Abstract

The time evolution of an atomic electron wave packet created by coherent superposition of Stark states in an electric field is calculated. If the electric field is modulated, the width and the oscillation time of the wave packet are not constant. It is shown that the resulting modulation of the atomic wave packet is analogous to the modulation that occurs for mode-locked laser pulses circulating in a modulated cavity.

© 1991 Optical Society of America

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References

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  1. C. V. Shank, R. L. Fork, R. Yen, and R. H. Stolen, Appl. Phys. Lett 40, 761 (1982); C. Rolland and P. B. Corkum, J. Opt. Soc. Am. B 5, 641 (1988).
    [Crossref]
  2. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer-Verlag, Berlin, 1957).
    [Crossref]
  3. A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 485 (1989); L. D. Noordam, A. ten Wolde, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 6999 (1989).
    [Crossref] [PubMed]
  4. L. D. Noordam, H. G. Muller, A. ten Wolde, and H. B. van Linden van den Heuvell, J. Phys. B 23, L115 (1990).
    [Crossref]
  5. E. Schrödinger, Naturwissenschaften 28, 664 (1926).
    [Crossref]
  6. M. J. Rosker, T. S. Rose, and A. H. Zewail, Chem. Phys. Lett. 146, 175 (1988).
    [Crossref]
  7. J. Parker and C. R. Stroud, Phys. Rev. Lett. 56, 716 (1986).
    [Crossref] [PubMed]
  8. G. Alber, H. Ritsch, and P. Zoller, Phys. Rev. A 34, 1058 (1986).
    [Crossref] [PubMed]
  9. A. ten Wolde, L. D. Noordam, H. G. Muller, and H. B. van Linden van den Heuvell, in Fundamentals of Laser Interactions II, Vol. 339 of Lecture Notes on Physics, F. Ehlotzky, ed. (Springer-Verlag, Berlin, 1989).
  10. A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. Lett. 61, 2099 (1988).
    [Crossref] [PubMed]
  11. J. A. Yeazell, M. Mallalieu, J. Parker, and C. R. Stroud, Phys. Rev. A 40, 5040 (1989).
    [Crossref] [PubMed]
  12. R. R. Alfano and P. P. Ho, IEEE J. Quantum Electron. 24, 351 (1988).
    [Crossref]

1990 (1)

L. D. Noordam, H. G. Muller, A. ten Wolde, and H. B. van Linden van den Heuvell, J. Phys. B 23, L115 (1990).
[Crossref]

1989 (2)

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 485 (1989); L. D. Noordam, A. ten Wolde, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 6999 (1989).
[Crossref] [PubMed]

J. A. Yeazell, M. Mallalieu, J. Parker, and C. R. Stroud, Phys. Rev. A 40, 5040 (1989).
[Crossref] [PubMed]

1988 (3)

R. R. Alfano and P. P. Ho, IEEE J. Quantum Electron. 24, 351 (1988).
[Crossref]

M. J. Rosker, T. S. Rose, and A. H. Zewail, Chem. Phys. Lett. 146, 175 (1988).
[Crossref]

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. Lett. 61, 2099 (1988).
[Crossref] [PubMed]

1986 (2)

J. Parker and C. R. Stroud, Phys. Rev. Lett. 56, 716 (1986).
[Crossref] [PubMed]

G. Alber, H. Ritsch, and P. Zoller, Phys. Rev. A 34, 1058 (1986).
[Crossref] [PubMed]

1982 (1)

C. V. Shank, R. L. Fork, R. Yen, and R. H. Stolen, Appl. Phys. Lett 40, 761 (1982); C. Rolland and P. B. Corkum, J. Opt. Soc. Am. B 5, 641 (1988).
[Crossref]

1926 (1)

E. Schrödinger, Naturwissenschaften 28, 664 (1926).
[Crossref]

Alber, G.

G. Alber, H. Ritsch, and P. Zoller, Phys. Rev. A 34, 1058 (1986).
[Crossref] [PubMed]

Alfano, R. R.

R. R. Alfano and P. P. Ho, IEEE J. Quantum Electron. 24, 351 (1988).
[Crossref]

Bethe, H. A.

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer-Verlag, Berlin, 1957).
[Crossref]

Fork, R. L.

C. V. Shank, R. L. Fork, R. Yen, and R. H. Stolen, Appl. Phys. Lett 40, 761 (1982); C. Rolland and P. B. Corkum, J. Opt. Soc. Am. B 5, 641 (1988).
[Crossref]

Ho, P. P.

R. R. Alfano and P. P. Ho, IEEE J. Quantum Electron. 24, 351 (1988).
[Crossref]

Lagendijk, A.

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 485 (1989); L. D. Noordam, A. ten Wolde, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 6999 (1989).
[Crossref] [PubMed]

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. Lett. 61, 2099 (1988).
[Crossref] [PubMed]

Mallalieu, M.

J. A. Yeazell, M. Mallalieu, J. Parker, and C. R. Stroud, Phys. Rev. A 40, 5040 (1989).
[Crossref] [PubMed]

Muller, H. G.

L. D. Noordam, H. G. Muller, A. ten Wolde, and H. B. van Linden van den Heuvell, J. Phys. B 23, L115 (1990).
[Crossref]

A. ten Wolde, L. D. Noordam, H. G. Muller, and H. B. van Linden van den Heuvell, in Fundamentals of Laser Interactions II, Vol. 339 of Lecture Notes on Physics, F. Ehlotzky, ed. (Springer-Verlag, Berlin, 1989).

Noordam, L. D.

L. D. Noordam, H. G. Muller, A. ten Wolde, and H. B. van Linden van den Heuvell, J. Phys. B 23, L115 (1990).
[Crossref]

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 485 (1989); L. D. Noordam, A. ten Wolde, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 6999 (1989).
[Crossref] [PubMed]

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. Lett. 61, 2099 (1988).
[Crossref] [PubMed]

A. ten Wolde, L. D. Noordam, H. G. Muller, and H. B. van Linden van den Heuvell, in Fundamentals of Laser Interactions II, Vol. 339 of Lecture Notes on Physics, F. Ehlotzky, ed. (Springer-Verlag, Berlin, 1989).

Parker, J.

J. A. Yeazell, M. Mallalieu, J. Parker, and C. R. Stroud, Phys. Rev. A 40, 5040 (1989).
[Crossref] [PubMed]

J. Parker and C. R. Stroud, Phys. Rev. Lett. 56, 716 (1986).
[Crossref] [PubMed]

Ritsch, H.

G. Alber, H. Ritsch, and P. Zoller, Phys. Rev. A 34, 1058 (1986).
[Crossref] [PubMed]

Rose, T. S.

M. J. Rosker, T. S. Rose, and A. H. Zewail, Chem. Phys. Lett. 146, 175 (1988).
[Crossref]

Rosker, M. J.

M. J. Rosker, T. S. Rose, and A. H. Zewail, Chem. Phys. Lett. 146, 175 (1988).
[Crossref]

Salpeter, E. E.

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer-Verlag, Berlin, 1957).
[Crossref]

Schrödinger, E.

E. Schrödinger, Naturwissenschaften 28, 664 (1926).
[Crossref]

Shank, C. V.

C. V. Shank, R. L. Fork, R. Yen, and R. H. Stolen, Appl. Phys. Lett 40, 761 (1982); C. Rolland and P. B. Corkum, J. Opt. Soc. Am. B 5, 641 (1988).
[Crossref]

Stolen, R. H.

C. V. Shank, R. L. Fork, R. Yen, and R. H. Stolen, Appl. Phys. Lett 40, 761 (1982); C. Rolland and P. B. Corkum, J. Opt. Soc. Am. B 5, 641 (1988).
[Crossref]

Stroud, C. R.

J. A. Yeazell, M. Mallalieu, J. Parker, and C. R. Stroud, Phys. Rev. A 40, 5040 (1989).
[Crossref] [PubMed]

J. Parker and C. R. Stroud, Phys. Rev. Lett. 56, 716 (1986).
[Crossref] [PubMed]

ten Wolde, A.

L. D. Noordam, H. G. Muller, A. ten Wolde, and H. B. van Linden van den Heuvell, J. Phys. B 23, L115 (1990).
[Crossref]

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 485 (1989); L. D. Noordam, A. ten Wolde, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 6999 (1989).
[Crossref] [PubMed]

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. Lett. 61, 2099 (1988).
[Crossref] [PubMed]

A. ten Wolde, L. D. Noordam, H. G. Muller, and H. B. van Linden van den Heuvell, in Fundamentals of Laser Interactions II, Vol. 339 of Lecture Notes on Physics, F. Ehlotzky, ed. (Springer-Verlag, Berlin, 1989).

van Linden van den Heuvell, H. B.

L. D. Noordam, H. G. Muller, A. ten Wolde, and H. B. van Linden van den Heuvell, J. Phys. B 23, L115 (1990).
[Crossref]

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 485 (1989); L. D. Noordam, A. ten Wolde, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 6999 (1989).
[Crossref] [PubMed]

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. Lett. 61, 2099 (1988).
[Crossref] [PubMed]

A. ten Wolde, L. D. Noordam, H. G. Muller, and H. B. van Linden van den Heuvell, in Fundamentals of Laser Interactions II, Vol. 339 of Lecture Notes on Physics, F. Ehlotzky, ed. (Springer-Verlag, Berlin, 1989).

Yeazell, J. A.

J. A. Yeazell, M. Mallalieu, J. Parker, and C. R. Stroud, Phys. Rev. A 40, 5040 (1989).
[Crossref] [PubMed]

Yen, R.

C. V. Shank, R. L. Fork, R. Yen, and R. H. Stolen, Appl. Phys. Lett 40, 761 (1982); C. Rolland and P. B. Corkum, J. Opt. Soc. Am. B 5, 641 (1988).
[Crossref]

Zewail, A. H.

M. J. Rosker, T. S. Rose, and A. H. Zewail, Chem. Phys. Lett. 146, 175 (1988).
[Crossref]

Zoller, P.

G. Alber, H. Ritsch, and P. Zoller, Phys. Rev. A 34, 1058 (1986).
[Crossref] [PubMed]

Appl. Phys. Lett (1)

C. V. Shank, R. L. Fork, R. Yen, and R. H. Stolen, Appl. Phys. Lett 40, 761 (1982); C. Rolland and P. B. Corkum, J. Opt. Soc. Am. B 5, 641 (1988).
[Crossref]

Chem. Phys. Lett. (1)

M. J. Rosker, T. S. Rose, and A. H. Zewail, Chem. Phys. Lett. 146, 175 (1988).
[Crossref]

IEEE J. Quantum Electron. (1)

R. R. Alfano and P. P. Ho, IEEE J. Quantum Electron. 24, 351 (1988).
[Crossref]

J. Phys. B (1)

L. D. Noordam, H. G. Muller, A. ten Wolde, and H. B. van Linden van den Heuvell, J. Phys. B 23, L115 (1990).
[Crossref]

Naturwissenschaften (1)

E. Schrödinger, Naturwissenschaften 28, 664 (1926).
[Crossref]

Phys. Rev. A (3)

G. Alber, H. Ritsch, and P. Zoller, Phys. Rev. A 34, 1058 (1986).
[Crossref] [PubMed]

J. A. Yeazell, M. Mallalieu, J. Parker, and C. R. Stroud, Phys. Rev. A 40, 5040 (1989).
[Crossref] [PubMed]

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 485 (1989); L. D. Noordam, A. ten Wolde, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. A 40, 6999 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. Lett. 61, 2099 (1988).
[Crossref] [PubMed]

J. Parker and C. R. Stroud, Phys. Rev. Lett. 56, 716 (1986).
[Crossref] [PubMed]

Other (2)

A. ten Wolde, L. D. Noordam, H. G. Muller, and H. B. van Linden van den Heuvell, in Fundamentals of Laser Interactions II, Vol. 339 of Lecture Notes on Physics, F. Ehlotzky, ed. (Springer-Verlag, Berlin, 1989).

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer-Verlag, Berlin, 1957).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Calculated time evolution of a Rydberg wave packet in an electric field of 200 V/cm. Laser excitation occurs at the peak of the modulating field, ϕ0 = π/2. The center of the n = 9 manifold is populated by a 7.5-psec pulse. The dashed line and curve show the result without a modulating field, and the solid curves show the result with a modulating field of 100 V/cm and an oscillation time of 289.3 psec. The phase difference is given in radians, and the horizontal lines indicate an integer ×2π.

Fig. 2
Fig. 2

Same as Fig. 1 except that the laser excitation occurs at a zero crossing of the modulating field, ϕ0 = 0.

Fig. 3
Fig. 3

Study of the asymmetric behavior of the second peak in Fig. 2. The modulating field values are 0 V/cm (dashed line and curve), 100 V/cm (solid curves), and 190 V/cm (dotted curves). The phase difference is given in radians.

Fig. 4
Fig. 4

Calculated time evolution of a Rydberg wave packet in an electric field of 200 V/cm. The center of the n = 9 manifold is populated by a 30-psec pulse. The modulating field (ϕ0 = 0, 800 V/cm) has a long oscillation time of 4.62 nsec. Note that the widths of the peaks become smaller than the duration of the excitation pulse.

Equations (10)

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ω n k = - 1 2 n 2 + 3 2 F n k - 1 16 n 4 F 2 ( 17 n 2 - 3 k 2 - 9 m 2 + 19 ) .
τ = 2 π / 3 F n .
ω n k ( t ) = - 1 2 n 2 + 3 2 F n k + 3 2 F m n k sin ( ω m t + ϕ 0 ) .
a k = b k , l 0 exp { [ - ( ω g - ω n k + ω L ) 2 t p 2 ] / 2 ln 2 } ,
Ψ ( r , t ) = k a k exp [ i ϕ k ( t ) ] k ,
ϕ k ( t ) = 0 t ω k ( t ) d t = ( - 1 2 n 2 + 3 2 F n k ) t + 3 F m n k 2 ω m [ cos ( ϕ 0 ) - cos ( ω m t + ϕ 0 ) ] .
Δ ϕ ( t ) = 3 F n t + ( 3 F m n / ω m ) [ cos ( ϕ 0 ) - cos ( ω m t + ϕ 0 ) ] .
Ψ ( r , t ) = l c l ( t ) l .
c l ( t ) = k a k b l k exp [ i ϕ k ( t ) ] .
P low - l ( t ) = l = 0 2 c l ( t ) 2

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