Singly Resonant Multiphoton Processes Involving Autoionizing States in the Be-like CIII Ion

Abstract

In this paper, we investigate the applicability of different theories on the intensity-dependent ionization rate for C²⁺ atomic targets at different laser wavelengths (frequency) and at linear polarization. We use the analytical formulas and draw conclusions, from numerical comparison with the results from ab initio ‘two-state model’ R-matrix Floquet calculation, on their correct predictions of the ionization rate. The single-photon ionization has been studied in the vicinity of the 1s² (²P^o)2pns (¹P^o), n = 5–12 autoionizing resonances at non-perturbative laser intensity. The results obtained from Perelomov–Popov–Terent’ev and Ammosov–Delone–Krainov models are compared in a region away from resonance where the two-state model description is not as good. To quantify the deviation between theoretical models, we analyze the ratio between different data sets as functions of the Keldysh parameter. We conclude that the results obtained with the model of Perelemov–Popov–Terent’ev are the closest to the ab initio R-matrix Floquet calculation.

1. Introduction

With the advance of laser technologies and the impact on the development of atomic physics, the study of atoms in strong laser fields has received special attention recently. The first such studies were based on the one-electron model of the atom. Reviews can be found in Refs. [1,2]. However, the electron correlation effect that occurs in atoms with two or more electrons in intense laser fields [3,4] have recently drawn some attention, with particular interest in the excitation of autoionizing resonances of the atom. While the time dependent Schrodinger equation allows one to follow the dynamics of the electron correlation during the laser pulse [5], the Floquet ansatz (see Ref. [6] for a review) is the ideal candidate for the study, using the adiabatic approach, of the atomic structure in the laser field.

Experiments on the ionization of atomic targets by intense pulses [7,8,9,10,11] have also been performed. The autoionizing states in the continuum manifest themselves as asymmetric peaks in the photon absorption spectra due to the mixing between their configurations. This phenomenon is called the Fano interference. In most cases, control over the Fano interference [12] has been achieved by an interplay between an intense high-frequency pump and a low-frequency probe pulse [13,14,15]. In modern laser facilities, where strong laser fields are produced with a duration comparable to the lifetime of the autoionization state, single- and multi-photon processes can be studied using the ab initio approaches [16,17]. For low-intensity electric fields, the autoionization states are defined using perturbation theory [18].

The electron and photon interactions with Be-like ions, including C²⁺, have been the subject of many detailed experimental and theoretical works. Two ionization channels attached to a 2p ionic state of C³⁺ are relevant for photoionization, namely 2pns(¹P^o) and 2pnd(¹P^o). Besides their spontaneous decay, the C²⁺ 1s²2pns (¹P₁^o) and 1s²2pnd (¹P₁^o) Rydberg states are also unstable against autoionization. Based on the combination of the Floquet theory and the R-matrix numerical method, ‘the R-matrix Floquet’ theory and code (RMF) [19], an impressive number of multiphoton processes have been studied, and a wide variety of interesting resonance effects in multiphoton ionization [20,21,22] have been predicted. One of them is the occurrence of laser-induced degenerate state (LIDS) phenomenon. During LIDS, the energies and widths of two states are made identical at certain laser frequencies and intensities [23]. The existence of LIDS has been observed in the RMF calculation for multiphoton transitions in C²⁺ and Al⁹⁺ [24]. In this earlier work, we carried out ab initio RMF calculation using the ‘two state model’ of Latinne et al. [23] when the two excited Rydberg states, namely the 1s²2sns (¹S^e) and 1s²2pn’s(¹P^o), Δn = n′ − n = 0, are resonantly coupled by one photon. We investigated this process in detail in the case of the carbon ion [25] and found the existence of LIDS in the region in which Δn = n′ − n = 2. The numerical results obtained have shown a good agreement between the two-state model and the full RMF calculations near the resonance frequency, but further away from this the agreement is less. Several quasi-energy spectra of the above-mentioned excited Rydberg ion states in laser fields have been compiled in our earlier works.

The main goal of this work is to compare results on the intensity-dependence ionization rate obtained from three theoretical models. The motivation is twofold. First, the Be-like C²⁺ ion will be the test case for a future pump-probe experiment based on the use of a table-top XUV laser in combination with a storage ring at GSI [26]. Secondly, because the two-state model used in our earlier analysis does not include the electron correlation effect and the nearby resonances, it gives results that are less accurate compared to the full RMF calculation. Therefore, we proposed in this paper to find the closest analytical model that can be used as a starting point in the analysis of future experiments. In the present paper, we complete the previous data sets with a focus on the total (energy and angle-integrated) ionization rate of Be-like C ions, for the Keldysh parameter γ ~ 1 and γ >> 1, including the Coulomb-correction. We use the analytical formulas and draw conclusions, from numerical comparison with the ab initio RMF calculation results, on their correct predictions of the ionization rate in the case of the 1s² (²P^o)2pns (¹P^o) Rydberg states in Be-like C²⁺. Section 2 summarizes the earlier analysis and its achievements. Section 3 reviews the analytical ionization rate formulas and provides comparison between results from present and earlier ab initio calculations. In Section 4, we discuss the numerical results and present concluding remarks.

Except where indicated otherwise, atomic units are used throughout the paper

2. One-Photon Ionization of the Excited 2pns ¹P^o States in Be-like C III Ion

In this section, we present the main achievements of our earlier RMF studies. The essential physical assumptions made when considering the process of one-photon ionization have been presented elsewhere. We recall here the main steps that are relevant for this work. Starting with the 1s²2s (²S) and 1s²2p (²P⁰) target states of the remaining Li-like target ion, the 1s²2s²(¹S) and 1s²2p²(¹S) states of Be-like ions were obtained by a configuration interaction expansion using the atomic structure code CIV3 [27]. RMATXII codes [28,29] were used to output energy values. To calculate the 1s²(²S^e)2sns (¹S^e) and 1s² (²P^o)2pns (¹P^o), 1s² (²P^o)2pnd (¹P^o), ‘bound’ and ‘excited’ Rydberg states, respectively, we used the quantum defect theory and the R-matrix numerical method. More details on the calculation, including quantum defects and energies, are given in Refs. [24,25,30,31]. In these works, attention was focused on the generation of LIDS involving the 2sns ¹S^e ‘bound’ excited Rydberg states, where the principal quantum number ranges from 5 to 12 and from 9 to 11 for C²⁺ and Al⁹⁺, respectively. We have shown that, by employing a single photon, these states can be made degenerate with the 2pns (¹P^o) states, when Δn = 0 and Δn = 2 for C²⁺, and Δn = 0 for Al⁹⁺.

The resonances were described in the ‘two state model’ to obtain the physical quantities as follows: position and width of the bound and autoionizing state, laser frequency at LIDS, Fano parameter. The laser intensity was varied only slightly above the first ionization threshold so that the atom remained in the Floquet eigenstate connected to the initial state. Detailed results are presented in Refs. [24,25]. In the present work, we complete previous data sets for laser frequency ω >> |E₀(α₀)|, where |E₀(α₀)| is the binding energy of the state of the atom in the field, and in which α₀ can take any value [32,33]. In this high-frequency regime, and for a linearly polarized field, RMF results show that as α₀ increases, the binding energy decreases. In Figure 1a,b, our results are plotted on the Floquet lifetime as a function of intensity, associated to the 1s²2p10s (¹P^o) and 1s²2p6s(¹P^o) Rydberg state, respectively, in C²⁺.The numbers adjacent to the points give the corresponding value of α₀, the classical amplitude of the oscillation of a free electron driven by the field.

The RMF calculations were performed for different laser frequencies and intensities. We used the laser field strength sets of data to estimate the Keldysh parameter at the barrier suppression ionization (BSI). This model occurs when the laser intensity is higher than the so-called barrier-suppression-ionization intensity, I_BSI = I_p⁴_/16 (with I_p being the ionization potential).

Table 1. Driving laser frequency in C²⁺, ω_tune (in atomic units, au), intensity I (W/cm²) ranges (I_min-I_max) of theoretical RMF data. The γ values at the barrier-suppression-ionization (BSI) intensity I_BSI and at the minimum laser intensity, I_min, are denoted as γ_BSI, and γ_max, respectively.

1s22pns (1Po) n	ωtune (au)	Imin (TW/cm2)	Imax (TW/cm2)	γBSI	γmax
5	0.3089	0.1	10.0	0.0425	0.2629
6	0.29785	0.45	10.0	0.0705	0.0949
7	0.29885	0.01	1.0	0.1155	0.5459
8	0.29877	0.01	1.0	0.1757	0.4759
9	0.29862	0.003	0.9	0.2533	0.7685
10	0.29849	0.005	0.2	0.3505	0.5328
11	0.29839	0.003	0.1	0.4699	0.6227

shows the driving laser frequencies and the corresponding laser intensity ranges used in our RMF calculation. The Keldysh parameter, γ, values at the barrier-suppression-ionization (BSI) intensity, I_BSI, denoted as γ_BSI, and at the minimum laser intensity, I_min, denoted as γ_max, are also presented in the last two columns, respectively.

3. Strong Field Ionization Rates

In this section, we investigate the applicability of different theories on the intensity-dependent ionization rate to our atomic system. The theoretical treatment of ionization of an atom in the radiation field is based on the work of Keldysh [34]. The Keldysh parameter, γ = ω√2I_p/F (where I_p is the ionization potential and F and ω are the laser-field strength and frequency, respectively) characterizes the degree of adiabaticity of the motion of active electrons through the barrier. In the region of small frequencies (ω << ω_t, where ω_t is the so-called tunnelling frequency), in which adiabatic approximation is correct, the Keldysh parameter is γ = ω/ω_t << 1. In this regime, only the initial and final states of the electron are significant, while the intermediate state places no rule. In this case, a simple ‘two state model’, similar to the one used in RMF calculations, can be applied. For energies of the emitted photon lower than the energy of the atomic state, and for laser strength lower than the atomic field strengths, the quasi-classic approximation becomes valid. Landau and Lifshitz [35] derived a formula for the ionization rate of hydrogen when the electron is in the ground state. The ionization rate is:

$$W_{}(t)=4\frac{{(2I_p)}^{5/2}}{E(t)}\exp \left(-\frac{2{\left(2I_p\right)}^{3/2}}{3E(t)}\right)$$

(1)

where E(t) is the laser’s electric field. Other treatments based on the quasi-static limit have been derived by Keldysh and Ammosov–Delone–Krainov (ADK) [36,37]. According to Keldysh’s theory, photoionization in weak or strong field regimes, respectively, is controlled by the Keldysh parameter, the approach resulting in a rate, w_K, expressed as the total ionization rate as a function of γ:

$$w_K=Q\left(\gamma ,I_p,\omega \right)\exp \left(-\xi \left(\gamma ,I_p,\omega \right)\right)$$

(2)

where Q (γ, I_p, ω) is a pre-exponential factor, and

$$\xi \left(\gamma ,I_p,\omega \right)=\frac{2}{\hslash \omega }{I}_p\left(1+\frac{1}{2{\gamma }^2}\right)\left[{\sinh }^{-1}\gamma -\gamma \frac{{\left(1+{\gamma }^2\right)}^{1/2}}{1+2{\gamma }^2}\right]$$

(3)

The pre-exponential factor in Equation (2) was obtained by Perelomov et al. [38,39] (the original PPT formula). For γ >> 1, Equations (2) and (3) lead to the scaling of the ionization rate depending on the intensity of the laser field and the minimum number of photons, K₀, required for photoionization, I^K₀. When γ:<< 1, Equations (2) and (3) yield w_K ∝ exp [−(4/3) (2)^1/2I_p^3/2E₀⁻¹ (1 − γ²/10) (see Equation (59) Ref. [36]). In this tunneling regime, these expressions describe the transmission of the electron’s wave functions through a triangular potential barrier formed by a rectangular potential step of height I_p and a dc electric field E₀ [40].

The Keldysh, PPT, and ADK theories are for short-range potentials and a weak laser field. The ADK model is related to the Keldysh–Faisal–Reiss (KFR) formula [41,42] for the tunneling limit. For the ionization rate of complex atoms (ions) from a state with energy, E, with orbital quantum number, l, and its projection, m, the ADK formula is:

$$\begin{array}{l}{W}_{ADK}=C_{n*l}^{2}f(l,m)\left|I_p\right|{\left(\frac{3F}{\pi {\left(2I_p\right)}^{3/2}}\right)}^{1/2}\times {\left(\frac{2}{F}{\left(2\left|I_p\right|\right)}^{3/2}\right)}^{2n*-\left|m\right|-1}\exp \left(-\frac{2{\left(2\left|I_p\right|\right)}^{3/2}}{3F}\right),\\ \end{array}$$

(4)

where F is the electric field strength and

$$\begin{array}{l}{C}_{n*l}^{}={\left(\frac{2e}{n*}\right)}^{n*}{\left(2\pi n*\right)}^{-1/2}\\ f(l,m)=\frac{\left(2l+1\right)\left(l+\left|m\right|\right)!}{2^{\left|m\right|}\left|m\right|!\left(l-\left|m\right|\right)!}\end{array}$$

In the above equations, n* = Z/(2I_p)^−1/2, Z is the nuclear charge of the atomic residue, I_p is the ionization potential of the state, and e = 2.71828. The validity of the ADK rate is expected to be best for F << 1, n* >> 1 and a driving laser frequency of ω << |I_p|. The ADK ionization rate does not depend on the laser frequency, so it is used only for the instantaneous rate, which in turn is dominated by m = 0.

Based on the ‘two-state model’ and single photon ionization, Krainov [43,44] suggested an extension of ADK theory to incorporate BSI. The rate is:

$$W_{Kr}=\frac{4\sqrt{3}}{\pi n*}\frac{F}{{(2F)}^{1/3}}{\left(\frac{4e{\left(\left|I_p\right|\right)}^{3/2}}{Fn*}\right)}^{2n*}\times {\displaystyle \underset{0}{\overset{\infty }{\int }}A{i}^2\left(x^2+\frac{2\left|F\right|}{{\left(2F\right)}^{3/2}}\right)}{x}^{2}dx$$

(5)

where Ai is the Airy function. This formula reduces to the ADK rate for a weak laser field (tunneling limit). In Figure 2 we plot, comparatively, the ionization rates versus laser intensity for ionization from the C²⁺ 1s²2pns (¹P^o) Rydberg states, n = 6, 7, 9, 10. For each state there are two curves, one associated with the ADK model (black circles) and the other (red diamonds) associated with Krainov’s formula, respectively. Qualitatively, similar results are obtained from calculation. The ADK model underestimates the ionization rate by an order of magnitude compared to Krainov’s formula.

The original PPT ionization rate, including Coulomb correction [37], is expressed as a sum over partial rates:

$$W_{PPT}(F,\omega )={\displaystyle \sum _{n\ge \nu }^{\infty }{w}_n}(F,\omega )$$

(6)

where F and ω are the amplitude and frequency of the laser field, respectively. The partial rate, w_n, is the probability of ionization with the absorption of n quanta, from a minimum value, ν, required to reach the effective ionization threshold. The PPT method can be applied under the condition F << F₀, ω << ω₀, with ω₀ = k²/2 and F₀ = k³ being the atomic quantities. In the case of ionization by a constant field and adiabatic approximation, the PPT formula for the probability of ionization is in the form (see Equation (4) in Ref. [37]:

$$w(F,\omega )={(3F/\pi F_0)}^{1/2}{w}_{stat}(F)$$

(7)

where F and ω are the external field strength and frequency, and F₀ is the intra-atomic electric field strength. We have compared the PPT results for ionization by a constant field with the ab initio RMF numerical values. Figure 3 presents, as an example, the calculated rate of ionization from the 1s²2p8s(¹P^o) Rydberg state by a constant field of intensity of I = 1·10¹² W/cm².

It has been mentioned that, when γ >> 1, scaling of the ionization rate with the incident laser-field intensity, I, is I^K₀ (with K₀ = [I_p/ω]). The PPT calculation carried out for all atomic states considered in this work led to the value K₀ = 1, confirming the number of Floquet blocks used in the RMF calculation. We exemplify this dependence in Figure 4 for the ionization from the 1s²2p8s(¹P^o) Rydberg state at a constant field intensity of 1 TW/cm².

The PPT formulas for w(F,ω) is applicable not only to the calculation of ionization from the ground state, but also from the excited states. Moreover, the PPT formula is applicable under the condition τ >> T_t (τ is the lifetime of the excited state, and T_t ~ 1/ω_t), from which it follows that this condition is satisfied by the range of values of field intensities when our investigation is performed. The original PPT formula, with the Coulomb correction factor (2/Fn*³) used in Figure 3, gives less accurate results. A new expression of the Coulomb correction factor [45] in the form (2/Fn*³) (1 + 2e⁻¹γ)⁻²ⁿ*³, e = 2. 718. (here F is the reduced electric field), which is applicable to any γ, led us to results close to the ab initio calculation for the corresponding laser intensity ranges used in our RMF calculation. In Figure 5a, we compare the results obtained with this ‘generalized PPT formula’ for the field of a plane-polarized wave (see Equation (54), Ref. [37]), the original PPT formula, and the results from RMF calculation. The graph refers to the calculated ionization rate vs. intensity at a driving laser frequency of ω = 0.3085 au for the 1s²2p5s(¹P^o) Rydberg state.

To quantify the deviation between the PPT, ADK, and the RMF theoretical approaches, we used the ratio between the corresponding ionization rate for each two data sets, and the results are plotted versus γ in Figure 5b and Figure 5c, respectively. We conclude that, for our atomic system, the results obtained with the ‘generalized’ model of Perelemov–Popov–Terent’ev are the closest to the ab initio R-matrix Floquet calculation.

In Figure 6a, we compare the frequency dependence of the ionization rate for the 1s²2p7s ¹P^o state, as outputs from the PPT, generalized PPT, and RMF calculations. In this figure, the RMF and generalized PPT calculations indicate similar rate values for the driving laser frequency (ω = 0.2982 au) close to the resonance frequency of 0.29885. The adiabaticity parameter is also indicated. The corresponding laser intensity (see Figure 6b) is 10¹⁰ W/cm², which is close to the estimated LIDS intensity [24]. Both curves display a sharp variation with increasing laser intensity (Figure 6b), as the laser frequency exceeds the resonance frequency. We plot the ratio between the two data sets versus γ in Figure 6c. The ionization rate vs. laser intensity and γ ~ 1 is shown in Figure 6d.

4. Discussion

In this article, we made use of our earlier reported works to extract those parameters that are important for the present investigation: the adiabaticity parameter values at the barrier suppression-ionization, and the maximum of the Keldysh parameter at the minimum laser intensity used in the ab initio calculation. The polarizability of these ions has also been previously reported [46]. Another reported result of this work is the lifetime of studied autoionizing Rydberg states as a function of laser intensity.

The present investigation revealed that the use of the PPT formula with Coulomb correction factor, valid for arbitrary values of γ, leads to values of ionization rates very close to those obtained from the ab initio calculation, near the resonance frequency. The closer the ratio is to 1, the better is the agreement between all these sets of data. This closeness of the results was to be expected considering the similarity of the hypotheses of both theoretical models: the single active electron approximation and ionization in a linearly polarized field. The fact that at intensities above the LIDS intensity this agreement is no longer valid is motivated by the essence of the LIDS phenomenon. At this laser intensity/frequency, the two states have identical energies and widths. At higher intensities, a real part of the energy remains adiabatically close to the real axis for one of these coupled states, while the autoionizing state has an imaginary part of the energy with a difference of zero. In addition, in the previous calculation, the LIDS frequency was close to the tuning frequency, given the small difference between the detuning and the ‘atomic’ frequency (which in this two state model is defined as the binding energy difference between the two states when the laser intensity goes to zero). From the preliminary study (not reported in this article) on another ion belonging to the beryllium isoelectronic sequence, we observed a similar behavior of ionization rates at laser intensities close to LIDS intensity. This study is in progress.

The ADK model has good agreement with the PPT for small γ values, but deviation becomes significant as γ approaches 1 and are of other orders of magnitude when γ is large.

Data sets are available from the author upon reasonable requests.