## Abstract

New features of the phenomenon of interference stabilization of Rydberg atoms are found to exist. The main of them are: (i) dynamical stabilization, which means that in case of pulses with a smooth envelope the time-dependent residual probability for an atom to survive in bound states remains almost constant in the middle part of a pulse (at the strongest fields); (ii) existence of the strong-field stabilization of the after-pulse residual probability in case of pulses longer than the classical Kepler period; and (iii) pulsation of the time-dependent Rydberg wave packet formed in the process of photoionization.

© Optical Society of America

## 1. Introduction

If an atom, excited initially into a high-energy (Rydberg) state, is photoionized by a laser field, the probability of its photoionization can be significantly suppressed if the field is strong enough. This is the key feature of the phenomenon known as Interference Stabilization (*IS*) of Rydberg atoms [1]. Qualitatively, this phenomenon is explained by field-induced Raman-type Rydberg-continuum-Rydberg transitions. In a sufficiently strong field, these transitions provide efficient coherent re-population of several neighboring Rydberg levels. Field-induced transitions from these levels to the continuum interfere and partially cancel each other. As the result, the rate of photoionization slows down and an atom gets stabilized.

There were many efforts to complete this qualitative picture by quantitative analysis (see, e.g., the book [2], Chapter 7, and references therein). The arising specific problems concern the conditions under which *IS* can occur, achievable degree of stabilization, behavior of the probability of strong-field photoionization from Rydberg levels in its dependence on the peak light intensity, pulse shape, and pulse duration, etc. We hope that this paper makes a step towards solution of some of these problems and to better understanding of the physics of *IS*.

Several approaches have been used to solve the problems of *IS* and strong-field photoionization from Rydberg levels. One of them is based on the expansion of the exact electron wave function in a series of the field-free atomic eigenfunctions. Coefficients of this expansion are the time-dependent probability amplitudes (*TDPA*) to find an atom in atomic states. Equations for *TDPA* follow directly from the Schrödinger equation. These equations are often simplified with the help of a procedure known as the adiabatic elimination of the continuum [1–5]. Though not quite rigorous (see a brief discussion in the following Section), this method appears to be very helpful for solving the problem of strong-field photoionization. In such a way, one excludes from consideration the continuum *TDPA* and reduces the problem to equations for Rydberg *TDPA* only, with the decay of an atom (its photoionization) taken into account via the tensor of ionization widths. In this paper, the approach of equations for Rydberg *TDPA* is used in a form improved in comparison with the earlier works [3–5]: realistic angular matrix elements are used and laser pulses with a smooth envelope are considered. The arising equations are solved numerically. Both cases of short and long light pulses (as compared to the classical Kepler period of a Rydberg electron) are considered. Dynamics of photoionization, stabilization of an atom, and the time-dependent structure of the Rydberg wave packet formed in the process of photoionization are investigated.

Another interesting and promising approach to the theory of *IS* is based on the idea of applying the quasi-classical (WKB) approximation directly to the Schrödinger equation for an electron in the Coulomb and strong light fields [6–8]. This approach does not use adiabatic elimination of the continuum and any related approximations, but it has its own limitations and, inevitably, some alternative simplifications have to be used. For these reasons, the method based on the quasi-classical approximation is not absolutely rigorous either. In particular, at present, the quasi-classical solution is obtained only for the case of laser pulses shorter than the classical Kepler period, and extension of such a theory upon the case of longer pulses is one of the problems which are interesting but are not solved yet.

At last, in the same short-pulse regime exact 3D numerical *ab-initio* solution of the Schrödinger equation was obtained [9], and the main results of such a solution appeared to be in a surprisingly good agreement with those of the analytical quasi-classical theory [6–8]. As a whole, each of the existing theoretical approaches has its weak and strong sides. None of them is absolutely rigorous. In such a situation, conclusions about quality and validity of theoretical methods and predictions can be made via comparison of their results with each other, as well as with the results of exact numerical solutions and experimental results, when they exist. The results obtained by different theoretical methods are often complementary to each other.

*IS* must be clearly differentiated from another type of strong-field stabilization known as the Kramers-Henneberger (*KH*) stabilization [10, 11]. These two phenomena differ from each other both in their physics and in the applicability conditions. In particular, if the *KH* stabilization is expected to occur, typically, at laser intensity *I* about 10^{15}–10^{16} W/cm^{2}, *IS* is estimated to take place at *I* ~ 10^{13} W/cm^{2}. This estimate agrees with the experimental data of Ref. [3]. This is one of the reasons why, we believe, the effect observed in [3] was just *IS*, though the interpretation given by its authors was slightly different. Qualitatively, the results of this experiment agree with the existing theory of *IS* [2, 4].

## 2. The model

In accordance with the above-described definitions, let us assume that an atom is excited initially to a state ψ_{n0l}(* r*) with a large value of the principal quantum number

*n*

_{0}>> 1 and, e.g., zero angular momentum

*l*= 0. Let such an atom be photoionized by a laser field with a frequency ω exceeding the electron binding energy, ω > |

*E*

_{n0}| = 1/2${n}_{0}^{2}$ (atomic units are used throughout the paper). Let the wave function of the electron Ψ(

*,*

**r***t*) be expanded in a series of the field-free Rydberg and continuum atomic wave functions ψ

_{nl}(

*) and ψ*

**r**_{El}(

*) with the expansion coefficients (*

**r***TDPA*)

*C*

_{nl}(

*t*) and

*C*

_{El}(

*t*), where

*E*≥ 0 is an electron energy in the continuum. With the help of the above-mentioned procedure of adiabatic elimination of the continuum the Schrödinger equation is reduced to the following set of equation for Rydberg

*TDPA C*

_{nl}(

*t*):

where Γ_{nl;n′l′} is the tensor of ionization widths,

ε_{0}(*t*) is the pulse envelope, ε_{0 max} is the peak field-strength amplitude, the constants *V*, β_{l}, and $\tilde{\beta}$
_{l}
are taken in the form

$${\beta}_{l}=\frac{4l\left(l-1\right)}{\left(2l-1\right)\sqrt{\left(2l+1\right)\left(2l-3\right)}},\phantom{\rule{.2em}{0ex}}{\tilde{\beta}}_{l}=4\frac{4{l}^{3}+6{l}^{2}-1}{\left(4{l}^{2}-1\right)\left(2l+3\right)},$$

and Γ(*x*) is the gamma-function. The constant *V* is obtained from the formula for the weak-field rate of photoionization calculated in the quasi-classical approximation (Eq. (33) of Ref. [8]) and the constants β
_{l}
and $\tilde{\beta}$
_{l}
are determined by angular matrix elements in the basis of spherical functions [12]. The calculations are carried out with the following smooth pulse envelope ε_{0}(*t*)

where 0 < *t* < τ and τ is proportional to the pulse duration. The initial conditions to Eqs. (1) are given by

As mentioned above, the procedure of adiabatic elimination of the continuum is not quite rigorous. In fact, this procedure is based on a series of other approximations, such as the well-known rotating-wave approximation, “pole” approximation, approximation of a flat continuum, etc. Many of them hardly can be justified rigorously in the case of a strong field. Not dwelling here upon any details, let us make only some short comments on some of these approximations and the model described.

Transitions taken into account in this model are shown schematically in Fig. 1. Continuum-continuum transitions are seen to be ignored. An attempt to describe them in the framework of a similar model would result in bound-bound hyper-Raman transitions between Rydberg levels with angular momentum changing by 4, 6, etc., rather than by 0 and 2 only, as in Eqs. (1) and (2). Such a generalization would require too many new unknown constants to be introduced in the definition of the tensor of ionization widths, which would become much more complicated than that of Eq. (2). In practice, this hardly would be useful. For this reason, we prefer to use the model in the form described above (Eqs. (2) and (3)). Its quality can be checked by comparison the results to be derived with calculations by other methods. On the other hand, it should be noted that, in the framework of the so-called model of essential states [13], the continuum-continuum transitions were taken into account in the theory of *IS* [1, 2] and they were shown to result in renormalization of the interaction constant *V* (3), whereas all the qualitative features of *IS* were shown to be conserved. This conclusion can be considered as an indication that, probably, *IS* is not too sensitive to the continuum-continuum transitions and, for qualitative conclusions, they can be ignored, as in the present model characterized by Eqs. (1) – (3) and Fig. 1.

As for the Raman-type transitions via the continuum shown in Fig. 1, in Eqs. (1) only the imaginary parts of the corresponding second-order matrix elements are taken into account. This simplification corresponds to the so called pole approximation, validity of which is, still, an open question (see the discussion in the review papers [14]). Very often [2, 4, 5], real parts of matrix elements are taken into account by a simple substitution of the imaginary unit *i* by *i* + α on the right-hand side of Eqs. (1), where α is a real constant. However, rigorously, this new constant α can be determined only via exact numerical calculations of the second-order bound-continuum-bound matrix elements for Rydberg states, and such data are not available now in sufficiently wide ranges of light frequencies ω and quantum numbers *n*, *n*′ and *l*, *l*′. On the other hand, calculations with arbitrarily chosen α [4, 5] show that the effect of *IS* can be only underestimated in the approximation α = 0 and, usually, at α ≠ 0 stabilization is stronger. For these reasons, in this paper we do not consider explicitly the case of nonzero α and restrict our consideration by Eqs. (1), which correspond to α = 0.

Eq. (3) for the interaction constant *V* is known to be valid at ω << 1 [2]. This restriction, together with the condition of one-photon transitions to the continuum, ω > |*E*_{n}
|, determines the range of frequencies, at which the present theory can pretend to be valid. An additional limitation can arise from the applicability condition of the dipole approximation, which gives λ = 2π*c*/ω > *r*_{max}
= 1/|*E*_{n}
| = 2*n*
^{2}, where *r*_{max}
is the size of the Rydberg orbit. For high Rydberg levels (*n* > 22) this restriction becomes stronger than ω << 1 and, then, the range of frequencies appears to be limited by 2π*c* |*E*_{n}
| ≈ 10^{3}|E_{n}| > ω > |*E*_{n}
|.

As it follows from all the earlier considerations of *IS*, as well as directly from Eqs. (1) and (2), stabilization of this type can occur at *V*≥1 or ε_{0}≥ω^{5/3}. Under this condition the Fermi-golden-rule ionization widths Γ_{nl, n′l′} (3) become larger than spacing between neighboring Rydberg levels *E*
_{n+1} - *E*_{n}
, and the perturbation theory becomes invalid. As mentioned above, for optical frequencies (ω ~ 0.1) the condition ε_{0} ~ ω^{5/3} corresponds to intensities below 10^{13} W/cm^{2}. This intensity is too low for the *KH* stabilization to take place. Indeed, the main characteristic parameter of the *KH* stabilization is the free-electron quiver motion amplitude α
_{KH}
= ε_{0}/ω^{2}. The *KH* stabilization can occur if α
_{KH}
is larger than a characteristic length of the system under consideration. For Rydberg atoms, at *V* ~ 1, the *KH* parameter α
_{KH}
is of the order of ω^{-1/3}. Hence, though much larger than one, α
_{KH}
remains smaller than both *r*_{max}
and the characteristic quasi-classical length *r*_{q}
= ω^{-2/3} of Refs. [7–9]: α
_{KH}
<< *r*_{q}
as long as ω << 1. Moreover, from the condition α
_{KH}
~ *r*_{q}
one can find limitations of the field ε_{0} up to which *IS* remains the only possible mechanism of stabilization: ε_{0} < ω^{4/3} or *V* < ω^{-1/3}, where ω^{-1/3} is assumed to be large, ω_{-1/3} >> 1. Under these conditions the *KH* stabilization does not compete with *IS*.

In the numerical solution described below, the set of equations (1) is truncated both in *n* and *l* by the conditions *n*
_{1} < *n* < *n*
_{2} and *l* < *l*_{max}
. In the calculations of this paper *n*
_{0} = 40, *n*
_{2, 1} = *n*
_{0} ± 6, and *l*_{max}
= 10. The dependence of the results on the truncation boundaries has been investigated in details but is not discussed in this paper (see also a brief discussion in Ref. 4). Here it should be mentioned only that convergence of the scheme with growing *n*
_{1} and *n*
_{2} is not too bad in the case of Eqs. (1) but it worsens when Eqs. (1) are “improved” by the above-discussed substitution of the imaginary unit *i* by *i* + α on the right-hand side to take into account real parts of the second-order Raman-type matrix elements. This is one of the reasons why here we do not consider such an improvement and put α = 0.

The *TDPA C*_{nl}
(*t*) found from the numerical solution of Eqs. (1) have been used to calculate many specific characteristics of the field-driven atom. Because of lack of space, not all of them are described in this paper. Here, the main attention is concentrated on the following three effects. First, the *TDPA C*_{nl}
(*t*) are used to calculate the total time-dependent residual probability to find an atom in any bound states

as well as the partial time-dependent residual probabilities to find an atom in manifolds of Rydberg states with given values of the angular momentum *l* but arbitrary values of the principal quantum number *n*

These calculations are used to demonstrate how stabilization can be seen in the dynamics of photoionization (dynamical stabilization), while the field is on. Second, the after-pulse total residual probability to find an atom in any bound states, *w*_{res}
(*t* = *τ*), is calculated with help of Eq. (6) in its dependence on the peak field-strength amplitude ε_{0 max} or the field-parameter *V* (3). The aim of these calculations is investigation of the conditions under which *IS* can occur at various pulse durations τ, either short or long in comparison with the Kepler period *t*_{K}
= 2π${n}_{0}^{3}$. At last, the third effect investigated below with the help of the *TDPA C*_{nl}
(*t*) is the evolution of the Rydberg wave packet formed in the process of photoionization in a strong field. The structure of the wave packet is characterized by the time-dependent Rydberg electron radial and angular density distributions determined as

$$\rho (\theta ,t)=2\pi \mathrm{sin}\left(\theta \right)\underset{0}{\overset{\infty}{\int}}{r}^{2}\mathit{dr}{\mid \sum _{\mathit{nl}}{C}_{\mathit{nl}}\left(t\right){\psi}_{\mathit{nl}}\left(r\right)\mid}^{2},$$

where *r* = |* r*|,

*d*Ω is an element of a solid angle in the direction of the electron position vector

*, and θ is the angle between*

**r***and*

**r****ε**

_{0}.

## 3. The results of calculations

The time-dependent total residual probability to find an atom in any bound states *w*_{res}
(*t*) (7) is shown in Fig. 2 for square and smooth (4) pulse envelopes with equal peak field strength amplitudes and τ = 5*t*_{K}
. A typical feature of the curve *w*_{res}
(*t*) corresponding to a rectangular pulse (blue) is a very fast decay of an atom at the initial stage (*t* close to zero), and this effect is seen to be missing in the case of a smooth pulse envelope (red). Besides, as a whole, at a given peak field strength ε_{0 max}, the residual probability is larger in the case of a rectangular pulse, i.e., the model of a suddenly turned on interaction overestimates the degree of achievable stabilization. In the case of a smooth pulse envelope (red) the main changes of the time-dependent residual probability are found to occur at the front and (much less) rear wings of the pulse. In the middle of the pulse (at 1.2 *t*_{K}
≤ *t* ≤ 3.5*t*_{K}
), there is a kind of a plateau, where *w*_{res}
(*t*) ≈ const. Such a plateau is an unambiguous indication of stabilization seen in the dynamics of photoionization (or the dynamical stabilization): the rate of transitions to the continuum (decay) is high at medium fields and low at high fields in the middle of the pulse.

The same effect of dynamical stabilization is seen pretty well in the pictures of Fig. 3, which show the time dependence of the relative partial residual probabilities *w*_{l}
(*t*)/*w*_{res}
(*t*) (7) to find an atom in Rydberg states with arbitrary values of the principal quantum number *n* but given values of the angular momentum *l*. These pictures show clearly that the efficiency of excitation of higher-*l* states is relatively low in a weak field (*a*), rather large in a medium field (*b*), and, again, very low in a strong field (*c*). This means that a strong field suppresses transitions to higher-*l* states. Moreover, the curves of Fig. 3c show that in a strong field more or less efficient transitions to higher-*l* states take place only at the front wing of a smooth pulse (at *t* < *t*_{K}
). After this, during the most part of a pulse, in agreement with the idea of a strong-field stabilization, the relative partial probabilities *w*_{l}
/*w*_{res}
remain more or less constant, and all of them are small in comparison with *w*
_{0}/*w*_{res}
.

In Fig. 4 the after-pulse residual probability *w*_{res}
(*t*=τ) is plotted in the dependence on the field-parameter *V* (3) for four different values of the pulse duration τ. These results show clearly that stabilization can take place at any pulse duration τ, either shorter or longer *t*_{K}
. Probably, the most interesting of these results is that corresponding to the longest pulse duration τ, τ = 7*t*_{K}
(the purple curve at Fig. 4). In this case stabilization is seen to arise at somewhat higher fields than in the cases of shorter pulses. At intermediate fields (*V* ≈ 1) and τ = 7*t*_{K}
, *w*_{res}
(*t*=τ) = 0, ionization is complete, and there is no stabilization. However, at higher fields a non-zero residual probability to find an atom in bound states arises again, and this is a clear manifestation of *IS* occurring at long pulse durations.

It should be noted that the difference in threshold fields for stabilization in cases of short and long pulses is not too large and, qualitatively, these threshold fields are determined by the same condition *V* ~ 1. Moreover, with the parameter *V* (3) written in natural units, this condition gives ε_{0}/ω^{5/3} ~ *ħ m*
^{1/3}/*e*
^{5/3}. Clearly, this is a quantum condition and, hence, *IS* is a quantum-mechanical phenomenon, which hardly can be imitated by any classical analogues. We do believe that in both cases of short and long pulses the described above stabilization has the same physical origin and can be explained in terms of re-population of Rydberg levels via Λ-type transitions and interference of transitions to the continuum [1].

The demonstrated effect of increasing *w*_{res}
(*t*=τ, *V*) at large *V* and long pulse durations τ can be considered as an answer to the objection against strong-field stabilization known as the “curse of Lambropoulos”. This objection consists of an assumption that in strong pulses with a smooth envelope complete ionization of atoms can take place at their front wings, at intermediate filed strength providing the fastest decay. As the result, no atoms are assumed to survive and to experience an action of a strong field in the middle of the pulse. A qualitative answer to this objection is that at high peak field strength of a pulse at a given pulse duration the field rises so quickly that the most dangerous region of intermediate fields becomes too short for a complete ionization of atoms, and some of them can survive to experience an action of a strong stabilizing field in the middle part of the pulse. Such a shortening of the part of a pulse, most dangerous for survival of neutral atoms, appears to be the stronger pronounced the higher the peak field strength is, and this is the reason of stabilization. Prediction of stabilization at a long pulse duration is one of the main conclusions of the present calculations. It would be interesting to confirm this result by calculations based on different approaches: quasi-classical approach of Refs. [8, 9] or direct and exact *ab-initio* solution of Ref. [10]. Unfortunately, at present, both of these alternative approaches provide results valid only in the case of short pulses, τ << *t*_{K}
. An extension of these methods to obtain solutions applicable in the case of long pulses is one of the main tasks of future investigations in this domain.

At last, the time-dependent normalized radial and angular distributions of the Rydberg electron density, ρ(*r*, *t*)/*w*_{res}
(*t*) and ρ(*r*, *t*)/*w*_{res}
(*t*) (8), are shown in the video picture (Fig. 5). This series of calculations corresponds to ionization by a pulse with the envelope (4), τ = 5*t*_{K}
, and ε_{0 max} corresponding to *V* = 2. The radial distribution is seen to take the form of a pulsating formation. The arrow under the *r*-axis characterizes the time-dependent average size of the Rydberg wave packet *r̄*(*t*). This function is shown explicitly in Fig. 6 together with the time-dependent rate of ionization *ẇ*
_{i}
(*t*) = -*ẇ*
_{res}
(*t*). Periodical pulsation of the wave packet and of its size are seen to proceed with the Kepler period *t*_{K}
. An origin of these oscillations can be related to the effect of a sudden change of the phase of the strong-field-driven quasi-classical Rydberg wave function predicted in the earlier analytical investigation [15]. The point at which such a change was found to occur was shown to move along the classical electron trajectory *r*_{cl}
(*t*). However, significant redistribution in time of the electron density was not obtained in the analytical theory of Ref. [15]. The curves of Fig. 6 show that the maxima of the time-dependent rate of ionization, correlate with the minima of *r̄*(*t*): the closer the pulsating wave packet comes to the nucleus, the larger the rate of ionization is.

## 4. Conclusion

In this paper, the model of coupled equations for *TDPA C*_{n}
(*t*) is used to describe photoionization and *IS* of Rydberg atoms in a strong laser field. In comparison with earlier investigations [2, 5], the model is improved; in particular, the case of a smooth pulse envelope is considered, and the following new results are derived.

- In the case of smooth pulses
*IS*is shown to be clearly seen in the dynamics of photoionization. In a strong field, the total and partial time-dependent residual probabilities to find an atom in bound states are shown to change significantly mainly at the front wing and to be almost constant in the middle part of a pulse. - The effect of
*IS*is shown to occur both at short and long pulse duration (as compared to the Kepler period). In the case of long pulses, ionization of an atom per pulse can be complete at intermediate fields and only in the region of stronger fields stabilization arises. - The Rydberg wave packet formed via strong-field Raman-type transitions is shown to have a form of a pulsating structure. The average size of the wave packet oscillates in time with the period equal to the classical Kepler period. The time-dependent rate of ionization is shown to have its peaks when the average size of the wave packet is minimal.

As mentioned above, the model used in this paper is not perfect. It uses many approximations which do not have rigorous justifications. Nevertheless, the results obtained are physically reasonable and qualitatively understandable. Comparison with results to be derived by different methods can be used for quality control of the present model and its predictions. This can be done only via significant extension of the known alternative methods, such as the quasi-classical analytical theory [8, 9] and direct exact solution of the Schrödinger equation [10]. We hope to return to these problems elsewhere.

## Acknowledgments

This work is supported partially by the Russian Fund of Basic Research (the grant # 9602-17649) and Civilian Research and Development Fund (the grant # RP1-244).

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