Interference stabilization (IS) of Rydberg atoms in a strong light field is the phenomenon which attracts interest of many physicists working in the field of laser-atom interactions [1–10]. By definition, “stabilization” means that the probability of ionization per pulse in its dependence on the laser intensity becomes a falling function or saturates at some level smaller than one. Such an effect is assumed to arise beginning from some critical intensity, which determines the threshold of stabilization. In this letter we report about the results of our theoretical investigation of competition between two channels of transitions providing IS: the Raman-type transitions between Rydberg levels via atomic states of the continuum (the so-called Λ-channel) and the Raman-type transitions via lower resonant levels (the V-channel). As a result, the conditions are formulated under which either the first or the second of these two channels predominates. In each of these two cases the conditions necessary for observation of IS are formulated explicitly.

 

Fig. 1. The general scheme of Λ- and V-type transitions taken into account in the model under consideration.

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Let us consider a scheme of laser-induced atomic transitions shown in Fig. 1, where both Rydberg–continuum and Rydberg–Rydberg resonance transitions are taken into account. Such a scheme is characterized by the following four parameters: the Rabi frequency Ω _R responsible for resonance Rydberg–Rydberg transitions, ionization width Γ describing the weak-field rate of Rydberg–continuum transitions, the gap ∆ between adjacent Rydberg levels, and the detuning from resonance δ. The first two of these four parameters are determined as

where V_a,b = 〈φ_a - ∣d ε∣ φ_b〉 are the matrix elements of the dipole laser-atom interaction, ε and ω are the field strength amplitude and frequency of a laser, n ₀ and n ₀ are, respectively, the principal quantum numbers of the initial Rydberg level E _n0 and the lower one E _ñ0 resonant to E _n0 (see Fig. 1). Let us assume E_n to be s-levels and E _ñ p-levels.

In the quasiclassical (WKB) approximation, Eqs. (1) are well-known [11–13] to give

The resonance detuning δ and the gap ∆ between adjacent Rydberg levels are given by:

Let us assume first that quantum numbers n ₀ and n ₀ are large and significantly different from each other:

In terms of the light frequency ω this means that the latter is assumed to be much larger than the binding energy of Rydberg levels but much smaller than the binding energy of the ground level E_g ,

 

Fig. 2. The Rabi frequency Ω_R and the ionization width Γ vs. the field strength parameter V = ε/ω^5/3; ∆ is the gap between the adjacent Rydberg levels (3).

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The Rabi frequency Ω_R and the ionization width Γ are plotted in Fig. 2 in the dependence on the parameter V = ε/ω^5/3. As Ω_R is a linear function of the field strength ε, whereas the ionization width Γ is proportional to ε², in a wide range of ε, Ω_R ≪ Γ. This means that perturbation of atomic spectrum due to the resonance interaction of Rydberg levels predominates perturbation caused by ionization broadening as long as ε < ε₃ where ε₃ is determined as the solution of the equation

Two other critical fields, ε₁ and ε₂, or critical values of the parameter V, V ₁ and V ₂, shown in Fig. 2, are determined as the solutions of equations

respectively. With the help of Eqs. (2) and (3) one gets the quasiclassical estimates of these critical fields:

As follows directly from (4) and (8), ε₁ ≪ ε₂ ≪ ε₃ and V ₁ ≪ V = 1 ≪ V ₃.

It should be noted that IS arising due to Λ-type transitions (with V-type transitions completely ignored) occurs at V > 1 or ε > ε². The estimates given above show that V-type transitions may give rise to non-perturbative effects (including IS) in much weaker fields, ε > ε₁ ≪ ε₂, or V > V ₁ ≪ 1. Numerically, e.g., at ñ ₀ = 5 and n ₀ = 25, Eqs. (8) give V ₁ ≈ 0.1 and V ₃ ≈ 10. For the frequency ω = 2×10^-2 a.u. this corresponds to

If, however, in contrast with the first inequality (5), ω ~ ∣E_n ∣, then ñ ₀ ~ n ₀ and ε₁ ~ ε₂ ~ ε₃ or V ₁ ~ V ₃ ~ 1. In this case, practically, there is no gap between ε₁ and ε₃ where effects arising from resonance interaction between Rydberg levels could be seen and nothing except the Λ-type IS is expected to occur.

Let us focus on the quantitative results we have obtained. From the Schrödinger equation we derived the differential equation for the projection ψ_bound of the wave function of the system under consideration on atomic laser-free bound states. According to the scheme of transitions of Fig. 1, the set of differential equations for expansion amplitudes of ψ_bound has the following form:

Because of irreversibility of ionization, the norm of ψ_bound is time-dependent, and the probability of ionization is determined as

after the laser pulse is off.

Eqs. (10) were solved analytically when the number of Rydberg levels involved was small, and numerically in the opposite case.

The results of calculations shown in Fig. 3 can be considered as the direct indication that the V-type transitions play really the significant role in non-linear effects in Rydberg atoms: the curve (a) presents the ionization probability vs. field strength with only the Λ-channel involved (with the V-channel completely ignored), and the curve (b) presents the ionization probability calculated with both channels taken into account. This calculations were made in the model of 15 Rydberg levels interacting with a lower resonant level. The behavior of the curves (a) and (b) is qualitatively different: with the V-type transitions involved, an atom shows much stronger stability than in the case of only the Λ-type transitions taken into account; the V-type stabilization arises at field strength ε ~ ε₁ much smaller than in the case of Λ-type stabilization (ε ~ ε₂). It should be noticed that the duration of laser pulse in the calculations was large compared to the Kepler period T _k of the electron′s motion on its classical orbit, T _k = 2${\pi n}_0^3$. Numerically, the pulse duration was in order of three Kepler periods. The necessity of such a long pulse duration is explained below.

 

Fig. 3. The ionization yield vs. the field strength parameter V calculated in the models with Ω_R = 0 (a) and Ω_R ≠ 0 (b).

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Fig.4. The ionization yield vs. detuning δ calculated for 4 different values of the field strength parameter V.

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The V-channel plays such an important role in the dynamics of photoionization because of a strong resonance coupling of the initial Rydberg state with the lower atomic states. Our consideration shows that the conditions under which the V-type transitions are crucially important are given by Eqs. (4), (5) completed with the condition of a not too large detuning δ:

where $\widetilde{∆}$ = E _ñ+1 - E_ñ is the gap between the two adjacent lower atomic levels (see Fig. 1). When, on the contrary, the detuning δ is on the order of a half of $\widetilde{∆}$ , δ ≈ $\widetilde{∆}$ /2, then, the contributions of each of these two lower levels closest to the resonance with the initially populated one suppress each other, and the role of the V-channel becomes negligible. By varying the detuning δ, one can provide a smooth transition from the V-type IS (when ∣δ∣ ≪ $\widetilde{∆}$ ) to the Λ-type IS occurring when δ ~ $\widetilde{∆}$ /2. The photoionization probability vs. the detuning δ calculated numerically is shown in Fig. 4, which confirms quantitatively the above-formulated qualitative conclusions. In the region of Λ-type IS the probability of ionization is much larger, than in the region of the V-type IS, and Λ-type IS occurs at much stronger fields. The calculations for Fig. 4 were made within the model of 15 upper Rydberg levels and 3 lower levels. The rectangular laser pulse profile was used in the calculations, the duration of the pulse t = 20 ∆^-1.

The conditions (4), (5) and (12) are necessary but not sufficient for the V-type IS to be observable. According to our analysis, in contrast with the Λ-type IS, the V-type IS can be observed only in the case of rather long laser pulses. Under condition Ω_R t > 1, where t is the pulse duration, and for the model 3-level system (two Rydberg levels E_n and a single level E_ñ , see Fig. 1) the following approximate expression for the ionization probability was derived:

As it is easily seen from (13), the ionization yield stops growing only when the parameter Λt approaches unity. Hence, the third condition under which the V-type IS at rather low fields can be observed has the form

This conditions can be rewritten as

where n ₀ ≫ ñ ₀ and, hence, the pulse duration has to be much larger than the classical Kepler period T _k = 2${\pi n}_0^3$.

In the case of a model with many levels taken into account, the numerical analysis was made to check whether the condition (15) is still essential or not. The curves of Fig. 5 describe the ionization probability in laser pulses of different duration, calculated within the model of 11 upper levels involved. As it can be easily noticed, the threshold of stabilization moves towards lower intensities while the pulse duration increases. However, the ionization yield appears to be less sensitive to the pulse duration than in the simplest 3-level model. Hence, the condition (15) seems to be too severe in the case of multilevel system. Nevertheless, the pulse duration still should be at least several Kepler periods for V-type IS to be observable.

Let us summarize the conditions of stabilization of each type: the V-type IS takes place when inequalities (4), (5), (12) are fulfilled and pulse duration is larger than the Kepler period. Under these conditions the atom shows very strong stability, and the stabilization arises at rather low fields ε ~ ε₁ (see Fig. 3, 5). If laser pulse is short, then the role of V-channel in redistribution of the atomic population at Rydberg levels is still important, but it seems hard to detect the V-type IS (in this case the ionization probability does not stop increasing at ε ~ ε₁).

The Λ-channel predominates when either the first of inequalities (4) is not satisfied or the detuning δ is large, δ ~ $\widetilde{∆}$ /2. In this case the well-known [1, 2] effect of the Λ-type IS arises.

 

Fig. 5. The ionization yield vs. the field strength parameter V calculated for 3 different pulse durations.

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Fig.6. The ionization yield vs. the field strength parameter V calculated for different pulse profiles.

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Experimental conditions under which the V-type IS is expected to be observable are given by the Eqs. (4), (5), (12), and t > T _k. A typical example of such parameters is: n ₀ = 25, ñ ₀ = 5, ω ≈ 8.10¹⁴ s ^-1, t > 15 ps, and ε ≥ 10⁶ V/cm (I ≥ 10⁹ W/cm²).

All the results described above were obtained for rectangular pulses. Consideration of smooth pulses shows that the V-type IS still occurs though it can be not as strong as in the case of rectangular pulses. The results of calculations for smooth and rectangular pulses are shown in Fig. 6. The model of two upper Rydberg levels and single lower level was used in the calculations.

It should be noted that formally in the above-considered model all the continuum-continuum (ATI) transitions were ignored. In principle, this is quite correct as long as V < 1 [1, 2], i.e., just in the region of fields where the V-type IS can occur. But even in stronger fields, V > 1, in the model of essential states, the ATI is known to only renormalization effect (V → (V/π)^1/2 [1, 2] which does not change qualitatively any predictions of the theory. However, more rigorous consideration of ATI, as well as level couplings dropped in the rotating wave approximation, require significant generalization of the used model, and we hope to return to these problems elsewhere.

We acknowledge the support of the Russian Fund of Basic Research (the grant #9602-17649) and of the Civilian Research and Development Fund (the grant #RP1-244).