Abstract

We present a method to determine the direction of a magnetic field in an experimental volume by analyzing the polarization of resonant optical radiation and data that demonstrate this method with a single trapped  40Ca+ ion. With a single trapped  40Ca+ ion, the results are insensitive to the magnitude of the magnetic field, the laser intensity, the detuning, and the bandwidth, provided the fluorescence signal is detectable. This method is applicable to many atomic systems in which optical pumping is possible.

© 2005 Optical Society of America

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References

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  1. R. Wynands, A. Nagel, S. Brandt, D. Meschede, and A. Weis, "Selection rules and line strengths of Zeeman-split dark resonances," Phys. Rev. A 58, 196-203 (1998).
    [CrossRef]
  2. H. Lee, M. Fleischhauer, and M. O. Scully, "Sensitive detection of magnetic fields including their orientation with a magnetometer based on atomic phase coherence," Phys. Rev. A 58, 2587-2595 (1998).
    [CrossRef]
  3. D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
    [CrossRef]
  4. W. M. Itano, "External-field shifts of the 199Hg+ optical frequency standard," J. Res. Natl. Inst. Stand. Technol. 105, 829-837 (2000).
    [CrossRef]
  5. E. N. Fortson, "Possibility of measuring parity nonconservation with a single trapped atomic ion," Phys. Rev. Lett. 70, 2383-2386 (1993).
    [CrossRef] [PubMed]
  6. W. M. Klipstein, S. K. Lamoreaux, and E. N. Fortson, "Observation of spontaneous spin polarization in an optically pumped cesium vapor," Phys. Rev. Lett. 76, 2266-2269 (1996).
    [CrossRef] [PubMed]
  7. D. Budker, D. J. Orlando, and V. Yashchuk, "Nonlinear la- ser spectroscopy and magneto-optics," Am. J. Phys. 67, 584-592 (1999).
    [CrossRef]
  8. "Searches for spatial anisotropy and a permanent electric dipole moment using optically pumped Hg," S. K. Lamoreaux, Ph.D. Thesis (University of Washington, Seattle, Wash., 1986), manuscript available from the author.
  9. S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and N. Fortson, "New constraints on time-reversal asymmetry from a search for a permanent electric dipole moment of 199Hg," Phys. Rev. Lett. 59, 2275-2278 (1987).
    [CrossRef] [PubMed]
  10. R. D. Cowen, The Theory of Atomic Structure and Spectra , (U. California, Berkeley, 1981), pp. 142-147.
  11. H. R. Gray, R. M. Whitley, and C. R. Stroud, Jr., "Coherent trapping of atomic populations," Opt. Lett. 3, 218-220 (1978).
    [CrossRef] [PubMed]

2002 (1)

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

2000 (1)

W. M. Itano, "External-field shifts of the 199Hg+ optical frequency standard," J. Res. Natl. Inst. Stand. Technol. 105, 829-837 (2000).
[CrossRef]

1999 (1)

D. Budker, D. J. Orlando, and V. Yashchuk, "Nonlinear la- ser spectroscopy and magneto-optics," Am. J. Phys. 67, 584-592 (1999).
[CrossRef]

1998 (2)

R. Wynands, A. Nagel, S. Brandt, D. Meschede, and A. Weis, "Selection rules and line strengths of Zeeman-split dark resonances," Phys. Rev. A 58, 196-203 (1998).
[CrossRef]

H. Lee, M. Fleischhauer, and M. O. Scully, "Sensitive detection of magnetic fields including their orientation with a magnetometer based on atomic phase coherence," Phys. Rev. A 58, 2587-2595 (1998).
[CrossRef]

1996 (1)

W. M. Klipstein, S. K. Lamoreaux, and E. N. Fortson, "Observation of spontaneous spin polarization in an optically pumped cesium vapor," Phys. Rev. Lett. 76, 2266-2269 (1996).
[CrossRef] [PubMed]

1993 (1)

E. N. Fortson, "Possibility of measuring parity nonconservation with a single trapped atomic ion," Phys. Rev. Lett. 70, 2383-2386 (1993).
[CrossRef] [PubMed]

1987 (1)

S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and N. Fortson, "New constraints on time-reversal asymmetry from a search for a permanent electric dipole moment of 199Hg," Phys. Rev. Lett. 59, 2275-2278 (1987).
[CrossRef] [PubMed]

1978 (1)

Brandt, S.

R. Wynands, A. Nagel, S. Brandt, D. Meschede, and A. Weis, "Selection rules and line strengths of Zeeman-split dark resonances," Phys. Rev. A 58, 196-203 (1998).
[CrossRef]

Budker, D.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

D. Budker, D. J. Orlando, and V. Yashchuk, "Nonlinear la- ser spectroscopy and magneto-optics," Am. J. Phys. 67, 584-592 (1999).
[CrossRef]

Fleischhauer, M.

H. Lee, M. Fleischhauer, and M. O. Scully, "Sensitive detection of magnetic fields including their orientation with a magnetometer based on atomic phase coherence," Phys. Rev. A 58, 2587-2595 (1998).
[CrossRef]

Fortson, E. N.

W. M. Klipstein, S. K. Lamoreaux, and E. N. Fortson, "Observation of spontaneous spin polarization in an optically pumped cesium vapor," Phys. Rev. Lett. 76, 2266-2269 (1996).
[CrossRef] [PubMed]

E. N. Fortson, "Possibility of measuring parity nonconservation with a single trapped atomic ion," Phys. Rev. Lett. 70, 2383-2386 (1993).
[CrossRef] [PubMed]

Fortson, N.

S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and N. Fortson, "New constraints on time-reversal asymmetry from a search for a permanent electric dipole moment of 199Hg," Phys. Rev. Lett. 59, 2275-2278 (1987).
[CrossRef] [PubMed]

Gawlik, W.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Gray, H. R.

Heckel, B. R.

S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and N. Fortson, "New constraints on time-reversal asymmetry from a search for a permanent electric dipole moment of 199Hg," Phys. Rev. Lett. 59, 2275-2278 (1987).
[CrossRef] [PubMed]

Itano, W. M.

W. M. Itano, "External-field shifts of the 199Hg+ optical frequency standard," J. Res. Natl. Inst. Stand. Technol. 105, 829-837 (2000).
[CrossRef]

Jacobs, J. P.

S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and N. Fortson, "New constraints on time-reversal asymmetry from a search for a permanent electric dipole moment of 199Hg," Phys. Rev. Lett. 59, 2275-2278 (1987).
[CrossRef] [PubMed]

Kimball, D. F.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Klipstein, W. M.

W. M. Klipstein, S. K. Lamoreaux, and E. N. Fortson, "Observation of spontaneous spin polarization in an optically pumped cesium vapor," Phys. Rev. Lett. 76, 2266-2269 (1996).
[CrossRef] [PubMed]

Lamoreaux, S. K.

W. M. Klipstein, S. K. Lamoreaux, and E. N. Fortson, "Observation of spontaneous spin polarization in an optically pumped cesium vapor," Phys. Rev. Lett. 76, 2266-2269 (1996).
[CrossRef] [PubMed]

S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and N. Fortson, "New constraints on time-reversal asymmetry from a search for a permanent electric dipole moment of 199Hg," Phys. Rev. Lett. 59, 2275-2278 (1987).
[CrossRef] [PubMed]

Lee, H.

H. Lee, M. Fleischhauer, and M. O. Scully, "Sensitive detection of magnetic fields including their orientation with a magnetometer based on atomic phase coherence," Phys. Rev. A 58, 2587-2595 (1998).
[CrossRef]

Meschede, D.

R. Wynands, A. Nagel, S. Brandt, D. Meschede, and A. Weis, "Selection rules and line strengths of Zeeman-split dark resonances," Phys. Rev. A 58, 196-203 (1998).
[CrossRef]

Nagel, A.

R. Wynands, A. Nagel, S. Brandt, D. Meschede, and A. Weis, "Selection rules and line strengths of Zeeman-split dark resonances," Phys. Rev. A 58, 196-203 (1998).
[CrossRef]

Orlando, D. J.

D. Budker, D. J. Orlando, and V. Yashchuk, "Nonlinear la- ser spectroscopy and magneto-optics," Am. J. Phys. 67, 584-592 (1999).
[CrossRef]

Raab, F. J.

S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and N. Fortson, "New constraints on time-reversal asymmetry from a search for a permanent electric dipole moment of 199Hg," Phys. Rev. Lett. 59, 2275-2278 (1987).
[CrossRef] [PubMed]

Rochester, S. M.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Scully, M. O.

H. Lee, M. Fleischhauer, and M. O. Scully, "Sensitive detection of magnetic fields including their orientation with a magnetometer based on atomic phase coherence," Phys. Rev. A 58, 2587-2595 (1998).
[CrossRef]

Stroud, C. R.

Weis, A.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

R. Wynands, A. Nagel, S. Brandt, D. Meschede, and A. Weis, "Selection rules and line strengths of Zeeman-split dark resonances," Phys. Rev. A 58, 196-203 (1998).
[CrossRef]

Whitley, R. M.

Wynands, R.

R. Wynands, A. Nagel, S. Brandt, D. Meschede, and A. Weis, "Selection rules and line strengths of Zeeman-split dark resonances," Phys. Rev. A 58, 196-203 (1998).
[CrossRef]

Yashchuk, V.

D. Budker, D. J. Orlando, and V. Yashchuk, "Nonlinear la- ser spectroscopy and magneto-optics," Am. J. Phys. 67, 584-592 (1999).
[CrossRef]

Yashchuk, V. V.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Am. J. Phys. (1)

D. Budker, D. J. Orlando, and V. Yashchuk, "Nonlinear la- ser spectroscopy and magneto-optics," Am. J. Phys. 67, 584-592 (1999).
[CrossRef]

J. Res. Natl. Inst. Stand. Technol. (1)

W. M. Itano, "External-field shifts of the 199Hg+ optical frequency standard," J. Res. Natl. Inst. Stand. Technol. 105, 829-837 (2000).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

R. Wynands, A. Nagel, S. Brandt, D. Meschede, and A. Weis, "Selection rules and line strengths of Zeeman-split dark resonances," Phys. Rev. A 58, 196-203 (1998).
[CrossRef]

H. Lee, M. Fleischhauer, and M. O. Scully, "Sensitive detection of magnetic fields including their orientation with a magnetometer based on atomic phase coherence," Phys. Rev. A 58, 2587-2595 (1998).
[CrossRef]

Phys. Rev. Lett. (3)

E. N. Fortson, "Possibility of measuring parity nonconservation with a single trapped atomic ion," Phys. Rev. Lett. 70, 2383-2386 (1993).
[CrossRef] [PubMed]

W. M. Klipstein, S. K. Lamoreaux, and E. N. Fortson, "Observation of spontaneous spin polarization in an optically pumped cesium vapor," Phys. Rev. Lett. 76, 2266-2269 (1996).
[CrossRef] [PubMed]

S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and N. Fortson, "New constraints on time-reversal asymmetry from a search for a permanent electric dipole moment of 199Hg," Phys. Rev. Lett. 59, 2275-2278 (1987).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Other (2)

R. D. Cowen, The Theory of Atomic Structure and Spectra , (U. California, Berkeley, 1981), pp. 142-147.

"Searches for spatial anisotropy and a permanent electric dipole moment using optically pumped Hg," S. K. Lamoreaux, Ph.D. Thesis (University of Washington, Seattle, Wash., 1986), manuscript available from the author.

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Figures (5)

Fig. 1
Fig. 1

Schematic of the four possible solutions from Eqs. (3). Small shifts. ·ΔB1 and ·ΔB2 (smaller arrows) can be used to determine the correct solution.

Fig. 2
Fig. 2

Partial Grotrian diagram for Ca+.

Fig. 3
Fig. 3

Magnetic field coils are aligned along {X, Y, Z}, whereas the 866-nm light is described in the {x, y, z} basis; kˆ=zˆ and is in the {x, y}-plane.

Fig. 4
Fig. 4

Relative calibration of the coils with linear polarization. These data are fitted to the curves Ii=(0.58/0.57)(μY/μX)IY+const.

Fig. 5
Fig. 5

Bi2 as a function of IX.

Tables (1)

Tables Icon

Table 1 Relative Calibration of the Coils with Linear Polarization

Equations (16)

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R=2πΩT2g(ω),
g(ω)=γ/2π(ω0-ω)2+γ2/4,
ΩT2=e242 m|η, F, m|E·r|η, F, m|2=e242 m,qEqF1F-mqmη, Frη, F2,
|η, Frη, F|2=γ(2F+1)3c34ω03e24π0-1
E+1E0E-1=E21-i00021i0U(θ)U(ϕ)xyeiα0,
S1=Ix-Iy=x2-y2=1-cos2(θ)1+cos2(θ) cos(2ϕ),
S2=Id-Id¯=2xy cos(α)=1-cos2(θ)1+cos2(θ) sin(2ϕ),
S3=I-1-I+1=2xy sin(α)=±2 cos(θ)1+cos2(θ),
S0=S12+S22+S32=1,
cos(θ)=±1-S12+S22S3,
cos(2ϕ)=S1S12+S22,
sin(2ϕ)=S2S12+S22.
Bi=μiIi+B0,i,i={X, Y, Z},
δθ=δBˆ,
δϕ=δBˆ|sin(θ)|.
B=μ·I+B0

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