## Stabilization and structure of wave packets in Rydberg atoms ionized by a strong light field

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

*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]

*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*.

*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]

*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]

*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]

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

## 2. The model

_{n0l}(

*) with a large value of the principal quantum number*

**r***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

*,*

**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*):

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

_{0}(

*t*) is the pulse envelope, ε

_{0 max}is the peak field-strength amplitude, the constants

*V*, β

_{l}, and

_{l}are taken in the form

*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]

_{l}and

_{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*)

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

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]

*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]

*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.

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]

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]

*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]

*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.

*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}|.

*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*.

*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

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]

*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.

*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

*l*but arbitrary values of the principal quantum number

*n*

*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

*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

*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

*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.

*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}.

*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.

*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

**21**, L155 (1988). [CrossRef]

*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

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]

*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]

*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.

## 4. Conclusion

*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.

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]

## Acknowledgments

## 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 |

2. | M. V. Fedorov, |

3. | J. Hoogenraad, R. B. Vrijen, and L. D. Noordam, “Ionization Suppression of Rydberg Atoms by Short Laser Pulses,” Phys. Rev. A |

4. | M. V. Fedorov, M.-M. Tegranchi, and S. M. Fedorov, “Interference Stabilization of Rydberg Atoms: Numerical Calculations and Physical Models,” J. Phys. B |

5. | R. Parzynski and A. Woiczik, “Interference Stabilization of Rydberg Atoms: an Analytical Model with Migration of Population to higher |

6. | M. V. Fedorov, “Quasiclassical Atomic Electron in a Strong Light Field,” J. Phys. B |

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. |

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 |

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 |

10. | M. Gavrila and J. Z. Kaminski, “Free-Free Transitions in Intense High-Frequency Fields,” Phys. Rev. Lett. |

11. | M. Gavrila, “Atomic Structure and Decay in High-Frequency Fields,” in |

12. | L. D. Landau and E. M. Lifshitz, |

13. | Z. Deng and J. H. Eberly, “Multiphoton Absorption Above Ionization Threshold by Atoms in Strong Laser Fields,” J. Opt. Soc. Am. B |

14. | N. B. Delone and M. V. Fedorov, “Above-Threshold Ionization,” Progr. Quant. Electron. |

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 |

**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

- 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]
- M. V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific (Singapore-New Jersey-London- Hong Kong, 1997).
- 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]
- 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]
- 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).
- M. V. Fedorov, "Quasiclassical Atomic Electron in a Strong Light Field," J. Phys. B 27, 4145-4167 (1994). [CrossRef]
- 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).
- 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]
- 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]
- M. Gavrila and J. Z. Kaminski, "Free-Free Transitions in Intense High-Frequency Fields," Phys. Rev. Lett. 52, 613 (1984). [CrossRef]
- 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.
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics, (Pergamon Press, New York, 1977).
- 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]
- 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]
- 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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