Abstract

We propose and investigate theoretically a novel scheme for transient slowing and cooling of two-level quantum systems with narrow transition linewidths by a sequence of counterpropagating, short, linearly polarized laser pulses with special frequency chirping. Both internal degrees of freedom and the motion of the center of mass of quantum systems are considered quantum mechanically. Interaction with a large number of laser pulses during the decay time permits a drastic decrease in the cooling time of such systems.

© 1996 Optical Society of America

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References

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  1. V. G. Minogin and V. S. Letokhov, Laser Light Pressure on Atoms (Gordon & Breach, New York, 1987).
  2. S. Chu and C. Wieman, eds., feature on laser cooling and trapping of Atoms, J. Opt. Soc. Am. B 6, 2020–2278 (1989).
  3. A. Aspect, Phys. Rep. 219, 141 (1992).
    [CrossRef]
  4. C. Cohen-Tannoudji, Phys. Rep. 219, 153 (1992).
    [CrossRef]
  5. G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Solomon, Phys. Rev. Lett. 70, 2249 (1993).
    [CrossRef] [PubMed]
  6. A. Hemmerich, C. Zimmermann, and T. W. Hänsch, Europhys. Lett. 22, 89 (1993).
    [CrossRef]
  7. T. W. Hänsch and A. Schawlow, Opt. Commun. 13, 68 (1975).
    [CrossRef]
  8. D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 699 (1975).
  9. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
    [CrossRef] [PubMed]
  10. M. Kasevich and S. Chu, Phys. Rev. Lett. 69, 1741 (1992).
    [CrossRef] [PubMed]
  11. A. P. Kazancev, Sov. Phys. Usp. 21, 58 (1978).
    [CrossRef]
  12. R. J. Cook, Phys. Rev. Lett. 41, 1788 (1978).
    [CrossRef]
  13. K. Molmer, Phys. Rev. Lett. 66, 2301 (1991).
    [CrossRef] [PubMed]
  14. S. Stenholm, Appl. Phys. 15, 287 (1978).
    [CrossRef]
  15. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
    [CrossRef]
  16. Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989).
    [CrossRef]
  17. Y. Castin and J. Dalibard, Europhys. Lett. 14, 761 (1991).
    [CrossRef]
  18. J. Guo and P. R. Berman, Phys. Rev. A 48, 3225 (1993).
    [CrossRef] [PubMed]
  19. C. Liedenbaum, S. Stolte, and J. Reuss, Phys. Rep. 178, (1989).
    [CrossRef]
  20. L. D. Landau and E. M. Lifshits, Quantum Mechanics. Nonrelativistic Theory (Pergamon, Oxford, 1958).
  21. J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, Phys. Rev. A 53, 2885 (1996).
    [CrossRef] [PubMed]
  22. J. S. Melinger, S. R. Ghandi, A. Hariharan, J. X. Tull, and W. S. Waren, Phys. Rev. Lett. 68, 2000 (1992).
    [CrossRef] [PubMed]
  23. R. J. Cook and A. F. Bernhardt, Phys. Rev. A 18, 2533 (1978).
    [CrossRef]
  24. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  25. M. D. Crisp, Phys. Rev. A 8, 2128 (1973).
    [CrossRef]
  26. G. P. Djotyan and J. S. Bakos, Opt. Commun. 97, 95 (1993).
    [CrossRef]

1996 (1)

J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, Phys. Rev. A 53, 2885 (1996).
[CrossRef] [PubMed]

1993 (4)

G. P. Djotyan and J. S. Bakos, Opt. Commun. 97, 95 (1993).
[CrossRef]

G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Solomon, Phys. Rev. Lett. 70, 2249 (1993).
[CrossRef] [PubMed]

A. Hemmerich, C. Zimmermann, and T. W. Hänsch, Europhys. Lett. 22, 89 (1993).
[CrossRef]

J. Guo and P. R. Berman, Phys. Rev. A 48, 3225 (1993).
[CrossRef] [PubMed]

1992 (4)

M. Kasevich and S. Chu, Phys. Rev. Lett. 69, 1741 (1992).
[CrossRef] [PubMed]

A. Aspect, Phys. Rep. 219, 141 (1992).
[CrossRef]

C. Cohen-Tannoudji, Phys. Rep. 219, 153 (1992).
[CrossRef]

J. S. Melinger, S. R. Ghandi, A. Hariharan, J. X. Tull, and W. S. Waren, Phys. Rev. Lett. 68, 2000 (1992).
[CrossRef] [PubMed]

1991 (2)

Y. Castin and J. Dalibard, Europhys. Lett. 14, 761 (1991).
[CrossRef]

K. Molmer, Phys. Rev. Lett. 66, 2301 (1991).
[CrossRef] [PubMed]

1989 (4)

1988 (1)

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

1978 (4)

S. Stenholm, Appl. Phys. 15, 287 (1978).
[CrossRef]

A. P. Kazancev, Sov. Phys. Usp. 21, 58 (1978).
[CrossRef]

R. J. Cook, Phys. Rev. Lett. 41, 1788 (1978).
[CrossRef]

R. J. Cook and A. F. Bernhardt, Phys. Rev. A 18, 2533 (1978).
[CrossRef]

1975 (2)

T. W. Hänsch and A. Schawlow, Opt. Commun. 13, 68 (1975).
[CrossRef]

D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 699 (1975).

1973 (1)

M. D. Crisp, Phys. Rev. A 8, 2128 (1973).
[CrossRef]

Allen, L.

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

Arimondo, E.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

Aspect, A.

A. Aspect, Phys. Rep. 219, 141 (1992).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

Bakos, J. S.

J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, Phys. Rev. A 53, 2885 (1996).
[CrossRef] [PubMed]

G. P. Djotyan and J. S. Bakos, Opt. Commun. 97, 95 (1993).
[CrossRef]

Berman, P. R.

J. Guo and P. R. Berman, Phys. Rev. A 48, 3225 (1993).
[CrossRef] [PubMed]

Bernhardt, A. F.

R. J. Cook and A. F. Bernhardt, Phys. Rev. A 18, 2533 (1978).
[CrossRef]

Castin, Y.

Chu, S.

M. Kasevich and S. Chu, Phys. Rev. Lett. 69, 1741 (1992).
[CrossRef] [PubMed]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, Phys. Rep. 219, 153 (1992).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

Cook, R. J.

R. J. Cook, Phys. Rev. Lett. 41, 1788 (1978).
[CrossRef]

R. J. Cook and A. F. Bernhardt, Phys. Rev. A 18, 2533 (1978).
[CrossRef]

Courtois, J.-Y.

G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Solomon, Phys. Rev. Lett. 70, 2249 (1993).
[CrossRef] [PubMed]

Crisp, M. D.

M. D. Crisp, Phys. Rev. A 8, 2128 (1973).
[CrossRef]

Dalibard, J.

Dehmelt, H.

D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 699 (1975).

Demeter, G.

J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, Phys. Rev. A 53, 2885 (1996).
[CrossRef] [PubMed]

Djotyan, G. P.

J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, Phys. Rev. A 53, 2885 (1996).
[CrossRef] [PubMed]

G. P. Djotyan and J. S. Bakos, Opt. Commun. 97, 95 (1993).
[CrossRef]

Eberly, J. H.

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

Ghandi, S. R.

J. S. Melinger, S. R. Ghandi, A. Hariharan, J. X. Tull, and W. S. Waren, Phys. Rev. Lett. 68, 2000 (1992).
[CrossRef] [PubMed]

Grynberg, G.

G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Solomon, Phys. Rev. Lett. 70, 2249 (1993).
[CrossRef] [PubMed]

Guo, J.

J. Guo and P. R. Berman, Phys. Rev. A 48, 3225 (1993).
[CrossRef] [PubMed]

Hänsch, T. W.

A. Hemmerich, C. Zimmermann, and T. W. Hänsch, Europhys. Lett. 22, 89 (1993).
[CrossRef]

T. W. Hänsch and A. Schawlow, Opt. Commun. 13, 68 (1975).
[CrossRef]

Hariharan, A.

J. S. Melinger, S. R. Ghandi, A. Hariharan, J. X. Tull, and W. S. Waren, Phys. Rev. Lett. 68, 2000 (1992).
[CrossRef] [PubMed]

Hemmerich, A.

A. Hemmerich, C. Zimmermann, and T. W. Hänsch, Europhys. Lett. 22, 89 (1993).
[CrossRef]

Kaiser, R.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

Kasevich, M.

M. Kasevich and S. Chu, Phys. Rev. Lett. 69, 1741 (1992).
[CrossRef] [PubMed]

Kazancev, A. P.

A. P. Kazancev, Sov. Phys. Usp. 21, 58 (1978).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshits, Quantum Mechanics. Nonrelativistic Theory (Pergamon, Oxford, 1958).

Letokhov, V. S.

V. G. Minogin and V. S. Letokhov, Laser Light Pressure on Atoms (Gordon & Breach, New York, 1987).

Liedenbaum, C.

C. Liedenbaum, S. Stolte, and J. Reuss, Phys. Rep. 178, (1989).
[CrossRef]

Lifshits, E. M.

L. D. Landau and E. M. Lifshits, Quantum Mechanics. Nonrelativistic Theory (Pergamon, Oxford, 1958).

Lounis, B.

G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Solomon, Phys. Rev. Lett. 70, 2249 (1993).
[CrossRef] [PubMed]

Melinger, J. S.

J. S. Melinger, S. R. Ghandi, A. Hariharan, J. X. Tull, and W. S. Waren, Phys. Rev. Lett. 68, 2000 (1992).
[CrossRef] [PubMed]

Minogin, V. G.

V. G. Minogin and V. S. Letokhov, Laser Light Pressure on Atoms (Gordon & Breach, New York, 1987).

Molmer, K.

K. Molmer, Phys. Rev. Lett. 66, 2301 (1991).
[CrossRef] [PubMed]

Reuss, J.

C. Liedenbaum, S. Stolte, and J. Reuss, Phys. Rep. 178, (1989).
[CrossRef]

Schawlow, A.

T. W. Hänsch and A. Schawlow, Opt. Commun. 13, 68 (1975).
[CrossRef]

Solomon, C.

G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Solomon, Phys. Rev. Lett. 70, 2249 (1993).
[CrossRef] [PubMed]

Sörlei, Zs.

J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, Phys. Rev. A 53, 2885 (1996).
[CrossRef] [PubMed]

Stenholm, S.

S. Stenholm, Appl. Phys. 15, 287 (1978).
[CrossRef]

Stolte, S.

C. Liedenbaum, S. Stolte, and J. Reuss, Phys. Rep. 178, (1989).
[CrossRef]

Tull, J. X.

J. S. Melinger, S. R. Ghandi, A. Hariharan, J. X. Tull, and W. S. Waren, Phys. Rev. Lett. 68, 2000 (1992).
[CrossRef] [PubMed]

Vansteenkiste, N.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

Verkerk, P.

G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Solomon, Phys. Rev. Lett. 70, 2249 (1993).
[CrossRef] [PubMed]

Wallis, H.

Waren, W. S.

J. S. Melinger, S. R. Ghandi, A. Hariharan, J. X. Tull, and W. S. Waren, Phys. Rev. Lett. 68, 2000 (1992).
[CrossRef] [PubMed]

Wineland, D.

D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 699 (1975).

Zimmermann, C.

A. Hemmerich, C. Zimmermann, and T. W. Hänsch, Europhys. Lett. 22, 89 (1993).
[CrossRef]

Appl. Phys. (1)

S. Stenholm, Appl. Phys. 15, 287 (1978).
[CrossRef]

Bull. Am. Phys. Soc. (1)

D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 699 (1975).

Europhys. Lett. (2)

Y. Castin and J. Dalibard, Europhys. Lett. 14, 761 (1991).
[CrossRef]

A. Hemmerich, C. Zimmermann, and T. W. Hänsch, Europhys. Lett. 22, 89 (1993).
[CrossRef]

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

Opt. Commun. (2)

T. W. Hänsch and A. Schawlow, Opt. Commun. 13, 68 (1975).
[CrossRef]

G. P. Djotyan and J. S. Bakos, Opt. Commun. 97, 95 (1993).
[CrossRef]

Phys. Rep. (3)

C. Liedenbaum, S. Stolte, and J. Reuss, Phys. Rep. 178, (1989).
[CrossRef]

A. Aspect, Phys. Rep. 219, 141 (1992).
[CrossRef]

C. Cohen-Tannoudji, Phys. Rep. 219, 153 (1992).
[CrossRef]

Phys. Rev. A (4)

J. Guo and P. R. Berman, Phys. Rev. A 48, 3225 (1993).
[CrossRef] [PubMed]

J. S. Bakos, G. P. Djotyan, G. Demeter, and Zs. Sörlei, Phys. Rev. A 53, 2885 (1996).
[CrossRef] [PubMed]

R. J. Cook and A. F. Bernhardt, Phys. Rev. A 18, 2533 (1978).
[CrossRef]

M. D. Crisp, Phys. Rev. A 8, 2128 (1973).
[CrossRef]

Phys. Rev. Lett. (6)

J. S. Melinger, S. R. Ghandi, A. Hariharan, J. X. Tull, and W. S. Waren, Phys. Rev. Lett. 68, 2000 (1992).
[CrossRef] [PubMed]

R. J. Cook, Phys. Rev. Lett. 41, 1788 (1978).
[CrossRef]

K. Molmer, Phys. Rev. Lett. 66, 2301 (1991).
[CrossRef] [PubMed]

G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Solomon, Phys. Rev. Lett. 70, 2249 (1993).
[CrossRef] [PubMed]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

M. Kasevich and S. Chu, Phys. Rev. Lett. 69, 1741 (1992).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

A. P. Kazancev, Sov. Phys. Usp. 21, 58 (1978).
[CrossRef]

Other (3)

L. D. Landau and E. M. Lifshits, Quantum Mechanics. Nonrelativistic Theory (Pergamon, Oxford, 1958).

V. G. Minogin and V. S. Letokhov, Laser Light Pressure on Atoms (Gordon & Breach, New York, 1987).

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

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Figures (5)

Fig. 1
Fig. 1

(a) Time dependence of the normalized Rabi frequency ΩR of a Gaussian laser pulse. (b) Time dependence of the normalized detuning (t)=ωL(t)-ω0 for a QS having zero velocity in the laboratory reference frame in the case of an asymmetrically chirped Gaussian laser pulse. (c) Time dependence of the normalized detuning (t) in the case of a symmetrically chirped Gaussian laser pulse.

Fig. 2
Fig. 2

Velocity distribution of the ensemble of two-level QS’s after the action of 10 symmetrically chirped counterpropagating laser pulses with Gaussian envelopes [see Eq. (14)] and with linear frequency chirp: ωL(t) = ω0 + 2c1kLvRt/τR. The dashed curve is the initial Maxwellian velocity distribution of the ensemble, described by Eq. (13). The parameters applied are c1 = 8π; ΩR(m) τR = 12π, τL = τR, q = 10, and vc/vR = 10.

Fig. 3
Fig. 3

Velocity distribution function of the ensemble of two-level QS’s in the ground (n1) and in the excited (n2) states: (a) After the action of the first asymmetrically chirped laser pulse [see Fig. 1(b)]. The dashed curve represents the velocity distribution of the QS obtained after the slowing process (see Fig. 2). (b) After the action of 14 subsequent counterpropagating symmetrically chirped laser pulses. (c) The intermediate quasi-equilibrium velocity distribution function of the ensemble after the first relaxation to the ground state. The dashed curve is the same as the solid curve in Fig. 2.

Fig. 4
Fig. 4

Velocity distribution function of the ensemble of two-level QS’s in the ground (n1) and in the excited (n2) states: (a) After the action of an asymmetrically chirped laser pulse applied after the first relaxation process, the 16th from the beginning of the cooling process. (b) After the action of 8 symmetrically chirped laser pulses applied after the first relaxation process, the 24th ones from the beginning of the laser cooling, with the 1st and the 16th asymmetrically chirped.

Fig. 5
Fig. 5

Final quasi-equilibrium velocity distribution of the ensemble after the second relaxation process (the first cooling cycle).

Equations (52)

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

itρ=(Ĥ-Ĥ*)ρ+iRˆρ,
Ĥ0(r,η)=Ĥ0(η)-22mΔ,
Ĥ0(η)ψn(η)=Enψn(η).
E(r, t)=1/2e{A+(t)exp[i(ωt-kLz)]+A-(t)×exp[i(ωt+kLz)]+c.c.},
ρ(rη, rη)= dκdκm,nanm(κ, κ, t)ψn*(η)ψm(η)×exp[-i/(En-Em)t]×exp[i/(κz-κz)].
t+i2m(κ2-κ2)alq(κ,κ)=i2n,mdln[anm(κ+kL,κ)A+ exp(iωt)+anm(κ-kL,κ)A+* exp(-iωt)+anm(κ-kL,κ)A- exp(iωt)+anm(κ+kL,κ)A-* exp(-iωt)]-n,mdqm*[anm(κ,κ+kL)A+* exp(-iωt)+anm(κ,κ-kL)A+ exp(iωt)+anm(κ,κ-kL)A-* exp(-iωt)+anm(κ,κ+kL)A- exp(iωt)]×exp{it/[(El-Eq)-(En-Em)]},
t+ξ12(κ, κ)ρ12(κ, κ)=i2d12{A+[n22(κ+kL, κ)-n11(κ, κ-kL)]+A-[n22(κ-kL, κ)-n11(κ, κ+kL)]},
t+ξ(κ, κ)n11(κ, κ)
=i2{d12[ρ21(κ+kL, κ)A++ρ21(κ-kL, κ)A-]-d12*[ρ12(κ, κ+kL)A+*+ρ12(κ, κ-kL)A-*]},
t+ξ(κ, κ)n22(κ, κ)=i2{d21[ρ12(κ-kL, κ)A+*+ρ12(κ+kL, κ)A-*]-d21*[ρ21(κ, κ-kL)A++ρ21(κ, κ+kL)A-]},
(t)=ω(t)-ω0,
ω0=(E2-E1)/,
ξ12(κ, κ)=i+i2m(κ2-κ2),
ξ(κ, κ)=i2m(κ2-κ2).
tn1(κ)=i2{d12[ρ21(κ+kL, κ)A++ρ21(κ-kL, κ)A--A]-d12*[ρ12(κ, κ+kL)A+*+ρ12(κ, κ-kL)A-*]},
tn2(κ)=i2{d21[ρ12(κ-kL, κ)A+*+ρ12(κ+kL, κ)A--A-A*]-d21*[ρ21(κ, κ-kL)A++ρ21(κ, κ+kL)A-,
t+ξ12(κ, κ+kL)ρ12(κ, κ+kL)=i2d12{A+[n2(κ+kL)-n1(κ)]+A-[n22(κ-kL, κ+kL)-n11(κ, κ+2kL)]}.
ρ12(κ, κ)=ρ21*(κ, κ).
t+ξ12(κ, κ+kL)ρ12(κ, κ+kL)=iΩ¯R[n22(κ-kL, κ+kL)-n11(κ, κ+2kL)],
tn1(κ)=n2(κ)T1,
tn2(κ)=-n2(κ)T1;
n2(κ)=-kLkLF(κ)n2(κ+κ)dk,
F(κ)=34kL1-κ2kL2.
n1(κ, t+)=n1(κ, 0)+-kLkLF(κ)n2(κ+κ, 0)dκ,
1,2(κ,t)=ωL-ω0+2γtkκ/m,
n0(v/vR)=1π1/2vRqexp[-(v/vR-vc/vR)2/q2],
ΩR(t)=ΩR(m) exp-12t/τRτL/τR2,
R±/t=Ω±×R±,
R±(t,κ)=eˆ1X±+eˆ2Y±+eˆ3Z±,
Ω±(t)=-eˆ1Ω±ΩR(t)+eˆ3θ±(t),
Ω±2(t)=Ω±R2(t)+θ±2(t);
X±(κ, t)=ρ12(κ, κ±k)+ρ12*(κ, κ±k),
Y±(κ, t)=i[ρ12*(κ, κ±k)-ρ12(κ, κ±k)],
Z±(κ, t)=n2(κ±k)-n1(κ),
θ±=ξ12(κ, κk)i[(t)±kv],  v=κ/m.
tX±=θ± Y±,
tY±=-θ±X+Ω±RZ±,
tZ±=-Ω±RY±.
|ν|=|θ˙Ω-θΩ˙|Ω2ΩR1,
X=±X cos[α(t)]±Z sin[α(t)],
Z=±X sin[α(t)]±Z cos[α(t)],
Y=Y,
β=t0tΩ(t)dt:
X˙+ν(β)Z=Y,
Y˙=±X,
Z˙=ν(β)X,
X=x0+x1+x2+,
Y=y0+y1+y2+,
Z=z0+z1+z2+,
X0(t)=±θ(t)Ω(t)[X(-)cos(β)Y(-)sin(β)]+ΩR(t)Ω(t)Z(-),
Y0(t)=±X(-)sin(β)+Y(-)cos(β),
Z0(t)=±θ(t)Ω(t)Z(-)±ΩR(t)Ω(t)[X(-)cos(β)Y(-)sin(β)].

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