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, S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964); Phys. Rev. 41, 391 (1966).
    [CrossRef]
  2. P. Ye, Y. R. Shen, Phys. Rev. A 25, 2183 (1982).
    [CrossRef]
  3. K. Duppen, D. P. Weitekamp, D. A. Wiersma, Chem. Phys. Lett. 106, 147 (1984); Chem. Phys. Lett. 108, 551 (1984).
    [CrossRef]
  4. W. H. Hesselink, D. A. Wiersma, Phys. Rev. Lett. 43, 1991 (1979); J. Chem. Phys. 75, 4192 (1981).
    [CrossRef]
  5. J. B. W. Morsink, W. H. Hesselink, D. A. Wiersma, Chem. Phys. Lett. 64, 1 (1979).
    [CrossRef]
  6. D. A. Wiersma, D. P. Weitekamp, K. Duppen, in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), p. 224.
    [CrossRef]
  7. H. Eichler, H. Stahl, J. Appl. Phys. 44, 3429 (1973), and references therein.
    [CrossRef]
  8. T. W. Mossberg, R. Kachru, S. R. Hartmann, 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, E. P. Ippen, J. Opt. Soc. Am. B 2, 6541985).
    [CrossRef]
  11. S. De Silvestri, A. M. Weiner, J. G. Fujimoto, 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, 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, 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, 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, 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, D. A. Wiersma, J. Chem. Phys. 73, 648 (1980).
    [CrossRef]
  26. W. H. Hesselink, 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, 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, D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, 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, E. P. Ippen, Chem. Phys. Lett. 112, 195 (1984).
[CrossRef]

K. Duppen, D. P. Weitekamp, 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, D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, 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, D. A. Wiersma, Chem. Phys. Lett. 102, 139 (1983).
[CrossRef]

1982 (2)

P. Ye, 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, D. A. Wiersma, J. Chem. Phys. 75, 4192 (1981).
[CrossRef]

1980 (1)

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

1979 (3)

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

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

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

1977 (1)

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

1973 (2)

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

H. Eichler, 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, Y. R. Shen, Phys. Rev. 133, A37 (1964).
[CrossRef]

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

Allen, L.

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

Arimondo, E.

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

Bloembergen, N.

N. Bloembergen, 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, C. Cohen-Tannoudji, Phys. Rev. Lett. 38, 757 (1977).
[CrossRef]

De Silvestri, S.

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

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

Duppen, K.

K. Duppen, D. P. Weitekamp, 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, D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, 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, D. A. Wiersma, Chem. Phys. Lett. 102, 139 (1983).
[CrossRef]

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

Eberly, J. H.

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

Eichler, H.

H. Eichler, 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, A. M. Flusberg, Phys. Rev. A 20, 1976 (1979).
[CrossRef]

Fujimoto, J. G.

S. De Silvestri, A. M. Weiner, J. G. Fujimoto, 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, A. M. Flusberg, Phys. Rev. A 20, 1976 (1979).
[CrossRef]

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

Hesselink, W. H.

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

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

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

J. B. W. Morsink, W. H. Hesselink, 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, E. P. Ippen, J. Opt. Soc. Am. B 2, 6541985).
[CrossRef]

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

Kachru, R.

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

Kurnit, N. A.

I. D. Abella, N. A. Kurnit, 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, D. A. Wiersma, Chem. Phys. Lett. 64, 1 (1979).
[CrossRef]

Moruzzi, G.

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

Mossberg, T. W.

T. W. Mossberg, R. Kachru, S. R. Hartmann, 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, 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, Y. R. Shen, Phys. Rev. A 25, 2183 (1982).
[CrossRef]

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

Stahl, H.

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

Weiner, A. M.

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

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

Weitekamp, D. P.

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

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

D. A. Wiersma, D. P. Weitekamp, K. Duppen, in Ultrafast Phenomena IV, D. H. Auston, 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, D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, 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, D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, 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, D. A. Wiersma, Chem. Phys. Lett. 106, 147 (1984); Chem. Phys. Lett. 108, 551 (1984).
[CrossRef]

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

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

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

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

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

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

Ye, P.

P. Ye, 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, E. P. Ippen, Chem. Phys. Lett. 112, 195 (1984).
[CrossRef]

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

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

D. P. Weitekamp, K. Duppen, 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, D. A. Wiersma, Chem. Phys. Lett. 108, 551 (1984); K. Duppen, L. W. Molenkamp, 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, H. Stahl, J. Appl. Phys. 44, 3429 (1973), and references therein.
[CrossRef]

J. Chem. Phys. (2)

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

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

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

J. Phys. B (1)

E. Arimondo, 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, Y. R. Shen, Phys. Rev. 133, A37 (1964).
[CrossRef]

Phys. Rev. A (2)

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

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

Phys. Rev. Lett. (3)

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

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

M. M. Salour, 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, K. Duppen, in Ultrafast Phenomena IV, D. H. Auston, 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, 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)

Equations on this page are rendered with MathJax. Learn more.

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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