Abstract

We study spin relaxation of optically trapped atoms that is due to light scattering from the trap laser. We observe relaxation times greater than 2 s for ground-state hyperfine-level populations of 85Rb atoms trapped in an optical dipole force trap operating as much as 65 nm to the red of the D1 line. The measured relaxation rate can be more than 100 times slower than the atoms’ total spontaneous scatter rate from the trap laser. This enhancement in atomic ground-state lifetime is due to an interference effect in spontaneous Raman scattering far from atomic resonance.

© 1994 Optical Society of America

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  1. D. E. Pritchard, Prog. Quantum Electron. 8, 209 (1984)
    [Crossref]
  2. D. E. Pritchard, in Electronic and Atomic Collisions, D. C. Lorents, W. E. Meyerhof, J. R. Peterson, eds. (Elsevier, New York, 1986), p. 593.
  3. T. Walker, P. Feng, “Ultracold collisions,”Adv. At. Mol. Opt. Phys. (to be published).
  4. C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993).
    [Crossref] [PubMed]
  5. N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
    [Crossref] [PubMed]
  6. W. M. Itano, J. C. Bergquist, D. J. Wineland, Science 237, 612 (1987); K. Gibble, S. Chu, Metrologia 29, 201 (1992).
    [Crossref] [PubMed]
  7. K. Gibble, S. Chu, Phys. Rev. Lett. 70, 1771 (1993).
    [Crossref] [PubMed]
  8. A. Migdall, J. V. Prodan, W. D. Phillips, T. H. Bergman, H. J. Metcalf, Phys. Rev. Lett. 54, 2596 (1985); A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, D. E. Pritchard, Phys. Rev. Lett. 61, 2431 (1988); E. A. Cornell, C. Monroe, C. E. Wieman, Phys. Rev. Lett. 67, 2439 (1991).
    [Crossref] [PubMed]
  9. E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, Phys. Rev. Lett. 59,2631 (1987); C. Monroe, W. Swann, H. Robinson, C. Wieman, Phys. Rev. Lett. 65, 1571 (1990).
    [Crossref] [PubMed]
  10. W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, D. E. Pritchard, Phys. Rev. Lett. 70, 2253 (1993). The dark MOT has a slower relaxation rate than the ordinary MOT but is still several orders of magnitude faster than the present trap.
    [Crossref] [PubMed]
  11. J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980); S. Chu, J. E. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).
    [Crossref] [PubMed]
  12. S. Rolston, C. Gertz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. C. Spreeuw, C. I. Westbrook, Proc. Soc. Photo-Opt. Instrum. Eng. 1726, 205 (1992).
  13. J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. A 47, R4567 (1993).
    [Crossref] [PubMed]
  14. R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, 1983), Chap. 8, p. 315.
  15. C. Cohen-Tannoudji, Ann. Phys. (Paris) 7, 423, 469 (1962); W. Happer, Rev. Mod. Phys. 44, 169 (1972).
    [Crossref]

1993 (4)

C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993).
[Crossref] [PubMed]

K. Gibble, S. Chu, Phys. Rev. Lett. 70, 1771 (1993).
[Crossref] [PubMed]

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, D. E. Pritchard, Phys. Rev. Lett. 70, 2253 (1993). The dark MOT has a slower relaxation rate than the ordinary MOT but is still several orders of magnitude faster than the present trap.
[Crossref] [PubMed]

J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. A 47, R4567 (1993).
[Crossref] [PubMed]

1992 (1)

S. Rolston, C. Gertz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. C. Spreeuw, C. I. Westbrook, Proc. Soc. Photo-Opt. Instrum. Eng. 1726, 205 (1992).

1988 (1)

N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[Crossref] [PubMed]

1987 (2)

W. M. Itano, J. C. Bergquist, D. J. Wineland, Science 237, 612 (1987); K. Gibble, S. Chu, Metrologia 29, 201 (1992).
[Crossref] [PubMed]

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, Phys. Rev. Lett. 59,2631 (1987); C. Monroe, W. Swann, H. Robinson, C. Wieman, Phys. Rev. Lett. 65, 1571 (1990).
[Crossref] [PubMed]

1985 (1)

A. Migdall, J. V. Prodan, W. D. Phillips, T. H. Bergman, H. J. Metcalf, Phys. Rev. Lett. 54, 2596 (1985); A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, D. E. Pritchard, Phys. Rev. Lett. 61, 2431 (1988); E. A. Cornell, C. Monroe, C. E. Wieman, Phys. Rev. Lett. 67, 2439 (1991).
[Crossref] [PubMed]

1984 (1)

D. E. Pritchard, Prog. Quantum Electron. 8, 209 (1984)
[Crossref]

1980 (1)

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980); S. Chu, J. E. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).
[Crossref] [PubMed]

1962 (1)

C. Cohen-Tannoudji, Ann. Phys. (Paris) 7, 423, 469 (1962); W. Happer, Rev. Mod. Phys. 44, 169 (1972).
[Crossref]

Ashkin, A.

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980); S. Chu, J. E. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).
[Crossref] [PubMed]

Bergman, T. H.

A. Migdall, J. V. Prodan, W. D. Phillips, T. H. Bergman, H. J. Metcalf, Phys. Rev. Lett. 54, 2596 (1985); A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, D. E. Pritchard, Phys. Rev. Lett. 61, 2431 (1988); E. A. Cornell, C. Monroe, C. E. Wieman, Phys. Rev. Lett. 67, 2439 (1991).
[Crossref] [PubMed]

Bergquist, J. C.

W. M. Itano, J. C. Bergquist, D. J. Wineland, Science 237, 612 (1987); K. Gibble, S. Chu, Metrologia 29, 201 (1992).
[Crossref] [PubMed]

Cable, A.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, Phys. Rev. Lett. 59,2631 (1987); C. Monroe, W. Swann, H. Robinson, C. Wieman, Phys. Rev. Lett. 65, 1571 (1990).
[Crossref] [PubMed]

Chu, S.

K. Gibble, S. Chu, Phys. Rev. Lett. 70, 1771 (1993).
[Crossref] [PubMed]

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, Phys. Rev. Lett. 59,2631 (1987); C. Monroe, W. Swann, H. Robinson, C. Wieman, Phys. Rev. Lett. 65, 1571 (1990).
[Crossref] [PubMed]

Cline, R. A.

J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. A 47, R4567 (1993).
[Crossref] [PubMed]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, Ann. Phys. (Paris) 7, 423, 469 (1962); W. Happer, Rev. Mod. Phys. 44, 169 (1972).
[Crossref]

Cornell, E. A.

C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993).
[Crossref] [PubMed]

Davis, K. B.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, D. E. Pritchard, Phys. Rev. Lett. 70, 2253 (1993). The dark MOT has a slower relaxation rate than the ordinary MOT but is still several orders of magnitude faster than the present trap.
[Crossref] [PubMed]

Doyle, J. M.

N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[Crossref] [PubMed]

Feng, P.

T. Walker, P. Feng, “Ultracold collisions,”Adv. At. Mol. Opt. Phys. (to be published).

Gertz, C.

S. Rolston, C. Gertz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. C. Spreeuw, C. I. Westbrook, Proc. Soc. Photo-Opt. Instrum. Eng. 1726, 205 (1992).

Gibble, K.

K. Gibble, S. Chu, Phys. Rev. Lett. 70, 1771 (1993).
[Crossref] [PubMed]

Gordon, J. P.

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980); S. Chu, J. E. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).
[Crossref] [PubMed]

Greytak, T. J.

N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[Crossref] [PubMed]

Heinzen, D. J.

J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. A 47, R4567 (1993).
[Crossref] [PubMed]

Helmerson, K.

S. Rolston, C. Gertz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. C. Spreeuw, C. I. Westbrook, Proc. Soc. Photo-Opt. Instrum. Eng. 1726, 205 (1992).

Hess, H. F.

N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[Crossref] [PubMed]

Itano, W. M.

W. M. Itano, J. C. Bergquist, D. J. Wineland, Science 237, 612 (1987); K. Gibble, S. Chu, Metrologia 29, 201 (1992).
[Crossref] [PubMed]

Jessen, P. S.

S. Rolston, C. Gertz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. C. Spreeuw, C. I. Westbrook, Proc. Soc. Photo-Opt. Instrum. Eng. 1726, 205 (1992).

Joffe, M. A.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, D. E. Pritchard, Phys. Rev. Lett. 70, 2253 (1993). The dark MOT has a slower relaxation rate than the ordinary MOT but is still several orders of magnitude faster than the present trap.
[Crossref] [PubMed]

Ketterle, W.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, D. E. Pritchard, Phys. Rev. Lett. 70, 2253 (1993). The dark MOT has a slower relaxation rate than the ordinary MOT but is still several orders of magnitude faster than the present trap.
[Crossref] [PubMed]

Kleppner, D.

N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[Crossref] [PubMed]

Kochanski, G. P.

N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[Crossref] [PubMed]

Lett, P. D.

S. Rolston, C. Gertz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. C. Spreeuw, C. I. Westbrook, Proc. Soc. Photo-Opt. Instrum. Eng. 1726, 205 (1992).

Loudon, R.

R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, 1983), Chap. 8, p. 315.

Martin, A.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, D. E. Pritchard, Phys. Rev. Lett. 70, 2253 (1993). The dark MOT has a slower relaxation rate than the ordinary MOT but is still several orders of magnitude faster than the present trap.
[Crossref] [PubMed]

Masuhara, N.

N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[Crossref] [PubMed]

Metcalf, H. J.

A. Migdall, J. V. Prodan, W. D. Phillips, T. H. Bergman, H. J. Metcalf, Phys. Rev. Lett. 54, 2596 (1985); A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, D. E. Pritchard, Phys. Rev. Lett. 61, 2431 (1988); E. A. Cornell, C. Monroe, C. E. Wieman, Phys. Rev. Lett. 67, 2439 (1991).
[Crossref] [PubMed]

Migdall, A.

A. Migdall, J. V. Prodan, W. D. Phillips, T. H. Bergman, H. J. Metcalf, Phys. Rev. Lett. 54, 2596 (1985); A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, D. E. Pritchard, Phys. Rev. Lett. 61, 2431 (1988); E. A. Cornell, C. Monroe, C. E. Wieman, Phys. Rev. Lett. 67, 2439 (1991).
[Crossref] [PubMed]

Miller, J. D.

J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. A 47, R4567 (1993).
[Crossref] [PubMed]

Monroe, C. R.

C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993).
[Crossref] [PubMed]

Myatt, C. J.

C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993).
[Crossref] [PubMed]

Phillips, W. D.

S. Rolston, C. Gertz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. C. Spreeuw, C. I. Westbrook, Proc. Soc. Photo-Opt. Instrum. Eng. 1726, 205 (1992).

A. Migdall, J. V. Prodan, W. D. Phillips, T. H. Bergman, H. J. Metcalf, Phys. Rev. Lett. 54, 2596 (1985); A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, D. E. Pritchard, Phys. Rev. Lett. 61, 2431 (1988); E. A. Cornell, C. Monroe, C. E. Wieman, Phys. Rev. Lett. 67, 2439 (1991).
[Crossref] [PubMed]

Prentiss, M.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, Phys. Rev. Lett. 59,2631 (1987); C. Monroe, W. Swann, H. Robinson, C. Wieman, Phys. Rev. Lett. 65, 1571 (1990).
[Crossref] [PubMed]

Pritchard, D. E.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, D. E. Pritchard, Phys. Rev. Lett. 70, 2253 (1993). The dark MOT has a slower relaxation rate than the ordinary MOT but is still several orders of magnitude faster than the present trap.
[Crossref] [PubMed]

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, Phys. Rev. Lett. 59,2631 (1987); C. Monroe, W. Swann, H. Robinson, C. Wieman, Phys. Rev. Lett. 65, 1571 (1990).
[Crossref] [PubMed]

D. E. Pritchard, Prog. Quantum Electron. 8, 209 (1984)
[Crossref]

D. E. Pritchard, in Electronic and Atomic Collisions, D. C. Lorents, W. E. Meyerhof, J. R. Peterson, eds. (Elsevier, New York, 1986), p. 593.

Prodan, J. V.

A. Migdall, J. V. Prodan, W. D. Phillips, T. H. Bergman, H. J. Metcalf, Phys. Rev. Lett. 54, 2596 (1985); A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, D. E. Pritchard, Phys. Rev. Lett. 61, 2431 (1988); E. A. Cornell, C. Monroe, C. E. Wieman, Phys. Rev. Lett. 67, 2439 (1991).
[Crossref] [PubMed]

Raab, E. L.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, Phys. Rev. Lett. 59,2631 (1987); C. Monroe, W. Swann, H. Robinson, C. Wieman, Phys. Rev. Lett. 65, 1571 (1990).
[Crossref] [PubMed]

Rolston, S.

S. Rolston, C. Gertz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. C. Spreeuw, C. I. Westbrook, Proc. Soc. Photo-Opt. Instrum. Eng. 1726, 205 (1992).

Sackett, C. A.

C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993).
[Crossref] [PubMed]

Sandberg, J. C.

N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[Crossref] [PubMed]

Spreeuw, R. J. C.

S. Rolston, C. Gertz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. C. Spreeuw, C. I. Westbrook, Proc. Soc. Photo-Opt. Instrum. Eng. 1726, 205 (1992).

Walker, T.

T. Walker, P. Feng, “Ultracold collisions,”Adv. At. Mol. Opt. Phys. (to be published).

Westbrook, C. I.

S. Rolston, C. Gertz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. C. Spreeuw, C. I. Westbrook, Proc. Soc. Photo-Opt. Instrum. Eng. 1726, 205 (1992).

Wieman, C. E.

C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993).
[Crossref] [PubMed]

Wineland, D. J.

W. M. Itano, J. C. Bergquist, D. J. Wineland, Science 237, 612 (1987); K. Gibble, S. Chu, Metrologia 29, 201 (1992).
[Crossref] [PubMed]

Ann. Phys. (1)

C. Cohen-Tannoudji, Ann. Phys. (Paris) 7, 423, 469 (1962); W. Happer, Rev. Mod. Phys. 44, 169 (1972).
[Crossref]

Phys. Rev. A (2)

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980); S. Chu, J. E. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).
[Crossref] [PubMed]

J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. A 47, R4567 (1993).
[Crossref] [PubMed]

Phys. Rev. Lett. (6)

C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993).
[Crossref] [PubMed]

N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[Crossref] [PubMed]

K. Gibble, S. Chu, Phys. Rev. Lett. 70, 1771 (1993).
[Crossref] [PubMed]

A. Migdall, J. V. Prodan, W. D. Phillips, T. H. Bergman, H. J. Metcalf, Phys. Rev. Lett. 54, 2596 (1985); A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, D. E. Pritchard, Phys. Rev. Lett. 61, 2431 (1988); E. A. Cornell, C. Monroe, C. E. Wieman, Phys. Rev. Lett. 67, 2439 (1991).
[Crossref] [PubMed]

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, Phys. Rev. Lett. 59,2631 (1987); C. Monroe, W. Swann, H. Robinson, C. Wieman, Phys. Rev. Lett. 65, 1571 (1990).
[Crossref] [PubMed]

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, D. E. Pritchard, Phys. Rev. Lett. 70, 2253 (1993). The dark MOT has a slower relaxation rate than the ordinary MOT but is still several orders of magnitude faster than the present trap.
[Crossref] [PubMed]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

S. Rolston, C. Gertz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. C. Spreeuw, C. I. Westbrook, Proc. Soc. Photo-Opt. Instrum. Eng. 1726, 205 (1992).

Prog. Quantum Electron. (1)

D. E. Pritchard, Prog. Quantum Electron. 8, 209 (1984)
[Crossref]

Science (1)

W. M. Itano, J. C. Bergquist, D. J. Wineland, Science 237, 612 (1987); K. Gibble, S. Chu, Metrologia 29, 201 (1992).
[Crossref] [PubMed]

Other (3)

D. E. Pritchard, in Electronic and Atomic Collisions, D. C. Lorents, W. E. Meyerhof, J. R. Peterson, eds. (Elsevier, New York, 1986), p. 593.

T. Walker, P. Feng, “Ultracold collisions,”Adv. At. Mol. Opt. Phys. (to be published).

R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, 1983), Chap. 8, p. 315.

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Figures (2)

Fig. 1
Fig. 1

Energy levels of 85Rb showing spontaneous transitions from a linearly polarized laser. The ground-state hyperfine splitting is 3 GHz, and the excited-state hyperfine splitting, not shown, is of the order of 0.3 GHz. The laser wavelength λL, is always to the red of both D lines, between 798 and 860 nm. For laser detunings greater than the excited-state fine- or hyperfine-structure splitting the scattering process can follow more than one path, as indicated by the solid lines.

Fig. 2
Fig. 2

Hyperfine population relaxation time as a function of trap laser wavelength. Experimental measurements are shown as filled circles. Calculated relaxation times are shown by the solid curve. The calculated mean time between spontaneous photon scattering events is shown by the dashed curve.

Equations (2)

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γ F M F M = 3 π c 2 ω L 3 I 2 h μ 4 | a F M F M ( 1 / 2 ) Δ 1 / 2 + a F M F M ( 3 / 2 ) Δ 3 / 2 | 2 ,
a F M F M ( J ) = Γ J ω J 3 q , F , M F M | μ q | F M F M | μ 0 | F M ,

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