## Coherent control of strong-field two-pulse ionization of Rydberg atoms

Optics Express, Vol. 6, Issue 5, pp. 117-123 (2000)

http://dx.doi.org/10.1364/OE.6.000117

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### Abstract

Strong-field ionization of Rydberg atoms is investigated in its dependence on
phase features of the initial coherent population of Rydberg levels. In the case
of a resonance between Rydberg levels and some lower-energy atomic level
(*V*-type transitions), this dependence is shown to be very
strong: by a proper choice of the initial population an atom can be made either
completely or very little ionized by a strong laser pulse. It is shown that
phase features of the initial coherent population of Rydberg levels and the
ionization yield can be efficiently controlled in a scheme of ionization by two
strong laser pulses with a varying delay time between them.

© Optical Society of America

*H*

_{0}is the field-free atomic Hamiltonian,

*V*(

*t*)=-

*(*

**F***t*)·

*is the dipole interaction energy,*

**r***(*

**F***t*)=

**F**_{0}cos(ω

*t*) is the electric field strength of a linearly polarized light wave (

**F**_{0}and ω are its filed-strength amplitude and frequency, respectively); here and everywhere below atomic units

*ħ*=

*m*=|

*e*|=1 are used.

*), lower-energy resonance atomic states (φ*

_{n}*) and continuum (φ*

_{m}*):*

_{E}*a*(

_{n}*t*) and

*a*(

_{m}*t*) only:

_{m,n}=

*V*

_{m,n}/2 is the tensor of Rabi frequencies and Γ

_{n,n′}=(π/2)

*V*

_{nE}V_{En}_{′}is the tensor of ionization widths;

*V*

_{α,β}=-

*·*

**F**

**r**_{α,β}and

*r*

_{α,β}=〈

*α*|

*|β〉 are the dipole matrix elements. By definition, components of the tensor of Rabi frequencies Ω*

**r**_{m,n}are proportional to the square root of laser intensity √

*I*, whereas components of the tensor of ionization widths Γ

_{n, n′}are proportional to

*I*. For this reason, up to very high intensities |Ω

_{m,n}|≫Γ

_{n,n′}[9

9. N.P. Poluektov and M.V. Fedorov, “Stabilization of a Rydberg atom and competition between the Λ and V transition channels”, JETP , **87**, 445 (1998). [CrossRef]

*n*(

*n*≫1) Rydberg levels are almost equidistant,

*E*

_{n+1}-

*E*≡Δ≈

_{n}*n*

^{-3}≈const., and matrix elements Ω

_{n,m}and Γ

_{n,n′}can be approximated [7–10, 12

12. N.B. Delone, S.P. Goreslavsky, and V.P. Krainov, “Quasiclassical dipole matrix elements for atomic continuum states”, J. Phys. B **22**, 2941 (1989). [CrossRef]

*n*-,

*n*

^{′}- and

*m*-independent constants Ω

*and Γ, correspondingly. Let us assume also that the light frequency ω is larger than binding energy |*

_{R}*E*| of all the initially populated Rydberg levels, where

_{n}*E*=-1/2

_{n}*n*

^{2}. Under this condition, inevitably, principal quantum numbers

*m*of lower-energy resonance levels are relatively small,

*m*≪

*n*(though we assume that

*m*≫1), and spacing between neighboring levels

*E*is relatively large,

_{m}*E*

_{m+1}-

*E*≈

_{m}*m*

^{-3}≫

*E*

_{n+1}-

*E*.≈

_{n}*n*

^{-3}. For this reason, if the Rabi frequency Ω

*obeys the conditions*

_{R}*m*

^{-3}≫Ω

*≥*

_{R}*n*

^{-3}, and if one of the levels

*a*(

_{m}*t*) except

*n*

_{0}is the principal quantum number of the level

*n*, for

*n*=

*n*

_{0},

*a*(0).

_{n}*b*and

_{n}*b*̃ are constants and

*γ*is the quasienergy to be found from equations

*n*

_{0}in the difference

*n*-

*n*

_{0}. In these notations,

*n*can take both positive and negative integer values and the “closest to resonance level”

*n*=0.

*γ*are eigenvalues of the set of equations (7) (j=0, ±1, ±2, …), the solution of the initial-value problem for Eqs. (5) can be presented in the form of a superposition

_{j}*C*

_{n,j}are constant and obey the same algebraic equations as the constants

*b*̃ and

*b*[Eqs. (7) with

_{n}*γ*=

*γ*] plus equations identical to (6) (initial conditions). Solutions of these equations can be found in a general form. In particular, from Eqs. (7) one can find the equation for quasienergies

_{j}*γ*

_{j}*a*(

_{n}*t*) can be shown to have the form

*and Γ [7, 12*

_{R}12. N.B. Delone, S.P. Goreslavsky, and V.P. Krainov, “Quasiclassical dipole matrix elements for atomic continuum states”, J. Phys. B **22**, 2941 (1989). [CrossRef]

*E*and

_{n}*E*, Ω

_{n}*≫Γ, and the strong-field criterion has the form Ω*

_{R}*≫Δ. In such a case, for strong fields, expansion in powers of Δ/Ω*

_{R}*and Γ/Ω*

_{R}*can be used to solve Eq.(9) approximately, and the solutions are given by*

_{R}*j*

_{0}-th quasienergy has a zero width,

*F*

_{0}(limited only by the applicability conditions of the used equations (4), (5)). Position of this stable quasienergy level coincides with the resonance detuning,

*j*

_{0}=0(δ=Δ/2). From Eqs. (3) and (10)–(12) we can find in this case a rather simple formula for the long-time limit of the ionization probability. The “long-time limit” means that all the quasienergy levels

*γ*with

_{j}*j*≠

*j*

_{0}are assumed to decay (to be ionized) completely and all the remaining bound-state population is concentrated at the stable quasienergy level

*γ*)|τ≫1 for

_{j}*j*≠

*j*

_{0}, has the form

*T*=2π/Δ=2π

_{K}*n*

^{3}is the classical Kepler period. Under these assumptions the long-time limit of the ionization probability is given by

*a*(0)=1 for

_{n}*n*=0 and 0 for

*n*≠0) Eq. (15) coincides with one of the results of Ref. [6

6. A. Wojcik and R. Parzinski, “Dark-state effect in Rydberg-atom stabilization” J.Opt.Soc.Am.B , **12**, 369 (1995). [CrossRef]

*n*=½ (i.e., if

*a*

_{-n}=

*a*

_{n+1}for

*n*=0, 1, 2, …) the sum on the right-hand side of Eq. (15) turns zero and

*w*(

_{ion}*τ*→∞)=1, i.e., in the long-time limit, ionization of an atom is complete. This is the case when the above-mentioned stable quasienergy state is not populated at all. For any other distributions of the initial probability amplitudes

*a*(0), population of the strong-field stable quasienergy state is different from zero and

_{n}*w*(

_{ion}*τ*→∞)≠1. In a general case, for a given realization of

*a*(0), in dependence on a growing field strength amplitude

_{n}*F*

_{0}, the probability of ionization

*w*(τ→∞,

_{ion}*F*

_{0}) falls and tends to its asymptotic value when

*F*

_{0}→∞,

*w*(

_{ion}*τ*→∞,

*F*

_{0}→∞)>0. However, in a special case when

*w*(

_{ion}*τ*→∞,

*F*

_{0}→∞)=0, i.e., asymptotically, an atom appears to be absolutely stable. This special case corresponds to a choice of the initial probability amplitudes

*a*(0) coinciding with amplitudes

_{n}*a*of the strong-field stable quasienergy state (10). For δ=Δ/2 Eqs. (10) take the form

_{n}*F*

_{0}→∞, this yields

*~Δ,*

_{R}*w*(

_{ion}*τ*→∞) (20) does not exceed 10%. This indicates a rather high degree of stabilization that occurs at moderate fields, if only the initial state of an atom is determined by Eqs. (19).

*F*

_{0}is characterized by the well known quasiclassical parameter

*V*=

*F*

_{0}/ω

^{5/3}[10, 12

12. N.B. Delone, S.P. Goreslavsky, and V.P. Krainov, “Quasiclassical dipole matrix elements for atomic continuum states”, J. Phys. B **22**, 2941 (1989). [CrossRef]

*δ*, which is assumed to obey the condition (13) of an exact resonance between the level

*E*+

_{n}*E*

_{n+1}) for some

*n*. The question is how strictly this assumption should be fulfilled? To answer this question, we have calculated numerically the probability of ionization

*w*vs.

_{ion}*δ*. The results of calculations (Fig. 3) show that in the limit of a strong field the dependence

*w*(

_{ion}*δ*) appears to be rather smooth and, hence, stabilization of an atom due to a proper choice of the initial coherent population can occur in a wide range of values of the detuning

*δ*.

*τ*. If an atom is excited initially to some Rydberg level

_{d}*E*. A mechanism of repopulation consists of Raman-type transitions via lower-energy resonance level

_{n}*a*(18). During the time between the first and second pulses, phases of these Rydberg states evolve in accordance with the law exp(-

_{n}*iE*), and at the initial probability amplitudes for the second pulse appear to be

_{n}t*τ*is equal to

_{d}*s*

*T*, where

_{K}*s*is an integer, the phase distribution of Eq. (21) repeats that of (18) and, hence, the probability of ionization of an atom by the second pulse is expected to be close to zero. If, however,

*τ*=(

_{d}*s*+½)

*T*, the probability of ionization by the second pulse is expected to be close to one. Such a periodical dependence of the probability of ionization on the delay time

_{K}*τ*is confirmed by the results of numerical calculations shown in Fig. 4. A specific experimental scheme for observation of such an effect can be similar to that of Ref. [13

_{d}13. M.B. Campbell, T.J. Bensky, and R.R. Jones, “Single-shot detection of wavepacket evolution”, Optics Express , **1**, 197 (1997). http://www.opticsexpress.org/oearchive/source/2217.htm [CrossRef] [PubMed]

*w*(

_{ion}*τ*) is a specific feature of a system with equidistant spectrum of levels. Rydberg atoms only approximately satisfy this condition, and the energy spectrum is not purely equidistant. This difference gives rise to complications and deviations of the dependence

_{d}*w*(

_{ion}*τ*) from a purely periodical one. However, it is possible to find such combinations of quantum numbers

_{d}*m*

_{0}and

*n*

_{0}and laser pulse parameters, for which the dependence

*w*(

_{ion}*τ*) is almost periodical (see Fig. 5). An example of parameters, corresponding to the picture of Fig. 5, is:

_{d}*n*

_{0}=25, ω=1,7×10

^{14}sec

^{-1},

*F*

_{0}=6×10

^{5}V/cm (

*I*=1×10

^{9}W/cm

^{2}), τ=50·

*T*=120 ps.

_{K}## 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. | L. Noordam, H. Stapelfeldt, D.I. Duncan, and T.F. Gallagher, “Redistribution of Rydberg States by Intense Picosecond Pulses”, Phys. Rev. Lett. |

3. | J.H. Hoogenraad, R.B. Vrijen, and L.D. Noordam, “Ionization suppression of Rydberg atoms by short laser pulses” Phys. Rev. A |

4. | M.Yu. Ivanov, “Suppression of resonant multiphoton ionization via Rydberg states” Phys.Rev.A , |

5. | A. Wojcik and R. Parzinski, “Rydberg-atom stabilization against photoionization: an analitically solvable model with resonance” Phys. Rev. A , |

6. | A. Wojcik and R. Parzinski, “Dark-state effect in Rydberg-atom stabilization” J.Opt.Soc.Am.B , |

7. | M.V. Fedorov and N.P. Poluektov, “Λ- and V-Type Transitions and Their Role in the Interference Stabilization of Rydberg Atoms”, Laser Physics , |

8. | M.V. Fedorov and N.P. Poluektov, “Competition between Λ- and V-type transitions in interference stabilization of Rydberg atoms”, Optics Express , |

9. | N.P. Poluektov and M.V. Fedorov, “Stabilization of a Rydberg atom and competition between the Λ and V transition channels”, JETP , |

10. | M.V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific: Singapore, 1997. |

11. | D.I. Duncan and R.R. Jones, “Interferometric characterization of Raman redistribution among perturbed Rydberg states of barium”, Phys. Rev. A , |

12. | N.B. Delone, S.P. Goreslavsky, and V.P. Krainov, “Quasiclassical dipole matrix elements for atomic continuum states”, J. Phys. B |

13. | M.B. Campbell, T.J. Bensky, and R.R. Jones, “Single-shot detection of wavepacket evolution”, Optics Express , |

**OCIS Codes**

(000.6800) General : Theoretical physics

(020.5780) Atomic and molecular physics : Rydberg states

(140.3550) Lasers and laser optics : Lasers, Raman

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 11, 2000

Published: February 28, 2000

**Citation**

Mikhail Fedorov and Nikolay Poluektov, "Coherent control of strong-field two-pulse ionization of Rydberg atoms," Opt. Express **6**, 117-123 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-6-5-117

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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]
- L. Noordam, H. Stapelfeldt, D.I. Duncan, and T.F. Gallagher, "Redistribution of Rydberg States by Intense Picosecond Pulses," Phys. Rev. Lett. 68, 1496 (1992). [CrossRef] [PubMed]
- J.H. 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.Yu. Ivanov, "Suppression of resonant multiphoton ionization via Rydberg states," Phys. Rev. A, 49, 1165 (1994). [CrossRef] [PubMed]
- A. Wojcik and R. Parzinski, "Rydberg-atom stabilization against photoionization: an analitically solvable model with resonance," Phys. Rev. A, 50, 2475 (1994). [CrossRef] [PubMed]
- A. Wojcik and R. Parzinski, "Dark-state effect in Rydberg-atom stabilization," J. Opt. Soc. Am. B, 12, 369 (1995). [CrossRef]
- M.V. Fedorov and N.P. Poluektov, "_- and V-Type Transitions and Their Role in the Interference Stabilization of Rydberg Atoms," Laser Physics, 7, 299 (1997).
- M.V. Fedorov and N.P. Poluektov, "Competition between _- and V-type transitions in interference stabilization of Rydberg atoms," Opt. Express, 2, 51 (1998). http://www.opticsexpress.org/oearchive/source/2982.htm [CrossRef] [PubMed]
- N.P. Poluektov and M.V. Fedorov, "Stabilization of a Rydberg atom and competition between the _ and V transition channels," JETP, 87, 445 (1998). [CrossRef]
- M.V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific: Singapore, 1997.
- D.I. Duncan and R.R. Jones, "Interferometric characterization of Raman redistribution among perturbed Rydberg states of barium," Phys. Rev. A, 53, 4338 (1996). [CrossRef] [PubMed]
- N.B. Delone, S.P. Goreslavsky, and V.P. Krainov, "Quasiclassical dipole matrix elements for atomic continuum states," J. Phys. B 22, 2941 (1989). [CrossRef]
- M.B. Campbell, T.J. Bensky, and R.R. Jones, "Single-shot detection of wavepacket evolution," Opt. Express, 1, 197 (1997). http://www.opticsexpress.org/oearchive/source/2217.htm [CrossRef] [PubMed]

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