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Optics Express

  • Editor: J. H. Eberly
  • Vol. 3, Iss. 7 — Sep. 28, 1998
  • pp: 271–279
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Stabilization and structure of wave packets in Rydberg atoms ionized by a strong light field

M. V. Fedorov and S. M. Fedorov  »View Author Affiliations


Optics Express, Vol. 3, Issue 7, pp. 271-279 (1998)
http://dx.doi.org/10.1364/OE.3.000271


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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

1. M. V. Fedorov and A. M. Movsesian, “Field-Induced Effects of Narrowing of Photoelectron Spectra and Stabilization of Rydberg Atoms,” J. Phys. B 21, L155 (1988). [CrossRef]

]. 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

2. M. V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific (Singapore-New Jersey-London-Hong Kong, 1997).

], 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

1. M. V. Fedorov and A. M. Movsesian, “Field-Induced Effects of Narrowing of Photoelectron Spectra and Stabilization of Rydberg Atoms,” J. Phys. B 21, L155 (1988). [CrossRef]

]. 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

3. J. Hoogenraad, R. B. Vrijen, and L. D. Noordam, “Ionization Suppression of Rydberg Atoms by Short Laser Pulses,” Phys. Rev. A 50, 4133 (1994). [CrossRef] [PubMed]

]: 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.

At last, in the same short-pulse regime exact 3D numerical ab-initio solution of the Schrödinger equation was obtained [9

9. A. M. Popov, E. A. Volkova, and O. V. Tikhonova, “Numerical Modeling of the Photoionization of Rydberg Atoms by the Field of an Electromagnetic Wave,” Sov. Phys. JETP 86, 328 (1998). [CrossRef]

], 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

6. M. V. Fedorov, “Quasiclassical Atomic Electron in a Strong Light Field,” J. Phys. B 27, 4145–4167 (1994). [CrossRef]

]. 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

10. M. Gavrila and J. Z. Kaminski, “Free-Free Transitions in Intense High-Frequency Fields,” Phys. Rev. Lett. 52, 613 (1984). [CrossRef]

, 11

11. M. Gavrila, “Atomic Structure and Decay in High-Frequency Fields,” in Atoms in Intense Laser Field, Ed. by M. Gavrila, Academic Press (New York, 1992), p.435.

]. 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 1015–1016 W/cm2, IS is estimated to take place at I ~ 1013 W/cm2. This estimate agrees with the experimental data of Ref. [3

3. J. Hoogenraad, R. B. Vrijen, and L. D. Noordam, “Ionization Suppression of Rydberg Atoms by Short Laser Pulses,” Phys. Rev. A 50, 4133 (1994). [CrossRef] [PubMed]

]. This is one of the reasons why, we believe, the effect observed in [3

3. J. Hoogenraad, R. B. Vrijen, and L. D. Noordam, “Ionization Suppression of Rydberg Atoms by Short Laser Pulses,” Phys. Rev. A 50, 4133 (1994). [CrossRef] [PubMed]

] 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

2. M. V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific (Singapore-New Jersey-London-Hong Kong, 1997).

, 4

4. M. V. Fedorov, M.-M. Tegranchi, and S. M. Fedorov, “Interference Stabilization of Rydberg Atoms: Numerical Calculations and Physical Models,” J. Phys. B 29, 2907–2924 (1996). [CrossRef]

].

2. The model

In accordance with the above-described definitions, let us assume that an atom is excited initially to a state ψn0l(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| = 1/2n02 (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 ψnl(r) and ψEl(r) with the expansion coefficients (TDPA) Cnl (t) and CEl (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 Cnl (t):

iC˙nl(t)EnCnl(t)=i2ε02(t)ε0max2nlΓnl;nlCnl(t),
(1)

where Γnl;nl is the tensor of ionization widths,

Γnl;nl=πV2(nn)32[βlδl,l2+βl+2δl,l+2+β˜lδl,l],
(2)

ε0(t) is the pulse envelope, ε0 max is the peak field-strength amplitude, the constants V, βl, and β˜ l are taken in the form

V=ε0maxω53Γ(2/3)π2733130.59ε0maxω53,
βl=4l(l1)(2l1)(2l+1)(2l3),β˜l=44l3+6l21(4l21)(2l+3),
(3)

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

8. M. V. Fedorov and O. V. Tikhonova, “Strong-Field Short-Pulse Photoionization of Rydberg Atoms: Interference Stabilization and Distribution of the Photoelectron Density in Space and Time,” Phys. Rev. A 58 (1998). [CrossRef]

]) and the constants β l and β˜ l are determined by angular matrix elements in the basis of spherical functions [12

12. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon Press (New York, 1977).

]. The calculations are carried out with the following smooth pulse envelope ε0(t)

ε0(t)=ε0maxsin2(πtτ),
(4)

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

Cnl(0)=δn,n0δl,0.
(5)

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

13. Z. Deng and J. H. Eberly, “Multiphoton Absorption Above Ionization Threshold by Atoms in Strong Laser Fields,” J. Opt. Soc. Am. B 2, 486 (1985). [CrossRef]

], the continuum-continuum transitions were taken into account in the theory of IS [1

1. M. V. Fedorov and A. M. Movsesian, “Field-Induced Effects of Narrowing of Photoelectron Spectra and Stabilization of Rydberg Atoms,” J. Phys. B 21, L155 (1988). [CrossRef]

, 2

2. M. V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific (Singapore-New Jersey-London-Hong Kong, 1997).

] 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.

Fig. 1. Raman-type transitions described by Eqs. (1), (2).

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

14. N. B. Delone and M. V. Fedorov, “Above-Threshold Ionization,” Progr. Quant. Electron. 13, 267 (1989); “Multiphoton Ionization of Atoms: New Effects,” Sov Phys. USPEKHI 32, 500 (1989). [CrossRef]

]). Very often [2

2. M. V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific (Singapore-New Jersey-London-Hong Kong, 1997).

, 4

4. M. V. Fedorov, M.-M. Tegranchi, and S. M. Fedorov, “Interference Stabilization of Rydberg Atoms: Numerical Calculations and Physical Models,” J. Phys. B 29, 2907–2924 (1996). [CrossRef]

, 5

5. R. Parzynski and A. Woiczik, “Interference Stabilization of Rydberg Atoms: an Analytical Model with Migration of Population to higher l-States” Laser Phys. 7, 551 (1997).

], 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

4. M. V. Fedorov, M.-M. Tegranchi, and S. M. Fedorov, “Interference Stabilization of Rydberg Atoms: Numerical Calculations and Physical Models,” J. Phys. B 29, 2907–2924 (1996). [CrossRef]

, 5

5. R. Parzynski and A. Woiczik, “Interference Stabilization of Rydberg Atoms: an Analytical Model with Migration of Population to higher l-States” Laser Phys. 7, 551 (1997).

] 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

2. M. V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific (Singapore-New Jersey-London-Hong Kong, 1997).

]. This restriction, together with the condition of one-photon transitions to the continuum, ω > |En |, 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 = 2πc/ω > rmax = 1/|En | = 2n 2, where rmax 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 2πc |En | ≈ 103|En| > ω > |En |.

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 ε0≥ω5/3. Under this condition the Fermi-golden-rule ionization widths Γnl, nl (3) become larger than spacing between neighboring Rydberg levels E n+1 - En , and the perturbation theory becomes invalid. As mentioned above, for optical frequencies (ω ~ 0.1) the condition ε0 ~ ω5/3 corresponds to intensities below 1013 W/cm2. 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 = ε02. 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 rmax and the characteristic quasi-classical length rq = ω-2/3 of Refs. [7–9

7. O. V. Tikhonova and M. V. Fedorov, “Quasicalassical Theory of Strong-Field Photoionization from Rydberg Levels of Atoms: Solution of the Initial-Value Problem,” Laser Phys. 7, 574–582 (1997).

]: α KH << rq as long as ω << 1. Moreover, from the condition α KH ~ rq one can find limitations of the field ε0 up to which IS remains the only possible mechanism of stabilization: ε0 < ω4/3 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 < lmax . In the calculations of this paper n 0 = 40, n 2, 1 = n 0 ± 6, and lmax = 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

4. M. V. Fedorov, M.-M. Tegranchi, and S. M. Fedorov, “Interference Stabilization of Rydberg Atoms: Numerical Calculations and Physical Models,” J. Phys. B 29, 2907–2924 (1996). [CrossRef]

). 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 Cnl (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 Cnl (t) are used to calculate the total time-dependent residual probability to find an atom in any bound states

wres(t)=nlCnl(t)2,
(6)

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

wl(t)=nCnl(t)2.
(7)

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, wres (t = τ), is calculated with help of Eq. (6) in its dependence on the peak field-strength amplitude ε0 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 tK = 2πn03. At last, the third effect investigated below with the help of the TDPA Cnl (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

ρ(r,t)=r20πnlCnl(t)ψnl(r)2dΩ,
ρ(θ,t)=2πsin(θ)0r2drnlCnl(t)ψnl(r)2,
(8)

where r = |r|, dΩ is an element of a solid angle in the direction of the electron position vector r, and θ is the angle between r and ε 0.

3. The results of calculations

The time-dependent total residual probability to find an atom in any bound states wres (t) (7) is shown in Fig. 2 for square and smooth (4) pulse envelopes with equal peak field strength amplitudes and τ = 5tK . A typical feature of the curve wres (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 ε0 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 tKt ≤ 3.5tK ), there is a kind of a plateau, where wres (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.

Fig. 2. The total residual probability to find an atom in bound states vs. time t (in Kepler periods) for rectangular (blue) and smooth (red) pulses; in both cases ε0 max corresponds to V = 3; the green curve is the envelope (4) of a smooth 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 wl (t)/wres (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 < tK ). After this, during the most part of a pulse, in agreement with the idea of a strong-field stabilization, the relative partial probabilities wl /wres remain more or less constant, and all of them are small in comparison with w 0/wres .

Fig. 3. The time-dependent partial residual probabilities wl vs. time t (in Kepler periods) for l = 0, 2, 4, and 6 (red, blue, green, and yellow), τ = 5tK , and V = 0.2 (a), 0.9 (b), and 3.6 (c).

In Fig. 4 the after-pulse residual probability wres (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 tK . Probably, the most interesting of these results is that corresponding to the longest pulse duration τ, τ = 7tK (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 = 7tK , wres (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.

Fig. 4. The after-pulse residual probability wres (τ) vs. the field-strength parameter V (3) for τ = 0.3tK (red), tK (blue), 5tK (green), and 7tK (purple).

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 ε05/3 ~ 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

1. M. V. Fedorov and A. M. Movsesian, “Field-Induced Effects of Narrowing of Photoelectron Spectra and Stabilization of Rydberg Atoms,” J. Phys. B 21, L155 (1988). [CrossRef]

].

The demonstrated effect of increasing wres (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

8. M. V. Fedorov and O. V. Tikhonova, “Strong-Field Short-Pulse Photoionization of Rydberg Atoms: Interference Stabilization and Distribution of the Photoelectron Density in Space and Time,” Phys. Rev. A 58 (1998). [CrossRef]

, 9

9. A. M. Popov, E. A. Volkova, and O. V. Tikhonova, “Numerical Modeling of the Photoionization of Rydberg Atoms by the Field of an Electromagnetic Wave,” Sov. Phys. JETP 86, 328 (1998). [CrossRef]

] or direct and exact ab-initio solution of Ref. [10

10. M. Gavrila and J. Z. Kaminski, “Free-Free Transitions in Intense High-Frequency Fields,” Phys. Rev. Lett. 52, 613 (1984). [CrossRef]

]. Unfortunately, at present, both of these alternative approaches provide results valid only in the case of short pulses, τ << tK . 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.

Fig. 5. The movie describing evolution of the Rydberg wave packet created in the process of ionization during the time when the field is on: the radial and angular (in the left upper corner) distributions of the electron density (8). [Media 1]
Fig. 6. The average size of the strong-field driven wave packet (red) and the time-dependent rate of ionization (blue).

4. Conclusion

In this paper, the model of coupled equations for TDPA Cn (t) is used to describe photoionization and IS of Rydberg atoms in a strong laser field. In comparison with earlier investigations [2

2. M. V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific (Singapore-New Jersey-London-Hong Kong, 1997).

, 5

5. R. Parzynski and A. Woiczik, “Interference Stabilization of Rydberg Atoms: an Analytical Model with Migration of Population to higher l-States” Laser Phys. 7, 551 (1997).

], the model is improved; in particular, the case of a smooth pulse envelope is considered, and the following new results are derived.

  1. 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.
  2. 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.
  3. 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

8. M. V. Fedorov and O. V. Tikhonova, “Strong-Field Short-Pulse Photoionization of Rydberg Atoms: Interference Stabilization and Distribution of the Photoelectron Density in Space and Time,” Phys. Rev. A 58 (1998). [CrossRef]

, 9

9. A. M. Popov, E. A. Volkova, and O. V. Tikhonova, “Numerical Modeling of the Photoionization of Rydberg Atoms by the Field of an Electromagnetic Wave,” Sov. Phys. JETP 86, 328 (1998). [CrossRef]

] and direct exact solution of the Schrödinger equation [10

10. M. Gavrila and J. Z. Kaminski, “Free-Free Transitions in Intense High-Frequency Fields,” Phys. Rev. Lett. 52, 613 (1984). [CrossRef]

]. 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).

References and links

1.

M. V. Fedorov and A. M. Movsesian, “Field-Induced Effects of Narrowing of Photoelectron Spectra and Stabilization of Rydberg Atoms,” J. Phys. B 21, L155 (1988). [CrossRef]

2.

M. V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific (Singapore-New Jersey-London-Hong Kong, 1997).

3.

J. Hoogenraad, R. B. Vrijen, and L. D. Noordam, “Ionization Suppression of Rydberg Atoms by Short Laser Pulses,” Phys. Rev. A 50, 4133 (1994). [CrossRef] [PubMed]

4.

M. V. Fedorov, M.-M. Tegranchi, and S. M. Fedorov, “Interference Stabilization of Rydberg Atoms: Numerical Calculations and Physical Models,” J. Phys. B 29, 2907–2924 (1996). [CrossRef]

5.

R. Parzynski and A. Woiczik, “Interference Stabilization of Rydberg Atoms: an Analytical Model with Migration of Population to higher l-States” Laser Phys. 7, 551 (1997).

6.

M. V. Fedorov, “Quasiclassical Atomic Electron in a Strong Light Field,” J. Phys. B 27, 4145–4167 (1994). [CrossRef]

7.

O. V. Tikhonova and M. V. Fedorov, “Quasicalassical Theory of Strong-Field Photoionization from Rydberg Levels of Atoms: Solution of the Initial-Value Problem,” Laser Phys. 7, 574–582 (1997).

8.

M. V. Fedorov and O. V. Tikhonova, “Strong-Field Short-Pulse Photoionization of Rydberg Atoms: Interference Stabilization and Distribution of the Photoelectron Density in Space and Time,” Phys. Rev. A 58 (1998). [CrossRef]

9.

A. M. Popov, E. A. Volkova, and O. V. Tikhonova, “Numerical Modeling of the Photoionization of Rydberg Atoms by the Field of an Electromagnetic Wave,” Sov. Phys. JETP 86, 328 (1998). [CrossRef]

10.

M. Gavrila and J. Z. Kaminski, “Free-Free Transitions in Intense High-Frequency Fields,” Phys. Rev. Lett. 52, 613 (1984). [CrossRef]

11.

M. Gavrila, “Atomic Structure and Decay in High-Frequency Fields,” in Atoms in Intense Laser Field, Ed. by M. Gavrila, Academic Press (New York, 1992), p.435.

12.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon Press (New York, 1977).

13.

Z. Deng and J. H. Eberly, “Multiphoton Absorption Above Ionization Threshold by Atoms in Strong Laser Fields,” J. Opt. Soc. Am. B 2, 486 (1985). [CrossRef]

14.

N. B. Delone and M. V. Fedorov, “Above-Threshold Ionization,” Progr. Quant. Electron. 13, 267 (1989); “Multiphoton Ionization of Atoms: New Effects,” Sov Phys. USPEKHI 32, 500 (1989). [CrossRef]

15.

M. V. Fedorov, M. Yu. Ivanov, and A. M. Movsesian, “Strong-Field Photoionization of an Initially Excited Hydrogen Atom: Formation of Rydberg Wavepacket, its Structure and Trapping of Population at Rydberg Levels,” J. Phys. B 23, 2245S–2257S (1990). [CrossRef]

OCIS Codes
(020.5780) Atomic and molecular physics : Rydberg states
(190.4180) Nonlinear optics : Multiphoton processes

ToC Category:
Research Papers

History
Original Manuscript: August 13, 1998
Revised Manuscript: August 11, 1998
Published: September 28, 1998

Citation
Mikhail Fedorov and Sergei Fedorov, "Stabilization and Structure of wave packets in Rydberg atoms ionized by a strong light field," Opt. Express 3, 271-279 (1998)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-7-271


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References

  1. M. V. Fedorov and A. M. Movsesian, "Field-Induced Effects of Narrowing of Photoelectron Spectra and Stabilization of Rydberg Atoms," J. Phys. B 21, L155 (1988). [CrossRef]
  2. M. V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific (Singapore-New Jersey-London- Hong Kong, 1997).
  3. J. Hoogenraad, R. B. Vrijen and L. D. Noordam, "Ionization Suppression of Rydberg Atoms by Short Laser Pulses," Phys. Rev. A 50, 4133 (1994). [CrossRef] [PubMed]
  4. M. V. Fedorov, M.-M. Tegranchi and S. M. Fedorov, "Interference Stabilization of Rydberg Atoms: Numerical Calculations and Physical Models," J. Phys. B 29, 2907-2924 (1996). [CrossRef]
  5. R. Parzynski and A. Woiczik, "Interference Stabilization of Rydberg Atoms: an Analytical Model with Migration of Population to higher l-States" Laser Phys. 7, 551 (1997).
  6. M. V. Fedorov, "Quasiclassical Atomic Electron in a Strong Light Field," J. Phys. B 27, 4145-4167 (1994). [CrossRef]
  7. O. V. Tikhonova and M. V. Fedorov, "Quasicalassical Theory of Strong-Field Photoionization from Rydberg Levels of Atoms: Solution of the Initial-Value Problem," Laser Phys. 7, 574-582 (1997).
  8. M. V. Fedorov and O. V. Tikhonova, "Strong-Field Short-Pulse Photoionization of Rydberg Atoms: Interference Stabilization and Distribution of the Photoelectron Density in Space and Time," Phys. Rev. A 58 (1998). [CrossRef]
  9. A. M. Popov, E. A. Volkova and O. V. Tikhonova, "Numerical Modeling of the Photoionization of Rydberg Atoms by the Field of an Electromagnetic Wave," Sov. Phys. JETP 86, 328 (1998). [CrossRef]
  10. M. Gavrila and J. Z. Kaminski, "Free-Free Transitions in Intense High-Frequency Fields," Phys. Rev. Lett. 52, 613 (1984). [CrossRef]
  11. M. Gavrila, "Atomic Structure and Decay in High-Frequency Fields," in Atoms in Intense Laser Field, Ed. by M. Gavrila, (Academic Press, New York, 1992), p. 435.
  12. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, (Pergamon Press, New York, 1977).
  13. Z. Deng and J. H. Eberly, "Multiphoton Absorption Above Ionization Threshold by Atoms in Strong Laser Fields," J. Opt. Soc. Am. B 2, 486 (1985). [CrossRef]
  14. N. B. Delone and M. V. Fedorov, "Above-Threshold Ionization," Progr. Quant. Electron. 13, 267 (1989); "Multiphoton Ionization of Atoms: New Effects," Sov Phys. USPEKHI 32, 500 (1989). [CrossRef]
  15. M. V. Fedorov, M. Yu. Ivanov and A. M. Movsesian, "Strong-Field Photoionization of an Initially Excited Hydrogen Atom: Formation of Rydberg Wavepacket, its Structure and Trapping of Population at Rydberg Levels," J. Phys. B 23, 2245S-2257S (1990). [CrossRef]

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