Abstract

In a recent experiment [ R. G. Devoe and R. G. Brewer, Phys. Rev. Lett. 50, 1269 ( 1983)], it was found that the optical Bloch equations could not satisfactorily explain the signal that was observed for free-induction decay in the impurity ion cyrstal Pr3+:LaF3. Several theories have been proposed to explain this failure of the Bloch equations. In this paper, the general validity conditions for the optical Bloch equations are examined within the limits of a Markovian relaxation model. The specific problem to be considered is the interaction of an optical field with two-level atoms. The atoms undergo relaxation as a result of coupling to a perturber bath that, itself, is negligibly affected by the relaxation process. First, a simple decay-parameter model is assumed for the relaxation of atomic density-matrix elements. Such a model is found to lead to a set of generalized Bloch equations of which the conventional Bloch equations form a subset. Subsequently, more-realistic models for relaxation in both vapors and solids are considered within the limits of the impact approximation (i.e., the duration of a fluctuation can be viewed as instantaneous with respect to all relevant time scales in the problem). It is found that, even in the impact (Markovian) approximation, the generalized Bloch equations cannot be expected to provide an adequate description of relaxation, owing to effects in which a fluctuation-induced change in the atomic transition frequency persists between fluctuations. In vapors this frequency shift is produced by velocity-changing collisions that change the atomic resonance frequency (as seen in the laboratory frame), whereas in solids it is produced by local-field fluctuations. The limiting conditions under which one can expect both the generalized and the conventional Bloch equations to retain their validity are explored.

© 1986 Optical Society of America

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  1. R. G. DeVoe, R. G. Brewer, “Experimental test of the optical Bloch equations for solids,” Phys. Rev. Lett. 50, 1269–1272 (1983).
    [CrossRef]
  2. F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
    [CrossRef]
  3. See, for example, M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Sec. 7.5; L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975), Chap. 2.
  4. A. Schenzle, M. Mitsunaga, R. G. De Voe, R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984); M. Yamanoi, J. H. Eberly, “Relaxation terms for strong field optical Bloch equations,” J. Opt. Soc. Am. B 1, 751–755 (1984); E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258–2266 (1983); E. Hanamaura, “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Opt. Soc. Jpn. 52, 3678–3684 (1983); J. Javanainen, “Free induction decay in a fluctuating two level system,” Opt. Commun. 50, 26–30 (1984); P. A. Apanasevich, S. Ya Kilin, A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984); K. Wódkiewicz, J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985); P. R. Berman, R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784–2796 (1985).
    [CrossRef] [PubMed]
  5. F. Bloch, “Generalized theory of relaxation,” Phys. Rev. 105, 1206–1222 (1957), and references therein.
    [CrossRef]
  6. A. G. Redfield, “Nuclear magnetic resonance saturation in solids,” Phys. Rev. 98, 1787–1809 (1955).
    [CrossRef]
  7. There are many texts that discuss this problem. See, for example, A. Abragam, Principles of Nuclear Magnetism (Oxford U. Press, London, 1961); C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, Berlin, 1980), and references therein.
  8. S. Mukamel, “Collisional broadening of spectral line shapes in two-photon and multiphoton processes,” Phys. Rep. 93, 1–60 (1982), and references therein.
    [CrossRef]
  9. See, for example, the paper by Shore [B. W. Shore, “Modeling noise by jump processes in strong laser–atom interactions,” J. Opt. Soc. Am. B1, 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, 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]
  10. For a review of relaxation in vapors in the impact approximation, see P. R. Berman, “Collisions in atomic vapors,” in New Trends in Atomic Physics, G. Grynberg, R. Stora, eds. (North-Holland, Amsterdam, 1984), pp. 453–514.
    [CrossRef]
  11. For simplicity, I have taken γ12t to be real for equations written in the “m” basis. For complex γ12t, one must replace γ12t and Δ appearing in these equations by Re(γ12t) and [Δ − Im(γ12t)], respectively. Thus a supplementary condition for the validity of the conventional Bloch equations is Im(γ12t) = 0.
  12. Conditions (4.6) and (4.7) are sufficient but may not be necessary for the validity of the simple decay-parameter model. For example, one might find a case in which −Γρ(v, t) + ∫ W(v′ → v)ρ(v′, t)dv′ ≃ −Γ′ρ(v, t). However, conditions (4.6) and (4.7) are the most likely way to ensure a reduction to Eqs. (3.2).
  13. If γ12ph= 0, the GBE(4) that one recovers in the limit |k· δv| ≪ χ has decay parameters determined by the spontaneous-emission rates. On the contrary, the decay rates observed in nonlinear spectroscopy in weaker fields may be larger than the spontaneous ones if |k· δv| χ. Thus one can observe a diminution of the effective decay rates with increasing field strength in certain experimental limits. When similar ideas are applied to relaxation in solids, they explain, in a qualitative manner, the results obtained by DeVoe and Brewer.1 A decrease of decay rates observed in vapors has also been explained using this type of argument [see A. G. Yodh, J. Golub, N. M. Carlson, T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–662 (1984)].
    [CrossRef]
  14. In the model of a state-independent collision interaction and the limit that Γ ≫ (Doppler width), one regains the GBE(4) in the limit that |k· δv| ≪ Γ, since Γ−1 serves as the effective coherence time in the problem. The decay rates are determined by spontaneous emission only, so that, in linear spectroscopy, the absorption line width is determined by the natural rather than the Doppler width—this is an example of motional narrowing [see, for example, P. R. Berman, “Theory of collision effects on atomic and molecular line shapes,” App. Phys. (Germany) 6, 183–296 (1975), and references to motional narrowing therein].
  15. Three recent papers containing references to earlier work are the following: A. P. Ghosh, C. D. Nabors, M. A. Attili, J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985); A. G. Yodh, J. Golub, T. W. Mossberg, “Collisional relaxation of excited-state Zeeman coherences in atomic ytterbium vapor,” Phys. Rev. A 32, 844–853 (1985); J. C. Keller, J. L. Le Gouët, “Stimulated photon echo for angular analysis of elastic and depolarizing Yb*-noble-gas collisions,” Phys. Rev. A 32, 1624–1642 (1985).
    [CrossRef] [PubMed]
  16. A. Schenzle, M. Mitsunaga, R. G. Devoe, R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984).
    [CrossRef]
  17. For an atomic vapor, the impact approximation is often associated with the instantaneous change in the phase of the atomic dipole resulting from a collision. The major part of this contribution is contained in the [−γ12phρ˜12] term in Eq. (4.5c). For the model of frequency fluctuations in solids that has been adopted, however, there is no such contribution to the decay of ρ˜12, since it is assumed that only the frequency and not the phase of the atomic dipole is changed as a result of the local-field fluctuations. As discussed in the text, this is equivalent to the case of a state-independent collisional interaction in the vapor. What is important for the validity of the impact approximation is that the collision duration or the jump time for frequency fluctuations be small compared with all relevant time scales in the problem. Note that, in general, collisions in a vapor result in an unseparable combination of instantaneous phase and velocity changes (see Ref. 10).
  18. The models for frequency shifts in vapors and solids are not totally analogous. In the vapor, the active atoms change their velocity as a result of collisions, which produce a change in the atomic transition frequency as viewed in the laboratory frame. In the solid, local-field fluctuations produce the frequency shifts ∊ shown in Fig. 2. This implies that the local environment changes and stays that way until the next fluctuation. Thus changes in the perturber bath are directly linked to changes in the active ion’s transition frequency. The net effect on the ions is the same as that for the vapor, however, provided thatl the local-field fluctuations can be modeled as a Markovian process.

1985 (1)

Three recent papers containing references to earlier work are the following: A. P. Ghosh, C. D. Nabors, M. A. Attili, J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985); A. G. Yodh, J. Golub, T. W. Mossberg, “Collisional relaxation of excited-state Zeeman coherences in atomic ytterbium vapor,” Phys. Rev. A 32, 844–853 (1985); J. C. Keller, J. L. Le Gouët, “Stimulated photon echo for angular analysis of elastic and depolarizing Yb*-noble-gas collisions,” Phys. Rev. A 32, 1624–1642 (1985).
[CrossRef] [PubMed]

1984 (3)

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

If γ12ph= 0, the GBE(4) that one recovers in the limit |k· δv| ≪ χ has decay parameters determined by the spontaneous-emission rates. On the contrary, the decay rates observed in nonlinear spectroscopy in weaker fields may be larger than the spontaneous ones if |k· δv| χ. Thus one can observe a diminution of the effective decay rates with increasing field strength in certain experimental limits. When similar ideas are applied to relaxation in solids, they explain, in a qualitative manner, the results obtained by DeVoe and Brewer.1 A decrease of decay rates observed in vapors has also been explained using this type of argument [see A. G. Yodh, J. Golub, N. M. Carlson, T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–662 (1984)].
[CrossRef]

A. Schenzle, M. Mitsunaga, R. G. De Voe, R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984); M. Yamanoi, J. H. Eberly, “Relaxation terms for strong field optical Bloch equations,” J. Opt. Soc. Am. B 1, 751–755 (1984); E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258–2266 (1983); E. Hanamaura, “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Opt. Soc. Jpn. 52, 3678–3684 (1983); J. Javanainen, “Free induction decay in a fluctuating two level system,” Opt. Commun. 50, 26–30 (1984); P. A. Apanasevich, S. Ya Kilin, A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984); K. Wódkiewicz, J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985); P. R. Berman, R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784–2796 (1985).
[CrossRef] [PubMed]

1983 (1)

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

1982 (1)

S. Mukamel, “Collisional broadening of spectral line shapes in two-photon and multiphoton processes,” Phys. Rep. 93, 1–60 (1982), and references therein.
[CrossRef]

1975 (1)

In the model of a state-independent collision interaction and the limit that Γ ≫ (Doppler width), one regains the GBE(4) in the limit that |k· δv| ≪ Γ, since Γ−1 serves as the effective coherence time in the problem. The decay rates are determined by spontaneous emission only, so that, in linear spectroscopy, the absorption line width is determined by the natural rather than the Doppler width—this is an example of motional narrowing [see, for example, P. R. Berman, “Theory of collision effects on atomic and molecular line shapes,” App. Phys. (Germany) 6, 183–296 (1975), and references to motional narrowing therein].

1957 (1)

F. Bloch, “Generalized theory of relaxation,” Phys. Rev. 105, 1206–1222 (1957), and references therein.
[CrossRef]

1955 (1)

A. G. Redfield, “Nuclear magnetic resonance saturation in solids,” Phys. Rev. 98, 1787–1809 (1955).
[CrossRef]

1946 (1)

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[CrossRef]

Abragam, A.

There are many texts that discuss this problem. See, for example, A. Abragam, Principles of Nuclear Magnetism (Oxford U. Press, London, 1961); C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, Berlin, 1980), and references therein.

Attili, M. A.

Three recent papers containing references to earlier work are the following: A. P. Ghosh, C. D. Nabors, M. A. Attili, J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985); A. G. Yodh, J. Golub, T. W. Mossberg, “Collisional relaxation of excited-state Zeeman coherences in atomic ytterbium vapor,” Phys. Rev. A 32, 844–853 (1985); J. C. Keller, J. L. Le Gouët, “Stimulated photon echo for angular analysis of elastic and depolarizing Yb*-noble-gas collisions,” Phys. Rev. A 32, 1624–1642 (1985).
[CrossRef] [PubMed]

Berman, P. R.

In the model of a state-independent collision interaction and the limit that Γ ≫ (Doppler width), one regains the GBE(4) in the limit that |k· δv| ≪ Γ, since Γ−1 serves as the effective coherence time in the problem. The decay rates are determined by spontaneous emission only, so that, in linear spectroscopy, the absorption line width is determined by the natural rather than the Doppler width—this is an example of motional narrowing [see, for example, P. R. Berman, “Theory of collision effects on atomic and molecular line shapes,” App. Phys. (Germany) 6, 183–296 (1975), and references to motional narrowing therein].

For a review of relaxation in vapors in the impact approximation, see P. R. Berman, “Collisions in atomic vapors,” in New Trends in Atomic Physics, G. Grynberg, R. Stora, eds. (North-Holland, Amsterdam, 1984), pp. 453–514.
[CrossRef]

Bloch, F.

F. Bloch, “Generalized theory of relaxation,” Phys. Rev. 105, 1206–1222 (1957), and references therein.
[CrossRef]

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[CrossRef]

Brewer, R. G.

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

A. Schenzle, M. Mitsunaga, R. G. De Voe, R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984); M. Yamanoi, J. H. Eberly, “Relaxation terms for strong field optical Bloch equations,” J. Opt. Soc. Am. B 1, 751–755 (1984); E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258–2266 (1983); E. Hanamaura, “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Opt. Soc. Jpn. 52, 3678–3684 (1983); J. Javanainen, “Free induction decay in a fluctuating two level system,” Opt. Commun. 50, 26–30 (1984); P. A. Apanasevich, S. Ya Kilin, A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984); K. Wódkiewicz, J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985); P. R. Berman, R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784–2796 (1985).
[CrossRef] [PubMed]

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

Carlson, N. M.

If γ12ph= 0, the GBE(4) that one recovers in the limit |k· δv| ≪ χ has decay parameters determined by the spontaneous-emission rates. On the contrary, the decay rates observed in nonlinear spectroscopy in weaker fields may be larger than the spontaneous ones if |k· δv| χ. Thus one can observe a diminution of the effective decay rates with increasing field strength in certain experimental limits. When similar ideas are applied to relaxation in solids, they explain, in a qualitative manner, the results obtained by DeVoe and Brewer.1 A decrease of decay rates observed in vapors has also been explained using this type of argument [see A. G. Yodh, J. Golub, N. M. Carlson, T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–662 (1984)].
[CrossRef]

De Voe, R. G.

A. Schenzle, M. Mitsunaga, R. G. De Voe, R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984); M. Yamanoi, J. H. Eberly, “Relaxation terms for strong field optical Bloch equations,” J. Opt. Soc. Am. B 1, 751–755 (1984); E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258–2266 (1983); E. Hanamaura, “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Opt. Soc. Jpn. 52, 3678–3684 (1983); J. Javanainen, “Free induction decay in a fluctuating two level system,” Opt. Commun. 50, 26–30 (1984); P. A. Apanasevich, S. Ya Kilin, A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984); K. Wódkiewicz, J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985); P. R. Berman, R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784–2796 (1985).
[CrossRef] [PubMed]

Devoe, R. G.

A. Schenzle, M. Mitsunaga, R. G. Devoe, 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, R. G. Brewer, “Experimental test of the optical Bloch equations for solids,” Phys. Rev. Lett. 50, 1269–1272 (1983).
[CrossRef]

Ghosh, A. P.

Three recent papers containing references to earlier work are the following: A. P. Ghosh, C. D. Nabors, M. A. Attili, J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985); A. G. Yodh, J. Golub, T. W. Mossberg, “Collisional relaxation of excited-state Zeeman coherences in atomic ytterbium vapor,” Phys. Rev. A 32, 844–853 (1985); J. C. Keller, J. L. Le Gouët, “Stimulated photon echo for angular analysis of elastic and depolarizing Yb*-noble-gas collisions,” Phys. Rev. A 32, 1624–1642 (1985).
[CrossRef] [PubMed]

Golub, J.

If γ12ph= 0, the GBE(4) that one recovers in the limit |k· δv| ≪ χ has decay parameters determined by the spontaneous-emission rates. On the contrary, the decay rates observed in nonlinear spectroscopy in weaker fields may be larger than the spontaneous ones if |k· δv| χ. Thus one can observe a diminution of the effective decay rates with increasing field strength in certain experimental limits. When similar ideas are applied to relaxation in solids, they explain, in a qualitative manner, the results obtained by DeVoe and Brewer.1 A decrease of decay rates observed in vapors has also been explained using this type of argument [see A. G. Yodh, J. Golub, N. M. Carlson, T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–662 (1984)].
[CrossRef]

Lamb, W. E.

See, for example, M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Sec. 7.5; L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975), Chap. 2.

Mitsunaga, M.

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

A. Schenzle, M. Mitsunaga, R. G. De Voe, R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984); M. Yamanoi, J. H. Eberly, “Relaxation terms for strong field optical Bloch equations,” J. Opt. Soc. Am. B 1, 751–755 (1984); E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258–2266 (1983); E. Hanamaura, “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Opt. Soc. Jpn. 52, 3678–3684 (1983); J. Javanainen, “Free induction decay in a fluctuating two level system,” Opt. Commun. 50, 26–30 (1984); P. A. Apanasevich, S. Ya Kilin, A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984); K. Wódkiewicz, J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985); P. R. Berman, R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784–2796 (1985).
[CrossRef] [PubMed]

Mossberg, T. W.

If γ12ph= 0, the GBE(4) that one recovers in the limit |k· δv| ≪ χ has decay parameters determined by the spontaneous-emission rates. On the contrary, the decay rates observed in nonlinear spectroscopy in weaker fields may be larger than the spontaneous ones if |k· δv| χ. Thus one can observe a diminution of the effective decay rates with increasing field strength in certain experimental limits. When similar ideas are applied to relaxation in solids, they explain, in a qualitative manner, the results obtained by DeVoe and Brewer.1 A decrease of decay rates observed in vapors has also been explained using this type of argument [see A. G. Yodh, J. Golub, N. M. Carlson, T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–662 (1984)].
[CrossRef]

Mukamel, S.

S. Mukamel, “Collisional broadening of spectral line shapes in two-photon and multiphoton processes,” Phys. Rep. 93, 1–60 (1982), and references therein.
[CrossRef]

Nabors, C. D.

Three recent papers containing references to earlier work are the following: A. P. Ghosh, C. D. Nabors, M. A. Attili, J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985); A. G. Yodh, J. Golub, T. W. Mossberg, “Collisional relaxation of excited-state Zeeman coherences in atomic ytterbium vapor,” Phys. Rev. A 32, 844–853 (1985); J. C. Keller, J. L. Le Gouët, “Stimulated photon echo for angular analysis of elastic and depolarizing Yb*-noble-gas collisions,” Phys. Rev. A 32, 1624–1642 (1985).
[CrossRef] [PubMed]

Redfield, A. G.

A. G. Redfield, “Nuclear magnetic resonance saturation in solids,” Phys. Rev. 98, 1787–1809 (1955).
[CrossRef]

Sargent, M.

See, for example, M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Sec. 7.5; L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975), Chap. 2.

Schenzle, A.

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

A. Schenzle, M. Mitsunaga, R. G. De Voe, R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984); M. Yamanoi, J. H. Eberly, “Relaxation terms for strong field optical Bloch equations,” J. Opt. Soc. Am. B 1, 751–755 (1984); E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258–2266 (1983); E. Hanamaura, “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Opt. Soc. Jpn. 52, 3678–3684 (1983); J. Javanainen, “Free induction decay in a fluctuating two level system,” Opt. Commun. 50, 26–30 (1984); P. A. Apanasevich, S. Ya Kilin, A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984); K. Wódkiewicz, J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985); P. R. Berman, R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784–2796 (1985).
[CrossRef] [PubMed]

Scully, M. O.

See, for example, M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Sec. 7.5; L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975), Chap. 2.

Shore, B. W.

See, for example, the paper by Shore [B. W. Shore, “Modeling noise by jump processes in strong laser–atom interactions,” J. Opt. Soc. Am. B1, 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, 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]

Thomas, J. E.

Three recent papers containing references to earlier work are the following: A. P. Ghosh, C. D. Nabors, M. A. Attili, J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985); A. G. Yodh, J. Golub, T. W. Mossberg, “Collisional relaxation of excited-state Zeeman coherences in atomic ytterbium vapor,” Phys. Rev. A 32, 844–853 (1985); J. C. Keller, J. L. Le Gouët, “Stimulated photon echo for angular analysis of elastic and depolarizing Yb*-noble-gas collisions,” Phys. Rev. A 32, 1624–1642 (1985).
[CrossRef] [PubMed]

Yodh, A. G.

If γ12ph= 0, the GBE(4) that one recovers in the limit |k· δv| ≪ χ has decay parameters determined by the spontaneous-emission rates. On the contrary, the decay rates observed in nonlinear spectroscopy in weaker fields may be larger than the spontaneous ones if |k· δv| χ. Thus one can observe a diminution of the effective decay rates with increasing field strength in certain experimental limits. When similar ideas are applied to relaxation in solids, they explain, in a qualitative manner, the results obtained by DeVoe and Brewer.1 A decrease of decay rates observed in vapors has also been explained using this type of argument [see A. G. Yodh, J. Golub, N. M. Carlson, T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–662 (1984)].
[CrossRef]

App. Phys. (Germany) (1)

In the model of a state-independent collision interaction and the limit that Γ ≫ (Doppler width), one regains the GBE(4) in the limit that |k· δv| ≪ Γ, since Γ−1 serves as the effective coherence time in the problem. The decay rates are determined by spontaneous emission only, so that, in linear spectroscopy, the absorption line width is determined by the natural rather than the Doppler width—this is an example of motional narrowing [see, for example, P. R. Berman, “Theory of collision effects on atomic and molecular line shapes,” App. Phys. (Germany) 6, 183–296 (1975), and references to motional narrowing therein].

Phys. Rep. (1)

S. Mukamel, “Collisional broadening of spectral line shapes in two-photon and multiphoton processes,” Phys. Rep. 93, 1–60 (1982), and references therein.
[CrossRef]

Phys. Rev. (3)

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[CrossRef]

F. Bloch, “Generalized theory of relaxation,” Phys. Rev. 105, 1206–1222 (1957), and references therein.
[CrossRef]

A. G. Redfield, “Nuclear magnetic resonance saturation in solids,” Phys. Rev. 98, 1787–1809 (1955).
[CrossRef]

Phys. Rev. A (2)

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

A. Schenzle, M. Mitsunaga, R. G. De Voe, R. G. Brewer, “Microscopic theory of optical line narrowing of a coherently driven solid,” Phys. Rev. A 30, 325–335 (1984); M. Yamanoi, J. H. Eberly, “Relaxation terms for strong field optical Bloch equations,” J. Opt. Soc. Am. B 1, 751–755 (1984); E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258–2266 (1983); E. Hanamaura, “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Opt. Soc. Jpn. 52, 3678–3684 (1983); J. Javanainen, “Free induction decay in a fluctuating two level system,” Opt. Commun. 50, 26–30 (1984); P. A. Apanasevich, S. Ya Kilin, A. P. Nizovtsev, “On ‘anomalous’ free induction decay rate,” Opt. Commun. 52, 279–282 (1984); K. Wódkiewicz, J. H. Eberly, “Random-telegraph-signal theory of optical resonance relaxation with applications to free induction decay,” Phys. Rev. A 32, 992–1001 (1985); P. R. Berman, R. G. Brewer, “Modified optical Bloch equations for solids,” Phys. Rev. A 32, 2784–2796 (1985).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

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

If γ12ph= 0, the GBE(4) that one recovers in the limit |k· δv| ≪ χ has decay parameters determined by the spontaneous-emission rates. On the contrary, the decay rates observed in nonlinear spectroscopy in weaker fields may be larger than the spontaneous ones if |k· δv| χ. Thus one can observe a diminution of the effective decay rates with increasing field strength in certain experimental limits. When similar ideas are applied to relaxation in solids, they explain, in a qualitative manner, the results obtained by DeVoe and Brewer.1 A decrease of decay rates observed in vapors has also been explained using this type of argument [see A. G. Yodh, J. Golub, N. M. Carlson, T. W. Mossberg, “Optically inhibited collisional dephasing,” Phys. Rev. Lett. 53, 659–662 (1984)].
[CrossRef]

Three recent papers containing references to earlier work are the following: A. P. Ghosh, C. D. Nabors, M. A. Attili, J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985); A. G. Yodh, J. Golub, T. W. Mossberg, “Collisional relaxation of excited-state Zeeman coherences in atomic ytterbium vapor,” Phys. Rev. A 32, 844–853 (1985); J. C. Keller, J. L. Le Gouët, “Stimulated photon echo for angular analysis of elastic and depolarizing Yb*-noble-gas collisions,” Phys. Rev. A 32, 1624–1642 (1985).
[CrossRef] [PubMed]

Other (8)

There are many texts that discuss this problem. See, for example, A. Abragam, Principles of Nuclear Magnetism (Oxford U. Press, London, 1961); C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, Berlin, 1980), and references therein.

For an atomic vapor, the impact approximation is often associated with the instantaneous change in the phase of the atomic dipole resulting from a collision. The major part of this contribution is contained in the [−γ12phρ˜12] term in Eq. (4.5c). For the model of frequency fluctuations in solids that has been adopted, however, there is no such contribution to the decay of ρ˜12, since it is assumed that only the frequency and not the phase of the atomic dipole is changed as a result of the local-field fluctuations. As discussed in the text, this is equivalent to the case of a state-independent collisional interaction in the vapor. What is important for the validity of the impact approximation is that the collision duration or the jump time for frequency fluctuations be small compared with all relevant time scales in the problem. Note that, in general, collisions in a vapor result in an unseparable combination of instantaneous phase and velocity changes (see Ref. 10).

The models for frequency shifts in vapors and solids are not totally analogous. In the vapor, the active atoms change their velocity as a result of collisions, which produce a change in the atomic transition frequency as viewed in the laboratory frame. In the solid, local-field fluctuations produce the frequency shifts ∊ shown in Fig. 2. This implies that the local environment changes and stays that way until the next fluctuation. Thus changes in the perturber bath are directly linked to changes in the active ion’s transition frequency. The net effect on the ions is the same as that for the vapor, however, provided thatl the local-field fluctuations can be modeled as a Markovian process.

See, for example, M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Sec. 7.5; L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975), Chap. 2.

See, for example, the paper by Shore [B. W. Shore, “Modeling noise by jump processes in strong laser–atom interactions,” J. Opt. Soc. Am. B1, 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, 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]

For a review of relaxation in vapors in the impact approximation, see P. R. Berman, “Collisions in atomic vapors,” in New Trends in Atomic Physics, G. Grynberg, R. Stora, eds. (North-Holland, Amsterdam, 1984), pp. 453–514.
[CrossRef]

For simplicity, I have taken γ12t to be real for equations written in the “m” basis. For complex γ12t, one must replace γ12t and Δ appearing in these equations by Re(γ12t) and [Δ − Im(γ12t)], respectively. Thus a supplementary condition for the validity of the conventional Bloch equations is Im(γ12t) = 0.

Conditions (4.6) and (4.7) are sufficient but may not be necessary for the validity of the simple decay-parameter model. For example, one might find a case in which −Γρ(v, t) + ∫ W(v′ → v)ρ(v′, t)dv′ ≃ −Γ′ρ(v, t). However, conditions (4.6) and (4.7) are the most likely way to ensure a reduction to Eqs. (3.2).

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Figures (2)

Fig. 1
Fig. 1

The two-level quantum system considered in this work.

Fig. 2
Fig. 2

A schematic representation of the changes in the ionic transition frequency produced by local-field fluctuations as a function of time t. The frequency jumps are assumed to occur instantaneously, and the statistics of this stochastic process are described by the kernel W(′ → ) discussed in the text. The average jump rate is Γ.

Equations (76)

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E ( R , t ) = ½ E ( t ) exp [ i ( k · R - Ω t ) ] + conjugate ,
ρ 12 ( R , t ) = ρ ˜ 12 ( t ) exp [ - i ( k · R - Ω t ) ] ,
ρ 11 ( R , t ) = ρ ˜ 11 ( t ) ,             ρ 22 ( R , t ) = ρ ˜ 22 ( t ) ,
ρ ˜ 21 ( t ) = [ ρ ˜ 12 ( t ) ] * ,
ρ ˜ ˙ 11 = - γ 11 ρ ˜ 11 + γ 2 ρ ˜ 22 - ½ ( i χ ρ ˜ 12 - i χ * ρ ˜ 21 ) + Λ 1 ,
ρ ˜ ˙ 22 = - γ 2 ρ ˜ 22 + ½ ( i χ ρ ˜ 12 - i χ * ρ ˜ 21 ) + Λ 2 ,
ρ ˜ ˙ 12 = - ( γ 12 - i Δ - i k · v ) ρ ˜ 12 - ½ i χ * ( ρ ˜ 11 - ρ ˜ 22 ) ,
ρ ˜ 21 = ρ ˜ 12 * ,
Δ = ω - Ω ,
χ ( t ) = 1 p · E ( t ) 2 / ,
γ 12 = ½ ( γ 1 + γ 2 ) .
m 1 = ρ ˜ 12 + ρ ˜ 21 ,
m 2 = i ( ρ ˜ 21 - ρ ˜ 12 ) ,
m 3 = ρ ˜ 22 - ρ ˜ 11 ,
m 4 = ρ ˜ 11 + ρ ˜ 22 .
m ˙ 1 = - γ 12 m 1 - ( Δ + k · v ) m 2 + [ Im ( χ ) ] m 3 ,
m ˙ 2 = - γ 12 m 2 + ( Ω + k · v ) m 1 + [ Re ( χ ) ] m 3 ,
m ˙ 3 - ½ ( γ 1 + γ 2 + γ 2 ) m 3 - ½ ( γ 2 + γ 2 - γ 1 ) m 4 - [ Im ( χ ) ] m 1 - [ Re ( χ ) ] m 2 + Λ 2 - Λ 1 ,
m ˙ 4 = - ½ ( γ 1 + γ 2 - γ 2 ) m 4 - ½ ( γ 2 - γ 2 - γ 1 ) m 3 + Λ 1 + Λ 2 .
γ 2 γ 2 + γ 1 , Λ 2 ~ 0 ,             Λ 1 ~ γ 1 ρ ˜ 11 eq , γ 1 ~ 0 ,
γ 1 γ 1 t = γ 1 + γ 1 r ,
γ 2 γ 2 t = γ 2 + γ 2 r ,
γ 12 γ 12 t = γ 12 + γ 12 r = ( γ 21 t ) * ,
ρ ˜ ˙ 11 = - γ 1 t ρ ˜ 11 + γ 2 ρ ˜ 22 - ½ ( i χ ρ ˜ 12 - i χ * ρ ˜ 21 ) + Λ 1 ,
ρ ˜ ˙ 22 = - γ 2 t ρ ˜ 22 + ½ ( i χ ρ ˜ 12 - i χ * ρ ˜ 21 ) + Λ 2 ,
ρ ˜ ˙ 12 = - ( γ 12 t - i Δ - i k · v ) ρ ˜ 12 - ½ i χ * ( ρ ˜ 11 - ρ ˜ 22 ) ,
ρ ˜ 21 = ρ ˜ 12 * ,
m ˙ 1 = - γ 12 t m 1 - ( Δ + k · v ) m 2 + [ Im ( χ ) ] m 3 ,
m ˙ 2 = - γ 12 t m 2 + ( Δ + k · v ) m 1 + [ Re ( χ ) ] m 3 ,
m ˙ 3 = - ½ ( γ 1 t + γ 2 t + γ 2 ) m 3 - ½ ( γ 2 t + γ 2 - γ 1 t ) m 4 - [ Im ( χ ) ] m 1 - [ Re ( χ ) ] m 2 + Λ 2 - Λ 1 ,
m ˙ 4 = - ½ ( γ 1 t + γ 2 t - γ 2 ) m 4 - ½ ( γ 2 t - γ 2 - γ 1 t ) m 3 + Λ 1 + Λ 2 .
γ 2 t - ( γ 1 t - γ 2 ) = 0
γ 2 t - γ 2 = γ 1 t ~ 0 ,             Λ 2 = 0 ,             Λ 1 = γ 1 t ρ ˜ 11 e q ~ 0.
m ˙ 1 = - ( 1 / T 2 ) m 1 - ( Δ + k · v ) m 2 + [ Im ( χ ) ] m 3 ,
m ˙ 2 = - ( 1 / T 2 ) m 2 + ( Δ + k · v ) m 1 + [ Re ( χ ) ] m 3 ,
m ˙ 3 = - ( 1 / T 1 ) m 3 - [ Im ( χ ) ] m 1 - [ Re ( χ ) ] m 2 - Λ ,
m ˙ 4 = - γ 2 t m 4 - γ 2 m 3 + Λ 1 + Λ 2 ,
1 / T 2 = γ 12 t ,
1 / T 1 = γ 1 t ,
Λ = Λ 1 - Λ 2 .
m ˙ 1 = - ( 1 / T 2 ) m 1 - ( Δ + k · v ) m 2 + [ Im ( χ ) ] m 3 ,
m ˙ 2 = - ( 1 / T 2 ) m 2 + ( Δ + k · v ) m 1 + [ Re ( χ ) ] m 3 ,
m ˙ 3 = - ( 1 / T 1 ) m 3 - [ Im ( χ ) ] m 1 - [ Re ( χ ) ] m 2 - Λ ,
m ˙ 4 = 0 ,
1 / T 2 = γ 12 t ,
1 / T 1 = γ 2 ,
Λ = m 4 e q / T 1 ,
M = ( m 1 m 2 m 3 )
Ω B = [ - Re ( χ ) IM ( χ ) Δ + k · v ] ,
d M d t = Ω B × M - ( 1 / T 2 0 0 1 / T 2 0 0 0 1 / T 1 ) M - ( 0 0 Λ ) ,
Γ τ c 1 ,
γ τ c 1 ,
Δ τ c 1 ,
χ τ c 1 ,
ρ _ ( R , v , t ) / t + v · ρ _ ( R , v , t ) = ( i ) - 1 [ H _ , ρ _ ] + [ ρ _ ( R , v , t ) / t ] sp + [ ρ _ ( R , v , t ) / t ] col + Λ _ ( R , v , t ) ,
ρ i j ( R , v , t ) / t ) col = - γ i j p h ( v ) ρ i j ( R , v , t ) - Γ i j ( v ) ρ i j ( R , v , t ) + W i j ( v v ) ρ i j ( R , v , t ) d v .
Γ i j ( v ) = W i j ( v v ) d v .
ρ ˜ 11 ( v , t ) / t = - γ 1 ρ ˜ 11 ( v , t ) + γ 2 ρ ˜ 22 ( v , t ) - ½ [ i χ ρ ˜ 12 ( v , t ) - i χ * ρ ˜ 21 ( v , t ) ] - Γ 11 ( v ) ρ ˜ 11 ( v , t ) + W 11 ( v v ) ρ ˜ 11 ( v , t ) d v + Λ 1 ( v ) ,
ρ ˜ 22 ( v , t ) / t = - γ 2 ρ ˜ 22 ( v , t ) + ½ [ i χ ρ ˜ 12 ( v , t ) - i χ * ρ ˜ 21 ( v , t ) ] - Γ 22 ( v ) ρ ˜ 22 ( v , t ) + W 22 ( v v ) ρ ˜ 22 ( v , t ) d v + Λ 2 ( v ) ,
ρ ˜ 12 ( v , t ) / t = - ( γ 12 - i Δ - i k · v ) ρ ˜ 12 ( v , t ) - ½ i χ * [ ρ ˜ 11 ( v , t ) - ρ ˜ 22 ( v , t ) ] - γ 12 p h ( v ) ρ ˜ 12 ( v , t ) - Γ 12 ( v ) ρ ˜ 12 ( v , t ) + W 12 ( v v ) ρ ˜ 12 ( v , t ) d v ,
ρ ˜ 21 ( v , t ) = [ ρ ˜ 12 ( v , t ) ] * .
- Γ i j ( v ) ρ ˜ i j ( v , t ) + W i j ( v v ) ρ ˜ i j ( v , t ) d v 0
W i j ( v v ) ρ ˜ i j ( v , t ) d v 0 .
γ 1 t = γ 1 ,             γ 2 t = γ 2 ,             γ 12 t = γ 12 + γ 12 p h ( v ) ,
γ 1 t = γ 1 + Γ 11 ( v ) ,             γ 2 t = γ 2 + Γ 22 ( v ) , γ 12 t = γ 12 + γ 12 p h ( v ) + Γ 12 ( v ) .
Γ 11 ( v ) = Γ 22 ( v ) = Γ 12 ( v ) = Γ 21 ( v ) Γ ( v ) ,
W 11 ( v v ) = W 22 ( v v ) = W 12 ( v v ) = W 21 ( v v ) W ( v v ) ,
γ 12 p h ( v ) = 0 ,
ω ω + ( t ) .
ρ ˜ ˙ 11 = - γ 1 ρ ˜ 11 + γ 2 ρ ˜ 22 - ½ ( i χ ρ ˜ 12 - i χ * ρ ˜ 21 ) + Λ 1 ( ω ) ,
ρ ˜ ˙ 22 = - γ 2 ρ ˜ 22 + ½ ( i χ ρ ˜ 12 - i χ * ρ ˜ 21 ) + Λ 2 ( ω ) ,
ρ ˜ ˙ 12 = - [ γ 12 - i ( ω - Ω ) - i ( t ) ] ρ ˜ 12 - ½ i χ * ( ρ ˜ 11 - ρ ˜ 22 ) ,
ρ ˜ 21 = ρ ˜ 12 * .
ρ ˜ 11 ( , ω , t ) / t = - γ 1 ρ ˜ 11 ( , ω , t ) + γ 2 ρ ˜ 22 ( , ω , t ) - ½ [ i χ ρ ˜ 12 ( , ω , t ) - i χ * ρ ˜ 21 ( , ω , t ) ] - Γ ρ ˜ 11 ( , ω , t ) + W ( ) ρ ˜ 11 ( , ω , t ) d + Λ 1 ( , ω ) , ρ ˜ 22 ( , ω , t ) / t = - γ 2 ρ ˜ 22 ( , ω , t ) + ½ [ i χ ρ ˜ 12 ( , ω , t ) - i χ * ρ ˜ 21 ( , ω , t ) ] - Γ ρ ˜ 22 ( , ω , t ) + W ( ) ρ ˜ 22 ( , ω , t ) d + Λ 2 ( , ω ) ,
ρ ˜ 12 ( , ω , t ) / t = - [ γ 12 - i ( ω - Ω ) - i ] ρ ˜ 12 ( , ω , t ) - ½ i χ * [ ρ ˜ 11 ( , ω , t ) - ρ ˜ 22 ( , ω , t ) ] - Γ ρ ˜ 12 ( , ω , t ) + W ( ) ρ ˜ 12 ( , ω , t ) d
ρ ˜ 21 ( , ω , t ) = [ ρ ˜ 12 ( , ω , t ) ] * ,

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