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, 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, V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 85, 852 (1983); A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, 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, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
    [CrossRef]
  10. V. G. Minogin, O. T. Serimaa, Opt. Commun. 30, 373 (1979).
    [CrossRef]
  11. A. P. Kazantsev, G. I. Surdutovich, 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, N. Ya. Shaparev, Opt. Commun. 27, 239 (1978); Zh. Eksp. Teor. Fiz. 77, 899 (1979).
    [CrossRef]
  14. V. S. Letokhov, V. G. Minogin, V. D. Pavlik, Opt. Commun. 19, 72 (1976); Zh. Eksp. Teor. Fiz. 72, 1328 (1977).
    [CrossRef]
  15. A. P. Kol’chenko, S. G. Rautian, R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).
  16. J. L. Hall, C. J. Borde, K. Jehara, Phys. Rev. Lett. 37, 1339 (1976).
    [CrossRef]
  17. V. S. Letokhov, V. P. Chebotaev, Nonlinear Laser Spectroscopy (Springer-Verlag, Berlin, 1977).
    [CrossRef]
  18. S. Stenholm, Phys. Rep. C43, 151 (1978).
    [CrossRef]
  19. J. Javanainen, S. Stenholm, Appl. Phys. 21, 163 (1980).
    [CrossRef]
  20. S. G. Rautian, G. I. Smirnov, 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, I. Yagofarov, preprint 5, Institute Optiki Atmosferi, Tomsk, 1982 (Russian); Opt. Spektr. 58, 500 (1985).
  23. D. A. Varshalovich, A. N. Moskalev, V. K. Hersonsky, Quantum Theory of Angular Moment (Nauka, Leningrad, 1975); A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, D. O. Chudesnikov, Opt. Commun. 52, 311 (1985).
    [CrossRef]

1983 (1)

A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 85, 852 (1983); A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, 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, 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, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[CrossRef]

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

1979 (1)

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

1978 (4)

N. V. Krasnov, 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, K. Jehara, Phys. Rev. Lett. 37, 1339 (1976).
[CrossRef]

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

1975 (1)

T. W. Hänsch, 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, R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).

Ashkin, A.

J. P. Gordon, 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, K. Jehara, Phys. Rev. Lett. 37, 1339 (1976).
[CrossRef]

Chebotaev, V. P.

V. S. Letokhov, 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, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[CrossRef]

Hall, J. L.

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

Hänsch, T. W.

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

Hersonsky, V. K.

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

Javanainen, J.

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

Jehara, K.

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

Kazantsev, A. P.

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

A. P. Kazantsev, G. I. Surdutovich, 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, 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, R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).

Krasnov, N. V.

N. V. Krasnov, 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, V. D. Pavlik, Opt. Commun. 19, 72 (1976); Zh. Eksp. Teor. Fiz. 72, 1328 (1977).
[CrossRef]

V. S. Letokhov, 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, O. T. Serimaa, Opt. Commun. 30, 373 (1979).
[CrossRef]

V. S. Letokhov, V. G. Minogin, 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, V. K. Hersonsky, Quantum Theory of Angular Moment (Nauka, Leningrad, 1975); A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, D. O. Chudesnikov, Opt. Commun. 52, 311 (1985).
[CrossRef]

Pavlik, V. D.

V. S. Letokhov, V. G. Minogin, 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, R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).

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

Schawlow, A. L.

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

Serimaa, O. T.

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

Shalagin, A. M.

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

Shaparev, N. Ya.

N. V. Krasnov, 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, 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, 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, R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).

Stenholm, S.

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

A. P. Kazantsev, G. I. Surdutovich, 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, I. Yagofarov, preprint 5, Institute Optiki Atmosferi, Tomsk, 1982 (Russian); Opt. Spektr. 58, 500 (1985).

Varshalovich, D. A.

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

Yagofarov, I.

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

Yakovlev, V. P.

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

A. P. Kazantsev, G. I. Surdutovich, 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, S. Stenholm, Appl. Phys. 21, 163 (1980).
[CrossRef]

Opt. Commun. (4)

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

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

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

V. S. Letokhov, V. G. Minogin, 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, 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, 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, R. I. Sokolovskii, Zh. Eksp. Teor. Fiz. 55, 1864 (1968).

A. P. Kazantsev, G. I. Surdutovich, 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, V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 85, 852 (1983); A. P. Kazantsev, G. I. Surdutovich, D. O. Chudesnikov, 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, I. Yagofarov, preprint 5, Institute Optiki Atmosferi, Tomsk, 1982 (Russian); Opt. Spektr. 58, 500 (1985).

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

S. G. Rautian, G. I. Smirnov, 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, 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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