Three-Photon Ionization with One-Photon Resonance between Excited Levels

: Within the framework of a three-level model, the process of three-photon ionization with one-photon resonance between two excited levels (with the lower one being initially unpopulated) is considered using the density matrix method. It is shown that such resonance can result in the appearance of a maximum in the three-photon ionization spectrum when detuning between the resonance wavenumber and the wavenumber of the transition responsible for the lower excited level being populated exceeds the laser radiation linewidth by more than three orders of magnitude.


Introduction
Interest in the study of multiphoton ionization has remained consistently high for more than half a century [1][2][3][4][5][6]. Such studies are important both for a deeper understanding of the nonlinear dynamics of quantum systems exposed to an intense electromagnetic field as well as for their relevance to a number of applied problems, in particular the physics of laser plasma. The resonantly enhanced three-photon ionization, being the most promising method for studying the energy structure of atoms and molecules, is of particular interest.
The resonant processes of three-photon ionization of an atom A in a laser field with a wavenumber ω are in the vast majority of cases due to one-photon A(1) +hω→A(2) + 2hω→A + + e (1) and two-photon A(1) + 2hω→A(3) +hω→A + + e excitation of atomic levels [1]. Here 1 is the ground level, while 2 and 3 are excited levels of the atom A. In the first case, the photon energyhω is coincident with that of level 2 ( ω = E 2 ) while in the second case the energy of two photons 2hω coincides with that of level 3 (2 ω = E 3 ). Note that in both cases the resonant transition is a one-or two-photon transition between the ground level and an excited level. The resonant transitions (1) and (2) result in the appearance of maxima in the threephoton ionization spectra due to a considerable increase (up to several orders of magnitude) in the probability of the A + ions' formation. One-and two-photon resonantly enhanced three-photon ionization has been extensively studied via experiment, and to a lesser extent theoretically (see for example Refs. [7][8][9][10][11][12]).
It is evident that in the case of three-photon ionization, a process is also possible where the resonant transition is the one between excited levels 2→3 ( ω = E 3 − E 2 ), while that between the ground and the lower excited levels 1→2 ( ω = E 2 )is non-resonant; it is important to note that level 2 is initially unpopulated. Experimental manifestation of transitions such as (3) has been observed at threephoton ionization of a number of atoms, in particular Na, Rb, Ba, and Yb (see for example Refs. [11,[13][14][15]). Thus, at three-photon ionization of the Ba atom [11] in the vicinity of the maxima related to the one-photon 6s 2 1 S 0 → 6s6p 1 P o 1 (ω = 18,060 cm −1 ) and twophoton 6s 2 1 S 0 →5d6d 3 D 2 (ω = 18,100 cm −1 ) transitions from the ground 6s 2 1 S 0 level (transitions 1→2 and 1→3, respectively), a clear maximum corresponding to the one-photon 6s6p 1 P o 1 → 5d6d 3 D 2 (ω = 18,140 cm −1 ) transition between the excited levels (transition 2→3) was also observed. The detuning for the first photon transition ( 6s 2 1 S 0 → 6s6p 1 P o 1 ) at the wavenumber ω = 18,140 cm −1 was 80 cm −1 which was more than an order of magnitude larger than the laser radiation linewidth (∆ω ≈ 2-3 cm −1 ). This indicates a substantially non-resonant character of the lower excited 6s6p 1 P o 1 level (level 2) population. The height of the maximum due to the one-photon 6s6p 1 P o 1 → 5d6d 3 D 2 (2→3) transition was less than an order of magnitude (A 12 /A 23 ≈ 60) lower than that corresponding to the one-photon 6s 2 1 S 0 →6s6p 1 P o 1 (1→2) transition and more than two orders of magnitude (A 13 /A 23 ≈ 560) lower than that related to the two-photon 6s 2 1 S 0 → 5d6d 3 D 2 (1→3) transition. Simultaneously, the height A 23 was higher by about an order of magnitude than the ion signal in the inter-resonance range. Note that here and elsewhere we use the transition wavenumbers in cm −1 . Recall that the conversion factor from eV to cm −1 is 1 eV = 8065.544 cm −1 [16].
It is worth noting that in Ref. [11] the maximum due to the 6s6p 1 P o 1 → 5d6d 3 D 2 (2→3) transition was observed at a low field strength of ε ≈ 2.5·10 4 V/cm (laser intensity ≈ 8.6·10 5 W/cm 2 ). This indicates that its nature is not related to the Stark shift of the excited 6s6p 1 P o 1 and 5d6d 3 D 2 levels in the laser radiation field, since at such a value of ε estimates show that their shifts are significantly less than the laser radiation linewidth.
In addition, both in the experiment [11] and in the above-mentioned works [13][14][15] the concentration of the studied atoms (in a beam or vapor) did not exceed 10 12 cm −3 . This allows one to exclude the processes of collisions and multiphoton scattering from the possible mechanisms of population of the lower excited level 2, in this case 6s6p 1 P o 1 . At present, practically the only mechanism that makes it possible to explain the resonant transitions of type (3) is non-resonant population of the lower excited level 2 in the field of pulsed laser radiation [14]. There are no quantitative calculations in the literature that confirm and describe this process.
In this work, the process of three-photon ionization in the presence of an intermediate one-photon resonance between two excited levels was investigated within the framework of a three-level model using the density matrix method.
Atoms 2021, 9,68 3 of 13 field of pulsed laser radiation [14]. There are no quantitative calculations in the literature that confirm and describe this process. In this work, the process of three-photon ionization in the presence of an intermediate one-photon resonance between two excited levels was investigated within the framework of a three-level model using the density matrix method.

Calculation Details
The three-level system studied in the present work is shown schematically in Figure  1. Depending on the detuning values Δ1 = ω-ω12, Δ2 = 2(ω-ω13), and Δ3 = ω-ω23 (for the 1→ 2, 1→3, and 2→3 transitions, respectively) in the laser field with a wavenumber ω, three types of resonances are possible in the system under consideration: a one-photon resonance between the ground level 1 and the excited level 2 (Δ1 = 0, Δ2 ≠ 0, Δ3 ≠ 0), a twophoton resonance between the ground level 1 and the excited level 3 (Δ1 ≠ 0, Δ2 = 0, Δ3 ≠ 0), and a one-photon resonance between the excited levels 2 and 3 (Δ1 ≠ 0, Δ2 ≠ 0, Δ3 = 0). Note that in the case of Δ1 ≠ 0, Δ2 ≠ 0, Δ3 ≠ 0 one has a direct three-photon ionization [1]. Behavior of such a three-level system in the laser field within the framework of the density matrix method is described by a differential equation system     Behavior of such a three-level system in the laser field within the framework of the density matrix method is described by a differential equation system 13 · s * 13 (t) , Here s ij (t) are slowly varying with time complex amplitudes of the density matrix elements ρ ij (t): ρ 11 13 ε 2 with r 12 , r 23 , and r (2) 13 being the dipole matrix elements of the one-photon 1→2, 2→3 transitions and composite matrix element of two-photon 1→3 transition, respectively, 22 , and V 33 being Stark shifts of the ground level 1 and the excited levels 2 and 3 caused by the laser field of the strength ε, and γ s3 being the natural widths of the excited levels 2 and 3 related to their spontaneous decay to lower levels, Atoms 2021, 9, 68 4 of 13 γ (2) i2 (ε) and γ i3 (ε) being the ionization widths of the levels 2 and 3 due to their one-and two-photon ionization, respectively, and γ L being the laser radiation linewidth.
It is important to note that the system (4) contains only transitions between levels 1, 2, and 3 with ionization from the excited levels 2 and 3 under laser radiation and does not include any additional processes that can result in an increase of the population of the lower excited level 2. Note also that it does not include three-photon ionization of the ground level 1 since the probability of such a process is well below those of the resonance processes under consideration.
For simplicity of calculation, the laser pulse profile was chosen to be Lorentzian for which the first-order correlation function is e(t 1 ) e(t 2 ) = ε 2 exp(−0.5 γ L t 1 − t 2 ) with e(t) being a stochastically fluctuating complex amplitude of the electric field. Note that the real laser profile is close to Gaussian. However, in this case the calculation becomes much more complicated without any substantial improvement. The laser pulse duration was chosen to be 2·10 −8 s −1 .
In this calculation we approximate the region of focused laser radiation by a cylinder with a Gaussian intensity distribution over the cross section ε 2 The number of ions produced by such a beam and normalized to the maximum number of ions that could be produced by a uniform beam of the same intensity (ε 2 (r) = ε 2 0 ) is determined as follows: The total probability of three-photon ionization P(ε(r), t) = 1 − s 11 (t) − s 22 (t) − s 33 (t) at each point of the cross-section r was determined by solving the differential equation system (4) numerically using the 4th and 5th order Runge-Kutta method. In addition, at each point r the ionization widths depending on the field strength ε were also determined: γ i3 (ε) = 8.5 · 10 −2 σ i3 ε 2 and γ (2) i2 (ε) = 2.98 · 10 14 σ 2 i2 ε 4 where σ i3 and σ 2 i2 are the crosssections of one-and two-photon ionization of the levels 3 and 2, respectively.

Three-Photon Ionization of the Ba Atom
To check the accuracy of the chosen model, we calculated the dependence of the single barium ion yield on the laser wavenumber N(ω) at three-photon ionization in the spectral range ω = 18,047-18,153 cm −1 which contains transitions with participation of the 6s6p 1 P o 1 (ω 12 = 18,060.3 cm −1 [17]) and 5d6d 3 D 2 (2ω 13 = 36,200.4 cm −1 [17]) levels, excitation of which was observed in Ref. [11]. In Figure 1 these are the levels 2 and 3, respectively. The resonance structure of the calculated N(ω) dependence ( Figure 2) is due to the one-photon 6s 2 1 S 0 → 6s6p 1 P o 1 transition (maximum 1 at ω = 18,060.3 cm −1 ), twophoton 6s 2 1 S 0 →5d6d 3 D 2 transition (maximum 2 at ω = 18,100.2 cm −1 ), and one-photon 6s6p 1 P o 1 →5d6d 3 D 2 transition between the excited levels (maximum 3 at ω = 18,140.1 cm −1 ). The N(ω) dependence shown in Figure 2 was calculated using the following values: 13 = 679 a. u., r 23 = 0.09 a. u., σ i3 = 10 −17 cm 2 , σ 2 i2 = 1.6·10 −46 cm 4 s, γ s2 = γ s3 = 10 −4 cm −1 , V 11 = V 22 = V 33 = 0, ε = 2.5·10 4 V/cm, and γ L = 0.2 cm −1 . The value γ L = 0.2 cm −1 was chosen from the consideration that at such a value, in our opinion, the influence of the Lorentzian "wings" of the laser radiation spectrum on the probabilities of the transitions under study, first of all that of the non-resonant population of the 6s6p 1 P o 1 level, can be neglected.  As already mentioned above, the Stark shift of the levels in a field of strength ε = 2.5·10 4 V/cm (laser intensity ≈ 8.6·10 5 W/cm 2 ) can be ignored. In particular, estimates show that even at the dynamic polarizability value of α = (2-3)·10 3 a. u. [18] the Stark shift ΔE = α·ε 2 /4 is not more than 4·10 -3 cm -1 . This is substantially less than the laser radiation linewidth L  = 0.2 cm -1 . For this reason, the Stark shift of the levels was not taken into account in the calculation ( 11  23 r and ) 2 ( 13 r we proceeded from the fact that the main contribution (not less than 95%) to the value of the composite matrix element ) 2 ( 13 r , because of the large value of 12 r and the relatively small detuning ( 12 13    ≈ 40 cm -1 ), comes from the matrix elements 12 r and 23 r : [1]. The value of 23 r was chosen in such a way that the ratio of the heights  As already mentioned above, the Stark shift of the levels in a field of strength ε = 2.5·10 4 V/cm (laser intensity ≈ 8.6·10 5 W/cm 2 ) can be ignored. In particular, estimates show that even at the dynamic polarizability value of α = (2-3)·10 3 a. u. [18] the Stark shift ∆E = α·ε 2 /4 is not more than 4·10 −3 cm −1 . This is substantially less than the laser radiation linewidth γ L = 0.2 cm −1 . For this reason, the Stark shift of the levels was not taken into account in the calculation (V 11 = V 22 = V 33 = 0).
The value of the matrix element r 12 = 5.5 a. u. was determined on the basis of the known oscillator strength f 12 = 1.64 of the one-photon 6s 2 1 S 0 → 6s6p 1 P o 1 transition [19]. Unfortunately, there are no data in the literature on the oscillator strength f 23 of the 6s6p 1 P o 1 →5d6d 3 D 2 transition. For this reason, when determining the value of the matrix elements r 23 and r (2) 13 we proceeded from the fact that the main contribution (not less than 95%) to the value of the composite matrix element r (2) 13 , because of the large value of r 12 and the relatively small detuning (ω 13 − ω 12 ≈ 40 cm −1 ), comes from the matrix elements r 12 and r 23 : r (2) 13 = r 12 · r 23 /4(ω 13 − ω 12 ) [1]. The value of r 23 was chosen in such a way that the ratio of the heights (A 2 /A 3 ) calc of the maxima 2 and 3 of the calculated N(ω) dependence ( Figure 2) coincided with the experimentally observed ratio (A 2 /A 3 ) exp [11].
In addition, the value of the composite matrix element r (2) 13 = 679 a. u. obtained using the value of r 23 = 0.09 a. u. is, in our opinion, consistent with the fact that the maximum due to the two-photon 6s2 1S0ß5d6d 3D2 transition belongs to the group of the most intense maxima, as it follows from the N(ω) dependence experimentally measured in Ref. [11]. Note also that variation in the σ i3 value within a range of 10 −17 -10 −18 cm 2 results in a change of the absolute heights of the maxima 2 and 3; however, their ratio (A 2 /A 3 ) calc ≈ 555 remains practically unchanged.
The value of the cross-section σ (2) i2 for the two-photon ionization of the 6s6p 1 P o 1 level was chosen in such a way that the ratio of the heights (A 2 /A 1 ) calc of the maxima 2 and 1 of the calculated N(ω) dependence ( Figure 2) coincided with the experimentally observed Atoms 2021, 9, 68 6 of 13 ratio (A 2 /A 1 ) exp [11]. At the above value σ (2) i2 = 1.6·10 −46 cm 4 s which, in our opinion, is quite reasonable [21,22], the ratio (A 2 /A 1 ) calc ≈ 209 is in good agreement with the experimentally observed one (A 2 /A 1 ) exp ≈ 210.
The natural widths γ s2 and γ s3 of the excited 6s6p 1 P o 1 and 5d6d 3D2 levels were chosen equal to 10 −4 cm −1 on the basis of the known lifetime τ ≈ 10 −8 s of the 6s6p 1 P o 1 level [23]. Note that variation in the γ s2 and γ s3 values within a range of 10 −3 -10 −5 cm −1 did not noticeably affect the result of the calculation.
A qualitative and quantitative comparison of the calculated N(ω) dependence (Figure 2) with the one experimentally observed in Ref. [11] allows one to make two main conclusions: first, the proposed three-level model and the differential equation system (4) well describe the ion signal ratios of the maxima 1, 2, and 3; second, since the system (4) does not include any additional processes that can result in an increase of the population of the lower excited level 2 (6s6p 1 P o 1 level in this case), one can argue that manifestation of the maximum 3 related to the one-photon 6s6p 1 P o 1 →5d6d 3 D 2 (2→3) transition in the N(ω) dependence is due to non-resonant population of the 6s6p 1 P o 1 (2) level with the detuning (∆ 1 ≈ 80 cm −1 ) significantly exceeding the laser line width (γ L = 0.2 cm −1 ).

Conditions of Effective Manifestation of the One-Photon Resonance between Excited Levels at Three-Photon Ionization
We return now to the process of one-photon resonance between the excited levels ( Figure 1, process c) and examine the influence of the values of the matrix elements r 12 , r 23 , ionization cross section σ i3 , and detuning ∆ 1 on the height of the maximum (denoted as A 23 ) due to the one-photon resonance 2→3 transition (∆ 3 = 0).
Since the differential equation system (4) is solved numerically, we fixed, for definiteness, the position of the upper excited level 3 by setting the resonance wavenumber ω 13 of the two-photon 1→3 transition equal to 18,000 cm −1 . The position of the lower excited level 2 changes depending on the specified detuning value ∆ 1 : ω 12 = (2ω 13 − ∆ 1 )/2. The wavenumber of the resonant one-photon 2→3 transition in this case is determined as follows: ω 23 = 2ω 13 − ω 12 . Note that the choice of the ω 13 value is formal and does not affect the calculation result which, as follows from the system (4), depends on the detuning values ∆ 1 , ∆ 2 , and ∆ 3 but not on the absolute values of the resonant wavenumbers ω 12 , ω 13 , and ω 23 . Thus, one can fix the position of the lower excited level 2 and vary the position of the upper level 3 or even fix the distance between the excited levels 2 and 3 (ω 23 = const) and vary their position towards the ground level 1. We also set the ionization cross-section σ i3 to 10 −17 cm 2 and the laser linewidth γ L to 0.2 cm −1 .
We restricted ourselves to the case of a weak field (ε = 2.5·10 5 V/cm), where the Stark shift of the levels 1, 2, and 3 and consequently a possible decrease of the detuning ∆ 1 can be neglected. Note that for a correct quantitative determination of the level shifts due to the dynamic Stark effect under the action of the laser field, one has to know the absolute dependences α(ω) for the levels 1, 2, and 3 in the vicinity of the resonance wavenumber ω 23 .
For the estimation, we proceed from the ratio A 2 /A 0 ≈ 10 4 (A 2 is the height of the maximum due to the two-photon 6s6p 1P1oß5d6d 3D2 transition, A 0 is the value of the ion signal in the inter-resonance range) experimentally observed in Ref. [11] and consider that the minimum height A min of the maximum related to the one-photon resonance 2→3 transition, at which it already manifests in the dependence N(ω), is equal to 2A 0 (or 2·10 −4 A 2 ). In our case (Figure 2), this value is A min ≈ 8·10 −4 arb. un. Note that the ratio A 2 /A 0 ≈ 10 4 , in our opinion, is rather typical for the strong maxima due to two-photon transitions observed in the experiments using atomic beams and electron multipliers. Figure 3 shows the dependences A 23 (r 12 ) calculated for different values of the matrix element r 23 and detuning ∆ 1 = 80 cm −1 (Figure 3a) and ∆ 1 = 160 cm −1 (Figure 3b). The horizontal dashed line marks the A min value. As can be seen, an increase of the matrix element r 12 results in an elevation of the A 23 height. Along with this, a slowdown in the rate of the A 23 height increase with r 12 is observed. It is also clearly seen that the minimum values of the matrix elements r 12 and r 23 , for which the condition A 23 = A min is fulfilled, increase with detuning ∆ 1 . In particular, the r 12 minimum value (at the maximum value r 23 = 0.2 a. u.) increases from 1.79 a. u. at ∆ 1 = 80 cm −1 (Figure 3a, curve 1) to 3.55 a. u. at ∆ 1 = 160 cm −1 (Figure 3b, curve 1) while the r 23 minimum value (at the maximum value r 12 = 5.5 a. u.) increases from 0.0087 a. u. at ∆ 1 = 80 cm −1 (Figure 3a, point 6) to 0.02 a. u. at ∆ 1 = 160 cm −1 (Figure 3b, point 4). Note that the maximum value of the matrix element r 12 , according to the available data on the oscillator strengths of the resonant S→P transitions in alkali and alkaline-earth metal atoms [23], is not greater than 5.5 a. u. trix element 23 r and detuning 1  = 80 сm -1 (Figure 3a) and 1  = 160 сm -1 (Figure 3, b). The horizontal dashed line marks the min A value. As can be seen, an increase of the matrix element r12 results in an elevation of the 23 A height. Along with this, a slowdown in the rate of the 23 A height increase with 12 r is observed. It is also clearly seen that the minimum values of the matrix elements 12 r and 23 r , for which the condition min 23 A A  is fulfilled, increase with detuning 1  . In particular, the 12 r minimum value (at the maximum value 23 r = 0.2 a. u.) increases from 1.79 a. u. at 1  = 80 сm -1 (Figure 3, a, curve 1) to 3.55 a. u. at 1  = 160 сm -1 (Figure 3b, curve 1) while the 23 r minimum value (at the maximum value 12 r = 5.5 a. u.) increases from 0.0087 a. u. at 1  = 80 сm -1 (Figure 3, a, point 6) to 0.02 a. u. at 1  = 160 сm -1 (Figure 3b, point 4). Note that the maximum value of the matrix element 12 r , according to the available data on the oscillator strengths of the resonant S→P transitions in alkali and alkaline-earth metal atoms [23], is not greater than 5.5 a. u.   Figure 4 shows dependences A 23 (r 23 ) calculated for different values of the matrix element r 12 and detuning ∆ 1 = 80 cm −1 (Figure 4a) and ∆ 1 = 160 cm −1 (Figure 4b). The dashed horizontal line marks the A min value. As can be seen, an increase of the matrix element r 23 also results in an increasing A 23 height. However in this case, in contrast to the A 23 (r 12 ) dependences (Figure 3), a more noticeable slowdown in the rate of the A 23 height increase with r 23 is observed (Figure 4). We restrict ourselves to the maximum value r 23 = 0.2 a. u. since the calculations show that a further practical increase of r 23 does not result in any further growth of the A 23 height. dashed horizontal line marks the min A value. As can be seen, an increase of the matrix element 23 r also results in an increasing 23 A height. However in this case, in contrast to the ) ( 12 23 r A dependences (Figure 3), a more noticeable slowdown in the rate of the 23 A height increase with 23 r is observed (Figure 4). We restrict ourselves to the maximum value 23 r = 0.2 a. u. since the calculations show that a further practical increase of 23 r does not result in any further growth of the 23 A height. It is clearly seen that similarly to the ) ( 12 23 r A dependences (Figure 3), the minimum values of the matrix elements 12 r and 23 r , for which the condition min 23 A A  is fulfilled, increase with detuning 1  (Figure 4). In particular, the 23 r minimum value (at the maximum value 12 r = 5.5 a. u.) increases from 0.0087 a. u. at 1  = 80 сm -1 (Figure 4, a, curve 1) to 0.0198 a. u. at 1  = 160 сm -1 (Figure 4, b, curve 1) while the 12 r minimum value (at the maximum value 23 r = 0.2 a. u.) increases from 1.79 a. u. at 1  = 80 сm -1 (Figure 4, a, point 9) to 3.55 a. u. at 1  = 160 сm -1 (Figure 3, b, point 10). The slowdown in the rate of the 23 A height increase with matrix elements 12 r and 23 r is, in our opinion, due to a saturation of the ion signal owing to a close to unity probability of two-photon resonance ionization from the level 2. The strong saturation of the ion signal at large values of 23 r is most likely explained by the fact the 2→3 transition is resonant while in the case of the non-resonant 1→2 transition the increase of the 12 r value results in a weaker saturation of the ion signal.
The above argument is confirmed by N(ω) dependences calculated for different values of the matrix elements 12 r , 23 r and detuning 1  = 80 сm -1 ( 23  = 18,040 сm -1 ). They are shown in Figure 5. It is clearly seen that with the 12 r variation from 3.0 a. u. (Figure 5, a) to 5.5 a. u. (Figure 5, b) the 23 A height of the maximum related to the 2→3 transition It is clearly seen that similarly to the A 23 (r 12 ) dependences (Figure 3), the minimum values of the matrix elements r 12 and r 23 , for which the condition A 23 = A min is fulfilled, increase with detuning ∆ 1 (Figure 4). In particular, the r 23 minimum value (at the maximum value r 12 = 5.5 a. u.) increases from 0.0087 a. u. at ∆ 1 = 80 cm −1 (Figure 4a, curve 1) to 0.0198 a. u. at ∆ 1 = 160 cm −1 (Figure 4b, curve 1) while the r 12 minimum value (at the maximum value r 23 = 0.2 a. u.) increases from 1.79 a. u. at ∆ 1 = 80 cm −1 (Figure 4a, point 9) to 3.55 a. u. at ∆ 1 = 160 cm −1 (Figure 3b, point 10).
The slowdown in the rate of the A 23 height increase with matrix elements r 12 and r 23 is, in our opinion, due to a saturation of the ion signal owing to a close to unity probability of two-photon resonance ionization from the level 2. The strong saturation of the ion signal at large values of r 23 is most likely explained by the fact the 2→3 transition is resonant while in the case of the non-resonant 1→2 transition the increase of the r 12 value results in a weaker saturation of the ion signal.
The above argument is confirmed by N(ω) dependences calculated for different values of the matrix elements r 12 , r 23 and detuning ∆ 1 = 80 cm −1 (ω 23 = 18,040 cm −1 ). They are shown in Figure 5. It is clearly seen that with the r 12 variation from 3.0 a. u. (Figure 5a) to 5.5 a. u. (Figure 5b) the A 23 height of the maximum related to the 2→3 transition increases but its width remains practically unchanged. In contrast, an increase of r 23 results in a growth of both the height and the width of the maximum. In particular, the maximum width increases from 0.26 cm −1 at r 23 = 0.02 a. u. (Figure 5a,b, curve 1) to 1.77 cm −1 at r 23 = 0.2 a. u. (Figure 5a,b, curve 3).
increases but its width remains practically unchanged. In contrast, an increase of 23 r results in a growth of both the height and the width of the maximum. In particular, the maximum width increases from 0.26 cm -1 at 23 r = 0.02 a. u. (Figure 5a,b, curve 1) to 1.77 cm -1 at 23 r = 0.2 a. u. (Figure 5a,b, curve 3).  On the basis of the A 23 (r 12 ) and A 23 (r 23 ) dependences calculated for different values of the detuning ∆ 1 , a relationship between the value of the maximum detuning ∆ 1max and the minimum values of the matrix elements r 12 and r 23 (for which the condition A 23 = A min is fulfilled) was found. As an example, Figure 6 shows dependences ∆ 1max (r 12min ) calculated for r 23min = 0.2, 0.06, and 0.02 a. u. (curves 1, 2, and 3, respectively) whereas Figure 7 shows dependences ∆ 1max (r 23min ) calculated for r 12min = 5.47, 3.0, and 0.5 a. u. (curves 1, 2, and 3, respectively). It is clearly seen that these dependences are of different nature. In particular, an increase of r 12min results in an almost linear growth of the ∆ 1max value while a significant slowdown in the ∆ 1max value increase with r 23min is observed. Such behavior of the ∆ 1max (r 12min ) and ∆ 1max (r 23min ) dependences agrees with the above A 23 (r 12 ) and A 23 (r 23 ) dependences and is caused by the stronger saturation of the ion signal in the case of the resonant 2→3 transition.    173  152  130  109  87  66  44  23  0.18  235  214  192  171  150  128  107  86  65  44  23  0.16  232  211  190  168  148  126  105  85  64  43  23  0.14  226  206  185  165  144  124  103  83  63  42  22  0.12  218  198  179  159  139  119  99  80  60  41  22  0.1  206  187  169  150  131  113  94  75  57  38  21  0.08  186  169  153  136  119  102  85  68  52  35  20  0.06  157  143  129  115  100  86  71  58  43  29  17  0.04  115  105  95  84  72  63  53  43  33  22  12  0.02  62  56  51  45  39  34  28  23  17  11  6 All the above dependences were obtained using the value σ i3 = 10 −17 cm 2 for ionization cross-section of the level 3. Study of the σ i3 value influence on the A 23 height shows that the ∆ 1max value decreases by about a factor of three with σ i3 decrease from 10 −17 cm 2 down to 10 −18 cm 2 regardless of the r 12min and r 23min values. As an example, Figure 8 shows the dependences ∆ 1max (σ i3 ) calculated for different r 12min and r 23min values whereas Table 2 presents the values of the coefficient k(σ i3 ) taking into account the ∆ 1max value decreasing with the cross-section σ i3 decrease: ∆ 1max (σ i3 ) = ∆ 1max (σ i3 = 10 −17 cm 2 )/k(σ i3 ).  The obtained results, in our opinion, are quite universal and can be used for the estimation of the conditions of manifestation of the resonance between two excited states at three-photon ionization of any atom. As an example, Table 3 contains data on possible one-photon transitions between two excited levels (2→3) at three-photon ionization of the barium, strontium, and calcium atoms. Table 3. Data on possible one-photon transitions between two excited levels at three-photon ionization of the Ba, Sr, and Ca atoms.    The obtained results, in our opinion, are quite universal and can be used for the estimation of the conditions of manifestation of the resonance between two excited states at three-photon ionization of any atom. As an example, Table 3 contains data on possible one-photon transitions between two excited levels (2→3) at three-photon ionization of the barium, strontium, and calcium atoms. Table 3. Data on possible one-photon transitions between two excited levels at three-photon ionization of the Ba, Sr, and Ca atoms. It is seen that in addition to the above 6s6p 1 P o 1 →5d6d 3 D 2 transition in the Ba atom, the transitions 5s5p 1 P o 1 → 5s10s 1 S 0 in the Sr atom and 4s4p 1 P o 1 → 4s10s 1 S 0 in the Ca atom are also possible. In both cases the one-photon 1 S 0 → 1 P o 1 transitions (1→2) are characterized by large matrix elements r 12 ≈ 5 a. u. [23]. Unfortunately, no data are available in the literature on the oscillator strengths of the 5s5p 1 P o 1 → 5s10s 1 S 0 (Sr) and 4s4p 1 P o 1 → 4s10s 1 S 0 (Ca) transitions, which does not allow for direct determination of the matrix elements for these transitions. As follows from the data presented in Table 1, in order for the transitions (2→3) under consideration to manifest in the N(ω) dependence (A 23 ≥ A min ) at the detuning values ∆ 1 ≈ 115-133 cm −1 , the r 23 value for these transitions should be not less than 0.05 a. u. Taking into account the fact that in contrast to the intercombination two-electron 6s6p 1 P o 1 →5d6d 3 D 2 transition in the Ba atom (r 23 ≈ 0.09 a. u.) the transitions under consideration are one-electron dipole-allowed ones, one can expect that the r 23 value for these transitions can reach 0.1-0.2 a. u. Therefore, there are good reasons to expect that the 5s5p 1 P o 1 → 5s10s 1 S 0 (Sr) and 4s4p 1 P o 1 → 4s10s 1 S 0 (Ca) transitions