Abstract

We have investigated the ac Stark shift of a two-photon transition induced by a model stochastic field by using a Monte Carlo technique. The model field has a stochastic single mode, which can possess both colored amplitude and colored frequency fluctuations. The stochastic field’s modification of the ac Stark shift is quantified by a parameter M, which is the ratio of the Stark shift in the model stochastic field to the Stark shift that would be obtained in a phase-diffusion field of equivalent linewidth (i.e., a field with frequency fluctuations but no amplitude fluctuations). In the regime of weak fields, below saturation of the bound-bound transition, we find that M is greater than unity: the Stark shift is enhanced by the stochastic field. Moreover, in this regime M is an increasing function of the field’s degree of photon bunching. In strong fields, where the bound–bound transition is nearly saturated, the enhancement of the Stark shift is diminished, so much so that in extremely strong fields M is actually less than unity. Our calculations indicate that the ac Stark shift’s modification occurs through two distinct processes, depending on the strength of the field. In weak fields enhancement of a multiphoton transition’s Stark shift is influenced primarily by the intrinsic correlation between ac-Stark-shift fluctuations and Rabi-frequency fluctuations. In strong fields the fluctuating Stark shifts give rise to an asymmetric resonance line shape in a fashion analogous to inhomogeneous broadening. The line shape’s peak position then has a sublinear dependence on the stochastic field’s intensity, and this yields a diminished value of M.

© 1992 Optical Society of America

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  1. M. Arditi, T. R. Carver, “Pressure, light, and temperature shifts in optical detection of 0–0 hyperfine resonance of alkali metals,” Phys. Rev. 124, 800 (1961).
    [CrossRef]
  2. J. P. Barrat, C. Cohen-Tannoudji, “Elargissement et deplacement des raies de resonance magnetique causes par une excitation optique,” J. Phys. 22, 443 (1961).
  3. B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11 (1968), and references therein.
    [CrossRef]
  4. See, for example, G. Busca, M. Tetu, J. Vanier, “Light shift and light broadening in the 87Rb maser,” Can. J. Phys. 13, 1379 (1973);C. H. Volk, R. P. Frueholz, “The role of long-term lamp fluctuations in the random walk of frequency behavior of the rubidium frequency standard: a case study,” J. Appl. Phys. 57, 980 (1985);J. C. Camparo, R. P. Frueholz, “Fundamental stability limits for the diode-laser-pumped atomic frequency standard,” J. Appl. Phys. 59, 3313 (1986).
    [CrossRef]
  5. J. S. Barkos, “ac Stark effect and multiphoton processes in atoms,” Phys. Rep. 31, 209 (1977).
    [CrossRef]
  6. C. Lecompte, G. Mainfray, C. Manus, F. Sanchez, “Laser temporal-coherence effects on multiphoton ionization processes,” Phys. Rev. A 11, 1009 (1975).
    [CrossRef]
  7. A. T. Georges, P. Lambropoulos, “Aspects of resonant multiphoton processes,” Adv. Electron. Electron Phys. 54, 191 (1980), and references therein.
    [CrossRef]
  8. N. B. Delone, V. A. Kovarskii, A. V. Massalov, N. F. Perel’man, “An atom in the radiation field of a multifrequency laser,” Sov. Phys. Usp. 23, 472 (1980).
    [CrossRef]
  9. R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973).
  10. L.-A. Lompre, G. Mainfray, C. Manus, J. P. Marinier, “Laser light statistics and bandwidth effects in resonant multiphoton ionisation of caesium atoms at 1.059 μ m,” J. Phys. B14, 4307 (1981).
    [CrossRef]
  11. J. Morellec, D. Normand, G. Petite, “Resonance shifts in the multiphoton ionization of cesium atoms,” Phys. Rev. A 14, 300 (1976).
    [CrossRef]
  12. P. Zoller, “Stark shifts and resonant multiphoton ionisation in multimode laser fields,” J. Phys. B 15, 2911 (1982).
    [CrossRef]
  13. Y. Gontier, M. Trahin, “Multiphoton ionisation and light statistics: application to the cesium atom,” J. Phys. B 12, 2123 (1979).
    [CrossRef]
  14. P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, M. D. Levenson, “Saturation effects in resonant three-photon ionisation of sodium with a nonmonochromatic field,” J. Phys. B 11, 1733 (1978).
    [CrossRef]
  15. P. Zoller, P. Lambropoulos, “Laser temporal coherence effects in two-photon resonant three-photon ionisation,” J. Phys. B 13, 69 (1980).
    [CrossRef]
  16. L. R. Brewer, F. Buchinger, M. Ligare, D. E. Kelleher, “Resonance-enhanced multiphoton ionization of atomic hydrogen,” Phys. Rev. A 39, 3912 (1989).
    [CrossRef] [PubMed]
  17. Most high-power pulsed lasers operate in many modes, and it is known that in the limit of many modes a multimode field becomes a chaotic field. On a more pragmatic level, a chaotic field’s nth-order coherence function may be written in terms of products of first-order coherence functions, which aids in the analytical evaluation of atomic averages.
  18. L. A. Westling, M. G. Raymer, “Intensity autocorrelation measurements and spontaneous FM phase locking in a multimode pulsed dye laser,” J. Opt. Soc. Am. B 3, 911 (1986).
    [CrossRef]
  19. J. C. Camparo, P. Lambropoulos, “Monte Carlo simulations of field fluctuations in strongly driven resonant transitions,” submitted to Phys. Rev. A.
  20. M. D. MacLaren, G. Marsaglia, “Uniform random number generators,” J. Assoc. Comput. Mach. 12, 83 (1965).
    [CrossRef]
  21. J. H. Ahrens, U. Dieter, “Computer methods for sampling from the exponential and normal distributions,” Commun. Assoc. Comput. Mach. 15, 873 (1972).
  22. M. J. Levin, “Generation of a sampled Gaussian time series having a specified correlation function,” IRE Trans. Inf. Theory IT-6, 545 (1960).
    [CrossRef]
  23. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).
  24. P. Zoller, P. Lambropoulos, “Non-Lorentzian laser lineshapes in intense field–atom interaction,” J. Phys. B 12, L547 (1979).
    [CrossRef]
  25. A. T. Georges, P. Lambropoulos, J. H. Marburger, “Theory of third-harmonic generation in metal vapors under two-photon resonance conditions,” Phys. Rev. A 15, 300 (1977).
    [CrossRef]
  26. P. W. Milonni, J. H. Eberly, “Temporal coherence in multi-photon absorption. Far off-resonance intermediate states,” J. Chem. Phys. 68, 1602 (1978).
    [CrossRef]
  27. Actually, this procedure is repeated twice: once for a positive value of κ and once for a negative value of κ. These two ac Stark shifts are then averaged in order to account for the stochastic realization shift, as discussed in Ref. 28.
  28. J. C. Camparo, P. Lambropoulos, “The stochastic realization shift,” Opt. Commun. 85, 213 (1991).
    [CrossRef]
  29. In the Monte Carlo simulations of the present study a random change in the field’s amplitude and frequency was made every 0.01 time units, and in the time between random changes the field characteristics were stable. If the time scale for some process was shorter than 0.01 time units, then for that process the field would have had monochromatic character. It might therefore be argued that with ω1= 56 and ω1= 100 and with random changes occurring every 0.01 time units, we were not simulating the stochastic field accurately enough for our simulation to be sensitive to the field’s degree of photon bunching. However, if this systematic effect were the explanation for the Stark shift enhancement’s ciritical |g2(0)| value, then there should have also been a critical |g2(0)| value for amplitude enhancement. Because this was not the case, it seems unlikely that a systematic effect of our Monte Carlo simulation can be the cause of the Stark shift enhancement’s critical |g2(0)| value.
  30. S. Jacobs, “How monochromatic is laser light?” Am. J. Phys. 47, 597 (1979).
    [CrossRef]
  31. J. G. Powles, “The adiabatic fast passage experiment in magnetic resonance,” Proc. Phys. Soc. London 71, 497 (1958).
    [CrossRef]
  32. J. C. Camparo, R. P. Frueholz, “Parameters of adiabatic rapid passage in the 0–0 hyperfine transition of 87Rb,” Phys. Rev. A 30, 803 (1984).
    [CrossRef]
  33. See, for example, A. Messiah, Quantum Mechanics (Wiley, New York, 1961), Vol. II;B. Holstein, “The adiabatic propagator,” Am. J. Phys. 57, 714 (1989).
    [CrossRef]
  34. A. T. Georges, P. Lambropoulos, “Saturation and stark splitting of an atomic transition in a stochastic field,” Phys. Rev. A 20, 991 (1979).
    [CrossRef]

1991 (1)

J. C. Camparo, P. Lambropoulos, “The stochastic realization shift,” Opt. Commun. 85, 213 (1991).
[CrossRef]

1989 (1)

L. R. Brewer, F. Buchinger, M. Ligare, D. E. Kelleher, “Resonance-enhanced multiphoton ionization of atomic hydrogen,” Phys. Rev. A 39, 3912 (1989).
[CrossRef] [PubMed]

1986 (1)

1984 (1)

J. C. Camparo, R. P. Frueholz, “Parameters of adiabatic rapid passage in the 0–0 hyperfine transition of 87Rb,” Phys. Rev. A 30, 803 (1984).
[CrossRef]

1982 (1)

P. Zoller, “Stark shifts and resonant multiphoton ionisation in multimode laser fields,” J. Phys. B 15, 2911 (1982).
[CrossRef]

1980 (3)

A. T. Georges, P. Lambropoulos, “Aspects of resonant multiphoton processes,” Adv. Electron. Electron Phys. 54, 191 (1980), and references therein.
[CrossRef]

N. B. Delone, V. A. Kovarskii, A. V. Massalov, N. F. Perel’man, “An atom in the radiation field of a multifrequency laser,” Sov. Phys. Usp. 23, 472 (1980).
[CrossRef]

P. Zoller, P. Lambropoulos, “Laser temporal coherence effects in two-photon resonant three-photon ionisation,” J. Phys. B 13, 69 (1980).
[CrossRef]

1979 (4)

S. Jacobs, “How monochromatic is laser light?” Am. J. Phys. 47, 597 (1979).
[CrossRef]

A. T. Georges, P. Lambropoulos, “Saturation and stark splitting of an atomic transition in a stochastic field,” Phys. Rev. A 20, 991 (1979).
[CrossRef]

Y. Gontier, M. Trahin, “Multiphoton ionisation and light statistics: application to the cesium atom,” J. Phys. B 12, 2123 (1979).
[CrossRef]

P. Zoller, P. Lambropoulos, “Non-Lorentzian laser lineshapes in intense field–atom interaction,” J. Phys. B 12, L547 (1979).
[CrossRef]

1978 (2)

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, M. D. Levenson, “Saturation effects in resonant three-photon ionisation of sodium with a nonmonochromatic field,” J. Phys. B 11, 1733 (1978).
[CrossRef]

P. W. Milonni, J. H. Eberly, “Temporal coherence in multi-photon absorption. Far off-resonance intermediate states,” J. Chem. Phys. 68, 1602 (1978).
[CrossRef]

1977 (2)

A. T. Georges, P. Lambropoulos, J. H. Marburger, “Theory of third-harmonic generation in metal vapors under two-photon resonance conditions,” Phys. Rev. A 15, 300 (1977).
[CrossRef]

J. S. Barkos, “ac Stark effect and multiphoton processes in atoms,” Phys. Rep. 31, 209 (1977).
[CrossRef]

1976 (1)

J. Morellec, D. Normand, G. Petite, “Resonance shifts in the multiphoton ionization of cesium atoms,” Phys. Rev. A 14, 300 (1976).
[CrossRef]

1975 (1)

C. Lecompte, G. Mainfray, C. Manus, F. Sanchez, “Laser temporal-coherence effects on multiphoton ionization processes,” Phys. Rev. A 11, 1009 (1975).
[CrossRef]

1973 (1)

See, for example, G. Busca, M. Tetu, J. Vanier, “Light shift and light broadening in the 87Rb maser,” Can. J. Phys. 13, 1379 (1973);C. H. Volk, R. P. Frueholz, “The role of long-term lamp fluctuations in the random walk of frequency behavior of the rubidium frequency standard: a case study,” J. Appl. Phys. 57, 980 (1985);J. C. Camparo, R. P. Frueholz, “Fundamental stability limits for the diode-laser-pumped atomic frequency standard,” J. Appl. Phys. 59, 3313 (1986).
[CrossRef]

1972 (1)

J. H. Ahrens, U. Dieter, “Computer methods for sampling from the exponential and normal distributions,” Commun. Assoc. Comput. Mach. 15, 873 (1972).

1968 (1)

B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11 (1968), and references therein.
[CrossRef]

1965 (1)

M. D. MacLaren, G. Marsaglia, “Uniform random number generators,” J. Assoc. Comput. Mach. 12, 83 (1965).
[CrossRef]

1961 (2)

M. Arditi, T. R. Carver, “Pressure, light, and temperature shifts in optical detection of 0–0 hyperfine resonance of alkali metals,” Phys. Rev. 124, 800 (1961).
[CrossRef]

J. P. Barrat, C. Cohen-Tannoudji, “Elargissement et deplacement des raies de resonance magnetique causes par une excitation optique,” J. Phys. 22, 443 (1961).

1960 (1)

M. J. Levin, “Generation of a sampled Gaussian time series having a specified correlation function,” IRE Trans. Inf. Theory IT-6, 545 (1960).
[CrossRef]

1958 (1)

J. G. Powles, “The adiabatic fast passage experiment in magnetic resonance,” Proc. Phys. Soc. London 71, 497 (1958).
[CrossRef]

Agostini, P.

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, M. D. Levenson, “Saturation effects in resonant three-photon ionisation of sodium with a nonmonochromatic field,” J. Phys. B 11, 1733 (1978).
[CrossRef]

Ahrens, J. H.

J. H. Ahrens, U. Dieter, “Computer methods for sampling from the exponential and normal distributions,” Commun. Assoc. Comput. Mach. 15, 873 (1972).

Arditi, M.

M. Arditi, T. R. Carver, “Pressure, light, and temperature shifts in optical detection of 0–0 hyperfine resonance of alkali metals,” Phys. Rev. 124, 800 (1961).
[CrossRef]

Barkos, J. S.

J. S. Barkos, “ac Stark effect and multiphoton processes in atoms,” Phys. Rep. 31, 209 (1977).
[CrossRef]

Barrat, J. P.

J. P. Barrat, C. Cohen-Tannoudji, “Elargissement et deplacement des raies de resonance magnetique causes par une excitation optique,” J. Phys. 22, 443 (1961).

Brewer, L. R.

L. R. Brewer, F. Buchinger, M. Ligare, D. E. Kelleher, “Resonance-enhanced multiphoton ionization of atomic hydrogen,” Phys. Rev. A 39, 3912 (1989).
[CrossRef] [PubMed]

Buchinger, F.

L. R. Brewer, F. Buchinger, M. Ligare, D. E. Kelleher, “Resonance-enhanced multiphoton ionization of atomic hydrogen,” Phys. Rev. A 39, 3912 (1989).
[CrossRef] [PubMed]

Busca, G.

See, for example, G. Busca, M. Tetu, J. Vanier, “Light shift and light broadening in the 87Rb maser,” Can. J. Phys. 13, 1379 (1973);C. H. Volk, R. P. Frueholz, “The role of long-term lamp fluctuations in the random walk of frequency behavior of the rubidium frequency standard: a case study,” J. Appl. Phys. 57, 980 (1985);J. C. Camparo, R. P. Frueholz, “Fundamental stability limits for the diode-laser-pumped atomic frequency standard,” J. Appl. Phys. 59, 3313 (1986).
[CrossRef]

Camparo, J. C.

J. C. Camparo, P. Lambropoulos, “The stochastic realization shift,” Opt. Commun. 85, 213 (1991).
[CrossRef]

J. C. Camparo, R. P. Frueholz, “Parameters of adiabatic rapid passage in the 0–0 hyperfine transition of 87Rb,” Phys. Rev. A 30, 803 (1984).
[CrossRef]

J. C. Camparo, P. Lambropoulos, “Monte Carlo simulations of field fluctuations in strongly driven resonant transitions,” submitted to Phys. Rev. A.

Carver, T. R.

M. Arditi, T. R. Carver, “Pressure, light, and temperature shifts in optical detection of 0–0 hyperfine resonance of alkali metals,” Phys. Rev. 124, 800 (1961).
[CrossRef]

Cohen-Tannoudji, C.

J. P. Barrat, C. Cohen-Tannoudji, “Elargissement et deplacement des raies de resonance magnetique causes par une excitation optique,” J. Phys. 22, 443 (1961).

Delone, N. B.

N. B. Delone, V. A. Kovarskii, A. V. Massalov, N. F. Perel’man, “An atom in the radiation field of a multifrequency laser,” Sov. Phys. Usp. 23, 472 (1980).
[CrossRef]

Dieter, U.

J. H. Ahrens, U. Dieter, “Computer methods for sampling from the exponential and normal distributions,” Commun. Assoc. Comput. Mach. 15, 873 (1972).

Eberly, J. H.

P. W. Milonni, J. H. Eberly, “Temporal coherence in multi-photon absorption. Far off-resonance intermediate states,” J. Chem. Phys. 68, 1602 (1978).
[CrossRef]

Frueholz, R. P.

J. C. Camparo, R. P. Frueholz, “Parameters of adiabatic rapid passage in the 0–0 hyperfine transition of 87Rb,” Phys. Rev. A 30, 803 (1984).
[CrossRef]

Georges, A. T.

A. T. Georges, P. Lambropoulos, “Aspects of resonant multiphoton processes,” Adv. Electron. Electron Phys. 54, 191 (1980), and references therein.
[CrossRef]

A. T. Georges, P. Lambropoulos, “Saturation and stark splitting of an atomic transition in a stochastic field,” Phys. Rev. A 20, 991 (1979).
[CrossRef]

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, M. D. Levenson, “Saturation effects in resonant three-photon ionisation of sodium with a nonmonochromatic field,” J. Phys. B 11, 1733 (1978).
[CrossRef]

A. T. Georges, P. Lambropoulos, J. H. Marburger, “Theory of third-harmonic generation in metal vapors under two-photon resonance conditions,” Phys. Rev. A 15, 300 (1977).
[CrossRef]

Gontier, Y.

Y. Gontier, M. Trahin, “Multiphoton ionisation and light statistics: application to the cesium atom,” J. Phys. B 12, 2123 (1979).
[CrossRef]

Happer, W.

B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11 (1968), and references therein.
[CrossRef]

Jacobs, S.

S. Jacobs, “How monochromatic is laser light?” Am. J. Phys. 47, 597 (1979).
[CrossRef]

Kelleher, D. E.

L. R. Brewer, F. Buchinger, M. Ligare, D. E. Kelleher, “Resonance-enhanced multiphoton ionization of atomic hydrogen,” Phys. Rev. A 39, 3912 (1989).
[CrossRef] [PubMed]

Kovarskii, V. A.

N. B. Delone, V. A. Kovarskii, A. V. Massalov, N. F. Perel’man, “An atom in the radiation field of a multifrequency laser,” Sov. Phys. Usp. 23, 472 (1980).
[CrossRef]

Lambropoulos, P.

J. C. Camparo, P. Lambropoulos, “The stochastic realization shift,” Opt. Commun. 85, 213 (1991).
[CrossRef]

A. T. Georges, P. Lambropoulos, “Aspects of resonant multiphoton processes,” Adv. Electron. Electron Phys. 54, 191 (1980), and references therein.
[CrossRef]

P. Zoller, P. Lambropoulos, “Laser temporal coherence effects in two-photon resonant three-photon ionisation,” J. Phys. B 13, 69 (1980).
[CrossRef]

P. Zoller, P. Lambropoulos, “Non-Lorentzian laser lineshapes in intense field–atom interaction,” J. Phys. B 12, L547 (1979).
[CrossRef]

A. T. Georges, P. Lambropoulos, “Saturation and stark splitting of an atomic transition in a stochastic field,” Phys. Rev. A 20, 991 (1979).
[CrossRef]

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, M. D. Levenson, “Saturation effects in resonant three-photon ionisation of sodium with a nonmonochromatic field,” J. Phys. B 11, 1733 (1978).
[CrossRef]

A. T. Georges, P. Lambropoulos, J. H. Marburger, “Theory of third-harmonic generation in metal vapors under two-photon resonance conditions,” Phys. Rev. A 15, 300 (1977).
[CrossRef]

J. C. Camparo, P. Lambropoulos, “Monte Carlo simulations of field fluctuations in strongly driven resonant transitions,” submitted to Phys. Rev. A.

Lecompte, C.

C. Lecompte, G. Mainfray, C. Manus, F. Sanchez, “Laser temporal-coherence effects on multiphoton ionization processes,” Phys. Rev. A 11, 1009 (1975).
[CrossRef]

Levenson, M. D.

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, M. D. Levenson, “Saturation effects in resonant three-photon ionisation of sodium with a nonmonochromatic field,” J. Phys. B 11, 1733 (1978).
[CrossRef]

Levin, M. J.

M. J. Levin, “Generation of a sampled Gaussian time series having a specified correlation function,” IRE Trans. Inf. Theory IT-6, 545 (1960).
[CrossRef]

Ligare, M.

L. R. Brewer, F. Buchinger, M. Ligare, D. E. Kelleher, “Resonance-enhanced multiphoton ionization of atomic hydrogen,” Phys. Rev. A 39, 3912 (1989).
[CrossRef] [PubMed]

Lompre, L.-A.

L.-A. Lompre, G. Mainfray, C. Manus, J. P. Marinier, “Laser light statistics and bandwidth effects in resonant multiphoton ionisation of caesium atoms at 1.059 μ m,” J. Phys. B14, 4307 (1981).
[CrossRef]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973).

MacLaren, M. D.

M. D. MacLaren, G. Marsaglia, “Uniform random number generators,” J. Assoc. Comput. Mach. 12, 83 (1965).
[CrossRef]

Mainfray, G.

C. Lecompte, G. Mainfray, C. Manus, F. Sanchez, “Laser temporal-coherence effects on multiphoton ionization processes,” Phys. Rev. A 11, 1009 (1975).
[CrossRef]

L.-A. Lompre, G. Mainfray, C. Manus, J. P. Marinier, “Laser light statistics and bandwidth effects in resonant multiphoton ionisation of caesium atoms at 1.059 μ m,” J. Phys. B14, 4307 (1981).
[CrossRef]

Manus, C.

C. Lecompte, G. Mainfray, C. Manus, F. Sanchez, “Laser temporal-coherence effects on multiphoton ionization processes,” Phys. Rev. A 11, 1009 (1975).
[CrossRef]

L.-A. Lompre, G. Mainfray, C. Manus, J. P. Marinier, “Laser light statistics and bandwidth effects in resonant multiphoton ionisation of caesium atoms at 1.059 μ m,” J. Phys. B14, 4307 (1981).
[CrossRef]

Marburger, J. H.

A. T. Georges, P. Lambropoulos, J. H. Marburger, “Theory of third-harmonic generation in metal vapors under two-photon resonance conditions,” Phys. Rev. A 15, 300 (1977).
[CrossRef]

Marinier, J. P.

L.-A. Lompre, G. Mainfray, C. Manus, J. P. Marinier, “Laser light statistics and bandwidth effects in resonant multiphoton ionisation of caesium atoms at 1.059 μ m,” J. Phys. B14, 4307 (1981).
[CrossRef]

Marsaglia, G.

M. D. MacLaren, G. Marsaglia, “Uniform random number generators,” J. Assoc. Comput. Mach. 12, 83 (1965).
[CrossRef]

Massalov, A. V.

N. B. Delone, V. A. Kovarskii, A. V. Massalov, N. F. Perel’man, “An atom in the radiation field of a multifrequency laser,” Sov. Phys. Usp. 23, 472 (1980).
[CrossRef]

Mathur, B. S.

B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11 (1968), and references therein.
[CrossRef]

Messiah, A.

See, for example, A. Messiah, Quantum Mechanics (Wiley, New York, 1961), Vol. II;B. Holstein, “The adiabatic propagator,” Am. J. Phys. 57, 714 (1989).
[CrossRef]

Milonni, P. W.

P. W. Milonni, J. H. Eberly, “Temporal coherence in multi-photon absorption. Far off-resonance intermediate states,” J. Chem. Phys. 68, 1602 (1978).
[CrossRef]

Morellec, J.

J. Morellec, D. Normand, G. Petite, “Resonance shifts in the multiphoton ionization of cesium atoms,” Phys. Rev. A 14, 300 (1976).
[CrossRef]

Normand, D.

J. Morellec, D. Normand, G. Petite, “Resonance shifts in the multiphoton ionization of cesium atoms,” Phys. Rev. A 14, 300 (1976).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

Perel’man, N. F.

N. B. Delone, V. A. Kovarskii, A. V. Massalov, N. F. Perel’man, “An atom in the radiation field of a multifrequency laser,” Sov. Phys. Usp. 23, 472 (1980).
[CrossRef]

Petite, G.

J. Morellec, D. Normand, G. Petite, “Resonance shifts in the multiphoton ionization of cesium atoms,” Phys. Rev. A 14, 300 (1976).
[CrossRef]

Powles, J. G.

J. G. Powles, “The adiabatic fast passage experiment in magnetic resonance,” Proc. Phys. Soc. London 71, 497 (1958).
[CrossRef]

Raymer, M. G.

Sanchez, F.

C. Lecompte, G. Mainfray, C. Manus, F. Sanchez, “Laser temporal-coherence effects on multiphoton ionization processes,” Phys. Rev. A 11, 1009 (1975).
[CrossRef]

Tang, H.

B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11 (1968), and references therein.
[CrossRef]

Tetu, M.

See, for example, G. Busca, M. Tetu, J. Vanier, “Light shift and light broadening in the 87Rb maser,” Can. J. Phys. 13, 1379 (1973);C. H. Volk, R. P. Frueholz, “The role of long-term lamp fluctuations in the random walk of frequency behavior of the rubidium frequency standard: a case study,” J. Appl. Phys. 57, 980 (1985);J. C. Camparo, R. P. Frueholz, “Fundamental stability limits for the diode-laser-pumped atomic frequency standard,” J. Appl. Phys. 59, 3313 (1986).
[CrossRef]

Trahin, M.

Y. Gontier, M. Trahin, “Multiphoton ionisation and light statistics: application to the cesium atom,” J. Phys. B 12, 2123 (1979).
[CrossRef]

Vanier, J.

See, for example, G. Busca, M. Tetu, J. Vanier, “Light shift and light broadening in the 87Rb maser,” Can. J. Phys. 13, 1379 (1973);C. H. Volk, R. P. Frueholz, “The role of long-term lamp fluctuations in the random walk of frequency behavior of the rubidium frequency standard: a case study,” J. Appl. Phys. 57, 980 (1985);J. C. Camparo, R. P. Frueholz, “Fundamental stability limits for the diode-laser-pumped atomic frequency standard,” J. Appl. Phys. 59, 3313 (1986).
[CrossRef]

Westling, L. A.

Wheatley, S. E.

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, M. D. Levenson, “Saturation effects in resonant three-photon ionisation of sodium with a nonmonochromatic field,” J. Phys. B 11, 1733 (1978).
[CrossRef]

Zoller, P.

P. Zoller, “Stark shifts and resonant multiphoton ionisation in multimode laser fields,” J. Phys. B 15, 2911 (1982).
[CrossRef]

P. Zoller, P. Lambropoulos, “Laser temporal coherence effects in two-photon resonant three-photon ionisation,” J. Phys. B 13, 69 (1980).
[CrossRef]

P. Zoller, P. Lambropoulos, “Non-Lorentzian laser lineshapes in intense field–atom interaction,” J. Phys. B 12, L547 (1979).
[CrossRef]

Adv. Electron. Electron Phys. (1)

A. T. Georges, P. Lambropoulos, “Aspects of resonant multiphoton processes,” Adv. Electron. Electron Phys. 54, 191 (1980), and references therein.
[CrossRef]

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S. Jacobs, “How monochromatic is laser light?” Am. J. Phys. 47, 597 (1979).
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[CrossRef]

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P. Zoller, “Stark shifts and resonant multiphoton ionisation in multimode laser fields,” J. Phys. B 15, 2911 (1982).
[CrossRef]

Y. Gontier, M. Trahin, “Multiphoton ionisation and light statistics: application to the cesium atom,” J. Phys. B 12, 2123 (1979).
[CrossRef]

P. Agostini, A. T. Georges, S. E. Wheatley, P. Lambropoulos, M. D. Levenson, “Saturation effects in resonant three-photon ionisation of sodium with a nonmonochromatic field,” J. Phys. B 11, 1733 (1978).
[CrossRef]

P. Zoller, P. Lambropoulos, “Laser temporal coherence effects in two-photon resonant three-photon ionisation,” J. Phys. B 13, 69 (1980).
[CrossRef]

P. Zoller, P. Lambropoulos, “Non-Lorentzian laser lineshapes in intense field–atom interaction,” J. Phys. B 12, L547 (1979).
[CrossRef]

Opt. Commun. (1)

J. C. Camparo, P. Lambropoulos, “The stochastic realization shift,” Opt. Commun. 85, 213 (1991).
[CrossRef]

Phys. Rep. (1)

J. S. Barkos, “ac Stark effect and multiphoton processes in atoms,” Phys. Rep. 31, 209 (1977).
[CrossRef]

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B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11 (1968), and references therein.
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M. Arditi, T. R. Carver, “Pressure, light, and temperature shifts in optical detection of 0–0 hyperfine resonance of alkali metals,” Phys. Rev. 124, 800 (1961).
[CrossRef]

Phys. Rev. A (6)

J. Morellec, D. Normand, G. Petite, “Resonance shifts in the multiphoton ionization of cesium atoms,” Phys. Rev. A 14, 300 (1976).
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C. Lecompte, G. Mainfray, C. Manus, F. Sanchez, “Laser temporal-coherence effects on multiphoton ionization processes,” Phys. Rev. A 11, 1009 (1975).
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A. T. Georges, P. Lambropoulos, “Saturation and stark splitting of an atomic transition in a stochastic field,” Phys. Rev. A 20, 991 (1979).
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Other (8)

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973).

L.-A. Lompre, G. Mainfray, C. Manus, J. P. Marinier, “Laser light statistics and bandwidth effects in resonant multiphoton ionisation of caesium atoms at 1.059 μ m,” J. Phys. B14, 4307 (1981).
[CrossRef]

Most high-power pulsed lasers operate in many modes, and it is known that in the limit of many modes a multimode field becomes a chaotic field. On a more pragmatic level, a chaotic field’s nth-order coherence function may be written in terms of products of first-order coherence functions, which aids in the analytical evaluation of atomic averages.

J. C. Camparo, P. Lambropoulos, “Monte Carlo simulations of field fluctuations in strongly driven resonant transitions,” submitted to Phys. Rev. A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

In the Monte Carlo simulations of the present study a random change in the field’s amplitude and frequency was made every 0.01 time units, and in the time between random changes the field characteristics were stable. If the time scale for some process was shorter than 0.01 time units, then for that process the field would have had monochromatic character. It might therefore be argued that with ω1= 56 and ω1= 100 and with random changes occurring every 0.01 time units, we were not simulating the stochastic field accurately enough for our simulation to be sensitive to the field’s degree of photon bunching. However, if this systematic effect were the explanation for the Stark shift enhancement’s ciritical |g2(0)| value, then there should have also been a critical |g2(0)| value for amplitude enhancement. Because this was not the case, it seems unlikely that a systematic effect of our Monte Carlo simulation can be the cause of the Stark shift enhancement’s critical |g2(0)| value.

Actually, this procedure is repeated twice: once for a positive value of κ and once for a negative value of κ. These two ac Stark shifts are then averaged in order to account for the stochastic realization shift, as discussed in Ref. 28.

See, for example, A. Messiah, Quantum Mechanics (Wiley, New York, 1961), Vol. II;B. Holstein, “The adiabatic propagator,” Am. J. Phys. 57, 714 (1989).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Coherence functions for a model stochastic field with γ = 3 and three different values of ω1: (a) first-order coherence functions g1(τ) and (b) absolute value of the second-order coherence functions |g2(τ)|. The decay times of the |g2(τ)| are ∼1/ω1.

Fig. 2
Fig. 2

(a) ac-Stark-shift modification factor as a function of the field’s degree of photon bunching with photon bunching quantified by |g2(0)| (Ω0 = 0.03 and κ = +0.5). Filled circles correspond to fields with γ = 3 (τcoh = 0.28 ± 15%) and ω1 = 5.6, 7.4, 10, 17, 30, 56, and 100. The open circle, indicated by the arrow, corresponds to a field with γ = 1 (τcoh = 0.86) and ω1 = 10. (b) Amplitude enhancement of the two-photon resonance as a function of the field’s degree of photon bunching. If A is defined as the amplitude of the two-photon resonance in the stochastic field and APD is defined as the resonance’s amplitude in a phase-diffusion field, the amplitude enhancement is given by A/APD.

Fig. 3
Fig. 3

ac Stark shift as a function of the nominal two-photon Rabi frequency Ω0 (i.e., field intensity) for a field with γ = 1 and ω1 = 10 [τcoh = 0.86 and |g2(0)| = 1.35]. As expected, the ac Stark shift is a linear function of field intensity for weak fields. However, in the presence of strong fields the ac Stark shift’s dependence on field intensity becomes sublinear.

Fig. 4
Fig. 4

ac-Stark-shift modification factor as a function of the nominal two-photon Rabi frequency Ω0 (i.e., field intensity). Squares correspond to a field with γ = 1 and ω1 = 10 [τcoh = 0.86 and |g2(0)| = 1.35], triangles correspond to a field with γ = 3 and ω1 = 10 [τcoh = 0.23 and |g2(0)| = 1.83], and circles again correspond to a field with γ = 3 and ω1 = 10, except that in this case there has been an artificial decorrelation between Rabi-frequency fluctuations and ac-Stark-shift fluctuations.

Fig. 5
Fig. 5

Line shapes for a two-photon resonance in a stochastic field: κ = −0.5, γ = 3, and ω1 = 10 [τcoh = 0.23 and |g2(0)| = 1.83]. (a) Ω0 = 0.03: the solid curve corresponds to the resonance induced by our model stochastic field; the dashed curve corresponds to the resonance that would be obtained in a phase-diffusion field with the same linewidth. (b) Ω0 = 30: the solid curve corresponds to the resonance induced by our model stochastic field; the dashed curve corresponds to the resonance that would be obtained in a phase-diffusion field with the same linewidth.

Equations (22)

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E ( t ) = [ E 0 + ( t ) ] cos { [ ω f + δ ω f ( t ) ] t } .
δ ω f ( τ ) δ ω f ( 0 ) = ( γ β ) exp ( β | τ | ) ,
( τ ) ( 0 ) = ( γ E 0 2 / ω 1 ) exp ( ω 1 | τ | ) .
σ ˙ 22 = Γ 1 σ 22 Ω Im ( σ 12 ) ,
σ ˙ 12 = i Δ σ 12 Γ 2 σ 12 + ( i Ω / 2 ) ( 2 σ 22 1 ) ,
Δ = 2 ω f ω 21 + 2 δ ω f ( t ) κ Ω 0 [ 1 + x ( t ) ] 2 ,
( 1 + x ) 2 = ( 1 + γ / ω 1 ) .
M δ ω ac δ ω ac .
σ 12 ( t ) Ω = i 2 0 t ( 2 σ 22 1 ) exp [ ( i Δ 0 + Γ 2 ) ( t t ) ] Ω ( t ) Ω ( t ) exp { i t t [ n δ ω f ( t ) δ ω ac ( t ) ] d t } d t .
σ 12 ( t ) Ω = i 2 0 t ( 2 σ 22 1 ) × exp [ ( i Δ 0 + Γ 2 + n 2 γ iM δ ω ac ) ( t t ) ] × Ω ( t ) Ω ( t ) d t ,
Ω ( t ) Ω ( t ) exp [ ( n 2 γ iM δ ω ac ) ( t t ) ] = Ω ( t ) Ω ( t ) exp [ i t t ( n δ ω f ( t ) δ ω ac ( t ) d t ]
M = i n 2 γ δ ω ac + i ( t t ) 1 δ ω ac In ( Ω ( t ) Ω ( t ) 1 × Ω ( t ) Ω ( t ) exp { i t t [ n δ ω f ( t ) δ ω ac ( t ) ] d t } ) .
M = i n 2 γ δ ω ac + i ( t t ) 1 δ ω ac In ( exp { in [ ϕ ( t ) ϕ ( t ) ] } × exp [ i t t δ ω ac ( t ) d t ] ) ,
M = i ( t t ) 1 δ ω ac In { exp [ i t t δ ω ac ( t ) d t ] } .
M = i ( t t ) 1 δ ω ac × In { 1 + j = 1 ( i ) j j ! t t δ ω ac ( t 1 ) δ ω ac ( t j ) d t 1 d t j } .
δ ω ac ( t 1 ) δ ω ac ( t m ) = δ ω ac m exp ( γ m | t 1 t 2 | ) exp ( γ m | t m 1 t m | ) ,
δ ω ac ( t 1 ) δ ω ac ( t m ) = δ ω ac m ( 2 / γ m ) m 1 δ ( t 1 t 2 ) δ ( t m 1 t m )
M = i ( t t ) 1 δ ω ac In [ 1 + j = 1 ( i ) j j ! δ ω ac j ( 2 γ j ) j 1 ( t t ) ] .
M = i / [ δ ω ac ( t t ) ] In [ 1 i δ ω ac ( t t ) ] .
M = 1 + ( i / 2 ) δ ω ac ( t t ) δ ω ac 2 ( t t ) 2 / 3 + ,
Re ( M ) 1.
δ ω res = i n 2 γ + i ( t t ) 1 In ( Ω ( t ) Ω ( t ) 1 Ω ( t ) Ω ( t ) × exp { in [ ϕ ( t ) ϕ ( t ) ] } ) .

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