## Dependence on frequency of strong-field atomic stabilization

Optics Express, Vol. 8, Issue 2, pp. 99-105 (2001)

http://dx.doi.org/10.1364/OE.8.000099

Acrobat PDF (99 KB)

### Abstract

It is shown that strong-field atomic stabilization can occur at any frequency, that analytical methods exist that can describe all essential features of stabilization, that relativistic effects enhance the stabilization phenomenon, and that a simple physical picture exists that explains these properties. A necessary prelude is to show that the frequency properties of the three methods often conjoined by the KFR (Keldysh-Faisal-Reiss) label are quite different. Applicability of the SFA (Strong-Field Approximation) to stabilization at any frequency is shown, and verified by exhibiting close correspondence to numerical predictions by Popov et al. that also span both low and high frequencies.

© Optical Society of America

## 1. Introduction

2. H. R. Reiss, “Atomic transitions in intense fields and the breakdown of perturbation theory,” Phys. Rev. Lett. **25**, 1149–1151 (1970). [CrossRef]

3. H. R. Reiss, “Nuclear beta decay induced by intense electromagnetic fields: forbidden transition examples,” Phys. Rev. C **27**, 1229–1243 (1983). [CrossRef]

5. F. H. M. Faisal, “Multiple absorption of laser photons by atoms,” J. Phys. B **6**, L89–L92 (1973). [CrossRef]

6. H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A **22**, 1786–1813 (1980). [CrossRef]

*ω*>>

*E*

_{B}, where

*ω*is the frequency of the laser field and

*E*

_{B}is the binding energy of the atom. Low frequency refers to ω<<

*E*

_{B}. Atomic units are used.

## 2. Fundamental transition amplitudes

*i*and

*f*will identify initial and final states. The overlap of the fully evolving state Ψ

_{i}onto some particular final non-interacting state Φ

_{f}gives the quantum probability amplitude that that final state will occur. One can write, using Eq. (3) for the second element below,

## 3. Strong-field analytical approximations

### 3.1 Keldysh approximation

### 3.2 Faisal approximation

5. F. H. M. Faisal, “Multiple absorption of laser photons by atoms,” J. Phys. B **6**, L89–L92 (1973). [CrossRef]

7. W. C. Henneberger, “Perturbation method for atoms in intense light beams,” Phys. Rev. Lett. **21**, 838–841 (1968). [CrossRef]

*U*

^{KH}(actually the inverse transformation) is applied to the initial state, giving

*α*⃗(t) is the classical displacement of the atomic electron from its center of oscillation in the field of the laser, treated here in the dipole approximation. The next step is to neglect this displacement,

### 3.3 Strong-field approximation (SFA)

6. H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A **22**, 1786–1813 (1980). [CrossRef]

8. H. R. Reiss, “Energetic electrons in strong-field ionization”, Phys. Rev. A **54**, R1765–R1768 (1996). [CrossRef] [PubMed]

9. M. Pont and M. Gavrila, “Stabilization of atomic hydrogen in superintense, high-frequency laser fields of circular polarization,” Phys. Rev. Lett. **65**, 2362–2365 (1990). [CrossRef] [PubMed]

10. H. R. Reiss, “High-frequency, high-intensity photoionization,” J. Opt. Soc. Am. B **13**, 355–362 (1996). [CrossRef]

## 4. Strong-field stabilization

### 4.1 Transition rates

*I*

_{stab}. Matters are less clear for an actual experiment, since ionization will saturate before

*I*

_{stab}is reached. Laboratory observation of stabilization will be considered elsewhere.

11. D. P. Crawford and H. R. Reiss, “Stabilization in relativistic photoionization with circularly polarized light,” Phys. Rev. A **50**, 1844–1850 (1994). [CrossRef] [PubMed]

12. H. R. Reiss, “Relativistic strong-field photoionization,” J. Opt. Soc. Am. B **7**, 574–586 (1990). [CrossRef]

^{†}γ

^{0}the γ

^{µ}are the Dirac gamma matrices; µ=0,1,2,3;

*A*

^{µ}is the four-vector potential of the laser field; and a time-favoring real metric is used. Under relativistic conditions, the neglect of Coulomb effects on the ionized electron is a well-justified approximation. One then uses the Dirac Volkov solution for the final-state wave function, uses known solutions (or superpositions thereof) to the hydrogen-atom Dirac equation for the initial-state wave function, and obtains thereby a completely relativistic Dirac transition amplitude. Details of this procedure are given in Ref. [12

12. H. R. Reiss, “Relativistic strong-field photoionization,” J. Opt. Soc. Am. B **7**, 574–586 (1990). [CrossRef]

11. D. P. Crawford and H. R. Reiss, “Stabilization in relativistic photoionization with circularly polarized light,” Phys. Rev. A **50**, 1844–1850 (1994). [CrossRef] [PubMed]

*I*

_{stab}is approximately the same for both circular and linear polarizations.

### 4.2 Comparison with alternative predictions of Istab

*I*

_{stab}have been calculated (by a method other than the SFA) for frequencies encompassing both the low and high frequency domains. Popov et al. [14

14. A. M. Popov, O. V. Tikhonova, and E. A. Volkova, “Applicability of the Kramers-Henneberger approximation in the theory of strong-field ionization,” J. Phys. B **32**, 3331–3345 (1999). [CrossRef]

*V*, or 0.0276

*a.u*. Theoretical studies of stabilization show that

*I*

_{stab}∝

*ω*

^{3}for high frequencies [15

15. H. R. Reiss, “Frequency and polarization effects in stabilization,” Phys. Rev. A **46**, 391–394 (1992). [CrossRef] [PubMed]

*I*

_{stab}∝

14. A. M. Popov, O. V. Tikhonova, and E. A. Volkova, “Applicability of the Kramers-Henneberger approximation in the theory of strong-field ionization,” J. Phys. B **32**, 3331–3345 (1999). [CrossRef]

*U*

_{p}>>

*E*

_{B}, where

*U*

_{p}is the ponderomotive energy. The values of

*I*

_{stab}predicted by the SFA in the neighborhood of ω=

*E*

_{B}do not satisfy that constraint. Only those stabilization intensities are plotted in Fig. 2 that satisfy SFA validity conditions, and hence a gap occurs. Plainly, the SFA and the results of Ref. [14

14. A. M. Popov, O. V. Tikhonova, and E. A. Volkova, “Applicability of the Kramers-Henneberger approximation in the theory of strong-field ionization,” J. Phys. B **32**, 3331–3345 (1999). [CrossRef]

### 4.3 Physical explanation for stabilization

*f*is approximately just

*t*) at any

*t*. Consider the nonrelativistic circular polarization case. The initial atom can be viewed as a spherical Bohr atom, and the final photoelectron will be in a circular orbit around the atom with essentially the classical energy and angular momentum of a free electron in the laser field. The cylindrical symmetry of the final orbit is imposed by the symmetry of the problem, and the number of photons absorbed to provide the initially bound electron with its final energy in the laser field is also appropriate to provide the final angular momentum. This is illustrated by the sketch in Fig. 3(a). As the field intensity increases, the classical radius of the orbit grows much larger than the size of the atom, the wave function overlap decreases, and stabilization sets in.

### 4.4 Relativistic effects on stabilization

**Φ**

_{i}and

**Ψ**

_{f}leading to the reduced relativistic rate seen in Fig. 1.

17. N. J. Kylstra, R. A. Worthington, A. Patel, P. L. Knight, J. R. V·zquez de Aldana, and L. Roso, “Breakdown of stabilization of atoms with intense, high-frequency laser pulses,” Phys. Rev. Lett. **85**, 1835–1838 (2000). [CrossRef] [PubMed]

18. H. R. Reiss, “Dipole-approximation magnetic fields in strong laser beams,” Phys. Rev. A **63**, 013409 (2001). [CrossRef]

11. D. P. Crawford and H. R. Reiss, “Stabilization in relativistic photoionization with circularly polarized light,” Phys. Rev. A **50**, 1844–1850 (1994). [CrossRef] [PubMed]

*I*

_{stab}) constraint of Eq. (14). Finally, we remark that the qualitative reasoning employed above and illustrated in Fig. 3 follows from the exact expression in Eq. (6), and is independent of any specific calculational method. Basically, Eq. (6) tells us that because the wave function overlap between the initial atom and a final relativistic free electron is reduced from the nonrelativistic situation, then as intensity increases the atom is denied an increasing portion of the probability of making a transition to an ionized state, and is thus more strongly stabilized relativistically.

## References and Links

1. | M. Gavrila, “Atomic structure and decay in high frequency fields,” in |

2. | H. R. Reiss, “Atomic transitions in intense fields and the breakdown of perturbation theory,” Phys. Rev. Lett. |

3. | H. R. Reiss, “Nuclear beta decay induced by intense electromagnetic fields: forbidden transition examples,” Phys. Rev. C |

4. | L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Zh. Eksp. Teor. Fiz.47, 1945–1957 (1964) [Sov. Phys. JETP20, 1307–1314 (1965)]. |

5. | F. H. M. Faisal, “Multiple absorption of laser photons by atoms,” J. Phys. B |

6. | H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A |

7. | W. C. Henneberger, “Perturbation method for atoms in intense light beams,” Phys. Rev. Lett. |

8. | H. R. Reiss, “Energetic electrons in strong-field ionization”, Phys. Rev. A |

9. | M. Pont and M. Gavrila, “Stabilization of atomic hydrogen in superintense, high-frequency laser fields of circular polarization,” Phys. Rev. Lett. |

10. | H. R. Reiss, “High-frequency, high-intensity photoionization,” J. Opt. Soc. Am. B |

11. | D. P. Crawford and H. R. Reiss, “Stabilization in relativistic photoionization with circularly polarized light,” Phys. Rev. A |

12. | H. R. Reiss, “Relativistic strong-field photoionization,” J. Opt. Soc. Am. B |

13. | H. R. Reiss and D. P. Crawford, “Relativistic photoionization,” in |

14. | A. M. Popov, O. V. Tikhonova, and E. A. Volkova, “Applicability of the Kramers-Henneberger approximation in the theory of strong-field ionization,” J. Phys. B |

15. | H. R. Reiss, “Frequency and polarization effects in stabilization,” Phys. Rev. A |

16. | H. R. Reiss, “Physical basis for strong-field stabilization of atoms against ionization,” Las. Phys. |

17. | N. J. Kylstra, R. A. Worthington, A. Patel, P. L. Knight, J. R. V·zquez de Aldana, and L. Roso, “Breakdown of stabilization of atoms with intense, high-frequency laser pulses,” Phys. Rev. Lett. |

18. | H. R. Reiss, “Dipole-approximation magnetic fields in strong laser beams,” Phys. Rev. A |

**OCIS Codes**

(020.4180) Atomic and molecular physics : Multiphoton processes

(270.4180) Quantum optics : Multiphoton processes

(270.6620) Quantum optics : Strong-field processes

**ToC Category:**

Focus Issue: Quantum control of photons and matter

**History**

Original Manuscript: November 9, 2000

Published: January 15, 2001

**Citation**

Howard Reiss, "Dependence on frequency of strong-field atomic stabilization," Opt. Express **8**, 99-105 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-99

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### References

- M. Gavrila, "Atomic structure and decay in high frequency fields," in Atoms in intense laser fields, M. Gavrila, ed. (Academic, Boston, MA 1992).
- H. R. Reiss, "Atomic transitions in intense fields and the breakdown of perturbation theory," Phys. Rev. Lett. 25, 1149-1151 (1970). [CrossRef]
- H. R. Reiss, "Nuclear beta decay induced by intense electromagnetic fields: forbidden transition examples," Phys. Rev. C 27, 1229-1243 (1983). [CrossRef]
- L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Zh. Eksp. Teor. Fiz. 47, 1945-1957 (1964) [Sov. Phys. JETP 20, 1307-1314 (1965)].
- F. H. M. Faisal, "Multiple absorption of laser photons by atoms," J. Phys. B 6, L89-L92 (1973). [CrossRef]
- H. R. Reiss, "Effect of an intense electromagnetic field on a weakly bound system," Phys. Rev. A 22, 1786-1813 (1980). [CrossRef]
- W. C. Henneberger, "Perturbation method for atoms in intense light beams," Phys. Rev. Lett. 21, 838-841 (1968). [CrossRef]
- H. R. Reiss, "Energetic electrons in strong-field ionization," Phys. Rev. A 54, R1765-R1768 (1996). [CrossRef] [PubMed]
- M. Pont and M. Gavrila, "Stabilization of atomic hydrogen in superintense, high-frequency laser fields of circular polarization," Phys. Rev. Lett. 65, 2362-2365 (1990). [CrossRef] [PubMed]
- H. R. Reiss, "High-frequency, high-intensity photoionization," J. Opt. Soc. Am. B 13, 355-362 (1996). [CrossRef]
- D. P. Crawford and H. R. Reiss, "Stabilization in relativistic photoionization with circularly polarized light," Phys. Rev. A 50, 1844-1850 (1994). [CrossRef] [PubMed]
- H. R. Reiss, "Relativistic strong-field photoionization," J. Opt. Soc. Am. B 7, 574-586 (1990). [CrossRef]
- H. R. Reiss and D. P. Crawford, "Relativistic photoionization," in Ultrafast phenomena and interaction of superstrong laser fields with matter: nonlinear optics and high-field physics, M. V.Fedorov, V. M. Gordienko, V. V. Shuvalov, V. D. Taranukhin, eds., Proc. SPIE 3735, 148-157 (1998).
- A. M. Popov, O. V. Tikhonova, and E. A. Volkova, "Applicability of the Kramers-Henneberger approximation in the theory of strong-field ionization," J. Phys. B 32, 3331-3345 (1999). [CrossRef]
- H. R. Reiss, "Frequency and polarization effects in stabilization," Phys. Rev. A 46, 391-394 (1992). [CrossRef] [PubMed]
- H. R. Reiss, "Physical basis for strong-field stabilization of atoms against ionization," Las. Phys. 7, 543-550 (1997).
- N. J. Kylstra, R. A. Worthington, A. Patel, P. L. Knight, J. R. V zquez de Aldana, L. Roso, "Breakdown of stabilization of atoms with intense, high-frequency laser pulses," Phys. Rev. Lett. 85, 1835-1838 (2000). [CrossRef] [PubMed]
- H. R. Reiss, "Dipole-approximation magnetic fields in strong laser beams," Phys. Rev. A 63, 013409 (2001). [CrossRef]

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