{'fraction_non_alphanumeric': 0.03912090745037381, 'fraction_numerical': 0.011278680072183552, 'mean_word_length': 4.615997104596453, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Kinetic of the Atomic Relaxation Induced by Laser Noise

We present a theoretical study of strong laser-atom interactions, when the laser field parameters are subjected to random processes. The atom is modelled by a two-level and three-level systems, while the statistical fluctuations of the laser field are described by a pre-Gaussian model.


Introduction
It is actually recognised that nearly all types of laser-atom interactions can be strongly affected by laser noise. In fact, real atoms experience a fluctuating environment of many perturbing interactions and ideal lasers exist only in theoretical models, while the used laser sources are subjected to many types of fluctuations notably in phase, amplitude and frequency. Stochastic variations of the Hamiltonian can be due not only to the field but also to the jump-type transitions of the atom from one energy state to another, jumps which are characteristic of condensed phases and which modulate its interaction with the medium [1]. Therefore, we cannot realise, without taking into account the statistical properties of the laser radiation, an exact comparison between theoretical predictions and experimental results. Incorporation of such stochastic properties into the Liouville equation by fully microscopic treatment (relaxation times and bandwidths) [2] give modest results, so we use a theoretical models based on the Markov pre-Gaussian processes [2][3][4][5]. These processes, composed of N-independent two-state jump processes (random telegraphs), form a non-Gaussian stochastic models, also called Markov chains. Such a Markov chain offers a detailed discussion of the atomic response. Our choice of the Markov chain is based on the simplicity of this model and the remarkable property of convergence to a Gaussian stochastic process when N → +∞. We derive the proper master equation for the density operator driven by a noise described by two-state stochastic telegraph process. This fundamental equation first introduced in quantum optic by Burshtein [2][3][4], contains all information concerning both the atomic transitions dynamic and the stochastic evolution of laser field fluctuations.
Several works have reported on the action of random process on a two-level system [1][2][3][4]6,7], particularly the evolution populations. In the present article we want to investigate the atomic response to a noise laser, i.e., to describe the relaxation of the atomic level populations ρ nn . It should be noted that the kinetic of the phase relaxation contains information on the width and shape of the spectral lines [1]. We also extend our study to a three-level system and we provide the ionisation influence for strong laser-atom interactions, when the laser field parameters are subjected to random process.
We are concerned here with an important theme of contemporary research, namely the interplay between quantum coherence and external noise. The destruction of quantum coherence by noise is central to many fields of physics and is reflected in the large number of papers recently published on this subject [8][9][10][11][12].
In this paper we are interested by a strong laser field whose frequency is resonant; time dependent perturbation theory and rotating wave approximation (RWA) [13] are therefore not adequate to resolve the interaction process. Indeed, the perturbation method is justified when the electric field strength F 0 remains much smaller than the atomic unit of field strength, namely F 0 << 5.10 9 V cm -1 and when the laser photon energy is not tuned close to an atomic transition energy. The second limitation is resolved by the RWA, however there is the Bloch-Sierget shift [13] which deteriorates the efficiency of this approximation, especially when the laser intensity increases. We explore the limits of validity of rotating wave approximation for multi-level system and/or for strong laser field. We have therefore developed a nonperturbative treatment of the laser-atom interactions, based on the Floquet theory [14][15][16][17][18]. This nonperturbative method allows to transform the time dependent Liouville or Schrödinger equation to a stationary problem of eigenvalues. This problem can be solved without restrictions on the laser parameters, i.e., its solution requires at most a finite matrix diagonalization. It is another advantage of Floquet method in the context of laser-atom interactions.
The paper is structured as follows. In section 2, we present a detailed description of the theoretical formalism, which is valid for the general case of multi-level atoms. By considering the case of noise strong laser-atom interactions, for which detailed Floquet calculations are feasible to obtain the density matrix elements, which are solutions of the master equation. The account of Floquet theory given here is rather brief, since the theory has been discussed at length in the recent literature (see e. g. refs. [14][15][16][17][18]).
Numerical results concerning the model of two-level and three-level systems are presented and discussed in section 3, where the phase and amplitude noises are described by random telegraph and Markov chain. The case of fluctuations due to collisions is analysed. We also discuss the noise effect on the ionisation process. At the end a summary of our results is given. 6

General formalism
We consider a model of multi-level atom excited by a classical purely monochromatic laser field described by an electric field linearly polarised, Here F 0 is the electric field amplitude (possibly fluctuating in magnitude) and φ(t) is the instantaneous phase of the laser ( fluctuating around the zero value ). The laser parameters are affected by a stochastic process of jumps. We represent these fluctuations by a Markovian Pre-Gaussian models. Thereafter, we denote the random telegraph and the Markov chain respectively [3]; by the processes [ x(t) = ±a ] and [ X(t) = {-Na, -(N-2)a, ..., (N-2)a, and Na}], with a is the amount of the jump assigned to the stochastic signal and N the number of random telegraph signals [3].
In order to incorporate a laser phase or amplitude noise in equations, treating the laser-atom interactions, we have therefore describe the system states considered in terms of the density matrix elements. The time dependent behaviour of the density operator ρ is given by the Liouville equation [18] [ where Γ and Λ are two diagonal matrix corresponding to the spontaneous emission process and defined by [18] Γ nn nn n n = ∑ < γ ' ' and Λ nn nn n n n n t = ∑ > γ ρ where γ nn' is the radiative decay rate. H(s) is the non-relativistic periodical Hamiltonian in the presence of the laser noisy and s labels any possible state of the random process.
In the case of the laser amplitude fluctuations, H(s) may be written as H(s) = H 0 + e F 0 (1 + x (t) ).R cos (ωt + φ(t)) (4) and, in the case of the laser phase fluctuations, it takes the following form H(s) = H 0 + e F 0 .R cos (ωt + φ(t) + x(t) ), where H 0 is the nonperturbative Hamiltonian, ω the frequency of laser field, e is the electric charge and R the dipole operator.
In the stationary eigenstates {|n〉} of the unperturbed system Hamiltonian, the equation (2) could be written in the following system of coupled differential equations for the time dependent matrix elements density ρ nn' (t) as where ω nn Since the process that we are considering here is Markovian, the conditioned probability density function associated with it, namely ( ) p s t s t , , 0 0 , is shown to satisfy the following Chapman-Kolmogorov equation [2][3][4]19,20] here s 0 is the initial state of Markov chain at the time t 0 .
In the simple case of random telegraph signal, the above equation reduces to ∂ ∂t p s t s t T p s t s t T p s t s t ( , , In the compact form , the equation (8) where g(s) is the initial probability distribution of the random process and ρ nn',s (t) the average value of ρ nn' (t) under the condition that x(t) is fixed at the value s at time t.
Combining the Chapman-Kolmogorov equation (9) for the probability density function and the dynamic equation (6) with Λ nn s nn n n n n s The equation (11) The Floquet coefficients C nn s M ', and the pseudo-energies ε can be found by solving numerically the eigenvalues problem where the dipole coupling matrix elements M nk s −1 ( ) and M nk s +1 ( ) are defined by In matrix representation, the system of equations (14) could be written in the following compact form, as where G is the infinite tridiagonal block Floquet matrix, ε and ψ the eigenvalues and the eigenvectors of G. In numerical calculations, the series expansion in function of M (expression (13)) is truncated to a finite M max number of terms which permits the convergence of eigenvalues and eigenvectors of G. In the absence of noise, the density operator satisfies a 9×9 matrix differential equation with constant coefficients ( case of three-level system ). The master equation allows stochastic telegraph type noise to be added at the sole expense to enlarge the matrix dimensionality from 9×9 to 9n s ×9n s; n s represents here the two states of random telegraph signal. When we consider the Floquet solution based on the development of Fourier, we enlarged also the matrix dimensionality from 9n s ×9n s to 9n s (2M max +1)×9n s (2M max +1), with (2M max +1) is the number of Floquet matrix blocs. We remark that dimensionality of G becomes very large [dim G = n 2 ×n s ×(2M max +1), with n is the number of atomic states] and depends of M max This latter is the truncated number of Fourier development permitting the convergence of eigenvalues and eigenvectors of Floquet matrix. It is important to remark the following fact, the Floquet pseudo-energies ε j are only defined modulo the photon energy, so the Floquet quasienergy are not unique. We extract the pseudo-energies ε j , which are physically independents, from the Floquet matrix eigenvalues.
The Floquet solution of the system (14) We remark that the Floquet method permits us to evaluate exactly atomic response functions in finite terms, and we can then examine the influence of laser fluctuations on these functions.

Results and discussion
By describing external sources of noise by Pre-Gaussian Markov chains composed of N-independent jump processes, we treated two different sources of pre-Gaussian noise (phase and amplitude) by using a simple general master equation soluble in finite terms.
Our application is carried out in two cases. In first, the model of two-level atom on resonance is considered and it results compared with those obtained by Eberly et al. and Wodkiewicz et al. [2,3]. We also provide some results for atomic response, when RWA approximation is not justified for strong laser fields. In second, the behaviour of threelevel atom response to laser noisy is investigated. We choose the inverse Rabi frequency Ω as time unit, (where Ω = d nk .F 0 , with d nk is the dipole matrix element between levels |n〉 and |k〉), in the aim to analyse the obtained results in terms of the noise strength .
In the absence of any noise source ( electric laser field is purely coherent) and if we neglect the other relaxation rates ( spontaneous decay and ionisation rate), the behaviour of the populations of two-level and three-level systems is illustrated in Figure 1., which shows a superposition of several undamped oscillation modes ( Rabi oscillations). The system remains indefinitely in this oscillatory state.

1. Two-level atom in presence of random telegraph laser noise.
As already noted by several authors [2][3][4], the behaviour of a two-level atom to pre-Gaussian noise in strong laser-atom interactions depends critically on a special telegraph noise. The agreement between the two methods (the numerical calculation within the framework of rotating wave approximation and the present nonperturbative method of calculation based on the Floquet theory) is good at resonant excitation and when the electric field strength F 0 remains much smaller than the atomic unit of field strength 5. 10 9 V/cm. This behaviour is illustrated in Figure 2.

2. Two-level atom in presence of Markov chain laser noise.
We generalise the case of random telegraph to a Markov chain, which treats the phase and amplitude fluctuations of strong laser field interacting resonantly with a two level system. (ΩT = 1000). Similar behaviour is also observed when we increase the number of phase telegraphs (seven telegraphs), but the damping becomes more intense and we clearly observe the convergence to a Gaussian limit.

Three-level system in presence of random telegraph laser noise.
Multilevel systems show a variety of interesting optical effects with laser fields [22][23]. It would be interesting to study how the incoherence sources affect the population evolution of multilevel atoms. For the sake of numerical simplicity (the Floquet matrix dimentionality largely increases from two-level to multi-level), we only extend our application to a three-level atom driven by stochastic strong laser field at resonance. Our model of three-level atom is a ladder system which contains three discrete bound states and a continuum. It may be considered as a generalisation of the so-called ''extended two-level'' model proposed by Yeh and Eberly [24], where they have assumed that the bound-continuum dipole moments are weakly energy dependent so that it is well justified to use the adiabatic following elimination of the continuum degrees of freedom [24]. In absence of ionisation, no loss out of the system occurs, but damping is supplied by spontaneous decay and laser noisy within the three-level system, so that non zero populations are maintained in a steady-state for large times. Figure 7 clearly illustrates this behaviour in the case of one phase telegraph, with a = 0.4π, three different phase switching rates (ΩT= 0.1, 1 and 10) and the electric field strength is F 0 = 5. 10 8 V/cm. The relaxation process is observed for (ΩT = 1) with strong damping.

Ionisation effects on field-atom interactions in presence of laser noise.
Since we have considered a strong laser field, we would have a large ionisation for all atoms. In order to take into account of the noise laser effects on the atomic populations and ionisation process, we will incorporate the responsible term of ionisation [τ EC = -R nc δ nn' -1/2 (R nc + R n'c ) (1 -δ nn' ), where R nc is the relaxation rate from the excited state |n〉 to the continuum |c〉] in the motion equations (2), (6), (11) and (14). Two ionisation probabilities are calculated respectively the instantaneous where R I and we represent the effect of different laser noises on the average ionisation probability P t I m ( ) . Same behaviour can be observed for the ionisation rate R I . Figure 9(a) displays the time evolution of the two-probabilities P I (t) and P t I m ( ) in absence of noise. Figure 9(b) illustrates the influence of one amplitude telegraph on the average ionisation probability P t I m ( ) . The same parameters as Figure 6 are used. The results depend on the fluctuations time scale compared to the other characteristic time scales of the problem. The minimum ionisation rate is obtained for (ΩT = 10). Figure 9(c) shows a weak effect of phase noise on the ionisation probability and minimum ionisation rate is obtained for (ΩT = 1).

Conclusion
In this paper, we have presented a general stochastic treatment to incoherence properties induced by laser fluctuations (phase and amplitude) and by collision effects