Abstract

We present two cooling mechanisms that lead to temperatures well below the Doppler limit. These mechanisms are based on laser polarization gradients and work at low laser power when the optical-pumping time between different ground-state sublevels becomes long. There is then a large time lag between the internal atomic response and the atomic motion, which leads to a large cooling force. In the simple case of one-dimensional molasses, we identify two types of polarization gradient that occur when the two counterpropagating waves have either orthogonal linear polarizations or orthogonal circular polarizations. In the first case, the light shifts of the ground-state Zeeman sublevels are spatially modulated, and optical pumping among them leads to dipole forces and to a Sisyphus effect analogous to the one that occurs in stimulated molasses. In the second case (σ+σ configuration), the cooling mechanism is radically different. Even at very low velocity, atomic motion produces a population difference among ground-state sublevels, which gives rise to unbalanced radiation pressures. From semiclassical optical Bloch equations, we derive for the two cases quantitative expressions for friction coefficients and velocity capture ranges. The friction coefficients are shown in both cases to be independent of the laser power, which produces an equilibrium temperature proportional to the laser power. The lowest achievable temperatures then approach the one-photon recoil energy. We briefly outline a full quantum treatment of such a limit.

© 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. D. Wineland, W. Itano, Phys. Rev. A 20, 1521 (1979);V. S. Letokhov, V. G. Minogin, Phys. Rev. 73, 1 (1981).
    [CrossRef]
  4. S. Stenholm, Rev. Mod. Phys. 58, 699 (1986).
    [CrossRef]
  5. S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, A. Ashkin, Phys. Rev. Lett. 55, 48 (1985).
    [CrossRef] [PubMed]
  6. D. Sesko, C. G. Fan, C. E. Wieman, J. Opt. Soc. Am. B 5, 1225 (1988).
    [CrossRef]
  7. P. Lett, R. Watts, C. Westbrook, W. D. Phillips, P. Gould, H. Metcalf, Phys. Rev. Lett. 61, 169 (1988).
    [CrossRef] [PubMed]
  8. Y. Shevy, D. S. Weiss, S. Chu, in Proceedings of the Conference on Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989);Y. Shevy, D. S. Weiss, P. J. Ungar, S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
    [CrossRef] [PubMed]
  9. J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Proceedings of the 11th Conference on Atomic Physics, S. Harsche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).
  10. S. Chu, D. S. Weiss, Y. Shevy, P. J. Ungar, in Proceedings of the 11th Conference on Atomic Physics, S. Harsche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).
  11. We restrict ourselves here to neutral atoms. Note that for trapped ions, mechanisms overcoming the Doppler limit were also proposed. They involve Raman two-photon processes:H. Dehmelt, G. Janik, W. Nagourney, Bull. Am. Phys. Soc. 30, 612 (1985);P. E. Toschek, Ann. Phys. (Paris) 10, 761 (1985);M. Lindberg, J. Javanainen, J. Opt. Soc. Am. B 3, 1008 (1986).
    [CrossRef]
  12. J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
    [CrossRef]
  13. J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
    [CrossRef]
  14. Other consequences of long atomic pumping times are described inW. Gawlik, J. Kowalski, F. Träger, M. Vollmer, J. Phys. B 20, 997 (1987).
    [CrossRef]
  15. J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980).
    [CrossRef]
  16. J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985).
    [CrossRef]
  17. A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, C. Cohen-Tannoudji, Phys. Rev. Lett. 57, 1688 (1986).
    [CrossRef] [PubMed]
  18. 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]
  19. Y. Castin, H. Wallis, J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989).
    [CrossRef]
  20. E. Arimondo, A. Bambini, S. Stenholm, Phys. Rev. A 24, 898 (1981).
    [CrossRef]
  21. J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 17, 4577 (1984).
    [CrossRef]
  22. J.-C. Lehmann, C. Cohen-Tannoudji, C. R. Acad. Sci. 258, 4463 (1964).
  23. J. Dupont-Roc, S. Haroche, C. Cohen-Tannoudji, Phys. Lett. 28A, 638 (1969).
  24. M. Lombardi, C. R. Acad. Sci. 265, 191 (1967);J. Phys. 30, 631 (1969).
  25. C. Cohen-Tannoudji, J. Dupont-Roc, Opt. Commun. 1, 184 (1969).
    [CrossRef]
  26. C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, R. Balian, S. Haroche, S. Liberman, eds. (North-Holland, Amsterdam, 1977).
  27. We also reincluded in the definition of the excited states the phase factors exp(±iπ/4) that appear when Eq. (2.7) is inserted into Eq. (4.1).
  28. J. Dalibard, A. Heidmann, C. Salomon, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer-Verlag, Berlin1987), p. 196.
    [CrossRef]
  29. K. Mølmer, Y. Castin, J. Phys. B (to be published).
  30. D. S. Weiss, E. Riis, Y. Shevy, P. J. Ungar, S. Chu, J. Opt. Soc. Am. B 6, 2072 (1989).
    [CrossRef]
  31. Y. Castin, K. Mølmer, J. Dalibard, C. Cohen-Tannoudji, in Proceedings of the Ninth International Conference on Laser Spectroscopy, M. Feld, A. Mooradian, J. Thomas, eds. (Springer-Verlag, Berlin, 1989).
  32. E. Arimondo, A. Aspect, R. Kaiser, C. Salomon, N. Vansteenkiste, Laboratoire de Spectroscopic Hertzienne Ecole Normale Supérieure, Université Paris VI, 24 Rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

1989 (2)

1988 (3)

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]

D. Sesko, C. G. Fan, C. E. Wieman, J. Opt. Soc. Am. B 5, 1225 (1988).
[CrossRef]

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

1987 (1)

Other consequences of long atomic pumping times are described inW. Gawlik, J. Kowalski, F. Träger, M. Vollmer, J. Phys. B 20, 997 (1987).
[CrossRef]

1986 (2)

A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, C. Cohen-Tannoudji, Phys. Rev. Lett. 57, 1688 (1986).
[CrossRef] [PubMed]

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

1985 (4)

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

We restrict ourselves here to neutral atoms. Note that for trapped ions, mechanisms overcoming the Doppler limit were also proposed. They involve Raman two-photon processes:H. Dehmelt, G. Janik, W. Nagourney, Bull. Am. Phys. Soc. 30, 612 (1985);P. E. Toschek, Ann. Phys. (Paris) 10, 761 (1985);M. Lindberg, J. Javanainen, J. Opt. Soc. Am. B 3, 1008 (1986).
[CrossRef]

J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985).
[CrossRef]

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

1984 (1)

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

1981 (1)

E. Arimondo, A. Bambini, S. Stenholm, Phys. Rev. A 24, 898 (1981).
[CrossRef]

1980 (2)

J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980).
[CrossRef]

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

1979 (1)

D. Wineland, W. Itano, Phys. Rev. A 20, 1521 (1979);V. S. Letokhov, V. G. Minogin, Phys. Rev. 73, 1 (1981).
[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).

1969 (2)

J. Dupont-Roc, S. Haroche, C. Cohen-Tannoudji, Phys. Lett. 28A, 638 (1969).

C. Cohen-Tannoudji, J. Dupont-Roc, Opt. Commun. 1, 184 (1969).
[CrossRef]

1967 (1)

M. Lombardi, C. R. Acad. Sci. 265, 191 (1967);J. Phys. 30, 631 (1969).

1964 (1)

J.-C. Lehmann, C. Cohen-Tannoudji, C. R. Acad. Sci. 258, 4463 (1964).

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]

E. Arimondo, A. Bambini, S. Stenholm, Phys. Rev. A 24, 898 (1981).
[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, S. Harsche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

E. Arimondo, A. Aspect, R. Kaiser, C. Salomon, N. Vansteenkiste, Laboratoire de Spectroscopic Hertzienne Ecole Normale Supérieure, Université Paris VI, 24 Rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

Ashkin, A.

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

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

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]

A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, C. Cohen-Tannoudji, Phys. Rev. Lett. 57, 1688 (1986).
[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, S. Harsche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

E. Arimondo, A. Aspect, R. Kaiser, C. Salomon, N. Vansteenkiste, Laboratoire de Spectroscopic Hertzienne Ecole Normale Supérieure, Université Paris VI, 24 Rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

J. Dalibard, A. Heidmann, C. Salomon, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer-Verlag, Berlin1987), p. 196.
[CrossRef]

Bambini, A.

E. Arimondo, A. Bambini, S. Stenholm, Phys. Rev. A 24, 898 (1981).
[CrossRef]

Bjorkholm, J. E.

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

Cable, A.

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

Castin, Y.

Y. Castin, H. Wallis, J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989).
[CrossRef]

K. Mølmer, Y. Castin, J. Phys. B (to be published).

Y. Castin, K. Mølmer, J. Dalibard, C. Cohen-Tannoudji, in Proceedings of the Ninth International Conference on Laser Spectroscopy, M. Feld, A. Mooradian, J. Thomas, eds. (Springer-Verlag, Berlin, 1989).

Chu, S.

D. S. Weiss, E. Riis, Y. Shevy, P. J. Ungar, S. Chu, J. Opt. Soc. Am. B 6, 2072 (1989).
[CrossRef]

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

Y. Shevy, D. S. Weiss, S. Chu, in Proceedings of the Conference on Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989);Y. Shevy, D. S. Weiss, P. J. Ungar, S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

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

Cohen-Tannoudji, C.

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]

A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, C. Cohen-Tannoudji, Phys. Rev. Lett. 57, 1688 (1986).
[CrossRef] [PubMed]

J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985).
[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. Dupont-Roc, S. Haroche, C. Cohen-Tannoudji, Phys. Lett. 28A, 638 (1969).

C. Cohen-Tannoudji, J. Dupont-Roc, Opt. Commun. 1, 184 (1969).
[CrossRef]

J.-C. Lehmann, C. Cohen-Tannoudji, C. R. Acad. Sci. 258, 4463 (1964).

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

C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, R. Balian, S. Haroche, S. Liberman, eds. (North-Holland, Amsterdam, 1977).

J. Dalibard, A. Heidmann, C. Salomon, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer-Verlag, Berlin1987), p. 196.
[CrossRef]

Y. Castin, K. Mølmer, J. Dalibard, C. Cohen-Tannoudji, in Proceedings of the Ninth International Conference on Laser Spectroscopy, M. Feld, A. Mooradian, J. Thomas, eds. (Springer-Verlag, Berlin, 1989).

Dalibard, J.

Y. Castin, H. Wallis, J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989).
[CrossRef]

A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, C. Cohen-Tannoudji, Phys. Rev. Lett. 57, 1688 (1986).
[CrossRef] [PubMed]

J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985).
[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, S. Harsche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

Y. Castin, K. Mølmer, J. Dalibard, C. Cohen-Tannoudji, in Proceedings of the Ninth International Conference on Laser Spectroscopy, M. Feld, A. Mooradian, J. Thomas, eds. (Springer-Verlag, Berlin, 1989).

J. Dalibard, A. Heidmann, C. Salomon, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer-Verlag, Berlin1987), p. 196.
[CrossRef]

Dehmelt, H.

We restrict ourselves here to neutral atoms. Note that for trapped ions, mechanisms overcoming the Doppler limit were also proposed. They involve Raman two-photon processes:H. Dehmelt, G. Janik, W. Nagourney, Bull. Am. Phys. Soc. 30, 612 (1985);P. E. Toschek, Ann. Phys. (Paris) 10, 761 (1985);M. Lindberg, J. Javanainen, J. Opt. Soc. Am. B 3, 1008 (1986).
[CrossRef]

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

Dupont-Roc, J.

J. Dupont-Roc, S. Haroche, C. Cohen-Tannoudji, Phys. Lett. 28A, 638 (1969).

C. Cohen-Tannoudji, J. Dupont-Roc, Opt. Commun. 1, 184 (1969).
[CrossRef]

Fan, C. G.

Gawlik, W.

Other consequences of long atomic pumping times are described inW. Gawlik, J. Kowalski, F. Träger, M. Vollmer, J. Phys. B 20, 997 (1987).
[CrossRef]

Gordon, J. P.

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

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]

Haroche, S.

J. Dupont-Roc, S. Haroche, C. Cohen-Tannoudji, Phys. Lett. 28A, 638 (1969).

Heidmann, A.

A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, C. Cohen-Tannoudji, Phys. Rev. Lett. 57, 1688 (1986).
[CrossRef] [PubMed]

J. Dalibard, A. Heidmann, C. Salomon, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer-Verlag, Berlin1987), p. 196.
[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);V. S. Letokhov, V. G. Minogin, Phys. Rev. 73, 1 (1981).
[CrossRef]

Janik, G.

We restrict ourselves here to neutral atoms. Note that for trapped ions, mechanisms overcoming the Doppler limit were also proposed. They involve Raman two-photon processes:H. Dehmelt, G. Janik, W. Nagourney, Bull. Am. Phys. Soc. 30, 612 (1985);P. E. Toschek, Ann. Phys. (Paris) 10, 761 (1985);M. Lindberg, J. Javanainen, J. Opt. Soc. Am. B 3, 1008 (1986).
[CrossRef]

Javanainen, J.

J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980).
[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, S. Harsche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

E. Arimondo, A. Aspect, R. Kaiser, C. Salomon, N. Vansteenkiste, Laboratoire de Spectroscopic Hertzienne Ecole Normale Supérieure, Université Paris VI, 24 Rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

Kowalski, J.

Other consequences of long atomic pumping times are described inW. Gawlik, J. Kowalski, F. Träger, M. Vollmer, J. Phys. B 20, 997 (1987).
[CrossRef]

Lehmann, J.-C.

J.-C. Lehmann, C. Cohen-Tannoudji, C. R. Acad. Sci. 258, 4463 (1964).

Lett, P.

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

Lombardi, M.

M. Lombardi, C. R. Acad. Sci. 265, 191 (1967);J. Phys. 30, 631 (1969).

Metcalf, H.

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

J. Dalibard, A. Heidmann, C. Salomon, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer-Verlag, Berlin1987), p. 196.
[CrossRef]

Mølmer, K.

K. Mølmer, Y. Castin, J. Phys. B (to be published).

Y. Castin, K. Mølmer, J. Dalibard, C. Cohen-Tannoudji, in Proceedings of the Ninth International Conference on Laser Spectroscopy, M. Feld, A. Mooradian, J. Thomas, eds. (Springer-Verlag, Berlin, 1989).

Nagourney, W.

We restrict ourselves here to neutral atoms. Note that for trapped ions, mechanisms overcoming the Doppler limit were also proposed. They involve Raman two-photon processes:H. Dehmelt, G. Janik, W. Nagourney, Bull. Am. Phys. Soc. 30, 612 (1985);P. E. Toschek, Ann. Phys. (Paris) 10, 761 (1985);M. Lindberg, J. Javanainen, J. Opt. Soc. Am. B 3, 1008 (1986).
[CrossRef]

Phillips, W. D.

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

Reynaud, S.

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

Riis, E.

Salomon, C.

A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, C. Cohen-Tannoudji, Phys. Rev. Lett. 57, 1688 (1986).
[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, S. Harsche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

J. Dalibard, A. Heidmann, C. Salomon, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer-Verlag, Berlin1987), p. 196.
[CrossRef]

E. Arimondo, A. Aspect, R. Kaiser, C. Salomon, N. Vansteenkiste, Laboratoire de Spectroscopic Hertzienne Ecole Normale Supérieure, Université Paris VI, 24 Rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

Schawlow, A.

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

Sesko, D.

Shevy, Y.

D. S. Weiss, E. Riis, Y. Shevy, P. J. Ungar, S. Chu, J. Opt. Soc. Am. B 6, 2072 (1989).
[CrossRef]

Y. Shevy, D. S. Weiss, S. Chu, in Proceedings of the Conference on Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989);Y. Shevy, D. S. Weiss, P. J. Ungar, S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

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

Stenholm, S.

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

E. Arimondo, A. Bambini, S. Stenholm, Phys. Rev. A 24, 898 (1981).
[CrossRef]

J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980).
[CrossRef]

Träger, F.

Other consequences of long atomic pumping times are described inW. Gawlik, J. Kowalski, F. Träger, M. Vollmer, J. Phys. B 20, 997 (1987).
[CrossRef]

Ungar, P. J.

D. S. Weiss, E. Riis, Y. Shevy, P. J. Ungar, S. Chu, J. Opt. Soc. Am. B 6, 2072 (1989).
[CrossRef]

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

E. Arimondo, A. Aspect, R. Kaiser, C. Salomon, N. Vansteenkiste, Laboratoire de Spectroscopic Hertzienne Ecole Normale Supérieure, Université Paris VI, 24 Rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

Vollmer, M.

Other consequences of long atomic pumping times are described inW. Gawlik, J. Kowalski, F. Träger, M. Vollmer, J. Phys. B 20, 997 (1987).
[CrossRef]

Wallis, H.

Watts, R.

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

Weiss, D. S.

D. S. Weiss, E. Riis, Y. Shevy, P. J. Ungar, S. Chu, J. Opt. Soc. Am. B 6, 2072 (1989).
[CrossRef]

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

Y. Shevy, D. S. Weiss, S. Chu, in Proceedings of the Conference on Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989);Y. Shevy, D. S. Weiss, P. J. Ungar, S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

Westbrook, C.

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

Wieman, C. E.

Wineland, D.

D. Wineland, W. Itano, Phys. Rev. A 20, 1521 (1979);V. S. Letokhov, V. G. Minogin, Phys. Rev. 73, 1 (1981).
[CrossRef]

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

Appl. Phys. (1)

J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980).
[CrossRef]

Bull. Am. Phys. Soc. (2)

We restrict ourselves here to neutral atoms. Note that for trapped ions, mechanisms overcoming the Doppler limit were also proposed. They involve Raman two-photon processes:H. Dehmelt, G. Janik, W. Nagourney, Bull. Am. Phys. Soc. 30, 612 (1985);P. E. Toschek, Ann. Phys. (Paris) 10, 761 (1985);M. Lindberg, J. Javanainen, J. Opt. Soc. Am. B 3, 1008 (1986).
[CrossRef]

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

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J.-C. Lehmann, C. Cohen-Tannoudji, C. R. Acad. Sci. 258, 4463 (1964).

M. Lombardi, C. R. Acad. Sci. 265, 191 (1967);J. Phys. 30, 631 (1969).

J. Opt. Soc. Am. B (4)

J. Phys. B (3)

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

Other consequences of long atomic pumping times are described inW. Gawlik, J. Kowalski, F. Träger, M. Vollmer, J. Phys. B 20, 997 (1987).
[CrossRef]

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

Opt. Commun. (2)

C. Cohen-Tannoudji, J. Dupont-Roc, Opt. Commun. 1, 184 (1969).
[CrossRef]

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

Phys. Lett. (1)

J. Dupont-Roc, S. Haroche, C. Cohen-Tannoudji, Phys. Lett. 28A, 638 (1969).

Phys. Rev. A (3)

E. Arimondo, A. Bambini, S. Stenholm, Phys. Rev. A 24, 898 (1981).
[CrossRef]

D. Wineland, W. Itano, Phys. Rev. A 20, 1521 (1979);V. S. Letokhov, V. G. Minogin, Phys. Rev. 73, 1 (1981).
[CrossRef]

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

Phys. Rev. Lett. (4)

A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, C. Cohen-Tannoudji, Phys. Rev. Lett. 57, 1688 (1986).
[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]

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]

Rev. Mod. Phys. (1)

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

Other (9)

Y. Shevy, D. S. Weiss, S. Chu, in Proceedings of the Conference on Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989);Y. Shevy, D. S. Weiss, P. J. Ungar, S. Chu, Phys. Rev. Lett. 62, 1118 (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, S. Harsche, 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, S. Harsche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989).

C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, R. Balian, S. Haroche, S. Liberman, eds. (North-Holland, Amsterdam, 1977).

We also reincluded in the definition of the excited states the phase factors exp(±iπ/4) that appear when Eq. (2.7) is inserted into Eq. (4.1).

J. Dalibard, A. Heidmann, C. Salomon, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer-Verlag, Berlin1987), p. 196.
[CrossRef]

K. Mølmer, Y. Castin, J. Phys. B (to be published).

Y. Castin, K. Mølmer, J. Dalibard, C. Cohen-Tannoudji, in Proceedings of the Ninth International Conference on Laser Spectroscopy, M. Feld, A. Mooradian, J. Thomas, eds. (Springer-Verlag, Berlin, 1989).

E. Arimondo, A. Aspect, R. Kaiser, C. Salomon, N. Vansteenkiste, Laboratoire de Spectroscopic Hertzienne Ecole Normale Supérieure, Université Paris VI, 24 Rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

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Figures (9)

Fig. 1
Fig. 1

The two types of polarization gradient in a 1-D molasses and the corresponding light-shifted ground-state sublevels for a Jg = 1/2 ↔ Je = 3/2 atomic transition, (a) σ+σ configuration: two counterpropagating waves, σ+and σ polarized, create a linear polarization that rotates in space, (b) lin ⊥ lin configuration: The two counterpropagating waves have orthogonal linear polarizations. The resulting polarization now has an ellipticity that varies in space: for z = 0 linear polarization along 1 = ( x + y ) / 2; for z = λ/8 σ polarization; for z = λ/4 linear polarization along 2 = ( x y ) / 2; for z = 3λ/8 σ+circular polarization … (c) Light-shifted ground-state sublevels for the σ+σ configuration: The light-shifted energies do not vary with z. (d) Light-shifted ground-state sublevels for the lin ⊥ lin configuration: The light-shifted energies oscillate in space with a period λ/2.

Fig. 2
Fig. 2

Atomic level scheme and Clebsh–Gordan coefficients for a Jg = 1/2 ↔ Je = 3/2 transition.

Fig. 3
Fig. 3

Light-shifted energies and steady-state populations (represented by filled circles) for a Jg = 1/2 ground state in the lin ⊥ lin configuration and for negative detuning. The lowest sublevel, having the largest negative light shift, is also the most populated one.

Fig. 4
Fig. 4

Atomic Sisyphus effect in the lin ⊥ lin configuration. Because of the time lag τp due to optical pumping, the atom sees on the average more uphill parts than downhill ones. The velocity of the atom represented here is such that υτpλ, in which case the atom travels overs λ in a relaxation time τp. The cooling force is then close to its maximal value.

Fig. 5
Fig. 5

Atomic level scheme and Clebsh–Gordan coefficients for a Jg = 1 ↔ Je = 2 transition.

Fig. 6
Fig. 6

Light-shifted ground-state sublevels of a Jg = 1 ↔ Je = 2 transition in the σ+σ configuration. The quantization axis Oy is chosen along the resulting linear laser polarization. The steady-state populations of these states (4/17, 9/17, 4/17) are represented by the filled circles. The double arrows represent couplings between Zeeman sublevels owing to the transformation to the moving rotating frame.

Fig. 7
Fig. 7

Variations with velocity υ of the force due to polarization gradients in the lin ⊥ lin configuration for a Jg = 1/2 ↔ Je = 3/2 transition (solid curve). The values of the parameters are Ω = 0.3Γ, δ = −Γ. The dotted curve shows sum of the two radiation pressure forces exerted independently by the two Doppler-shifted counterpropagating waves. The force due to polarization gradients leads to a much higher friction coefficient (slope at υ = 0) but acts on a much narrower velocity range.

Fig. 8
Fig. 8

Variations with velocity of the steady-state radiative force for a Jg = 1 ↔ Je = 2 transition in the σ+σ configuration (Ω = 0.25 Γ; δ = −0.5 Γ). The slope of the force near υ = 0 is very high (see also inset), showing that there is polarization gradient cooling. This new cooling force acts in the velocity range ∼ Δ′. Outside this range, the force is nearly equal to the Doppler force (shown by the dotted curve) calculated by neglecting all coherences between ground-state sublevels |gmz.

Fig. 9
Fig. 9

(a) Fictitious W atom: This atom is the simplest atomic-level scheme leading to extra cooling in the σ+σ configuration (b) Time evolution of the atomic velocity distribution for a W transition (δ = −Γ, Ω = 0.2 Γ, Γ = 200ℏh2k2/M as for a sodium atom). The initial velocity distribution is chosen to be Gaussian with a rms width of 16 recoil velocities. The evolution is calculated via the generalized optical Bloch equations including recoil, which permit a full quantum treatment of atomic motion. During the evolution, the velocity distribution becomes non-Gaussian, with a very narrow peak (HWHM ≃ 2 recoil velocities) superimposed upon a much broader background.

Equations (158)

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k Δ υ Γ ,
k B T D = Γ 2 .
k B T R = 2 k 2 2 M ,
= υ τ λ = k υ τ .
τ R = 1 Γ ,
= k υ τ R = k υ Γ .
τ p = 1 Γ ,
= k υ τ p = k υ Γ .
Ω Γ Γ Γ .
E ( z , t ) = + ( z ) exp ( i ω L t ) + c . c . ,
+ ( z ) = 0 e ikz + 0 e ikz .
= + = 1 2 ( x + i y ) ,
= = 1 2 ( x i y ) .
+ ( z ) = 1 2 ( 0 0 ) X i 2 ( 0 + 0 ) Y ,
X = x cos k z y sin k z ,
Y = x sin k z + y cos k z .
= x ,
= y .
+ ( z ) = 0 2 ( cos k z x + y 2 i sin k z y + x 2 ) .
δ = ω L ω A
F = α υ
k υ Γ ,
d W d t Δ τ p = Δ Γ .
d W d t F υ .
d W d t α υ 2 .
α k 2 Δ Γ .
Γ Ω 2 Γ / δ 2 ,
Δ Ω 2 / δ ,
α k 2 δ Γ .
Δ 0 = 4 / 3 Δ 1 .
z = υ t .
φ = k z = k υ t .
V rot = k υ J z .
Γ | Δ | .
k υ | Δ | ,
| g 0 y ¯ = | g 0 y + k υ 2 ( Δ 0 Δ 1 ) | g + 1 y + k υ 2 ( Δ 0 Δ 1 ) | g 1 y .
| g + 1 y ¯ = | g + 1 y + k υ 2 ( Δ 1 Δ 0 ) | g 0 y ,
| g 1 y ¯ = | g 1 y + k υ 2 ( Δ 1 Δ 0 ) | g 0 y .
g 0 | ¯ y ρ st | g 0 y ¯ = 9 / 17 ,
g + 1 | ¯ y ρ st | g + 1 y ¯ = g 1 | ¯ y ρ st | g 1 y ¯ = 4 / 17 .
g 0 | ¯ y J z | g 0 y ¯ = 2 k υ Δ 0 Δ 1 ,
g + 1 | ¯ y J z | g + 1 y ¯ = g 1 | ¯ y J z | g 1 y ¯ = k υ Δ 1 Δ 0 .
J z st = 2 k υ Δ 0 Δ 1 ( 9 17 2 17 2 17 ) = 40 17 k υ Δ 0 ,
Π + 1 Π 1 = 40 17 k υ Δ 0 .
υ > 0 , δ < 0 Π 1 > Π + 1 .
( Π + 1 Π 1 ) Γ k υ Γ Δ ,
F k 2 Γ Δ υ .
α k 2 Γ Δ ,
d W d t Γ ( Π + 1 Π 1 ) k υ Γ Δ h k 2 υ 2 ,
V = [ D + · + ( r ) exp ( i ω L t ) + D · ( r ) exp ( i ω L t ) ] .
V = Ω 2 sin k z [ | e 3 / 2 g 1 / 2 | + 1 3 | e 1 / 2 g 1 / 2 | ] × exp ( i ω L t ) + Ω 2 cos k z [ | e 3 / 2 g 1 / 2 | + 1 3 | e 1 / 2 g 1 / 2 | ] exp ( i ω L t ) + h . c .
Ω = 2 d 0 .
f = d V d z = k Ω 2 cos k z [ ρ ( g 1 / 2 , e 3 / 2 + 1 3 ρ ( g 1 / 2 , e 1 / 2 ) + c . c . ] + k Ω 2 sin k z [ ρ ( g 1 / 2 , e 3 / 2 ) + 1 3 ρ ( g 1 / 2 , e 1 / 2 ) + c . c . ,
ρ ( g i , e j ) = g i | ρ | e j exp ( i ω L t ) ,
ρ ˙ ( g 1 / 2 , e 3 / 2 ) = ( i δ + Γ 2 ) ρ ( g 1 / 2 , e 3 / 2 ) i Ω 6 cos k z e 1 / 2 | ρ | e 3 / 2 + i Ω 2 sin k z [ g 1 / 2 | ρ | g 1 / 2 e 3 / 2 | ρ | e 3 / 2 ] .
ρ ( g 1 / 2 , e 3 / 2 ) = Ω / 2 δ i Γ 2 Π 1 / 2 sin k z ,
Π ± 1 / 2 = g ± 1 / 2 | ρ | g ± 1 / 2 .
e 3 / 2 | ρ ˙ | e 3 / 2 = Γ e 3 / 2 | ρ | e 3 / 2 + i Ω 2 sin k z [ ρ ( e 3 / 2 , g 1 / 2 ) ρ ( g 1 / 2 , e 3 / 2 ) ] .
e 3 / 2 | ρ | e 3 / 2 = s 0 Π 1 / 2 sin 2 k z ,
s 0 = Ω 2 / 2 δ 2 + Γ 2 4 .
Π · 1 / 2 = Γ [ e 3 / 2 | ρ | e 3 / 2 + 2 3 e 1 / 2 | ρ | e 1 / 2 + 1 3 e 1 / 2 | ρ | e 1 / 2 ] + [ i Ω 2 ρ ( g 1 / 2 , e 3 / 2 ) sin k z + i Ω 6 ρ ( g 1 / 2 , e 1 / 2 ) cos k z + c . c . ] .
Π · i = 1 τ p [ Π i Π i st ( z ) ] ,
1 / τ p = Γ = 2 Γ s 0 / 9 ,
Π 1 / 2 st ( z ) = sin 2 k z , Π 1 / 2 st ( z ) = cos 2 k z .
f = 2 3 k δ s 0 ( Π 1 / 2 Π 1 / 2 ) sin 2 k z .
Δ E 1 / 2 = Δ + = δ s 0 ( sin 2 k z + 1 3 cos 2 k z ) = E 0 δ s 0 3 cos 2 k z , Δ E 1 / 2 = Δ = δ s 0 ( cos 2 k z + 1 3 sin 2 k z ) = E 0 + δ s 0 3 cos 2 k z ,
E 0 = 2 3 δ s 0 .
f ± 1 / 2 = d d z Δ E ± 1 / 2 = 2 3 k δ s 0 sin 2 k z ,
f = f 1 / 2 Π 1 / 2 + f 1 / 2 Π 1 / 2 .
f ( z , υ = 0 ) = 2 3 k δ s 0 sin 2 k z cos 2 k z = d U d z ,
U ( z ) = 1 6 δ s 0 sin 2 2 k z .
Π i ( z , υ ) = Π i st ( z ) υ τ p d Π i st d z + .
f ( z , υ ) = f ( z , υ = 0 ) υ τ p i = ± 1 / 2 f i d Π i st d z = f ( z , υ = 0 ) + 4 3 k 2 δ s 0 υ τ p sin 2 ( 2 k z ) .
f ¯ ( υ ) = α υ ,
α = 2 3 k 2 δ s 0 τ p ( δ < 0 )
α = 3 k 2 δ Γ ( δ < 0 ) .
Π ± 1 / 2 ( z , υ ) = 1 2 ( 1 cos 2 k z + 2 k υ τ p sin 2 k z 1 + 4 υ 2 τ p 2 ) .
f ¯ ( υ ) = α υ 1 + ( υ 2 / υ c 2 ) ,
k υ c = 1 / 2 τ p .
k B T = D p α .
D p ¯ = 7 10 2 k 2 Γ s 0 .
D p = 0 d τ [ f ( t ) f ( t + τ ) ¯ f ¯ 2 ] ,
f ( t ) f ( t + τ ) ¯ = i = ± 1 / 2 j = ± 1 / 2 f i f j P ( i , t ; j , t + τ ) ,
D p = 4 [ f 1 / 2 ( z ) ] 2 Π 1 / 2 st ( z ) Π 1 / 2 st ( z ) τ p = 2 2 k 2 δ 2 Γ s 0 sin 4 ( 2 k z ) .
D p ¯ = 3 4 2 k 2 δ 2 Γ s 0 .
| δ | Γ k B T = D p α | δ | 4 s 0
| δ | Γ k B T Ω 2 8 | δ | .
υ rms υ c Ω k 2 M | δ | 3 Γ 2 ,
υ rms k M | δ | Γ .
V = Ω 2 [ | g 1 e 2 | + 1 2 | g 0 e 1 | + 1 6 | g 1 e 0 | ] × exp [ i ( ω t k z ) ] + h . c . + Ω 2 [ | g 1 e 2 | + 1 2 | g 0 e 1 | + 1 6 | g 1 e 0 | ] × exp [ i ( ω + t k z ) ] + h . c . ,
Ω = 2 d 0 / [ identical to Eq . ( 4.3 ) ] , ω ± = ω L ± k υ .
f = i k Ω [ ρ ( e 2 , g 1 ) + 1 2 ρ ( e 1 , g 0 ) + 1 6 ρ ( e 0 , g 1 ) ] + c . c . i k Ω [ ρ ( e 2 , g 1 ) + 1 2 ρ ( e 1 , g 0 ) + 1 6 ρ ( e 0 , g 1 ) ] + c . c . ,
ρ ( e i ± 1 , g i ) = e i ± 1 | ρ | g i exp ( i ω t ) .
ρ · ( e 1 , g 0 ) = [ i ( δ k υ ) Γ 2 ] ρ ( e 1 , g 0 ) + i Ω 2 2 [ e 1 | ρ | e 1 + e 1 | ρ | e 1 exp ( 2 i k υ t ) g 0 | ρ | g 0 ] .
ρ ( e 1 , g 0 ) = Ω / 2 2 δ k υ + i Γ 2 Π 0 ,
Π i = g i | ρ | g i .
C r = Re [ g 1 | ρ | g 1 exp ( 2 i k υ t ) ] , C i = Im [ g 1 | ρ | g 1 exp ( 2 i k υ t ) ] .
f = k Γ 2 [ Π 1 ( s + s 6 ) + Π 0 ( s + s 2 ) + Π 1 ( s + 6 s ) + C r ( s + s 6 ) 1 3 C i ( s + δ k υ Γ + s δ + k υ Γ ) ] ,
s ± = Ω 2 / 2 ( δ k υ ) 2 + Γ 2 4 1
Π ± 1 = 1 34 ( 13 ± 240 k υ s 0 Γ δ Γ 5 Γ 2 + 4 δ 2 ) , Π 0 = 4 17 , C r = 5 34 , C i = 60 17 k υ s 0 Γ Γ 2 5 Γ 2 + 4 δ 2 ,
s 0 = Ω 2 δ 2 + ( Γ 2 / 4 ) .
f = k Γ 2 s 0 [ 5 6 ( Π 1 Π 1 ) 2 3 δ Γ C i ] .
f = α υ , α = 120 17 δ Γ 5 Γ 2 + 4 δ 2 k 2 .
Π ± 1 = 3 s ± ( s + 5 s ± ) 15 ( s + 2 + s 2 ) + 14 s + s , Π 0 = 8 s + s 15 ( s + 2 + s 2 ) + 14 s + s .
D p = D 1 + D 2 ,
D 1 = 18 170 2 k 2 Γ s 0 , D 2 = [ 36 17 1 1 + ( 4 δ 2 / 5 Γ 2 ) + 4 17 ] 2 k 2 Γ s 0 .
k B T = Ω 2 | δ | [ 29 300 + 254 75 Γ 2 / 4 δ 2 + ( Γ 2 / 4 ) ] .
k υ ¯ | Δ | Ω k 2 M δ υ ¯ k M .
( p ) = { | e 2 , p 2 k , | g 1 , p k , | e 0 , p , | g 1 , p + k , | e 2 , p + 2 k }
Doppler cooling { friction proportional to power capture range independent of power , Polarization gradient cooling { friction independent to power capture range proportional of power
V D · y = D x sin k υ t + D y cos k υ t .
T ( t ) = exp ( i k υ t J z / ) .
[ J z , D x ] = i D y , [ J z , D y ] = i D x ,
T ( t ) [ D x sin k υ t + D y cos k υ t ] T + ( t ) = D y .
i [ d T ( t ) d t ] T + ( t ) = k υ J z ,
| g ± 1 y = ± 1 2 | g + 1 z + i 2 | g 0 z 1 2 | g 1 z ,
| g 0 y = 1 2 | g + 1 z + 1 2 | g 1 z .
g + 1 | y V rot | g 0 y = g 0 | y V rot | g + 1 y = k υ / 2 ,
g 1 | y V rot | g 0 y = g 0 | y V rot | g 1 y = k υ / 2 .
| g ± 1 y ¯ = | g ± 1 y + | g 0 y g 0 | y V rot | g ± 1 y ( Δ 1 Δ 0 ) ,
| g 0 y ¯ = | g 0 y + | g + 1 y g + 1 | y V rot | g 0 y ( Δ 0 Δ 1 ) + | g 1 y g 1 | y V rot | g 0 y ( Δ 0 Δ 1 ) .
δ ¯ ± = δ ± k υ + i Γ 2 , δ ¯ 3 ± = δ ± 3 k υ + i Γ 2 .
ρ ( e 1 , g 0 ) = Ω 2 2 δ Π 0 , ρ ( e 1 , g 0 ) = Ω 2 2 δ + Π 0 .
ρ ( e 2 , g 1 ) = Ω 2 δ ¯ Π 1 , ρ ( e 2 , g 1 ) = Ω 2 δ ¯ + Π 1 , ρ ( e 0 , g 1 ) = Ω 2 6 δ ¯ + ( Π 1 + C r i C i ) , ρ ( e 0 , g 1 ) = Ω 2 6 δ ¯ ( Π 1 + C r + i C i ) , ρ ( e 2 , g 1 ) = Ω 2 δ ¯ 3 ( C r + i C i ) , ρ ( e 2 , g 1 ) = Ω 2 δ ¯ 3 + ( C r i C i )
ρ · ( e 2 , e 2 ) = Γ ρ ( e 2 , e 2 ) + i Ω 2 [ ρ ( e 2 , g 1 ) ρ ( g 1 , e 2 ) ] .
ρ ( e 2 , e 2 ) = s + 2 Π 1 ,
ρ ( e 1 , e 1 ) = s + 4 Π 0 ρ ( e 1 , e 1 ) = s 4 Π 0 , ρ ( e 2 , e 2 ) = s 2 Π 1 , ρ ( e 0 , e 0 ) = s 12 Π 1 + s + 12 Π 1 + ν 1 C r + μ 1 C i ,
μ 1 = [ ( δ + k υ ) s ( δ k υ ) s + ] / 6 Γ , ν 1 = ( s + + s ) / 12 .
ρ ( e 1 , e 1 ) = e 1 | ρ | e 1 exp ( 2 i k υ t )
ρ · ( e 1 , e 1 ) = ( Γ + 2 i k υ ) ρ ( e 1 , e 1 ) + i Ω 2 2 [ ρ ( e 1 , g 0 ) ρ ( g 0 , e 1 ) ] .
ρ ( e 1 , e 1 ) = ( μ 2 + i ν 2 ) Π 0 ,
μ 2 = 3 Γ 2 ν 1 Γ 2 μ 1 k υ Γ 2 + 4 k 2 υ 2 , ν 2 = 3 Γ 2 μ 1 Γ + 2 ν 1 k υ Γ 2 + 4 k 2 υ 2 .
ρ ( e 2 , e 0 ) + ρ ( e 0 , e 2 ) = ( e 2 | ρ | e 0 + e 0 | ρ | e 2 ) exp ( 2 i k υ t ) ,
ρ ( e 2 , e 0 ) + ρ ( e 0 , e 2 ) = 2 3 ( μ 2 + i ν 2 ) ( Π 1 + Π 1 ) + ( μ 4 + i ν 4 ) ( C r + i C i ) ,
μ 4 = 3 2 Γ Γ 2 + 4 k 2 υ 2 [ Γ ( ν 1 + ν 3 ) 2 k υ ( μ 1 + μ 3 ) ] , ν 4 = 3 2 Γ Γ 2 + 4 k 2 υ 2 [ 2 k υ ( ν 1 + ν 3 ) + Γ ( μ 1 + μ 3 ) ]
μ 3 = [ ( δ + 3 k υ ) s 3 ( δ 3 k υ ) s 3 + ] / 6 Γ , ν 3 = ( s 3 + + s 3 ) / 12 .
Π · 1 = Γ ρ ( e 2 , e 2 ) + Γ 2 ρ ( e 1 , e 1 ) + Γ 6 ρ ( e 0 , e 0 ) + i Ω 2 [ ρ ( g 1 , e 2 ) ρ ( e 2 , g 1 ) ] + i Ω 2 6 [ ρ ( g 1 , e 0 ) ρ ( e 0 , g 1 ) ] .
0 = 5 6 s Π 1 + 3 2 s + Π 0 + 1 6 s + Π 1 + ( 2 ν 1 s ) C r + 2 ( μ 1 δ + k υ Γ s ) C i .
0 = 1 6 s Π 1 + 3 2 s Π 0 5 6 s + Π 1 + ( 2 ν 1 s + ) C r + 2 ( μ 1 + δ k υ Γ s + ) C i .
1 = Π 1 + Π 0 + Π 1
ρ ( g 1 , g 1 ) = g 1 | ρ | g 1 exp ( 2 i k υ t ) = C r + i C i ,
ρ · ( g 1 , g 1 ) = 2 i k υ ρ ( g 1 , g 1 ) + Γ 2 ρ ( e 1 , e 1 ) + Γ 6 [ ρ ( e 2 , e 0 ) + ρ ( e 0 , e 2 ) ] + i Ω 2 6 [ ρ ( g 1 , e 0 ) ρ ( e 0 , g 1 ) ] + i Ω 2 [ ρ ( g 1 , e 2 ) ρ ( e 2 , g 1 ) ] .
0 = ( μ 2 s 8 ) Π 1 + 3 2 μ 2 Π 0 + ( μ 2 s + 8 ) Π 1 + μ 5 C r ν 5 C i ,
0 = ( ν 2 + s δ + k υ 4 Γ ) Π 1 + 3 2 ν 2 Π 0 + ( ν 2 s + δ k υ 4 Γ ) Π 1 + ν 5 C r + μ 5 C i ,
μ 5 = 3 2 μ 4 3 4 ν 1 9 ν 3 ,
ν 5 = 6 k υ Γ + 3 2 ν 4 + 3 2 μ 1 + 9 μ 3 .
s + = s = s 0 ,
0 = 5 6 Π 1 + 3 2 Π 0 + 1 6 Π 1 2 3 C r 2 δ Γ C i , 0 = 1 6 Π 1 + 3 2 Π 0 5 6 Π 1 2 3 C r + 2 δ Γ C i , 1 = Π 1 + Π 0 + Π 1 , 0 = 1 8 Π 1 + 3 8 Π 0 + 1 8 Π 1 5 4 C r + 6 k υ Γ s 0 C i , 0 = δ 4 Γ Π 1 δ 4 Γ Π 1 6 k υ Γ s 0 C r 5 4 C i .
5 s Π 1 + 9 s + Π 0 + s + Π 1 = 0 , s Π 1 + 9 s Π 0 5 s + Π 1 = 0 , Π 1 + Π 0 + Π 1 = 1 .
g 0 , p | ρ ˙ | g 0 , p = Γ 2 e 1 , p | ρ | e 1 , p ¯ + 2 Γ 3 e 0 , p | ρ | e 0 , p ¯ ¯ + Γ 2 e 1 , p | ρ | e 1 , p ¯ + i Ω 2 2 [ g 0 , p | ρ | e 1 , p + k + g 0 , p | ρ | e 1 , p k ] exp ( i ω L t ) + c . c .
σ ± : e 1 , p | ρ | e 1 , p ¯ = d p 3 8 k ( 1 + p 2 2 k 2 ) × e 1 , p + p | ρ | e 1 , p + p , π : e 0 , p | ρ | e 0 , p ¯ ¯ = d p 3 4 k ( 1 p 2 2 k 2 ) × e 0 , p + p | ρ | e 0 , p + p .
{ | e m , p + m k ; | g n , p + n k } , m ˙ = 0 , ± 1 , ± 2 , n = 0 , ± 1
Π m ( p ) = g m , p + m k | ρ | g m , p + m k , m = 0 , ± 1 , ( C r + i C i ) ( p ) = g 1 , p + k | ρ | g 1 , p k ,
Π ˙ 0 ( p ) = Γ s 0 2 { 1 9 Π 1 ( p ) ¯ ¯ + 1 4 [ Π 0 ( p k ) ¯ + Π 0 ( p + k ) ¯ ] Π 0 ( p ) + 1 9 Π 1 ( p ) ¯ ¯ + 2 9 C r ( p ) } , Π ˙ 1 ( p ) = Γ s 0 2 { Π 1 ( p k ) ¯ + 1 36 Π 1 ( p + k ) ¯ 7 6 Π 1 ( p ) + 1 4 Π 0 ( p ) ¯ ¯ + 1 36 Π 1 ( p + k ) ¯ + 1 18 C 1 ( p + k ) ¯ 1 6 C r ( p ) δ 3 Γ C i ( p ) } , C ˙ i ( p ) = Γ s 0 2 { δ 6 Γ [ Π 1 ( p ) Π 1 ( p ) ] + 1 6 [ C i ( p k ) ¯ + C i ( p + k ) ¯ ] 7 6 C i ( p ) } .
p C i = d p p C i ( p ) = 2 δ 5 Γ p Π 1 , p Π 1 = 36 17 k 1 1 + ( 4 δ 2 / 5 Γ 2 ) = p Π 1 ;
p Π 0 = p C r = 0 .
d d t p 2 = d d t ( p 2 Π 1 + p 2 Π 0 + p 2 Π 1 ) .
d d t p 2 = 2 k 2 Γ s 0 { 72 17 1 1 + ( 4 δ 2 / 5 Γ 2 ) + 58 85 } ,

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