Abstract

Optical magnetometers measure magnetic fields with extremely high precision and without cryogenics. However, at geomagnetic fields, important for applications from landmine removal to archaeology, they suffer from nonlinear Zeeman splitting, leading to systematic dependence on sensor orientation. We present experimental results on a method of eliminating this systematic error, using the hexadecapole atomic polarization moment. In particular, we demonstrate selective production of the atomic hexadecapole moment at Earth’s magnetic field and verify its immunity to nonlinear Zeeman splitting. This technique promises to eliminate directional errors in all-optical atomic magnetometers, potentially improving their measurement accuracy by several orders of magnitude.

© 2008 Optical Society of America

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  1. J. Vanier and C. Audoin, The quantum physics of atomic frequency standards (A. Hilger, Bristol; Philadelphia, 1989).
    [CrossRef]
  2. D. Budker and M. V. Romalis, "Optical Magnetometry," Nat. Phys. 3, 227-234 (2007).
    [CrossRef]
  3. P. S. J. Ivan H. Deutsch, Gavin K. Brennen, "Quantum Computing with Neutral Atoms in an Optical Lattice," Fortschritte der Physik 48, 925-943 (2000).
    [CrossRef]
  4. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
    [CrossRef] [PubMed]
  5. G. Yusa, K. Muraki, K. Takashina, K. Hashimoto, and Y. Hirayama, "Controlled multiple quantum coherences of nuclear spins in a nanometre-scale device," Nature 434, 1001-1005 (2005).
    [CrossRef] [PubMed]
  6. E. B. Alexandrov, "Recent Progress in Optically Pumped Magnetometers," Phys. Scr. T105, 27-30 (2003).
    [CrossRef]
  7. I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, "A subfemtotesla multichannel atomic magnetometer," Nature 422, 596-599 (2003).
    [CrossRef] [PubMed]
  8. V. V. Yashchuk, D. Budker,W. Gawlik, D. F. Kimball, Y. P. Malakyan, and S. M. Rochester, "Selective addressing of high-rank atomic polarization moments," Phys. Rev. Lett. 90, 253,001 (2003).
    [CrossRef]
  9. S. Pustelny, D. F. Jackson Kimball, S. M. Rochester, V. V. Yashchuk, W. Gawlik, and D. Budker, "Pump-probe nonlinear magneto-optical rotation with frequency-modulated light," Phys. Rev. A 73, 023,817 (2006).
    [CrossRef]
  10. G. Xu and D. J. Heinzen, "State-selective Rabi and Ramsey magnetic resonance line shapes," Phys. Rev. A 59, R922-R925 (1999).
    [CrossRef]
  11. D. Heinzen and G. Xu, "Application of quantum control theory to manipulate the Zeeman states of an atom," Quantum Electronics and Laser Science Conference Technical Digest. 177-178 (1999).
  12. C. Chin, V. Leiber, V. Vuleti´c, A. J. Kerman, and S. Chu, "Measurement of an electron??s electric dipole moment using Cs atoms trapped in optical lattices," Phys. Rev. A 63, 033,401 (2001).
    [CrossRef]
  13. A. K. V. E. B. Aleksandrov, M. V. Balabas, and A. S. Pazgalev, "Multiple-quantum radio-frequency spectroscopy of atoms: Application to the metrology of geomagnetic fields," J. Tech. Phys. 44 (1999).
    [CrossRef]
  14. E. B. Alexandrov, A. S. Pazgalev, and J. L. Rasson, "Observation of four-quantum resonance in the Zeeman structure of the ground-state of 39K," Opt. Spectrosk. 82, 14-20 (1997).
  15. A. I. Okunevich, "On the possibility of detecting the transverse component of the hexadecapole moment of atoms in fluorescent emission," Opt. Spectrosk. 91, 177-83 (2001).
    [CrossRef]
  16. A. Omont, Irreducible Components of the Density Matrix: Application to Optical Pumping (Pergamon Press, 1977).
  17. M. Auzinsh, "Angular momenta dynamics in magnetic and electric field: classical and quantum approach," Can. J. Phys. 75, 853-72 (1997).
    [CrossRef]
  18. S. M. Rochester and D. Budker, "Atomic polarization visualized," Am. J. Phys. 69, 450-4 (2001).
    [CrossRef]
  19. E. B. Alexandrov, M. Auzinsh, D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Dynamic effects in nonlinear magneto-optics of atoms and molecules: review," J. Opt. Soc. Am. B 22, 7-20 (2005).
    [CrossRef]
  20. E. B. Alexandrov, M. V. Balabas, D. Budker, D. English, D. F. Kimball, C. H. Li, and V. V. Yashchuk, "Lightinduced desorption of alkali-metal atoms from paraffin coating," Phys. Rev. A  66, 042,903/1-12 [Erratum: Phys. Rev. A 70, 049,902(E) (2004)] (2002).
  21. V. Acosta, M. P. Ledbetter, S. M. Rochester, D. Budker, D. F. Jackson-Kimball, D. C. Hovde, W. Gawlik, S. Pustelny, and J. Zachorowski, "Nonlinear magneto-optical rotation with frequency-modulated light in the geophysical field range," Phys. Rev. A  73, 053,404 (2006).
    [CrossRef]
  22. G. W. Series, "Theory of the modulation of light in optical pumping experiments," Proc. Phys. Soc. 88, 957-968 (1966).
    [CrossRef]
  23. M. Ducloy, "Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory," Phys. Rev. A 8, 1844-59 (1973).
    [CrossRef]
  24. M. P. Auzinsh, M. Y. Tamanis, and R. S. Ferber, "Zeeman quantum beats after optical depopulation of the ground electronic state of diatomic molecules," Sov. Phys. JETP 63, 688-693 (1986).
  25. S. Pustelny, D. F. Jackson Kimball, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Influence of magnetic-field inhomogeneity on nonlinear magneto-optical resonances," Phys. Rev. A 74, 063,406 (2006).
    [CrossRef]

2007 (1)

D. Budker and M. V. Romalis, "Optical Magnetometry," Nat. Phys. 3, 227-234 (2007).
[CrossRef]

2005 (2)

G. Yusa, K. Muraki, K. Takashina, K. Hashimoto, and Y. Hirayama, "Controlled multiple quantum coherences of nuclear spins in a nanometre-scale device," Nature 434, 1001-1005 (2005).
[CrossRef] [PubMed]

E. B. Alexandrov, M. Auzinsh, D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Dynamic effects in nonlinear magneto-optics of atoms and molecules: review," J. Opt. Soc. Am. B 22, 7-20 (2005).
[CrossRef]

2004 (1)

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

2003 (2)

E. B. Alexandrov, "Recent Progress in Optically Pumped Magnetometers," Phys. Scr. T105, 27-30 (2003).
[CrossRef]

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, "A subfemtotesla multichannel atomic magnetometer," Nature 422, 596-599 (2003).
[CrossRef] [PubMed]

2001 (2)

A. I. Okunevich, "On the possibility of detecting the transverse component of the hexadecapole moment of atoms in fluorescent emission," Opt. Spectrosk. 91, 177-83 (2001).
[CrossRef]

S. M. Rochester and D. Budker, "Atomic polarization visualized," Am. J. Phys. 69, 450-4 (2001).
[CrossRef]

2000 (1)

P. S. J. Ivan H. Deutsch, Gavin K. Brennen, "Quantum Computing with Neutral Atoms in an Optical Lattice," Fortschritte der Physik 48, 925-943 (2000).
[CrossRef]

1999 (1)

G. Xu and D. J. Heinzen, "State-selective Rabi and Ramsey magnetic resonance line shapes," Phys. Rev. A 59, R922-R925 (1999).
[CrossRef]

1997 (2)

E. B. Alexandrov, A. S. Pazgalev, and J. L. Rasson, "Observation of four-quantum resonance in the Zeeman structure of the ground-state of 39K," Opt. Spectrosk. 82, 14-20 (1997).

M. Auzinsh, "Angular momenta dynamics in magnetic and electric field: classical and quantum approach," Can. J. Phys. 75, 853-72 (1997).
[CrossRef]

1986 (1)

M. P. Auzinsh, M. Y. Tamanis, and R. S. Ferber, "Zeeman quantum beats after optical depopulation of the ground electronic state of diatomic molecules," Sov. Phys. JETP 63, 688-693 (1986).

1973 (1)

M. Ducloy, "Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory," Phys. Rev. A 8, 1844-59 (1973).
[CrossRef]

1966 (1)

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

Alexandrov, E. B.

E. B. Alexandrov, M. Auzinsh, D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Dynamic effects in nonlinear magneto-optics of atoms and molecules: review," J. Opt. Soc. Am. B 22, 7-20 (2005).
[CrossRef]

E. B. Alexandrov, "Recent Progress in Optically Pumped Magnetometers," Phys. Scr. T105, 27-30 (2003).
[CrossRef]

E. B. Alexandrov, A. S. Pazgalev, and J. L. Rasson, "Observation of four-quantum resonance in the Zeeman structure of the ground-state of 39K," Opt. Spectrosk. 82, 14-20 (1997).

Allred, J. C.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, "A subfemtotesla multichannel atomic magnetometer," Nature 422, 596-599 (2003).
[CrossRef] [PubMed]

Auzinsh, M.

Auzinsh, M. P.

M. P. Auzinsh, M. Y. Tamanis, and R. S. Ferber, "Zeeman quantum beats after optical depopulation of the ground electronic state of diatomic molecules," Sov. Phys. JETP 63, 688-693 (1986).

Budker, D.

D. Budker and M. V. Romalis, "Optical Magnetometry," Nat. Phys. 3, 227-234 (2007).
[CrossRef]

E. B. Alexandrov, M. Auzinsh, D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Dynamic effects in nonlinear magneto-optics of atoms and molecules: review," J. Opt. Soc. Am. B 22, 7-20 (2005).
[CrossRef]

S. M. Rochester and D. Budker, "Atomic polarization visualized," Am. J. Phys. 69, 450-4 (2001).
[CrossRef]

Cirac, J. I.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

Ducloy, M.

M. Ducloy, "Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory," Phys. Rev. A 8, 1844-59 (1973).
[CrossRef]

Ferber, R. S.

M. P. Auzinsh, M. Y. Tamanis, and R. S. Ferber, "Zeeman quantum beats after optical depopulation of the ground electronic state of diatomic molecules," Sov. Phys. JETP 63, 688-693 (1986).

Fiurasek, J.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

Hashimoto, K.

G. Yusa, K. Muraki, K. Takashina, K. Hashimoto, and Y. Hirayama, "Controlled multiple quantum coherences of nuclear spins in a nanometre-scale device," Nature 434, 1001-1005 (2005).
[CrossRef] [PubMed]

Heinzen, D. J.

G. Xu and D. J. Heinzen, "State-selective Rabi and Ramsey magnetic resonance line shapes," Phys. Rev. A 59, R922-R925 (1999).
[CrossRef]

Hirayama, Y.

G. Yusa, K. Muraki, K. Takashina, K. Hashimoto, and Y. Hirayama, "Controlled multiple quantum coherences of nuclear spins in a nanometre-scale device," Nature 434, 1001-1005 (2005).
[CrossRef] [PubMed]

Julsgaard, B.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

Kimball, D. F.

Kominis, I. K.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, "A subfemtotesla multichannel atomic magnetometer," Nature 422, 596-599 (2003).
[CrossRef] [PubMed]

Kornack, T. W.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, "A subfemtotesla multichannel atomic magnetometer," Nature 422, 596-599 (2003).
[CrossRef] [PubMed]

Muraki, K.

G. Yusa, K. Muraki, K. Takashina, K. Hashimoto, and Y. Hirayama, "Controlled multiple quantum coherences of nuclear spins in a nanometre-scale device," Nature 434, 1001-1005 (2005).
[CrossRef] [PubMed]

Okunevich, A. I.

A. I. Okunevich, "On the possibility of detecting the transverse component of the hexadecapole moment of atoms in fluorescent emission," Opt. Spectrosk. 91, 177-83 (2001).
[CrossRef]

Pazgalev, A. S.

E. B. Alexandrov, A. S. Pazgalev, and J. L. Rasson, "Observation of four-quantum resonance in the Zeeman structure of the ground-state of 39K," Opt. Spectrosk. 82, 14-20 (1997).

Polzik, E. S.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

Rasson, J. L.

E. B. Alexandrov, A. S. Pazgalev, and J. L. Rasson, "Observation of four-quantum resonance in the Zeeman structure of the ground-state of 39K," Opt. Spectrosk. 82, 14-20 (1997).

Rochester, S. M.

Romalis, M. V.

D. Budker and M. V. Romalis, "Optical Magnetometry," Nat. Phys. 3, 227-234 (2007).
[CrossRef]

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, "A subfemtotesla multichannel atomic magnetometer," Nature 422, 596-599 (2003).
[CrossRef] [PubMed]

Series, G. W.

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

Sherson, J.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

Takashina, K.

G. Yusa, K. Muraki, K. Takashina, K. Hashimoto, and Y. Hirayama, "Controlled multiple quantum coherences of nuclear spins in a nanometre-scale device," Nature 434, 1001-1005 (2005).
[CrossRef] [PubMed]

Tamanis, M. Y.

M. P. Auzinsh, M. Y. Tamanis, and R. S. Ferber, "Zeeman quantum beats after optical depopulation of the ground electronic state of diatomic molecules," Sov. Phys. JETP 63, 688-693 (1986).

Xu, G.

G. Xu and D. J. Heinzen, "State-selective Rabi and Ramsey magnetic resonance line shapes," Phys. Rev. A 59, R922-R925 (1999).
[CrossRef]

Yashchuk, V. V.

Yusa, G.

G. Yusa, K. Muraki, K. Takashina, K. Hashimoto, and Y. Hirayama, "Controlled multiple quantum coherences of nuclear spins in a nanometre-scale device," Nature 434, 1001-1005 (2005).
[CrossRef] [PubMed]

Am. J. Phys. (1)

S. M. Rochester and D. Budker, "Atomic polarization visualized," Am. J. Phys. 69, 450-4 (2001).
[CrossRef]

Can. J. Phys. (1)

M. Auzinsh, "Angular momenta dynamics in magnetic and electric field: classical and quantum approach," Can. J. Phys. 75, 853-72 (1997).
[CrossRef]

Fortschritte der Physik (1)

P. S. J. Ivan H. Deutsch, Gavin K. Brennen, "Quantum Computing with Neutral Atoms in an Optical Lattice," Fortschritte der Physik 48, 925-943 (2000).
[CrossRef]

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

Nat. Phys. (1)

D. Budker and M. V. Romalis, "Optical Magnetometry," Nat. Phys. 3, 227-234 (2007).
[CrossRef]

Nature (3)

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, "A subfemtotesla multichannel atomic magnetometer," Nature 422, 596-599 (2003).
[CrossRef] [PubMed]

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

G. Yusa, K. Muraki, K. Takashina, K. Hashimoto, and Y. Hirayama, "Controlled multiple quantum coherences of nuclear spins in a nanometre-scale device," Nature 434, 1001-1005 (2005).
[CrossRef] [PubMed]

Opt. Spectrosk. (2)

E. B. Alexandrov, A. S. Pazgalev, and J. L. Rasson, "Observation of four-quantum resonance in the Zeeman structure of the ground-state of 39K," Opt. Spectrosk. 82, 14-20 (1997).

A. I. Okunevich, "On the possibility of detecting the transverse component of the hexadecapole moment of atoms in fluorescent emission," Opt. Spectrosk. 91, 177-83 (2001).
[CrossRef]

Phys. Rev. A (2)

G. Xu and D. J. Heinzen, "State-selective Rabi and Ramsey magnetic resonance line shapes," Phys. Rev. A 59, R922-R925 (1999).
[CrossRef]

M. Ducloy, "Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory," Phys. Rev. A 8, 1844-59 (1973).
[CrossRef]

Phys. Scr. (1)

E. B. Alexandrov, "Recent Progress in Optically Pumped Magnetometers," Phys. Scr. T105, 27-30 (2003).
[CrossRef]

Proc. Phys. Soc. (1)

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

Sov. Phys. JETP (1)

M. P. Auzinsh, M. Y. Tamanis, and R. S. Ferber, "Zeeman quantum beats after optical depopulation of the ground electronic state of diatomic molecules," Sov. Phys. JETP 63, 688-693 (1986).

Other (10)

S. Pustelny, D. F. Jackson Kimball, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Influence of magnetic-field inhomogeneity on nonlinear magneto-optical resonances," Phys. Rev. A 74, 063,406 (2006).
[CrossRef]

V. V. Yashchuk, D. Budker,W. Gawlik, D. F. Kimball, Y. P. Malakyan, and S. M. Rochester, "Selective addressing of high-rank atomic polarization moments," Phys. Rev. Lett. 90, 253,001 (2003).
[CrossRef]

S. Pustelny, D. F. Jackson Kimball, S. M. Rochester, V. V. Yashchuk, W. Gawlik, and D. Budker, "Pump-probe nonlinear magneto-optical rotation with frequency-modulated light," Phys. Rev. A 73, 023,817 (2006).
[CrossRef]

J. Vanier and C. Audoin, The quantum physics of atomic frequency standards (A. Hilger, Bristol; Philadelphia, 1989).
[CrossRef]

D. Heinzen and G. Xu, "Application of quantum control theory to manipulate the Zeeman states of an atom," Quantum Electronics and Laser Science Conference Technical Digest. 177-178 (1999).

C. Chin, V. Leiber, V. Vuleti´c, A. J. Kerman, and S. Chu, "Measurement of an electron??s electric dipole moment using Cs atoms trapped in optical lattices," Phys. Rev. A 63, 033,401 (2001).
[CrossRef]

A. K. V. E. B. Aleksandrov, M. V. Balabas, and A. S. Pazgalev, "Multiple-quantum radio-frequency spectroscopy of atoms: Application to the metrology of geomagnetic fields," J. Tech. Phys. 44 (1999).
[CrossRef]

A. Omont, Irreducible Components of the Density Matrix: Application to Optical Pumping (Pergamon Press, 1977).

E. B. Alexandrov, M. V. Balabas, D. Budker, D. English, D. F. Kimball, C. H. Li, and V. V. Yashchuk, "Lightinduced desorption of alkali-metal atoms from paraffin coating," Phys. Rev. A  66, 042,903/1-12 [Erratum: Phys. Rev. A 70, 049,902(E) (2004)] (2002).

V. Acosta, M. P. Ledbetter, S. M. Rochester, D. Budker, D. F. Jackson-Kimball, D. C. Hovde, W. Gawlik, S. Pustelny, and J. Zachorowski, "Nonlinear magneto-optical rotation with frequency-modulated light in the geophysical field range," Phys. Rev. A  73, 053,404 (2006).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

States, energies, and layout of experiment. Part A shows the states involved in the four-quantum coherence. Part B shows the linearity of the m=±2 states’ energies as a function of magnetic field (shown for purposes of illustration over a much larger range of fields than are experimentally relevant ). Part C shows angular momentum probability surfaces (see text) for quadrupole (left) and hexadecapole (right) for F=2. The atomic polarizations are transverse to and precess around the magnetic-field quantization axis. Part D shows the respective directions of laser polarizations (two-headed arrows), propagation directions, and magnetic field.

Fig. 2.
Fig. 2.

Magnetic-field dependence of optical rotation amplitudes for quadrupole (blue diamonds), hexadecapole pumped with light modulated at 4Ω L (maroon squares), and hexadecapole pumped at 2Ω L (green triangles). The quadrupole and hexadecapole, pumped at 2Ω L decrease much more slowly with magnetic field compared to hexadecapole pumped at 4Ω L . The solid lines are fits by ad-hoc functions.

Fig. 3.
Fig. 3.

Comparison of the demodulated quadrupole signal (top) without phase flip (see text) and of the hexadecapole signal obtained with phase flip (bottom). For the case of pumping without phase flip (top), the signal demodulated at 2Ω L is plotted alongside the raw optical rotation signal. The inset on the top plot is a magnification of the raw optical-rotation signal during a revival stage of the quadrupole-signal beats occurring due to the nonlinear Zeeman effect (NLZ). This signal on the bottom plot (with phase flip) is demodulated at 4Ω L , and the resulting curve shows the absence of the beats related to NLZ. The inset on the right shows the details of the pumping pulses near the phase flip, along with the angular-momentum probability surfaces characterizing the ensemble at different stages of pumping. The magnetic field around which these surfaces precess is normal to the page.

Fig. 4.
Fig. 4.

The dependence of the observed relaxation rates for the quadrupole and hexadecapole signals on the magnetic-field gradient applied in the direction of the magnetic field. In both cases, pumping was with light pulsed at a repetition rate of 2Ω L /(2π) at B≈108 mG, and the relaxation was determined from the optical rotation after the pump light was shut off. Solid lines show fits to fourth-order polynomials; the ratio of curvatures near the vertex is consistent with the expected four times higher sensitivity of the hexadecapole to gradients [25].

Equations (3)

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ρ q ( κ ) = m , m = F F ( 1 ) F m F , m , F m κq ρ m , m .
ρ + 4 ( 4 ) = ρ 2 , 2
ρ 4 ( 4 ) = ρ 2,2 .

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