Abstract

The resonant interaction of optical radiation with atomic vapors can excite the atoms into anisotropic states with long lifetimes. The conventional description of these states uses an expansion in a basis of multipole moments, which are represented by irreducible tensor operators. We demonstrate an experimental method that permits the excitation of these atomic multipole moments in specific, experimentally controllable spatial orientations. Complementary optical methods permit the observation of multipole moments of arbitrary order and a direct measurement of their spatial orientation. We demonstrate these procedures in the ground state of atomic sodium, using optical pumping to polarize the angular-momentum substates. We show how the experimental results can be used to separate signal contributions that originate from different multipole moments.

© 1994 Optical Society of America

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  1. A. Omont, “Irreducible components of the density matrix. Application to optical pumping,” Prog. Quantum Electron. 5, 69–138 (1977).
    [CrossRef]
  2. G. Herman and A. Scharmann, “Untersuchungen zur Zeeman Spektroskopie mit Hilfe nichtlinearer Resonanzen eines multimoden Lasers,” Z. Phys. 254, 46–56 (1972).
    [CrossRef]
  3. M. Ducloy, M. P. Gorza, and B. Decomps, “Higher-order nonlinear effects in a gas laser: creation and detection of a hexadecupole moment in the neon 2p4level,” Opt. Commun. 8, 21–25 (1973).
    [CrossRef]
  4. W. Gawlik, J. Kowalski, R. Neumann, and F. Träger, “Observation of the electric hexadecupole moment of free Na atoms in a forward scattering experiment,” Opt. Commun. 12, 400–404 (1974).
    [CrossRef]
  5. S. Giraud-Cotton, V. P. Kaftandjian, and L. Klein, “Magnetic optical activity in intense laser fields. I. Self-rotation and Verdet-constant,” Phys. Rev. A 32, 2211–2222 (1985).
    [CrossRef] [PubMed]
  6. K. H. Drake, W. Lange, and J. Mlynek, “Nonlinear Faraday and Voigt effect in a J= 1 to J′= 0 transition in atomic samarium vapour,” Opt. Commun. 66, 315–320 (1988).
    [CrossRef]
  7. L. M. Barkov, D. A. Melik-Pasheayev, and M. S. Zolotorev, “Nonlinear Faraday rotation in samarium vapor,” Opt. Commun. 70, 467–472 (1989).
    [CrossRef]
  8. R. Bernheim, Optical pumping (Benjamin, New York, 1965).
  9. A. Kastler, “Optical methods for studying Hertzian resonances,” Science 158, 214–221 (1967).
    [CrossRef] [PubMed]
  10. W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
    [CrossRef]
  11. W. E. Bell and A. L. Bloom, “Optically driven spin precession,” Phys. Rev. Lett. 6, 280–281 (1961).
    [CrossRef]
  12. G. W. Series, “Theory of the modulation of light in optical pumping experiments,” Proc. Phys. Soc. 88, 957–968 (1966).
    [CrossRef]
  13. H. G. Dehmelt, “Modulation of a light beam by precessing absorbing atoms,” Phys. Rev. 105, 1924–1925 (1957).
    [CrossRef]
  14. R. B. Partridge and G. W. Series, “The modulated absorption of light in an optical pumping experiment on 4He,” Proc. Phys. Soc. 88, 969–982 (1966).
    [CrossRef]
  15. W. Happer and B. S. Mathur, “Off-resonant light as a probe of optically pumped alkali vapors,” Phys. Rev. Lett. 18, 577–580 (1967).
    [CrossRef]
  16. S. Pancharatnam, “Modulated birefringence of a spin-assembly in magnetic resonance,” Phys. Lett. 27A, 509–510 (1968).
  17. J. Mlynek, N. C. Wong, R. G. DeVoe, E. S. Kintzer, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. Lett. 50, 993–996 (1983).
    [CrossRef]
  18. N. C. Wong, E. S. Kintzer, J. Mlynek, R. G. DeVoe, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. B 28, 4993–5010 (1983).
    [CrossRef]
  19. P. Kulina and R. H. Rinkleff, “Determination of the tensor polarizabilities in Yb and Sm by quantum beat spectroscopy,” Z. Phys. A 304, 371–372 (1982).
    [CrossRef]
  20. A. P. Ghosh, C. D. Nabors, M. A. Attili, and J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985).
    [CrossRef] [PubMed]
  21. R. M. Lowe, D. S. Gough, R. J. McLean, and P. Hannaford, “Determination of ground-state depolarization rates by transmission Zeeman beat spectroscopy: application to Sm i 4f66s2 7F1,” Phys. Rev. A 36, 5490–5493 (1987).
    [CrossRef] [PubMed]
  22. R. J. McLean, P. Hannaford, and R. M. Lowe, “Transmission Zeeman beat spectroscopy: application to the 7F ground level term of Sm i,” Phys. Rev. A 42, 6616–6628 (1990).
    [CrossRef] [PubMed]
  23. H. G. Dehmelt, “Slow spin relaxation of optically polarized sodium atoms,” Phys. Rev. 105, 1487–1489 (1957).
    [CrossRef]
  24. C. Cohen-Tannoudji, “Observation d’un déplacement de raie de résonance magnetique causé par l’exicitation optique,” C. R. 252, 394–396 (1961).
  25. W. Lange and J. Mlynek, “Quantum beats in transmission by time-resolved polarization spectroscopy,” Phys. Rev. Lett. 40, 1373–1375 (1978).
    [CrossRef]
  26. T. Mishina, Y. Fukuda, and T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
    [CrossRef]
  27. S. Appelt, P. Scheufler, and M. Mehring, “Direct observation of single- and double-quantum sublevel coherence in rubidium vapor by optical Raman beat detection,” Opt. Commun. 74, 110–114 (1989).
    [CrossRef]
  28. C. Cohen-Tannoudji, “Theorie quantique du cycle de pompage optique,” Ann. Phys. 7, 423–460 (1962).
  29. D. Suter and H. Klepel, “Indirect observation of ‘forbidden’ Raman transitions by laser-induced coherence transfer,” Europhys. Lett. 19, 469–474 (1992).
    [CrossRef]
  30. D. Suter, T. Marty, and H. Klepel. “Rotation properties of multipole moments in atomic sublevel spectroscopy,” Opt. Lett. 18, 531–533 (1993).
    [CrossRef] [PubMed]
  31. R. L. Shoemaker and R. G. Brewer, “Two-photon superradi-ance,” Phys. Rev. Lett. 28, 1430–1433 (1972).
    [CrossRef]
  32. D. Sutter, M. Rosatzin, and J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
    [CrossRef]
  33. D. Suter and T. Marty, “Laser induced dynamics of atomic sublevel coherences,” Opt. Commun. 100, 443–450 (1993).
    [CrossRef]
  34. J. Jeener, Lecture presented at the Ampére International Summer School II, Basko Polje, Yugoslavia, 1971 (Université Libre de Bruxelles, B-1050 Bruxelles, Belgium).
  35. J. Jeener, B. H. Meier, P. Bachmann, and R. R. Ernst, “Investigation of exchange processes by two-dimensional NMR spectroscopy,” J. Chem. Phys. 71, 4546–4553 (1979).
    [CrossRef]
  36. R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987).
  37. D. Suter, H. Klepel, and J. Mlynek, “Time-resolved two-dimensional spectroscopy of optically driven atomic sublevel coherences,” Phys. Rev. Lett. 67, 2001–2004 (1991).
    [CrossRef] [PubMed]
  38. D. Suter and J. Mlynek, “Dynamics of atomic sublevel coherences during modulated optical pumping,” Phys. Rev. A 43, 6124–6134 (1991).
    [CrossRef] [PubMed]
  39. H. Klepel and D. Suter, “Transverse optical pumping with polarization-modulated light,” Opt. Commun. 90, 46–50 (1992).
    [CrossRef]
  40. M. Rosatzin, D. Suter, W. Lange, and J. Mlynek, “Phase and amplitude variations of optically induced spin transients,” J. Opt. Soc. Am. B 7, 1231–1238 (1990).
    [CrossRef]
  41. D. Suter and J. Mlynek, “Laser excitation and detection of magnetic resonance,” in Adv. Magn. Opt. Reson. 16, 1–83 (1991).
  42. D. Suter and J. G. Pearson, “Experimental classification of multi-spin coherence under the full rotation group,” Chem. Phys. Lett. 144, 328–332 (1988).
    [CrossRef]

1993 (2)

D. Suter and T. Marty, “Laser induced dynamics of atomic sublevel coherences,” Opt. Commun. 100, 443–450 (1993).
[CrossRef]

D. Suter, T. Marty, and H. Klepel. “Rotation properties of multipole moments in atomic sublevel spectroscopy,” Opt. Lett. 18, 531–533 (1993).
[CrossRef] [PubMed]

1992 (2)

H. Klepel and D. Suter, “Transverse optical pumping with polarization-modulated light,” Opt. Commun. 90, 46–50 (1992).
[CrossRef]

D. Suter and H. Klepel, “Indirect observation of ‘forbidden’ Raman transitions by laser-induced coherence transfer,” Europhys. Lett. 19, 469–474 (1992).
[CrossRef]

1991 (3)

D. Suter, H. Klepel, and J. Mlynek, “Time-resolved two-dimensional spectroscopy of optically driven atomic sublevel coherences,” Phys. Rev. Lett. 67, 2001–2004 (1991).
[CrossRef] [PubMed]

D. Suter and J. Mlynek, “Dynamics of atomic sublevel coherences during modulated optical pumping,” Phys. Rev. A 43, 6124–6134 (1991).
[CrossRef] [PubMed]

D. Suter and J. Mlynek, “Laser excitation and detection of magnetic resonance,” in Adv. Magn. Opt. Reson. 16, 1–83 (1991).

1990 (3)

M. Rosatzin, D. Suter, W. Lange, and J. Mlynek, “Phase and amplitude variations of optically induced spin transients,” J. Opt. Soc. Am. B 7, 1231–1238 (1990).
[CrossRef]

D. Sutter, M. Rosatzin, and J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
[CrossRef]

R. J. McLean, P. Hannaford, and R. M. Lowe, “Transmission Zeeman beat spectroscopy: application to the 7F ground level term of Sm i,” Phys. Rev. A 42, 6616–6628 (1990).
[CrossRef] [PubMed]

1989 (2)

S. Appelt, P. Scheufler, and M. Mehring, “Direct observation of single- and double-quantum sublevel coherence in rubidium vapor by optical Raman beat detection,” Opt. Commun. 74, 110–114 (1989).
[CrossRef]

L. M. Barkov, D. A. Melik-Pasheayev, and M. S. Zolotorev, “Nonlinear Faraday rotation in samarium vapor,” Opt. Commun. 70, 467–472 (1989).
[CrossRef]

1988 (3)

K. H. Drake, W. Lange, and J. Mlynek, “Nonlinear Faraday and Voigt effect in a J= 1 to J′= 0 transition in atomic samarium vapour,” Opt. Commun. 66, 315–320 (1988).
[CrossRef]

T. Mishina, Y. Fukuda, and T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
[CrossRef]

D. Suter and J. G. Pearson, “Experimental classification of multi-spin coherence under the full rotation group,” Chem. Phys. Lett. 144, 328–332 (1988).
[CrossRef]

1987 (1)

R. M. Lowe, D. S. Gough, R. J. McLean, and P. Hannaford, “Determination of ground-state depolarization rates by transmission Zeeman beat spectroscopy: application to Sm i 4f66s2 7F1,” Phys. Rev. A 36, 5490–5493 (1987).
[CrossRef] [PubMed]

1985 (2)

A. P. Ghosh, C. D. Nabors, M. A. Attili, and J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985).
[CrossRef] [PubMed]

S. Giraud-Cotton, V. P. Kaftandjian, and L. Klein, “Magnetic optical activity in intense laser fields. I. Self-rotation and Verdet-constant,” Phys. Rev. A 32, 2211–2222 (1985).
[CrossRef] [PubMed]

1983 (2)

J. Mlynek, N. C. Wong, R. G. DeVoe, E. S. Kintzer, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. Lett. 50, 993–996 (1983).
[CrossRef]

N. C. Wong, E. S. Kintzer, J. Mlynek, R. G. DeVoe, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. B 28, 4993–5010 (1983).
[CrossRef]

1982 (1)

P. Kulina and R. H. Rinkleff, “Determination of the tensor polarizabilities in Yb and Sm by quantum beat spectroscopy,” Z. Phys. A 304, 371–372 (1982).
[CrossRef]

1979 (1)

J. Jeener, B. H. Meier, P. Bachmann, and R. R. Ernst, “Investigation of exchange processes by two-dimensional NMR spectroscopy,” J. Chem. Phys. 71, 4546–4553 (1979).
[CrossRef]

1978 (1)

W. Lange and J. Mlynek, “Quantum beats in transmission by time-resolved polarization spectroscopy,” Phys. Rev. Lett. 40, 1373–1375 (1978).
[CrossRef]

1977 (1)

A. Omont, “Irreducible components of the density matrix. Application to optical pumping,” Prog. Quantum Electron. 5, 69–138 (1977).
[CrossRef]

1974 (1)

W. Gawlik, J. Kowalski, R. Neumann, and F. Träger, “Observation of the electric hexadecupole moment of free Na atoms in a forward scattering experiment,” Opt. Commun. 12, 400–404 (1974).
[CrossRef]

1973 (1)

M. Ducloy, M. P. Gorza, and B. Decomps, “Higher-order nonlinear effects in a gas laser: creation and detection of a hexadecupole moment in the neon 2p4level,” Opt. Commun. 8, 21–25 (1973).
[CrossRef]

1972 (3)

G. Herman and A. Scharmann, “Untersuchungen zur Zeeman Spektroskopie mit Hilfe nichtlinearer Resonanzen eines multimoden Lasers,” Z. Phys. 254, 46–56 (1972).
[CrossRef]

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
[CrossRef]

R. L. Shoemaker and R. G. Brewer, “Two-photon superradi-ance,” Phys. Rev. Lett. 28, 1430–1433 (1972).
[CrossRef]

1968 (1)

S. Pancharatnam, “Modulated birefringence of a spin-assembly in magnetic resonance,” Phys. Lett. 27A, 509–510 (1968).

1967 (2)

W. Happer and B. S. Mathur, “Off-resonant light as a probe of optically pumped alkali vapors,” Phys. Rev. Lett. 18, 577–580 (1967).
[CrossRef]

A. Kastler, “Optical methods for studying Hertzian resonances,” Science 158, 214–221 (1967).
[CrossRef] [PubMed]

1966 (2)

R. B. Partridge and G. W. Series, “The modulated absorption of light in an optical pumping experiment on 4He,” Proc. Phys. Soc. 88, 969–982 (1966).
[CrossRef]

G. W. Series, “Theory of the modulation of light in optical pumping experiments,” Proc. Phys. Soc. 88, 957–968 (1966).
[CrossRef]

1962 (1)

C. Cohen-Tannoudji, “Theorie quantique du cycle de pompage optique,” Ann. Phys. 7, 423–460 (1962).

1961 (2)

C. Cohen-Tannoudji, “Observation d’un déplacement de raie de résonance magnetique causé par l’exicitation optique,” C. R. 252, 394–396 (1961).

W. E. Bell and A. L. Bloom, “Optically driven spin precession,” Phys. Rev. Lett. 6, 280–281 (1961).
[CrossRef]

1957 (2)

H. G. Dehmelt, “Modulation of a light beam by precessing absorbing atoms,” Phys. Rev. 105, 1924–1925 (1957).
[CrossRef]

H. G. Dehmelt, “Slow spin relaxation of optically polarized sodium atoms,” Phys. Rev. 105, 1487–1489 (1957).
[CrossRef]

Appelt, S.

S. Appelt, P. Scheufler, and M. Mehring, “Direct observation of single- and double-quantum sublevel coherence in rubidium vapor by optical Raman beat detection,” Opt. Commun. 74, 110–114 (1989).
[CrossRef]

Attili, M. A.

A. P. Ghosh, C. D. Nabors, M. A. Attili, and J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985).
[CrossRef] [PubMed]

Bachmann, P.

J. Jeener, B. H. Meier, P. Bachmann, and R. R. Ernst, “Investigation of exchange processes by two-dimensional NMR spectroscopy,” J. Chem. Phys. 71, 4546–4553 (1979).
[CrossRef]

Barkov, L. M.

L. M. Barkov, D. A. Melik-Pasheayev, and M. S. Zolotorev, “Nonlinear Faraday rotation in samarium vapor,” Opt. Commun. 70, 467–472 (1989).
[CrossRef]

Bell, W. E.

W. E. Bell and A. L. Bloom, “Optically driven spin precession,” Phys. Rev. Lett. 6, 280–281 (1961).
[CrossRef]

Bernheim, R.

R. Bernheim, Optical pumping (Benjamin, New York, 1965).

Bloom, A. L.

W. E. Bell and A. L. Bloom, “Optically driven spin precession,” Phys. Rev. Lett. 6, 280–281 (1961).
[CrossRef]

Bodenhausen, G.

R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987).

Brewer, R. G.

J. Mlynek, N. C. Wong, R. G. DeVoe, E. S. Kintzer, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. Lett. 50, 993–996 (1983).
[CrossRef]

N. C. Wong, E. S. Kintzer, J. Mlynek, R. G. DeVoe, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. B 28, 4993–5010 (1983).
[CrossRef]

R. L. Shoemaker and R. G. Brewer, “Two-photon superradi-ance,” Phys. Rev. Lett. 28, 1430–1433 (1972).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, “Theorie quantique du cycle de pompage optique,” Ann. Phys. 7, 423–460 (1962).

C. Cohen-Tannoudji, “Observation d’un déplacement de raie de résonance magnetique causé par l’exicitation optique,” C. R. 252, 394–396 (1961).

Decomps, B.

M. Ducloy, M. P. Gorza, and B. Decomps, “Higher-order nonlinear effects in a gas laser: creation and detection of a hexadecupole moment in the neon 2p4level,” Opt. Commun. 8, 21–25 (1973).
[CrossRef]

Dehmelt, H. G.

H. G. Dehmelt, “Modulation of a light beam by precessing absorbing atoms,” Phys. Rev. 105, 1924–1925 (1957).
[CrossRef]

H. G. Dehmelt, “Slow spin relaxation of optically polarized sodium atoms,” Phys. Rev. 105, 1487–1489 (1957).
[CrossRef]

DeVoe, R. G.

N. C. Wong, E. S. Kintzer, J. Mlynek, R. G. DeVoe, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. B 28, 4993–5010 (1983).
[CrossRef]

J. Mlynek, N. C. Wong, R. G. DeVoe, E. S. Kintzer, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. Lett. 50, 993–996 (1983).
[CrossRef]

Drake, K. H.

K. H. Drake, W. Lange, and J. Mlynek, “Nonlinear Faraday and Voigt effect in a J= 1 to J′= 0 transition in atomic samarium vapour,” Opt. Commun. 66, 315–320 (1988).
[CrossRef]

Ducloy, M.

M. Ducloy, M. P. Gorza, and B. Decomps, “Higher-order nonlinear effects in a gas laser: creation and detection of a hexadecupole moment in the neon 2p4level,” Opt. Commun. 8, 21–25 (1973).
[CrossRef]

Ernst, R. R.

J. Jeener, B. H. Meier, P. Bachmann, and R. R. Ernst, “Investigation of exchange processes by two-dimensional NMR spectroscopy,” J. Chem. Phys. 71, 4546–4553 (1979).
[CrossRef]

R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987).

Fukuda, Y.

T. Mishina, Y. Fukuda, and T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
[CrossRef]

Gawlik, W.

W. Gawlik, J. Kowalski, R. Neumann, and F. Träger, “Observation of the electric hexadecupole moment of free Na atoms in a forward scattering experiment,” Opt. Commun. 12, 400–404 (1974).
[CrossRef]

Ghosh, A. P.

A. P. Ghosh, C. D. Nabors, M. A. Attili, and J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985).
[CrossRef] [PubMed]

Giraud-Cotton, S.

S. Giraud-Cotton, V. P. Kaftandjian, and L. Klein, “Magnetic optical activity in intense laser fields. I. Self-rotation and Verdet-constant,” Phys. Rev. A 32, 2211–2222 (1985).
[CrossRef] [PubMed]

Gorza, M. P.

M. Ducloy, M. P. Gorza, and B. Decomps, “Higher-order nonlinear effects in a gas laser: creation and detection of a hexadecupole moment in the neon 2p4level,” Opt. Commun. 8, 21–25 (1973).
[CrossRef]

Gough, D. S.

R. M. Lowe, D. S. Gough, R. J. McLean, and P. Hannaford, “Determination of ground-state depolarization rates by transmission Zeeman beat spectroscopy: application to Sm i 4f66s2 7F1,” Phys. Rev. A 36, 5490–5493 (1987).
[CrossRef] [PubMed]

Hannaford, P.

R. J. McLean, P. Hannaford, and R. M. Lowe, “Transmission Zeeman beat spectroscopy: application to the 7F ground level term of Sm i,” Phys. Rev. A 42, 6616–6628 (1990).
[CrossRef] [PubMed]

R. M. Lowe, D. S. Gough, R. J. McLean, and P. Hannaford, “Determination of ground-state depolarization rates by transmission Zeeman beat spectroscopy: application to Sm i 4f66s2 7F1,” Phys. Rev. A 36, 5490–5493 (1987).
[CrossRef] [PubMed]

Happer, W.

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
[CrossRef]

W. Happer and B. S. Mathur, “Off-resonant light as a probe of optically pumped alkali vapors,” Phys. Rev. Lett. 18, 577–580 (1967).
[CrossRef]

Hashi, T.

T. Mishina, Y. Fukuda, and T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
[CrossRef]

Herman, G.

G. Herman and A. Scharmann, “Untersuchungen zur Zeeman Spektroskopie mit Hilfe nichtlinearer Resonanzen eines multimoden Lasers,” Z. Phys. 254, 46–56 (1972).
[CrossRef]

Jeener, J.

J. Jeener, B. H. Meier, P. Bachmann, and R. R. Ernst, “Investigation of exchange processes by two-dimensional NMR spectroscopy,” J. Chem. Phys. 71, 4546–4553 (1979).
[CrossRef]

J. Jeener, Lecture presented at the Ampére International Summer School II, Basko Polje, Yugoslavia, 1971 (Université Libre de Bruxelles, B-1050 Bruxelles, Belgium).

Kaftandjian, V. P.

S. Giraud-Cotton, V. P. Kaftandjian, and L. Klein, “Magnetic optical activity in intense laser fields. I. Self-rotation and Verdet-constant,” Phys. Rev. A 32, 2211–2222 (1985).
[CrossRef] [PubMed]

Kastler, A.

A. Kastler, “Optical methods for studying Hertzian resonances,” Science 158, 214–221 (1967).
[CrossRef] [PubMed]

Kintzer, E. S.

J. Mlynek, N. C. Wong, R. G. DeVoe, E. S. Kintzer, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. Lett. 50, 993–996 (1983).
[CrossRef]

N. C. Wong, E. S. Kintzer, J. Mlynek, R. G. DeVoe, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. B 28, 4993–5010 (1983).
[CrossRef]

Klein, L.

S. Giraud-Cotton, V. P. Kaftandjian, and L. Klein, “Magnetic optical activity in intense laser fields. I. Self-rotation and Verdet-constant,” Phys. Rev. A 32, 2211–2222 (1985).
[CrossRef] [PubMed]

Klepel, H.

D. Suter, T. Marty, and H. Klepel. “Rotation properties of multipole moments in atomic sublevel spectroscopy,” Opt. Lett. 18, 531–533 (1993).
[CrossRef] [PubMed]

H. Klepel and D. Suter, “Transverse optical pumping with polarization-modulated light,” Opt. Commun. 90, 46–50 (1992).
[CrossRef]

D. Suter and H. Klepel, “Indirect observation of ‘forbidden’ Raman transitions by laser-induced coherence transfer,” Europhys. Lett. 19, 469–474 (1992).
[CrossRef]

D. Suter, H. Klepel, and J. Mlynek, “Time-resolved two-dimensional spectroscopy of optically driven atomic sublevel coherences,” Phys. Rev. Lett. 67, 2001–2004 (1991).
[CrossRef] [PubMed]

Kowalski, J.

W. Gawlik, J. Kowalski, R. Neumann, and F. Träger, “Observation of the electric hexadecupole moment of free Na atoms in a forward scattering experiment,” Opt. Commun. 12, 400–404 (1974).
[CrossRef]

Kulina, P.

P. Kulina and R. H. Rinkleff, “Determination of the tensor polarizabilities in Yb and Sm by quantum beat spectroscopy,” Z. Phys. A 304, 371–372 (1982).
[CrossRef]

Lange, W.

M. Rosatzin, D. Suter, W. Lange, and J. Mlynek, “Phase and amplitude variations of optically induced spin transients,” J. Opt. Soc. Am. B 7, 1231–1238 (1990).
[CrossRef]

K. H. Drake, W. Lange, and J. Mlynek, “Nonlinear Faraday and Voigt effect in a J= 1 to J′= 0 transition in atomic samarium vapour,” Opt. Commun. 66, 315–320 (1988).
[CrossRef]

W. Lange and J. Mlynek, “Quantum beats in transmission by time-resolved polarization spectroscopy,” Phys. Rev. Lett. 40, 1373–1375 (1978).
[CrossRef]

Lowe, R. M.

R. J. McLean, P. Hannaford, and R. M. Lowe, “Transmission Zeeman beat spectroscopy: application to the 7F ground level term of Sm i,” Phys. Rev. A 42, 6616–6628 (1990).
[CrossRef] [PubMed]

R. M. Lowe, D. S. Gough, R. J. McLean, and P. Hannaford, “Determination of ground-state depolarization rates by transmission Zeeman beat spectroscopy: application to Sm i 4f66s2 7F1,” Phys. Rev. A 36, 5490–5493 (1987).
[CrossRef] [PubMed]

Marty, T.

D. Suter, T. Marty, and H. Klepel. “Rotation properties of multipole moments in atomic sublevel spectroscopy,” Opt. Lett. 18, 531–533 (1993).
[CrossRef] [PubMed]

D. Suter and T. Marty, “Laser induced dynamics of atomic sublevel coherences,” Opt. Commun. 100, 443–450 (1993).
[CrossRef]

Mathur, B. S.

W. Happer and B. S. Mathur, “Off-resonant light as a probe of optically pumped alkali vapors,” Phys. Rev. Lett. 18, 577–580 (1967).
[CrossRef]

McLean, R. J.

R. J. McLean, P. Hannaford, and R. M. Lowe, “Transmission Zeeman beat spectroscopy: application to the 7F ground level term of Sm i,” Phys. Rev. A 42, 6616–6628 (1990).
[CrossRef] [PubMed]

R. M. Lowe, D. S. Gough, R. J. McLean, and P. Hannaford, “Determination of ground-state depolarization rates by transmission Zeeman beat spectroscopy: application to Sm i 4f66s2 7F1,” Phys. Rev. A 36, 5490–5493 (1987).
[CrossRef] [PubMed]

Mehring, M.

S. Appelt, P. Scheufler, and M. Mehring, “Direct observation of single- and double-quantum sublevel coherence in rubidium vapor by optical Raman beat detection,” Opt. Commun. 74, 110–114 (1989).
[CrossRef]

Meier, B. H.

J. Jeener, B. H. Meier, P. Bachmann, and R. R. Ernst, “Investigation of exchange processes by two-dimensional NMR spectroscopy,” J. Chem. Phys. 71, 4546–4553 (1979).
[CrossRef]

Melik-Pasheayev, D. A.

L. M. Barkov, D. A. Melik-Pasheayev, and M. S. Zolotorev, “Nonlinear Faraday rotation in samarium vapor,” Opt. Commun. 70, 467–472 (1989).
[CrossRef]

Mishina, T.

T. Mishina, Y. Fukuda, and T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
[CrossRef]

Mlynek, J.

D. Suter, H. Klepel, and J. Mlynek, “Time-resolved two-dimensional spectroscopy of optically driven atomic sublevel coherences,” Phys. Rev. Lett. 67, 2001–2004 (1991).
[CrossRef] [PubMed]

D. Suter and J. Mlynek, “Dynamics of atomic sublevel coherences during modulated optical pumping,” Phys. Rev. A 43, 6124–6134 (1991).
[CrossRef] [PubMed]

D. Suter and J. Mlynek, “Laser excitation and detection of magnetic resonance,” in Adv. Magn. Opt. Reson. 16, 1–83 (1991).

M. Rosatzin, D. Suter, W. Lange, and J. Mlynek, “Phase and amplitude variations of optically induced spin transients,” J. Opt. Soc. Am. B 7, 1231–1238 (1990).
[CrossRef]

D. Sutter, M. Rosatzin, and J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
[CrossRef]

K. H. Drake, W. Lange, and J. Mlynek, “Nonlinear Faraday and Voigt effect in a J= 1 to J′= 0 transition in atomic samarium vapour,” Opt. Commun. 66, 315–320 (1988).
[CrossRef]

N. C. Wong, E. S. Kintzer, J. Mlynek, R. G. DeVoe, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. B 28, 4993–5010 (1983).
[CrossRef]

J. Mlynek, N. C. Wong, R. G. DeVoe, E. S. Kintzer, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. Lett. 50, 993–996 (1983).
[CrossRef]

W. Lange and J. Mlynek, “Quantum beats in transmission by time-resolved polarization spectroscopy,” Phys. Rev. Lett. 40, 1373–1375 (1978).
[CrossRef]

Nabors, C. D.

A. P. Ghosh, C. D. Nabors, M. A. Attili, and J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985).
[CrossRef] [PubMed]

Neumann, R.

W. Gawlik, J. Kowalski, R. Neumann, and F. Träger, “Observation of the electric hexadecupole moment of free Na atoms in a forward scattering experiment,” Opt. Commun. 12, 400–404 (1974).
[CrossRef]

Omont, A.

A. Omont, “Irreducible components of the density matrix. Application to optical pumping,” Prog. Quantum Electron. 5, 69–138 (1977).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, “Modulated birefringence of a spin-assembly in magnetic resonance,” Phys. Lett. 27A, 509–510 (1968).

Partridge, R. B.

R. B. Partridge and G. W. Series, “The modulated absorption of light in an optical pumping experiment on 4He,” Proc. Phys. Soc. 88, 969–982 (1966).
[CrossRef]

Pearson, J. G.

D. Suter and J. G. Pearson, “Experimental classification of multi-spin coherence under the full rotation group,” Chem. Phys. Lett. 144, 328–332 (1988).
[CrossRef]

Rinkleff, R. H.

P. Kulina and R. H. Rinkleff, “Determination of the tensor polarizabilities in Yb and Sm by quantum beat spectroscopy,” Z. Phys. A 304, 371–372 (1982).
[CrossRef]

Rosatzin, M.

D. Sutter, M. Rosatzin, and J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
[CrossRef]

M. Rosatzin, D. Suter, W. Lange, and J. Mlynek, “Phase and amplitude variations of optically induced spin transients,” J. Opt. Soc. Am. B 7, 1231–1238 (1990).
[CrossRef]

Scharmann, A.

G. Herman and A. Scharmann, “Untersuchungen zur Zeeman Spektroskopie mit Hilfe nichtlinearer Resonanzen eines multimoden Lasers,” Z. Phys. 254, 46–56 (1972).
[CrossRef]

Scheufler, P.

S. Appelt, P. Scheufler, and M. Mehring, “Direct observation of single- and double-quantum sublevel coherence in rubidium vapor by optical Raman beat detection,” Opt. Commun. 74, 110–114 (1989).
[CrossRef]

Series, G. W.

R. B. Partridge and G. W. Series, “The modulated absorption of light in an optical pumping experiment on 4He,” Proc. Phys. Soc. 88, 969–982 (1966).
[CrossRef]

G. W. Series, “Theory of the modulation of light in optical pumping experiments,” Proc. Phys. Soc. 88, 957–968 (1966).
[CrossRef]

Shoemaker, R. L.

R. L. Shoemaker and R. G. Brewer, “Two-photon superradi-ance,” Phys. Rev. Lett. 28, 1430–1433 (1972).
[CrossRef]

Suter, D.

D. Suter and T. Marty, “Laser induced dynamics of atomic sublevel coherences,” Opt. Commun. 100, 443–450 (1993).
[CrossRef]

D. Suter, T. Marty, and H. Klepel. “Rotation properties of multipole moments in atomic sublevel spectroscopy,” Opt. Lett. 18, 531–533 (1993).
[CrossRef] [PubMed]

H. Klepel and D. Suter, “Transverse optical pumping with polarization-modulated light,” Opt. Commun. 90, 46–50 (1992).
[CrossRef]

D. Suter and H. Klepel, “Indirect observation of ‘forbidden’ Raman transitions by laser-induced coherence transfer,” Europhys. Lett. 19, 469–474 (1992).
[CrossRef]

D. Suter and J. Mlynek, “Laser excitation and detection of magnetic resonance,” in Adv. Magn. Opt. Reson. 16, 1–83 (1991).

D. Suter, H. Klepel, and J. Mlynek, “Time-resolved two-dimensional spectroscopy of optically driven atomic sublevel coherences,” Phys. Rev. Lett. 67, 2001–2004 (1991).
[CrossRef] [PubMed]

D. Suter and J. Mlynek, “Dynamics of atomic sublevel coherences during modulated optical pumping,” Phys. Rev. A 43, 6124–6134 (1991).
[CrossRef] [PubMed]

M. Rosatzin, D. Suter, W. Lange, and J. Mlynek, “Phase and amplitude variations of optically induced spin transients,” J. Opt. Soc. Am. B 7, 1231–1238 (1990).
[CrossRef]

D. Suter and J. G. Pearson, “Experimental classification of multi-spin coherence under the full rotation group,” Chem. Phys. Lett. 144, 328–332 (1988).
[CrossRef]

Sutter, D.

D. Sutter, M. Rosatzin, and J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
[CrossRef]

Thomas, J. E.

A. P. Ghosh, C. D. Nabors, M. A. Attili, and J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985).
[CrossRef] [PubMed]

Träger, F.

W. Gawlik, J. Kowalski, R. Neumann, and F. Träger, “Observation of the electric hexadecupole moment of free Na atoms in a forward scattering experiment,” Opt. Commun. 12, 400–404 (1974).
[CrossRef]

Wokaun, A.

R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987).

Wong, N. C.

J. Mlynek, N. C. Wong, R. G. DeVoe, E. S. Kintzer, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. Lett. 50, 993–996 (1983).
[CrossRef]

N. C. Wong, E. S. Kintzer, J. Mlynek, R. G. DeVoe, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. B 28, 4993–5010 (1983).
[CrossRef]

Zolotorev, M. S.

L. M. Barkov, D. A. Melik-Pasheayev, and M. S. Zolotorev, “Nonlinear Faraday rotation in samarium vapor,” Opt. Commun. 70, 467–472 (1989).
[CrossRef]

Adv. Magn. Opt. Reson. (1)

D. Suter and J. Mlynek, “Laser excitation and detection of magnetic resonance,” in Adv. Magn. Opt. Reson. 16, 1–83 (1991).

Ann. Phys. (1)

C. Cohen-Tannoudji, “Theorie quantique du cycle de pompage optique,” Ann. Phys. 7, 423–460 (1962).

C. R. (1)

C. Cohen-Tannoudji, “Observation d’un déplacement de raie de résonance magnetique causé par l’exicitation optique,” C. R. 252, 394–396 (1961).

Chem. Phys. Lett. (1)

D. Suter and J. G. Pearson, “Experimental classification of multi-spin coherence under the full rotation group,” Chem. Phys. Lett. 144, 328–332 (1988).
[CrossRef]

Europhys. Lett. (1)

D. Suter and H. Klepel, “Indirect observation of ‘forbidden’ Raman transitions by laser-induced coherence transfer,” Europhys. Lett. 19, 469–474 (1992).
[CrossRef]

J. Chem. Phys. (1)

J. Jeener, B. H. Meier, P. Bachmann, and R. R. Ernst, “Investigation of exchange processes by two-dimensional NMR spectroscopy,” J. Chem. Phys. 71, 4546–4553 (1979).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (8)

H. Klepel and D. Suter, “Transverse optical pumping with polarization-modulated light,” Opt. Commun. 90, 46–50 (1992).
[CrossRef]

D. Suter and T. Marty, “Laser induced dynamics of atomic sublevel coherences,” Opt. Commun. 100, 443–450 (1993).
[CrossRef]

T. Mishina, Y. Fukuda, and T. Hashi, “Optical generation and detection of Δm= 2 Zeeman coherence in the Cs ground state with a diode laser,” Opt. Commun. 66, 25–30 (1988).
[CrossRef]

S. Appelt, P. Scheufler, and M. Mehring, “Direct observation of single- and double-quantum sublevel coherence in rubidium vapor by optical Raman beat detection,” Opt. Commun. 74, 110–114 (1989).
[CrossRef]

M. Ducloy, M. P. Gorza, and B. Decomps, “Higher-order nonlinear effects in a gas laser: creation and detection of a hexadecupole moment in the neon 2p4level,” Opt. Commun. 8, 21–25 (1973).
[CrossRef]

W. Gawlik, J. Kowalski, R. Neumann, and F. Träger, “Observation of the electric hexadecupole moment of free Na atoms in a forward scattering experiment,” Opt. Commun. 12, 400–404 (1974).
[CrossRef]

K. H. Drake, W. Lange, and J. Mlynek, “Nonlinear Faraday and Voigt effect in a J= 1 to J′= 0 transition in atomic samarium vapour,” Opt. Commun. 66, 315–320 (1988).
[CrossRef]

L. M. Barkov, D. A. Melik-Pasheayev, and M. S. Zolotorev, “Nonlinear Faraday rotation in samarium vapor,” Opt. Commun. 70, 467–472 (1989).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. (1)

S. Pancharatnam, “Modulated birefringence of a spin-assembly in magnetic resonance,” Phys. Lett. 27A, 509–510 (1968).

Phys. Rev. (2)

H. G. Dehmelt, “Modulation of a light beam by precessing absorbing atoms,” Phys. Rev. 105, 1924–1925 (1957).
[CrossRef]

H. G. Dehmelt, “Slow spin relaxation of optically polarized sodium atoms,” Phys. Rev. 105, 1487–1489 (1957).
[CrossRef]

Phys. Rev. A (5)

D. Sutter, M. Rosatzin, and J. Mlynek, “Optically driven spin nutations in the ground state of atomic sodium,” Phys. Rev. A 41, 1634–1644 (1990).
[CrossRef]

R. M. Lowe, D. S. Gough, R. J. McLean, and P. Hannaford, “Determination of ground-state depolarization rates by transmission Zeeman beat spectroscopy: application to Sm i 4f66s2 7F1,” Phys. Rev. A 36, 5490–5493 (1987).
[CrossRef] [PubMed]

R. J. McLean, P. Hannaford, and R. M. Lowe, “Transmission Zeeman beat spectroscopy: application to the 7F ground level term of Sm i,” Phys. Rev. A 42, 6616–6628 (1990).
[CrossRef] [PubMed]

S. Giraud-Cotton, V. P. Kaftandjian, and L. Klein, “Magnetic optical activity in intense laser fields. I. Self-rotation and Verdet-constant,” Phys. Rev. A 32, 2211–2222 (1985).
[CrossRef] [PubMed]

D. Suter and J. Mlynek, “Dynamics of atomic sublevel coherences during modulated optical pumping,” Phys. Rev. A 43, 6124–6134 (1991).
[CrossRef] [PubMed]

Phys. Rev. B (1)

N. C. Wong, E. S. Kintzer, J. Mlynek, R. G. DeVoe, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. B 28, 4993–5010 (1983).
[CrossRef]

Phys. Rev. Lett. (7)

J. Mlynek, N. C. Wong, R. G. DeVoe, E. S. Kintzer, and R. G. Brewer, “Raman heterodyne detection of nuclear magnetic resonance,” Phys. Rev. Lett. 50, 993–996 (1983).
[CrossRef]

W. E. Bell and A. L. Bloom, “Optically driven spin precession,” Phys. Rev. Lett. 6, 280–281 (1961).
[CrossRef]

W. Happer and B. S. Mathur, “Off-resonant light as a probe of optically pumped alkali vapors,” Phys. Rev. Lett. 18, 577–580 (1967).
[CrossRef]

A. P. Ghosh, C. D. Nabors, M. A. Attili, and J. E. Thomas, “3P1-orientation velocity-changing collision kernels studied by isolated multipole echoes,” Phys. Rev. Lett. 54, 1794–1797 (1985).
[CrossRef] [PubMed]

W. Lange and J. Mlynek, “Quantum beats in transmission by time-resolved polarization spectroscopy,” Phys. Rev. Lett. 40, 1373–1375 (1978).
[CrossRef]

R. L. Shoemaker and R. G. Brewer, “Two-photon superradi-ance,” Phys. Rev. Lett. 28, 1430–1433 (1972).
[CrossRef]

D. Suter, H. Klepel, and J. Mlynek, “Time-resolved two-dimensional spectroscopy of optically driven atomic sublevel coherences,” Phys. Rev. Lett. 67, 2001–2004 (1991).
[CrossRef] [PubMed]

Proc. Phys. Soc. (2)

G. W. Series, “Theory of the modulation of light in optical pumping experiments,” Proc. Phys. Soc. 88, 957–968 (1966).
[CrossRef]

R. B. Partridge and G. W. Series, “The modulated absorption of light in an optical pumping experiment on 4He,” Proc. Phys. Soc. 88, 969–982 (1966).
[CrossRef]

Prog. Quantum Electron. (1)

A. Omont, “Irreducible components of the density matrix. Application to optical pumping,” Prog. Quantum Electron. 5, 69–138 (1977).
[CrossRef]

Rev. Mod. Phys. (1)

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
[CrossRef]

Science (1)

A. Kastler, “Optical methods for studying Hertzian resonances,” Science 158, 214–221 (1967).
[CrossRef] [PubMed]

Z. Phys. (1)

G. Herman and A. Scharmann, “Untersuchungen zur Zeeman Spektroskopie mit Hilfe nichtlinearer Resonanzen eines multimoden Lasers,” Z. Phys. 254, 46–56 (1972).
[CrossRef]

Z. Phys. A (1)

P. Kulina and R. H. Rinkleff, “Determination of the tensor polarizabilities in Yb and Sm by quantum beat spectroscopy,” Z. Phys. A 304, 371–372 (1982).
[CrossRef]

Other (3)

R. Bernheim, Optical pumping (Benjamin, New York, 1965).

J. Jeener, Lecture presented at the Ampére International Summer School II, Basko Polje, Yugoslavia, 1971 (Université Libre de Bruxelles, B-1050 Bruxelles, Belgium).

R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987).

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Figures (10)

Fig. 1
Fig. 1

a, Energy-level system of the sodium ground state and b, calculated sublevel spectrum in a magnetic field of 0.7 mG. The numbers above the resonance lines indicate the assignment to the sublevel transitions in the form ( F , m F F , m F ). The origin of the frequency axis is at 5.2 MHz.

Fig. 2
Fig. 2

Principle of the pump–probe experiment used to create and observe atomic multipole moments. B, Magnetic field.

Fig. 3
Fig. 3

Example of time-domain sublevel spectroscopy. The top trace shows the time dependence of the pump laser intensity. The second trace represents the signal recorded by polarization-selective detection of the transmitted probe beam. The bottom trace contains the sublevel spectrum, which was obtained by Fourier transformation and contains signal contributions from q = ±1 multipole moments. B, Magnetic field strength; PSD, phase-sensitive detection.

Fig. 4
Fig. 4

Principle of two-dimensional time-resolved spectroscopy. The insets indicate the evolution of the system. The bottom trace corresponds to the pump laser intensity as a function of time.

Fig. 5
Fig. 5

Larmor precession of multipoles and observable signal for dipoles (upper part) and second-order tensor elements (lower part).

Fig. 6
Fig. 6

Line shapes as functions of the signal phase for a complex Lorentzian line-shape function.

Fig. 7
Fig. 7

Setup used for the experimental implementation. P, polarizer; BS, beam splitter, AOM, acousto-optic modulator; EOM, electro-optic modulator; rf, radio-frequency synthesizer; B, magnetic field; PBS, polarizing beam splitter; PD1,2, photodiodes; M, mixer.

Fig. 8
Fig. 8

Raman spectra of the Zeeman transitions in the sodium ground state recorded with various modulation phases during the first pulse. Each column represents one resonance line for four different phases. Above each column the assignment to the individual sublevel transitions is given as well as the order of the corresponding multipole moment.

Fig. 9
Fig. 9

Phase of the resonance lines as a function of the excitation phase. The solid lines represent the theoretical behavior; the circles, squares, and triangles show the experimental values.

Fig. 10
Fig. 10

Sublevel spectra for individual multipole orders obtained by linear combination of four spectra measured with the orientation of the atomic state rotated by 0, π/2, π, and 3π/2. For details, see text.

Equations (31)

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H g = A J · I μ J B · J μ I B · I
H 2 = μ F B F z μ F ( 2 ) B 2 F z 2 ,
ρ = k q a k q T q ( k ) ,
ρ ( t ) = exp ( i μ F ( 2 ) B 2 t F z 2 ) exp ( i Ω L t F z ) ρ ( 0 ) × exp ( i Ω L t F z ) exp ( i μ F ( 2 ) B 2 t F z 2 ) .
ρ ( t ) = exp ( i μ F ( 2 ) B 2 t F z 2 ) [ k q exp ( i q Ω L t ) a k q T q ( k ) ] × exp ( i μ F ( 2 ) B 2 t F z 2 ) .
A q , r = k = | q | 2 F α k q r T q ( k ) .
exp ( i ϕ F z ) A q , r exp ( i ϕ F z ) = exp ( i q ϕ ) A q , r ,
exp ( i H t ) A q , r exp ( i H t ) = exp ( i ω q , r t ) A q , r .
ω q , r = q Ω L + μ F ( 2 ) B 2 ( m F 2 m F 2 ) .
ρ ( 0 ) = q , r d q , r A q , r
ρ ( t ) = q , r exp ( i ω q , r t ) d q , r A q , r .
H ( r ) = U H U i U U , ρ ( r ) = U ρ U ,
U = exp ( i ω t F z ) .
H 2 ( r ) = ( Ω L ω ) F z μ F ( 2 ) B 2 F z 2 ,
ρ ( r ) ( t ) = q , r exp [ i ω q , r ( r ) t ] d q , r A q , r , ω q , r ( r ) = q ( Ω L ω ) + μ F ( 2 ) B 2 ( m F 2 m F 2 ) .
H mod = exp ( i ω mod t F z ) H int exp ( i ω mod t F z ) .
A ( ϕ ) = exp ( i ϕ F z ) A ( 0 ) exp ( i ϕ F z ) .
H mod ( ϕ ) = exp [ i ( ω mod t + ϕ ) F z ] H int × exp [ i ( ω mod t + ϕ ) F z ] .
H mod ( r ) ( ϕ ) = exp ( i ϕ F z ) H int exp ( i ϕ F z ) ;
ρ ( t ; ϕ ) = exp [ H mod ( r ) ( ϕ ) t ] ρ eq exp [ H mod ( r ) ( ϕ ) t ] = exp ( i ϕ F z ) ρ ( t ; ϕ = 0 ) exp ( i ϕ F z ) ,
ρ ( r ) ( t 1 ) = exp [ i H ( r ) t 1 ] ρ 0 exp [ i H ( r ) t 1 ] = q , r d q , r exp [ i ω q , r ( r ) t 1 ] A q , r .
ρ ( t 1 , 0 ) = q , r A q , r q , r r q r q r d q , r exp ( i ω q r t 1 ) .
ρ ( t 1 , t 2 ) = q , r exp ( i ω q r t 2 ) A q , r × q , r r q r q r d q , r exp ( i ω q r t 1 ) .
T 1 ( 1 ) = r = 1 2 F b 1 r A 1 r .
S ( t 1 , t 2 ) = Tr [ T 1 ( 1 ) ρ ( t 1 , t 2 ) ] = r b 1 r exp ( i ω 1 r t 2 ) q , r r 1 r q r d q , r exp ( i ω q r t 1 ) ,
S ( t 1 ) = q , r c q , r d q , r exp ( i ω q r t 1 ) = Tr [ A eff ρ ( t 1 ) ] ,
c q , r = r b 1 r exp ( i ω 1 r t 2 ) r 1 r q r
A eff ( t 2 ) = q , r c q , r A q , r = r b 1 r exp ( i ω 1 r t 2 ) q , r r 1 r q r A q , r .
A eff ( ω 2 ) = g ( ω 2 ) q , r r 1 r q r A q , r , g ( ω 2 ) = F [ r b 1 r exp ( i ω 1 r t 2 ) ] ,
ρ ( t 1 ; ϕ ) = exp ( i ϕ F z ) ρ ( t 1 , 0 ) exp ( i ϕ F z ) = q , r exp ( i q ϕ ) d q , r exp ( i ω q r t 1 ) A q , r .
S ( t 1 , ϕ ) = q , r c q , r d q , r exp ( i q ϕ ) exp ( i ω q r t 1 ) .

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