Abstract

We present a quantum theory of one-dimensional laser cooling of free atoms using a transition with a J = 0 ground state and a J = 1 excited state. This treatment is valid both for broad lines (recoil energy small compared with the energy width Γ of the excited level) and for narrow lines. For broad lines we recover the well-known cooling limit for a two-level transition (∼Γ/2), whereas for a narrow line the cooling limit is found to be of the order of the recoil energy. The stationary momentum distribution is obtained for both cases and is found to be close to the one obtained by Monte Carlo simulations.

© 1989 Optical Society of America

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  1. T. W. Hänsch, A. Schawlow, Opt. Commun. 13, 68 (1975).
    [CrossRef]
  2. D. Wineland, H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).
  3. See, e.g., S. Stenholm, Rev. Mod. Phys. 58, 699 (1986).
    [CrossRef]
  4. J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
    [CrossRef]
  5. S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, A. Ashkin, Phys. Rev. Lett. 55, 48 (1985).
    [CrossRef] [PubMed]
  6. P. Lett, R. Watts, C. Westbrook, W. D. Phillips, P. Gould, H. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
    [CrossRef] [PubMed]
  7. J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).
  8. S. Chu, D. S. Weiss, Y. Shevy, P. J. Ungar, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).
  9. W. D. Phillips, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 (personal communication, 1988).
  10. J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
    [CrossRef]
  11. J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 17, 4577 (1984).
    [CrossRef]
  12. S. Stenholm, Appl. Phys. 15, 287 (1978).
    [CrossRef]
  13. C. Bordé, in Advances in Laser Spectroscopy, S. Arecchi, F. Strumia, eds. (Plenum, New York, 1983).
  14. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988);J. Opt. Soc. Am. B 6, 2112 (1989).
    [CrossRef] [PubMed]
  15. D. Wineland, W. Itano, Phys. Rev. A 20, 1521 (1979).
    [CrossRef]
  16. R. Blatt, G. Lafyatis, W. Phillips, S. Stenholm, D. Wineland, Phys. Scr. T22, 216 (1988).
    [CrossRef]
  17. H. Wallis, W. Ertmer, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989);J. Opt. Soc. Am. B. 26, 2111 (1989).
  18. This result is apparently in contradiction to the limit derived in Ref. 16. Actually, this limit was obtained by an energy balance in a single absorption–spontaneous-emission cycle, without any reference to the excitation rate. The argument is therefore valid only if the excitation rate is nearly constant with respect to atomic velocity. This condition is violated for the cooling with a broadband laser of Ref. 17, in which the excitation rate for an atom at rest is much smaller than for an atom with a velocity larger than ℏk/m.
  19. P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
    [CrossRef]
  20. One could be tempted to obtain this result with a simple reasoning dealing only with energy balance in a single-absorption– spontaneous-emission cycle. In such reasoning the probability of absorption r±of one σ±photon is obtained from r±= γ±/γ++ γ−). One would therefore start from 〈p(r−− r+)〉 = 7ℏk/10 instead of Eq. (4.5) and would obtain after a similar algebra Ēκ= (Ēκ)cl+ Er/400 instead of expression (4.17). The error in such simple reasoning lies in the fact that one would neglect the variation with velocity of the time interval between successive cycles. Taking this variation into account correctly enhances the contribution to 〈p2〉 of low-velocity atoms and gives back the correct result [expression (4.17)].

1989 (2)

1988 (3)

R. Blatt, G. Lafyatis, W. Phillips, S. Stenholm, D. Wineland, Phys. Scr. T22, 216 (1988).
[CrossRef]

P. Lett, R. Watts, C. Westbrook, W. D. Phillips, P. Gould, H. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988);J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef] [PubMed]

1986 (1)

See, e.g., S. Stenholm, Rev. Mod. Phys. 58, 699 (1986).
[CrossRef]

1985 (2)

J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
[CrossRef]

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, A. Ashkin, Phys. Rev. Lett. 55, 48 (1985).
[CrossRef] [PubMed]

1984 (1)

J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 17, 4577 (1984).
[CrossRef]

1979 (1)

D. Wineland, W. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

1978 (1)

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

1975 (2)

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

D. Wineland, H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).

Arimondo, E.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988);J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef] [PubMed]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Ashkin, A.

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, A. Ashkin, Phys. Rev. Lett. 55, 48 (1985).
[CrossRef] [PubMed]

Aspect, A.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988);J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef] [PubMed]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Bjorkholm, J. E.

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, A. Ashkin, Phys. Rev. Lett. 55, 48 (1985).
[CrossRef] [PubMed]

Blatt, R.

R. Blatt, G. Lafyatis, W. Phillips, S. Stenholm, D. Wineland, Phys. Scr. T22, 216 (1988).
[CrossRef]

Bordé, C.

C. Bordé, in Advances in Laser Spectroscopy, S. Arecchi, F. Strumia, eds. (Plenum, New York, 1983).

Cable, A.

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, A. Ashkin, Phys. Rev. Lett. 55, 48 (1985).
[CrossRef] [PubMed]

Chu, S.

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, A. Ashkin, Phys. Rev. Lett. 55, 48 (1985).
[CrossRef] [PubMed]

S. Chu, D. S. Weiss, Y. Shevy, P. J. Ungar, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Cohen-Tannoudji, C.

J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988);J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef] [PubMed]

J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
[CrossRef]

J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 17, 4577 (1984).
[CrossRef]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Dalibard, J.

J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
[CrossRef]

J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
[CrossRef]

J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 17, 4577 (1984).
[CrossRef]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Dehmelt, H.

D. Wineland, H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).

Ertmer, W.

H. Wallis, W. Ertmer, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989);J. Opt. Soc. Am. B. 26, 2111 (1989).

Gould, P.

P. Lett, R. Watts, C. Westbrook, W. D. Phillips, P. Gould, H. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

Hänsch, T. W.

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

Hollberg, L.

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, A. Ashkin, Phys. Rev. Lett. 55, 48 (1985).
[CrossRef] [PubMed]

Itano, W.

D. Wineland, W. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

Kaiser, R.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988);J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef] [PubMed]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Lafyatis, G.

R. Blatt, G. Lafyatis, W. Phillips, S. Stenholm, D. Wineland, Phys. Scr. T22, 216 (1988).
[CrossRef]

Lett, P.

P. Lett, R. Watts, C. Westbrook, W. D. Phillips, P. Gould, H. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

Lett, P. D.

Metcalf, H.

P. Lett, R. Watts, C. Westbrook, W. D. Phillips, P. Gould, H. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

Phillips, W.

R. Blatt, G. Lafyatis, W. Phillips, S. Stenholm, D. Wineland, Phys. Scr. T22, 216 (1988).
[CrossRef]

Phillips, W. D.

P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
[CrossRef]

P. Lett, R. Watts, C. Westbrook, W. D. Phillips, P. Gould, H. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

W. D. Phillips, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 (personal communication, 1988).

Reynaud, S.

J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 17, 4577 (1984).
[CrossRef]

Rolston, S. L.

Salomon, C.

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Schawlow, A.

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

Shevy, Y.

S. Chu, D. S. Weiss, Y. Shevy, P. J. Ungar, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Stenholm, S.

R. Blatt, G. Lafyatis, W. Phillips, S. Stenholm, D. Wineland, Phys. Scr. T22, 216 (1988).
[CrossRef]

See, e.g., S. Stenholm, Rev. Mod. Phys. 58, 699 (1986).
[CrossRef]

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

Tanner, C. E.

Ungar, P. J.

S. Chu, D. S. Weiss, Y. Shevy, P. J. Ungar, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Vansteenkiste, N.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988);J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef] [PubMed]

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Wallis, H.

H. Wallis, W. Ertmer, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989);J. Opt. Soc. Am. B. 26, 2111 (1989).

Watts, R.

P. Lett, R. Watts, C. Westbrook, W. D. Phillips, P. Gould, H. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

Watts, R. N.

Weiss, D. S.

S. Chu, D. S. Weiss, Y. Shevy, P. J. Ungar, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Westbrook, C.

P. Lett, R. Watts, C. Westbrook, W. D. Phillips, P. Gould, H. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

Westbrook, C. I.

Wineland, D.

R. Blatt, G. Lafyatis, W. Phillips, S. Stenholm, D. Wineland, Phys. Scr. T22, 216 (1988).
[CrossRef]

D. Wineland, W. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

D. Wineland, H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).

Appl. Phys. (1)

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

Bull. Am. Phys. Soc. (1)

D. Wineland, H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).

J. Opt. Soc. Am. B (2)

J. Phys. B (2)

J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 17, 4577 (1984).
[CrossRef]

J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
[CrossRef]

Opt. Commun. (1)

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

Phys. Rev. A (1)

D. Wineland, W. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

Phys. Rev. Lett. (3)

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, A. Ashkin, Phys. Rev. Lett. 55, 48 (1985).
[CrossRef] [PubMed]

P. Lett, R. Watts, C. Westbrook, W. D. Phillips, P. Gould, H. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
[CrossRef] [PubMed]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988);J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef] [PubMed]

Phys. Scr. (1)

R. Blatt, G. Lafyatis, W. Phillips, S. Stenholm, D. Wineland, Phys. Scr. T22, 216 (1988).
[CrossRef]

Rev. Mod. Phys. (1)

See, e.g., S. Stenholm, Rev. Mod. Phys. 58, 699 (1986).
[CrossRef]

Other (7)

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

S. Chu, D. S. Weiss, Y. Shevy, P. J. Ungar, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

W. D. Phillips, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 (personal communication, 1988).

H. Wallis, W. Ertmer, in Proceedings of the 11th Conference on Atomic Physics, Paris, July 1988, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989);J. Opt. Soc. Am. B. 26, 2111 (1989).

This result is apparently in contradiction to the limit derived in Ref. 16. Actually, this limit was obtained by an energy balance in a single absorption–spontaneous-emission cycle, without any reference to the excitation rate. The argument is therefore valid only if the excitation rate is nearly constant with respect to atomic velocity. This condition is violated for the cooling with a broadband laser of Ref. 17, in which the excitation rate for an atom at rest is much smaller than for an atom with a velocity larger than ℏk/m.

One could be tempted to obtain this result with a simple reasoning dealing only with energy balance in a single-absorption– spontaneous-emission cycle. In such reasoning the probability of absorption r±of one σ±photon is obtained from r±= γ±/γ++ γ−). One would therefore start from 〈p(r−− r+)〉 = 7ℏk/10 instead of Eq. (4.5) and would obtain after a similar algebra Ēκ= (Ēκ)cl+ Er/400 instead of expression (4.17). The error in such simple reasoning lies in the fact that one would neglect the variation with velocity of the time interval between successive cycles. Taking this variation into account correctly enhances the contribution to 〈p2〉 of low-velocity atoms and gives back the correct result [expression (4.17)].

C. Bordé, in Advances in Laser Spectroscopy, S. Arecchi, F. Strumia, eds. (Plenum, New York, 1983).

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Figures (7)

Fig. 1
Fig. 1

A (g, J = 0) → (e, J = 1) atom irradiated by two counterpropagating σ+ and σ polarized waves.

Fig. 2
Fig. 2

Time evolution of the momentum distribution W(p) for the case of a narrow line (Γ = Er) with a Rabi frequency Ω = Γ and a detuning δ ¯ = 2.5 Γ (discretization 200 points, 10 points per recoil momentum). The last distribution (for t = ∞) was obtained directly by solving the set of equations deduced from ρ ˙ = 0.

Fig. 3
Fig. 3

(a) Variation of the stationary rms momentum with Rabi frequency Ω for the two narrow lines Γ = Er and Γ= 0.1Er. For each Rabi frequency we took the detuning δ ¯ that minimizes the rms momentum. (b) Variation with the Rabi frequency Ω of the detuning δ ¯ that minimizes the rms momentum for the two lines Γ = Er and Γ = 0.1Er.

Fig. 4
Fig. 4

Stationary momentum distributions W(p) for various Rabi frequencies Ω, in the case of a narrow line Γ = E r ( δ ¯ = 2.5 Γ ).

Fig. 5
Fig. 5

Comparison of the stationary momentum distribution obtained by (a) the quantum method and (b) a Monte Carlo approach (Γ = Er, Ω = 0.2Γ, δ ¯ = 2.5 Γ).

Fig. 6
Fig. 6

Variations (for Ω ≪ Γ) of the stationary kinetic energy Ēκ with detuning δ for various recoil shifts. The curve with Er = 0 represents the predictions of usual semiclassical molasses theory [Eq. (4.9)].

Fig. 7
Fig. 7

Variations for Ω ≪ Γ of the stationary kinetic energy Ēκ with the ratio Γ/Er. The detuning is always chosen in order to minimize Ēκ.

Equations (93)

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

Ē κ = 1 2 m v 2 ¯ 1 4 Γ E r = 2 k 2 2 m .
δ = ω L ω A 1.7 k 2 m ,
Ē κ = p 2 ¯ 2 m 0.5 E r Γ .
( p ) = { | g , p , | e + , p + k , | e , p k } .
π g ( p ) = g , p | ρ | g , p , π + ( p ) = e + , p + k | ρ | e + , p + k , ρ g + ( p ) = g , p | ρ | e + , p + k exp ( i ω L t ) , ρ + ( p ) = e + , p + k | ρ | e , p k ,
π g ( p ) = g , p | ρ | g , p = + d p x d p y g , p x , p y , p z = p | ρ | g , p x , p y , p z = p .
H 0 = P z 2 2 m + ω A i = 0 , + , | e i e i | ;
V = [ D + + ( Z ) exp ( i ω L t ) + D ( Z ) exp ( i ω L t ) ] ;
[ ρ ˙ g + ( p ) ] H 0 = i ( ω L ω A k p m k 2 2 m ) ρ g + ( p ) .
δ = ω L ω A
δ ¯ = ω L ω A k 2 2 m .
[ ρ ˙ + ( p ) ] H 0 = 2 i k p m ρ + ( p ) , ρ + ( p ) = [ ρ + ( p ) ] * .
D = d ( | g e + | u + + | g e | u + | g e 0 | u z ) , D + = ( D ) + ,
u + = 1 2 ( u x + i u y ) , u = 1 2 ( u x i u y ) .
+ ( Z ) = 0 2 ( u + e i k z + u e i k z ) , ( Z ) = [ + ( Z ) ] + ,
Ω = d 0 ,
e ikz | p = | p + k ,
V | g , p = Ω 2 ( | e + , p + k + | e , p k ) , V | e ± , p ± k = Ω 2 | g , p ,
[ π ˙ + ( p ) ] V = i Ω 2 [ ρ + g ( p ) ρ g + ( p ) ] , [ π ˙ + ( p ) ] V = i Ω 2 [ ρ + g ( p ) ρ g + ( p ) ] i Ω 2 [ ρ g ( p ) ρ g ( p ) ] , [ ρ ˙ g + ( p ) ] V = i Ω 2 [ π g ( p ) π + ( p ) ] i Ω 2 ρ + ( p ) , [ ρ ˙ + ( p ) ] V = i Ω 2 [ ρ + g ( p ) ρ g ( p ) ] .
[ π ˙ g ( p ) ] SE = Γ k k d p N ( p ) [ π + ( p + p k ) + π ( p + p + k ) ] = Γ [ π ¯ + ( p k ) + π ¯ ( p + k ) ] ,
N ( p ) = 3 8 k ( 1 + p 2 2 k 2 ) ,
π ¯ ± ( p ) = k k d p N ( p ) π ± ( p + p ) .
[ π ˙ + ( p ) ] SE = Γ π + ( p ) , [ ρ ˙ + ( p ) ] SE = Γ ρ + ( p ) , [ ρ ˙ g + ( p ) ] SE = Γ 2 ρ g + ( p ) .
π ˙ g ( p ) = Γ [ π ¯ + ( p k ) + π ¯ ( p + k ) ] i Ω 2 [ ρ + g ( p ) ρ g + ( p ) ] i Ω 2 [ ρ g ( p ) ρ g ( p ) ] , π ˙ + ( p ) = Γ π + ( p ) + i Ω 2 [ ρ + g ( p ) ρ g + ( p ) ] ρ ˙ g + ( p ) = [ i ( δ ¯ k p m ) + Γ 2 ] ρ g + ( p ) + i Ω 2 [ π g ( p ) π + ( p ) ] i Ω 2 ρ + ( p ) , ρ ˙ + ( p ) = ( 2 i k p m + Γ ) ρ + ( p ) + i Ω 2 [ ρ + g ( p ) ρ g ( p ) ] ,
π ˙ g ( p ) + π ˙ + ( p ) + π ˙ ( p ) = Γ [ π + ( p ) + π ( p ) ] + Γ [ π ¯ + ( p k ) + π ¯ ( p + k ) ] .
W ( p ) = π g ( p ) + π + ( p k ) + π ( p + k ) .
Ē κ 7 Γ 40
Ē κ 0.5 E r .
C m k 2 Γ 2 Ω 2
C proportional to E r 2 Γ ( Ω ) 2
t p 1 γ γ + k p 3 2 Γ m 2 Ω 2 ,
γ ± = Γ Ω 2 / 4 ( δ ¯ k p / m ) 2 + Γ 2 / 4
ϕ = t p + t p k + t p 2 k + + t k 1 2 Γ ( p 2 / 2 m ) 2 ( Ω ) 2 .
π ± ( p ) = Ω 2 / 4 Γ 2 / 4 + ( δ ¯ k p m ) 2 π g ( p ) .
π + ( p ) + π ( p ) = π ¯ + ( p k ) + π ¯ ( p + k ) .
π g ( p ) = π g ( p ) ,
+ d p p 2 [ π + ( p ) + π ( p ) ] = + d p p 2 k k d p N ( p ) × [ π + ( p + p k ) + π ( p + p + k ) ] .
p ( γ γ + ) ( p ) = 7 10 k ( γ + + γ ) ( p ) ,
k k d p p 2 N ( p ) = 2 5 2 k 2 .
γ ( p ) γ + ( p ) Γ Ω 2 δ ¯ k p / m ( δ ¯ 2 + Γ 2 / 4 ) 2 ,
γ ( p ) + γ + ( p ) Γ Ω 2 / 2 δ ¯ 2 + Γ 2 / 4 .
Ē κ = p 2 2 m = 7 40 δ ¯ 2 + Γ 2 / 4 δ ¯ .
δ ¯ < 7 20 k 2 m ( 3 + 2 n ) .
π g normalizable if δ ¯ < 21 20 k 2 m ,
p 2 exists if δ ¯ < 7 4 k 2 m .
γ ( p ) γ + ( p ) Γ Ω 2 δ ¯ k p / m ( δ ¯ 2 + Γ 2 / 4 ) 2 ( 1 + A p 2 ) , γ ( p ) + γ + ( p ) Γ Ω 2 / 2 δ ¯ 2 + Γ 2 / 4 ( 1 + B p 2 ) ,
A = k 2 m 2 2 δ ¯ 2 Γ 2 / 2 ( δ ¯ 2 + Γ 2 / 4 ) 2 ,
B = k 2 m 2 3 δ ¯ 2 Γ 2 / 4 ( δ ¯ 2 + Γ 2 / 4 ) 2 .
Ē κ = ( Ē κ ) cl 2 m ( 3 A B ) ( Ē κ ) cl 2 ,
Ē κ ( Ē κ ) cl 147 400 E r .
π ¯ ± ( p k ) = k k d p N ( p ) π ± ( p + p k ) = π ± ( p ) k ( π ± p ) ( p ) + 7 2 k 2 10 ( 2 π ± p 2 ) ( p ) + ,
p G ( p ) = 0 ,
G ( p ) = π ( p ) π + ( p ) + 7 k 10 p [ π ( p ) + π + ( p ) ] .
π g ( p ) = π 0 [ Γ 2 / 4 + ( δ ¯ k p / m ) 2 ] [ Γ 2 / 4 + ( δ ¯ + k p / m ) 2 ] ( Γ 2 / 4 + δ ¯ 2 + k 2 p 2 / m ) α ,
α = 1 10 δ ¯ 7 k 2 / m .
( Γ 2 / 4 + δ ¯ 2 + k 2 p 2 / m 2 ) α = exp [ α In ( Γ 2 / 4 + δ ¯ 2 + k 2 p 2 / m 2 ) ] .
ln ( Γ 2 / 4 + δ ¯ 2 + k 2 p 2 / m 2 ) = ln ( Γ 2 / 4 + δ ¯ 2 ) + k 2 p 2 / m 2 δ ¯ 2 + Γ 2 / 4 + .
π ( p ) = π ¯ 0 exp ( E κ 2 Ē κ ) ,
( d ρ / d t ) S . E . = Γ 2 [ ( Δ + Δ ) ρ + ρ ( Δ + Δ ) ] + 3 Γ 8 π d 2 Ω n ( Δ + ) + exp ( i k n R ) ρ × exp ( i k n R ) ( Δ + )
Δ = | g e | u + | g e + | u + + | g e 0 | u z , Δ + = ( Δ ) + .
( t g , p | ρ | g , p ) S . E . = 3 Γ 8 π d 2 Ω × n g , p + k n | ( Δ + ) + ρ ( Δ + ) | g , p + k n .
= + 1 u + + 1 u + 0 u z ,
( t g , p | ρ | g , p ) S . E . = 3 Γ 8 π q q d 2 Ω n q q * , e q , p + k n | ρ | e q , p + k n ,
( π g t ) SE ( p ) = 3 Γ 8 π q q d 2 Ω n q q * , e q , p z = p + k cos θ | ρ | e q , p z = p + k cos θ ,
n q q * = δ q q n q n q * ,
0 2 π d ϕ n q q * = 8 π 3 δ q q N q ( θ ) ,
N ± ( θ ) = 3 4 1 + cos 2 θ 2 , N 0 ( θ ) = ¾ sin 2 θ .
p = k cos θ .
( π g t ) SE ( p ) = Γ k k d p N ( p ) × [ π + ( p k + p ) + π ( p + k + p ) ] ,
N ( p ) = N ± ( θ ) k = 3 8 k [ 1 + ( p k ) 2 ] ,
p 2 p 2 g = d p p 2 π g ( p ) d p π g ( p ) .
R g ( q ) = k π g ( k q ) , R ± ( q ) = k π ± ( k q ) .
D = δ ¯ k 2 / 2 m ( renormalized detuning measured in units of recoil shift ) , G = 1 2 δ ¯ + i Γ / 2 k 2 / 2 m ,
( Ω 4 E r ) 2 R g ( q ) = | G q | 2 R + ( q ) = | G + q | 2 R ( q ) .
R + ( q ) + R ( q ) = 1 + 1 d q N ¯ ( q ) [ R + ( q + q 1 ) + R ( q + q + 1 ) ] ,
N ¯ ( q ) = ( 1 + q 2 )
R g ( q ) = R g ( q ) , R + ( q ) = R ( q ) .
R n = d q R + ( q ) q n / n ! d q R + ( q ) .
p 2 = 2 k 2 4 ! R 4 3 ! D R 3 + 2 ! | G | 2 R 2 2 ! R 2 D R 1 + | G | 2 .
( 2 n + 1 ) R 2 n + 1 D R 2 n + | G | 2 2 n R 2 n 1 = 0 .
j = 0 2 n + 1 R j L 2 n j + 2 = 0 ,
L n = 1 n ! 1 + 1 ( 1 + q ) n N ¯ ( q ) d q .
( 2 n + 1 ) R 2 n + 1 D R 2 n = | G | 2 2 n R 2 n 1 , R 2 n + 1 + L 2 R 2 n = j = 0 2 n 1 R j L 2 n j + 2 .
R 1 = L 2 .
R 2 = | 3 L 2 | G | 2 / 2 1 L 2 L 3 L 4 | 3 L 2 + D .
p 2 = 1 4 ( L 2 k ) 2 [ D 3 + 1 + g 2 D 5 + 24 10 α ( D 3 ) R ( D , g ) D 5 ] .
R ( D , g ) = 4 α [ 2 g 2 + 3 ( D 2 ) 2 ] + 20 β ( D 2 ) [ g 2 + ( D 2 ) 2 ] + 24 α
D = D / L 2 , g = Γ / ( 2 E r L 2 )
α = 1 / 3 ( L 2 L 3 L 4 ) / L 2 3 , β = 1 / 5 + [ L 2 L 5 L 6 + ( L 2 2 L 3 ) ( L 2 L 3 L 4 ) ] / L 2 5 .
Ē κ 1 4 L 2 2 E r ( D 3 + 1 + g 2 D 5 ) .
δ ¯ = 6 L 2 E r = 4.2 E r
( Ē κ ) inf approx = L 2 2 E r = 0.49 E r .
δ ¯ 4.45 E r , ( Ē κ ) inf 0.53 E r .

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