Abstract

The simultaneous excitation of adjacent atomic transitions by two light waves equally detuned from their resonances gives rise to destructive interference of the transitions and causes a narrow dark line, or coherent dip, to appear in both light scattering and absorption. With one of the light fields detuned from resonance, the line is asymmetric and is accompanied by a shifted bright line; the two are represented by a Fano line profile. The spectral separation of the bright and dark lines is closely related—although not identical—to the net shift that is found in the two-photon-resonant denominator of the susceptibility, to the shift of the normal modes of the atom’s two internal excitations, and to the net difference of the energy separations of dressed and bare atomic levels. We report measurement of the resonance shift of the 2S1/22D3/2 transition of a single cooled and trapped Ba+ ion, and we analyze its relationship to those identified light shifts.

© 1998 Optical Society of America

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References

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  1. V. P. Chebotayev, “Three-level laser spectroscopy,” in High-Resolution Laser Spectroscopy, K. Shimoda, ed., Vol. 13 of Topics in Applied Physics (Springer-Verlag, Berlin, 1976). For the shift of molecular radio-frequency transitions see S. H. Autler and C. H. Townes, Phys. Rev. 100, 707 (1955).
    [CrossRef]
  2. H.-I. Yoo and J. H. Eberly, Phys. Rep. 118, 239 (1985).
    [CrossRef]
  3. Th. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, and P. E. Toschek, Z. Phys. 226, 293 (1969).
    [CrossRef]
  4. T. Hänsch and P. Toschek, Ann. Phys. (Leipzig) 23, 271 (1969).
    [CrossRef]
  5. Th. Hänsch and P. E. Toschek, Z. Phys. 236, 213 (1970).
    [CrossRef]
  6. A. Schabert, R. Keil, and P. E. Toschek, Appl. Phys. 6, 181 (1975); Opt. Commun. 13, 265 (1975).
    [CrossRef]
  7. M. Sargent III and P. E. Toschek, Appl. Phys. 11, 107 (1976).
    [CrossRef]
  8. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Processus d’Interaction entre Photons et Atoms (Editions CNRS, Paris, 1988).
  9. G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento B 36, 5 (1976).
    [CrossRef]
  10. U. Fano, Phys. Rev. 124, 1866 (1961).
    [CrossRef]
  11. B. Lounis and C. Cohen-Tannoudji, J. Phys. (France) II 2, 579 (1992).
    [CrossRef]
  12. R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975); G. Orriols, Nuovo Cimento B 53, 1 (1979).
    [CrossRef]
  13. J. Bialas, W. J. Firth, and P. E. Toschek, Opt. Commun. 36, 317 (1981); 37, 451 (1981).
    [CrossRef]
  14. W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Phys. Rev. A 22, 1137 (1980).
    [CrossRef]
  15. I. Siemers, M. Schubert, R. Blatt, W. Neuhauser, and P. E. Toschek, Europhys. Lett. 18, 139 (1992).
    [CrossRef]
  16. D. J. Wineland, W. M. Itano, and J. C. Bergquist, Opt. Lett. 12, 389 (1987); J. Bialas, R. Blatt, W. Neuhauser, and P. E. Toschek, Opt. Commun. 59, 27 (1986).
    [CrossRef] [PubMed]
  17. See, e.g., I. Siemers, V. Enders, R. Blatt, W. Neuhauser, and P. E. Toschek, in Proceedings of the 6th European Frequency and Time Forum, ESA SP-340 (European Space Agency, Paris, 1992).
  18. See, e.g., R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
    [CrossRef]
  19. G. Janik, W. Nagourney, and H. Dehmelt, J. Opt. Soc. Am. B 2, 1251 (1985).
    [CrossRef]
  20. See, e.g., P. R. Hemmer and M. G. Prentiss, J. Opt. Soc. Am. B 5, 1613 (1988).
    [CrossRef]
  21. P. M. Radmore and P. L. Knight, J. Phys. B 15, 561 (1982).
    [CrossRef]
  22. I. Steiner, V. Enders, F. Elsner, W. Neuhauser, P. E. Toschek, R. Blatt, and J. Helmcke, Appl. Phys. B 49, 251 (1989).
    [CrossRef]
  23. D. A. Allan, Proc. IEEE 54, 221 (1966); J. A. Barnes, A. R. Chi, L. S. Cutter, D. L. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, Jr., W. L. Smith, R. L. Sydot, R. F. C. Vessot, and G. M. R. Winkler, IEEE Trans. Instrum. Meas. IM-20, 105 (1971).
    [CrossRef]
  24. Th. Sauter, H. Gilhaus, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Z. Phys. D 10, 153 (1988).
    [CrossRef]
  25. M. Schubert, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Phys. Rev. Lett. 68, 3016 (1992).
    [CrossRef] [PubMed]
  26. W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Appl. Phys. 17, 123 (1978).
    [CrossRef]
  27. W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).
    [CrossRef]
  28. H. Gilhaus, Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Opt. Commun. 69, 25 (1988).
    [CrossRef]

1992

B. Lounis and C. Cohen-Tannoudji, J. Phys. (France) II 2, 579 (1992).
[CrossRef]

I. Siemers, M. Schubert, R. Blatt, W. Neuhauser, and P. E. Toschek, Europhys. Lett. 18, 139 (1992).
[CrossRef]

M. Schubert, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Phys. Rev. Lett. 68, 3016 (1992).
[CrossRef] [PubMed]

1989

I. Steiner, V. Enders, F. Elsner, W. Neuhauser, P. E. Toschek, R. Blatt, and J. Helmcke, Appl. Phys. B 49, 251 (1989).
[CrossRef]

1988

Th. Sauter, H. Gilhaus, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Z. Phys. D 10, 153 (1988).
[CrossRef]

See, e.g., P. R. Hemmer and M. G. Prentiss, J. Opt. Soc. Am. B 5, 1613 (1988).
[CrossRef]

H. Gilhaus, Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Opt. Commun. 69, 25 (1988).
[CrossRef]

1985

1982

P. M. Radmore and P. L. Knight, J. Phys. B 15, 561 (1982).
[CrossRef]

1980

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Phys. Rev. A 22, 1137 (1980).
[CrossRef]

1978

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Appl. Phys. 17, 123 (1978).
[CrossRef]

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

1976

See, e.g., R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

M. Sargent III and P. E. Toschek, Appl. Phys. 11, 107 (1976).
[CrossRef]

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento B 36, 5 (1976).
[CrossRef]

1970

Th. Hänsch and P. E. Toschek, Z. Phys. 236, 213 (1970).
[CrossRef]

1969

Th. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, and P. E. Toschek, Z. Phys. 226, 293 (1969).
[CrossRef]

T. Hänsch and P. Toschek, Ann. Phys. (Leipzig) 23, 271 (1969).
[CrossRef]

1961

U. Fano, Phys. Rev. 124, 1866 (1961).
[CrossRef]

Alzetta, G.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento B 36, 5 (1976).
[CrossRef]

Blatt, R.

I. Siemers, M. Schubert, R. Blatt, W. Neuhauser, and P. E. Toschek, Europhys. Lett. 18, 139 (1992).
[CrossRef]

M. Schubert, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Phys. Rev. Lett. 68, 3016 (1992).
[CrossRef] [PubMed]

I. Steiner, V. Enders, F. Elsner, W. Neuhauser, P. E. Toschek, R. Blatt, and J. Helmcke, Appl. Phys. B 49, 251 (1989).
[CrossRef]

Th. Sauter, H. Gilhaus, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Z. Phys. D 10, 153 (1988).
[CrossRef]

H. Gilhaus, Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Opt. Commun. 69, 25 (1988).
[CrossRef]

Cohen-Tannoudji, C.

B. Lounis and C. Cohen-Tannoudji, J. Phys. (France) II 2, 579 (1992).
[CrossRef]

Dehmelt, H.

G. Janik, W. Nagourney, and H. Dehmelt, J. Opt. Soc. Am. B 2, 1251 (1985).
[CrossRef]

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Phys. Rev. A 22, 1137 (1980).
[CrossRef]

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

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Appl. Phys. 17, 123 (1978).
[CrossRef]

Eberly, J. H.

H.-I. Yoo and J. H. Eberly, Phys. Rep. 118, 239 (1985).
[CrossRef]

Elsner, F.

I. Steiner, V. Enders, F. Elsner, W. Neuhauser, P. E. Toschek, R. Blatt, and J. Helmcke, Appl. Phys. B 49, 251 (1989).
[CrossRef]

Enders, V.

I. Steiner, V. Enders, F. Elsner, W. Neuhauser, P. E. Toschek, R. Blatt, and J. Helmcke, Appl. Phys. B 49, 251 (1989).
[CrossRef]

Fano, U.

U. Fano, Phys. Rev. 124, 1866 (1961).
[CrossRef]

Gilhaus, H.

Th. Sauter, H. Gilhaus, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Z. Phys. D 10, 153 (1988).
[CrossRef]

H. Gilhaus, Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Opt. Commun. 69, 25 (1988).
[CrossRef]

Gozzini, A.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento B 36, 5 (1976).
[CrossRef]

Hänsch, T.

T. Hänsch and P. Toschek, Ann. Phys. (Leipzig) 23, 271 (1969).
[CrossRef]

Hänsch, Th.

Th. Hänsch and P. E. Toschek, Z. Phys. 236, 213 (1970).
[CrossRef]

Th. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, and P. E. Toschek, Z. Phys. 226, 293 (1969).
[CrossRef]

Helmcke, J.

I. Steiner, V. Enders, F. Elsner, W. Neuhauser, P. E. Toschek, R. Blatt, and J. Helmcke, Appl. Phys. B 49, 251 (1989).
[CrossRef]

Hemmer, P. R.

Hohenstatt, M.

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Phys. Rev. A 22, 1137 (1980).
[CrossRef]

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

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Appl. Phys. 17, 123 (1978).
[CrossRef]

Janik, G.

Keil, R.

Th. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, and P. E. Toschek, Z. Phys. 226, 293 (1969).
[CrossRef]

Knight, P. L.

P. M. Radmore and P. L. Knight, J. Phys. B 15, 561 (1982).
[CrossRef]

Lounis, B.

B. Lounis and C. Cohen-Tannoudji, J. Phys. (France) II 2, 579 (1992).
[CrossRef]

Moi, L.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento B 36, 5 (1976).
[CrossRef]

Nagourney, W.

Neuhauser, W.

M. Schubert, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Phys. Rev. Lett. 68, 3016 (1992).
[CrossRef] [PubMed]

I. Siemers, M. Schubert, R. Blatt, W. Neuhauser, and P. E. Toschek, Europhys. Lett. 18, 139 (1992).
[CrossRef]

I. Steiner, V. Enders, F. Elsner, W. Neuhauser, P. E. Toschek, R. Blatt, and J. Helmcke, Appl. Phys. B 49, 251 (1989).
[CrossRef]

Th. Sauter, H. Gilhaus, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Z. Phys. D 10, 153 (1988).
[CrossRef]

H. Gilhaus, Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Opt. Commun. 69, 25 (1988).
[CrossRef]

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Phys. Rev. A 22, 1137 (1980).
[CrossRef]

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Appl. Phys. 17, 123 (1978).
[CrossRef]

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

Orriols, G.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento B 36, 5 (1976).
[CrossRef]

Prentiss, M. G.

Radmore, P. M.

P. M. Radmore and P. L. Knight, J. Phys. B 15, 561 (1982).
[CrossRef]

Sargent III, M.

M. Sargent III and P. E. Toschek, Appl. Phys. 11, 107 (1976).
[CrossRef]

Sauter, Th.

H. Gilhaus, Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Opt. Commun. 69, 25 (1988).
[CrossRef]

Th. Sauter, H. Gilhaus, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Z. Phys. D 10, 153 (1988).
[CrossRef]

Schabert, A.

Th. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, and P. E. Toschek, Z. Phys. 226, 293 (1969).
[CrossRef]

Schmelzer, Ch.

Th. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, and P. E. Toschek, Z. Phys. 226, 293 (1969).
[CrossRef]

Schubert, M.

I. Siemers, M. Schubert, R. Blatt, W. Neuhauser, and P. E. Toschek, Europhys. Lett. 18, 139 (1992).
[CrossRef]

M. Schubert, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Phys. Rev. Lett. 68, 3016 (1992).
[CrossRef] [PubMed]

Siemers, I.

M. Schubert, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Phys. Rev. Lett. 68, 3016 (1992).
[CrossRef] [PubMed]

I. Siemers, M. Schubert, R. Blatt, W. Neuhauser, and P. E. Toschek, Europhys. Lett. 18, 139 (1992).
[CrossRef]

Th. Sauter, H. Gilhaus, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Z. Phys. D 10, 153 (1988).
[CrossRef]

Steiner, I.

I. Steiner, V. Enders, F. Elsner, W. Neuhauser, P. E. Toschek, R. Blatt, and J. Helmcke, Appl. Phys. B 49, 251 (1989).
[CrossRef]

Stroud, C. R.

See, e.g., R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

Toschek, P.

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

T. Hänsch and P. Toschek, Ann. Phys. (Leipzig) 23, 271 (1969).
[CrossRef]

Toschek, P. E.

I. Siemers, M. Schubert, R. Blatt, W. Neuhauser, and P. E. Toschek, Europhys. Lett. 18, 139 (1992).
[CrossRef]

M. Schubert, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Phys. Rev. Lett. 68, 3016 (1992).
[CrossRef] [PubMed]

I. Steiner, V. Enders, F. Elsner, W. Neuhauser, P. E. Toschek, R. Blatt, and J. Helmcke, Appl. Phys. B 49, 251 (1989).
[CrossRef]

Th. Sauter, H. Gilhaus, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Z. Phys. D 10, 153 (1988).
[CrossRef]

H. Gilhaus, Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Opt. Commun. 69, 25 (1988).
[CrossRef]

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Phys. Rev. A 22, 1137 (1980).
[CrossRef]

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Appl. Phys. 17, 123 (1978).
[CrossRef]

M. Sargent III and P. E. Toschek, Appl. Phys. 11, 107 (1976).
[CrossRef]

Th. Hänsch and P. E. Toschek, Z. Phys. 236, 213 (1970).
[CrossRef]

Th. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, and P. E. Toschek, Z. Phys. 226, 293 (1969).
[CrossRef]

Whitley, R. M.

See, e.g., R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

Yoo, H.-I.

H.-I. Yoo and J. H. Eberly, Phys. Rep. 118, 239 (1985).
[CrossRef]

Ann. Phys. (Leipzig)

T. Hänsch and P. Toschek, Ann. Phys. (Leipzig) 23, 271 (1969).
[CrossRef]

Appl. Phys.

M. Sargent III and P. E. Toschek, Appl. Phys. 11, 107 (1976).
[CrossRef]

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Appl. Phys. 17, 123 (1978).
[CrossRef]

Appl. Phys. B

I. Steiner, V. Enders, F. Elsner, W. Neuhauser, P. E. Toschek, R. Blatt, and J. Helmcke, Appl. Phys. B 49, 251 (1989).
[CrossRef]

Europhys. Lett.

I. Siemers, M. Schubert, R. Blatt, W. Neuhauser, and P. E. Toschek, Europhys. Lett. 18, 139 (1992).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. (France) II

B. Lounis and C. Cohen-Tannoudji, J. Phys. (France) II 2, 579 (1992).
[CrossRef]

J. Phys. B

P. M. Radmore and P. L. Knight, J. Phys. B 15, 561 (1982).
[CrossRef]

Nuovo Cimento B

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cimento B 36, 5 (1976).
[CrossRef]

Opt. Commun.

H. Gilhaus, Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Opt. Commun. 69, 25 (1988).
[CrossRef]

Phys. Rep.

H.-I. Yoo and J. H. Eberly, Phys. Rep. 118, 239 (1985).
[CrossRef]

Phys. Rev.

U. Fano, Phys. Rev. 124, 1866 (1961).
[CrossRef]

Phys. Rev. A

W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Phys. Rev. A 22, 1137 (1980).
[CrossRef]

See, e.g., R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

Phys. Rev. Lett.

M. Schubert, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Phys. Rev. Lett. 68, 3016 (1992).
[CrossRef] [PubMed]

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

Z. Phys.

Th. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, and P. E. Toschek, Z. Phys. 226, 293 (1969).
[CrossRef]

Th. Hänsch and P. E. Toschek, Z. Phys. 236, 213 (1970).
[CrossRef]

Z. Phys. D

Th. Sauter, H. Gilhaus, I. Siemers, R. Blatt, W. Neuhauser, and P. E. Toschek, Z. Phys. D 10, 153 (1988).
[CrossRef]

Other

D. A. Allan, Proc. IEEE 54, 221 (1966); J. A. Barnes, A. R. Chi, L. S. Cutter, D. L. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, Jr., W. L. Smith, R. L. Sydot, R. F. C. Vessot, and G. M. R. Winkler, IEEE Trans. Instrum. Meas. IM-20, 105 (1971).
[CrossRef]

A. Schabert, R. Keil, and P. E. Toschek, Appl. Phys. 6, 181 (1975); Opt. Commun. 13, 265 (1975).
[CrossRef]

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Processus d’Interaction entre Photons et Atoms (Editions CNRS, Paris, 1988).

V. P. Chebotayev, “Three-level laser spectroscopy,” in High-Resolution Laser Spectroscopy, K. Shimoda, ed., Vol. 13 of Topics in Applied Physics (Springer-Verlag, Berlin, 1976). For the shift of molecular radio-frequency transitions see S. H. Autler and C. H. Townes, Phys. Rev. 100, 707 (1955).
[CrossRef]

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975); G. Orriols, Nuovo Cimento B 53, 1 (1979).
[CrossRef]

J. Bialas, W. J. Firth, and P. E. Toschek, Opt. Commun. 36, 317 (1981); 37, 451 (1981).
[CrossRef]

D. J. Wineland, W. M. Itano, and J. C. Bergquist, Opt. Lett. 12, 389 (1987); J. Bialas, R. Blatt, W. Neuhauser, and P. E. Toschek, Opt. Commun. 59, 27 (1986).
[CrossRef] [PubMed]

See, e.g., I. Siemers, V. Enders, R. Blatt, W. Neuhauser, and P. E. Toschek, in Proceedings of the 6th European Frequency and Time Forum, ESA SP-340 (European Space Agency, Paris, 1992).

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Figures (11)

Fig. 1
Fig. 1

Two coupled atomic transitions connecting three energy levels that include metastable level |3〉 (Λ-shaped system).

Fig. 2
Fig. 2

Excited-state population, or rate of resonance scattering, of an atom with a Λ-shaped level system for three different values of detuning of light 1: a, Δ1=0 MHz; b, -20 MHz; c, -40 MHz. Ratio of Rabi frequencies, Ω1/Ω2=50.

Fig. 3
Fig. 3

Resonance shift (solid curves) and two-photon light shifts δ0 (short-dashed curves), δL (long-dashed curves), and δS (long-and-short-dashed curves) calculated from steady-state solution of Bloch’s equation, versus detuning of light 1. Here is defined as spectral separation of a dark line from its associated bright line in the resonance scattering that varies as the population of the resonance level (see Fig. 2). For the definitions of δ0, δL, and δS see Eqs. (10), (17), and (21), respectively. (a) Ω1=0.1 Γ1, Ω2=0.01 Γ2 (top) and Ω1=Γ1, Ω2=0.01 Γ2 (bottom). (b) Ω1=Γ1, Ω2=Ω2 (top) and Ω1=10 Γ1, Ω2=Γ2 (bottom). (c) Ω=Γ1, Ω2=10 Γ2 (top) and Ω1=10 Γ1, Ω2=10 Γ2 (bottom).

Fig. 4
Fig. 4

Resonance shift and two-photon light shifts δ0, δL, and δS calculated from steady-state solution of Bloch’s equation versus Rabi frequency Ω1 of light 1 at resonance (Δ1=0). (a) Ω2=0.01 Γ2, (b) Ω2=Γ2, (c) Ω2=10 Γ2.

Fig. 5
Fig. 5

Calculated eigenvalues of the atomic (dashed curves) and the effective (solid curves) Hamiltonian; detuning of first light Δ1=-10 MHz. (a) Rabi frequencies Ω1=0.5 Γ1 and Ω2=0.5 Γ2. The effective levels cross, and the light shift shows dispersive character (see the expanded detail at the right). (b) Rabi frequencies Ω1=0.5 Γ1 and Ω2=1.2 Γ2. The effective levels repel each other, and the light shift is maximum on two-photon resonance Δ1-Δ2=δ1-δ3.

Fig. 6
Fig. 6

Light shift δS of dressed levels versus detuning and Rabi frequencies Ω1 of strong light 1. Ω2=0.1 Γ2.

Fig. 7
Fig. 7

Relevant energy levels of Ba+ including Zeeman splitting.

Fig. 8
Fig. 8

Experimental arrangement: pm, photomultiplier.

Fig. 9
Fig. 9

Excitation spectra of a single Ba+ ion and χ2 fit for three different values of the green-light (1) detuning. Detuning Δ1 as shown. Rabi frequencies (means of the values—scattering by a few percent—that have been implemented with individual spectra): Ω1=0.58 Γ1 and Ω2=0.67 Γ2.

Fig. 10
Fig. 10

Measured values of resonance shift (filled circles) and corresponding calculated data of (solid curves) and of the two-photon light shifts δ0 (short-dashed curves), δL (long-dashed curves), and δS (short–long-dashed curves). Rabi frequencies Ω1, Ω2, as shown.

Fig. 11
Fig. 11

Observed bright and dark resonance (filled circles) made to fit in with solutions of the optical Bloch equations for an eight-level system (solid curve), for a three-level system (short-dashed curve), and by the scattering rate |T1|2 representing a Fano profile (long-dashed curve).

Tables (3)

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Table 1 Ba+ Transitions, Their Rates of Relaxation Γi, and Wavelength Values of Resonant Light

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Table 2 Characteristics of the Miniaturized Electrodynamic Ion Trap

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Table 3 Zeeman-Split Fano Line Componentsa

Equations (45)

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ρ=m,n=13ρmn|mn|,
H=HA+HI,
HA=-Δ1|22|+(Δ2-Δ1)|33|
HI=(Ω1|21|+Ω2|23|+H.c.)
Ei(ωi)=iEi cos ωit,
ρ˙=-i [H,ρ]+R,
ρ˙22=-Γρ22-i(ρ12-ρ21)Ω1/2-i(ρ32-ρ23)Ω2/2,
ρ˙11=Γ1ρ22+i(ρ12-ρ21)Ω1/2,
ρ˙33=Γ3ρ22+i(ρ32-ρ23)Ω2/2,
ρ˙12=(-Γ/2-iΔ1)ρ12-i(ρ22-ρ11)Ω1/2+iρ13Ω2/2,
ρ˙32=(-Γ/2-iΔ2)ρ32-i(ρ22-ρ33)Ω2/2+iρ31Ω1/2,
ρ˙13=i(Δ3-Δ1)ρ13+iρ12Ω2/2-iρ23Ω1/2-Γ13ρ13,
(Δ1=0)=±(Ω12/Γ1+Ω22/Γ2)(Ω12+Ω22)216Ω12/Γ11/4.
(Δ1)=(Ω12+Ω22)2 Ω12/Γ1+Ω22/Γ2Ω24/Γ2-Ω14/Γ1 14Δ1.
D=Δ12Δ23Δ13-Δ12Ω12/4-Δ23Ω22/4=Δ12Δ23(Δ13+δ+iγ),
γ=Ω12Δ22+(Γ/2)2+Ω22Δ12+(Γ/2)2 Γ2,
δ=Δ1Ω224Δ12+Γ2-Δ2Ω124Δ22+Γ2
δ0Δ4Δ2+Γ2 (Ω22-Ω12).
δ0(0)=±12Ω12-Γ2,
=Ω22+Ω12Ω22-Ω122δ0,
Δ2(max Im ρ23)=Δ2(max ρ22),
χ±=Δ2(max Re ρ13)=12 Ω11±Γ/Ω1.
(χ+2+χ-2)/2=(Γ/2)2+δ02.
D=Δ12(Δ2-δ1+iκ1)(Δ2-δ2+iκ2)=Δ12D2.
δ1,2(Δ1=0)=±1/2Ω12-Γ˜2/4=±δL(0),
κ1,2(Δ1=0)=Γ˜/4.
δL2(0)+Γ˜42=δ02(0)+Γ22.
Heff(3)=0Ω1/20Ω1/2-Δ1-iΓ/2Ω2/20Ω2/2Δ2-Δ1.
δS=Δ1s1/2,Γβ=Γs1/2,Γα=Γ(1-s1/2),
δS=12 (Ω˜-|Δ1|),Γα=Γ2 1-Δ1Ω˜,
Γβ=Γ2 1+Δ1Ω˜,
T1=(1/4)Ω2Ω(Δ1-Δ2)D2-1,
=p2+p22-q+Δg4,
p=-w2-b23,2q=b2+p2-b1/p,
w=-(2/3)il3/2/sin 2φ,
φ=-arctantanθ21/3,
θ=arcsin2k -l3/27,
k=-b12-23 b2(b22-4b0)+1627 b23,
l=b22-4b0-43 b22;
b0=-3256 Δ14-14 mΔ12+ma,
b1=18 Δ12+mΔ1,
b2=-38 Δ12,
m=(GΩ12-RΩ22)/4a,
a=(Ω12+Ω22)2(G+R),
G=Ω12Γ2,R=Ω22Γ1.

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