Abstract

We present a summary of the results of a simple two-level theory of Doppler cooling in optical molasses and contrast it with the recent theories of multilevel, polarization-gradient cooling. The effects of single-photon recoil and of trapping in microscopic optical potential wells are also considered. Experiments are described in which the temperature of sodium atoms released from optical molasses is measured and found to be well below the Doppler-cooling limit. Measurements of the temperature dependence on many experimental parameters are found to be in good qualitative agreement with the new theories of polarization-gradient cooling.

© 1989 Optical Society of America

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    [CrossRef]
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  47. We set ℏω0 equal to the internal energy plus one recoil energy. Wineland and Itano3 use a different convention: our ω0 is equivalent to their ω0′.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  57. A. Anderson, M. Boshier, S. Harocke, E. Hinds, W. Jhe, D. Meschede, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 626.
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    [CrossRef]
  59. V. Balykin, V. Letokhov, A. Sidorov, Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984) [JETP Lett. 40, 1026 (1984)];V. Balykin, V. Letokhov, V. Minogin, Yu. Rozhdestvenskii, A. Sidorov, Zh. Eksp. Teor. Fiz. 90, 871 (1986) [Sov. Phys. JETP 63, 508 (1986)].
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  61. This technique was suggested to us by H. Metcalf.
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  63. The analysis presented in the next two paragraphs was not used in assigning TOF temperatures in Ref. 25. Instead, we assigned an uncertainty based on our then limited knowledge of the probe effect.
  64. S. Chu, Department of Physics, Stanford University, Stanford, California 94305 (personal communication);J. Dalibard, Laboratoire de Spectroscopie Hertzienne de l’Ecole Normale Supérieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).
  65. A. DeMarchi, Metrologia 18, 103 (1982);R. Beausoleil, T. Hänsch, Phys. Rev. A 33, 1661 (1986).Such an experiment was recently performed by the Stanford group [M. Kasevich, E. Riis, S. Chu, and R. DeVoe, Phys. Rev. Lett. 63, 612 (1989)], using atoms from a Zeeman-assisted radiation-pressure trap but without the launch mechanism that we suggest here.
    [CrossRef] [PubMed]

1989 (4)

Y. Shevy, D. Weiss, P. Ungar, S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

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

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

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

1988 (12)

C. Tanner, B. Masterson, C. Wieman, Opt. Lett. 13, 357 (1988).
[CrossRef] [PubMed]

V. Balykin, V. Letokhov, Yu. Ovchinnikov, A. Sidorov, S. Shul’ga, Opt. Lett. 13, 958 (1988).
[CrossRef] [PubMed]

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

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

P. Lett, P. Gould, W. Phillips, Hyperfine Interact. 44, 335 (1988).
[CrossRef]

E. Liang, C. Dermer, Opt. Commun. 65, 419 (1988).
[CrossRef]

C. Salomon, J. Dalibard, C. R. Acad. Sci. Paris 306, 1319 (1988).

H. Wallis, W. Ertmer, J. Phys. B 21, 2999 (1988).
[CrossRef]

P. Gould, P. Lett, P. Julienne, W. Phillips, H. Thorsheim, J. Weiner, Phys. Rev. Lett. 60, 788 (1988).
[CrossRef] [PubMed]

A. P. Kazantsev, G. Surdutovich, V. Yakovlev, Opt. Commun. 68, 103 (1988).
[CrossRef]

W. Phillips, P. Gould, P. Lett, Science 239, 877 (1988).
[CrossRef] [PubMed]

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

1987 (2)

W. Phillips, H. Metcalf, Sci. Am. 256, 50 (1987).
[CrossRef]

C. Salomon, J. Dalibard, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, Phys. Rev. Lett. 59, 1659 (1987).
[CrossRef] [PubMed]

1986 (3)

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

S. Chu, J. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).
[CrossRef] [PubMed]

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

1985 (6)

1984 (3)

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

V. Balykin, V. Letokhov, A. Sidorov, Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984) [JETP Lett. 40, 1026 (1984)];V. Balykin, V. Letokhov, V. Minogin, Yu. Rozhdestvenskii, A. Sidorov, Zh. Eksp. Teor. Fiz. 90, 871 (1986) [Sov. Phys. JETP 63, 508 (1986)].

Prog. Quantum Electron. 8 (3 & 4) (1984).

1982 (2)

W. Itano, D. Wineland, Phys. Rev. A 25, 35 (1982).
[CrossRef]

A. DeMarchi, Metrologia 18, 103 (1982);R. Beausoleil, T. Hänsch, Phys. Rev. A 33, 1661 (1986).Such an experiment was recently performed by the Stanford group [M. Kasevich, E. Riis, S. Chu, and R. DeVoe, Phys. Rev. Lett. 63, 612 (1989)], using atoms from a Zeeman-assisted radiation-pressure trap but without the launch mechanism that we suggest here.
[CrossRef] [PubMed]

1980 (3)

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

R. Cook, Phys. Rev. A 22, 1078 (1980).
[CrossRef]

A. Ashkin, Science 210, 1081 (1980).
[CrossRef] [PubMed]

1979 (2)

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

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

1978 (3)

W. Neuhauser, M. Hohenstatt, P. Toschek, H. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

D. Wineland, R. Drullinger, F. Walls, Phys. Rev. Lett. 40, 1639 (1978).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 40, 729 (1978).
[CrossRef]

1977 (1)

V. Letokhov, V. Minogin, B. Pavlik, Sov. Phys. JETP 45, 698 (1977).

1975 (2)

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

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

1921 (1)

See, for example, E. Buchwald, Ann. Phys. 66, 1 (1921).
[CrossRef]

Anderson, A.

A. Anderson, M. Boshier, S. Harocke, E. Hinds, W. Jhe, D. Meschede, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 626.

Arimondo, E.

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

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 199, and personal communications.

Ashkin, A.

S. Chu, J. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).
[CrossRef] [PubMed]

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

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

A. Ashkin, Science 210, 1081 (1980).
[CrossRef] [PubMed]

A. Ashkin, Phys. Rev. Lett. 40, 729 (1978).
[CrossRef]

Aspect, A.

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

C. Salomon, J. Dalibard, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, Phys. Rev. Lett. 59, 1659 (1987).
[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 Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 199, and personal communications.

Balykin, V.

V. Balykin, V. Letokhov, Yu. Ovchinnikov, A. Sidorov, S. Shul’ga, Opt. Lett. 13, 958 (1988).
[CrossRef] [PubMed]

V. Balykin, V. Letokhov, A. Sidorov, Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984) [JETP Lett. 40, 1026 (1984)];V. Balykin, V. Letokhov, V. Minogin, Yu. Rozhdestvenskii, A. Sidorov, Zh. Eksp. Teor. Fiz. 90, 871 (1986) [Sov. Phys. JETP 63, 508 (1986)].

Bjorkholm, J.

S. Chu, J. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).
[CrossRef] [PubMed]

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

S. Chu, M. Prentiss, A. Cable, J. Bjorkholm, in Laser Spectroscopy VIII, W. Persson, S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.

Blatt, R.

W. Ertmer, R. Blatt, J. Hall, M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef] [PubMed]

Boshier, M.

A. Anderson, M. Boshier, S. Harocke, E. Hinds, W. Jhe, D. Meschede, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 626.

Buchwald, E.

See, for example, E. Buchwald, Ann. Phys. 66, 1 (1921).
[CrossRef]

Cable, A.

S. Chu, J. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).
[CrossRef] [PubMed]

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

S. Chu, M. Prentiss, A. Cable, J. Bjorkholm, in Laser Spectroscopy VIII, W. Persson, S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.

Castin, Y.

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

Chu, S.

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

Y. Shevy, D. Weiss, P. Ungar, S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

S. Chu, J. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).
[CrossRef] [PubMed]

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

S. Chu, Department of Physics, Stanford University, Stanford, California 94305 (personal communication);J. Dalibard, Laboratoire de Spectroscopie Hertzienne de l’Ecole Normale Supérieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

S. Chu, M. Prentiss, A. Cable, J. Bjorkholm, in Laser Spectroscopy VIII, W. Persson, S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.

S. Chu, Y. Shevy, D. Weiss, P. Ungar, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 636.

Y. Shevy, D. Weiss, S. Chu, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 287.

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).
[CrossRef] [PubMed]

C. Salomon, J. Dalibard, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, Phys. Rev. Lett. 59, 1659 (1987).
[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, 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 Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 199, and personal communications.

Cook, R.

R. Cook, Phys. Rev. A 22, 1078 (1980).
[CrossRef]

Dalibard, J.

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

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

C. Salomon, J. Dalibard, C. R. Acad. Sci. Paris 306, 1319 (1988).

C. Salomon, J. Dalibard, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, Phys. Rev. Lett. 59, 1659 (1987).
[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, 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 Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 199, and personal communications.

J. Dalibard, Thèse de doctorat d’étatès Sciences Physique (Université de Paris, Paris, 1986).

J. Dalibard, Laboratoire de Spectroscopie Hertzienne de l’Ecole Normale Supérieure 24 rue Lhomond, F-75231 Paris Cedex 05, France (personal communication);D. Weiss, E. Riis, Y. Shevy, P. Ungar, S. Chu, J. Opt. Soc. Am. B 6, 2072 (1989);H. Metcalf, Department of Physics, State University of New York at Stony Brook, Stony Brook, New York11794 (personal communication).
[CrossRef]

Dehmelt, H.

W. Neuhauser, M. Hohenstatt, P. Toschek, H. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

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

DeMarchi, A.

A. DeMarchi, Metrologia 18, 103 (1982);R. Beausoleil, T. Hänsch, Phys. Rev. A 33, 1661 (1986).Such an experiment was recently performed by the Stanford group [M. Kasevich, E. Riis, S. Chu, and R. DeVoe, Phys. Rev. Lett. 63, 612 (1989)], using atoms from a Zeeman-assisted radiation-pressure trap but without the launch mechanism that we suggest here.
[CrossRef] [PubMed]

Dermer, C.

E. Liang, C. Dermer, Opt. Commun. 65, 419 (1988).
[CrossRef]

Drullinger, R.

D. Wineland, R. Drullinger, F. Walls, Phys. Rev. Lett. 40, 1639 (1978).
[CrossRef]

Ertmer, W.

H. Wallis, W. Ertmer, J. Phys. B 21, 2999 (1988).
[CrossRef]

W. Ertmer, R. Blatt, J. Hall, M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef] [PubMed]

Fan, C.

Feynman, R.

R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1963), Vol. I.

Gordon, J.

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

Gould, P.

P. Lett, P. Gould, W. Phillips, Hyperfine Interact. 44, 335 (1988).
[CrossRef]

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

W. Phillips, P. Gould, P. Lett, Science 239, 877 (1988).
[CrossRef] [PubMed]

P. Gould, P. Lett, P. Julienne, W. Phillips, H. Thorsheim, J. Weiner, Phys. Rev. Lett. 60, 788 (1988).
[CrossRef] [PubMed]

P. Gould, P. Lett, W. Phillips, in Laser Spectroscopy VIII, W. Persson, S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 64.

Hall, J.

W. Ertmer, R. Blatt, J. Hall, M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef] [PubMed]

Hänsch, T.

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

Harocke, S.

A. Anderson, M. Boshier, S. Harocke, E. Hinds, W. Jhe, D. Meschede, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 626.

Heidmann, A.

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

Hijmans, T.

T. Hijmans, O. Luiten, I. Setija, J. Walraven, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 275.

Hinds, E.

A. Anderson, M. Boshier, S. Harocke, E. Hinds, W. Jhe, D. Meschede, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 626.

Hohenstatt, M.

W. Neuhauser, M. Hohenstatt, P. Toschek, H. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

Hollberg, L.

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

Itano, W.

W. Itano, D. Wineland, Phys. Rev. A 25, 35 (1982).
[CrossRef]

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

Javanainen, J.

Jhe, W.

A. Anderson, M. Boshier, S. Harocke, E. Hinds, W. Jhe, D. Meschede, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 626.

Julienne, P.

P. Gould, P. Lett, P. Julienne, W. Phillips, H. Thorsheim, J. Weiner, Phys. Rev. Lett. 60, 788 (1988).
[CrossRef] [PubMed]

Kaiser, R.

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

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 199, and personal communications.

Kaivola, M.

Kazantsev, A. P.

A. P. Kazantsev, G. Surdutovich, V. Yakovlev, Opt. Commun. 68, 103 (1988).
[CrossRef]

A. P. Kazantsev, Phys. Rep.129, 75 (1985).
[CrossRef]

Leighton, R.

R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1963), Vol. I.

Letokhov, V.

V. Balykin, V. Letokhov, Yu. Ovchinnikov, A. Sidorov, S. Shul’ga, Opt. Lett. 13, 958 (1988).
[CrossRef] [PubMed]

V. Balykin, V. Letokhov, A. Sidorov, Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984) [JETP Lett. 40, 1026 (1984)];V. Balykin, V. Letokhov, V. Minogin, Yu. Rozhdestvenskii, A. Sidorov, Zh. Eksp. Teor. Fiz. 90, 871 (1986) [Sov. Phys. JETP 63, 508 (1986)].

V. Letokhov, V. Minogin, B. Pavlik, Sov. Phys. JETP 45, 698 (1977).

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

V. Letokhov, V. Minogin, Phys. Rep.73, 1 (1981).
[CrossRef]

Lett, P.

P. Lett, P. Gould, W. Phillips, Hyperfine Interact. 44, 335 (1988).
[CrossRef]

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

P. Gould, P. Lett, P. Julienne, W. Phillips, H. Thorsheim, J. Weiner, Phys. Rev. Lett. 60, 788 (1988).
[CrossRef] [PubMed]

W. Phillips, P. Gould, P. Lett, Science 239, 877 (1988).
[CrossRef] [PubMed]

P. Gould, P. Lett, W. Phillips, in Laser Spectroscopy VIII, W. Persson, S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 64.

Liang, E.

E. Liang, C. Dermer, Opt. Commun. 65, 419 (1988).
[CrossRef]

Luiten, O.

T. Hijmans, O. Luiten, I. Setija, J. Walraven, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 275.

Masterson, B.

Meschede, D.

A. Anderson, M. Boshier, S. Harocke, E. Hinds, W. Jhe, D. Meschede, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 626.

Metcalf, H.

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

C. Salomon, J. Dalibard, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, Phys. Rev. Lett. 59, 1659 (1987).
[CrossRef] [PubMed]

W. Phillips, H. Metcalf, Sci. Am. 256, 50 (1987).
[CrossRef]

W. Phillips, J. Prodan, H. Metcalf, J. Opt. Soc. Am. B 2, 1751 (1985).
[CrossRef]

This technique was suggested to us by H. Metcalf.

Minogin, V.

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

V. Letokhov, V. Minogin, B. Pavlik, Sov. Phys. JETP 45, 698 (1977).

V. Letokhov, V. Minogin, Phys. Rep.73, 1 (1981).
[CrossRef]

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

Neuhauser, W.

W. Neuhauser, M. Hohenstatt, P. Toschek, H. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

Nielsen, U.

Ovchinnikov, Yu.

Pavlik, B.

V. Letokhov, V. Minogin, B. Pavlik, Sov. Phys. JETP 45, 698 (1977).

Phillips, W.

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

P. Gould, P. Lett, P. Julienne, W. Phillips, H. Thorsheim, J. Weiner, Phys. Rev. Lett. 60, 788 (1988).
[CrossRef] [PubMed]

W. Phillips, P. Gould, P. Lett, Science 239, 877 (1988).
[CrossRef] [PubMed]

P. Lett, P. Gould, W. Phillips, Hyperfine Interact. 44, 335 (1988).
[CrossRef]

W. Phillips, H. Metcalf, Sci. Am. 256, 50 (1987).
[CrossRef]

W. Phillips, J. Prodan, H. Metcalf, J. Opt. Soc. Am. B 2, 1751 (1985).
[CrossRef]

P. Gould, P. Lett, W. Phillips, in Laser Spectroscopy VIII, W. Persson, S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 64.

Poulsen, O.

Prentiss, M.

S. Chu, M. Prentiss, A. Cable, J. Bjorkholm, in Laser Spectroscopy VIII, W. Persson, S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.

Prodan, J.

Purcell, E.

E. Purcell, Harvard University, Cambridge, Massachusetts 02138 (personal communication), cited in Ref. 3.

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).

Reynaud, S.

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

Riis, E.

Salomon, C.

C. Salomon, J. Dalibard, C. R. Acad. Sci. Paris 306, 1319 (1988).

C. Salomon, J. Dalibard, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, Phys. Rev. Lett. 59, 1659 (1987).
[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 Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 199, and personal communications.

Sands, M.

R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1963), Vol. I.

Schawlow, A.

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

Serimaa, O.

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

Sesko, D.

Setija, I.

T. Hijmans, O. Luiten, I. Setija, J. Walraven, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 275.

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. Weiss, P. Ungar, S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

Y. Shevy, D. Weiss, S. Chu, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 287.

S. Chu, Y. Shevy, D. Weiss, P. Ungar, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 636.

Shul’ga, S.

Sidorov, A.

V. Balykin, V. Letokhov, Yu. Ovchinnikov, A. Sidorov, S. Shul’ga, Opt. Lett. 13, 958 (1988).
[CrossRef] [PubMed]

V. Balykin, V. Letokhov, A. Sidorov, Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984) [JETP Lett. 40, 1026 (1984)];V. Balykin, V. Letokhov, V. Minogin, Yu. Rozhdestvenskii, A. Sidorov, Zh. Eksp. Teor. Fiz. 90, 871 (1986) [Sov. Phys. JETP 63, 508 (1986)].

Stenholm, S.

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

Surdutovich, G.

A. P. Kazantsev, G. Surdutovich, V. Yakovlev, Opt. Commun. 68, 103 (1988).
[CrossRef]

Tanner, C.

Thorsheim, H.

P. Gould, P. Lett, P. Julienne, W. Phillips, H. Thorsheim, J. Weiner, Phys. Rev. Lett. 60, 788 (1988).
[CrossRef] [PubMed]

Toschek, P.

W. Neuhauser, M. Hohenstatt, P. Toschek, H. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

Ungar, P.

Y. Shevy, D. Weiss, P. Ungar, S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

S. Chu, Y. Shevy, D. Weiss, P. Ungar, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 636.

Ungar, P. J.

Vansteenkiste, N.

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

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 199, and personal communications.

Wallis, H.

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

H. Wallis, W. Ertmer, J. Phys. B 21, 2999 (1988).
[CrossRef]

Walls, F.

D. Wineland, R. Drullinger, F. Walls, Phys. Rev. Lett. 40, 1639 (1978).
[CrossRef]

Walraven, J.

T. Hijmans, O. Luiten, I. Setija, J. Walraven, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 275.

Watts, R.

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

Weiner, J.

P. Gould, P. Lett, P. Julienne, W. Phillips, H. Thorsheim, J. Weiner, Phys. Rev. Lett. 60, 788 (1988).
[CrossRef] [PubMed]

Weiss, D.

Y. Shevy, D. Weiss, P. Ungar, S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

Y. Shevy, D. Weiss, S. Chu, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 287.

S. Chu, Y. Shevy, D. Weiss, P. Ungar, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 636.

Weiss, D. S.

Westbrook, C.

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

Wieman, C.

Wineland, D.

W. Itano, D. Wineland, Phys. Rev. A 25, 35 (1982).
[CrossRef]

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

D. Wineland, R. Drullinger, F. Walls, Phys. Rev. Lett. 40, 1639 (1978).
[CrossRef]

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

Yakovlev, V.

A. P. Kazantsev, G. Surdutovich, V. Yakovlev, Opt. Commun. 68, 103 (1988).
[CrossRef]

Zhu, M.

W. Ertmer, R. Blatt, J. Hall, M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef] [PubMed]

Ann. Phys. (1)

See, for example, E. Buchwald, Ann. Phys. 66, 1 (1921).
[CrossRef]

Bull. Am. Phys. Soc. (1)

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

C. R. Acad. Sci. Paris (1)

C. Salomon, J. Dalibard, C. R. Acad. Sci. Paris 306, 1319 (1988).

Hyperfine Interact. (1)

P. Lett, P. Gould, W. Phillips, Hyperfine Interact. 44, 335 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (7)

J. Phys. B (2)

H. Wallis, W. Ertmer, J. Phys. B 21, 2999 (1988).
[CrossRef]

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

Metrologia (1)

A. DeMarchi, Metrologia 18, 103 (1982);R. Beausoleil, T. Hänsch, Phys. Rev. A 33, 1661 (1986).Such an experiment was recently performed by the Stanford group [M. Kasevich, E. Riis, S. Chu, and R. DeVoe, Phys. Rev. Lett. 63, 612 (1989)], using atoms from a Zeeman-assisted radiation-pressure trap but without the launch mechanism that we suggest here.
[CrossRef] [PubMed]

Opt. Commun. (4)

A. P. Kazantsev, G. Surdutovich, V. Yakovlev, Opt. Commun. 68, 103 (1988).
[CrossRef]

E. Liang, C. Dermer, Opt. Commun. 65, 419 (1988).
[CrossRef]

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

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

Opt. Lett. (2)

Phys. Rev. A (4)

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

W. Itano, D. Wineland, Phys. Rev. A 25, 35 (1982).
[CrossRef]

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

R. Cook, Phys. Rev. A 22, 1078 (1980).
[CrossRef]

Phys. Rev. Lett. (12)

D. Wineland, R. Drullinger, F. Walls, Phys. Rev. Lett. 40, 1639 (1978).
[CrossRef]

W. Neuhauser, M. Hohenstatt, P. Toschek, H. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 40, 729 (1978).
[CrossRef]

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

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

Y. Shevy, D. Weiss, P. Ungar, S. Chu, Phys. Rev. Lett. 62, 1118 (1989).
[CrossRef] [PubMed]

S. Chu, J. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).
[CrossRef] [PubMed]

C. Salomon, J. Dalibard, A. Aspect, H. Metcalf, C. Cohen-Tannoudji, Phys. Rev. Lett. 59, 1659 (1987).
[CrossRef] [PubMed]

P. Gould, P. Lett, P. Julienne, W. Phillips, H. Thorsheim, J. Weiner, Phys. Rev. Lett. 60, 788 (1988).
[CrossRef] [PubMed]

W. Ertmer, R. Blatt, J. Hall, M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef] [PubMed]

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

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

Pis’ma Zh. Eksp. Teor. Fiz. (1)

V. Balykin, V. Letokhov, A. Sidorov, Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984) [JETP Lett. 40, 1026 (1984)];V. Balykin, V. Letokhov, V. Minogin, Yu. Rozhdestvenskii, A. Sidorov, Zh. Eksp. Teor. Fiz. 90, 871 (1986) [Sov. Phys. JETP 63, 508 (1986)].

Prog. Quantum Electron. (1)

Prog. Quantum Electron. 8 (3 & 4) (1984).

Rev. Mod. Phys. (1)

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

Sci. Am. (1)

W. Phillips, H. Metcalf, Sci. Am. 256, 50 (1987).
[CrossRef]

Science (2)

W. Phillips, P. Gould, P. Lett, Science 239, 877 (1988).
[CrossRef] [PubMed]

A. Ashkin, Science 210, 1081 (1980).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. Letokhov, V. Minogin, B. Pavlik, Sov. Phys. JETP 45, 698 (1977).

Other (21)

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

A. P. Kazantsev, Phys. Rep.129, 75 (1985).
[CrossRef]

V. Letokhov, V. Minogin, Phys. Rep.73, 1 (1981).
[CrossRef]

E. Purcell, Harvard University, Cambridge, Massachusetts 02138 (personal communication), cited in Ref. 3.

One could imagine a two-level atom achieved with a J= 0 → J= 1 transition with a magnetic field applied to split the J= 1 levels so much that only one transition, say the 0 → 0 transition, was in resonance with the laser. Two counterpropagating waves, polarized at ±45° to the magnetic field, would both drive the two-level transition, but they would create no standing wave in the usual sense of an intensity modulation. There would be, however, a periodic spatial variation in the coupling between the atom and the field since the local polarization from the superposition of the two waves would vary in space even though the intensity of the light would not. This variation would have the same kind of effect as would a true standing wave.

J. Dalibard, Thèse de doctorat d’étatès Sciences Physique (Université de Paris, Paris, 1986).

J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 199, and personal communications.

S. Chu, Y. Shevy, D. Weiss, P. Ungar, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 636.

R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1963), Vol. I.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).

P. Gould, P. Lett, W. Phillips, in Laser Spectroscopy VIII, W. Persson, S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 64.

S. Chu, M. Prentiss, A. Cable, J. Bjorkholm, in Laser Spectroscopy VIII, W. Persson, S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.

D. Wineland, W. Itano, J. Bergquist, J. Bollinger, eds., Trapped Ions and Laser Cooling, Natl. Bur. Stand. (U.S.) Tech Note1086 (U.S. Government Printing Office, Washington, D.C., 1985);Trapped Ions and Laser Cooling II, Natl. Inst. Stand. Technol. Technical Note1324 (U.S. Government Printing Office, Washington, D.C., 1988).

This technique was suggested to us by H. Metcalf.

Y. Shevy, D. Weiss, S. Chu, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 287.

The analysis presented in the next two paragraphs was not used in assigning TOF temperatures in Ref. 25. Instead, we assigned an uncertainty based on our then limited knowledge of the probe effect.

S. Chu, Department of Physics, Stanford University, Stanford, California 94305 (personal communication);J. Dalibard, Laboratoire de Spectroscopie Hertzienne de l’Ecole Normale Supérieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

We set ℏω0 equal to the internal energy plus one recoil energy. Wineland and Itano3 use a different convention: our ω0 is equivalent to their ω0′.

J. Dalibard, Laboratoire de Spectroscopie Hertzienne de l’Ecole Normale Supérieure 24 rue Lhomond, F-75231 Paris Cedex 05, France (personal communication);D. Weiss, E. Riis, Y. Shevy, P. Ungar, S. Chu, J. Opt. Soc. Am. B 6, 2072 (1989);H. Metcalf, Department of Physics, State University of New York at Stony Brook, Stony Brook, New York11794 (personal communication).
[CrossRef]

A. Anderson, M. Boshier, S. Harocke, E. Hinds, W. Jhe, D. Meschede, in Atomic Physics 11, S. Haroche, J. C. Gay, G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 626.

T. Hijmans, O. Luiten, I. Setija, J. Walraven, in Spin Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 275.

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Figures (24)

Fig. 1
Fig. 1

Force versus velocity for Δ = − Γ/2 and low intensity. The dashed curves are the individual forces due to the two counterpropagating beams, and the solid curve is the net force. Note the linear region around υ = 0.

Fig. 2
Fig. 2

Damping coefficient α versus detuning red of resonance for classical molasses (solid), standing wave (dashed), and σ+σ (dotted), for different intensities. a, I/I0 = 0.3; b, I/I0 = 1.0; c, I/I0 = 3.0; d, I/I0 = 10.0

Fig. 3
Fig. 3

Velocity distribution from a Monte Carlo simulation of three-dimensional sodium molasses with Δ = −Γ/2 and I/I0 ≪ 1. The straight line represents an exponential fit to the data, giving a temperature of 239 μK.

Fig. 4
Fig. 4

Vrms versus number of scattered photons for η = 104, Δ = −ηΓ, and υR = 3 cm/sec.

Fig. 5
Fig. 5

Velocity distributions for η = 0.1, 1, 5, 100, showing the appearance of a hole for large η. In all cases υR = 3 cm/sec. Δ = −1.1ηΓ, except for η = 0.1, where Δ = −Γ/2.

Fig. 6
Fig. 6

Minimum temperature versus recoil parameter. Note that the minimum temperature approaches TR for η > 1, and Γ/2 (solid line) for η < 1. For η ≫ 1, the temperature was derived by using only the low-velocity atoms of the distribution.

Fig. 7
Fig. 7

Fluorescence intensity of the optical molasses versus time in the R&R method (see text). The molasses beams are turned off for a period toff lasting, in this case, 20 msec. The data are from R&R measurements taken in our laboratory.

Fig. 8
Fig. 8

Molasses fluorescence versus time as the source of slow atoms is chopped on and off. The molasses is allowed to load for ∼4 sec, whereupon the atomic beam and its cooling laser are chopped off, allowing the molasses to decay.

Fig. 9
Fig. 9

Molasses lifetime versus laser detuning to the red of resonance. The open boxes are early experimental measurements made in our laboratory, while the solid curve is the prediction of Eq. (30) with I/I0 = 0.5.

Fig. 10
Fig. 10

Added decay rate of the molasses due to imbalance in one pair of beams versus the percent intensity imbalance. The filled circles and open boxes indicate two independent experimental data sets. The solid curve indicates the classical molasses predictions of Subsection 2.E with Δ = − 2Γ and I/I0 = 0.5. The dashed curves are from similar calculations made using the damping coefficients of the polarization-gradient forces of Section 3. The upper dashed curve is from the polarization-rotation damping of Eq. (34), and the lower dashed curve is from the ellipticity-gradient damping of Eq. (35).

Fig. 11
Fig. 11

TOF data and results of numerical calculations at 25 and 250 μK, all normalized to the same height. Δ = −2.5Γ. The data points show fluorescence intensity versus time, and the solid curves indicate the expected signal.

Fig. 12
Fig. 12

Block diagram of the optical system. The spatial filter was not present for most of the data. An electro-optic modulator is designated by EOM.

Fig. 13
Fig. 13

Sketch of the apparatus, showing the relative position of the Zeeman tuning magnet and the molasses region (not to scale).

Fig. 14
Fig. 14

Sodium energy-level diagram (not to scale) indicating the cycling transition used for laser cooling.

Fig. 15
Fig. 15

Temperature measured by TOF as a function of turn-off time tr. The quantity tr is the time that it takes to ramp down the molasses laser beam intensity with an AOM.

Fig. 16
Fig. 16

Temperature measured by TOF versus molasses laser power. The power was measured before the beginning of the shutoff. The AOM shutoff took 200 nsec (squares); the mechanical shutoff took 20–30 μsec (filled circles). The power indicated is the power in the molasses laser beam before it was split into thirds and retroreflected. Δ = −2Γ.

Fig. 17
Fig. 17

a, Molasses laser power as a function of time for the data in b. The mechanical shutter shuts off approximately 10 μsec after the AOM. b, Temperature versus Plow showing that low-power molasses beams cool the atoms despite a fast shut-off. Δ = −2Γ.

Fig. 18
Fig. 18

a, Molasses laser power versus time for the data in Fig. b. b, Temperature versus th, showing rapid heating of the atoms in very-low-power (0.14-mW) molasses.

Fig. 19
Fig. 19

Temperature versus magnetic field. The data were taken with a mechanical shutter. The two powers displayed were measured before the shutoff. Δ = − 2Γ.

Fig. 20
Fig. 20

Measured temperature versus laser detuning Δ, to the red of resonance. Also shown is the prediction of classical, low-power, one-dimensional molasses (solid curve), Eq. (12).

Fig. 21
Fig. 21

Molasses decay rate versus magnetic field. Δ = −2.5Γ; power, 10 mW.

Fig. 22
Fig. 22

Molasses decay rate versus laser power. Δ = −2.5Γ.

Fig. 23
Fig. 23

Time integral of the fluorescence intensity versus molasses temperature showing the differential sensitivity to velocity of the TOF method.

Fig. 24
Fig. 24

Fluorescence intensity from a probe versus time after a collective launch of 1 m/sec has been given to a 30-μK sample of molasses 2 cm below the probe.

Equations (44)

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F ± = ± k Γ 2 I / I 0 1 + I / I 0 + [ 2 ( Δ k υ ) / Γ ] 2 .
F = k Γ 2 I I 0 k υ Γ 16 Δ / Γ 1 + 8 Γ 2 ( Δ 2 + k 2 υ 2 ) + 16 Γ 4 ( Δ 2 k 2 υ 2 ) 2 .
F = 4 k I I 0 k υ ( 2 Δ / Γ ) [ 1 + ( 2 Δ / Γ ) 2 ] 2 .
α = 4 k 2 I I 0 ( 2 Δ / Γ ) [ 1 + ( 2 Δ / Γ ) 2 ] 2 .
( d E / d t ) cool = F υ = α υ 2 .
τ cool = E ( d E / d t ) cool = M / 2 α
τ damp = υ ( d υ / d t ) cool = M / α .
d ( p 2 ) / d t = 2 2 k 2 R = 2 D p ,
( d E / d t ) heat = 2 k 2 R / M = D p / M .
( d E / d t ) heat = 2 k 2 M Γ I / I 0 1 + ( 2 Δ / Γ ) 2 .
υ 2 = Γ 4 M 1 + ( 2 Δ / Γ ) 2 2 | Δ | / Γ .
k B T = D p α = Γ 4 1 + ( 2 Δ / Γ ) 2 2 | Δ | / Γ .
k B T min = Γ / 2 .
α = 4 k 2 I I 0 2 Δ / Γ [ 1 + 2 NI / I 0 + ( 2 Δ / Γ ) 2 ] 2 .
With fixed detuning : I I 0 = ( 2 Δ / Γ ) 2 + 1 2 N ,
With fixed intensity : 2 Δ Γ = ( 1 + 2 NI / I 0 3 ) 1 / 2 .
α = k 2 4 N for 2 Δ / Γ = 1 , I / I 0 = 1 / N .
R = Γ 2 2 NI / I 0 1 + 2 NI / I 0 + ( 2 Δ / Γ ) 2 ,
k B T = Γ 4 1 + 2 NI / I 0 + ( 2 Δ / Γ ) 2 2 | Δ | / Γ .
F = k Γ 2 I / I 0 1 + I / I 0 + { 2 [ Δ + k ( V + υ ) ] / Γ } 2 ,
F = M a k Γ 2 I / I 0 1 + I / I 0 + [ 2 ( Δ + k υ ) / Γ ] 2 ,
F = 2 k Γ I I 0 k υ ( 2 Δ / Γ ) [ 1 + I / I 0 + ( 2 Δ / Γ ) 2 ] 2 .
k B T = Γ 4 1 + I / I 0 + ( 2 Δ / Γ ) 2 2 | Δ | / Γ ,
α = k 2 ( 1 + 4 s ) 3 / 2 { 4 s ( 2 Δ / Γ ) 1 + ( 2 Δ / Γ ) 2 ( 1 + 2 s ) 2 Δ Γ [ 1 + 6 s + 6 s 2 ( 1 + 4 s ) 3 / 2 ] }
s = I / I 0 1 + ( 2 Δ / Γ ) 2 .
α = k 2 2 s 1 + 2 s 2 ( 2 Δ / Γ ) ( 2 Δ / Γ ) 2 + { 1 + ½ s [ ( 2 Δ / Γ ) 2 + 1 ] } 2 ,
D x = k B T α = D p N α 2 ,
r 2 = N x 2 = 2 N t D D x = 2 t D D p α 2 .
t D = 8 k 2 r 2 N Γ I I 0 ( 2 Δ / Γ ) 2 [ 1 + 2 NI / I 0 + ( 2 Δ / Γ ) 2 ] 3 .
t D = 4 k 2 r 2 27 N 2 Γ for 2 Δ / Γ = 1 , I / I 0 = 1 / 2 N .
n ( t ) = n 0 exp ( t / τ M ) , τ M = r 2 / π 2 D x .
υ drift = F ext / α .
F im = k Γ 2 I / I 0 1 + 2 NI / I 0 + ( 2 Δ / Γ ) 2 .
υ drift = Γ 8 k 1 + 2 NI / I 0 + ( 2 Δ / Γ ) 2 2 | Δ | / Γ .
α = 60 17 k 2 2 Δ / Γ 5 + ( 2 Δ / Γ ) 2 .
α = 3 2 k 2 2 Δ Γ .
k B T = Γ I / I 0 ( 2 | Δ | / Γ ) [ 29 300 + 254 75 1 1 + ( 2 Δ / Γ ) 2 ] ,
k B T = Γ I / I 0 8 ( 2 | Δ | / Γ ) .
δ ~ ( I / I 0 ) Γ 2 / Δ .
M υ rms 2 ~ ( I / I 0 ) ( Γ 2 / | Δ | ) .
M υ rms k .
M υ rms k ( 2 | Δ | / Γ ) .
F = α υ 1 + υ 2 / υ c 2 ,
k υ min / Γ = k / 2 M υ min = υ R / 2 υ min 1 ,

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