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We numerically investigate the ionization mechanism in a real hydrogen atom under intense fem to second chirped laser pulses. The central carrier frequency of the pulses is chosen to be 6.2 eV (λ = 200 nm), which corresponds to the fourth-harmonic of the Ti:Sapphire laser. Our simulation of the laser-atom interaction consists on numerically solving the three-dimensional time-dependent Schrodinger equation with a spectral method. The unperturbed wave functions and electronic energies of the atomic system were found by using an L^{2} discretization technique based on the expansion of the wave functions on B-spline functions. The presented results of kinetic energy spectra of the emitted electrons show the sensitivity of the ionization process to the chirp parameter. Particular attention is paid to the important role of the excited bound states involved in the ionization paths.

The continuous progress in intense and short laser pulses generation techniques has led to extensive experimental and theoretical investigations on how to use these pulses as tools to control the ultra fast electron dynamic in atomic and molecular systems [1-3]. Recently, physicists took an interest in using chirped laser pulses to investigate laser-matter interactions. These pulses are efficient to control the ionization and excitation mechanisms when compared with unchirped laser pulses, i.e. transformlimited laser pulses [4-9]. The ability to control population transfer from an initial state to a desired final state is of crucial importance as it will open the way to the control of the product yields in chemical reactions.

The most known nonlinear processes that arise in ultra fast laser-matter interactions will change when intense and short chirped laser pulses are used instead of the transform-limited laser pulses [10-14]. The above-threshold ionization (ATI) process is the most important and well-understood nonlinear process by physicists. It is the signature that atoms or molecules can absorb more photons than is strictly necessary for the ionization to occur. The shape of the ATI spectrum, i.e., the kinetic energy spectrum of the emitted electrons, exhibits a series of peaks equally separated by the absorbed incident photon frequency [15-18].

In this paper, we point out the effect of the laser chirp parameter on the kinetic energy spectra of the emitted electrons from the hydrogen atom. We used intense femtosecond chirped laser pulses of central carrier frequency, which corresponded to the fourth-harmonic of the Ti:Sapphire laser. A chirped laser pulse is experimentally produced by chirp filtering of a transform-limited laser pulse. We consider, in this paper, a realistic experimental representation of a linearly polarized chirped laser pulse, in which the chirp parameter increases the time duration of the original transform-limited laser pulse and decreases its intensity. The frequency bandwidth of the resulting chirped pulse remains the same as that of the corresponding transform-limited laser pulse. The theoretical approach we used to calculate the unperturbed energetic structure of the hydrogen atom is based on an discretization technique using B-spline functions [19,20]. The kinetic energy spectra of the emitted electrons were generated by numerically solving the threedimensional time-dependent Schrödinger equation using a spectral method [21-23]. We follow in time the transfer of the population to the intermediate bound states involved in the ionization paths. We examine then the sensitivity of the excitation mechanism to the sign of the chirp parameter and how that affects the kinetic energy spectra of the emitted electrons.

Atomic units (a.u.) are used throughout the paper unless otherwise mentioned.

The time-independent Schrödinger equation (TISE) describing the electron motion around the atomic nucleus is given by:

is the nonrelativistic field-free Hamiltonian in spherical coordinates, which reads

For a given electron angular momentum l and projection m, the solution of Equation (1) can be written as follows:

is the spherical harmonic functions depending on angular coordinates. The B-spline function of order k denoted by is a piecewise polynomial of degree k-1 [

Within the electric dipole approximation, the time-dependent Schrödinger equation (TDSE) describing the electron motion in the presence of the laser field is given by:

describes the interaction of the electron with the laser field. It could be expressed in different gauges as for example the length or velocity gauges. As we adopt here the velocity gauge, is written as the scalar product of the potential vector and the electron momentum:

where is the vector potential of the chirped laser pulse linearly polarized along the z axis. The timedependent total wave function in Equation (4) is expanded on the basis of the field-free atomic eigenstates, normalized to unity,

By substituting Equation (6) into Equation (4), we obtain a set of coupled integro-differential equations, which we solved using an explicit fifth-order Runge-Kutta numerical method [

The vector potential of the chirped laser pulse considered here is given by:

With ξ is the chirp parameter, the pulse carrierenvelope phase (CEP), the peak amplitude, the instantaneous frequency and the Gaussian time envelope. We note that, in experiment, chirp filters are implemented by use of dispersive optical systems. Upon transmission through a filter characterized by a b chirp coefficient, an initially transform-limited pulse becomes chirped, i.e., with frequency that varies in time and depends on the chirp parameter (for the experimental details see [

The expression of is given by :

where is the electric field amplitude of the chirped pulse, with W·cm^{−2}the atomic unit of intensity and the central carrier frequency. The intensity of the chirped pulse is related to the intensity of the transform-limited pulse by:

The instantaneous frequency and the Gaussiantime envelope are given, respectively, by :

and

The FWHM duration of the chirped pulse is given by:

At the switch-off of the laser pulse, the probability of finding the systemin the electronic eigenstate is given by:

The kinetic energy spectrum of the emitted electrons is calculated as follows:

with the density of continuum states [

For the numerical calculations, we have considered a set of B-splines of order defined on the radial box of maximum length a.u.. We have fixed the maximum electron angular momentum to. As we choose the laser pulse to be polarized linearly along the z axis, only electronic transitions between states of magnetic quantum number are permitted. We kept in all our calculations the CEP constant and set to. The convergence of our numerical results has been checked by increasing the number of B-sp line functions, the radial box size and the number of angular moment a. We have also checked that the results are gauge independent.

We consider in this paper a transform-limited pulse (ξ = 0) of central carrier frequency ω_{0} = 0.228 a.u. (6.2 eV), which corresponds to the fourth-harmonic of Ti:Sa laser (λ = 200 nm). The pulse has an intensity W·cm^{−2} and a FWHM duration τ_{0} = 137.79 a.u. =3.33 fs.

We plot in ^{−2} and a FWHM duration a.u. = 4.7 fs. The form of the vector potential used in this paper (see Equation (7)) simulates a realistic experimental case in which the chirp filtering decreases the intensity of the transform-limited pulse and increases its total duration. We also present in

We submit the hydrogen atom to each of three pulses showed in _{P} = 0.5 a.u. = 13.6 eV). The many peaks observed in kinetic energy spectra of the emitted electrons have relatively narrow widths, and are centered on electronic energies E^{S} given by:

where is the electron ponderomotive energy. The hydrogen atom under a strong pulse of intensity I and photon frequency ω experiences an ionization potential increase of U_{P}.

We notice in

how the sign of the chirp influences the time variation of the instantaneous frequency, which affects the energy shift of the excited states and then the energy of emitted electrons.

Let us now observe more precisely if the sensitivity of the ionization process will persist for other chirp parameter values. In Figures 3(a)-(d), we present in nonlogarithmic scale the first peak of the emitted electron energy spectrum which is due to the ionization of the atom with three photons. The results were obtained at the end of the laser pulses with the following chirp parameters We compare in each case the chirped pulses results with that obtained with the unchirped pulse. In the first case, the intensity of the upand down-chirped pulses is W∙cm^{−2} and their FWHM duration is. In the second case, the intensity and the FWHM duration of the chirped pulses are the same used previously. In the third case, W∙cm^{−2} and. Finally, in the fourth case,

the intensity decreases to W∙cm^{−2} and the duration increases to.

When we compare between the results presented in

The significant change in the shape of the emitted electron energy spectra, particularly with adown-chirped laser pulse, results from the sensitivity of the excitation process to the sign of the chirp parameter. The latter affects the average of the population transferred to the intermediate bound states involved in the ionization paths. To prove the role of the excited bound states in this study, we removed them all from the atomic basis set and then we solved the TDSE. We have found that the chirp influence on ATI spectra has totally disappeared.

_{0} = 0.228 a.u.. To understand the influence of the chirp parameter on the excitation mechanism, we have examined how the population of the excited states by the up-chirped laser pulse takes place compared to the down-chirped pulses. We followed in time the population of the 2s, 2p, 3s, 4s, 3d and 4d bound states in the presence of the chirped pulses of used previously. We found that the 2s and 2p bound electronic states, off-resonance, are not affected by the chirp in contrast with the 3s, 4s, 3d and 4d states that are close to resonance by two-photon transition from the 1s. We also found that the 3d and 4d excited states are much more populated than the 3s and 4s states. The ionization occurs mostly by electrons transition from the states to the continuum channel and feebly from the states to the continuum channel.

So, we only report in

resonant transition from the 1s to the 4d first and from 1s to the 3d later in time. That explains why in

The double peaks structure observed in Figures 3(c) and (d), when we fix the value of the chirp parameter to and, is essentially due to two factors. The first is by increasing, we decrease the laser intensity, so the population of the 3d state decreases as it becomes then less coupled to 1s by resonant two-photon transition. The second is a chirped pulse with a negative ξ increases the population of the 4d state. Ionization, in this case, occurs by the transfer of the population from both the 3d and 4d states to the continuum channel causing the appearance of the two narrow peaks observed in the emitted electron energy spectra. In contrast with the case when we decrease (increase) the chirp parameter (laser intensity), see Figures 3(a) and (b), here the transfer of the population occurs predominantly from the most populated 3d state to the continuum. Here, no double narrow peaks structure is observed.

The excitation mechanism could be controlled by chirped laser pulses. By varying the sign of the chirp parameter, the instantaneous frequency changes in time differently, which induces different excitation scenarios. With chirped laser pulses, we could determine which level will have a bigger average population during the laser-atom interaction and thus control the ionization yield. We have investigated the sensitivity of the ionization process to the chirp parameter for various other harmonic frequencies of Ti:Sa laser. We have observed that the sensitivity is generally more accentuated when the chirp

induces resonant transitions in the excitation mechanism.

We have presented in this paper the results of numerical calculations of the emitted electron energy spectra in the hydrogen atom under intense fem to second chirped laser pulses. We have observed that the ionization and excitation dynamics are very sensitive to the chirp parameter. Intense ultra short chirped laser pulses could be an important tool to study image and control the electron dynamic in atoms and molecules. The study presented in this paper could be applicable to control the transfer or the inversion of the population in another atomic target. The chirp dependence of the emitted electron energy spectra observed here may prove useful for experimental characterization of chirped laser pulses.

The present research was supported by the NSERC and by the New Brunswick Innovation Foundation (NBIF). Allocation of CPU time and assistance with the computer facilities from the Atlantic Computational Excellence Network (ACE net, St-John’s, NL, Canada) and from the “Réseau Québecois de Calcul de Haute Performance” are acknowledged.