## Competition between Λ- and *V*-type transitions in interference stabilization of Rydberg atoms

Optics Express, Vol. 2, Issue 2, pp. 51-57 (1998)

http://dx.doi.org/10.1364/OE.2.000051

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### Abstract

The problem of Interference Stabilization of Rydberg atoms is considered. Two kinds of Raman-type transitions can be responsible for the effect: Λ-type transitions via the continuum and *V*-type transitions via lower resonant atomic levels. The main distinctions between Λ- and *V*-stabilization are described. The conditions under which each of these two effects can exist are found and discussed.

© Optical Society of America

1. M.V. Fedorov and A.M. Movsesian, J. Phys. B **21**, L155 (1988). [CrossRef]

*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.

_{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

*V*

_{a,b}= 〈φ

_{a}- ∣

**∣ φ**

*d*ε_{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.

*n*

_{0}and

*n*

_{0}are large and significantly different from each other:

*E*

_{g},

_{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 ε

^{2}, 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 < ε

_{3}where ε

_{3}is determined as the solution of the equation

_{1}and ε

_{2}, or critical values of the parameter

*V*,

*V*

_{1}and

*V*

_{2}, shown in Fig. 2, are determined as the solutions of equations

_{1}≪ ε

_{2}≪ ε

_{3}and

*V*

_{1}≪

*V*= 1 ≪

*V*

_{3}.

*V*-type transitions completely ignored) occurs at

*V*> 1 or ε > ε

^{2}. The estimates given above show that

*V*-type transitions may give rise to non-perturbative effects (including IS) in much weaker fields, ε > ε

_{1}≪ ε

_{2}, 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

*E*

_{n}∣, then

*ñ*

_{0}~

*n*

_{0}and ε

_{1}~ ε

_{2}~ ε

_{3}or

*V*

_{1}~

*V*

_{3}~ 1. In this case, practically, there is no gap between ε

_{1}and ε

_{3}where effects arising from resonance interaction between Rydberg levels could be seen and nothing except the Λ-type IS is expected to occur.

_{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:

_{bound}is time-dependent, and the probability of ionization is determined as

*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 ~ ε

_{1}much smaller than in the case of Λ-type stabilization (ε ~ ε

_{2}). 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

*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:

*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

*V*-channel becomes negligible. By varying the detuning δ, one can provide a smooth transition from the

*V*-type IS (when ∣δ∣ ≪

*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}.

*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:

*t*approaches unity. Hence, the third condition under which the

*V*-type IS at rather low fields can be observed has the form

*n*

_{0}≫

*ñ*

_{0}and, hence, the pulse duration has to be much larger than the classical Kepler period

*T*

_{k}= 2

*V*-type IS to be observable.

*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 ~ ε

_{1}(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 ~ ε

_{1}).

1. M.V. Fedorov and A.M. Movsesian, J. Phys. B **21**, L155 (1988). [CrossRef]

*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 ≈ 8.10

^{14}

*s*

^{-1},

*t*> 15 ps, and ε ≥ 10

^{6}V/cm (

*I*≥ 10

^{9}W/cm

^{2}).

*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.

*V*< 1 [1

1. M.V. Fedorov and A.M. Movsesian, J. Phys. B **21**, L155 (1988). [CrossRef]

*V*> 1, in the model of essential states, the ATI is known to only renormalization effect (

*V*→ (

*V*/

*π*)

^{1/2}[1

1. M.V. Fedorov and A.M. Movsesian, J. Phys. B **21**, L155 (1988). [CrossRef]

## References and links

1. | M.V. Fedorov and A.M. Movsesian, J. Phys. B |

2. | M.V. Fedorov, Com. At. Mol. Phys. |

3. | L. Roso-Franco, G. Orriols, and J.H. Eberly, Laser Phys. |

4. | L.D. Noordam, H. Stapelfeldt, and D.I. Duncan, Phys. Rev. Lett. |

5. | A. Wojcik and R. Parzinski, Phys. Rev. A |

6. | Ivanov M.Yu, Phys. Rev. A |

7. | J.H. Hoogenraad, R.B. Vrijen, and L.D. Noordam, Phys. Rev. A |

8. | A. Wojcik and R. ParzinskiJ. Opt. Soc. Am. B 12 , |

9. | R.B. Vrijen, J.H. Hoogenraad, and L.D. NoordamPhys. Rev. A |

10. | M.V. Fedorov and N.P. Poluektov, Laser Phys. |

11. | I.Ya. Bersons, JETP |

12. | N.B. Delone, S.P. Goreslavsky, and K.P. Krainov, J. Phys. B. |

13. | M.S. Adams, M.V. Fedorov, V.P. Krainov, and D.D. Meyerhofer, Phys. Rev. A |

14. | Ya.B. Zeldovich, JETP |

15. | V.I. Ritus, JETP |

**OCIS Codes**

(020.5780) Atomic and molecular physics : Rydberg states

(140.3550) Lasers and laser optics : Lasers, Raman

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 12, 1997

Revised Manuscript: November 7, 1997

Published: January 19, 1998

**Citation**

Mikhail Fedorov and Nikolay Poluektov, "Competition between L- and V-type transitions in interference stabilization of Rydberg atoms," Opt. Express **2**, 51-57 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-2-51

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### References

- M.V. Fedorov and A.M. Movsesian, J. Phys. B 21, L155 (1988). [CrossRef]
- M.V. Fedorov, Com. At. Mol. Phys. 27, 203 (1992).
- L. Roso-Franco, G. Orriols, and J.H. Eberly, Laser Phys. 2, 741 (1992).
- L.D. Noordam, H. Stapelfeldt, and D.I. Duncan, Phys. Rev. Lett. 68, 1496 (1992). [CrossRef] [PubMed]
- A. Wojcik and R. Parzinski, Phys. Rev. A 50, 2475 (1994). [CrossRef] [PubMed]
- M. Yu. Ivanov, Phys. Rev. A 49, 1165 (1994). [CrossRef]
- J.H. Hoogenraad, R.B. Vrijen, and L.D. Noordam, Phys. Rev. A 50, 4133 (1994). [CrossRef] [PubMed]
- A. Wojcik and R. Parzinski, J. Opt. Soc. Am. B 12, 3, 369 (1995). [CrossRef]
- R.B. Vrijen, J.H. Hoogenraad, and L.D. Noordam Phys. Rev. A 52, 2279 (1995). [CrossRef] [PubMed]
- M.V. Fedorov and N.P. Poluektov, Laser Phys. 7, 299 (1997).
- I. Ya. Bersons, JETP 53, 891 (1981).
- N.B. Delone, S.P. Goreslavsky, and V.P. Krainov, J. Phys. B. 22, 2941 (1989). [CrossRef]
- M.S. Adams, M.V. Fedorov, V.P. Krainov, and D.D. Meyerhofer, Phys. Rev. A 52, 125 (1995). [CrossRef] [PubMed]
- Ya.B. Zeldovich, JETP 24, 1006 (1967).
- V.I. Ritus, JETP 24, 1041 (1967).

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