 leur largeur et V 0 leur hauteur, n le rang de la fonction super-Gaussienne, et a le paramètre de régularisation du potentiel pseudo-Coulombien. L'introduction des barrières structure le continuum avec des états quasi-liés, dont le premier (le plus bas en énergie) est ajusté pour correspondre (énergie et durée de vie) à la résonance de forme dans N 2 . On utilise, en u.a., a = 0.8975, L 1 = 1.25, L 2 = 2.395, V 0 = 0.8 and n = 4.

La distribution d'états résonants peut être obtenue soit en calculant une densité d'états, par diagonalisation de l'Hamiltonien sur une grille, soit par la méthode du shooting qui nous permet d'obtenir les états du continuum et leur déphasage par rapport à une référence [198]. Ces états sont normalisés en énergie par la procédure de Strömgren [199] et le déphasage est calculé à l'aide d'un Wronskien [198,200,201]. Cette dernière méthode nous permet alors de calculer l'élément de matrice dipolaire de transition et de le comparer à la phase spectrale obtenue dans les spectres de GHOE.

Tout d'abord, on observe une augmentation du rendement harmonique à l'énergie de la résonance, en accord avec les travaux de Tudorovskaya et al [196] où des potentiels avec barrières ont été étudiés de façon similaire. La phase des harmoniques est ensuite calculée par rapport à une référence atomique afin d'enlever la contribution quadratique propre à la GHOE qui masque les petites contributions que l'on souhaite étudier. On observe un saut de π, adouci et couvrant plusieurs harmoniques. Il correspond exactement à la phase de l'élément de matrice dipolaire de transition. Après séparation des trajectoires courtes et longues, par la méthode de l'absorbeur et par propagation macroscopique dans le gaz générateur, on montre que ce saut de phase se retrouve de façon identique dans les trajectoires longues que dans les trajectoires courtes, comme on le voit en figure S.7. Une étude de la dépendance de ce saut de phase avec l'intensité laser montre qu'il reste inchangé, mis à part pour les trajectoires courtes à des ordres harmoniques proches du seuil d'ionisation (le "plongement" observé en figure S.7a&b).

Afin d'étudier finement l'évolution de la position en énergie de la résonance en fonction de l'intensité laser, nous avons effectué des calculs R-box [205]. Nous observons un déplacement de la résonance de l'ordre du Stark-shift de l'état fondamental (∼ 0.4 eV à une intensité de 5 × 10 14 W.cm -2 ). Ce déplacement est donc négligeable devant la largeur en énergie de la résonance (∼ 10 eV). De plus, ces calculs nous dévoilent l'existence de résonances induites par le champ, qui pourraient expliquer

Alain de m'avoir accueilli dans son

(5.6.28), black lines) and the spectra obtained with the fully correlated model (green lines).

VIBRATING DIATOMIC MOLECULES

To this end, the expressions of the dipoles should be:

(5.6.27)

in the R-approach, and:

(5.6.28)

in the v-approach, where c v is the coefficient of state v of energy ε v populated at initial time t i = 0. The first approach requires the resolution of the nuclear TDSE given by Eq. (5.6.5) to obtain ξ (R,t i (ω)), which gives simply:

(5.6.29)

Unfortunately, we did not have time to perform the simulations.

A.3. 4TH ORDER RUNGE-KUTTA ALGORITHM

Using Euler's method to approach the kinetic energy operator, as presented in Sec. 3.2.4, the TISE is solved as:

In the box, the expression of Ψ is:

with the modes:

Hence, the TISE reads:

When ∆x → 0,

and then we retrieve the expression:

[Recalling Eq. (3.2.30)]

A.3 4th order Runge-Kutta algorithm

The Runge-Kutta algorithm of 4th order (RK4) is a numerical scheme to solve the following initial value problem:

where y is an unknown scalar (vector) function and f a given scalar (matrix) operator on time t and function y. Knowing y 0 = y(t 0 ), the value of y at initial time t 0 , and given a time-step ∆t and a time-grid

, we compute y n = y(t n ) at all times t n with the recurrence:

n + 2k

where:

n ∆t .

Numerical solution

The solution χ E of the TISE obtained numerically (here with the shooting method, using RK4 algorithm) is generally not normalized in energy scale.

In the asymptotic region, we assume that we can write (for an odd state):

where the constant C is a priori unknown and depends on the initial conditions used in the shooting algorithm. We can determine it by fitting the numerical solution with the asymptotic form χ (a) E given by Eq. (A.4.2). To do so, one need to determine the asymptotic forms of θ and k. Since θ is the primitive of k, the procedure requires integration over a finite interval in the asymptotic region [x 0 , x 1 ] ⊂ [x a , +∞[. Let us define the intermediate quantities:

Therefore:

sin(α) = sin(θ (x 1 )) cos(θ (x 0 ))sin(θ (x 0 )) cos(θ (x 1 )) cos(α) = cos(θ (x 0 )) sin(θ (x 1 )) + sin(θ (x 0 )) cos(θ (x 1 ))

Finally, we obtain:

.

(A.4.9)

To calculate C we thus need to evaluate quantities a 0 , a 1 and α. Hence we need to determine k in the asymptotic region.

Determination of k Let us start by inserting the analytical asymptotic wave-function χ Yet,

(A.6.5)

Hence, the condition given by Eq. (A.6.4) leads to:

The first condition gives x q = (2q + 1)π, q ∈ Z, and thus |z(x q )| = 1, i.e. we found the maxima. The second condition does not have simple solutions, however we can study its asymptotic behavior. If α does no longer dependent on x, and α = 0, we retrieve the condition cos(x) = 0, i.e. x q = (2q + 1)π/2. In the case where α depends slowly on x (allowing to neglect its second derivative), we can search the deviation δ x = xx q to solutions x q with the help of first order Taylor expansions around x q : cos(x q + δ x ) α(x q ) + δ x α (x q ) 2 -1 + α (x q ) α(x q ) + δ x α (x q ) sin(x q + δ x ) =sin(x q )δ x α 2 (x q ) + 2δ x α(x q )α (x q ) -1 + α (x q )α(x q ) + δ x α 2 (x q ) sin(x q ) = α (x q )α(x q ) + δ x α 2 (x q ) -1 + α 2 (x q ) = 0, (A.6.7)

which leads to:

Hence, the minima of |z(x)| are encountered approximately at: xq = x q + α (x q )α(x q ) α 2 (x q )α 2 (x q ) + 1 .

(A.6.9)

Recalling that |α(x q )| < 1, α 2 (x q )α 2 (x q ) + 1 > 0 and thus the position of the minimum is shifted from ideal case x q by a quantity which sign depends on the sign of α (x q )α(x q ). In Sec. 5.4.5 we found an expression of a modified recombination dipole matrix element of the form of z(x), where x = kR/2 and:

With the use of a Gaussian function for the symmetric atomic state, we find that (similarly to the calculus of γ app in Sec. 5.4.8):

where we neglected, for simplicity, the dependence of δ on x. Finally:

The first minimum of z(x) is encountered at x 0 = π/2, for which we have:

where we used H 2 parameters (in a.u., R = 1.425 and α 0 = I p = 0.567). At high intensity, such as I L = 5 × 10 14 W.cm -2 , the maximum electric amplitude that can be achieved is about 0.12 a.u., leading to:

that is a displacement of 1.2% of the position of the minimum amplitude. It corresponds to a maximum shift of about 1 harmonic order. In most of the cases, laser intensity is lower, and recombination does not occur at the maximum of the electric field. Hence this shift is much smaller than ω L , and consequently can be neglected.

Slope of the phase-jump As explained previously, this complex function describes deviated path on the complex plane, associated with smoothed π-jumps of the phase θ (x) around the minima of |z(x)|. We search an expression of the slope of the phasejump at minima positions x q . The first derivative of θ (x) is:

x) .

(A.6.15)

Let us start in the case where α is independent on x, i.e.

The second derivative is:

(A.6.17)

The inflexion point of the phase-jump is given by the condition:

which is impossible by choice of α. Hence, the only possibility for which θ (x) is zero is when the denominator diverges to ±∞, i.e. when x → x q = (2q + 1)π/2. Hence, the inflexion point corresponds exactly to the minimum of |z(x)|. The value of the slope γ of the phase-jump at x q is:

α .

(A.6.19)

If now α depends on x, we can evaluate the slope γ at position x q (since we have seen that the shift δ x is small). From Eq. (A.6.15) we easily find: γ = 1 α(x q ) + α (x q ) α 2 (x q ) = γ 1 + α (x q ) α(x q )

(A.6.20)

As previously, we can evaluate this correction in the physical case of two-center interferences in H 2 . We find:

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