Abstract

It is shown that the dependence of relaxational processes on radiation intensity associated with the finiteness of correlation time τc of relaxational perturbations is to a great extent defined by the statistics of these perturbations. Generalized master equations (GME’s) that take into account the nonvanishing correlation time τc are obtained by using the characteristic operator method. With the Gaussian statistics assumption for adiabatic perturbations causing a stochastic transition-frequency modulation, these GME’s are used to reveal the main features of the free-induction-decay rate dependence on radiation power. Good agreement with the experiment of DeVoe and Brewer [ Phys. Rev. Lett. 50, 1263 ( 1983)] is obtained. Our preceding theory [ Opt. Commun. 52, 279 ( 1984)] based on the correlation (Born) approximation closely agrees with the results of this paper at τc/T2 ≪ 1 and T1/T2 < 3.67.

© 1986 Optical Society of America

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References

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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. I. L. Carlsten, A. Szöke, M. G. Raymer, “Collisional redistribution and saturation of near-resonance scattered light,” Phys. Rev. A 15, 1029–1045 (1977).
    [CrossRef]
  3. A. M. Bonch-Bruevich, T. A. Vartanyan, V. V. Khromov, “Experimental observation of the Landau–Zener nonlinearity at the optical excitation of atom,” Zh. Eksp. Teor. Fiz. 78, 538–544 (1980).
  4. P. A. Apanasevich, A. P. Nizovtsev, “Some features of the relaxation processes manifestation in optics,” Kvantovaya Elektron. (Moscow) 2, 1654–1664 (1975).
  5. E. Hanamura, “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Phys. Soc. Jpn. 52, 3678–3684 (1983).
    [CrossRef]
  6. 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]
  7. M. Yamanoi, J. M. Eberly, “Optical Bloch equations for low-temperature solids,” Phys. Rev. Lett. 52, 1353 (1984).
    [CrossRef]
  8. M. Yamanoi, J. M. Eberly, “Relaxation terms for strong-field optical Bloch equations,” J. Opt. Soc. Am. B 1, 751–755 (1984).
    [CrossRef]
  9. P. A. Apanasevich, S. Ya. Kilin, A. P. Nizovtsev, N. S. Onishchenko, “On “anomalous” free induction decay rate,” Opt. Commun. 52, 279–282 (1984).
    [CrossRef]
  10. P. A. Apanasevich, S. Ya. Kilin, A. P. Nizovtsev, N. S. Onishchenko, “Generalized non-linear optics master equations taking into account the correlation time of relaxational perturbations,” Izv. Akad. Nauk SSSR Ser. Fiz. 49, 541–547 (1985).
  11. S. Mukamel, “Non-Markovian theory of molecular relaxation. I. Vibrational relaxation and dephasing in condensed phases,” Chem. Phys. 37, 33–47 (1979).
    [CrossRef]
  12. M. Loéve, “Fonctions aléatoires de second ordre,” Rev. Sci. 84, 195–206 (1946).
  13. K. Karhunen, “Über lineare Methoden in der Wakrscheinlich-keitstrechnung,” Ann. Acad. Sci. Fenn. Ser. A 1, 3–79 (1947).
  14. G. S. Agarwal, “Master equations for time correlation functions of a quantum system interacting with stochastic perturbations and applications to emission and absorption line shapes,” Z. Phys. 33B, 111–124 (1979).
  15. S. Ya. Kilin, A. P. Nizovtsev, “Generalized nonlinear optics master equations taking into account the correlation time of relaxational perturbations,” submitted to J. Phys. B.
  16. It should be noted that Eq. (24) differs from the expression for Dw(t) that could be obtained from Eq. (7) by using the second-order cumulant expansion by the total time ordering. For the general explanation of the second-order cumulant-expansion procedure, see, e.g., N. G. Van Kampen, in Fundamental Problems in Statistical Mechanics III, E. G. D. Cohen, ed. (North-Holland, Amsterdam, 1975), p. 257–276. For its application to coherent optical transients, see E. Hanamura, J. Phys. Soc. Jpn. 52, 2258–2266 (1983) and also Refs. 8 and 10.
    [CrossRef]
  17. The renormalization of system–field interaction (of Rabi frequency) by collisions was discussed also by Zaidi (Can. J. Phys. 59, 750–767 (1981), using the Feynman-diagrain technique for the GME derivation.
  18. Equations (26) in the limit of t ≫ τc were also obtained in Ref. 8 [see Eq. (9) of Ref. 8] using the second-order cumulant expansion and assumption K0τc, ≪ 1. Our derivation employing the only assumption of Gaussian statistics for relaxational perturbations is free from such limitations.
  19. The possibility of three-exponential FID’s was pointed out in Ref. 8, but, unfortunately, a detailed analysis of such a FID regime was not made in this paper.
  20. A. G. Redfield, “Nuclear magnetic resonance saturation and rotary saturation in solids,” Phys. Rev. 98, 1787–1809 (1955).
    [CrossRef]
  21. M. J. Weber, “Spontaneous emission probabilities and quantum efficiencies for excited states of Pr3+ in LaF3,” J. Chem. Phys. 48, 4774–4780 (1968).
    [CrossRef]
  22. R. M. Macfarlane, R. M. Shelby, R. L. Shoemaker, “Ultra-high-resolution spectroscopy: photon echoes in YAlO3:Pr3+ and LaF3:Pr3+,” Phys. Rev. Lett. 43, 1726–1730 (1979).
    [CrossRef]
  23. J. Javanainen, “Free-induction decay in a fluctuating two-level system,” Opt. Commun. 50, 26–30 (1984).
    [CrossRef]
  24. R. Boscaino, F. M. Gelardi, G. Messina, “Second-harmonic free-induction decay in a two-level spin system,” Phys. Rev. A 28, 495–497 (1983).
    [CrossRef]

1985 (1)

P. A. Apanasevich, S. Ya. Kilin, A. P. Nizovtsev, N. S. Onishchenko, “Generalized non-linear optics master equations taking into account the correlation time of relaxational perturbations,” Izv. Akad. Nauk SSSR Ser. Fiz. 49, 541–547 (1985).

1984 (5)

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]

M. Yamanoi, J. M. Eberly, “Optical Bloch equations for low-temperature solids,” Phys. Rev. Lett. 52, 1353 (1984).
[CrossRef]

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

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

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

1983 (3)

R. Boscaino, F. M. Gelardi, G. Messina, “Second-harmonic free-induction decay in a two-level spin system,” Phys. Rev. A 28, 495–497 (1983).
[CrossRef]

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

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

1981 (1)

The renormalization of system–field interaction (of Rabi frequency) by collisions was discussed also by Zaidi (Can. J. Phys. 59, 750–767 (1981), using the Feynman-diagrain technique for the GME derivation.

1980 (1)

A. M. Bonch-Bruevich, T. A. Vartanyan, V. V. Khromov, “Experimental observation of the Landau–Zener nonlinearity at the optical excitation of atom,” Zh. Eksp. Teor. Fiz. 78, 538–544 (1980).

1979 (3)

S. Mukamel, “Non-Markovian theory of molecular relaxation. I. Vibrational relaxation and dephasing in condensed phases,” Chem. Phys. 37, 33–47 (1979).
[CrossRef]

G. S. Agarwal, “Master equations for time correlation functions of a quantum system interacting with stochastic perturbations and applications to emission and absorption line shapes,” Z. Phys. 33B, 111–124 (1979).

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

1977 (1)

I. L. Carlsten, A. Szöke, M. G. Raymer, “Collisional redistribution and saturation of near-resonance scattered light,” Phys. Rev. A 15, 1029–1045 (1977).
[CrossRef]

1975 (1)

P. A. Apanasevich, A. P. Nizovtsev, “Some features of the relaxation processes manifestation in optics,” Kvantovaya Elektron. (Moscow) 2, 1654–1664 (1975).

1968 (1)

M. J. Weber, “Spontaneous emission probabilities and quantum efficiencies for excited states of Pr3+ in LaF3,” J. Chem. Phys. 48, 4774–4780 (1968).
[CrossRef]

1955 (1)

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

1947 (1)

K. Karhunen, “Über lineare Methoden in der Wakrscheinlich-keitstrechnung,” Ann. Acad. Sci. Fenn. Ser. A 1, 3–79 (1947).

1946 (1)

M. Loéve, “Fonctions aléatoires de second ordre,” Rev. Sci. 84, 195–206 (1946).

Agarwal, G. S.

G. S. Agarwal, “Master equations for time correlation functions of a quantum system interacting with stochastic perturbations and applications to emission and absorption line shapes,” Z. Phys. 33B, 111–124 (1979).

Apanasevich, P. A.

P. A. Apanasevich, S. Ya. Kilin, A. P. Nizovtsev, N. S. Onishchenko, “Generalized non-linear optics master equations taking into account the correlation time of relaxational perturbations,” Izv. Akad. Nauk SSSR Ser. Fiz. 49, 541–547 (1985).

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

P. A. Apanasevich, A. P. Nizovtsev, “Some features of the relaxation processes manifestation in optics,” Kvantovaya Elektron. (Moscow) 2, 1654–1664 (1975).

Bonch-Bruevich, A. M.

A. M. Bonch-Bruevich, T. A. Vartanyan, V. V. Khromov, “Experimental observation of the Landau–Zener nonlinearity at the optical excitation of atom,” Zh. Eksp. Teor. Fiz. 78, 538–544 (1980).

Boscaino, R.

R. Boscaino, F. M. Gelardi, G. Messina, “Second-harmonic free-induction decay in a two-level spin system,” Phys. Rev. A 28, 495–497 (1983).
[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]

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

Carlsten, I. L.

I. L. Carlsten, A. Szöke, M. G. Raymer, “Collisional redistribution and saturation of near-resonance scattered light,” Phys. Rev. A 15, 1029–1045 (1977).
[CrossRef]

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]

Eberly, J. M.

M. Yamanoi, J. M. Eberly, “Optical Bloch equations for low-temperature solids,” Phys. Rev. Lett. 52, 1353 (1984).
[CrossRef]

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

Gelardi, F. M.

R. Boscaino, F. M. Gelardi, G. Messina, “Second-harmonic free-induction decay in a two-level spin system,” Phys. Rev. A 28, 495–497 (1983).
[CrossRef]

Hanamura, E.

E. Hanamura, “Stochastic theory of coherent optical transients. II. Free induction decay in Pr3+:LaF3,” J. Phys. 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]

Karhunen, K.

K. Karhunen, “Über lineare Methoden in der Wakrscheinlich-keitstrechnung,” Ann. Acad. Sci. Fenn. Ser. A 1, 3–79 (1947).

Khromov, V. V.

A. M. Bonch-Bruevich, T. A. Vartanyan, V. V. Khromov, “Experimental observation of the Landau–Zener nonlinearity at the optical excitation of atom,” Zh. Eksp. Teor. Fiz. 78, 538–544 (1980).

Kilin, S. Ya.

P. A. Apanasevich, S. Ya. Kilin, A. P. Nizovtsev, N. S. Onishchenko, “Generalized non-linear optics master equations taking into account the correlation time of relaxational perturbations,” Izv. Akad. Nauk SSSR Ser. Fiz. 49, 541–547 (1985).

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

S. Ya. Kilin, A. P. Nizovtsev, “Generalized nonlinear optics master equations taking into account the correlation time of relaxational perturbations,” submitted to J. Phys. B.

Loéve, M.

M. Loéve, “Fonctions aléatoires de second ordre,” Rev. Sci. 84, 195–206 (1946).

Macfarlane, R. M.

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

Messina, G.

R. Boscaino, F. M. Gelardi, G. Messina, “Second-harmonic free-induction decay in a two-level spin system,” Phys. Rev. A 28, 495–497 (1983).
[CrossRef]

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]

Mukamel, S.

S. Mukamel, “Non-Markovian theory of molecular relaxation. I. Vibrational relaxation and dephasing in condensed phases,” Chem. Phys. 37, 33–47 (1979).
[CrossRef]

Nizovtsev, A. P.

P. A. Apanasevich, S. Ya. Kilin, A. P. Nizovtsev, N. S. Onishchenko, “Generalized non-linear optics master equations taking into account the correlation time of relaxational perturbations,” Izv. Akad. Nauk SSSR Ser. Fiz. 49, 541–547 (1985).

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

P. A. Apanasevich, A. P. Nizovtsev, “Some features of the relaxation processes manifestation in optics,” Kvantovaya Elektron. (Moscow) 2, 1654–1664 (1975).

S. Ya. Kilin, A. P. Nizovtsev, “Generalized nonlinear optics master equations taking into account the correlation time of relaxational perturbations,” submitted to J. Phys. B.

Onishchenko, N. S.

P. A. Apanasevich, S. Ya. Kilin, A. P. Nizovtsev, N. S. Onishchenko, “Generalized non-linear optics master equations taking into account the correlation time of relaxational perturbations,” Izv. Akad. Nauk SSSR Ser. Fiz. 49, 541–547 (1985).

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

Raymer, M. G.

I. L. Carlsten, A. Szöke, M. G. Raymer, “Collisional redistribution and saturation of near-resonance scattered light,” Phys. Rev. A 15, 1029–1045 (1977).
[CrossRef]

Redfield, A. G.

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

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]

Shelby, R. M.

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

Szöke, A.

I. L. Carlsten, A. Szöke, M. G. Raymer, “Collisional redistribution and saturation of near-resonance scattered light,” Phys. Rev. A 15, 1029–1045 (1977).
[CrossRef]

Van Kampen, N. G.

It should be noted that Eq. (24) differs from the expression for Dw(t) that could be obtained from Eq. (7) by using the second-order cumulant expansion by the total time ordering. For the general explanation of the second-order cumulant-expansion procedure, see, e.g., N. G. Van Kampen, in Fundamental Problems in Statistical Mechanics III, E. G. D. Cohen, ed. (North-Holland, Amsterdam, 1975), p. 257–276. For its application to coherent optical transients, see E. Hanamura, J. Phys. Soc. Jpn. 52, 2258–2266 (1983) and also Refs. 8 and 10.
[CrossRef]

Vartanyan, T. A.

A. M. Bonch-Bruevich, T. A. Vartanyan, V. V. Khromov, “Experimental observation of the Landau–Zener nonlinearity at the optical excitation of atom,” Zh. Eksp. Teor. Fiz. 78, 538–544 (1980).

Weber, M. J.

M. J. Weber, “Spontaneous emission probabilities and quantum efficiencies for excited states of Pr3+ in LaF3,” J. Chem. Phys. 48, 4774–4780 (1968).
[CrossRef]

Yamanoi, M.

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

M. Yamanoi, J. M. Eberly, “Optical Bloch equations for low-temperature solids,” Phys. Rev. Lett. 52, 1353 (1984).
[CrossRef]

Ann. Acad. Sci. Fenn. Ser. A (1)

K. Karhunen, “Über lineare Methoden in der Wakrscheinlich-keitstrechnung,” Ann. Acad. Sci. Fenn. Ser. A 1, 3–79 (1947).

Can. J. Phys. (1)

The renormalization of system–field interaction (of Rabi frequency) by collisions was discussed also by Zaidi (Can. J. Phys. 59, 750–767 (1981), using the Feynman-diagrain technique for the GME derivation.

Chem. Phys. (1)

S. Mukamel, “Non-Markovian theory of molecular relaxation. I. Vibrational relaxation and dephasing in condensed phases,” Chem. Phys. 37, 33–47 (1979).
[CrossRef]

Izv. Akad. Nauk SSSR Ser. Fiz. (1)

P. A. Apanasevich, S. Ya. Kilin, A. P. Nizovtsev, N. S. Onishchenko, “Generalized non-linear optics master equations taking into account the correlation time of relaxational perturbations,” Izv. Akad. Nauk SSSR Ser. Fiz. 49, 541–547 (1985).

J. Chem. Phys. (1)

M. J. Weber, “Spontaneous emission probabilities and quantum efficiencies for excited states of Pr3+ in LaF3,” J. Chem. Phys. 48, 4774–4780 (1968).
[CrossRef]

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

J. Phys. Soc. Jpn. (1)

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

Kvantovaya Elektron. (Moscow) (1)

P. A. Apanasevich, A. P. Nizovtsev, “Some features of the relaxation processes manifestation in optics,” Kvantovaya Elektron. (Moscow) 2, 1654–1664 (1975).

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, A. P. Nizovtsev, N. S. Onishchenko, “On “anomalous” free induction decay rate,” Opt. Commun. 52, 279–282 (1984).
[CrossRef]

Phys. Rev. (1)

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

Phys. Rev. A (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]

R. Boscaino, F. M. Gelardi, G. Messina, “Second-harmonic free-induction decay in a two-level spin system,” Phys. Rev. A 28, 495–497 (1983).
[CrossRef]

I. L. Carlsten, A. Szöke, M. G. Raymer, “Collisional redistribution and saturation of near-resonance scattered light,” Phys. Rev. A 15, 1029–1045 (1977).
[CrossRef]

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]

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

M. Yamanoi, J. M. Eberly, “Optical Bloch equations for low-temperature solids,” Phys. Rev. Lett. 52, 1353 (1984).
[CrossRef]

Rev. Sci. (1)

M. Loéve, “Fonctions aléatoires de second ordre,” Rev. Sci. 84, 195–206 (1946).

Z. Phys. (1)

G. S. Agarwal, “Master equations for time correlation functions of a quantum system interacting with stochastic perturbations and applications to emission and absorption line shapes,” Z. Phys. 33B, 111–124 (1979).

Zh. Eksp. Teor. Fiz. (1)

A. M. Bonch-Bruevich, T. A. Vartanyan, V. V. Khromov, “Experimental observation of the Landau–Zener nonlinearity at the optical excitation of atom,” Zh. Eksp. Teor. Fiz. 78, 538–544 (1980).

Other (4)

Equations (26) in the limit of t ≫ τc were also obtained in Ref. 8 [see Eq. (9) of Ref. 8] using the second-order cumulant expansion and assumption K0τc, ≪ 1. Our derivation employing the only assumption of Gaussian statistics for relaxational perturbations is free from such limitations.

The possibility of three-exponential FID’s was pointed out in Ref. 8, but, unfortunately, a detailed analysis of such a FID regime was not made in this paper.

S. Ya. Kilin, A. P. Nizovtsev, “Generalized nonlinear optics master equations taking into account the correlation time of relaxational perturbations,” submitted to J. Phys. B.

It should be noted that Eq. (24) differs from the expression for Dw(t) that could be obtained from Eq. (7) by using the second-order cumulant expansion by the total time ordering. For the general explanation of the second-order cumulant-expansion procedure, see, e.g., N. G. Van Kampen, in Fundamental Problems in Statistical Mechanics III, E. G. D. Cohen, ed. (North-Holland, Amsterdam, 1975), p. 257–276. For its application to coherent optical transients, see E. Hanamura, J. Phys. Soc. Jpn. 52, 2258–2266 (1983) and also Refs. 8 and 10.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Regions of the specific FID signal behavior as a function of laser intensity and system relaxational parameters. Region I is characterized by three-exponential decay with FID rates Re Γ0, Re Γ+, and Re Γ; Re Γ is the smallest. In region II the beat frequencies of the FID signal are changed. In regions III and IV, the FID is monoexponential, with a maximum weight of exp(−Γ0t) in region III and exp(−Γt) in region IV (where Redfield’s theory is valid).

Fig. 2
Fig. 2

The boundaries of regions I and II for different values of parameter γ1 = τc/T0: 0.05 (1), 0.012 (2), 0.01 (3). The dashed line shows the boundary of region A from Ref. 9 (curve D = 0). At γ1 → 0, the boundary of region I approaches this curve. The dashed–dotted line shows the boundary of region II for τc = 0.

Fig. 3
Fig. 3

Field dependence of real parts of Γ0 Γ+, and Γ. Thick lines denote the rates for maximum-weight exponents. Points X, X1, and X2 correspond to the boundary crossings of region I; points Y and Y1 correspond to those of region II.

Fig. 4
Fig. 4

FID rates as a function of laser intensity for T1 = 500 μsec, T2 = 21.7 μsec, and τc = 6 μsec. The points correspond to the experimental data of Ref. 1.

Equations (54)

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K ( t , τ ) = U ^ ( t ) U ^ ( τ )
r ˙ = - i ( t J z + 2 v J x ) r ,
r = [ Re ρ b a Im ρ b a ( ρ a a - ρ b b ) / 2 ] = ( x y z ) ,
J x = ( 0 0 0 0 0 - i 0 i 0 ) ,             J y = ( 0 0 i 0 0 0 - i 0 0 ) , J z = ( 0 - i 0 i 0 0 0 0 0 )
[ J x , J y ] = i J z ,             [ J x , J z ] = - i J y ,             [ J y , J z ] = i J x .
R ˙ ( t ) = - i Ω J z R ( t ) + exp ( - i Ω J z t ) D ˙ w ( t ) D w - 1 ( t ) × exp ( i Ω J z t ) R ( t ) ,
J z = exp ( - i θ J y ) J z exp ( i θ J y ) = J z cos θ + J y sin θ , cos θ = 0 / Ω ,             sin θ = 2 v / Ω ,             Ω = 0 ( 1 + 4 v 2 / 0 2 ) 1 / 2
D w ( t ) = T exp { 0 t [ - i U ( τ ) J z ( τ ) ] d τ } ,
J z ( t ) = exp ( i Ω J z t ) J z ( 0 ) exp ( - i Ω J z t ) = cos θ J z - ( 1 / 2 ) sin θ J + e i Ω t - ( 1 / 2 ) sin θ J - e - i Ω t ,
U ( t ) = k a k φ k ( t ) λ k ,
φ ( t ) = λ U ( t ) U ( τ ) φ ( τ ) d τ ;
U ( t ) U ( τ ) = k φ k ( t ) φ k ( τ ) / λ k .
F ( t , { ξ } ) = exp ( k i ξ k a k ) ρ ( t ) ,
Φ ( t , { ξ } ) = exp ( k i ξ k a k ) r ( t )
Φ t = - i Ω J z Φ - i k φ k ( t ) λ k J z Φ i ξ k ,
Φ ( t = 0 ) = Φ 0 ( { ξ } ) R ( 0 ) ,
R t = - i Ω J z R - i k φ k ( t ) λ k J z Φ k ( 1 ) , Φ k ( 1 ) t = - i Ω J z Φ k ( 1 ) - i k 1 φ k 1 ( t ) λ k J z Φ k 1 , k ( 2 ) ,
R = r ,             Φ k ( 1 ) = Φ i ξ k | { ξ } = 0 a k r , Φ k , k 1 ( 2 ) = 2 Φ i ξ k i ξ k 1 | { ξ } = 0 a k a k 1 r .
Φ = exp ( - i Ω J z t ) Φ ,
Φ t = - i k φ k ( t ) λ k J z ( t ) Φ i ξ k ,
Φ ( t , { ξ } ) = { T exp [ - i 0 t k φ k ( τ ) λ k J z ( τ ) i ξ k Φ 0 d τ ] } R ( 0 ) .
D w ( t , 0 ) = { T exp [ - i 0 t k φ k ( τ ) λ k J z ( τ ) i ξ k Φ 0 d τ ] } | { ξ } = 0 .
Φ 0 ( { ξ } ) = k Φ 0 k ( ξ k ) ,
D w ( t , 0 ) = T { k exp [ - i 0 t d τ φ k ( τ ) λ k × J z ( τ ) ξ k Φ 0 k ( i ξ k ) ] } { ξ } = 0 = T k Φ 0 k [ - i 0 t d τ φ k ( τ ) λ k J z ( τ ) ] .
Φ 0 k ( ξ k ) = exp ( i ξ k ) 2 ,
D w ( t , 0 ) = T exp [ - 0 t d τ 0 τ d τ k φ k ( τ ) φ k ( τ ) λ k J z ( τ ) J z ( τ ) ] = T exp [ - 0 t d τ 0 τ d τ U ( τ ) U ( τ ) J z ( τ ) J z ( τ ) ] .
R ˙ ( t ) = - i Ω J z R ( t ) - 0 t d τ U ( t ) U ( τ ) J z ( 0 ) J z ( τ - t ) R ( t ) .
σ ˙ a b = - i ( 0 + γ a b t ) σ a b + i v ˜ a b t ( σ a a - σ b b ) ,
σ ˙ a a = - σ ˙ b b = i v b a 0 σ a b - i v a b 0 σ b a - A σ a a ,
γ a b t = A / 2 + 0 t d τ K ( t , t - τ ) [ c 2 + s 2 2 exp ( - i Ω τ ) + s 2 2 exp ( i Ω τ ) ]
v ˜ a b t = ( v ˜ b a t ) * = v a b 0 { 1 - i Ω 0 t d τ K ( t , t - τ ) × [ c - c + 2 exp ( - i Ω τ ) + c - 2 exp ( i Ω τ ) ] }
γ a b t γ a b = A 2 + K ˜ ( 0 ) c 2 = Re K ˜ ( i Ω ) s 2 ,
v ˜ a b t v ˜ a b = v a b 0 ( 1 - 1 Ω { Im K ˜ ( i Ω ) + i × [ K ˜ ( 0 ) - Re K ˜ ( i Ω ) ] c } ) ,
K ( τ ) = K 0 γ c exp ( - γ c τ ) ,
σ ¯ a b = - i v ¯ a b ( γ a b - i 0 ) γ a b 2 + 0 2 + 4 ( v a b 0 / A ) Re [ v ¯ a b ( γ a b - i 0 ) ] .
I signal ( t ) ~ d 0 P ( 0 ) Re [ E a b ( t ) E ( t ) ] .
I signal ( t ) ~ cos [ ( ω 0 - ω ) t ] exp ( - t / T 2 ) - d 0 σ ¯ a b ( 0 ) × exp ( i 0 t ) + c . c . = cos [ ( ω 0 - ω ) t ] exp ( - t / T 2 ) { g 0 × exp ( - Γ 0 t ) + g + exp ( - Γ + t ) + g - exp ( - Γ - t ) } + c . c . ,
Q ( z ) = z 3 - a z 2 + b z - c ( z - Γ 0 2 ) ( z - Γ + 2 ) ( z - Γ - 2 ) ,
a τ c 2 = m + 2 n , b τ c 4 = ( 2 m + n ) n + K 0 T 1 α φ , c τ c 6 = m n 2 + K 0 T 1 α φ n ,
φ = ( 2 v τ c ) 2 ,             γ 1 = τ c / T 1 ,             γ 2 = τ c / T 2 .
( z - Γ 1 2 ) ( z - Γ 2 ) 2 = z 3 - a z 2 + b z - c .
c ( 27 c - 2 a b ) - ( a 2 - 4 b ) ( b 2 - 4 a c ) = 0.
( z - Γ 2 ) 3 = z 3 - a z 2 + b z - c
a 2 = 3 b ,             a 3 = 27 c ,
( v τ c ) 2 = [ 1 + γ 2 ( 1 - 2 γ 1 ) ] 3 54 γ 1 ( 1 - γ 1 ) 2 ( 2 γ 2 - γ 1 ) ,
[ 1 + γ 2 ( 1 - 2 γ 1 ) ] 3 9 γ 1 ( 1 - γ 1 ) 2 ( 2 γ 2 - γ 1 ) = 3 ( γ 2 2 - 1 ) + [ 1 + γ 2 ( 1 - 2 γ 1 ) ] 2 γ 1 ( 1 - γ 1 ) .
T 2 17 τ c ,             ( v τ c ) 2 54 T 1 / ( 17 ) 2 τ c .
( z - Γ 0 2 ) { [ z - Γ 0 2 + ( Im Γ ± ) 2 ] 2 + 4 Γ 0 2 ( Im Γ ± ) 2 } = z 3 - a z 2 + b z - c .
{ a + 2 [ ( 7 b - a 2 ) / 3 ] 1 / 2 } { a - 5 [ ( 7 b - a 2 ) / 3 ] 1 / 2 } 2 - 7 3 c = 0.
γ 1 = ( a 0 + 2 s 0 ) ( a 0 - 5 s 0 ) 2 - 7 3 ( b 0 - a 0 + 1 ) 7 d ( 5 + 4 a 0 / 3 s 0 ) ( a 0 - 5 s 0 ) - 7 3 ( c 0 - d ) ,
s 0 = [ ( 7 b 0 - a 0 2 ) / 3 ] 1 / 2 ;             a 0 = φ / 2 ( φ + 1 ) + 2 ; b 0 = φ / ( φ + 1 ) + K 0 T 1 φ / ( φ + 1 ) 2 + 1 ; c 0 = ( K 0 T 1 ) 2 φ 2 / ( φ + 1 ) 3 ; d = K 0 T 1 ( K 0 T 1 + 1 / 2 ) φ / ( φ + 1 ) 2 .
( a 0 + 2 s 0 ) ( a 0 - 5 s 0 ) 2 - 7 3 ( b 0 - a 0 + 1 ) = 0.
K 0 T 1 = ( v τ c ) - 2 ,
K 0 T 1 = ( 17 / 4 ) ( v τ c ) 2 .

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