Abstract

The concept of a grating in real and frequency space is examined in the context of a three-pulse optical excitation cycle applied to a pseudo two-level model system. The calculations are done analytically using the Liouville-operator formalism in matrix form. It is shown that a continuous transition occurs from a grating in real space to a grating in frequency space when the first two excitation pulses separate in time. During this transition, the role of the population-relaxation time constant (T1) is taken over by the dephasing time constant (T2) bringing out the irreversible nature of the loss of coherence in an excited state. The underlying space–time transformation when moving from a grating in real space to a grating in frequency space further clarifies the loss in symmetry of the scattering pattern induced by a probe pulse by attributing it to the law of causality. It is finally concluded that the generalized grating concept is a powerful means of analyzing or predicting the effects of multiple-pulse multicolor optical-coherence experiments.

© 1986 Optical Society of America

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References

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  1. I. D. Abella, N. A. Kurnit, and S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964); Phys. Rev. 41, 391 (1966).
    [Crossref]
  2. P. Ye and Y. R. Shen, Phys. Rev. A 25, 2183 (1982).
    [Crossref]
  3. K. Duppen, D. P. Weitekamp, and D. A. Wiersma, Chem. Phys. Lett. 106, 147 (1984); Chem. Phys. Lett. 108, 551 (1984).
    [Crossref]
  4. W. H. Hesselink and D. A. Wiersma, Phys. Rev. Lett. 43, 1991 (1979); J. Chem. Phys. 75, 4192 (1981).
    [Crossref]
  5. J. B. W. Morsink, W. H. Hesselink, and D. A. Wiersma, Chem. Phys. Lett. 64, 1 (1979).
    [Crossref]
  6. D. A. Wiersma, D. P. Weitekamp, and K. Duppen, in Ultrafast Phenomena IV, D. H. Auston and K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 224.
    [Crossref]
  7. H. Eichler and H. Stahl, J. Appl. Phys. 44, 3429 (1973), and references therein.
    [Crossref]
  8. T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, Phys. Rev. A 20, 1976 (1979).
    [Crossref]
  9. M. D. Fayer, Ann. Rev. Phys. Chem. 33, 63 (1982).
    [Crossref]
  10. A. M. Weiner, S. De Silvestri, and E. P. Ippen, J. Opt. Soc. Am. B 2, 6541985).
    [Crossref]
  11. S. De Silvestri, A. M. Weiner, J. G. Fujimoto, and E. P. Ippen, Chem. Phys. Lett. 112, 195 (1984).
    [Crossref]
  12. E. L. Hahn, Phys. Rev. 80, 580 (1950).
    [Crossref]
  13. For a recent discussion of this approximation, see Ph. de Bree, Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1981).
  14. N. Bloembergen and Y. R. Shen, Phys. Rev. 133, A37 (1964).
    [Crossref]
  15. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), Chap. 2.
  16. R. L. Schoemaker, in Laser and Coherence Spectroscopy, J. I. Steinfeld, ed. (Plenum, New York, 1978).
  17. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  18. W. H. Hesselink, Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1980).
  19. D. P. Weitekamp, K. Duppen, and D. A. Wiersma, Chem. Phys. Lett. 102, 139 (1983).
    [Crossref]
  20. Not the interaction picture. The connection between this rotating frame picture and the conventional interaction picture is ρ˜12 = exp[+i(ω21− ω)t]ρ12INT and ρ˜21 = exp[−i(ω21− ω)t]ρ21INT.
  21. E. Arimondo and G. Moruzzi, J. Phys. B 6, 2382 (1973).
    [Crossref]
  22. See, for instance, R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1983), Chap. 2.
  23. U. Fano, Phys. Rev. 131, 259 (1963).
    [Crossref]
  24. E. J. Putzer, Am. Math. Monthly 73, 2 (1966).
    [Crossref]
  25. W. H. Hesselink and D. A. Wiersma, J. Chem. Phys. 73, 648 (1980).
    [Crossref]
  26. W. H. Hesselink and D. A. Wiersma, J. Chem. Phys. 75, 4192 (1981).
    [Crossref]
  27. In this definition, the decay of the amplitude of an optical FID is PFID(t) ~ exp(−t2/T2)exp(−t/2T2′2).
  28. M. M. Salour and C. Cohen-Tannoudji, Phys. Rev. Lett. 38, 757 (1977).
    [Crossref]
  29. In some published papers ρ11(t12+) was written for a situation in which population relaxation is negligible on the time scale of the pulse separation t12. This limitation was not always pointed out, however [e.g., K. Duppen, D. P. Weitzkamp, and D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, and D. A. Wiersma, Physica 127B, 349 (1984)]. The correct, general, expression for ρ11isρ11(t12+)=¼{2-β exp(-k31t12+β exp(-t12/T1)+cos θ1× [β exp(-k31t12)-β exp(-t12/T1]+cos θ2× [2-β exp(-k31t12)+(β-2)exp(-t12/T1)]+ cos θ1 cos θ2[β exp(-k31t12)-(β-2)× exp(-t12/T1)]-2 sin θ1 sin θ2× exp(-t12/T2)cos(Δt12-k12·r+ϕ12)]}.
    [Crossref]

1985 (1)

1984 (3)

S. De Silvestri, A. M. Weiner, J. G. Fujimoto, and E. P. Ippen, Chem. Phys. Lett. 112, 195 (1984).
[Crossref]

K. Duppen, D. P. Weitekamp, and D. A. Wiersma, Chem. Phys. Lett. 106, 147 (1984); Chem. Phys. Lett. 108, 551 (1984).
[Crossref]

In some published papers ρ11(t12+) was written for a situation in which population relaxation is negligible on the time scale of the pulse separation t12. This limitation was not always pointed out, however [e.g., K. Duppen, D. P. Weitzkamp, and D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, and D. A. Wiersma, Physica 127B, 349 (1984)]. The correct, general, expression for ρ11isρ11(t12+)=¼{2-β exp(-k31t12+β exp(-t12/T1)+cos θ1× [β exp(-k31t12)-β exp(-t12/T1]+cos θ2× [2-β exp(-k31t12)+(β-2)exp(-t12/T1)]+ cos θ1 cos θ2[β exp(-k31t12)-(β-2)× exp(-t12/T1)]-2 sin θ1 sin θ2× exp(-t12/T2)cos(Δt12-k12·r+ϕ12)]}.
[Crossref]

1983 (1)

D. P. Weitekamp, K. Duppen, and D. A. Wiersma, Chem. Phys. Lett. 102, 139 (1983).
[Crossref]

1982 (2)

P. Ye and Y. R. Shen, Phys. Rev. A 25, 2183 (1982).
[Crossref]

M. D. Fayer, Ann. Rev. Phys. Chem. 33, 63 (1982).
[Crossref]

1981 (1)

W. H. Hesselink and D. A. Wiersma, J. Chem. Phys. 75, 4192 (1981).
[Crossref]

1980 (1)

W. H. Hesselink and D. A. Wiersma, J. Chem. Phys. 73, 648 (1980).
[Crossref]

1979 (3)

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, Phys. Rev. A 20, 1976 (1979).
[Crossref]

W. H. Hesselink and D. A. Wiersma, Phys. Rev. Lett. 43, 1991 (1979); J. Chem. Phys. 75, 4192 (1981).
[Crossref]

J. B. W. Morsink, W. H. Hesselink, and D. A. Wiersma, Chem. Phys. Lett. 64, 1 (1979).
[Crossref]

1977 (1)

M. M. Salour and C. Cohen-Tannoudji, Phys. Rev. Lett. 38, 757 (1977).
[Crossref]

1973 (2)

E. Arimondo and G. Moruzzi, J. Phys. B 6, 2382 (1973).
[Crossref]

H. Eichler and H. Stahl, J. Appl. Phys. 44, 3429 (1973), and references therein.
[Crossref]

1966 (1)

E. J. Putzer, Am. Math. Monthly 73, 2 (1966).
[Crossref]

1964 (2)

N. Bloembergen and Y. R. Shen, Phys. Rev. 133, A37 (1964).
[Crossref]

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964); Phys. Rev. 41, 391 (1966).
[Crossref]

1963 (1)

U. Fano, Phys. Rev. 131, 259 (1963).
[Crossref]

1950 (1)

E. L. Hahn, Phys. Rev. 80, 580 (1950).
[Crossref]

Abella, I. D.

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964); Phys. Rev. 41, 391 (1966).
[Crossref]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Arimondo, E.

E. Arimondo and G. Moruzzi, J. Phys. B 6, 2382 (1973).
[Crossref]

Bloembergen, N.

N. Bloembergen and Y. R. Shen, Phys. Rev. 133, A37 (1964).
[Crossref]

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), Chap. 2.

Cohen-Tannoudji, C.

M. M. Salour and C. Cohen-Tannoudji, Phys. Rev. Lett. 38, 757 (1977).
[Crossref]

De Silvestri, S.

A. M. Weiner, S. De Silvestri, and E. P. Ippen, J. Opt. Soc. Am. B 2, 6541985).
[Crossref]

S. De Silvestri, A. M. Weiner, J. G. Fujimoto, and E. P. Ippen, Chem. Phys. Lett. 112, 195 (1984).
[Crossref]

Duppen, K.

K. Duppen, D. P. Weitekamp, and D. A. Wiersma, Chem. Phys. Lett. 106, 147 (1984); Chem. Phys. Lett. 108, 551 (1984).
[Crossref]

In some published papers ρ11(t12+) was written for a situation in which population relaxation is negligible on the time scale of the pulse separation t12. This limitation was not always pointed out, however [e.g., K. Duppen, D. P. Weitzkamp, and D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, and D. A. Wiersma, Physica 127B, 349 (1984)]. The correct, general, expression for ρ11isρ11(t12+)=¼{2-β exp(-k31t12+β exp(-t12/T1)+cos θ1× [β exp(-k31t12)-β exp(-t12/T1]+cos θ2× [2-β exp(-k31t12)+(β-2)exp(-t12/T1)]+ cos θ1 cos θ2[β exp(-k31t12)-(β-2)× exp(-t12/T1)]-2 sin θ1 sin θ2× exp(-t12/T2)cos(Δt12-k12·r+ϕ12)]}.
[Crossref]

D. P. Weitekamp, K. Duppen, and D. A. Wiersma, Chem. Phys. Lett. 102, 139 (1983).
[Crossref]

D. A. Wiersma, D. P. Weitekamp, and K. Duppen, in Ultrafast Phenomena IV, D. H. Auston and K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 224.
[Crossref]

Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Eichler, H.

H. Eichler and H. Stahl, J. Appl. Phys. 44, 3429 (1973), and references therein.
[Crossref]

Fano, U.

U. Fano, Phys. Rev. 131, 259 (1963).
[Crossref]

Fayer, M. D.

M. D. Fayer, Ann. Rev. Phys. Chem. 33, 63 (1982).
[Crossref]

Flusberg, A. M.

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, Phys. Rev. A 20, 1976 (1979).
[Crossref]

Fujimoto, J. G.

S. De Silvestri, A. M. Weiner, J. G. Fujimoto, and E. P. Ippen, Chem. Phys. Lett. 112, 195 (1984).
[Crossref]

Hahn, E. L.

E. L. Hahn, Phys. Rev. 80, 580 (1950).
[Crossref]

Hartmann, S. R.

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, Phys. Rev. A 20, 1976 (1979).
[Crossref]

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964); Phys. Rev. 41, 391 (1966).
[Crossref]

Hesselink, W. H.

W. H. Hesselink and D. A. Wiersma, J. Chem. Phys. 75, 4192 (1981).
[Crossref]

W. H. Hesselink and D. A. Wiersma, J. Chem. Phys. 73, 648 (1980).
[Crossref]

W. H. Hesselink and D. A. Wiersma, Phys. Rev. Lett. 43, 1991 (1979); J. Chem. Phys. 75, 4192 (1981).
[Crossref]

J. B. W. Morsink, W. H. Hesselink, and D. A. Wiersma, Chem. Phys. Lett. 64, 1 (1979).
[Crossref]

W. H. Hesselink, Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1980).

Ippen, E. P.

A. M. Weiner, S. De Silvestri, and E. P. Ippen, J. Opt. Soc. Am. B 2, 6541985).
[Crossref]

S. De Silvestri, A. M. Weiner, J. G. Fujimoto, and E. P. Ippen, Chem. Phys. Lett. 112, 195 (1984).
[Crossref]

Kachru, R.

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, Phys. Rev. A 20, 1976 (1979).
[Crossref]

Kurnit, N. A.

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964); Phys. Rev. 41, 391 (1966).
[Crossref]

Loudon, R.

See, for instance, R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1983), Chap. 2.

Morsink, J. B. W.

J. B. W. Morsink, W. H. Hesselink, and D. A. Wiersma, Chem. Phys. Lett. 64, 1 (1979).
[Crossref]

Moruzzi, G.

E. Arimondo and G. Moruzzi, J. Phys. B 6, 2382 (1973).
[Crossref]

Mossberg, T. W.

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, Phys. Rev. A 20, 1976 (1979).
[Crossref]

Putzer, E. J.

E. J. Putzer, Am. Math. Monthly 73, 2 (1966).
[Crossref]

Salour, M. M.

M. M. Salour and C. Cohen-Tannoudji, Phys. Rev. Lett. 38, 757 (1977).
[Crossref]

Schoemaker, R. L.

R. L. Schoemaker, in Laser and Coherence Spectroscopy, J. I. Steinfeld, ed. (Plenum, New York, 1978).

Shen, Y. R.

P. Ye and Y. R. Shen, Phys. Rev. A 25, 2183 (1982).
[Crossref]

N. Bloembergen and Y. R. Shen, Phys. Rev. 133, A37 (1964).
[Crossref]

Stahl, H.

H. Eichler and H. Stahl, J. Appl. Phys. 44, 3429 (1973), and references therein.
[Crossref]

Weiner, A. M.

A. M. Weiner, S. De Silvestri, and E. P. Ippen, J. Opt. Soc. Am. B 2, 6541985).
[Crossref]

S. De Silvestri, A. M. Weiner, J. G. Fujimoto, and E. P. Ippen, Chem. Phys. Lett. 112, 195 (1984).
[Crossref]

Weitekamp, D. P.

K. Duppen, D. P. Weitekamp, and D. A. Wiersma, Chem. Phys. Lett. 106, 147 (1984); Chem. Phys. Lett. 108, 551 (1984).
[Crossref]

D. P. Weitekamp, K. Duppen, and D. A. Wiersma, Chem. Phys. Lett. 102, 139 (1983).
[Crossref]

D. A. Wiersma, D. P. Weitekamp, and K. Duppen, in Ultrafast Phenomena IV, D. H. Auston and K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 224.
[Crossref]

Weitzkamp, D. P.

In some published papers ρ11(t12+) was written for a situation in which population relaxation is negligible on the time scale of the pulse separation t12. This limitation was not always pointed out, however [e.g., K. Duppen, D. P. Weitzkamp, and D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, and D. A. Wiersma, Physica 127B, 349 (1984)]. The correct, general, expression for ρ11isρ11(t12+)=¼{2-β exp(-k31t12+β exp(-t12/T1)+cos θ1× [β exp(-k31t12)-β exp(-t12/T1]+cos θ2× [2-β exp(-k31t12)+(β-2)exp(-t12/T1)]+ cos θ1 cos θ2[β exp(-k31t12)-(β-2)× exp(-t12/T1)]-2 sin θ1 sin θ2× exp(-t12/T2)cos(Δt12-k12·r+ϕ12)]}.
[Crossref]

Wiersma, D. A.

In some published papers ρ11(t12+) was written for a situation in which population relaxation is negligible on the time scale of the pulse separation t12. This limitation was not always pointed out, however [e.g., K. Duppen, D. P. Weitzkamp, and D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, and D. A. Wiersma, Physica 127B, 349 (1984)]. The correct, general, expression for ρ11isρ11(t12+)=¼{2-β exp(-k31t12+β exp(-t12/T1)+cos θ1× [β exp(-k31t12)-β exp(-t12/T1]+cos θ2× [2-β exp(-k31t12)+(β-2)exp(-t12/T1)]+ cos θ1 cos θ2[β exp(-k31t12)-(β-2)× exp(-t12/T1)]-2 sin θ1 sin θ2× exp(-t12/T2)cos(Δt12-k12·r+ϕ12)]}.
[Crossref]

K. Duppen, D. P. Weitekamp, and D. A. Wiersma, Chem. Phys. Lett. 106, 147 (1984); Chem. Phys. Lett. 108, 551 (1984).
[Crossref]

D. P. Weitekamp, K. Duppen, and D. A. Wiersma, Chem. Phys. Lett. 102, 139 (1983).
[Crossref]

W. H. Hesselink and D. A. Wiersma, J. Chem. Phys. 75, 4192 (1981).
[Crossref]

W. H. Hesselink and D. A. Wiersma, J. Chem. Phys. 73, 648 (1980).
[Crossref]

W. H. Hesselink and D. A. Wiersma, Phys. Rev. Lett. 43, 1991 (1979); J. Chem. Phys. 75, 4192 (1981).
[Crossref]

J. B. W. Morsink, W. H. Hesselink, and D. A. Wiersma, Chem. Phys. Lett. 64, 1 (1979).
[Crossref]

D. A. Wiersma, D. P. Weitekamp, and K. Duppen, in Ultrafast Phenomena IV, D. H. Auston and K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 224.
[Crossref]

Ye, P.

P. Ye and Y. R. Shen, Phys. Rev. A 25, 2183 (1982).
[Crossref]

Am. Math. Monthly (1)

E. J. Putzer, Am. Math. Monthly 73, 2 (1966).
[Crossref]

Ann. Rev. Phys. Chem. (1)

M. D. Fayer, Ann. Rev. Phys. Chem. 33, 63 (1982).
[Crossref]

Chem. Phys. Lett. (5)

S. De Silvestri, A. M. Weiner, J. G. Fujimoto, and E. P. Ippen, Chem. Phys. Lett. 112, 195 (1984).
[Crossref]

K. Duppen, D. P. Weitekamp, and D. A. Wiersma, Chem. Phys. Lett. 106, 147 (1984); Chem. Phys. Lett. 108, 551 (1984).
[Crossref]

J. B. W. Morsink, W. H. Hesselink, and D. A. Wiersma, Chem. Phys. Lett. 64, 1 (1979).
[Crossref]

D. P. Weitekamp, K. Duppen, and D. A. Wiersma, Chem. Phys. Lett. 102, 139 (1983).
[Crossref]

In some published papers ρ11(t12+) was written for a situation in which population relaxation is negligible on the time scale of the pulse separation t12. This limitation was not always pointed out, however [e.g., K. Duppen, D. P. Weitzkamp, and D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, and D. A. Wiersma, Physica 127B, 349 (1984)]. The correct, general, expression for ρ11isρ11(t12+)=¼{2-β exp(-k31t12+β exp(-t12/T1)+cos θ1× [β exp(-k31t12)-β exp(-t12/T1]+cos θ2× [2-β exp(-k31t12)+(β-2)exp(-t12/T1)]+ cos θ1 cos θ2[β exp(-k31t12)-(β-2)× exp(-t12/T1)]-2 sin θ1 sin θ2× exp(-t12/T2)cos(Δt12-k12·r+ϕ12)]}.
[Crossref]

J. Appl. Phys. (1)

H. Eichler and H. Stahl, J. Appl. Phys. 44, 3429 (1973), and references therein.
[Crossref]

J. Chem. Phys. (2)

W. H. Hesselink and D. A. Wiersma, J. Chem. Phys. 73, 648 (1980).
[Crossref]

W. H. Hesselink and D. A. Wiersma, J. Chem. Phys. 75, 4192 (1981).
[Crossref]

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

J. Phys. B (1)

E. Arimondo and G. Moruzzi, J. Phys. B 6, 2382 (1973).
[Crossref]

Phys. Rev. (3)

U. Fano, Phys. Rev. 131, 259 (1963).
[Crossref]

E. L. Hahn, Phys. Rev. 80, 580 (1950).
[Crossref]

N. Bloembergen and Y. R. Shen, Phys. Rev. 133, A37 (1964).
[Crossref]

Phys. Rev. A (2)

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, Phys. Rev. A 20, 1976 (1979).
[Crossref]

P. Ye and Y. R. Shen, Phys. Rev. A 25, 2183 (1982).
[Crossref]

Phys. Rev. Lett. (3)

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964); Phys. Rev. 41, 391 (1966).
[Crossref]

W. H. Hesselink and D. A. Wiersma, Phys. Rev. Lett. 43, 1991 (1979); J. Chem. Phys. 75, 4192 (1981).
[Crossref]

M. M. Salour and C. Cohen-Tannoudji, Phys. Rev. Lett. 38, 757 (1977).
[Crossref]

Other (9)

See, for instance, R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1983), Chap. 2.

Not the interaction picture. The connection between this rotating frame picture and the conventional interaction picture is ρ˜12 = exp[+i(ω21− ω)t]ρ12INT and ρ˜21 = exp[−i(ω21− ω)t]ρ21INT.

In this definition, the decay of the amplitude of an optical FID is PFID(t) ~ exp(−t2/T2)exp(−t/2T2′2).

D. A. Wiersma, D. P. Weitekamp, and K. Duppen, in Ultrafast Phenomena IV, D. H. Auston and K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 224.
[Crossref]

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), Chap. 2.

R. L. Schoemaker, in Laser and Coherence Spectroscopy, J. I. Steinfeld, ed. (Plenum, New York, 1978).

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

W. H. Hesselink, Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1980).

For a recent discussion of this approximation, see Ph. de Bree, Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1981).

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Figures (5)

Fig. 1
Fig. 1

Level scheme of a three-level system. Wavy arrows indicate decay channels. Level |1〉 and |2〉 are coupled by a radiation field at frequency ω.

Fig. 2
Fig. 2

Schematic diagram showing the various photon echoes and optical FID’s produced by a sequence of three pulses. See Appendix A.

Fig. 3
Fig. 3

Modulation of the population in states |1〉 and |2〉 after application of two resonant π/2 pulses separated by 100 psec. The horizontal axis gives the detuning from the line center. It does not indicate the absolute energy in either the ground or the excited state. The envelope of the modulation represents a line width of 1.5 cm−1. The phase of the modulation was chosen to be zero.

Fig. 4
Fig. 4

Schematic representation of the interference pattern of two crossed monochromatic beams. A third beam can scatter from the resulting transient hologram.

Fig. 5
Fig. 5

(a) Grating scattering experiment. The two scattering directions are denoted by 3PSE and 3PVE. The observed intensity of the two signals is equal when t12 = 0. PMT, photomultiplier tube. (b) Scattering intensity as a function of the delay t12 for an inhomogeneously broadened transition (T2 > T2′) The delay t23 is assumed to be large (t23t12). The decay on one side is determined predominantly by T2 and on the other side by T2′.

Equations (38)

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P ( t ) = N μ = N Tr [ ρ ( t ) μ ] ,
i ρ t = [ H 0 , ρ ] + [ H I , ρ ] + i ( ρ t ) random ,
H 0 = n E n n n
H I = - 1 2 i n , m μ n , m · E 0 , i ( r , t ) × exp { - i [ ω i t - k i · r + ϕ i ] } n m .
( ρ n n t ) r a n d o m = m w m n ρ m m - m w n m ρ n n ,
( ρ n m t ) random = - Γ n m ρ n m .
ρ ˙ 11 = i 12 2 { ρ ˜ 21 exp [ - i ( k · r - ϕ ) ] - ρ ˜ 12 exp [ + i ( k · r - ϕ ) ] } + k 21 ρ 22 + k 31 ρ 33 , ρ ˜ ˙ 12 = i 12 2 ( ρ 22 - ρ 11 ) exp [ - i ( k · r - ϕ ) ] - ( 1 T 2 - i Δ ) ρ ˜ 12 , ρ ˜ ˙ 21 = i 12 2 ( ρ 11 - ρ 22 ) exp [ + i ( k · r - ϕ ) ] - ( 1 T 2 + i Δ ) ρ ˜ 21 , ρ ˙ 22 = i 12 2 { ρ ˜ 12 exp [ + i ( k · r - ϕ ) ] - ρ ˜ 21 exp [ - i ( k · r - ϕ ) ] } - ( k 21 + k 23 ) ρ 22 , ρ ˙ 33 = k 23 ρ 22 - k 31 ρ 33 .
ρ ˜ 12 = ρ 12 e - i ω t ,             ρ ˜ 21 = ρ 21 e + i ω t .
ρ ˙ = i L ρ ,
ρ ( t ) = e i L t ρ ( 0 ) .
e i L t = j = 0 n - 1 r j + 1 ( t ) P j ,
r ˙ 1 = λ 1 r 1 ,         r ˙ j = r j - 1 + λ j r j ,
θ = μ 12 E 0 ( r , t ) d t .
A ( 12 t ) = A ( θ ) = 1 2 [ 1 + cos θ - i α * sin θ i α sin θ 1 - cos θ 0 - i α sin θ 1 + cos θ ( 1 - cos θ ) α 2 i α sin θ 0 i α * sin θ ( 1 - cos θ ) ( α * ) 2 1 + cos θ - i α * sin θ 0 1 - cos θ i α sin θ - i α sin θ 1 + cos θ 0 0 0 0 0 2 ]
B ( t ) = [ 1 0 0 1 - β exp ( - k 31 t ) + ( β - 1 ) exp ( - t / T 1 ) 1 - exp ( - k 31 t ) 0 exp [ ( i Δ - 1 T 2 ) t ] 0 0 0 0 0 exp [ ( i Δ - 1 T 2 ) t ] 0 0 0 0 0 exp ( - t / T 1 ) 0 0 0 0 β exp ( - k 31 t ) - β exp ( - t / T 1 ) exp ( - k 31 t ) ] ,
ρ ( t ) = B ( t ) A ( θ 3 ) B ( t 23 ) A ( θ 2 ) B ( t 12 ) A ( θ 1 ) ρ ( 0 ) .
P ( Δ , t ) = 2 N μ 12 Re [ ρ 12 ( Δ , t ) ] .
P ( t ) = - + g ( Δ ) P ( Δ , t ) d Δ .
g ( Δ ) = T 2 2 π exp ( - Δ 2 T 2 2 / 2 ) ,
ρ 12 ( Δ , t ) = - i 8 sin θ 1 sin θ 2 sin θ 3 [ ( β - 2 ) exp ( - t 23 / T 1 ) - β exp ( - k 31 t 23 ) ] exp [ - ( t 12 + t ) / T 2 ] × ( exp [ i Δ ( t - t 12 ) ] exp { + i [ ω t - ( k 3 + k 2 - k 1 ) · r + ( ϕ 3 + ϕ 2 - ϕ 1 ) ] }
+ exp [ i Δ ( t + t 12 ) ] exp { + i [ ω t - ( k 3 + k 1 - k 2 ) · r + ( ϕ 3 + ϕ 1 - ϕ 2 ) ] } ) .
P ( t ) ~ exp [ - ( t 12 + t ) / T 2 ] exp [ - ( t - t 12 ) 2 / 2 T 2 2 ] .
Δ t = 2 T 2 2 ln 2 .
P ( t ) ~ exp [ - ( t 12 + t ) / T 2 ] exp [ - ( t + t 12 ) 2 / 2 T 2 2 ] .
P ( t ) ~ exp ( - t 23 / T 1 ) .
P ( t ) ~ 1 + exp ( - t 23 / T 1 ) .
( ρ 22 - ρ 11 ) = ½ { cos θ 2 [ β exp ( - k 31 t 12 ) - ( β - 2 ) exp ( - t 12 / T 1 ) - 2 ] + cos θ 1 cos θ 2 × [ ( β - 2 ) exp ( - t 12 / T 1 ) - β exp ( - k 31 t 12 ) ] + 2 sin θ 1 sin θ 2 exp ( - t 12 / T 2 ) × cos ( Δ t 12 - k 12 · r + ϕ 12 ) } .
Λ grating = λ light 2 sin θ .
P ( t ) ~ exp ( - t / T 2 ) exp ( - t 2 / 2 T 2 2 ) .
ρ 12 ( t 12 + t 23 + t ) = - i 8 sin θ 1 ( cos θ 2 + 1 ) ( cos θ 3 + 1 ) × exp [ - ( t 12 + t 23 + t ) / T 2 ] × exp [ + i Δ ( t 12 + t 23 + t ) ] × exp [ + i ( ω t - k 1 · r + ϕ 1 ) ] ,
- i 8 sin θ 2 ( cos θ 3 + 1 ) { 2 + ( cos θ 1 - 1 ) × [ β exp ( - k 31 t 12 ) - ( β - 2 ) × exp ( - t 12 / T 1 ) ] } × exp [ - ( t 23 + t ) / T 2 ] exp [ + i Δ ( t 23 + t ) ] × exp [ + i ( ω t - k 2 · r + ϕ 2 ) ] ,
- i 8 sin θ 3 ( 4 + ( cos θ 1 - 1 ) × { 2 β exp [ - k 31 ( t 12 + t 23 ) ] - cos θ 2 × ( 2 β - 4 ) exp [ - ( t 12 + t 23 ) / T 1 ] } + ( cos θ 2 - 1 ) [ 2 β exp ( - k 31 t 23 ) - ( 2 β - 4 ) exp ( - t 23 / T 1 ) + ( cos θ 1 - 1 ) ( cos θ 2 - 1 ) × { β 2 exp [ - k 31 ( t 12 + t 23 ) ] + ( β 2 - 2 β ) exp [ - ( t 12 + t 23 ) / T 1 ] - ( β 2 - 2 β ) exp ( - t 12 / T 1 ) exp ( - k 31 t 23 ) - ( β 2 - 2 β ) exp ( - k 31 t 12 ) exp ( - t 23 / T 1 ) } ) × exp ( - t / T 2 ) exp ( + i Δ t ) × exp [ + i ( ω t - k 3 · r + ϕ 3 ) ] ,
- i 8 sin θ 1 ( cos θ 2 - 1 ) ( cos θ 3 + 1 ) × exp [ - ( t 12 + t 23 + t ) / T 2 ] × exp [ + i Δ ( t 23 + t - t 12 ) ] × exp { + i [ ω t - ( 2 k 2 - k 1 ) · r + ( 2 ϕ 2 - ϕ 1 ) ] } ,
- i 8 sin θ 1 ( cos θ 2 + 1 ) ( cos θ 3 - 1 ) × exp [ - ( t 12 + t 23 + t ) / T 2 ] × exp [ + i Δ ( t - t 12 - t 23 ) ] × exp { + i [ ω t - ( 2 k 3 - k 1 ) · r + ( 2 ϕ 3 - ϕ 1 ) ] } ,
- i 8 sin θ 2 ( cos θ 3 - 1 ) { 2 - ( 1 - cos θ 1 ) × [ β exp ( - k 31 t 12 ) - ( β - 2 ) exp ( - t 12 / T 1 ) ] } × exp [ - ( t 23 + t ) / T 2 ] exp [ + i Δ ( t - t 23 ) ] × exp { + i [ ω t - ( 2 k 3 - k 2 ) · r + ( 2 ϕ 3 - ϕ 2 ) ] } ,
- i 8 sin θ 1 ( cos θ 2 - 1 ) ( cos θ 3 - 1 ) × exp [ - ( t 12 + t 23 + t ) / T 2 ] × exp [ + i Δ ( t - t 23 + t 12 ) ] × exp { + i [ ω t - ( 2 k 3 - 2 k 2 + k 1 ) · r + ( 2 ϕ 3 - 2 ϕ 2 + ϕ 1 ) ] } ,
- i 8 sin θ 1 sin θ 2 sin θ 3 [ ( β - 2 ) × exp ( - t 23 / T 1 ) - β exp ( - k 31 t 23 ) ] × exp [ - ( t 12 + t ) / T 2 ] × ( exp [ i Δ ( t - t 12 ) ] exp { + i [ ω t - ( k 3 - k 1 + k 2 ) · r + ( ϕ 3 - ϕ 1 + ϕ 2 ) ] } + exp [ i Δ ( t + t 12 ) ] exp { + i [ ω t - ( k 3 - k 2 + k 1 ) · r + ( ϕ 3 - ϕ 2 + ϕ 1 ) ] } ) ; β = k 23 k 21 + k 23 - k 31 , ( T 1 ) - 1 = k 21 + k 23 , ( T 2 ) - 1 = ( 2 T 1 ) - 1 + ( T 2 * ) - 1 .
ρ11(t12+)=¼{2-βexp(-k31t12+βexp(-t12/T1)+cosθ1×[βexp(-k31t12)-βexp(-t12/T1]+cosθ2×[2-βexp(-k31t12)+(β-2)exp(-t12/T1)]+cosθ1cosθ2[βexp(-k31t12)-(β-2)×exp(-t12/T1)]-2sinθ1sinθ2×exp(-t12/T2)cos(Δt12-k12·r+ϕ12)]}.

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