Abstract

In a previous paper [ J. Opt. Soc. Am. B 3, 564 ( 1986)], the validity conditions for the optical Bloch equations were reviewed. It was shown that, even within the limits of an impact or Markovian approximation, the optical Bloch equations fail to account properly for fluctuation-induced changes in atomic transition frequencies. Such changes are properly incorporated in a quantum-mechanical transport equation (QMTE) in which the fluctuation-induced frequency shifts are totally characterized by kernels W(′ → ) that give the probability density per unit time for a fluctuation to change the frequency shift from ′ to . The QMTE describes the interaction of atoms with both an external radiation field and the perturber bath producing the fluctuations. A general method for solving the QMTE as a perturbation series in the external field is presented. Specific calculations are carried out for strong-redistribution, difference [W(′ → ) is a function of (′) only], and Brownian motion kernels. It is shown that, although the kernels possess fundamental differences, they can yield similar results in certain limits. As an example, a perturbation calculation is performed for the free-induction decay (FID) of atoms prepared by a cw laser field and then allowed to radiate when the field is suddenly removed. Radical departures from the predictions of the conventional Bloch equations are found in certain limits, including a first-order contribution to FID in vapors and a nonexponential FID decay for atoms in vapors or solids. The implications of these results to a consistent interpretation of a recent experiment [ Phys. Rev. Lett. 50, 1269 ( 1983)] on FID in the impurity ion crystal Pr3+:LaF3 are explored.

© 1986 Optical Society of America

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References

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  1. P. R. Berman, “Validity conditions for the optical Bloch equations,” J. Opt. Soc. Am. B 3, 564–571 (1986).
    [Crossref]
  2. See, for example, B. W. Shore, “Modeling noise by jump processes in strong laser–atom interactions,” J. Opt. Soc. Am. B 1, 176–188 (1984), which contains extensive references to earlier work. The modeling of frequency fluctuations by jump processes is often attributed to Anderson and Kubo [P. W. Anderson, “A mathematical model for the narrowing of spectral lines by exchange or motion,” J. Phys. Soc. Jpn. 9, 316–339 (1954); R. Kubo, “Note on the stochastic theory of resonance absorption,” J. Phys. Soc. Jpn 9, 935–944 (1954)]. In the optical domain, Burshtein and Oseledchik [A. I. Burshtein and Y. S. Oseledchik, “Relaxation in a system subjected to suddenly changing perturbations in the presence of correlation between successive values of the perturbation,” Sov. Phys. JETP 24, 716–724 (1967)] derive an equation for atomic relaxation resulting from jump processes.
    [Crossref]
  3. G. E. Uhlenbeck and L. S. Ornstein, “On the theory of Brownian motion,” Phys. Rev. 36, 823–841 (1930); see also N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981); Fluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds. (North-Holland, Amsterdam, 1979).
    [Crossref]
  4. See, for example, P. R. Berman, “Theory of collision effects on atomic and molecular lineshapes,” Appl. Phys. (Germany) 6, 283–296 (1975), and references therein; “Collisions in atomic vapors,” in New Trends in Atomic Physics, Les Houches, Session38, 1982, G. Grynberg and R. Stora, eds. (North-Holland, Amsterdam, 1984), pp. 451–514, and references therein; V. P. Kochanov, S. G. Rautian, and A. M. Shalagin, “Broadening of nonlinear resonances by velocity-changing collisions,” Sov. Phys. JETP 45, 714–722 (1977); A. G. Kofman and A. I. Burshtein, “Kinetics of Doppler-spectrum saturation,” Sov. Phys. JETP 49, 1019–1026 (1979).
    [Crossref]
  5. A. Schenzle, M. Mitsunaga, R. G. DeVoe, and R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984).
    [Crossref]
  6. M. Yamanoi and J. H. Eberly, “Relaxation terms for strong field optical Bloch equations,” J. Opt. Soc. Am. B 1, 751–755 (1984).
    [Crossref]
  7. E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258–2266 (1983); “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Opt. Soc. Jpn. 52, 3678–3684 (1983).
    [Crossref]
  8. J. Javanainen, “Free induction decay in a fluctuating two level system,” Opt. Commun. 50, 26–30 (1984).
    [Crossref]
  9. P. A. Apanasevich, S. Ya. Kilin, and A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984).
    [Crossref]
  10. K. Wodkiewicz and J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985).
    [Crossref] [PubMed]
  11. P. R. Berman and R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784 (1985).
    [Crossref] [PubMed]
  12. R. G. DeVoe and R. G. Brewer, “Experimental test of the optical Bloch equations for solids,” Phys. Rev. Lett. 50, 1269–1272 (1983).
    [Crossref]
  13. The resonance approximation consists of neglecting terms that vary as exp[+i(Ω + ω)t]. The field-interaction representation consists of writing ρ12(R, v, t) = ρ12(v, t)exp[−i(k· R− Ωt)].
  14. S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943).
    [Crossref]
  15. Note that expression (5.68) lacks the delta-function contribution found in expression (5.23) [compare also expressions (5.58) and (5.7) for Γ ≫ γi]. The Brownian motion model cannot reproduce this delta-function contribution [see A. P. Kolchenko, A. A. Pukhov, S. G. Rautian, and A. M. Shalagin, “Effect of selective collisions on the velocity distribution of atoms and on nonlinear interference effects,” Sov. Phys. JETP 36, 619–628 (1973); P. R. Berman, “Brownian motion of atomic systems: Fokker–Planck limit of the transport equation,” Phys. Rev. A 9, 2170–2176 (1974)]. When integrals of Eq. (5.7) or (5.8) or expression (5.23) are taken, the delta-function contribution is negligible, provided that Γ ≫ γi and Γ ≫ ∊0(i.e., in the strict Brownian motion limit when Γ → ∞).
    [Crossref]
  16. See, for example, P. R. Berman, “Theory of collision effects on atomic and molecular lineshapes,” Appl. Phys (Germany) 6, 283–296 (1975), and references therein to motional narrowing in atomic and molecular vapors. See also K. Shimoda, “Line broadening and narrowing effects,” in High Resolution Spectroscopy, Vol. 13 of Topics in Applied Physics, K. Shimoda, ed. (Springer-Verlag, Berlin, 1976), pp. 11–49. Motional narrowing in nuclear magnetic resonance dates from earlier times: see, for example, A. Abragam, Principles of Nuclear Magnetism (Oxford U. Press, Oxford, 1961); C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, Berlin, 1980), and references therein.
    [Crossref]
  17. P. R. Berman and R. G. Brewer, “Modified Bloch equations for solids,” in Laser Spectroscopy VII, T. W. Hänsch and Y. R. Shen, eds. (Springer-Verlag, Berlin, 1985).
    [Crossref]
  18. R. M. Macfarlane, R. M. Shelby, and R. L. Shoemaker, “Ultra-high-resolution spectroscopy: photon echoes in YAlO3:Pr3+ and LaF3: Pr3+,” Phys. Rev. Lett. 43, 1726–1730 (1979).
    [Crossref]
  19. P. R. Berman, J. M. Levy, and R. G. Brewer, “Coherent optical transient study of molecular collisions: theory and observations,” Phys. Rev. A 11, 1668–1688 (1975).
    [Crossref]
  20. A. G. Yodh, J. Golub, N. W. Carlson, and T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–667 (1984).
    [Crossref]

1986 (1)

1985 (2)

K. Wodkiewicz and J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985).
[Crossref] [PubMed]

P. R. Berman and R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784 (1985).
[Crossref] [PubMed]

1984 (6)

J. Javanainen, “Free induction decay in a fluctuating two level system,” Opt. Commun. 50, 26–30 (1984).
[Crossref]

P. A. Apanasevich, S. Ya. Kilin, and A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984).
[Crossref]

A. G. Yodh, J. Golub, N. W. Carlson, and T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–667 (1984).
[Crossref]

A. Schenzle, M. Mitsunaga, R. G. DeVoe, and R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984).
[Crossref]

See, for example, B. W. Shore, “Modeling noise by jump processes in strong laser–atom interactions,” J. Opt. Soc. Am. B 1, 176–188 (1984), which contains extensive references to earlier work. The modeling of frequency fluctuations by jump processes is often attributed to Anderson and Kubo [P. W. Anderson, “A mathematical model for the narrowing of spectral lines by exchange or motion,” J. Phys. Soc. Jpn. 9, 316–339 (1954); R. Kubo, “Note on the stochastic theory of resonance absorption,” J. Phys. Soc. Jpn 9, 935–944 (1954)]. In the optical domain, Burshtein and Oseledchik [A. I. Burshtein and Y. S. Oseledchik, “Relaxation in a system subjected to suddenly changing perturbations in the presence of correlation between successive values of the perturbation,” Sov. Phys. JETP 24, 716–724 (1967)] derive an equation for atomic relaxation resulting from jump processes.
[Crossref]

M. Yamanoi and J. H. Eberly, “Relaxation terms for strong field optical Bloch equations,” J. Opt. Soc. Am. B 1, 751–755 (1984).
[Crossref]

1983 (2)

E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258–2266 (1983); “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Opt. Soc. Jpn. 52, 3678–3684 (1983).
[Crossref]

R. G. DeVoe and R. G. Brewer, “Experimental test of the optical Bloch equations for solids,” Phys. Rev. Lett. 50, 1269–1272 (1983).
[Crossref]

1979 (1)

R. M. Macfarlane, R. M. Shelby, and R. L. Shoemaker, “Ultra-high-resolution spectroscopy: photon echoes in YAlO3:Pr3+ and LaF3: Pr3+,” Phys. Rev. Lett. 43, 1726–1730 (1979).
[Crossref]

1975 (3)

P. R. Berman, J. M. Levy, and R. G. Brewer, “Coherent optical transient study of molecular collisions: theory and observations,” Phys. Rev. A 11, 1668–1688 (1975).
[Crossref]

See, for example, P. R. Berman, “Theory of collision effects on atomic and molecular lineshapes,” Appl. Phys (Germany) 6, 283–296 (1975), and references therein to motional narrowing in atomic and molecular vapors. See also K. Shimoda, “Line broadening and narrowing effects,” in High Resolution Spectroscopy, Vol. 13 of Topics in Applied Physics, K. Shimoda, ed. (Springer-Verlag, Berlin, 1976), pp. 11–49. Motional narrowing in nuclear magnetic resonance dates from earlier times: see, for example, A. Abragam, Principles of Nuclear Magnetism (Oxford U. Press, Oxford, 1961); C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, Berlin, 1980), and references therein.
[Crossref]

See, for example, P. R. Berman, “Theory of collision effects on atomic and molecular lineshapes,” Appl. Phys. (Germany) 6, 283–296 (1975), and references therein; “Collisions in atomic vapors,” in New Trends in Atomic Physics, Les Houches, Session38, 1982, G. Grynberg and R. Stora, eds. (North-Holland, Amsterdam, 1984), pp. 451–514, and references therein; V. P. Kochanov, S. G. Rautian, and A. M. Shalagin, “Broadening of nonlinear resonances by velocity-changing collisions,” Sov. Phys. JETP 45, 714–722 (1977); A. G. Kofman and A. I. Burshtein, “Kinetics of Doppler-spectrum saturation,” Sov. Phys. JETP 49, 1019–1026 (1979).
[Crossref]

1973 (1)

Note that expression (5.68) lacks the delta-function contribution found in expression (5.23) [compare also expressions (5.58) and (5.7) for Γ ≫ γi]. The Brownian motion model cannot reproduce this delta-function contribution [see A. P. Kolchenko, A. A. Pukhov, S. G. Rautian, and A. M. Shalagin, “Effect of selective collisions on the velocity distribution of atoms and on nonlinear interference effects,” Sov. Phys. JETP 36, 619–628 (1973); P. R. Berman, “Brownian motion of atomic systems: Fokker–Planck limit of the transport equation,” Phys. Rev. A 9, 2170–2176 (1974)]. When integrals of Eq. (5.7) or (5.8) or expression (5.23) are taken, the delta-function contribution is negligible, provided that Γ ≫ γi and Γ ≫ ∊0(i.e., in the strict Brownian motion limit when Γ → ∞).
[Crossref]

1943 (1)

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943).
[Crossref]

1930 (1)

G. E. Uhlenbeck and L. S. Ornstein, “On the theory of Brownian motion,” Phys. Rev. 36, 823–841 (1930); see also N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981); Fluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds. (North-Holland, Amsterdam, 1979).
[Crossref]

Apanasevich, P. A.

P. A. Apanasevich, S. Ya. Kilin, and A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984).
[Crossref]

Berman, P. R.

P. R. Berman, “Validity conditions for the optical Bloch equations,” J. Opt. Soc. Am. B 3, 564–571 (1986).
[Crossref]

P. R. Berman and R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784 (1985).
[Crossref] [PubMed]

See, for example, P. R. Berman, “Theory of collision effects on atomic and molecular lineshapes,” Appl. Phys (Germany) 6, 283–296 (1975), and references therein to motional narrowing in atomic and molecular vapors. See also K. Shimoda, “Line broadening and narrowing effects,” in High Resolution Spectroscopy, Vol. 13 of Topics in Applied Physics, K. Shimoda, ed. (Springer-Verlag, Berlin, 1976), pp. 11–49. Motional narrowing in nuclear magnetic resonance dates from earlier times: see, for example, A. Abragam, Principles of Nuclear Magnetism (Oxford U. Press, Oxford, 1961); C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, Berlin, 1980), and references therein.
[Crossref]

P. R. Berman, J. M. Levy, and R. G. Brewer, “Coherent optical transient study of molecular collisions: theory and observations,” Phys. Rev. A 11, 1668–1688 (1975).
[Crossref]

See, for example, P. R. Berman, “Theory of collision effects on atomic and molecular lineshapes,” Appl. Phys. (Germany) 6, 283–296 (1975), and references therein; “Collisions in atomic vapors,” in New Trends in Atomic Physics, Les Houches, Session38, 1982, G. Grynberg and R. Stora, eds. (North-Holland, Amsterdam, 1984), pp. 451–514, and references therein; V. P. Kochanov, S. G. Rautian, and A. M. Shalagin, “Broadening of nonlinear resonances by velocity-changing collisions,” Sov. Phys. JETP 45, 714–722 (1977); A. G. Kofman and A. I. Burshtein, “Kinetics of Doppler-spectrum saturation,” Sov. Phys. JETP 49, 1019–1026 (1979).
[Crossref]

P. R. Berman and R. G. Brewer, “Modified Bloch equations for solids,” in Laser Spectroscopy VII, T. W. Hänsch and Y. R. Shen, eds. (Springer-Verlag, Berlin, 1985).
[Crossref]

Brewer, R. G.

P. R. Berman and R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784 (1985).
[Crossref] [PubMed]

A. Schenzle, M. Mitsunaga, R. G. DeVoe, and R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984).
[Crossref]

R. G. DeVoe and R. G. Brewer, “Experimental test of the optical Bloch equations for solids,” Phys. Rev. Lett. 50, 1269–1272 (1983).
[Crossref]

P. R. Berman, J. M. Levy, and R. G. Brewer, “Coherent optical transient study of molecular collisions: theory and observations,” Phys. Rev. A 11, 1668–1688 (1975).
[Crossref]

P. R. Berman and R. G. Brewer, “Modified Bloch equations for solids,” in Laser Spectroscopy VII, T. W. Hänsch and Y. R. Shen, eds. (Springer-Verlag, Berlin, 1985).
[Crossref]

Carlson, N. W.

A. G. Yodh, J. Golub, N. W. Carlson, and T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–667 (1984).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943).
[Crossref]

DeVoe, R. G.

A. Schenzle, M. Mitsunaga, R. G. DeVoe, and R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984).
[Crossref]

R. G. DeVoe and R. G. Brewer, “Experimental test of the optical Bloch equations for solids,” Phys. Rev. Lett. 50, 1269–1272 (1983).
[Crossref]

Eberly, J. H.

K. Wodkiewicz and J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985).
[Crossref] [PubMed]

M. Yamanoi and J. H. Eberly, “Relaxation terms for strong field optical Bloch equations,” J. Opt. Soc. Am. B 1, 751–755 (1984).
[Crossref]

Golub, J.

A. G. Yodh, J. Golub, N. W. Carlson, and T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–667 (1984).
[Crossref]

Hanamura, E.

E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258–2266 (1983); “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Opt. Soc. Jpn. 52, 3678–3684 (1983).
[Crossref]

Javanainen, J.

J. Javanainen, “Free induction decay in a fluctuating two level system,” Opt. Commun. 50, 26–30 (1984).
[Crossref]

Kilin, S. Ya.

P. A. Apanasevich, S. Ya. Kilin, and A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984).
[Crossref]

Kolchenko, A. P.

Note that expression (5.68) lacks the delta-function contribution found in expression (5.23) [compare also expressions (5.58) and (5.7) for Γ ≫ γi]. The Brownian motion model cannot reproduce this delta-function contribution [see A. P. Kolchenko, A. A. Pukhov, S. G. Rautian, and A. M. Shalagin, “Effect of selective collisions on the velocity distribution of atoms and on nonlinear interference effects,” Sov. Phys. JETP 36, 619–628 (1973); P. R. Berman, “Brownian motion of atomic systems: Fokker–Planck limit of the transport equation,” Phys. Rev. A 9, 2170–2176 (1974)]. When integrals of Eq. (5.7) or (5.8) or expression (5.23) are taken, the delta-function contribution is negligible, provided that Γ ≫ γi and Γ ≫ ∊0(i.e., in the strict Brownian motion limit when Γ → ∞).
[Crossref]

Levy, J. M.

P. R. Berman, J. M. Levy, and R. G. Brewer, “Coherent optical transient study of molecular collisions: theory and observations,” Phys. Rev. A 11, 1668–1688 (1975).
[Crossref]

Macfarlane, R. M.

R. M. Macfarlane, R. M. Shelby, and R. L. Shoemaker, “Ultra-high-resolution spectroscopy: photon echoes in YAlO3:Pr3+ and LaF3: Pr3+,” Phys. Rev. Lett. 43, 1726–1730 (1979).
[Crossref]

Mitsunaga, M.

A. Schenzle, M. Mitsunaga, R. G. DeVoe, and R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984).
[Crossref]

Mossberg, T. W.

A. G. Yodh, J. Golub, N. W. Carlson, and T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–667 (1984).
[Crossref]

Nizovtsev, A. P.

P. A. Apanasevich, S. Ya. Kilin, and A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984).
[Crossref]

Ornstein, L. S.

G. E. Uhlenbeck and L. S. Ornstein, “On the theory of Brownian motion,” Phys. Rev. 36, 823–841 (1930); see also N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981); Fluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds. (North-Holland, Amsterdam, 1979).
[Crossref]

Pukhov, A. A.

Note that expression (5.68) lacks the delta-function contribution found in expression (5.23) [compare also expressions (5.58) and (5.7) for Γ ≫ γi]. The Brownian motion model cannot reproduce this delta-function contribution [see A. P. Kolchenko, A. A. Pukhov, S. G. Rautian, and A. M. Shalagin, “Effect of selective collisions on the velocity distribution of atoms and on nonlinear interference effects,” Sov. Phys. JETP 36, 619–628 (1973); P. R. Berman, “Brownian motion of atomic systems: Fokker–Planck limit of the transport equation,” Phys. Rev. A 9, 2170–2176 (1974)]. When integrals of Eq. (5.7) or (5.8) or expression (5.23) are taken, the delta-function contribution is negligible, provided that Γ ≫ γi and Γ ≫ ∊0(i.e., in the strict Brownian motion limit when Γ → ∞).
[Crossref]

Rautian, S. G.

Note that expression (5.68) lacks the delta-function contribution found in expression (5.23) [compare also expressions (5.58) and (5.7) for Γ ≫ γi]. The Brownian motion model cannot reproduce this delta-function contribution [see A. P. Kolchenko, A. A. Pukhov, S. G. Rautian, and A. M. Shalagin, “Effect of selective collisions on the velocity distribution of atoms and on nonlinear interference effects,” Sov. Phys. JETP 36, 619–628 (1973); P. R. Berman, “Brownian motion of atomic systems: Fokker–Planck limit of the transport equation,” Phys. Rev. A 9, 2170–2176 (1974)]. When integrals of Eq. (5.7) or (5.8) or expression (5.23) are taken, the delta-function contribution is negligible, provided that Γ ≫ γi and Γ ≫ ∊0(i.e., in the strict Brownian motion limit when Γ → ∞).
[Crossref]

Schenzle, A.

A. Schenzle, M. Mitsunaga, R. G. DeVoe, and R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984).
[Crossref]

Shalagin, A. M.

Note that expression (5.68) lacks the delta-function contribution found in expression (5.23) [compare also expressions (5.58) and (5.7) for Γ ≫ γi]. The Brownian motion model cannot reproduce this delta-function contribution [see A. P. Kolchenko, A. A. Pukhov, S. G. Rautian, and A. M. Shalagin, “Effect of selective collisions on the velocity distribution of atoms and on nonlinear interference effects,” Sov. Phys. JETP 36, 619–628 (1973); P. R. Berman, “Brownian motion of atomic systems: Fokker–Planck limit of the transport equation,” Phys. Rev. A 9, 2170–2176 (1974)]. When integrals of Eq. (5.7) or (5.8) or expression (5.23) are taken, the delta-function contribution is negligible, provided that Γ ≫ γi and Γ ≫ ∊0(i.e., in the strict Brownian motion limit when Γ → ∞).
[Crossref]

Shelby, R. M.

R. M. Macfarlane, R. M. Shelby, and R. L. Shoemaker, “Ultra-high-resolution spectroscopy: photon echoes in YAlO3:Pr3+ and LaF3: Pr3+,” Phys. Rev. Lett. 43, 1726–1730 (1979).
[Crossref]

Shoemaker, R. L.

R. M. Macfarlane, R. M. Shelby, and R. L. Shoemaker, “Ultra-high-resolution spectroscopy: photon echoes in YAlO3:Pr3+ and LaF3: Pr3+,” Phys. Rev. Lett. 43, 1726–1730 (1979).
[Crossref]

Shore, B. W.

Uhlenbeck, G. E.

G. E. Uhlenbeck and L. S. Ornstein, “On the theory of Brownian motion,” Phys. Rev. 36, 823–841 (1930); see also N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981); Fluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds. (North-Holland, Amsterdam, 1979).
[Crossref]

Wodkiewicz, K.

K. Wodkiewicz and J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985).
[Crossref] [PubMed]

Yamanoi, M.

Yodh, A. G.

A. G. Yodh, J. Golub, N. W. Carlson, and T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–667 (1984).
[Crossref]

Appl. Phys (Germany) (1)

See, for example, P. R. Berman, “Theory of collision effects on atomic and molecular lineshapes,” Appl. Phys (Germany) 6, 283–296 (1975), and references therein to motional narrowing in atomic and molecular vapors. See also K. Shimoda, “Line broadening and narrowing effects,” in High Resolution Spectroscopy, Vol. 13 of Topics in Applied Physics, K. Shimoda, ed. (Springer-Verlag, Berlin, 1976), pp. 11–49. Motional narrowing in nuclear magnetic resonance dates from earlier times: see, for example, A. Abragam, Principles of Nuclear Magnetism (Oxford U. Press, Oxford, 1961); C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, Berlin, 1980), and references therein.
[Crossref]

Appl. Phys. (Germany) (1)

See, for example, P. R. Berman, “Theory of collision effects on atomic and molecular lineshapes,” Appl. Phys. (Germany) 6, 283–296 (1975), and references therein; “Collisions in atomic vapors,” in New Trends in Atomic Physics, Les Houches, Session38, 1982, G. Grynberg and R. Stora, eds. (North-Holland, Amsterdam, 1984), pp. 451–514, and references therein; V. P. Kochanov, S. G. Rautian, and A. M. Shalagin, “Broadening of nonlinear resonances by velocity-changing collisions,” Sov. Phys. JETP 45, 714–722 (1977); A. G. Kofman and A. I. Burshtein, “Kinetics of Doppler-spectrum saturation,” Sov. Phys. JETP 49, 1019–1026 (1979).
[Crossref]

J. Opt. Soc. Am. B (3)

J. Phys. Soc. Jpn. (1)

E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258–2266 (1983); “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Opt. Soc. Jpn. 52, 3678–3684 (1983).
[Crossref]

Opt. Commun. (2)

J. Javanainen, “Free induction decay in a fluctuating two level system,” Opt. Commun. 50, 26–30 (1984).
[Crossref]

P. A. Apanasevich, S. Ya. Kilin, and A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984).
[Crossref]

Phys. Rev. (1)

G. E. Uhlenbeck and L. S. Ornstein, “On the theory of Brownian motion,” Phys. Rev. 36, 823–841 (1930); see also N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981); Fluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds. (North-Holland, Amsterdam, 1979).
[Crossref]

Phys. Rev. A (4)

A. Schenzle, M. Mitsunaga, R. G. DeVoe, and R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984).
[Crossref]

K. Wodkiewicz and J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985).
[Crossref] [PubMed]

P. R. Berman and R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784 (1985).
[Crossref] [PubMed]

P. R. Berman, J. M. Levy, and R. G. Brewer, “Coherent optical transient study of molecular collisions: theory and observations,” Phys. Rev. A 11, 1668–1688 (1975).
[Crossref]

Phys. Rev. Lett. (3)

A. G. Yodh, J. Golub, N. W. Carlson, and T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–667 (1984).
[Crossref]

R. M. Macfarlane, R. M. Shelby, and R. L. Shoemaker, “Ultra-high-resolution spectroscopy: photon echoes in YAlO3:Pr3+ and LaF3: Pr3+,” Phys. Rev. Lett. 43, 1726–1730 (1979).
[Crossref]

R. G. DeVoe and R. G. Brewer, “Experimental test of the optical Bloch equations for solids,” Phys. Rev. Lett. 50, 1269–1272 (1983).
[Crossref]

Rev. Mod. Phys. (1)

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943).
[Crossref]

Sov. Phys. JETP (1)

Note that expression (5.68) lacks the delta-function contribution found in expression (5.23) [compare also expressions (5.58) and (5.7) for Γ ≫ γi]. The Brownian motion model cannot reproduce this delta-function contribution [see A. P. Kolchenko, A. A. Pukhov, S. G. Rautian, and A. M. Shalagin, “Effect of selective collisions on the velocity distribution of atoms and on nonlinear interference effects,” Sov. Phys. JETP 36, 619–628 (1973); P. R. Berman, “Brownian motion of atomic systems: Fokker–Planck limit of the transport equation,” Phys. Rev. A 9, 2170–2176 (1974)]. When integrals of Eq. (5.7) or (5.8) or expression (5.23) are taken, the delta-function contribution is negligible, provided that Γ ≫ γi and Γ ≫ ∊0(i.e., in the strict Brownian motion limit when Γ → ∞).
[Crossref]

Other (2)

P. R. Berman and R. G. Brewer, “Modified Bloch equations for solids,” in Laser Spectroscopy VII, T. W. Hänsch and Y. R. Shen, eds. (Springer-Verlag, Berlin, 1985).
[Crossref]

The resonance approximation consists of neglecting terms that vary as exp[+i(Ω + ω)t]. The field-interaction representation consists of writing ρ12(R, v, t) = ρ12(v, t)exp[−i(k· R− Ωt)].

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Figures (1)

Fig. 1
Fig. 1

The two-level quantum system considered in this work.

Tables (1)

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Table 1 Comparison of Kernels

Equations (171)

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E ( R , t ) = ½ E exp [ i ( k · R - Ω t ) ] + conjugate ,
ω ω + ,
Γ = W ( ) d .
ρ 11 ( , t ) / t = - γ 1 ρ 11 ( , t ) + γ 2 ρ 22 ( , t ) - ½ [ i χ ρ 12 ( , t ) - i χ * ρ 21 ( , t ) ] - Γ ρ 11 ( , t ) + W ( ) ρ 11 ( , t ) d + Λ 1 ,
ρ 22 ( , t ) / t = - γ 2 ρ 22 ( , t ) + ½ [ i χ ρ 12 ( , t ) - i χ * ρ 21 ( , t ) ] - Γ ρ 22 ( , t ) + W ( ) ρ 22 ( , t ) d + Λ 2 ,
ρ 12 ( , t ) / t = - [ γ 12 - i ( Δ + ) ] ρ 12 ( , t ) - ½ i χ * [ ρ 11 ( , t ) - ρ 22 ( , t ) ] - Γ ρ 12 ( , t ) + W ( ) ρ 12 ( , t ) d ,
ρ 21 ( , t ) = [ ρ 12 ( , t ) ] * ,
Δ = ω - Ω ,
γ 12 = ½ ( γ 1 + γ 2 ) ,
χ = 1 p · E 2 / ,
Λ i ( Δ , ) = λ i g ( Δ ) f ( )             ( i = 1 , 2 ) ,
γ 2 γ 2 + γ 1 , Λ 2 ~ 0 ,             Λ 1 ~ γ 1 g ( Δ ) f ( ) , γ 1 ~ 0.
ρ ( , t ) / t = - A ( ) ρ ( , t ) + B ρ ( , t ) + W ( ) ρ ( , t ) d + Λ ( Δ , ) ,
ρ = ( ρ 11 ρ 22 ρ 12 ρ 21 ) ,
A ( ) = ( γ 1 + Γ - γ 2 0 0 0 γ 2 + Γ 0 0 0 0 γ 12 + Γ - i ( Δ + ) 0 0 0 0 γ 12 + Γ + i ( Δ + ) )
B = 1 2 i ( 0 0 - χ χ * 0 0 χ - χ * - χ * χ * 0 0 χ - χ 0 0 ) ,
Λ ( Δ , ) = g ( Δ ) f ( ) ( λ 1 λ 2 0 0 ) .
ρ ( , t ) = G ( , , t - t o ) ρ ( , t o ) d + d t o t d t G ( , , t - t ) [ Λ ( Δ , ) + B ρ ( , t ) ] ,
G ( , , τ ) / τ = - A ( ) G ( , , τ ) + W ( ) G ( , , τ ) d ,
G ( , , 0 ) = δ ( - ) 1 .
ρ ( ) = G ( , ) [ Λ ( Δ , ) + B ρ ( ) ] d ,
- A ( ) G ( , ) + W ( ) G ( , ) d = - δ ( - ) 1 .
G ( , ) = 0 G ( , , τ ) d τ .
G 11 ; 11 G 11 , G 22 ; 22 G 22 , G 12 ; 12 G 12 , G 21 ; 21 = G 21 . G 11 ; 22 G 1 ; 2 .
f ( ) W ( ) = W ( ) f ( ) ,
W ( ) f ( ) d = Γ f ( ) .
G i i ( , , τ ) d = exp ( - γ i τ ) ,
G 1 ; 2 ( , , τ ) d = γ 2 γ 1 - γ 2 [ exp ( - γ 2 τ ) - exp ( - γ 1 τ ) ] ,
G i i ( , ) d = 1 / γ i ,
G 1 ; 2 ( , ) d = γ 2 γ 2 1 γ 1 ,
G i i ( , , τ ) f ( ) d = f ( ) exp ( - γ i τ ) ,
G 1 ; 2 ( , , τ ) f ( ) d = f ( ) γ 2 γ 1 - γ 2 [ exp ( - γ 2 τ ) - exp ( - γ 1 τ ) ] ,
G i i ( , ) f ( ) d = f ( ) / γ i ,
G 1 ; 2 ( , ) f ( ) d = f ( ) γ 2 γ 2 1 γ 1 .
Γ = W ( ) d
W ( ) = W ( - α ) ,
( - ) = ( 1 / Γ ) ( - ) W ( ) d = - ( 1 - α )
( δ ) 2 = ( 1 / Γ ) ( - ) 2 W ( ) d = σ 2 + ( 1 - α ) 2 2 ,
σ 2 = ( 1 / Γ ) x 2 W ( x ) d x .
W ( - α ) = W ( ) = Γ f ( ) .
( δ ) 2 = ½ 0 2 + 2 ,
0 2 = 2 2 = 2 2 f ( ) d .
W ( - α ) W ( - ) .
δ σ 0 ,
( Γ T c ) 1 / 2 δ 0 ,
( δ ) 2 = κ ( 1 - α ) 0 2
β = Γ ( 1 - α )
- Γ G ( , , τ ) + W ( ) G ( , , τ ) d β G ( , , τ ) + β G ( , , τ ) + q 2 G ( , , τ ) 2 ,
q = ½ Γ ( δ ) 2 = ½ β κ 0 2 .
f ( ) = lim γ 2 0 G 22 ( , , ) = ( π 0 2 ) - 1 / 2 exp ( - 2 / 0 2 ) ,
0 2 = 2 q / β .
( δ ) 2 = ( 1 - α ) 0 2 = ( β / Γ ) 0 2 ~ 0.
( δ ) T c 1 ,
G i i ( , , τ ) / τ = - ( γ i + Γ ) G i i ( , , τ ) + W ( ) G i i ( , , τ ) d ,
G 1 ; 2 ( , , τ ) / τ = - ( γ 1 + Γ ) G 1 ; 2 ( , , τ ) + γ 2 G 22 ( , , τ ) + W ( ) G 1 ; 2 ( , , τ ) d ,
G 12 ( , , τ ) / τ = - [ γ 12 + Γ - i ( Δ + ) ] G 12 ( , , τ ) + W ( ) G 12 ( , , τ ) d ,
G 21 ( , , τ ) = [ G 12 ( , , τ ) ] * ,
G i i ( , , 0 ) = G 12 ( , , 0 ) = δ ( - ) ,             G 1 ; 2 ( , , 0 ) = 0 ;
- ( γ i + Γ ) G i i ( , ) + W ( ) G i i ( , ) d = - δ ( - ) ,
- ( γ 1 + Γ ) G 1 ; 2 ( , ) + W ( ) G 1 ; 2 ( , ) d = - γ 2 G 22 ( , ) ,
- [ γ 12 + Γ - i ( Δ + ) ] G 12 ( , ) + W ( ) G 12 ( , ) d = - δ ( - ) ,
G 21 ( , ) = [ G 12 ( , ) ] * .
G i i ( , , τ ) / τ = - ( γ i + Γ ) G i i ( , , τ ) + Γ f ( ) G i i ( , , τ ) d ,
G 1 ; 2 ( , , τ ) / τ = - ( γ 1 + Γ ) G 1 ; 2 ( , , τ ) + Γ f ( ) G 1 ; 2 ( , , τ ) d + γ 2 G 22 ( , , τ ) ,
G i i ( , , τ ) = exp ( - γ i τ ) [ e - Γ τ δ ( - ) + ( 1 - e - Γ τ ) f ( ) ]
G 1 ; 2 ( , , τ ) = γ 2 γ 1 - γ 2 [ exp ( - γ 2 T τ ) - exp ( - γ 1 T τ ) ] × [ δ ( - ) - f ( ) ] + γ 2 γ 1 T - γ 2 f ( ) × [ exp ( - γ 2 τ ) - exp ( - γ 1 T τ ) ] ( 1 + Γ γ 1 - γ 2 ) - γ 2 γ 1 - γ 2 [ exp ( - γ 1 τ ) - exp ( - γ 1 T τ ) ] ,
γ i T = γ i + Γ .
G i i ( , ) = δ ( - ) γ i T + Γ γ i γ i T f ( )
G 1 ; 2 ( , ) = γ 2 γ 1 T γ 2 T δ ( - ) + γ 2 Γ γ 1 T γ 2 ( 1 γ 1 + 1 γ 2 T ) f ( ) .
G 12 ( , , τ ) / τ = - [ γ 12 T - i ( Δ + ) ] G 12 ( , , τ ) + f ( ) G 12 ( , , τ ) d ,
γ 12 T = γ 12 + Γ
I ( , τ ) = G 12 ( , , τ ) d ,
G 12 ( , , τ ) = exp { - [ γ 12 T - i ( Δ + ) ] τ } δ ( - ) + Γ f ( ) 0 τ exp { - [ γ 12 T - i ( Δ + ) ] ( τ - τ ) } × I ( , τ ) d τ ,
I ( , τ ) = exp { - [ γ 12 T - i ( Δ + ) ] τ } + 0 τ K ( τ - τ ) I ( , τ ) d τ ,
K ( τ ) = Γ f ( ) exp { - [ γ 12 T - i ( Δ + ) ] τ } d
I ( , 0 ) = 1.
I ( , τ ) = 1 2 π - d ν e - i ν τ [ 1 - K ˜ ( ν ) ] [ γ 12 T - i ( Δ + + ν ) ] ,
K ˜ ( ν ) = 0 K ( τ ) e i ν τ d τ = Γ d f ( ) γ 12 T - i ( Δ + + ν ) .
I ( , τ ) ~ exp { - [ γ 12 T - i ( Δ + ) ] τ } ,             Γ 0 .
[ γ 12 T - i ( Δ + + ν ) ] [ 1 - K ˜ ( ν ) ] ~ γ 12 - i ( Δ + ν ) + Γ 2 [ Γ - i ( Δ + ν ) ] 2
I ( , τ ) ~ exp [ - ( γ 12 + 2 Γ - i Δ ) τ ] ,             Γ 0 , γ 12 .
G 12 ( , ) = 1 γ 12 T - i ( Δ + ) { δ ( - ) + Γ f ( ) [ 1 - K ˜ ( 0 ) ] [ γ 12 T - i ( Δ + ) ] } .
G 12 ( , ) ~ 1 γ 12 T - i ( Δ + ) [ δ ( - ) + Γ f ( ) γ 12 T - i ( Δ + ) ] ,             Γ 0 .
G 12 ( , ) ~ 1 γ 12 T - i Δ [ δ ( - ) + Γ f ( ) γ 12 + Γ 2 ( Γ - i Δ ) 2 - i Δ ] ,             Γ 0 , γ 12 .
G i i ( , , τ ) / τ = - ( γ i + Γ ) G i i ( , , τ ) + W ( - ) G i i ( , , τ ) d ,
G 1 ; 2 ( , , τ ) / τ = - ( γ 1 + Γ ) G 1 ; 2 ( , , τ ) + γ 2 G 22 ( , , τ ) + W ( - ) G 1 ; 2 ( , , τ ) d ,
G i i ( , , τ ) = 1 2 π - exp [ i ( - ) τ ] exp [ - Γ i ( τ ) τ ] d τ ,
G 1 ; 2 ( , , τ ) = γ 2 2 π - exp [ i ( - ) τ ] × exp [ - Γ 1 ( τ ) τ ] - exp [ - Γ 2 ( τ ) τ ] Γ 1 ( τ ) - Γ 2 ( τ ) d τ ,
Γ i ( τ ) = γ i + Γ - W ( x ) exp ( - i x τ ) d x = γ i + Γ - W ( x ) cos ( x τ ) d x
G i i ( , ) = 1 2 π - exp [ i ( - ) τ ] Γ i ( τ ) d τ ,
G 1 ; 2 ( , ) = γ 2 2 π - exp [ i ( - ) τ ] Γ 1 ( τ ) Γ 2 ( τ ) d τ .
W ( x ) = Γ [ 2 π ( δ ) 2 ] - 1 / 2 exp [ - x 2 / 2 ( δ ) 2 ] .
Γ i ( τ ) = γ i + Γ ( 1 - exp { - ½ [ ( δ ) τ ] 2 } ) .
G i i ( , ) ~ δ ( - ) γ i , G 1 ; 2 ( , ) ~ γ 2 γ 1 γ 2 δ ( - ) ,             Γ γ i .
G i i ( , ) , G 1 ; 2 ( , ) ~ 0             for - 0 .
τ max 1 / 0 ,
[ ( Γ / γ i ) ( δ / 0 ) 2 ] 1 / 2 1.
Γ i ( τ ) ~ γ i + q τ 2
q = ½ Γ ( δ ) 2 .
( β / γ i ) 1 / 2 1.
Γ i ( τ ) ~ γ i + Γ ,             ( δ ) τ 1.
G 12 ( , , τ ) = exp { - [ γ 12 T - i ( Δ + ) ] τ } G ¯ 12 ( , , τ ) ,
G ¯ 12 ( , , τ ) / τ = W ( - ) × exp [ - i ( - ) τ ] G ¯ 12 ( , , τ ) d ,
G 12 ( , , τ ) = exp [ i ( Δ + ) τ ] 2 π × - exp [ i ( - ) τ ] exp [ - Γ 12 ( τ , τ ) τ ] d τ ,
Γ 12 ( τ , τ ) = γ 12 T - - exp ( i x τ ) [ 1 - exp ( - i x τ ) i x τ ] W ( x ) d x .
I ( , τ ) = G 12 ( , , τ ) d = exp { - [ Γ 12 ( τ , τ ) - i ( Δ + ) ] τ } ,
Γ 12 ( τ , τ ) = γ 12 T - - sin ( x τ ) x τ W ( x ) d x .
G 12 ( , ) = 1 2 π 0 d τ exp [ i ( Δ + ) τ ] - d τ × exp [ i ( - ) τ ] exp [ - Γ 12 ( τ , τ ) τ ] = 1 2 π - d τ - τ d τ exp ( i τ ) × exp [ i Δ ( τ - τ ) ] exp [ - d ( τ , τ ) ( τ - τ ) ] ,
d ( τ , τ ) = Γ 12 ( - τ , τ - τ ) = γ 12 T + - [ sin ( x τ ) - sin ( x τ ) ] x ( τ - τ ) W ( x ) d x .
Γ 12 ( τ , τ ) ~ γ 12 T ,             ( δ ) τ 1 ,
d ( τ , τ ) ~ γ 12 T ,             ( δ ) τ - τ 1.
Γ 12 ( τ , τ ) ~ γ 12 + q τ 2 ,
d ( τ , τ ) ~ γ 12 + 1 3 q ( τ 3 - τ 3 τ - τ ) .
[ ( Γ / γ 12 ) ( δ / 0 ) 2 ] 1 / 2 1 ,
( β / γ 12 ) 1 / 2 1.
[ Γ ( δ / 0 ) 2 τ ] 1 / 2 1 ,
( β τ ) 1 / 2 1.
G i i ( , , τ ) / τ = - γ i G i i ( , , τ ) + β G i i ( , , τ ) + β G i i ( , , τ ) + q 2 G i i ( , , τ ) 2 ,
G 1 ; 2 ( , , τ ) / τ = - γ 1 G 1 ; 2 ( , , τ ) + γ 2 G 22 ( , , τ ) + β G 1 ; 2 ( , , τ ) + β G 1 ; 2 ( , , τ ) + q 2 G 1 ; 2 ( , , τ ) 2 ,
G i i ( , , τ ) = ( 2 π c ) - 1 / 2 exp ( - γ i τ ) exp ( - η 2 / 2 c ) ,
η = - e - β τ ,             c = q β ( 1 - e - 2 β τ ) .
G i i ( , ) ~ 1 γ i 1 ( π 0 2 ) 1 / 2 exp [ - ( 0 ) 2 ] ,             β γ i .
G 1 ; 2 ( , ) ~ γ 2 γ 1 γ 2 1 ( π 0 2 ) 1 / 2 exp [ - ( 0 ) 2 ] ,             β γ i .
q 2 G i i ( , ) 2 + β G i i ( , ) + β G i i ( , ) - γ i G i i ( , ) = - δ ( - ) ,
q 2 G 1 ; 2 ( , ) 2 + β G 1 ; 2 ( , ) + β G 1 ; 2 ( , ) - γ 1 G 1 ; 2 ( , ) = - γ 2 G 22 ( , ) .
G i i ( , ) ~ 1 2 π - exp [ i ( - ) τ ] γ i + q τ 2 d τ ,             ( β / γ i ) 1 / 2 1 ,
G 1 ; 2 ( , ) ~ 1 2 π - exp [ i ( - ) τ ] ( γ 1 + q τ 2 ) ( γ 2 + q τ 2 ) d τ ,             ( β / γ i ) 1 / 2 1.
G 12 ( , , τ ) = ( 2 π c ) - 1 / 2 exp [ - ( γ 12 - i Δ ) τ ] × exp [ i ( / β ) ( 1 - e - β τ ) ] × exp { - [ η 2 + 2 i b η - ( a c - b 2 ) 2 c ] } ,
a = 2 q β 3 [ β τ - ( 1 - e - β τ ) - 1 2 ( 1 - e - β τ ) 2 ] ,
b = q β 2 ( 1 - e - β τ ) 2 ,
c = q β ( 1 - e - 2 β τ ) ,
η = - e - β τ .
I ( , τ ) = G 12 ( , , τ ) d = exp [ - ( γ 12 - i Δ ) τ ] exp [ i ( / β ) ( 1 - e - β τ ) ] e - a / 2 .
2 q / β 3 = 0 2 / β 2 1 ,
( / β ) ~ 0 ,             1 2 a ~ 0 2 2 β τ = 2 τ β
I ( , τ ) ~ exp [ - ( γ 12 - i Δ ) τ ] exp [ - ( 2 / β ) τ ] ,             β 0 .
I ( , τ ) ~ exp { - [ γ 12 - i ( Δ + ) ] τ } exp ( - 2 β τ 3 ) ,             ( β τ ) 1 / 2 1 ,
G 12 ( , ) ~ f ( ) γ 12 + q β 2 - i Δ = f ( ) γ 12 + 2 β - i Δ ,             0 β ,
ρ 12 ( t ) = d Δ d ρ 12 ( Δ , , t ) ,             t > 0.
ρ 12 ( t ) = d Δ d d G 12 ( , , t ) ρ 12 ( Δ , ) = d Δ d I ( , t ) ρ 12 ( Δ , ) ,
I ( , t ) = G 12 ( , , t ) d
ρ 12 ( 1 ) ( Δ , ) = 1 2 i χ * N 21 g ( Δ ) G 12 ( , ) f ( ) d
ρ 12 ( 3 ) ( Δ , ) = - 1 8 i χ * χ 2 N 21 g ( Δ ) d d d i v × G 12 ( , ) [ G 11 ( , ) + G 22 ( , ) - G 1 ; 2 ( , ) ] [ G 12 ( i v , ) * + G 12 ( i v , ) ] f ( i v ) ,
N 21 = λ 2 γ 2 - ( λ 1 γ 1 + γ 2 λ 2 γ 1 γ 2 )
Δ 0 = ω 0 - Ω .
g v ( Δ ) = δ ( Δ - Δ 0 ) ,
g s ( Δ ) = [ π ( Δ * ) 2 ] - 1 / 2 exp { - [ ( Δ - Δ 0 ) / Δ * ] 2 } .
f ( ) = ( π 0 2 ) - 1 / 2 exp [ - ( / 0 ) 2 ] .
0 γ 1 , γ 2
Δ 0 0             ( vapor ) ,
Δ 0 Δ *             ( solid ) .
ρ 12 ( 1 ) ( t ) = 1 2 i χ * N 21 exp [ - ( γ 12 + 2 Γ ) - i Δ 0 t ] γ 12 + 2 Γ - i Δ 0             ( vapor ) .
ρ 12 ( 3 ) ( t ) = - 1 8 i χ * χ 2 g s ( 0 ) 1 γ L 1 ( 2 / Γ ) × exp [ - 2 ( γ 12 + 2 Γ ) t ] ,
1 γ L = 1 γ 2 + 1 γ 1 ( 1 - γ 2 γ 2 ) .
ρ 12 ( 3 ) ( t ) = - 1 8 i π χ * χ 2 N ¯ 21 1 γ L T 1 γ 12 T exp ( - 2 γ 12 T t ) ,
N ¯ 21 = { f v ( Δ 0 )             ( vapor ) g s ( 0 )             ( solid )
1 γ L T = 1 γ 2 T + 1 γ 1 T ( 1 - γ 2 γ 2 T ) .
ρ 12 ( 3 ) ( t ) = - 1 8 ( 2 π i ) χ * χ 2 N ¯ 21 t exp [ - 2 Γ 12 ( t , t ) t ] L ( t ) d t ,
1 L ( t ) = 1 Γ 2 ( t ) + 1 Γ 1 ( t ) [ 1 - γ 2 Γ 2 ( t ) ] ,
( δ ) t 1 ,
ρ 12 ( 3 ) ( t ) = - 1 8 i π χ * χ 2 N ¯ 21 1 γ 12 T γ L T exp ( - 2 γ 12 T t ) ,
Γ ( δ ) 2 / γ i 3 1 ,
ρ 12 ( 3 ) ( t ) = - 1 8 ( 2 π i ) χ * χ 2 N ¯ 21 t exp [ - 2 ( γ 12 τ + 1 3 q τ 3 ) ] × [ 1 γ 2 + q τ 2 + 1 γ 1 + q τ 2 ( 1 - γ 2 γ 2 + q τ 2 ) ] d τ ,
q = ½ Γ ( δ ) 2 .
τ max ( γ i / q ) 1 / 2 .
q τ max 3 γ i ( γ i / q ) 1 / 2 = [ 2 γ i 3 / Γ ( δ ) 2 ] 1 / 2 1 ,
ρ 12 ( 3 ) ( t ) = - 1 8 ( 2 π i ) χ * χ 2 N ¯ 21 × { ( q γ 2 ) - 1 / 2 [ 1 2 π - tan - 1 ( q / γ 2 t ) ] ( 1 - γ 2 γ 2 - γ 1 ) + ( q γ 1 ) - 1 / 2 [ 1 2 π - tan - 1 ( q / γ 1 t ) ] ( 1 + γ 2 γ 2 - γ 1 ) } .
( δ ) t max = ( γ i / Γ ) 1 / 2 1.
[ Γ ( δ ) 2 / γ i ] 1 / 2 0 ,
[ Γ ( δ / 0 ) 2 t max ] 1 / 2 = [ 2 Γ ( δ ) 2 γ i ] 1 / 4 / 0 ( γ i / 0 ) 1 / 2 1 ,
δ Γ ,             δ ( γ i / Γ ) 1 / 2 0 ,             δ ( γ i / Γ ) 1 / 2 γ i .

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