Abstract

A set of three classical coupled pendulums is used to model the stimulated resonance Raman interaction. This model provides a simple, intuitive, physical description of the resonance Raman process and can also be used to interpret experimental observations, including the dynamics of Raman-induced transparency, the physical nature of Ramsey fringes in separated-field excitation, and the effects of off-resonant laser excitation. The model has also been extended to suggest what might be observed for strong laser fields.

© 1988 Optical Society of America

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  1. 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 I2 vapor using a two-photon resonant scattering technique,” Chem. Phys. Lett. 96, 253 (1983);K. Takagi, R. F. Curl, and R. T. M. Su, “Spectroscopy with modulation sidebands,” Appl. Phys. 7, 181 (1975).
    [Crossref] [PubMed]
  2. E. Buhr and J. Mlynek, “Collison-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]
  3. 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]
  4. 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 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]
  5. 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);P. F. Fracassi, L. Angeloni, and R. G. DellaValle, “Effects of dampings and dephasings in resonance Raman spectroscopy,” Chem. Phys. Lett. 115, 428 (1985);P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriaca 56, 103 (1984);V. Zadkov, N. Koroteev, M. Rychev, and A. Fedorov, “Saturated coherent Raman scattering spectroscopy for molecular gasses,” Moscow Univ. Phys. Bull. 39, 30 (1984);N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984);F. H. Mies, “Resonance fluorescence and Raman line shapes produced by monochromatic laser fields: effects of branching ratio and homogeneous broadening,” J. Quant.Spectrosc. Radiat. Transfer  29, 237 (1983);P. M. Radmore, “Population trapping in a multilevel system,” Phys. Rev. A 26, 2252 (1982);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);B. Halperin and J. A. Koninfstein, “Conditions for excited-state Raman and absorption processes during optical pumping,” Can. J. Chem. 59, 2792 (1981);F. H. Miew and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981);J. E. Thomas and W. W. Quivers, “Transit-time effects in optically pumped coupled three-level systems,” Phys. Rev. A 22, 2115, (1980);G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento 53B, 1 (1979);H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978);C. Cohen-Tannoudji and S. Reynaud, “Modification of resonance Raman scattering in very intense laser fields,” J. Phys. B 10, 365 (1977);E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977);R. M. Whitley and C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498 (1976);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);A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
    [Crossref] [PubMed]
  6. P. Kumar and J. H. Shapiro, “Observation of Raman-shifted oscillation near the sodium D lines,” Opt. Lett. 10, 226 (1985);M. Kaivola, P. Thorsen, and O. Poulsen, “Dispersive line shapes and optical pumping in a three-level system,” Phys. Rev. A 32, 207 (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 multimode dye-laser,” Nuovo Cimento 52B, 209 (1979);R. P. Hackel and S. Ezekiel, “Observation of sub-natural linewidths by two-step resonant scattering in I2 vapor,” Phys. Rev. Lett. 42, 1736 (1979);R. L. Shoemaker and R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430 (1972).
    [Crossref] [PubMed]
  7. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap.15.
  8. R. A. Becker, Introduction to Theoretical Mechanics (McGraw-Hill, New York, 1954), Chaps. 7 and 14.
  9. C. Cohen-Tannoudji, “Atoms 〈〈Habilles〉〉 per des photons optiques ou de radiofrequence,” J. Phys. C 5a, 11 (1971);C. Cohen-Tannoudji and S. Haroche, “Absorption et diffusion de photons optiques par un atome en interaction avec des photons de radiofrequence,” J. Phys. (Paris) 30, 153 (1969).
    [Crossref]
  10. P. R. Berman and R. Salomaa, “Comparison between dressed-atom and bare-atom pictures in laser spectroscopy,” Phys. Rev. A 25, 2667 (1982);P. R. Berman and J. Ziegler, “Generalized dressed-atom approach to atom-strong-field interactions—application to the theory of lasers and Bloch–Siegert shifts,” Phys. Rev. A 15, 2042 (1977).
    [Crossref]
  11. A. S. Davydov, Quantum Mechanics (Pergamon, Oxford, 1965), Chap. 9.
  12. N. F. Ramsey, Molecular Beams (Oxford U. Press, London, 1963), Chap. 5.

1986 (4)

E. Buhr and J. Mlynek, “Collison-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]

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 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]

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);P. F. Fracassi, L. Angeloni, and R. G. DellaValle, “Effects of dampings and dephasings in resonance Raman spectroscopy,” Chem. Phys. Lett. 115, 428 (1985);P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriaca 56, 103 (1984);V. Zadkov, N. Koroteev, M. Rychev, and A. Fedorov, “Saturated coherent Raman scattering spectroscopy for molecular gasses,” Moscow Univ. Phys. Bull. 39, 30 (1984);N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984);F. H. Mies, “Resonance fluorescence and Raman line shapes produced by monochromatic laser fields: effects of branching ratio and homogeneous broadening,” J. Quant.Spectrosc. Radiat. Transfer  29, 237 (1983);P. M. Radmore, “Population trapping in a multilevel system,” Phys. Rev. A 26, 2252 (1982);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);B. Halperin and J. A. Koninfstein, “Conditions for excited-state Raman and absorption processes during optical pumping,” Can. J. Chem. 59, 2792 (1981);F. H. Miew and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981);J. E. Thomas and W. W. Quivers, “Transit-time effects in optically pumped coupled three-level systems,” Phys. Rev. A 22, 2115, (1980);G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento 53B, 1 (1979);H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978);C. Cohen-Tannoudji and S. Reynaud, “Modification of resonance Raman scattering in very intense laser fields,” J. Phys. B 10, 365 (1977);E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977);R. M. Whitley and C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498 (1976);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);A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
[Crossref] [PubMed]

1985 (2)

P. Kumar and J. H. Shapiro, “Observation of Raman-shifted oscillation near the sodium D lines,” Opt. Lett. 10, 226 (1985);M. Kaivola, P. Thorsen, and O. Poulsen, “Dispersive line shapes and optical pumping in a three-level system,” Phys. Rev. A 32, 207 (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 multimode dye-laser,” Nuovo Cimento 52B, 209 (1979);R. P. Hackel and S. Ezekiel, “Observation of sub-natural linewidths by two-step resonant scattering in I2 vapor,” Phys. Rev. Lett. 42, 1736 (1979);R. L. Shoemaker and R. G. Brewer, “Two-photon superradiance,” Phys. Rev. Lett. 28, 1430 (1972).
[Crossref] [PubMed]

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 I2 vapor using a two-photon resonant scattering technique,” Chem. Phys. Lett. 96, 253 (1983);K. Takagi, R. F. Curl, and R. T. M. Su, “Spectroscopy with modulation sidebands,” Appl. Phys. 7, 181 (1975).
[Crossref] [PubMed]

1982 (1)

P. R. Berman and R. Salomaa, “Comparison between dressed-atom and bare-atom pictures in laser spectroscopy,” Phys. Rev. A 25, 2667 (1982);P. R. Berman and J. Ziegler, “Generalized dressed-atom approach to atom-strong-field interactions—application to the theory of lasers and Bloch–Siegert shifts,” Phys. Rev. A 15, 2042 (1977).
[Crossref]

1971 (1)

C. Cohen-Tannoudji, “Atoms 〈〈Habilles〉〉 per des photons optiques ou de radiofrequence,” J. Phys. C 5a, 11 (1971);C. Cohen-Tannoudji and S. Haroche, “Absorption et diffusion de photons optiques par un atome en interaction avec des photons de radiofrequence,” J. Phys. (Paris) 30, 153 (1969).
[Crossref]

Becker, R. A.

R. A. Becker, Introduction to Theoretical Mechanics (McGraw-Hill, New York, 1954), Chaps. 7 and 14.

Berman, P. R.

P. R. Berman and R. Salomaa, “Comparison between dressed-atom and bare-atom pictures in laser spectroscopy,” Phys. Rev. A 25, 2667 (1982);P. R. Berman and J. Ziegler, “Generalized dressed-atom approach to atom-strong-field interactions—application to the theory of lasers and Bloch–Siegert shifts,” Phys. Rev. A 15, 2042 (1977).
[Crossref]

Buhr, E.

E. Buhr and J. Mlynek, “Collison-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]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, “Atoms 〈〈Habilles〉〉 per des photons optiques ou de radiofrequence,” J. Phys. C 5a, 11 (1971);C. Cohen-Tannoudji and S. Haroche, “Absorption et diffusion de photons optiques par un atome en interaction avec des photons de radiofrequence,” J. Phys. (Paris) 30, 153 (1969).
[Crossref]

Dalton, B. J.

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);P. F. Fracassi, L. Angeloni, and R. G. DellaValle, “Effects of dampings and dephasings in resonance Raman spectroscopy,” Chem. Phys. Lett. 115, 428 (1985);P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriaca 56, 103 (1984);V. Zadkov, N. Koroteev, M. Rychev, and A. Fedorov, “Saturated coherent Raman scattering spectroscopy for molecular gasses,” Moscow Univ. Phys. Bull. 39, 30 (1984);N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984);F. H. Mies, “Resonance fluorescence and Raman line shapes produced by monochromatic laser fields: effects of branching ratio and homogeneous broadening,” J. Quant.Spectrosc. Radiat. Transfer  29, 237 (1983);P. M. Radmore, “Population trapping in a multilevel system,” Phys. Rev. A 26, 2252 (1982);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);B. Halperin and J. A. Koninfstein, “Conditions for excited-state Raman and absorption processes during optical pumping,” Can. J. Chem. 59, 2792 (1981);F. H. Miew and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981);J. E. Thomas and W. W. Quivers, “Transit-time effects in optically pumped coupled three-level systems,” Phys. Rev. A 22, 2115, (1980);G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento 53B, 1 (1979);H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978);C. Cohen-Tannoudji and S. Reynaud, “Modification of resonance Raman scattering in very intense laser fields,” J. Phys. B 10, 365 (1977);E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977);R. M. Whitley and C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498 (1976);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);A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
[Crossref] [PubMed]

Davydov, A. S.

A. S. Davydov, Quantum Mechanics (Pergamon, Oxford, 1965), Chap. 9.

Ezekiel, S.

Hemmer, P. R.

Kieu, T. D.

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);P. F. Fracassi, L. Angeloni, and R. G. DellaValle, “Effects of dampings and dephasings in resonance Raman spectroscopy,” Chem. Phys. Lett. 115, 428 (1985);P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriaca 56, 103 (1984);V. Zadkov, N. Koroteev, M. Rychev, and A. Fedorov, “Saturated coherent Raman scattering spectroscopy for molecular gasses,” Moscow Univ. Phys. Bull. 39, 30 (1984);N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984);F. H. Mies, “Resonance fluorescence and Raman line shapes produced by monochromatic laser fields: effects of branching ratio and homogeneous broadening,” J. Quant.Spectrosc. Radiat. Transfer  29, 237 (1983);P. M. Radmore, “Population trapping in a multilevel system,” Phys. Rev. A 26, 2252 (1982);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);B. Halperin and J. A. Koninfstein, “Conditions for excited-state Raman and absorption processes during optical pumping,” Can. J. Chem. 59, 2792 (1981);F. H. Miew and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981);J. E. Thomas and W. W. Quivers, “Transit-time effects in optically pumped coupled three-level systems,” Phys. Rev. A 22, 2115, (1980);G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento 53B, 1 (1979);H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978);C. Cohen-Tannoudji and S. Reynaud, “Modification of resonance Raman scattering in very intense laser fields,” J. Phys. B 10, 365 (1977);E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977);R. M. Whitley and C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498 (1976);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);A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
[Crossref] [PubMed]

Knight, P. L.

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);P. F. Fracassi, L. Angeloni, and R. G. DellaValle, “Effects of dampings and dephasings in resonance Raman spectroscopy,” Chem. Phys. Lett. 115, 428 (1985);P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriaca 56, 103 (1984);V. Zadkov, N. Koroteev, M. Rychev, and A. Fedorov, “Saturated coherent Raman scattering spectroscopy for molecular gasses,” Moscow Univ. Phys. Bull. 39, 30 (1984);N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984);F. H. Mies, “Resonance fluorescence and Raman line shapes produced by monochromatic laser fields: effects of branching ratio and homogeneous broadening,” J. Quant.Spectrosc. Radiat. Transfer  29, 237 (1983);P. M. Radmore, “Population trapping in a multilevel system,” Phys. Rev. A 26, 2252 (1982);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);B. Halperin and J. A. Koninfstein, “Conditions for excited-state Raman and absorption processes during optical pumping,” Can. J. Chem. 59, 2792 (1981);F. H. Miew and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981);J. E. Thomas and W. W. Quivers, “Transit-time effects in optically pumped coupled three-level systems,” Phys. Rev. A 22, 2115, (1980);G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento 53B, 1 (1979);H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978);C. Cohen-Tannoudji and S. Reynaud, “Modification of resonance Raman scattering in very intense laser fields,” J. Phys. B 10, 365 (1977);E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977);R. M. Whitley and C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498 (1976);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);A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
[Crossref] [PubMed]

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]

Kumar, P.

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.

E. Buhr and J. Mlynek, “Collison-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]

Ontai, G. P.

Ramsey, N. F.

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

Salomaa, R.

P. R. Berman and R. Salomaa, “Comparison between dressed-atom and bare-atom pictures in laser spectroscopy,” Phys. Rev. A 25, 2667 (1982);P. R. Berman and J. Ziegler, “Generalized dressed-atom approach to atom-strong-field interactions—application to the theory of lasers and Bloch–Siegert shifts,” Phys. Rev. A 15, 2042 (1977).
[Crossref]

Shapiro, J. H.

Shimizu, F.

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 I2 vapor using a two-photon resonant scattering technique,” Chem. Phys. Lett. 96, 253 (1983);K. Takagi, R. F. Curl, and R. T. M. Su, “Spectroscopy with modulation sidebands,” Appl. Phys. 7, 181 (1975).
[Crossref] [PubMed]

Shimizu, K.

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 I2 vapor using a two-photon resonant scattering technique,” Chem. Phys. Lett. 96, 253 (1983);K. Takagi, R. F. Curl, and R. T. M. Su, “Spectroscopy with modulation sidebands,” Appl. Phys. 7, 181 (1975).
[Crossref] [PubMed]

Takuma, H.

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 I2 vapor using a two-photon resonant scattering technique,” Chem. Phys. Lett. 96, 253 (1983);K. Takagi, R. F. Curl, and R. T. M. Su, “Spectroscopy with modulation sidebands,” Appl. Phys. 7, 181 (1975).
[Crossref] [PubMed]

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]

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap.15.

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)

J. Phys. C (1)

C. Cohen-Tannoudji, “Atoms 〈〈Habilles〉〉 per des photons optiques ou de radiofrequence,” J. Phys. C 5a, 11 (1971);C. Cohen-Tannoudji and S. Haroche, “Absorption et diffusion de photons optiques par un atome en interaction avec des photons de radiofrequence,” J. Phys. (Paris) 30, 153 (1969).
[Crossref]

Opt. Acta (1)

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);P. F. Fracassi, L. Angeloni, and R. G. DellaValle, “Effects of dampings and dephasings in resonance Raman spectroscopy,” Chem. Phys. Lett. 115, 428 (1985);P. L. Knight, M. A. Lauder, P. M. Radmore, and B. J. Dalton, “Making atoms transparent: trapped superpositions,” Acta Phys. Austriaca 56, 103 (1984);V. Zadkov, N. Koroteev, M. Rychev, and A. Fedorov, “Saturated coherent Raman scattering spectroscopy for molecular gasses,” Moscow Univ. Phys. Bull. 39, 30 (1984);N. I. Shamrov, “Induced transparency in resonant induced Raman scattering,” Zh. Prikl. Spektrosk. 40, 346 (1984);F. H. Mies, “Resonance fluorescence and Raman line shapes produced by monochromatic laser fields: effects of branching ratio and homogeneous broadening,” J. Quant.Spectrosc. Radiat. Transfer  29, 237 (1983);P. M. Radmore, “Population trapping in a multilevel system,” Phys. Rev. A 26, 2252 (1982);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);B. Halperin and J. A. Koninfstein, “Conditions for excited-state Raman and absorption processes during optical pumping,” Can. J. Chem. 59, 2792 (1981);F. H. Miew and Y. B. Aryeh, “Kinetics and spectroscopy of near-resonant optical pumping in intense fields,” J. Chem. Phys. 74, 53 (1981);J. E. Thomas and W. W. Quivers, “Transit-time effects in optically pumped coupled three-level systems,” Phys. Rev. A 22, 2115, (1980);G. Orriols, “Nonabsorption resonances by nonlinear coherent effects in a three-level system,” Nuovo Cimento 53B, 1 (1979);H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978);C. Cohen-Tannoudji and S. Reynaud, “Modification of resonance Raman scattering in very intense laser fields,” J. Phys. B 10, 365 (1977);E. Courtens and S. Szoke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588 (1977);R. M. Whitley and C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498 (1976);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);A. Szoke and E. Courtens, “Time-resolved resonance fluorescence and resonance Raman scattering,” Phys. Rev. Lett. 34, 1053 (1975).
[Crossref] [PubMed]

Opt. Lett. (1)

Phys. Rev. A (2)

P. R. Berman and R. Salomaa, “Comparison between dressed-atom and bare-atom pictures in laser spectroscopy,” Phys. Rev. A 25, 2667 (1982);P. R. Berman and J. Ziegler, “Generalized dressed-atom approach to atom-strong-field interactions—application to the theory of lasers and Bloch–Siegert shifts,” Phys. Rev. A 15, 2042 (1977).
[Crossref]

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 I2 vapor using a two-photon resonant scattering technique,” Chem. Phys. Lett. 96, 253 (1983);K. Takagi, R. F. Curl, and R. T. M. Su, “Spectroscopy with modulation sidebands,” Appl. Phys. 7, 181 (1975).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

E. Buhr and J. Mlynek, “Collison-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 (4)

A. S. Davydov, Quantum Mechanics (Pergamon, Oxford, 1965), Chap. 9.

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

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap.15.

R. A. Becker, Introduction to Theoretical Mechanics (McGraw-Hill, New York, 1954), Chaps. 7 and 14.

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Figures (18)

Fig. 1
Fig. 1

a, Schematic diagram of stimulated resonance Raman interaction. b, Experimental data showing the Raman interaction as a decrease in fluorescence.

Fig. 2
Fig. 2

Coupled-pendulum model for the stimulated resonance Raman interaction.

Fig. 3
Fig. 3

Plot of Raman-system composite-state amplitudes versus time in the case of equal Rabi frequencies, Ω = 0.2γ2.

Fig. 4
Fig. 4

a, Motion of the coupled-pendulum model in the case of equal coupling-spring strengths,Ω′ = 0.2γ2′. b, Trapped-mode contribution to motion in a. c, Damped-mode contribution to motion in a.

Fig. 5
Fig. 5

a, Separated-field excitation of the Raman interaction. b, Experimental data showing Ramsey fringes for a 15-cm interaction zone separation in a sodium atomic beam.

Fig. 6
Fig. 6

Uncoupled-pendulum model for the dark-zone Raman interaction.

Fig. 7
Fig. 7

Motion of the uncoupled-pendulum system corresponding to a, short zone A interaction times; b, long zone A interaction times.

Fig. 8
Fig. 8

Schematic diagrams illustrating a, correlated laser detuning δ and b, laser difference-frequency detuning Δ.

Fig. 9
Fig. 9

a, Coupled-pendulum model to an off-resonance Raman interaction, δ′ = 2 γ2′: coupling-spring strength, Ω′ = 0.2 γ2′. b, Trapped-mode contribution to motion in a. c, Damped-mode contribution to motion in a.

Fig. 10
Fig. 10

Plot of average pendulum oscillation amplitudes versus time for the off-resonance model of Fig. 9a.

Fig. 11
Fig. 11

Experimental data showing single-zone line-shape asymmetries obtained for correlated laser detunings of a, δ = 0.5γ2 and b, δ = γ2. Rabi frequencies in both cases are Ω1 = 0.4γ2 and Ω2= 0.13γ2+ = (1/2)(Ω12 + Ω22)1/2}.

Fig. 12
Fig. 12

Coupled-pendulum models showing the physical basis of the line-shape asymmetries in Fig. 11: a, pendulum model corresponding to δ′ = 0.5γ2 and Δ = +0.65(Ω′2/γ2); b, instantaneous damped-mode contribution to motion in a; c, pendulum model corresponding to δ′ − 0.5γ2 and Δ = −0.65(Ω′2/γ2); d, instantaneous damped-mode contribution to motion in c.

Fig. 13
Fig. 13

a, Plot of average pendulum oscillation amplitudes for the line-shape asymmetry model of Fig. 12a. b, The same as a but for the model of Fig. 12d.

Fig. 14
Fig. 14

Pendulum models showing asymmetry reversals: a, opposite correlated detuning case, δ = −0.5γ2 and Δ = +0.65(Ω2/γ2); b, pendulum 3 oscillating initially instead of pendulum 1 for δ = + 0.5γ2 and Δ = +0.65(Ω2/γ2).

Fig. 15
Fig. 15

Effects of correlated detuning on the uncoupled (dark-zone) pendulum model for δ′ = 2γ2: a, initial conditions obtained from row two of Fig. 9a, (Ω′/2δ′)τ = π/4; b, instantaneous trapped-mode contribution to motion in a; c, instantaneous damped-mode contribution to motion in a.

Fig. 16
Fig. 16

Experimental data showing Ramsey-fringe phase shifts Δ versus correlated laser detuning δ for zone A laser intensities such that a, (Ω2/γ2) τ = π/8 and b, Ω2/γ2τ = π/2.

Fig. 17
Fig. 17

a, Motion of modified coupled-pendulum model in the limit of strong coupling springs (compared with pendulum-2 damping). b, Trapped-mode contribution to motion in a. c, Damped-mode contribution to motion in a. d and e, Normal-mode contributions to damped mode of c.

Fig. 18
Fig. 18

a, Motion of modified coupled-pendulum model for unequal coupling springs. b, Motion for initial conditions resembling a trapped mode from the equal spring model.

Tables (1)

Tables Icon

Table 1 Summary of Correspondences between the Stimulated Resonance Raman Interaction (Three-Level Atom) and a Set of Three Classical Coupled Pendulumsa

Equations (15)

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Ψ ( t ) = [ a 1 ( t ) | 1 | ω 1 exp ( i δ t ) exp ( ½ i Δ t ) + a 2 ( t ) | 2 + a 3 ( t ) | 3 | ω 2 exp ( i δ t ) exp ( ½ i Δ t ) ] exp [ i ( 2 / ) t ] ,
δ = ½ ( ω 1 + ω 2 ) [ 2 ½ ( 1 + 3 ) ] / , Δ = ( ω 1 ω 2 ) ( 3 1 ) / .
( A ˙ 1 A ˙ 2 A ˙ 3 ) = ( ½ i Δ ½ i Ω 1 * 0 ½ i Ω 1 ½ ( γ 2 2 i δ ) ½ i Ω 2 0 ½ i Ω 2 * ½ i Δ ) ( A 1 A 2 A 3 ) ,
A 1 = a 1 exp ( ½ i Δ t ) , A 2 = a 2 exp ( i δ t ) , A 3 = a 3 exp ( ½ i Δ t ) .
η ¨ 1 + [ ( ω 0 + 1 2 Δ ) 2 + k 1 M ] η 1 k 1 M η 2 = 0 , η ¨ 2 + [ ( ω 0 δ ) 2 + k 1 M + k 2 M ] η 2 k 1 M η 1 k 2 M η 3 = γ 2 η ˙ 2 , η ¨ 3 + [ ( ω 0 + 1 2 Δ ) 2 + k 2 M ] η 3 k 2 M η 2 = 0.
η ( t ) = [ N ( t ) exp ( i ω 0 t ) + c . c ] .
( 1 2 3 ) = { ½ i ( Δ + s 11 ) ½ i Ω 1 0 ½ i Ω 1 ½ [ γ 2 2 i ( δ ½ s 22 ) ] ½ i Ω 2 0 ½ i Ω 2 ½ i ( Δ s 33 ) } ( N 1 N 2 N 3 ) ,
Ω 1 = 1 ω 0 k 1 M , Ω 2 = 1 ω 0 k 2 M , s 11 = Ω 1 , s 22 = Ω 1 + Ω 2 , s 33 = Ω 2 .
| = ½ [ | 1 | ω 1 | 3 ω 2 ] , | + = 1 2 [ | 1 | ω 1 + | 3 | ω 2 + 2 i ( Ω γ 2 2 i δ ) | 2 ] .
B ( T ) = B ( τ ) cos ( ½ Δ T ) i B + ( τ ) sin ( ½ Δ T ) , B + ( T ) = i B ( τ ) sin ( ½ Δ T ) + B + ( τ ) cos ( ½ Δ T ) .
Ω 1 = k 1 M ω 1 , Ω 2 = k 2 M ω 2
Ω 1 = μ 21 E 1 , Ω 2 = μ 23 E 2
1 , 2 , 3
1 2 i ω 0 N ¯ 1 + ( ω 0 Δ + 1 4 Δ 2 + k 1 M ) N 1 k 1 M N 2 = 0 . N ¨ 2 ( 2 i ω 0 γ 2 ) 2 ( 2 ω 0 δ δ 2 k 1 M k 2 M i ω 0 γ 2 ) N 2 k 1 M N 1 k 2 M N 3 = 0 , N ¨ 3 2 i ω 0 3 + ( ω 0 Δ + 1 4 Δ 2 + k 2 M ) N 3 k 2 M N 2 = 0 ,
ω 0 N , N ¨ ω 0 , γ 2 ω 0 , δ ω 0 , Δ ω 0 .

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