Abstract

We have measured the ac Stark effect in a two-zone stimulated Raman interaction in a sodium atomic beam. In addition, we have derived simple theoretical expressions for the ac Stark shift, based on a closed three-level system, in the Λ configuration, and have achieved qualitative agreement with the data. In particular, the magnitude and sense of the ac Stark shifts are found to depend on laser intensities as well as on the initial populations of the two low-lying levels of the Λ configuration. Specifically, the observed ac Stark shifts are smaller for larger laser intensities and also for smaller initial population differences between the two low-lying levels. The ac Stark effect must be considered for many potential applications of the Raman effect, such as a Raman clock. In this connection, we have identified conditions for the reduction of the ac Stark shift in our sodium Raman clock.

© 1989 Optical Society of America

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  1. For example, B. J. Dalton, T. D. Kieu, and P. L. Knight, “Theory of ultra-high-resolution optical Raman Ramsey spectroscopy,” Opt. Acta 33, 459 (1986); D. Pegg, “Interaction of three-level atoms with modulated lasers,” Opt. Acta 33, 363 (1986); N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984); S. Swain, “Conditions for population trapping in a three-level system,” J. Phys. B 15, 3405 (1982); P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a three-level system,” J. Phys. B 15, 561 (1982); G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento B 53, 1 (1979); H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978); A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
    [CrossRef] [PubMed]
  2. P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriasca 56, 103 (1984); F. H. Mies and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981); E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977); M. Sargent and P. Horwitz, “Three-level Rabi flopping,” Phys. Rev. A 13, 1962 (1976); R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady state cases,” Phys. Rev. A 11, 1641 (1975).
    [CrossRef]
  3. J. Mlynek and R. Grimm, “Raman heterodyne Ramsey spectroscopy in a samarium atomic beam” Appl. Phys. B 45, 77 (1988); M. Kaivola, P. Thorsen, and O. Poulsen, “Dispersive line shapes and optical pumping in a three-level system,” Phys. Rev. A 32, 207 (1985); F. Shimizu, K. Shimizu, and H. Takuma, “Selective vibrational pumping of a molecular beam by a stimulated Raman process,” Phys. Rev. A 31, 3132 (1985); A. Sharma, W. Happar, and Y. Q. Lu, “Sub-Doppler-broadened magnetic field resonances in the resonant stimulated electronic Raman scattering of multimode laser light,” Phys. Rev. A 29, 749 (1984); R. E. Tench and S. Ezekiel, “Precision measurements of hyperfine predissociation in I2vapor using a two-photon resonant scattering technique,” Chem. Phys. Lett. 96, 253 (1983); R. E. Tench, B. W. Peuse, P. R. Hemmer, J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Two laser Raman difference frequency technique applied to high precision spectroscopy,” J. Phys. Colloq. 42, 45 (1981); P. Kumar and J. H. Shapiro, “Observation of Raman-shifted oscillation near the sodium D lines,” Opt. Lett. 10, 226 (1985); M. S. Feld, M. M. Burns, T. U. Kuhl, P. G. Pappas, and D. E. Murnick, “Laser-saturation spectroscopy with optical pumping,” Opt. Lett. 5, 79 (1980); G. Alzetta, L. Moi, and G. Orriols, “Nonabsorption hyperfine resonances in a sodium vapor irradiated by a multiniode dye-laser,” Nuovo Cimento B 52, 209 (1979); R. P. Hackel and S. Ezekiel, “Observation of subnatural linewidths by two-step resonant scattering in I2vapor,” Phys. Rev. Lett. 42, 1736 (1979); K. Takagi, R. F. Curl, and R. T. M. Su, “Spectroscopy with modulation sidebands,” Appl. Phys. 7, 181 (1975); R. L. Shoemaker and R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430, (1972).
    [CrossRef] [PubMed]
  4. E. Buhr and J. Mlynek, “Collision-induced Ramsey resonances in Sm vapor,” Phys. Rev. Lett. 57, 1300 (1986); A. A. Dabagyan, M. E. Movsesyan, T. O. Ovakimyan, and S. V. Shmavonyan, “Stimulated processes in potassium vapor in the presence of a buffer gas,” Sov. Phys. JETP 58, 700 (1983).
    [CrossRef] [PubMed]
  5. D. Krokel, K. Ludewigt, and H. Welling, “Frequency up-conversion by stimulated hyper-Raman scattering,” IEEE J. Quantum Electron. QE-22, 489 (1986); R. S. F. Chang, M. T. Duignan, R. H. Lehmberg, and N. Djeu, “Use of stimulated Raman scattering for reducing the divergence of severely aberrated laser beams,” in Excimer Lasers: Their Applications and New Frontiers in Lasers, R. W. Waynank, ed., Proc. Soc. Photo-Opt. Eng.476, 81 (1984); J. C. White, “Up-conversion of excimer lasers via stimulated anti-Stokes Raman scattering,” IEEE J. Quantum Electron. QE-20, 185 (1984); N. V. Znamenskii and V. I. Odintsov, “Infrared stimulated Raman scattering in rubidium vapor with a tunable pump frequency,” Opt. Spectrosc. (USSR) 54, 55 (1983); R. Wyatt, N. P. Ernsting, and W. G. Wrobel, “Tunable electronic Raman laser at 16 microns,” Appl. Phys. B 27, 175 (1982); M. L. Steyn-Ross and D. F. Walls, “Quantum theory of a Raman laser,” Opt. Acta 28, 201 (1981).
    [CrossRef]
  6. P. R. Hemmer, G. P. Ontai, and S. Ezekiel, “Precision studies of stimulated resonance Raman interactions in an atomic beam,” J. Opt. Soc. Am. B 3, 219 (1986); P. R. Hemmer, S. Ezekiel, and C. C. Leiby, “Stabilization of a microwave oscillator using a resonance Raman transition in a sodium beam,” Opt. Lett. 8, 440 (1983); P. Knight, “New frequency standards from ultra-narrow Raman resonances,” Nature (London) 297, 16 (1982); J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Observation of Ramsey fringes using a stimulated resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867 (1982); J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Ultrahigh resolution spectroscopy and frequency standards in the microwave and far-infrared region using optical lasers,” Opt. Lett. 6, 298 (1981).
    [CrossRef] [PubMed]
  7. P. R. Hemmer, V. D. Natoli, M. S. Shahriar, B. Bernacki, H. Lamela-Rivera, S. P. Smith, and S. Ezekiel, in 41st Annual Symposium on Frequency Control (Institute of Electrical and Electronics Engineers, New York, 1987), p. 42.
    [CrossRef]
  8. E. De Clercq and P. Cerez, “Evaluation of the light shift in a frequency standard based on Raman induced Ramsey resonance,” Opt. Comm. 45, 91 (1983); B. J. Dalton and P. L. Knight, “The effects of laser field fluctuations on coherent population trapping,” J. Phys. B. 15, 3997 (1982).
    [CrossRef]
  9. N. F. Ramsey, Molecular Beams (Oxford U. Press, London, 1963), Chap. 5, Sec. 3.
  10. Average laser intensity is defined here by the following:(Ω2)average=1τ1/2∫-∞∞Ω2(t)dt,where τ1/2is the atom transit time corresponding to the half-intensity positions on the actual laser beam profile and Ω2= ½(|Ω1|2+ |Ω2|2. Here, |Ω1|2 and |Ω2|2 are the laser intensities at ω1and ω2, respectively, in units of Rabi frequency squared.
  11. For a nonrectangular laser beam profile, Ω is a function of time for a moving atom in the atomic beam. In that case, Eqs. (6a) and (6b) are modified by making the making the following replacements:Ω2τ→∫-∞∞Ω2(t)dt,         fΩ2τ→∫-∞∞f(t)Ω2(t)dt,where it is assumed that the two interaction regions do not overlap, so that∫(ΩA2+ΩB2)dt=∫ΩA2dt+∫ΩB2dt.For the intensities used in our experiments, f is nearly a constant (unity) and can be pulled outside the integral.
  12. Interaction times and transit times are computed by using the thermal velocity v=2kT/M characteristic of a sodium beam produced by a 400°C oven.
  13. In general, velocity averaging is accomplished by simply performing a weighted average over all the velocities present in the atomic beam. However, in the present case the ac Stark shift was measured by locking to a minimum of a velocity averaged Ramsey-fringe line shape. This ac Stark shift is not the same as the weighted average of the ac Stark shifts for each atomic velocity. Nevertheless, over the limited ranges of common-mode laser detuning where ϕ and ΔT are both small (so that ϕ≅ sin ϕ and ΔT≅ sin ΔT) for all velocities included in the average, a relatively simple numerical calculation is possible and is found to agree well with the single-velocity results presented here. Therefore it is anticipated that the complete numerical calculation of the Raman two-zone ac Stark effect, including velocity averaging in a thermal sodium beam, will probably give results that are qualitatively similar to those presented in this paper.
  14. It should be pointed out that the effective value of r as indicated is merely an estimate of the first-order correction to the closed three-level-system results. The exact solution of the sodium system is more complex than simply using an effective value of r and is currently in progress.

1988 (1)

J. Mlynek and R. Grimm, “Raman heterodyne Ramsey spectroscopy in a samarium atomic beam” Appl. Phys. B 45, 77 (1988); M. Kaivola, P. Thorsen, and O. Poulsen, “Dispersive line shapes and optical pumping in a three-level system,” Phys. Rev. A 32, 207 (1985); F. Shimizu, K. Shimizu, and H. Takuma, “Selective vibrational pumping of a molecular beam by a stimulated Raman process,” Phys. Rev. A 31, 3132 (1985); A. Sharma, W. Happar, and Y. Q. Lu, “Sub-Doppler-broadened magnetic field resonances in the resonant stimulated electronic Raman scattering of multimode laser light,” Phys. Rev. A 29, 749 (1984); R. E. Tench and S. Ezekiel, “Precision measurements of hyperfine predissociation in I2vapor using a two-photon resonant scattering technique,” Chem. Phys. Lett. 96, 253 (1983); R. E. Tench, B. W. Peuse, P. R. Hemmer, J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Two laser Raman difference frequency technique applied to high precision spectroscopy,” J. Phys. Colloq. 42, 45 (1981); P. Kumar and J. H. Shapiro, “Observation of Raman-shifted oscillation near the sodium D lines,” Opt. Lett. 10, 226 (1985); M. S. Feld, M. M. Burns, T. U. Kuhl, P. G. Pappas, and D. E. Murnick, “Laser-saturation spectroscopy with optical pumping,” Opt. Lett. 5, 79 (1980); G. Alzetta, L. Moi, and G. Orriols, “Nonabsorption hyperfine resonances in a sodium vapor irradiated by a multiniode dye-laser,” Nuovo Cimento B 52, 209 (1979); R. P. Hackel and S. Ezekiel, “Observation of subnatural linewidths by two-step resonant scattering in I2vapor,” Phys. Rev. Lett. 42, 1736 (1979); K. Takagi, R. F. Curl, and R. T. M. Su, “Spectroscopy with modulation sidebands,” Appl. Phys. 7, 181 (1975); R. L. Shoemaker and R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430, (1972).
[CrossRef] [PubMed]

1986 (4)

E. Buhr and J. Mlynek, “Collision-induced Ramsey resonances in Sm vapor,” Phys. Rev. Lett. 57, 1300 (1986); A. A. Dabagyan, M. E. Movsesyan, T. O. Ovakimyan, and S. V. Shmavonyan, “Stimulated processes in potassium vapor in the presence of a buffer gas,” Sov. Phys. JETP 58, 700 (1983).
[CrossRef] [PubMed]

D. Krokel, K. Ludewigt, and H. Welling, “Frequency up-conversion by stimulated hyper-Raman scattering,” IEEE J. Quantum Electron. QE-22, 489 (1986); R. S. F. Chang, M. T. Duignan, R. H. Lehmberg, and N. Djeu, “Use of stimulated Raman scattering for reducing the divergence of severely aberrated laser beams,” in Excimer Lasers: Their Applications and New Frontiers in Lasers, R. W. Waynank, ed., Proc. Soc. Photo-Opt. Eng.476, 81 (1984); J. C. White, “Up-conversion of excimer lasers via stimulated anti-Stokes Raman scattering,” IEEE J. Quantum Electron. QE-20, 185 (1984); N. V. Znamenskii and V. I. Odintsov, “Infrared stimulated Raman scattering in rubidium vapor with a tunable pump frequency,” Opt. Spectrosc. (USSR) 54, 55 (1983); R. Wyatt, N. P. Ernsting, and W. G. Wrobel, “Tunable electronic Raman laser at 16 microns,” Appl. Phys. B 27, 175 (1982); M. L. Steyn-Ross and D. F. Walls, “Quantum theory of a Raman laser,” Opt. Acta 28, 201 (1981).
[CrossRef]

For example, B. J. Dalton, T. D. Kieu, and P. L. Knight, “Theory of ultra-high-resolution optical Raman Ramsey spectroscopy,” Opt. Acta 33, 459 (1986); D. Pegg, “Interaction of three-level atoms with modulated lasers,” Opt. Acta 33, 363 (1986); N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984); S. Swain, “Conditions for population trapping in a three-level system,” J. Phys. B 15, 3405 (1982); P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a three-level system,” J. Phys. B 15, 561 (1982); G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento B 53, 1 (1979); H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978); A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
[CrossRef] [PubMed]

P. R. Hemmer, G. P. Ontai, and S. Ezekiel, “Precision studies of stimulated resonance Raman interactions in an atomic beam,” J. Opt. Soc. Am. B 3, 219 (1986); P. R. Hemmer, S. Ezekiel, and C. C. Leiby, “Stabilization of a microwave oscillator using a resonance Raman transition in a sodium beam,” Opt. Lett. 8, 440 (1983); P. Knight, “New frequency standards from ultra-narrow Raman resonances,” Nature (London) 297, 16 (1982); J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Observation of Ramsey fringes using a stimulated resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867 (1982); J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Ultrahigh resolution spectroscopy and frequency standards in the microwave and far-infrared region using optical lasers,” Opt. Lett. 6, 298 (1981).
[CrossRef] [PubMed]

1984 (1)

P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriasca 56, 103 (1984); F. H. Mies and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981); E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977); M. Sargent and P. Horwitz, “Three-level Rabi flopping,” Phys. Rev. A 13, 1962 (1976); R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady state cases,” Phys. Rev. A 11, 1641 (1975).
[CrossRef]

1983 (1)

E. De Clercq and P. Cerez, “Evaluation of the light shift in a frequency standard based on Raman induced Ramsey resonance,” Opt. Comm. 45, 91 (1983); B. J. Dalton and P. L. Knight, “The effects of laser field fluctuations on coherent population trapping,” J. Phys. B. 15, 3997 (1982).
[CrossRef]

Bernacki, B.

P. R. Hemmer, V. D. Natoli, M. S. Shahriar, B. Bernacki, H. Lamela-Rivera, S. P. Smith, and S. Ezekiel, in 41st Annual Symposium on Frequency Control (Institute of Electrical and Electronics Engineers, New York, 1987), p. 42.
[CrossRef]

Buhr, E.

E. Buhr and J. Mlynek, “Collision-induced Ramsey resonances in Sm vapor,” Phys. Rev. Lett. 57, 1300 (1986); A. A. Dabagyan, M. E. Movsesyan, T. O. Ovakimyan, and S. V. Shmavonyan, “Stimulated processes in potassium vapor in the presence of a buffer gas,” Sov. Phys. JETP 58, 700 (1983).
[CrossRef] [PubMed]

Cerez, P.

E. De Clercq and P. Cerez, “Evaluation of the light shift in a frequency standard based on Raman induced Ramsey resonance,” Opt. Comm. 45, 91 (1983); B. J. Dalton and P. L. Knight, “The effects of laser field fluctuations on coherent population trapping,” J. Phys. B. 15, 3997 (1982).
[CrossRef]

Dalton, B. J.

For example, B. J. Dalton, T. D. Kieu, and P. L. Knight, “Theory of ultra-high-resolution optical Raman Ramsey spectroscopy,” Opt. Acta 33, 459 (1986); D. Pegg, “Interaction of three-level atoms with modulated lasers,” Opt. Acta 33, 363 (1986); N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984); S. Swain, “Conditions for population trapping in a three-level system,” J. Phys. B 15, 3405 (1982); P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a three-level system,” J. Phys. B 15, 561 (1982); G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento B 53, 1 (1979); H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978); A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
[CrossRef] [PubMed]

P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriasca 56, 103 (1984); F. H. Mies and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981); E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977); M. Sargent and P. Horwitz, “Three-level Rabi flopping,” Phys. Rev. A 13, 1962 (1976); R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady state cases,” Phys. Rev. A 11, 1641 (1975).
[CrossRef]

De Clercq, E.

E. De Clercq and P. Cerez, “Evaluation of the light shift in a frequency standard based on Raman induced Ramsey resonance,” Opt. Comm. 45, 91 (1983); B. J. Dalton and P. L. Knight, “The effects of laser field fluctuations on coherent population trapping,” J. Phys. B. 15, 3997 (1982).
[CrossRef]

Ezekiel, S.

Grimm, R.

J. Mlynek and R. Grimm, “Raman heterodyne Ramsey spectroscopy in a samarium atomic beam” Appl. Phys. B 45, 77 (1988); M. Kaivola, P. Thorsen, and O. Poulsen, “Dispersive line shapes and optical pumping in a three-level system,” Phys. Rev. A 32, 207 (1985); F. Shimizu, K. Shimizu, and H. Takuma, “Selective vibrational pumping of a molecular beam by a stimulated Raman process,” Phys. Rev. A 31, 3132 (1985); A. Sharma, W. Happar, and Y. Q. Lu, “Sub-Doppler-broadened magnetic field resonances in the resonant stimulated electronic Raman scattering of multimode laser light,” Phys. Rev. A 29, 749 (1984); R. E. Tench and S. Ezekiel, “Precision measurements of hyperfine predissociation in I2vapor using a two-photon resonant scattering technique,” Chem. Phys. Lett. 96, 253 (1983); R. E. Tench, B. W. Peuse, P. R. Hemmer, J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Two laser Raman difference frequency technique applied to high precision spectroscopy,” J. Phys. Colloq. 42, 45 (1981); P. Kumar and J. H. Shapiro, “Observation of Raman-shifted oscillation near the sodium D lines,” Opt. Lett. 10, 226 (1985); M. S. Feld, M. M. Burns, T. U. Kuhl, P. G. Pappas, and D. E. Murnick, “Laser-saturation spectroscopy with optical pumping,” Opt. Lett. 5, 79 (1980); G. Alzetta, L. Moi, and G. Orriols, “Nonabsorption hyperfine resonances in a sodium vapor irradiated by a multiniode dye-laser,” Nuovo Cimento B 52, 209 (1979); R. P. Hackel and S. Ezekiel, “Observation of subnatural linewidths by two-step resonant scattering in I2vapor,” Phys. Rev. Lett. 42, 1736 (1979); K. Takagi, R. F. Curl, and R. T. M. Su, “Spectroscopy with modulation sidebands,” Appl. Phys. 7, 181 (1975); R. L. Shoemaker and R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430, (1972).
[CrossRef] [PubMed]

Hemmer, P. R.

Kieu, T. D.

For example, B. J. Dalton, T. D. Kieu, and P. L. Knight, “Theory of ultra-high-resolution optical Raman Ramsey spectroscopy,” Opt. Acta 33, 459 (1986); D. Pegg, “Interaction of three-level atoms with modulated lasers,” Opt. Acta 33, 363 (1986); N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984); S. Swain, “Conditions for population trapping in a three-level system,” J. Phys. B 15, 3405 (1982); P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a three-level system,” J. Phys. B 15, 561 (1982); G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento B 53, 1 (1979); H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978); A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
[CrossRef] [PubMed]

Knight, P. L.

For example, B. J. Dalton, T. D. Kieu, and P. L. Knight, “Theory of ultra-high-resolution optical Raman Ramsey spectroscopy,” Opt. Acta 33, 459 (1986); D. Pegg, “Interaction of three-level atoms with modulated lasers,” Opt. Acta 33, 363 (1986); N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984); S. Swain, “Conditions for population trapping in a three-level system,” J. Phys. B 15, 3405 (1982); P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a three-level system,” J. Phys. B 15, 561 (1982); G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento B 53, 1 (1979); H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978); A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
[CrossRef] [PubMed]

P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriasca 56, 103 (1984); F. H. Mies and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981); E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977); M. Sargent and P. Horwitz, “Three-level Rabi flopping,” Phys. Rev. A 13, 1962 (1976); R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady state cases,” Phys. Rev. A 11, 1641 (1975).
[CrossRef]

Krokel, D.

D. Krokel, K. Ludewigt, and H. Welling, “Frequency up-conversion by stimulated hyper-Raman scattering,” IEEE J. Quantum Electron. QE-22, 489 (1986); R. S. F. Chang, M. T. Duignan, R. H. Lehmberg, and N. Djeu, “Use of stimulated Raman scattering for reducing the divergence of severely aberrated laser beams,” in Excimer Lasers: Their Applications and New Frontiers in Lasers, R. W. Waynank, ed., Proc. Soc. Photo-Opt. Eng.476, 81 (1984); J. C. White, “Up-conversion of excimer lasers via stimulated anti-Stokes Raman scattering,” IEEE J. Quantum Electron. QE-20, 185 (1984); N. V. Znamenskii and V. I. Odintsov, “Infrared stimulated Raman scattering in rubidium vapor with a tunable pump frequency,” Opt. Spectrosc. (USSR) 54, 55 (1983); R. Wyatt, N. P. Ernsting, and W. G. Wrobel, “Tunable electronic Raman laser at 16 microns,” Appl. Phys. B 27, 175 (1982); M. L. Steyn-Ross and D. F. Walls, “Quantum theory of a Raman laser,” Opt. Acta 28, 201 (1981).
[CrossRef]

Lamela-Rivera, H.

P. R. Hemmer, V. D. Natoli, M. S. Shahriar, B. Bernacki, H. Lamela-Rivera, S. P. Smith, and S. Ezekiel, in 41st Annual Symposium on Frequency Control (Institute of Electrical and Electronics Engineers, New York, 1987), p. 42.
[CrossRef]

Lauder, M. A.

P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriasca 56, 103 (1984); F. H. Mies and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981); E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977); M. Sargent and P. Horwitz, “Three-level Rabi flopping,” Phys. Rev. A 13, 1962 (1976); R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady state cases,” Phys. Rev. A 11, 1641 (1975).
[CrossRef]

Ludewigt, K.

D. Krokel, K. Ludewigt, and H. Welling, “Frequency up-conversion by stimulated hyper-Raman scattering,” IEEE J. Quantum Electron. QE-22, 489 (1986); R. S. F. Chang, M. T. Duignan, R. H. Lehmberg, and N. Djeu, “Use of stimulated Raman scattering for reducing the divergence of severely aberrated laser beams,” in Excimer Lasers: Their Applications and New Frontiers in Lasers, R. W. Waynank, ed., Proc. Soc. Photo-Opt. Eng.476, 81 (1984); J. C. White, “Up-conversion of excimer lasers via stimulated anti-Stokes Raman scattering,” IEEE J. Quantum Electron. QE-20, 185 (1984); N. V. Znamenskii and V. I. Odintsov, “Infrared stimulated Raman scattering in rubidium vapor with a tunable pump frequency,” Opt. Spectrosc. (USSR) 54, 55 (1983); R. Wyatt, N. P. Ernsting, and W. G. Wrobel, “Tunable electronic Raman laser at 16 microns,” Appl. Phys. B 27, 175 (1982); M. L. Steyn-Ross and D. F. Walls, “Quantum theory of a Raman laser,” Opt. Acta 28, 201 (1981).
[CrossRef]

Mlynek, J.

J. Mlynek and R. Grimm, “Raman heterodyne Ramsey spectroscopy in a samarium atomic beam” Appl. Phys. B 45, 77 (1988); M. Kaivola, P. Thorsen, and O. Poulsen, “Dispersive line shapes and optical pumping in a three-level system,” Phys. Rev. A 32, 207 (1985); F. Shimizu, K. Shimizu, and H. Takuma, “Selective vibrational pumping of a molecular beam by a stimulated Raman process,” Phys. Rev. A 31, 3132 (1985); A. Sharma, W. Happar, and Y. Q. Lu, “Sub-Doppler-broadened magnetic field resonances in the resonant stimulated electronic Raman scattering of multimode laser light,” Phys. Rev. A 29, 749 (1984); R. E. Tench and S. Ezekiel, “Precision measurements of hyperfine predissociation in I2vapor using a two-photon resonant scattering technique,” Chem. Phys. Lett. 96, 253 (1983); R. E. Tench, B. W. Peuse, P. R. Hemmer, J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Two laser Raman difference frequency technique applied to high precision spectroscopy,” J. Phys. Colloq. 42, 45 (1981); P. Kumar and J. H. Shapiro, “Observation of Raman-shifted oscillation near the sodium D lines,” Opt. Lett. 10, 226 (1985); M. S. Feld, M. M. Burns, T. U. Kuhl, P. G. Pappas, and D. E. Murnick, “Laser-saturation spectroscopy with optical pumping,” Opt. Lett. 5, 79 (1980); G. Alzetta, L. Moi, and G. Orriols, “Nonabsorption hyperfine resonances in a sodium vapor irradiated by a multiniode dye-laser,” Nuovo Cimento B 52, 209 (1979); R. P. Hackel and S. Ezekiel, “Observation of subnatural linewidths by two-step resonant scattering in I2vapor,” Phys. Rev. Lett. 42, 1736 (1979); K. Takagi, R. F. Curl, and R. T. M. Su, “Spectroscopy with modulation sidebands,” Appl. Phys. 7, 181 (1975); R. L. Shoemaker and R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430, (1972).
[CrossRef] [PubMed]

E. Buhr and J. Mlynek, “Collision-induced Ramsey resonances in Sm vapor,” Phys. Rev. Lett. 57, 1300 (1986); A. A. Dabagyan, M. E. Movsesyan, T. O. Ovakimyan, and S. V. Shmavonyan, “Stimulated processes in potassium vapor in the presence of a buffer gas,” Sov. Phys. JETP 58, 700 (1983).
[CrossRef] [PubMed]

Natoli, V. D.

P. R. Hemmer, V. D. Natoli, M. S. Shahriar, B. Bernacki, H. Lamela-Rivera, S. P. Smith, and S. Ezekiel, in 41st Annual Symposium on Frequency Control (Institute of Electrical and Electronics Engineers, New York, 1987), p. 42.
[CrossRef]

Ontai, G. P.

Radmore, P. M.

P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriasca 56, 103 (1984); F. H. Mies and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981); E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977); M. Sargent and P. Horwitz, “Three-level Rabi flopping,” Phys. Rev. A 13, 1962 (1976); R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady state cases,” Phys. Rev. A 11, 1641 (1975).
[CrossRef]

Ramsey, N. F.

N. F. Ramsey, Molecular Beams (Oxford U. Press, London, 1963), Chap. 5, Sec. 3.

Shahriar, M. S.

P. R. Hemmer, V. D. Natoli, M. S. Shahriar, B. Bernacki, H. Lamela-Rivera, S. P. Smith, and S. Ezekiel, in 41st Annual Symposium on Frequency Control (Institute of Electrical and Electronics Engineers, New York, 1987), p. 42.
[CrossRef]

Smith, S. P.

P. R. Hemmer, V. D. Natoli, M. S. Shahriar, B. Bernacki, H. Lamela-Rivera, S. P. Smith, and S. Ezekiel, in 41st Annual Symposium on Frequency Control (Institute of Electrical and Electronics Engineers, New York, 1987), p. 42.
[CrossRef]

Welling, H.

D. Krokel, K. Ludewigt, and H. Welling, “Frequency up-conversion by stimulated hyper-Raman scattering,” IEEE J. Quantum Electron. QE-22, 489 (1986); R. S. F. Chang, M. T. Duignan, R. H. Lehmberg, and N. Djeu, “Use of stimulated Raman scattering for reducing the divergence of severely aberrated laser beams,” in Excimer Lasers: Their Applications and New Frontiers in Lasers, R. W. Waynank, ed., Proc. Soc. Photo-Opt. Eng.476, 81 (1984); J. C. White, “Up-conversion of excimer lasers via stimulated anti-Stokes Raman scattering,” IEEE J. Quantum Electron. QE-20, 185 (1984); N. V. Znamenskii and V. I. Odintsov, “Infrared stimulated Raman scattering in rubidium vapor with a tunable pump frequency,” Opt. Spectrosc. (USSR) 54, 55 (1983); R. Wyatt, N. P. Ernsting, and W. G. Wrobel, “Tunable electronic Raman laser at 16 microns,” Appl. Phys. B 27, 175 (1982); M. L. Steyn-Ross and D. F. Walls, “Quantum theory of a Raman laser,” Opt. Acta 28, 201 (1981).
[CrossRef]

Acta Phys. Austriasca (1)

P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriasca 56, 103 (1984); F. H. Mies and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981); E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977); M. Sargent and P. Horwitz, “Three-level Rabi flopping,” Phys. Rev. A 13, 1962 (1976); R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady state cases,” Phys. Rev. A 11, 1641 (1975).
[CrossRef]

Appl. Phys. B (1)

J. Mlynek and R. Grimm, “Raman heterodyne Ramsey spectroscopy in a samarium atomic beam” Appl. Phys. B 45, 77 (1988); M. Kaivola, P. Thorsen, and O. Poulsen, “Dispersive line shapes and optical pumping in a three-level system,” Phys. Rev. A 32, 207 (1985); F. Shimizu, K. Shimizu, and H. Takuma, “Selective vibrational pumping of a molecular beam by a stimulated Raman process,” Phys. Rev. A 31, 3132 (1985); A. Sharma, W. Happar, and Y. Q. Lu, “Sub-Doppler-broadened magnetic field resonances in the resonant stimulated electronic Raman scattering of multimode laser light,” Phys. Rev. A 29, 749 (1984); R. E. Tench and S. Ezekiel, “Precision measurements of hyperfine predissociation in I2vapor using a two-photon resonant scattering technique,” Chem. Phys. Lett. 96, 253 (1983); R. E. Tench, B. W. Peuse, P. R. Hemmer, J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Two laser Raman difference frequency technique applied to high precision spectroscopy,” J. Phys. Colloq. 42, 45 (1981); P. Kumar and J. H. Shapiro, “Observation of Raman-shifted oscillation near the sodium D lines,” Opt. Lett. 10, 226 (1985); M. S. Feld, M. M. Burns, T. U. Kuhl, P. G. Pappas, and D. E. Murnick, “Laser-saturation spectroscopy with optical pumping,” Opt. Lett. 5, 79 (1980); G. Alzetta, L. Moi, and G. Orriols, “Nonabsorption hyperfine resonances in a sodium vapor irradiated by a multiniode dye-laser,” Nuovo Cimento B 52, 209 (1979); R. P. Hackel and S. Ezekiel, “Observation of subnatural linewidths by two-step resonant scattering in I2vapor,” Phys. Rev. Lett. 42, 1736 (1979); K. Takagi, R. F. Curl, and R. T. M. Su, “Spectroscopy with modulation sidebands,” Appl. Phys. 7, 181 (1975); R. L. Shoemaker and R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430, (1972).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

D. Krokel, K. Ludewigt, and H. Welling, “Frequency up-conversion by stimulated hyper-Raman scattering,” IEEE J. Quantum Electron. QE-22, 489 (1986); R. S. F. Chang, M. T. Duignan, R. H. Lehmberg, and N. Djeu, “Use of stimulated Raman scattering for reducing the divergence of severely aberrated laser beams,” in Excimer Lasers: Their Applications and New Frontiers in Lasers, R. W. Waynank, ed., Proc. Soc. Photo-Opt. Eng.476, 81 (1984); J. C. White, “Up-conversion of excimer lasers via stimulated anti-Stokes Raman scattering,” IEEE J. Quantum Electron. QE-20, 185 (1984); N. V. Znamenskii and V. I. Odintsov, “Infrared stimulated Raman scattering in rubidium vapor with a tunable pump frequency,” Opt. Spectrosc. (USSR) 54, 55 (1983); R. Wyatt, N. P. Ernsting, and W. G. Wrobel, “Tunable electronic Raman laser at 16 microns,” Appl. Phys. B 27, 175 (1982); M. L. Steyn-Ross and D. F. Walls, “Quantum theory of a Raman laser,” Opt. Acta 28, 201 (1981).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Acta (1)

For example, B. J. Dalton, T. D. Kieu, and P. L. Knight, “Theory of ultra-high-resolution optical Raman Ramsey spectroscopy,” Opt. Acta 33, 459 (1986); D. Pegg, “Interaction of three-level atoms with modulated lasers,” Opt. Acta 33, 363 (1986); N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984); S. Swain, “Conditions for population trapping in a three-level system,” J. Phys. B 15, 3405 (1982); P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a three-level system,” J. Phys. B 15, 561 (1982); G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento B 53, 1 (1979); H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978); A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
[CrossRef] [PubMed]

Opt. Comm. (1)

E. De Clercq and P. Cerez, “Evaluation of the light shift in a frequency standard based on Raman induced Ramsey resonance,” Opt. Comm. 45, 91 (1983); B. J. Dalton and P. L. Knight, “The effects of laser field fluctuations on coherent population trapping,” J. Phys. B. 15, 3997 (1982).
[CrossRef]

Phys. Rev. Lett. (1)

E. Buhr and J. Mlynek, “Collision-induced Ramsey resonances in Sm vapor,” Phys. Rev. Lett. 57, 1300 (1986); A. A. Dabagyan, M. E. Movsesyan, T. O. Ovakimyan, and S. V. Shmavonyan, “Stimulated processes in potassium vapor in the presence of a buffer gas,” Sov. Phys. JETP 58, 700 (1983).
[CrossRef] [PubMed]

Other (7)

P. R. Hemmer, V. D. Natoli, M. S. Shahriar, B. Bernacki, H. Lamela-Rivera, S. P. Smith, and S. Ezekiel, in 41st Annual Symposium on Frequency Control (Institute of Electrical and Electronics Engineers, New York, 1987), p. 42.
[CrossRef]

N. F. Ramsey, Molecular Beams (Oxford U. Press, London, 1963), Chap. 5, Sec. 3.

Average laser intensity is defined here by the following:(Ω2)average=1τ1/2∫-∞∞Ω2(t)dt,where τ1/2is the atom transit time corresponding to the half-intensity positions on the actual laser beam profile and Ω2= ½(|Ω1|2+ |Ω2|2. Here, |Ω1|2 and |Ω2|2 are the laser intensities at ω1and ω2, respectively, in units of Rabi frequency squared.

For a nonrectangular laser beam profile, Ω is a function of time for a moving atom in the atomic beam. In that case, Eqs. (6a) and (6b) are modified by making the making the following replacements:Ω2τ→∫-∞∞Ω2(t)dt,         fΩ2τ→∫-∞∞f(t)Ω2(t)dt,where it is assumed that the two interaction regions do not overlap, so that∫(ΩA2+ΩB2)dt=∫ΩA2dt+∫ΩB2dt.For the intensities used in our experiments, f is nearly a constant (unity) and can be pulled outside the integral.

Interaction times and transit times are computed by using the thermal velocity v=2kT/M characteristic of a sodium beam produced by a 400°C oven.

In general, velocity averaging is accomplished by simply performing a weighted average over all the velocities present in the atomic beam. However, in the present case the ac Stark shift was measured by locking to a minimum of a velocity averaged Ramsey-fringe line shape. This ac Stark shift is not the same as the weighted average of the ac Stark shifts for each atomic velocity. Nevertheless, over the limited ranges of common-mode laser detuning where ϕ and ΔT are both small (so that ϕ≅ sin ϕ and ΔT≅ sin ΔT) for all velocities included in the average, a relatively simple numerical calculation is possible and is found to agree well with the single-velocity results presented here. Therefore it is anticipated that the complete numerical calculation of the Raman two-zone ac Stark effect, including velocity averaging in a thermal sodium beam, will probably give results that are qualitatively similar to those presented in this paper.

It should be pointed out that the effective value of r as indicated is merely an estimate of the first-order correction to the closed three-level-system results. The exact solution of the sodium system is more complex than simply using an effective value of r and is currently in progress.

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Figures (11)

Fig. 1
Fig. 1

(a) Schematic of stimulated resonance Raman interaction, (b) single-zone resonance Raman line shape, (c) schematic of off-resonance Raman interaction, and (d) single-zone off-resonance Raman line shape.

Fig. 2
Fig. 2

(a) Schematic of separated-field Raman excitation, (b) two-zone Ramsey-fringe line shape observed in zone B fluorescence, (c) theoretical fringe shape resulting from velocity averaging, and (d) theoretical velocity-averaged fringe shape for π/2 phase-shifted fringes.

Fig. 3
Fig. 3

Observed Ramsey-fringe asymmetries for a fixed common-mode detuning of δ = 0.8γ2. Initial populations: (a) ρ110 = N, ρ330 = 0; (b) ρ110 = ρ330 = N/2; (c) ρ110 = 0, ρ330 = N.

Fig. 4
Fig. 4

Schematic of experimental setup for measuring the ac Stark shift as a function of common-mode laser detuning. DET’s, detectors.

Fig. 5
Fig. 5

Data showing the ac Stark shift as a function of common-mode laser detuning, for several combinations of laser intensity and initial populations, as labeled.

Fig. 6
Fig. 6

Calculated ac Stark shift, for conditions corresponding to the data presented in Fig. 5.

Fig. 7
Fig. 7

Illustration of the insensitivity of the ac Stark shift to differences in Rabi frequencies. (Top trace) Data showing the ac Stark shift for unequal Rabi frequencies, Ω 1 / Ω 2 = 2. (Middle trace) Theoretical ac Stark shift for Ω 1 / Ω 2 = 2. (Bottom trace) Theoretical ac Stark shift for |Ω1| = |Ω2| but same (|Ω1|2 + |Ω2|2).

Fig. 8
Fig. 8

Illustration of insensitivity of the ac Stark shift to the exact laser beam profile. Zone A transit times, τ are as indicated (Ω2τ remains constant). Top traces are experimental, bottom traces are theory.

Fig. 9
Fig. 9

Sodium hyperfine states involved in Raman interaction, for circularly polarized light. Solid lines are possible Raman transitions. Heavy solid lines show the m = 0, Δm = 0 Raman transitions used in these two-zone studies. Dotted lines are spontaneous decay paths. Number are relative matrix elements squared.

Fig. 10
Fig. 10

Revised theoretical plots of the ac Stark shift, for conditions corresponding to the data presented in Fig. 5. Effects of nearby magnetic sublevels are included to first order by using an effective normalized difference of within-system decay rates, r′ = 0.275.

Fig. 11
Fig. 11

High-resolution ac Stark shift versus laser detuning: (a) experimental trace for Ω = 0.13γ2, (ρ110ρ330) = 0.7N, and (b) corresponding theoretical best fit, with r′ = 0.27.

Tables (4)

Tables Icon

Table 1 Definitions Used in Eq. (1)

Tables Icon

Table 2 Approximations Used in Eq. (3)

Tables Icon

Table 3 Definitions Used in Eq. (3)

Tables Icon

Table 4 Approximations Used in Eq. (5)

Equations (30)

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ρ ˙ 11 = - ( ½ i Ω 1 α 12 + c . c . ) + Γ 21 ρ 22 ,
ρ ˙ 22 = ( ½ i Ω 1 α 12 + c . c . ) + ( ½ i Ω 2 α 32 + c . c . ) - γ 2 ρ 22 ,
ρ ˙ 33 = - ( ½ i Ω 2 α 32 + c . c . ) + Γ 23 ρ 22 ,
α ˙ 12 = ½ i Ω 1 * ( ρ 22 - ρ 11 ) - ½ i Ω 2 * α 13 - ( ½ γ 2 + i δ 1 ) α 12 ,
α ˙ 32 = ½ i Ω 2 * ( ρ 22 - ρ 33 ) - ½ i Ω 1 * α 13 * - ( ½ γ 2 + i δ 2 ) α 32 ,
α ˙ 13 = ½ i Ω 1 * α 32 * - ½ i Ω 2 α 12 - i ( δ 1 - δ 2 ) α 13 ,
E ( r , t ) = ½ ( E 1 ( r ) exp ( - i ω 1 t ) + c . c . ) + ½ ( E 2 ( r ) exp ( - i ω 2 t ) + c . c . ) .
d d t [ 2 Re α 13 2 Im α 13 ρ 11 - ρ 33 ] = A [ 2 Re α 13 2 Im α 13 ρ 11 - ρ 33 ] + B ,
ρ 22 = Ω 2 S f γ 2 [ N + d ( ρ 11 - ρ 33 ) + g ( 2 Re α 13 ) ] .
A = [ - [ 1 - g 2 ( 1 - f ) ] Ω 2 S ( Δ - d Ω 2 D ) g d ( 1 - f ) Ω 2 S - ( Δ - d Ω 2 D ) - Ω 2 S - g Ω 2 D [ r d f + g d ( 1 - f ) ] Ω 2 S g Ω 2 D [ r d f - 1 + d 2 ( 1 - f ) ] Ω 2 S ] ,
B = N Ω 2 S [ - g f 0 ( r - d ) f ] .
d d t [ 2 Re α 13 2 Im α 13 ρ 11 - ρ 33 ] = [ 0 Δ 0 - Δ 0 0 0 0 0 ] [ 2 Re α 13 2 Im α 13 ρ 11 - ρ 33 ]
ρ 22 = 0.
d d t [ 2 Re α 13 2 Im α 13 ρ 11 - ρ 33 ] = [ f Ω 2 S 0 0 0 - Ω 2 S - Ω 2 D 0 Ω 2 D - Ω 2 S ] [ 2 Re α 13 2 Im α 13 ρ 11 - ρ 33 ] + N [ - f Ω 2 S 0 0 ]
ρ 22 = Ω 2 S f γ 2 ( N + 2 Re α 13 ) .
T T + T B γ 2 ρ 22 d t = N [ 1 - exp ( - f Ω 2 S B τ B ) ] × { 1 + [ 1 - exp ( - f Ω 2 S A τ A ) ] sec ϕ cos ( Δ T - ϕ ) } ,
tan ϕ = Im α 13 ( τ A ) Re α 13 ( τ A ) = - ( ρ 11 0 - ρ 33 0 ) sin ( Ω 2 D A τ A ) exp ( - Ω 2 S A τ A ) N [ 1 - exp ( - f Ω 2 S A τ A ) ] .
tan ϕ = - ( ρ 11 0 - ρ 33 0 ) sin ( Ω 2 D τ ) exp ( - Ω 2 S τ ) N [ 1 - exp ( - f Ω 2 S τ ) ] + ξ d ,
ξ = ( ρ 11 0 - ρ 33 0 ) { cos ( Ω 2 D τ ) - exp [ ( 1 - f ) Ω 2 S τ ] } exp ( - Ω 2 S τ )
tan ϕ = - ( ρ 11 0 - ρ 33 0 ) [ sin ( Ω 2 D τ ) ] exp ( - Ω 2 S τ ) + r ζ N { 1 - exp [ - ( f - r d f ) Ω 2 S τ ] } + d ξ - r η ,
ζ = ½ f Ω 2 S [ E H - F G ] [ N + d ( ρ 11 0 - ρ 33 0 ) ] ,
η = ½ d f Ω 2 S [ E G + F H ] [ N - d ( ρ 11 0 - ρ 33 0 ) ] ,
ξ = ( ρ 11 0 - ρ 33 0 ) E ,
E = { cos ( Ω 2 D τ ) - exp [ ( 1 - f + r d f ) Ω 2 S τ ] } exp ( - Ω 2 S τ ) ,
F = [ sin ( Ω 2 D τ ) ] exp ( - Ω 2 S τ ) ,
G = 2 Ω 2 S ( 1 - f + r d f ) [ Ω 2 S ( 1 - f + r d f ) ] 2 + [ Ω 2 D ] 2 ,
H = - 2 Ω 2 D [ Ω 2 S ( 1 - f + r d f ) ] 2 + [ Ω 2 D ] 2 ,
(Ω2)average=1τ1/2-Ω2(t)dt,
Ω2τ-Ω2(t)dt,         fΩ2τ-f(t)Ω2(t)dt,
(ΩA2+ΩB2)dt=ΩA2dt+ΩB2dt.

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