Abstract

Kinetic phenomena of resonant-particle motion in a standing-light-wave field are theoretically discussed. The light-pressure force and its fluctuations are found in a wide range of the parameters: the atomic velocity, the intensity, and the detuning of the field. There are two characteristic regions of the detunings in a strong field: the adiabatic region and the region of Landau–Zener resonances. The quantum fluctuations of the inner atomic state that are due to the Landau–Zener transitions result, specifically, in the interference effect in the mean light-pressure force. The effects of the spatial grating of the cooled atoms, the particle velocity bunching, and the recoil-effect dependence on the nonlinear absorption are considered for slow particles. The effect of optical pumping by linearly polarized light taking into account the recoil effect is fundamentally new. Because of this effect, the anisotropic resonance medium becomes gyrotropic.

© 1985 Optical Society of America

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  1. A. Ashkin, Phys. Rev. Lett. 24, 156 (1970); Phys. Rev. Lett. 25, 1321 (1970).
    [Crossref]
  2. A. P. Kazantsev, Zh. Eksp. Teor. Fiz. 66, 1599 (1974) [Sov. Phys. JETP 39, 784 (1974)].
  3. T. W. Hänsch and A. L. Schawlow, Opt. Commun. 13, 68 (1975).
    [Crossref]
  4. A. P. Kazantsev, Zh. Eksp. Teor. Fiz. 67, 1606 (1974) [Sov. Phys. JETP 40, 825 (1974)].
  5. A. P. Kazantsev, Usp. Fiz. Nauk 124, 113 (1978) [Sov. Phys. Usp. 21, 58 (1978)].
  6. R. J. Cook, Phys. Rev. Lett. 41, 1788 (1978); Phys. Rev. A 20, 224 (1979).
    [Crossref]
  7. A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 85, 852 (1983); A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, and V. P. Yakovlev, At. Mol. Phys. 18, 2619 (1985).
    [Crossref]
  8. R. J. Cook, Phys. Rev. A 21, 268 (1980); Phys. Rev. A 22, 1078 (1980).
    [Crossref]
  9. J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).
    [Crossref]
  10. V. G. Minogin and O. T. Serimaa, Opt. Commun. 30, 373 (1979).
    [Crossref]
  11. A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 80, 541 (1981); J. Physique (Paris) 42, 1231 (1981); Appl. Phys. 25, 165 (1981).
  12. For the Schrödinger equation with the Hamiltonian [Eq. (2)] this symmetry has been discussed in Ref. 7.
  13. N. V. Krasnov and N. Ya. Shaparev, Opt. Commun. 27, 239 (1978); Zh. Eksp. Teor. Fiz. 77, 899 (1979).
    [Crossref]
  14. V. S. Letokhov, V. G. Minogin, and V. D. Pavlik, Opt. Commun. 19, 72 (1976); Zh. Eksp. Teor. Fiz. 72, 1328 (1977).
    [Crossref]
  15. A. P. Kol’chenko, S. G. Rautian, and R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).
  16. J. L. Hall, C. J. Borde, and K. Jehara, Phys. Rev. Lett. 37, 1339 (1976).
    [Crossref]
  17. V. S. Letokhov and V. P. Chebotaev, Nonlinear Laser Spectroscopy (Springer-Verlag, Berlin, 1977).
    [Crossref]
  18. S. Stenholm, Phys. Rep. C43, 151 (1978).
    [Crossref]
  19. J. Javanainen and S. Stenholm, Appl. Phys. 21, 163 (1980).
    [Crossref]
  20. S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, Nonlinear Resonances in Atomic and Molecular Spectra (Nauka, Novosibirsk, 1979).
  21. T. K. Melik-Barkhudarov, Zh. Eksp. Teor. Fiz. 83, 1241 (1982).
  22. A. P. Kazantsev, V. S. Smirnov, A. M. Tumaikin, and I. Yagofarov, preprint 5, Institute Optiki Atmosferi, Tomsk, 1982 (Russian); Opt. Spektr. 58, 500 (1985).
  23. D. A. Varshalovich, A. N. Moskalev, and V. K. Hersonsky, Quantum Theory of Angular Moment (Nauka, Leningrad, 1975); A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, and D. O. Chudesnikov, Opt. Commun. 52, 311 (1985).
    [Crossref]

1983 (1)

A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 85, 852 (1983); A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, and V. P. Yakovlev, At. Mol. Phys. 18, 2619 (1985).
[Crossref]

1982 (1)

T. K. Melik-Barkhudarov, Zh. Eksp. Teor. Fiz. 83, 1241 (1982).

1981 (1)

A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 80, 541 (1981); J. Physique (Paris) 42, 1231 (1981); Appl. Phys. 25, 165 (1981).

1980 (3)

R. J. Cook, Phys. Rev. A 21, 268 (1980); Phys. Rev. A 22, 1078 (1980).
[Crossref]

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[Crossref]

J. Javanainen and S. Stenholm, Appl. Phys. 21, 163 (1980).
[Crossref]

1979 (1)

V. G. Minogin and O. T. Serimaa, Opt. Commun. 30, 373 (1979).
[Crossref]

1978 (4)

N. V. Krasnov and N. Ya. Shaparev, Opt. Commun. 27, 239 (1978); Zh. Eksp. Teor. Fiz. 77, 899 (1979).
[Crossref]

A. P. Kazantsev, Usp. Fiz. Nauk 124, 113 (1978) [Sov. Phys. Usp. 21, 58 (1978)].

R. J. Cook, Phys. Rev. Lett. 41, 1788 (1978); Phys. Rev. A 20, 224 (1979).
[Crossref]

S. Stenholm, Phys. Rep. C43, 151 (1978).
[Crossref]

1976 (2)

J. L. Hall, C. J. Borde, and K. Jehara, Phys. Rev. Lett. 37, 1339 (1976).
[Crossref]

V. S. Letokhov, V. G. Minogin, and V. D. Pavlik, Opt. Commun. 19, 72 (1976); Zh. Eksp. Teor. Fiz. 72, 1328 (1977).
[Crossref]

1975 (1)

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

1974 (2)

A. P. Kazantsev, Zh. Eksp. Teor. Fiz. 67, 1606 (1974) [Sov. Phys. JETP 40, 825 (1974)].

A. P. Kazantsev, Zh. Eksp. Teor. Fiz. 66, 1599 (1974) [Sov. Phys. JETP 39, 784 (1974)].

1970 (1)

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970); Phys. Rev. Lett. 25, 1321 (1970).
[Crossref]

1968 (1)

A. P. Kol’chenko, S. G. Rautian, and R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).

Ashkin, A.

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[Crossref]

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970); Phys. Rev. Lett. 25, 1321 (1970).
[Crossref]

Borde, C. J.

J. L. Hall, C. J. Borde, and K. Jehara, Phys. Rev. Lett. 37, 1339 (1976).
[Crossref]

Chebotaev, V. P.

V. S. Letokhov and V. P. Chebotaev, Nonlinear Laser Spectroscopy (Springer-Verlag, Berlin, 1977).
[Crossref]

Cook, R. J.

R. J. Cook, Phys. Rev. A 21, 268 (1980); Phys. Rev. A 22, 1078 (1980).
[Crossref]

R. J. Cook, Phys. Rev. Lett. 41, 1788 (1978); Phys. Rev. A 20, 224 (1979).
[Crossref]

Gordon, J. P.

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[Crossref]

Hall, J. L.

J. L. Hall, C. J. Borde, and K. Jehara, Phys. Rev. Lett. 37, 1339 (1976).
[Crossref]

Hänsch, T. W.

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

Hersonsky, V. K.

D. A. Varshalovich, A. N. Moskalev, and V. K. Hersonsky, Quantum Theory of Angular Moment (Nauka, Leningrad, 1975); A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, and D. O. Chudesnikov, Opt. Commun. 52, 311 (1985).
[Crossref]

Javanainen, J.

J. Javanainen and S. Stenholm, Appl. Phys. 21, 163 (1980).
[Crossref]

Jehara, K.

J. L. Hall, C. J. Borde, and K. Jehara, Phys. Rev. Lett. 37, 1339 (1976).
[Crossref]

Kazantsev, A. P.

A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 85, 852 (1983); A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, and V. P. Yakovlev, At. Mol. Phys. 18, 2619 (1985).
[Crossref]

A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 80, 541 (1981); J. Physique (Paris) 42, 1231 (1981); Appl. Phys. 25, 165 (1981).

A. P. Kazantsev, Usp. Fiz. Nauk 124, 113 (1978) [Sov. Phys. Usp. 21, 58 (1978)].

A. P. Kazantsev, Zh. Eksp. Teor. Fiz. 67, 1606 (1974) [Sov. Phys. JETP 40, 825 (1974)].

A. P. Kazantsev, Zh. Eksp. Teor. Fiz. 66, 1599 (1974) [Sov. Phys. JETP 39, 784 (1974)].

A. P. Kazantsev, V. S. Smirnov, A. M. Tumaikin, and I. Yagofarov, preprint 5, Institute Optiki Atmosferi, Tomsk, 1982 (Russian); Opt. Spektr. 58, 500 (1985).

Kol’chenko, A. P.

A. P. Kol’chenko, S. G. Rautian, and R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).

Krasnov, N. V.

N. V. Krasnov and N. Ya. Shaparev, Opt. Commun. 27, 239 (1978); Zh. Eksp. Teor. Fiz. 77, 899 (1979).
[Crossref]

Letokhov, V. S.

V. S. Letokhov, V. G. Minogin, and V. D. Pavlik, Opt. Commun. 19, 72 (1976); Zh. Eksp. Teor. Fiz. 72, 1328 (1977).
[Crossref]

V. S. Letokhov and V. P. Chebotaev, Nonlinear Laser Spectroscopy (Springer-Verlag, Berlin, 1977).
[Crossref]

Melik-Barkhudarov, T. K.

T. K. Melik-Barkhudarov, Zh. Eksp. Teor. Fiz. 83, 1241 (1982).

Minogin, V. G.

V. G. Minogin and O. T. Serimaa, Opt. Commun. 30, 373 (1979).
[Crossref]

V. S. Letokhov, V. G. Minogin, and V. D. Pavlik, Opt. Commun. 19, 72 (1976); Zh. Eksp. Teor. Fiz. 72, 1328 (1977).
[Crossref]

Moskalev, A. N.

D. A. Varshalovich, A. N. Moskalev, and V. K. Hersonsky, Quantum Theory of Angular Moment (Nauka, Leningrad, 1975); A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, and D. O. Chudesnikov, Opt. Commun. 52, 311 (1985).
[Crossref]

Pavlik, V. D.

V. S. Letokhov, V. G. Minogin, and V. D. Pavlik, Opt. Commun. 19, 72 (1976); Zh. Eksp. Teor. Fiz. 72, 1328 (1977).
[Crossref]

Rautian, S. G.

A. P. Kol’chenko, S. G. Rautian, and R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).

S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, Nonlinear Resonances in Atomic and Molecular Spectra (Nauka, Novosibirsk, 1979).

Schawlow, A. L.

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

Serimaa, O. T.

V. G. Minogin and O. T. Serimaa, Opt. Commun. 30, 373 (1979).
[Crossref]

Shalagin, A. M.

S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, Nonlinear Resonances in Atomic and Molecular Spectra (Nauka, Novosibirsk, 1979).

Shaparev, N. Ya.

N. V. Krasnov and N. Ya. Shaparev, Opt. Commun. 27, 239 (1978); Zh. Eksp. Teor. Fiz. 77, 899 (1979).
[Crossref]

Smirnov, G. I.

S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, Nonlinear Resonances in Atomic and Molecular Spectra (Nauka, Novosibirsk, 1979).

Smirnov, V. S.

A. P. Kazantsev, V. S. Smirnov, A. M. Tumaikin, and I. Yagofarov, preprint 5, Institute Optiki Atmosferi, Tomsk, 1982 (Russian); Opt. Spektr. 58, 500 (1985).

Sokolovskii, R. I.

A. P. Kol’chenko, S. G. Rautian, and R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).

Stenholm, S.

J. Javanainen and S. Stenholm, Appl. Phys. 21, 163 (1980).
[Crossref]

S. Stenholm, Phys. Rep. C43, 151 (1978).
[Crossref]

Surdutovich, G. I.

A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 85, 852 (1983); A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, and V. P. Yakovlev, At. Mol. Phys. 18, 2619 (1985).
[Crossref]

A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 80, 541 (1981); J. Physique (Paris) 42, 1231 (1981); Appl. Phys. 25, 165 (1981).

Tumaikin, A. M.

A. P. Kazantsev, V. S. Smirnov, A. M. Tumaikin, and I. Yagofarov, preprint 5, Institute Optiki Atmosferi, Tomsk, 1982 (Russian); Opt. Spektr. 58, 500 (1985).

Varshalovich, D. A.

D. A. Varshalovich, A. N. Moskalev, and V. K. Hersonsky, Quantum Theory of Angular Moment (Nauka, Leningrad, 1975); A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, and D. O. Chudesnikov, Opt. Commun. 52, 311 (1985).
[Crossref]

Yagofarov, I.

A. P. Kazantsev, V. S. Smirnov, A. M. Tumaikin, and I. Yagofarov, preprint 5, Institute Optiki Atmosferi, Tomsk, 1982 (Russian); Opt. Spektr. 58, 500 (1985).

Yakovlev, V. P.

A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 85, 852 (1983); A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, and V. P. Yakovlev, At. Mol. Phys. 18, 2619 (1985).
[Crossref]

A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 80, 541 (1981); J. Physique (Paris) 42, 1231 (1981); Appl. Phys. 25, 165 (1981).

Appl. Phys. (1)

J. Javanainen and S. Stenholm, Appl. Phys. 21, 163 (1980).
[Crossref]

Opt. Commun. (4)

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

V. G. Minogin and O. T. Serimaa, Opt. Commun. 30, 373 (1979).
[Crossref]

N. V. Krasnov and N. Ya. Shaparev, Opt. Commun. 27, 239 (1978); Zh. Eksp. Teor. Fiz. 77, 899 (1979).
[Crossref]

V. S. Letokhov, V. G. Minogin, and V. D. Pavlik, Opt. Commun. 19, 72 (1976); Zh. Eksp. Teor. Fiz. 72, 1328 (1977).
[Crossref]

Phys. Rep. (1)

S. Stenholm, Phys. Rep. C43, 151 (1978).
[Crossref]

Phys. Rev. A (2)

R. J. Cook, Phys. Rev. A 21, 268 (1980); Phys. Rev. A 22, 1078 (1980).
[Crossref]

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[Crossref]

Phys. Rev. Lett. (3)

R. J. Cook, Phys. Rev. Lett. 41, 1788 (1978); Phys. Rev. A 20, 224 (1979).
[Crossref]

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970); Phys. Rev. Lett. 25, 1321 (1970).
[Crossref]

J. L. Hall, C. J. Borde, and K. Jehara, Phys. Rev. Lett. 37, 1339 (1976).
[Crossref]

Usp. Fiz. Nauk (1)

A. P. Kazantsev, Usp. Fiz. Nauk 124, 113 (1978) [Sov. Phys. Usp. 21, 58 (1978)].

Zh. Eksp. Teor. Fiz. (6)

A. P. Kol’chenko, S. G. Rautian, and R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).

A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 80, 541 (1981); J. Physique (Paris) 42, 1231 (1981); Appl. Phys. 25, 165 (1981).

A. P. Kazantsev, Zh. Eksp. Teor. Fiz. 66, 1599 (1974) [Sov. Phys. JETP 39, 784 (1974)].

A. P. Kazantsev, Zh. Eksp. Teor. Fiz. 67, 1606 (1974) [Sov. Phys. JETP 40, 825 (1974)].

A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 85, 852 (1983); A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, and V. P. Yakovlev, At. Mol. Phys. 18, 2619 (1985).
[Crossref]

T. K. Melik-Barkhudarov, Zh. Eksp. Teor. Fiz. 83, 1241 (1982).

Other (5)

A. P. Kazantsev, V. S. Smirnov, A. M. Tumaikin, and I. Yagofarov, preprint 5, Institute Optiki Atmosferi, Tomsk, 1982 (Russian); Opt. Spektr. 58, 500 (1985).

D. A. Varshalovich, A. N. Moskalev, and V. K. Hersonsky, Quantum Theory of Angular Moment (Nauka, Leningrad, 1975); A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, and D. O. Chudesnikov, Opt. Commun. 52, 311 (1985).
[Crossref]

S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, Nonlinear Resonances in Atomic and Molecular Spectra (Nauka, Novosibirsk, 1979).

For the Schrödinger equation with the Hamiltonian [Eq. (2)] this symmetry has been discussed in Ref. 7.

V. S. Letokhov and V. P. Chebotaev, Nonlinear Laser Spectroscopy (Springer-Verlag, Berlin, 1977).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1

Frequency dependence of the friction force 〈fr〉 [Eq. (39)] for slow atoms averaged over the field period: a, V0/ħγ = 0.35 (curve 1), V0/ħγ = 1.79 (curve 2), V0/ħγ = 2.5 (curve 3). b, V0/ħγ = 3.54 (curve 1), V0/ħγ = 7.9 (curve 2). For negative detuning 〈fr〉 is prolonged oddly.

Fig. 2
Fig. 2

The distribution function of slow atoms for large interaction time η ≫ 1: a, δ = 1 (heating); b, δ = −1 (cooling).

Equations (75)

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i ( / t + Γ ^ ) ρ ( 1 , 2 ) = H ( 1 ) ρ ( 1 , 2 ) - ρ ( 1 , 2 ) H ( 2 ) , H = p ^ 2 / 2 m + H ,
H = - Δ 2 σ 3 = σ 1 V ( x ) ,             V ( x ) = V 0 sin k x , V 0 = d E 0 ,             γ = 4 3 d 2 k 3 / ,
Γ ^ ρ = [ γ ρ b b γ 2 ρ b a γ 2 ρ a b - γ ^ ( 1 - 2 ) ρ b b ] ,             γ ^ ( 1 - 2 ) = γ ^ ( r 1 - r 2 ) , γ ^ ( r ) = γ d n φ ( n ) exp ( i nr k ) ,
φ ( n ) = 3 π 8 [ 1 - ( dn ) 2 / d 2 ] ,
d F d t = - V p ( P + P * ) + 1 2 δ γ ^ ( F + Q ) ,
( d d t + γ ) Q = - 2 i V ( P - P * ) - γ F ,
( d d t - i ν ) P = - i V Q - 1 2 V F p ,             ν = Δ + i γ / 2 ,
δ γ ^ = γ k 2 5 [ ( / p ) 2 - 1 2 ( d d p ) 2 ] ,
p 2 / 2 m V ,
( / t + γ ) Q = - 2 i V ( t ) ( P - P * ) - γ ,
( / t - i ν ) P = - i V ( t ) Q ,             V ( t ) V [ x ( t ) ] .
f = f x ( t ) = 2 v d V d t Re P ( t ) .
f ( x ) = tr ( d ^ ρ ) d E d x = 2 Re P ( x ) d V d x .
γ , k v Δ .
f g = - x V 2 ( x ) Δ .
f r = γ v k 2 V 0 2 Δ 3 { 1 - V 0 4 4 Δ 2 [ γ 2 + ( 2 k v ) 2 ] } .
V 0 2 / 2 Δ > γ ,
V 0 / ω , γ
ρ ( t ) = U ^ ( t ) ρ ˜ ( t ) U ^ + ( t ) ,             U ^ = ( u * v - v u ) .
ψ 1 = ( v u ) ,             ψ 2 = ( u * - v * ) , u ( t ) = [ 1 2 ( 1 + 1 / ) ] 1 / 2 exp ( - i Δ 2 d t ) , v ( t ) = - sign ( V / Δ ) [ 1 2 ( - 1 / ) ] 1 / 2 exp ( - i Δ 2 d t ) , ( t ) = [ 1 + ( 2 V / Δ ) 2 ] 1 / 2 .
d ρ ˜ d t = - U ^ + ( Γ ^ ρ ) U ^ ,             ρ ˜ = ( ρ 22 ρ 21 ρ 12 ρ 11 ) ,
d A d t = - γ 2 + 1 2 2 A + γ / , d B d t = - γ 2 ( 1 + 2 - 1 2 2 ) B .
Δ Δ 0 = ( V 0 ω / ) 1 / 2 V 0 / h ,
γ × δ t 1 ,             i . e . , k v Δ γ / V 0 ,
ρ ˜ ( t < 0 ) U ^ L ρ ˜ ( t < 0 ) U ^ L + , U ^ L = [ I - R 2 exp ( i χ ) R - R I - R 2 exp ( - i χ ) ] , R = exp ( - π ξ ) , ξ = 1 8 Δ 2 Δ 0 2 , χ = π 4 + arg Γ ( 1 - i ξ ) - ξ ln e / ξ .
ρ ( t + π / ω ) = σ 3 ρ ( t ) σ 3
A ( t ) = γ × [ Φ ( π / ω ) L exp ( - μ 1 ) 1 - L exp ( - μ 1 ) + Φ ( t ) ] exp [ - μ 1 ( t ) ] , Φ ( t ) = 0 t d t ( t ) exp [ μ 1 ( t ) ] , μ 1 ( t ) = γ 2 0 t ( 1 + 1 / 2 ) d t ,             μ 2 ( t ) = γ 4 0 t ( 3 - 1 / 2 ) d t , μ 1 = μ 1 ( π / ω ) ,             μ 2 = μ 2 ( π / ω ) , L = ( 1 - R 2 ) ( c h μ 2 + cos 2 φ ) - R 2 s h μ 2 ( 1 - R 2 ) ( c h μ 2 + cos 2 φ ) + R 2 s h μ 2 , φ = χ + Δ 2 0 π / ω ( t ) d t , B ( t ) = γ 2 Φ ( π / ω ) exp ( - μ 2 ) 1 - L exp ( - μ 1 ) × R 1 - R 2 exp ( i χ ) [ exp ( μ 2 ) + exp ( - 2 i φ ) ] ( 1 - R 2 ) ( c h μ 2 + cos 2 φ ) + R 2 s h μ 2 × exp [ - μ 2 ( t ) ] .
A ( t ) = γ 1 - exp ( - μ 1 ) 0 π / ω d τ ( t - τ ) × exp [ - γ 2 0 τ d τ 2 ( t - τ ) + 1 2 ( t - τ ) ] .
f = U x A ( x ) ,             U ( x ) = Δ 2 ( x ) ,             x = v t .
f g = - U 1 x ,             U 1 ( x ) = Δ 2 ln [ 1 + 2 ( x ) ] , f r = - 2 Δ v γ 2 ( 2 - 1 ) ( 1 + 2 ) 3 ( d d x ) 2 .
f g = - U 2 x ,             U 2 ( x ) = C g U ( x ) , C g = 2 - 1 - 2 = 4 cos α K ( sin α ) π ( 1 + cos α ) , f r = - C r Δ γ / v ,             sin α = V 0 [ V 0 2 + ( Δ 2 ) 2 ] 1 / 2 , C r = 1 2 [ - 1 + - 1 2 1 + - 2 ] - 1 = 2 K ( sin α ) [ E ( sin α ) + cos 2 α K ( sin α ) ] π 2 ( 1 + cos α ) - 1 2 ,
d d t ρ 22 + γ × ρ 22 [ V ( t ) / Δ ] 4 .
γ Δ / V 0 k v γ
f = - 2 π sign v × k Δ ( I - R 2 ) ln ( β Δ 0 2 / γ Δ ) , β = 8 exp ( - C ) ,
f = - γ Δ 2 v [ 4 π 2 ln ( 8 V 0 Δ ) cos 2 φ cos 2 φ + 3 R 2 / 2 ( 1 - R 2 ) - 1 ] .
f = γ Δ 2 v [ 1 - 2 3 π ( Δ Δ 0 ) 2 ln 8 V 0 Δ × cos 2 φ ]
k v x γ
P 0 = V ν Q 0 ,             Q 0 = - F 1 + χ ,             χ = 2 V 2 2 ν 2 , V = V 0 sin k x .
ν P 1 - V Q 1 = i 2 d V d x F p x - i d P 0 d t , γ Q 1 + 2 i V ( P 1 - P * 1 ) = - d Q 0 d t .
d F d t + p x [ ( f g + f r ) F ] = ( 2 p x 2 D x x + 2 p y 2 D y y + 2 p z 2 D z z ) F , f g = - U 1 x ,             U 1 = Δ 2 ln ( 1 + χ ) , f r = 2 Δ γ v x ν 4 × ( d V d x ) 2 1 - χ - 2 ν 2 χ 2 / γ 2 ( 1 + χ ) 3 , D x x = D s + γ 2 ν 2 ( d V d x ) 2 [ 1 + 4 Δ 2 χ ( χ 2 - γ 2 / ν 2 ) γ 2 ( 1 + χ ) 3 ] ,             D y y = 2 D z z = D s = γ ( k ) 2 10 × χ 1 + χ .
f r ~ k Δ ,
v c = γ 2 k [ ( V 0 / V c ) 4 - 1 ] 1 / 2 .
f r ( v ) = 0 ,             d d v f r ( v ) < 0
v F x + p [ ( f g + f r ) ] = 2 p 2 ( D F ) .
F x + { v ( , x ) [ X ( x ) F - D ( x ) F ] } = 0 , v ( , x ) = ± { 2 m [ - U 1 ( x ) ] } 1 / 2 ,             X ( x ) = f r / v .
F ( x , ) = F 0 ( ) + F 1 ( x , ) ,             F 1 F 0 .
F 1 x + { v ( , x ) [ X ( x ) F 0 - D ( x ) d F 0 d ] } = 0.
d d { v ( , x ) X ( x ) F 0 - v ( , x ) D ( x ) d F 0 d } = 0.
F 0 ( ) = const . exp ( min U 1 d v X / v D ) .
F 0 ( ) = const . exp ( - / T ) ,             T = - D X ,
T = 7 20 ( Δ + γ 2 / 4 Δ )
D ( x ) 1 2 γ ( k V 0 / Δ ) 2 ,             U 1 ( x ) = ( k V 0 x ) 2 / Δ , X ( x ) 2 γ Δ ( k V 0 Δ ) 2 [ 1 - 8 ( k V 0 x ) 4 ( 2 γ Δ ) 2 ] .
F 0 ( ) = const exp [ - ( / 0 ) 3 ] , 0 = ( 3 γ 2 Δ / 4 ) 1 / 3 ,             = p 2 / 2 m + ( k V 0 x ) 2 / Δ .
F η + ξ [ f ( ξ ) F ] = 0 ,             f ( ξ ) = 1 1 + ( δ - ξ ) 2 - 1 1 + ( δ + ξ ) 2 , η = 4 t r V 0 2 ( γ ) 2 ,             t = y / v y ,             ξ = 2 k v x / γ ,             δ = 2 Δ / γ .
F ( ξ , η ) = F 0 [ ξ 0 ( ξ , η ) ] f [ ξ 0 ( ξ , ν ) ] / f ( ξ ) , η = ξ 0 ξ d ξ / f ( ξ ) = 1 16 [ ξ 4 - ξ 0 4 + 4 ( 1 - δ 2 ) ( ξ 2 - ξ 0 2 ) + 2 ( 1 + δ 2 ) 2 ln ( ξ 2 / ξ 0 2 ) ] ,
f ( ξ 0 ) f ( ξ ) = 1 - 4 η δ × ( 1 + δ 2 ) 2 + 2 ( δ 2 - 1 ) ξ 2 - 3 ξ 4 [ ( ξ 2 + 1 - δ 2 ) 2 + 4 δ 2 ] .
X ~ X 0 { 1 - V 0 2 2 γ 2 [ 1 + 1 1 + δ 2 + r τ × 2 δ ( 1 + δ 2 ) 2 ] } .
F ( ξ , η ) = F 0 { ( 1 + ξ 4 / 4 ) exp ( - η + ξ 4 / 16 ) , ξ < 2 η 1 / 4 ( 1 - 16 η ) / ξ 4 ) - 3 / 4 , ξ > 2 η 1 / 4 ,
F ( ξ , η ) = F 0 { 1 / 2 ξ η 3 / 4 , exp ( - η ) < ξ 1 ( 1 + 16 η / ξ 4 ) - 3 / 4 , ξ 1 .
P = V ν Q - i 2 Δ d V d x F p - i Δ 2 d d t ( V Q ) ,             p = m v = p x .
d F d t - U x G p = 0 ,
d G d t + γ 2 + 1 2 2 G = γ F + U x F p , = [ 1 + ( 2 V / Δ ) 2 ] 1 / 2 .
G = 0 d τ { γ F ( x - v τ , t - τ , p ) ( x - v τ ) + U ( x - v τ ) x × F ( x - v τ , t - τ , p ) p } × exp { - γ 2 0 τ [ 1 + 1 / 2 ( x - v τ ) ] d τ } .
d d t ρ 11 - U x p ρ 11 + γ 1 ( x ) ρ 11 = γ 2 ( x ) ρ 22 , d d t ρ 22 + U x p ρ 22 + γ 2 ( x ) ρ 22 = γ 1 ( x ) ρ 11 , γ I , 2 = γ ( ± 1 2 ) 2 .
[ d d t ρ μ μ ( a a ) ( r , p ) ] rel = k 3 2 π m m i j d μ m i d m μ j × d n ( δ i j - n i n j ) ρ m m ( b b ) ( r , p + n k ) .
ρ X q ( a a ) = μ μ ( - 1 ) j a - μ C j a μ , j a - μ X q ρ μ μ ( a a ) ,
[ d d t ρ X q ( a a ) ] rel = γ K X d n ρ X q ( b b ) ( r , p + n k ) + γ X 1 q 1 d n K X 1 q 1 X q ( n ) ρ x 1 q 1 ( b b ) ( r , p + n k ) , K X = ( - 1 ) j a + j b + X + 1 ( 2 j b + 1 ) { j b j b X j a j a 1 } , γ = 4 3 k 3 h ( 2 j b + 1 ) j a d j b 2 , K X 1 q 1 X q ( n ) = [ 3 π ( 2 X 1 + 1 ) ] 1 / 2 ( 2 j b + 1 ) × q Y 2 q * ( n ) C 2 q , X 1 q 1 X q { j a j a X 1 j b j b X 1 1 2 } ,
[ d d t ρ μ μ ( a a ) ( rp ) ] rel = 4 k 3 3 i j m m d μ m i d m μ j × { δ i j + ( k ) 2 5 [ δ i j l 2 p l 2 - 1 2 2 p i p j ] } ρ m m ( b b ) ( rp ) .
i d d t ρ i j ( a a ) ( rp t ) = ρ i ( a b ) ( r , p + k 2 , t ) V j / - V i * ρ j ( b a ) ( r , p + k / 2 , t ) + i γ 3 × d n 3 8 π ( δ i j - n i n j ) ρ ( b b ) ( r , p + n k ) , i ( d d t + γ ) ρ ( b b ) ( r , p , t ) = i [ V i * h ρ i ( b a ) ( r , p - k 2 , t ) - V i h ρ i ( a b ) ( r , p - k 2 , t ) ] , [ i d d t + ν ( p - k 2 ) ] ρ i ( b a ) ( r , p , t ) = V i ρ ( b b ) ( r , p + k 2 , t ) - j V j ρ j i ( a a ) ( r , p - k 2 , t ) ν ( p - k 2 ) = Δ - kv + i γ / 2 ,             V i = d E 0 i , d = 0 d 1 3 .
i d d t ρ i j ( a a ) ( r , p , t ) = [ l V i * V l 2 ν ( p ) ρ l j ( a a ) ( r , p , t ) - V l * V j 2 ν ( p ) * ρ i l ( a a ) ( r , p , t ) ] + i γ 3 d n 3 8 π ( δ i j - n i n j ) × l s V l V s * 2 ν ( p - k + n k ) 2 ρ l s ( a a ) ( p - k + n k ) .
M i ( r , p , t ) = i j l i j l ρ j l ( a a ) .
ρ i j ( a a ) = 1 3 δ i j f ( p ) + ρ i j ( 1 ) + ρ i j ( 2 ) + ,
ρ i j ( 1 ) = - τ ( p ) γ k 2 90 · V 2 2 p i p j · f ( p ) ν ( p ) 2 .
M = M ( p ) d 3 p = - f 0 k 2 90 V 2 2 ν ( p 0 2 × [ V × p 0 ] ( V · p 0 ) τ 2 ( p 0 ) Re 1 ν ( p 0 ) .
α = 1 15 x l γ τ 2 ( Δ - k v 0 ) ( k p 0 ) 2 | V ν ( p 0 ) | 4 sin 2 θ sin θ sin 2 φ ,

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