Abstract

We report measurements of nonlinear magneto-optical rotation (NMOR) for the D2 line of 87Rb atoms in an antirelaxation-coated vapor cell in the presence of a radio-frequency (rf) field. The experimental NMOR signals as a function of rf field frequency for various rf field powers are compared to a theoretical model based on the density-matrix formalism. The comparison between experiment and theory enables understanding of the ground-state atomic spin polarization dynamics, illustrated using plots of the probability distribution of the atomic angular momentum.

© 2010 Optical Society of America

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  1. A. Kastler, "Quelques suggestions concernant la production optique et la d’etection optique dúne inégalité de population des niveaux de quantifigation spatiale des atomes. Application á l’expérience de Stern et Gerlach et á la résonance magnétique," J. Phys. Radium 11, 255-265 (1950).
    [CrossRef]
  2. C. Cohen-Tannoudji, Atoms in Electromagnetic Fields (World Scientific, Singapore, 1994).
  3. T. G. Walker, and W. Happer, "Spin-exchange optical pumping of noble-gas nuclei," Rev. Mod. Phys. 69, 629 (1997).
    [CrossRef]
  4. A. Okunevich, "Laser pumping and magneto-optical rotation of the light polarization plane in a cell with an antirelaxation coating of the walls: I. Statement and solution of the problem," Opt. Spectrosc. 97, 834-841 (2004).
    [CrossRef]
  5. D. F. Jackson Kimball, L. R. Jacome, S. Guttikonda, E. J. Bahr, and L. F. Chan, "Magnetometric sensitivity optimization for nonlinear optical rotation with frequency-modulated light: Rubidium D2 line," J. Appl. Phys. 106, 063113 (2009).
    [CrossRef]
  6. D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
    [CrossRef]
  7. 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 (2002).
    [CrossRef]
  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, 253001 (2003).
    [CrossRef] [PubMed]
  9. S. Pustelny, D. F. J. 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, 023817 (2006).
    [CrossRef]
  10. V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
    [CrossRef] [PubMed]
  11. S. Xu, S. M. Rochester, V. V. Yashchuk, M. H. Donaldson, and D. Budker, "Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation," Rev. Sci. Instrum. 77, 083106 (2006).
    [CrossRef]
  12. A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001).
    [CrossRef]
  13. S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, "Magnetic resonance imaging with an optical atomic magnetometer," Proc. Natl. Acad. Sci. U.S.A. 103, 12668-12671 (2006).
    [CrossRef]
  14. R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003).
    [CrossRef]
  15. M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, "Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation," Phys. Rev. A 75, 023405 (2007).
    [CrossRef]
  16. W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, "Quantum noise limited and entanglement-assisted magnetometry," Phys. Rev. Lett. 104, 133601 (2010).
    [CrossRef] [PubMed]
  17. W. Chalupczak, P. Josephs-Franks, S. Pustelny, and W. Gawlik, "Optical-radio-frequency resonances free from power broadening," Phys. Rev. A 81, 013422 (2010).
    [CrossRef]
  18. J. Brossel, and F. Bitter, "A new "double resonance" method for investigating atomic energy levels. Application to Hg 3P1*," Phys. Rev. 86, 308-316 (1952).
    [CrossRef]
  19. H. Friedmann, and A. D. Wilson-Gordon, "Dispersion profiles of the absorptive response of a two-level system interacting with two intense fields," Phys. Rev. A 36, 1333-1341 (1987).
    [CrossRef] [PubMed]
  20. S. Chakmakjian, K. Koch, and J. C. R. Stroud, "Observation of resonances at subharmonics of the rabi frequency in the saturated absorption of a 100%amplitude-modulated laser beam," J. Opt. Soc. Am. B 5, 2015-2020 (1988).
    [CrossRef]
  21. G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006).
    [CrossRef]
  22. A. Weis, G. Bison, and A. S. Pazgalev, "Theory of double resonance magnetometers based on atomic alignment," Phys. Rev. A 74, 033401 (2006).
    [CrossRef]
  23. S. M. Rochester, and D. Budker, "Atomic polarization visualized," Am. J. Phys. 69, 450-454 (2001).
    [CrossRef]
  24. D. F. Jackson Kimball, O. Neitzke, E. J. Bahr, S. Guttikonda, S. M. Rochester, M. P. Ledbetter, I. Novikova, B. Coste, S. A. Rangwala, J. M. Higbie, A. I. Okunevich, V. V. Yashchuk, and D. Budker (in preparation).
  25. M. A. Bouchiat, and J. Brossel, "Relaxation of optically pumped Rb atoms on paraffin-coated walls," Phys. Rev. 147, 41-54 (1966).
    [CrossRef]
  26. D. Budker, V. Yashchuk, and M. Zolotorev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
    [CrossRef]
  27. M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
    [CrossRef]
  28. K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, "Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor," Appl. Opt. 37, 3295 (1998).
    [CrossRef]
  29. V. Yashchuk, D. Budker, and J. Davis, "Laser frequency stabilization using linear magneto-optics," Rev. Sci. Instrum. 71, 341 (2000).
    [CrossRef]
  30. D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088 (2000).
    [CrossRef] [PubMed]
  31. M. Auzinsh, "Angular momenta dynamics in magnetic and electric field: Classical and quantum approach," Can. J. Phys. 75, 853-872 (1997).
    [CrossRef]
  32. M. Auzinsh, D. Budker, and S. M. Rochester, Optically Polarized Atoms: Understanding Light-Atom Interactions (Oxford University Press, Oxford, 2010).
  33. S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hänsch, "Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
    [CrossRef] [PubMed]
  34. 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 (2005).
    [CrossRef]

2010 (2)

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, "Quantum noise limited and entanglement-assisted magnetometry," Phys. Rev. Lett. 104, 133601 (2010).
[CrossRef] [PubMed]

W. Chalupczak, P. Josephs-Franks, S. Pustelny, and W. Gawlik, "Optical-radio-frequency resonances free from power broadening," Phys. Rev. A 81, 013422 (2010).
[CrossRef]

2009 (1)

D. F. Jackson Kimball, L. R. Jacome, S. Guttikonda, E. J. Bahr, and L. F. Chan, "Magnetometric sensitivity optimization for nonlinear optical rotation with frequency-modulated light: Rubidium D2 line," J. Appl. Phys. 106, 063113 (2009).
[CrossRef]

2008 (1)

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

2007 (1)

M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, "Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation," Phys. Rev. A 75, 023405 (2007).
[CrossRef]

2006 (5)

S. Xu, S. M. Rochester, V. V. Yashchuk, M. H. Donaldson, and D. Budker, "Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation," Rev. Sci. Instrum. 77, 083106 (2006).
[CrossRef]

S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, "Magnetic resonance imaging with an optical atomic magnetometer," Proc. Natl. Acad. Sci. U.S.A. 103, 12668-12671 (2006).
[CrossRef]

S. Pustelny, D. F. J. 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, 023817 (2006).
[CrossRef]

G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006).
[CrossRef]

A. Weis, G. Bison, and A. S. Pazgalev, "Theory of double resonance magnetometers based on atomic alignment," Phys. Rev. A 74, 033401 (2006).
[CrossRef]

2005 (2)

M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
[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 (2005).
[CrossRef]

2004 (1)

A. Okunevich, "Laser pumping and magneto-optical rotation of the light polarization plane in a cell with an antirelaxation coating of the walls: I. Statement and solution of the problem," Opt. Spectrosc. 97, 834-841 (2004).
[CrossRef]

2003 (2)

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, 253001 (2003).
[CrossRef] [PubMed]

R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003).
[CrossRef]

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 (2002).
[CrossRef]

2001 (2)

A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001).
[CrossRef]

S. M. Rochester, and D. Budker, "Atomic polarization visualized," Am. J. Phys. 69, 450-454 (2001).
[CrossRef]

2000 (3)

V. Yashchuk, D. Budker, and J. Davis, "Laser frequency stabilization using linear magneto-optics," Rev. Sci. Instrum. 71, 341 (2000).
[CrossRef]

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088 (2000).
[CrossRef] [PubMed]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

1998 (2)

K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, "Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor," Appl. Opt. 37, 3295 (1998).
[CrossRef]

D. Budker, V. Yashchuk, and M. Zolotorev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
[CrossRef]

1997 (2)

M. Auzinsh, "Angular momenta dynamics in magnetic and electric field: Classical and quantum approach," Can. J. Phys. 75, 853-872 (1997).
[CrossRef]

T. G. Walker, and W. Happer, "Spin-exchange optical pumping of noble-gas nuclei," Rev. Mod. Phys. 69, 629 (1997).
[CrossRef]

1993 (1)

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hänsch, "Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

1988 (1)

S. Chakmakjian, K. Koch, and J. C. R. Stroud, "Observation of resonances at subharmonics of the rabi frequency in the saturated absorption of a 100%amplitude-modulated laser beam," J. Opt. Soc. Am. B 5, 2015-2020 (1988).
[CrossRef]

1987 (1)

H. Friedmann, and A. D. Wilson-Gordon, "Dispersion profiles of the absorptive response of a two-level system interacting with two intense fields," Phys. Rev. A 36, 1333-1341 (1987).
[CrossRef] [PubMed]

1966 (1)

M. A. Bouchiat, and J. Brossel, "Relaxation of optically pumped Rb atoms on paraffin-coated walls," Phys. Rev. 147, 41-54 (1966).
[CrossRef]

1952 (1)

J. Brossel, and F. Bitter, "A new "double resonance" method for investigating atomic energy levels. Application to Hg 3P1*," Phys. Rev. 86, 308-316 (1952).
[CrossRef]

1950 (1)

A. Kastler, "Quelques suggestions concernant la production optique et la d’etection optique dúne inégalité de population des niveaux de quantifigation spatiale des atomes. Application á l’expérience de Stern et Gerlach et á la résonance magnétique," J. Phys. Radium 11, 255-265 (1950).
[CrossRef]

Acosta, V. M.

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, "Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation," Phys. Rev. A 75, 023405 (2007).
[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 (2005).
[CrossRef]

M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
[CrossRef]

Auzinsh, M.

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[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 (2005).
[CrossRef]

M. Auzinsh, "Angular momenta dynamics in magnetic and electric field: Classical and quantum approach," Can. J. Phys. 75, 853-872 (1997).
[CrossRef]

Bahr, E. J.

D. F. Jackson Kimball, L. R. Jacome, S. Guttikonda, E. J. Bahr, and L. F. Chan, "Magnetometric sensitivity optimization for nonlinear optical rotation with frequency-modulated light: Rubidium D2 line," J. Appl. Phys. 106, 063113 (2009).
[CrossRef]

Balabas, M. V.

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, "Quantum noise limited and entanglement-assisted magnetometry," Phys. Rev. Lett. 104, 133601 (2010).
[CrossRef] [PubMed]

M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
[CrossRef]

Barrall, G.

A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001).
[CrossRef]

Bibber, K.

R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003).
[CrossRef]

Bison, G.

G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006).
[CrossRef]

A. Weis, G. Bison, and A. S. Pazgalev, "Theory of double resonance magnetometers based on atomic alignment," Phys. Rev. A 74, 033401 (2006).
[CrossRef]

Bitter, F.

J. Brossel, and F. Bitter, "A new "double resonance" method for investigating atomic energy levels. Application to Hg 3P1*," Phys. Rev. 86, 308-316 (1952).
[CrossRef]

Bouchiat, M. A.

M. A. Bouchiat, and J. Brossel, "Relaxation of optically pumped Rb atoms on paraffin-coated walls," Phys. Rev. 147, 41-54 (1966).
[CrossRef]

Bradley, R.

R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003).
[CrossRef]

Brossel, J.

M. A. Bouchiat, and J. Brossel, "Relaxation of optically pumped Rb atoms on paraffin-coated walls," Phys. Rev. 147, 41-54 (1966).
[CrossRef]

J. Brossel, and F. Bitter, "A new "double resonance" method for investigating atomic energy levels. Application to Hg 3P1*," Phys. Rev. 86, 308-316 (1952).
[CrossRef]

Budker, D.

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, "Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation," Phys. Rev. A 75, 023405 (2007).
[CrossRef]

S. Xu, S. M. Rochester, V. V. Yashchuk, M. H. Donaldson, and D. Budker, "Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation," Rev. Sci. Instrum. 77, 083106 (2006).
[CrossRef]

S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, "Magnetic resonance imaging with an optical atomic magnetometer," Proc. Natl. Acad. Sci. U.S.A. 103, 12668-12671 (2006).
[CrossRef]

S. Pustelny, D. F. J. 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, 023817 (2006).
[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 (2005).
[CrossRef]

M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
[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, 253001 (2003).
[CrossRef] [PubMed]

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 (2002).
[CrossRef]

S. M. Rochester, and D. Budker, "Atomic polarization visualized," Am. J. Phys. 69, 450-454 (2001).
[CrossRef]

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088 (2000).
[CrossRef] [PubMed]

V. Yashchuk, D. Budker, and J. Davis, "Laser frequency stabilization using linear magneto-optics," Rev. Sci. Instrum. 71, 341 (2000).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

D. Budker, V. Yashchuk, and M. Zolotorev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
[CrossRef]

Buess, M.

A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001).
[CrossRef]

Burnett, L.

A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001).
[CrossRef]

Chakmakjian, S.

S. Chakmakjian, K. Koch, and J. C. R. Stroud, "Observation of resonances at subharmonics of the rabi frequency in the saturated absorption of a 100%amplitude-modulated laser beam," J. Opt. Soc. Am. B 5, 2015-2020 (1988).
[CrossRef]

Chalupczak, W.

W. Chalupczak, P. Josephs-Franks, S. Pustelny, and W. Gawlik, "Optical-radio-frequency resonances free from power broadening," Phys. Rev. A 81, 013422 (2010).
[CrossRef]

Chan, L. F.

D. F. Jackson Kimball, L. R. Jacome, S. Guttikonda, E. J. Bahr, and L. F. Chan, "Magnetometric sensitivity optimization for nonlinear optical rotation with frequency-modulated light: Rubidium D2 line," J. Appl. Phys. 106, 063113 (2009).
[CrossRef]

Clarke, J.

R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003).
[CrossRef]

Corwin, K. L.

K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, "Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor," Appl. Opt. 37, 3295 (1998).
[CrossRef]

Davis, J.

V. Yashchuk, D. Budker, and J. Davis, "Laser frequency stabilization using linear magneto-optics," Rev. Sci. Instrum. 71, 341 (2000).
[CrossRef]

Di Domenico, G.

G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006).
[CrossRef]

Donaldson, M. H.

S. Xu, S. M. Rochester, V. V. Yashchuk, M. H. Donaldson, and D. Budker, "Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation," Rev. Sci. Instrum. 77, 083106 (2006).
[CrossRef]

S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, "Magnetic resonance imaging with an optical atomic magnetometer," Proc. Natl. Acad. Sci. U.S.A. 103, 12668-12671 (2006).
[CrossRef]

Epstein, R. J.

K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, "Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor," Appl. Opt. 37, 3295 (1998).
[CrossRef]

Friedmann, H.

H. Friedmann, and A. D. Wilson-Gordon, "Dispersion profiles of the absorptive response of a two-level system interacting with two intense fields," Phys. Rev. A 36, 1333-1341 (1987).
[CrossRef] [PubMed]

Garroway, A.

A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001).
[CrossRef]

Gawlik, W.

W. Chalupczak, P. Josephs-Franks, S. Pustelny, and W. Gawlik, "Optical-radio-frequency resonances free from power broadening," Phys. Rev. A 81, 013422 (2010).
[CrossRef]

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

S. Pustelny, D. F. J. 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, 023817 (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, 253001 (2003).
[CrossRef] [PubMed]

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 (2002).
[CrossRef]

Graf, M. T.

M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
[CrossRef]

Grisins, P.

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

Groeger, S.

G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006).
[CrossRef]

Guttikonda, S.

D. F. Jackson Kimball, L. R. Jacome, S. Guttikonda, E. J. Bahr, and L. F. Chan, "Magnetometric sensitivity optimization for nonlinear optical rotation with frequency-modulated light: Rubidium D2 line," J. Appl. Phys. 106, 063113 (2009).
[CrossRef]

Hand, C. F.

K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, "Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor," Appl. Opt. 37, 3295 (1998).
[CrossRef]

Hänsch, T. W.

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hänsch, "Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

Happer, W.

T. G. Walker, and W. Happer, "Spin-exchange optical pumping of noble-gas nuclei," Rev. Mod. Phys. 69, 629 (1997).
[CrossRef]

Hibbs, A.

A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001).
[CrossRef]

Higbie, J. M.

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

Jackson Kimball, D. F.

D. F. Jackson Kimball, L. R. Jacome, S. Guttikonda, E. J. Bahr, and L. F. Chan, "Magnetometric sensitivity optimization for nonlinear optical rotation with frequency-modulated light: Rubidium D2 line," J. Appl. Phys. 106, 063113 (2009).
[CrossRef]

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

Jacome, L. R.

D. F. Jackson Kimball, L. R. Jacome, S. Guttikonda, E. J. Bahr, and L. F. Chan, "Magnetometric sensitivity optimization for nonlinear optical rotation with frequency-modulated light: Rubidium D2 line," J. Appl. Phys. 106, 063113 (2009).
[CrossRef]

Jensen, K.

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, "Quantum noise limited and entanglement-assisted magnetometry," Phys. Rev. Lett. 104, 133601 (2010).
[CrossRef] [PubMed]

Josephs-Franks, P.

W. Chalupczak, P. Josephs-Franks, S. Pustelny, and W. Gawlik, "Optical-radio-frequency resonances free from power broadening," Phys. Rev. A 81, 013422 (2010).
[CrossRef]

Kanorsky, S. I.

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hänsch, "Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

Kastler, A.

A. Kastler, "Quelques suggestions concernant la production optique et la d’etection optique dúne inégalité de population des niveaux de quantifigation spatiale des atomes. Application á l’expérience de Stern et Gerlach et á la résonance magnétique," J. Phys. Radium 11, 255-265 (1950).
[CrossRef]

Kerner, K.

M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
[CrossRef]

Kimball, D.

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088 (2000).
[CrossRef] [PubMed]

Kimball, D. F.

M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
[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 (2005).
[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, 253001 (2003).
[CrossRef] [PubMed]

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 (2002).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

Kimball, D. F. J.

S. Pustelny, D. F. J. 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, 023817 (2006).
[CrossRef]

Kinion, D.

R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003).
[CrossRef]

Knowles, P.

G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006).
[CrossRef]

Koch, K.

S. Chakmakjian, K. Koch, and J. C. R. Stroud, "Observation of resonances at subharmonics of the rabi frequency in the saturated absorption of a 100%amplitude-modulated laser beam," J. Opt. Soc. Am. B 5, 2015-2020 (1988).
[CrossRef]

Krauter, H.

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, "Quantum noise limited and entanglement-assisted magnetometry," Phys. Rev. Lett. 104, 133601 (2010).
[CrossRef] [PubMed]

Krzemien, L.

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

Ledbetter, M. P.

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, "Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation," Phys. Rev. A 75, 023405 (2007).
[CrossRef]

Lu, Z.-T.

K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, "Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor," Appl. Opt. 37, 3295 (1998).
[CrossRef]

Malakyan, Y. P.

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, 253001 (2003).
[CrossRef] [PubMed]

Matsuki, S.

R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003).
[CrossRef]

Matthews, R.

A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001).
[CrossRef]

Miller, J.

A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001).
[CrossRef]

Muck, M.

R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003).
[CrossRef]

Okunevich, A.

A. Okunevich, "Laser pumping and magneto-optical rotation of the light polarization plane in a cell with an antirelaxation coating of the walls: I. Statement and solution of the problem," Opt. Spectrosc. 97, 834-841 (2004).
[CrossRef]

Pazgalev, A. S.

G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006).
[CrossRef]

A. Weis, G. Bison, and A. S. Pazgalev, "Theory of double resonance magnetometers based on atomic alignment," Phys. Rev. A 74, 033401 (2006).
[CrossRef]

Pines, A.

S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, "Magnetic resonance imaging with an optical atomic magnetometer," Proc. Natl. Acad. Sci. U.S.A. 103, 12668-12671 (2006).
[CrossRef]

Polzik, E. S.

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, "Quantum noise limited and entanglement-assisted magnetometry," Phys. Rev. Lett. 104, 133601 (2010).
[CrossRef] [PubMed]

Pustelny, S.

W. Chalupczak, P. Josephs-Franks, S. Pustelny, and W. Gawlik, "Optical-radio-frequency resonances free from power broadening," Phys. Rev. A 81, 013422 (2010).
[CrossRef]

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, "Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation," Phys. Rev. A 75, 023405 (2007).
[CrossRef]

S. Pustelny, D. F. J. 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, 023817 (2006).
[CrossRef]

Rebetez, M.

G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006).
[CrossRef]

Renema, J. J.

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, "Quantum noise limited and entanglement-assisted magnetometry," Phys. Rev. Lett. 104, 133601 (2010).
[CrossRef] [PubMed]

Rochester, S.

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088 (2000).
[CrossRef] [PubMed]

Rochester, S. M.

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, "Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation," Phys. Rev. A 75, 023405 (2007).
[CrossRef]

S. Xu, S. M. Rochester, V. V. Yashchuk, M. H. Donaldson, and D. Budker, "Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation," Rev. Sci. Instrum. 77, 083106 (2006).
[CrossRef]

S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, "Magnetic resonance imaging with an optical atomic magnetometer," Proc. Natl. Acad. Sci. U.S.A. 103, 12668-12671 (2006).
[CrossRef]

S. Pustelny, D. F. J. 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, 023817 (2006).
[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 (2005).
[CrossRef]

M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
[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, 253001 (2003).
[CrossRef] [PubMed]

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 (2002).
[CrossRef]

S. M. Rochester, and D. Budker, "Atomic polarization visualized," Am. J. Phys. 69, 450-454 (2001).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

Rosenberg, L. J.

R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003).
[CrossRef]

Saudan, H.

G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006).
[CrossRef]

Sikivie, P.

R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003).
[CrossRef]

Stroud, J. C. R.

S. Chakmakjian, K. Koch, and J. C. R. Stroud, "Observation of resonances at subharmonics of the rabi frequency in the saturated absorption of a 100%amplitude-modulated laser beam," J. Opt. Soc. Am. B 5, 2015-2020 (1988).
[CrossRef]

Suits, B.

A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001).
[CrossRef]

Walker, T. G.

T. G. Walker, and W. Happer, "Spin-exchange optical pumping of noble-gas nuclei," Rev. Mod. Phys. 69, 629 (1997).
[CrossRef]

Wasilewski, W.

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, "Quantum noise limited and entanglement-assisted magnetometry," Phys. Rev. Lett. 104, 133601 (2010).
[CrossRef] [PubMed]

Weis, A.

A. Weis, G. Bison, and A. S. Pazgalev, "Theory of double resonance magnetometers based on atomic alignment," Phys. Rev. A 74, 033401 (2006).
[CrossRef]

G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006).
[CrossRef]

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 (2002).
[CrossRef]

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hänsch, "Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

Wieman, C. E.

K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, "Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor," Appl. Opt. 37, 3295 (1998).
[CrossRef]

Wilson-Gordon, A. D.

H. Friedmann, and A. D. Wilson-Gordon, "Dispersion profiles of the absorptive response of a two-level system interacting with two intense fields," Phys. Rev. A 36, 1333-1341 (1987).
[CrossRef] [PubMed]

Wong, C.

M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
[CrossRef]

Wurster, J.

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hänsch, "Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

Xu, S.

S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, "Magnetic resonance imaging with an optical atomic magnetometer," Proc. Natl. Acad. Sci. U.S.A. 103, 12668-12671 (2006).
[CrossRef]

S. Xu, S. M. Rochester, V. V. Yashchuk, M. H. Donaldson, and D. Budker, "Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation," Rev. Sci. Instrum. 77, 083106 (2006).
[CrossRef]

Yashchuk, V.

V. Yashchuk, D. Budker, and J. Davis, "Laser frequency stabilization using linear magneto-optics," Rev. Sci. Instrum. 71, 341 (2000).
[CrossRef]

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088 (2000).
[CrossRef] [PubMed]

D. Budker, V. Yashchuk, and M. Zolotorev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
[CrossRef]

Yashchuk, V. V.

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, "Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation," Phys. Rev. A 75, 023405 (2007).
[CrossRef]

S. Xu, S. M. Rochester, V. V. Yashchuk, M. H. Donaldson, and D. Budker, "Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation," Rev. Sci. Instrum. 77, 083106 (2006).
[CrossRef]

S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, "Magnetic resonance imaging with an optical atomic magnetometer," Proc. Natl. Acad. Sci. U.S.A. 103, 12668-12671 (2006).
[CrossRef]

S. Pustelny, D. F. J. 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, 023817 (2006).
[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 (2005).
[CrossRef]

M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
[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, 253001 (2003).
[CrossRef] [PubMed]

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 (2002).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

Zolotorev, M.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

D. Budker, V. Yashchuk, and M. Zolotorev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
[CrossRef]

Am. J. Phys. (1)

S. M. Rochester, and D. Budker, "Atomic polarization visualized," Am. J. Phys. 69, 450-454 (2001).
[CrossRef]

Appl. Opt. (1)

K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, "Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor," Appl. Opt. 37, 3295 (1998).
[CrossRef]

Can. J. Phys. (1)

M. Auzinsh, "Angular momenta dynamics in magnetic and electric field: Classical and quantum approach," Can. J. Phys. 75, 853-872 (1997).
[CrossRef]

IEEE Trans. Geosci. Rem. Sens. (1)

A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001).
[CrossRef]

J. Appl. Phys. (1)

D. F. Jackson Kimball, L. R. Jacome, S. Guttikonda, E. J. Bahr, and L. F. Chan, "Magnetometric sensitivity optimization for nonlinear optical rotation with frequency-modulated light: Rubidium D2 line," J. Appl. Phys. 106, 063113 (2009).
[CrossRef]

J. Opt. Soc. Am. B (2)

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 (2005).
[CrossRef]

S. Chakmakjian, K. Koch, and J. C. R. Stroud, "Observation of resonances at subharmonics of the rabi frequency in the saturated absorption of a 100%amplitude-modulated laser beam," J. Opt. Soc. Am. B 5, 2015-2020 (1988).
[CrossRef]

J. Phys. Radium (1)

A. Kastler, "Quelques suggestions concernant la production optique et la d’etection optique dúne inégalité de population des niveaux de quantifigation spatiale des atomes. Application á l’expérience de Stern et Gerlach et á la résonance magnétique," J. Phys. Radium 11, 255-265 (1950).
[CrossRef]

Opt. Express (1)

V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008).
[CrossRef] [PubMed]

Opt. Spectrosc. (1)

A. Okunevich, "Laser pumping and magneto-optical rotation of the light polarization plane in a cell with an antirelaxation coating of the walls: I. Statement and solution of the problem," Opt. Spectrosc. 97, 834-841 (2004).
[CrossRef]

Phys. Rev. (2)

J. Brossel, and F. Bitter, "A new "double resonance" method for investigating atomic energy levels. Application to Hg 3P1*," Phys. Rev. 86, 308-316 (1952).
[CrossRef]

M. A. Bouchiat, and J. Brossel, "Relaxation of optically pumped Rb atoms on paraffin-coated walls," Phys. Rev. 147, 41-54 (1966).
[CrossRef]

Phys. Rev. A (9)

W. Chalupczak, P. Josephs-Franks, S. Pustelny, and W. Gawlik, "Optical-radio-frequency resonances free from power broadening," Phys. Rev. A 81, 013422 (2010).
[CrossRef]

M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005).
[CrossRef]

G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006).
[CrossRef]

A. Weis, G. Bison, and A. S. Pazgalev, "Theory of double resonance magnetometers based on atomic alignment," Phys. Rev. A 74, 033401 (2006).
[CrossRef]

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hänsch, "Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

H. Friedmann, and A. D. Wilson-Gordon, "Dispersion profiles of the absorptive response of a two-level system interacting with two intense fields," Phys. Rev. A 36, 1333-1341 (1987).
[CrossRef] [PubMed]

M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, "Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation," Phys. Rev. A 75, 023405 (2007).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

S. Pustelny, D. F. J. 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, 023817 (2006).
[CrossRef]

Phys. Rev. Lett. (4)

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, 253001 (2003).
[CrossRef] [PubMed]

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, "Quantum noise limited and entanglement-assisted magnetometry," Phys. Rev. Lett. 104, 133601 (2010).
[CrossRef] [PubMed]

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088 (2000).
[CrossRef] [PubMed]

D. Budker, V. Yashchuk, and M. Zolotorev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, "Magnetic resonance imaging with an optical atomic magnetometer," Proc. Natl. Acad. Sci. U.S.A. 103, 12668-12671 (2006).
[CrossRef]

Rev. Mod. Phys. (3)

R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003).
[CrossRef]

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 (2002).
[CrossRef]

T. G. Walker, and W. Happer, "Spin-exchange optical pumping of noble-gas nuclei," Rev. Mod. Phys. 69, 629 (1997).
[CrossRef]

Rev. Sci. Instrum. (2)

S. Xu, S. M. Rochester, V. V. Yashchuk, M. H. Donaldson, and D. Budker, "Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation," Rev. Sci. Instrum. 77, 083106 (2006).
[CrossRef]

V. Yashchuk, D. Budker, and J. Davis, "Laser frequency stabilization using linear magneto-optics," Rev. Sci. Instrum. 71, 341 (2000).
[CrossRef]

Other (3)

M. Auzinsh, D. Budker, and S. M. Rochester, Optically Polarized Atoms: Understanding Light-Atom Interactions (Oxford University Press, Oxford, 2010).

D. F. Jackson Kimball, O. Neitzke, E. J. Bahr, S. Guttikonda, S. M. Rochester, M. P. Ledbetter, I. Novikova, B. Coste, S. A. Rangwala, J. M. Higbie, A. I. Okunevich, V. V. Yashchuk, and D. Budker (in preparation).

C. Cohen-Tannoudji, Atoms in Electromagnetic Fields (World Scientific, Singapore, 1994).

Supplementary Material (3)

» Media 1: MOV (245 KB)     
» Media 2: MOV (677 KB)     
» Media 3: MOV (829 KB)     

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Figures (7)

Fig. 1
Fig. 1

A laser beam linearly polarized along the z axis propagates through a paraffin-coated vapor cell, producing ground-state alignment in the 87Rb atoms. A constant magnetic field B0 and an oscillating rf magnetic field Brf are applied within the magnetic shield that surrounds the cell. A balanced polarimeter and lock-in amplifier are used to analyze polarization rotation of the light induced by the atoms. A resonance is observed when the rf frequency is equal to the ground-state Larmor frequency of the atoms.

Fig. 2
Fig. 2

Experimental spectra (dotted lines) and theoretical predictions (solid lines) for three different values of the rf field amplitude. The experimental values (obtained from calibrated measurements of the coil current) in terms of the Rabi frequency Ωrf are (a) Ωrf/(2π) = 3 Hz, (b) Ωrf/(2π) = 63 Hz, (c) Ωrf/(2π) = 158 Hz. These values are used in the theory, along with the parameters natural width Γ = 38 μs−1 = 2π× (6.1 MHz) (chosen equal to the natural width of the Rb D2 transition), light power 5 μW, ground-state relaxation rate γt = 50 s−1 = 2π × (8 Hz), and bias-field Larmor frequency ΩL/(2π) = 554.75 Hz. These last three parameters are chosen for optimal agreement with the data—they are reasonably close to the experimental values. The value for light power is applied to the theoretical model using the method described in Sec. 2.2. At low rf-field strengths, as in row (a), the spectra are Lorentzians. An additional central feature appears in the resonances when Ωrf exceeds γt, as in rows (b) and (c). Boxes in row (c) indicate regions plotted on expanded scales in Fig. 3.

Fig. 3
Fig. 3

Row (c) of Fig. 2 (boxed regions) plotted on expanded scales. (The vertical scale is different for each plot.) A resonance due to high-field effects can be seen near ω rf / ( 2 π ) = 1 3 Ω L / ( 2 π ) 185 Hz ..

Fig. 4
Fig. 4

Angular-momentum-probability surfaces illustrating the behavior of the system with a resonant (Δrf = 0), weak rf field. (a) The light, with electric field E, produces atomic alignment along the z-axis. (b) In the rotating frame, the atoms precess around the x-axis. Due to relaxation, a steady state is reached. (c) In the laboratory frame, the atoms precess around the z-axis, with the linear dichroism of the ensemble inducing oscillating optical rotation in the z-polarized, ŷ-propagating light field. ( Media 1 animates the precession and optical rotation as a function of time.) The transmission axis of the polarized ensemble, in this case parallel to the atomic alignment axis, is indicated with a line drawn on the surface.

Fig. 5
Fig. 5

As Fig. 4, but with an off-resonant rf field. (a) In the rotating frame, the atoms precess around the total effective field, which lies in the xz plane. As a result, the polarization is no longer entirely in the yz plane. (b) Far off resonance, the effective magnetic field is large enough that the polarization is completely averaged about the magnetic-field direction. (The arrow representing Btot is truncated, as it is too long to fit entirely on the plot.) Media 2 shows the polarization, along with the in-phase component of the optical-rotation signal, as the rf-field detuning is swept through resonant and off-resonant conditions

Fig. 6
Fig. 6

Rotating-frame AMPS for intermediate rf field strength, in the ideal case in which γtΩrf ≪ |ΩL|. Optical rotation is indicated for the instant in which the rotating frame coincides with the laboratory frame—this means that the generation of the in-phase component of the signal is shown. (a) When the rf field is on resonance, the atomic polarization (created along the z-axis) is averaged about the x-axis. The resulting polarization is a “doughnut” aligned along the x-axis. The transmission axis, perpendicular to the alignment axis, is marked with a line on the surface. (b) For small detunings, the alignment axis follows the effective magnetic field direction. (c) When the effective magnetic field is at the magic angle θ m = arccos ( 1 / 3 ) to the light polarization direction, the atomic polarization is completely averaged out due to precession. (d) For larger detunings, the polarization regains its original “peanut” shape, and the transmission axis is along the alignment axis. Media 3 shows the atomic polarization and in-phase optical rotation as the rf-field detuning is swept through resonance.

Fig. 7
Fig. 7

Cross section of the AMPS for pure alignment along the z-axis, described by Y2,0 (θ, ϕ) ∝ 3cos2θ – 1. Positive function values are shown as solid lines, negative function values as dashed lines. The magic angle θ m = arccos ( 1 / 3 ) is indicated.

Equations (19)

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H = ( Ω L Ω rf 2 2 ( e i ω rf t + e i ω rf t ) 0 0 Ω rf 2 2 ( e i ω rf t + e i ω rf t ) 0 Ω rf 2 2 ( e i ω rf t + e i ω rf t ) Ω R 2 3 0 Ω rf 2 2 ( e i ω rf t + e i ω rf t ) Ω L 0 0 Ω R 2 3 0 Δ ) ,
U ( t ) = ( e i ω rf t 0 0 0 0 1 0 0 0 0 e i ω rf t 0 0 0 0 1 ) ,
H = ( Δ rf Ω rf 2 2 0 0 Ω rf 2 2 0 Ω rf 2 2 Ω R 2 3 0 Ω rf 2 2 Δ rf 0 0 Ω R 2 3 0 Δ ) ,
ρ ˙ = i [ H , ρ ] 1 2 { ζ , ρ } + Λ ,
ζ = ( γ t 0 0 0 0 γ t 0 0 0 0 γ t 0 0 0 0 Γ + γ t ) ,
Λ = ( γ t 3 + Γ 3 ρ e 0 e 0 0 0 0 0 γ t 3 + Γ 3 ρ e 0 e 0 0 0 0 0 γ t 3 + Γ 3 ρ e 0 e 0 0 0 0 0 0 ) ,
φ in = 3 2 N Γ λ 2 4 π Ω R Im ( ρ e 0 g 1 ρ e 0 g 1 ) ,
φ out = 3 2 N Γ λ 2 4 π Ω R Re ( ρ e 0 g 1 + ρ e 0 g 1 ) ,
φ abs = ( φ in ) 2 + ( φ out ) 2 = 3 2 N Γ λ 2 4 π Ω R { [ Im ( ρ e 0 g 1 ρ e 0 g 1 ) ] 2 + [ Re ( ρ e 0 g 1 + ρ e 0 g 1 ) ] 2 } 1 / 2 .
φ in = N Δ rf λ 2 Ω rf ( 2 γ t 2 + 8 Δ rf 2 Ω rf 2 ) Ω R 2 36 π Γ γ t ( γ t 2 + 4 Δ rf 2 + Ω rt 2 ) [ 4 ( γ t 2 + Δ rf 2 ) + Ω rf 2 ] ,
φ out = N λ 2 Ω rf ( 4 γ t 2 + 16 Δ rf 2 + Ω rf 2 ) Ω R 2 72 π Γ ( γ t 2 + 4 Δ rf 2 + Ω rf 2 ) [ 4 ( γ t 2 + Δ rf 2 ) + Ω rf 2 ] ,
φ in = N Δ rf λ 2 Ω R 2 72 π Γ γ t ( γ t 2 + Δ rf 2 ) Ω rf N Δ rf λ 2 Ω R 2 ( 7 γ t 2 + 10 Δ rf 2 ) 288 π Γ γ t ( γ t 2 + Δ rf 2 ) 2 ( γ t 2 + 4 Δ rf 2 ) Ω rf 3 + O [ Ω rf ] 5 ,
φ out = N λ 2 Ω R 2 72 π Γ ( γ t 2 + Δ rf 2 ) Ω rf N λ 2 Ω R 2 ( 4 γ t 2 + 7 Δ rf 2 ) 288 π Γ ( γ t 2 + Δ rf 2 ) 2 ( γ t 2 + 4 Δ rf 2 ) Ω rf 3 + O [ Ω rf ] 5 ,
φ abs = N λ 2 Ω R 2 Ω rf 144 π Γ γ t γ t 2 + Δ rf 2 = N λ 2 ( e γ t 2 + 5 Δ rf 2 ) Ω R 2 Ω rf 3 288 π Γ γ t ( γ t 2 + Δ rf 2 ) 3 / 2 ( γ t 2 + 4 Δ rf 2 ) + O [ Ω rf ] 5 .
ρ ˙ ( t ) = A 1 ρ ( t ) + A 2 ρ ( t ) ( e i ω rf t + e i ω rf t ) + b ,
ρ ( t ) = Σ a n e in ω rf ,
in ω rf a n e in ω rf t = A 1 a n e in ω rf t + A 2 a n ( e i ( n + 1 ) ω rf t + e i ( n 1 ) ω rf t ) + b ,
[ A 1 a n in ω rf a n + A 2 ( a n 1 + a n + 1 ) ] e in ω rf t = b ,
A 2 a n 1 + ( A 1 in ω rf ) a n + A 2 a n + 1 = b δ n , 0 ,

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