Abstract

We demonstrate a novel method of measuring magnetic field based on the transient signal of the K-Rb-21Ne co-magnetometer operating in nuclear spin magnetization self-compensation magnetic field regime. The operation condition for self-compensation magnetic field by nuclear spin magnetization of 21Ne in steady state is presented. We characterize the dynamics of the coupled spin ensembles by a set of Bloch equations, and formulate the expression of transient output signal. After verifying the stability of this method, the measurement range and error are studied. This method is also verified to be valid in various temperature and pumping light power density.

© 2017 Optical Society of America

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  1. P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
    [Crossref]
  2. M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).
  3. G. Bison, R. Wynands, and A. Weis, “Dynamical mapping of the human cardiomagnetic field with a room-temperature, laser-optical sensor,” Opt. Express 11(8), 904 (2003).
    [Crossref] [PubMed]
  4. H. J. Lee, J. H. Shim, H. S. Moon, and K. Kim, “Flat-response spin-exchange relaxation free atomic magnetometer under negative feedback,” Opt. Express 22(17), 19887 (2014).
    [Crossref] [PubMed]
  5. D. Drung, “High-performance DC SQUID read-out electronics,” Physica C 368(1–4), 134–140 (2002).
    [Crossref]
  6. I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
    [Crossref] [PubMed]
  7. H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
    [Crossref]
  8. D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
    [Crossref] [PubMed]
  9. D. Budker and M. V. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
    [Crossref]
  10. M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
    [Crossref] [PubMed]
  11. T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
    [Crossref] [PubMed]
  12. G. Vasilakis, “Precision measurements of spin interactions with high density atomic vapors,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, USA, (2011).
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    [Crossref]
  14. Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
    [Crossref]
  15. E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
    [Crossref] [PubMed]
  16. M. V. Romalis, “Hybrid optical pumping of optically dense alkali-metal vapor without quenching gas,” Phys. Rev. Lett. 105(24), 243001 (2010).
    [Crossref]
  17. T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. 69(2), 629–642 (1997).
    [Crossref]
  18. R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
    [Crossref]
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    [Crossref] [PubMed]
  20. S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
    [Crossref]
  21. R. K. Ghosh and M. V. Romalis, “Measurement of spin-exchange and relaxation parameters for polarizing 21Ne with K and Rb,” Phys. Rev. A 81(4), 043415 (2010).
    [Crossref]
  22. I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005).
    [Crossref]
  23. J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
    [Crossref] [PubMed]
  24. Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
    [Crossref]
  25. L. Duan, J. Fang, R. Li, L. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express,  23(25), 32481 (2015).
    [Crossref] [PubMed]

2016 (4)

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

2015 (1)

2014 (1)

2013 (1)

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

2011 (1)

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

2010 (3)

M. V. Romalis, “Hybrid optical pumping of optically dense alkali-metal vapor without quenching gas,” Phys. Rev. Lett. 105(24), 243001 (2010).
[Crossref]

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
[Crossref]

R. K. Ghosh and M. V. Romalis, “Measurement of spin-exchange and relaxation parameters for polarizing 21Ne with K and Rb,” Phys. Rev. A 81(4), 043415 (2010).
[Crossref]

2009 (1)

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

2007 (1)

D. Budker and M. V. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
[Crossref]

2005 (3)

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
[Crossref] [PubMed]

I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005).
[Crossref]

2003 (3)

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

G. Bison, R. Wynands, and A. Weis, “Dynamical mapping of the human cardiomagnetic field with a room-temperature, laser-optical sensor,” Opt. Express 11(8), 904 (2003).
[Crossref] [PubMed]

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

2002 (3)

D. Drung, “High-performance DC SQUID read-out electronics,” Physica C 368(1–4), 134–140 (2002).
[Crossref]

T. W. Kornack and M. V. Romalis, “Dynamics of two overlapping spin ensembles interacting by spin exchange,” Phys. Rev. Lett. 89(25), 253002 (2002).
[Crossref] [PubMed]

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

1997 (1)

T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. 69(2), 629–642 (1997).
[Crossref]

1989 (1)

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Allred, J. C.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

Anderson, L. W.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Babcock, E.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Bison, G.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

G. Bison, R. Wynands, and A. Weis, “Dynamical mapping of the human cardiomagnetic field with a room-temperature, laser-optical sensor,” Opt. Express 11(8), 904 (2003).
[Crossref] [PubMed]

Brown, J. M.

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

Budker, D.

D. Budker and M. V. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
[Crossref]

Castagna, N.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Cates, G. D.

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Chen, Y.

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

Cheuk, L. W.

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

Chien, T. R.

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Dang, H. B.

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
[Crossref]

Ding, M.

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

L. Duan, J. Fang, R. Li, L. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express,  23(25), 32481 (2015).
[Crossref] [PubMed]

Driehuys, B.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Drung, D.

D. Drung, “High-performance DC SQUID read-out electronics,” Physica C 368(1–4), 134–140 (2002).
[Crossref]

Duan, L.

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

L. Duan, J. Fang, R. Li, L. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express,  23(25), 32481 (2015).
[Crossref] [PubMed]

Dural, N.

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

Fan, W.

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

Fang, J.

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

L. Duan, J. Fang, R. Li, L. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express,  23(25), 32481 (2015).
[Crossref] [PubMed]

Ghosh, R. K.

R. K. Ghosh and M. V. Romalis, “Measurement of spin-exchange and relaxation parameters for polarizing 21Ne with K and Rb,” Phys. Rev. A 81(4), 043415 (2010).
[Crossref]

T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
[Crossref] [PubMed]

Gonatas, D.

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Grauch, V. J. S.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Hansen, R. O.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Happer, W.

T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. 69(2), 629–642 (1997).
[Crossref]

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Hersman, F. W.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Hofer, A.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Hu, Z.

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Jiang, L.

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

L. Duan, J. Fang, R. Li, L. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express,  23(25), 32481 (2015).
[Crossref] [PubMed]

Kadlecek, S.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Kim, K.

Knowles, P.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Kominis, I. K.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

Kornack, T. W.

T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
[Crossref] [PubMed]

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

T. W. Kornack and M. V. Romalis, “Dynamics of two overlapping spin ensembles interacting by spin exchange,” Phys. Rev. Lett. 89(25), 253002 (2002).
[Crossref] [PubMed]

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

LaFehr, T. R.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Lee, H. J.

Li, R.

Li, S.

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

Li, Y.

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Liu, X.

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Lu, Y.

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

Lyman, R. N.

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

Maloof, A. C.

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
[Crossref]

Moon, H. S.

Mtchedlishvili, A.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Nabighian, M. N.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Nelson, I.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Pazgalev, A.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Peirce, J. W.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Phillips, J. D.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Quan, W.

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Romalis, M. V.

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
[Crossref]

R. K. Ghosh and M. V. Romalis, “Measurement of spin-exchange and relaxation parameters for polarizing 21Ne with K and Rb,” Phys. Rev. A 81(4), 043415 (2010).
[Crossref]

M. V. Romalis, “Hybrid optical pumping of optically dense alkali-metal vapor without quenching gas,” Phys. Rev. Lett. 105(24), 243001 (2010).
[Crossref]

D. Budker and M. V. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
[Crossref]

T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
[Crossref] [PubMed]

I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005).
[Crossref]

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

T. W. Kornack and M. V. Romalis, “Dynamics of two overlapping spin ensembles interacting by spin exchange,” Phys. Rev. Lett. 89(25), 253002 (2002).
[Crossref] [PubMed]

Ruder, M. E.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Savukov, I. M.

I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005).
[Crossref]

Schaefer, S. R.

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Sheng, D.

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

Shim, J. H.

Smiciklas, M.

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

Smullin, S. J.

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

Vasilakis, G.

G. Vasilakis, “Precision measurements of spin interactions with high density atomic vapors,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, USA, (2011).

Walker, T. G.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. 69(2), 629–642 (1997).
[Crossref]

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Wang, W.

Weis, A.

Weisa, A.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Wynands, R.

Yuan, H.

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Zhang, H.

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

Zou, S.

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Appl. Phys. Lett. (1)

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
[Crossref]

Geophysics (1)

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

J. Phys. B (1)

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Nat. Phys. (1)

D. Budker and M. V. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
[Crossref]

Nature (1)

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

Nucl. Instrum. Methods Phys. Res., Sect. A (1)

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Opt. Express (3)

Phys. Rev. A (5)

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

R. K. Ghosh and M. V. Romalis, “Measurement of spin-exchange and relaxation parameters for polarizing 21Ne with K and Rb,” Phys. Rev. A 81(4), 043415 (2010).
[Crossref]

I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005).
[Crossref]

Phys. Rev. Lett. (7)

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

M. V. Romalis, “Hybrid optical pumping of optically dense alkali-metal vapor without quenching gas,” Phys. Rev. Lett. 105(24), 243001 (2010).
[Crossref]

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
[Crossref] [PubMed]

T. W. Kornack and M. V. Romalis, “Dynamics of two overlapping spin ensembles interacting by spin exchange,” Phys. Rev. Lett. 89(25), 253002 (2002).
[Crossref] [PubMed]

Physica C (1)

D. Drung, “High-performance DC SQUID read-out electronics,” Physica C 368(1–4), 134–140 (2002).
[Crossref]

Rev. Mod. Phys. (1)

T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. 69(2), 629–642 (1997).
[Crossref]

Sci. Rep. (1)

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

Other (1)

G. Vasilakis, “Precision measurements of spin interactions with high density atomic vapors,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, USA, (2011).

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Figures (6)

Fig. 1
Fig. 1

Intuitive compensation principle of self-compensation co-magnetometer. (a) Without transverse magnetic field By, Pe and Pn direct along z-axis. (b) When applying By, Pe and Pn would precess around the total magnetic field in the transient process. (c) Pn would stabilize at the direction in y–z plane in a short time, resulting in the projection of Bn with respect to y-axis compensating By, hence Pe returns back to z-axis unaffected by By in steady state.

Fig. 2
Fig. 2

The schematic of the co-magnetometer. GT, Glan-Thompson polarizer; PD, photo detector; PBS, polarization beam splitter; ECDL, external cavity diode laser; TA, tapered amplifier; BE, beam expander; DFB, distributed feedback laser; PEM, photo elastic modulator.

Fig. 3
Fig. 3

The responses to transverse magnetic field By0 = 0.32 nT with various bias magnetic field. (a) The bias magnetic field Bz is set to the compensation point Bc = −312.7 nT. (b) Bz is set to the non-compensation point Bc = −362.7 nT. (c) The difference between the initial offset value and the steady state value Δ S x 0 is measured with same input magnetic field By0 = 0.32 nT and various bias magnetic field δ Bz.

Fig. 4
Fig. 4

The stability of the system parameters. (a) The variation of the four fitted system parameters over time. The response signals to By0 = 0.32nT at every half hour are continuously measured five times, of which the standard deviations of the fitted system parameters are plotted as the error bars. (b) The R-square of the fitting curves with different input magnetic field. The response signals to each input magnetic field are measured five times, of which the standard deviations of the fitted R-square are regarded as the error bars.

Fig. 5
Fig. 5

(a) The fitted By as function of the input By. The red dots denote the measured data, while the function of the black line is y = x. The response signals to each input signal are measured five times, of which the standard deviations of the fitted magnetic fields are illustrated as the error bars. (b) The corresponding residuals between the fitted By and the input By. The horizontal axis is in logarithmic coordinates. (c) The measured and fitted responses to small magnetic field. The black solid curve is the measured probe signal without input magnetic field. The circle point, asterisk point and upward-pointing triangle point are the measured signals to 1.6 pT, 8 pT, and 16 pT respectively, while the red solid curve, blue solid curve and green solid curve are the fitted curves to 1.6 pT, 8 pT, and 16 pT respectively.

Fig. 6
Fig. 6

The fitting results of response signals to various input magnetic field in different conditions. (a) and (b) denote the fitted R-square and By with different pumping light power density respectively. (c) and (d) designate the fitted R-square and By with different operation temperature respectively. The function of the black line in (b), identical to the one in (d), is y = x. The insets in (b) and (d) show the magnification of the fitted By with the input magnetic field of about 0.288 nT and 0.672 nT respectively. In (a) and (b), the response signals to each input magnetic field are measured five times under different pumping light power density, of which the standard deviations of the R-square and the fitted magnetic fields are plotted as the error bars respectively. While in (c) and (d), the definitions of the error bars are the same as (a) and (b) except for the response signals are measured under different operation temperature.

Equations (16)

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P e t = γ e Q ( B + λ M n P n + L ) × P e Ω × P e + R m S m + R p S p + R se ne P n Q P e Q { T 1 , T 2 , T 2 } , P n t = γ n ( B + λ M e P e ) × P n Ω × P n + R se en P e R tot n P n .
S x = K d P x e = e λ 1 r t ( P 1 r cos λ 1 i t P 1 i sin λ 1 i t ) + e λ 2 r t ( P 2 r cos λ 2 i t P 2 i sin λ 2 i t ) + S x steady , λ 1 = λ 1 r + i λ 1 i = R tot e 2 Q + a 2 + b 2 + a 2 2 + i [ ( γ e ( L z B z e ) 2 Q γ n B z n 2 ) + Sign [ b ] a 2 + b 2 a 2 2 ] , λ 2 = λ 2 r + i λ 2 i = R tot e 2 Q + a 2 + b 2 + a 2 2 + i [ ( γ e ( L z B z e ) 2 Q γ n B z n 2 ) Sign [ b ] a 2 + b 2 a 2 2 ] , a = ( R tot e Q ) 2 ( γ e L z B z e Q + γ n B z n ) 2 4 γ e B z e γ n B z n Q , b = 2 R tot e Q ( γ e L z B z e Q + γ n B z n ) 4 γ e B z e γ n B z n Q ,
P 1 r = K r B y 0 + K 1 r 0 Ω y , P 1 i = K i B y 0 + K 1 i 0 Ω y , P 2 r = K r B y 0 + K 2 r 0 Ω y , P 2 i = K i B y 0 + K 2 i 0 Ω y , K r = λ 1 r λ 2 r ( λ 2 r λ 1 r ) 2 + ( λ 2 i λ 1 i ) 2 K d γ e P z e Q , K i = λ 2 i λ 1 i ( λ 2 r λ 1 r ) 2 + ( λ 2 i λ 1 i ) 2 K d γ e P z e Q , S x steady = K steady ( δ B z B n B y + Ω y γ n ) , K steady = K d γ e P z e R tot e R tot e + γ e 2 ( L z + δ B z ) 2 .
P ˜ e t = { R tot e Q + i [ γ e Q ( B z + λ M n P z n + L z ) Ω z ] } P ˜ e + ( R se ne Q i γ e λ M n P z e Q ) P ˜ n + P z e γ e ( B y i B x ) / Q ,
P ˜ n t = ( R se en i γ n λ M e P z n ) P ˜ e + { R tot n + i [ γ n ( B z + λ M e P z e ) Ω z ] } P ˜ n + P z n γ n ( B y Ω y i B x + i Ω x ) .
P ˜ e ( t ) = P 1 e λ 1 t + P 2 e λ 2 t + P e steady ,
P x e ( t ) = Re [ P ˜ e ] = Re [ P 1 e λ 1 t + P 2 e λ 2 t ] + P x e steady ,
λ 1 = λ 1 r + i λ 1 i = R tot e 2 Q + a 2 + b 2 + a 2 2 + i [ ( γ e ( L z B z e ) 2 Q γ n B z n 2 ) + Sign [ b ] a 2 + b 2 a 2 2 ] ,
λ 2 = λ 2 r + i λ 2 i = R tot e 2 Q + a 2 + b 2 + a 2 2 + i [ ( γ e ( L z B z e ) 2 Q γ n B z n 2 ) Sign [ b ] a 2 + b 2 a 2 2 ] ,
a = ( R tot e Q ) 2 ( γ e L z B z e Q + γ n B z n ) 2 4 γ e B z e γ n B z n Q ,
b = 2 R tot e Q ( γ e L z B z e Q + γ n B z n ) 4 γ e B z e γ n B z n Q ,
S x = K d P x e = e λ 1 r t ( P 1 r cos λ 1 i t P 1 i sin λ 1 i t ) + e λ 2 r t ( P 2 r cos λ 2 i t P 2 i sin λ 2 i t ) + S x steady .
S x steady = K d P x e steady = K d γ e P z e R tot e R tot e 2 + γ e 2 ( L z + δ B z ) 2 ( δ B z B n B y + Ω y γ n ) ,
P 1 r = K r B y 0 + K 1 r 0 Ω y , P 1 i = K i B y 0 + K 1 i 0 Ω y ,
P 2 r = K r B y 0 + K 2 r 0 Ω y , P 2 i = K i B y 0 + K 2 i 0 Ω y ,
K r = λ 1 r λ 2 r ( λ 2 r λ 1 r ) 2 + ( λ 2 i λ 1 i ) 2 K d γ e P z e Q , K i = λ 2 i λ 1 i ( λ 2 r λ 1 r ) 2 + ( λ 2 i λ 1 i ) 2 K d γ e P z e Q .

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