Abstract

We present a method for monitoring the atomic density number on site based on atomic spin exchange relaxation. When the spin polarization P ≪ 1, the atomic density numbers could be estimated by measuring magnetic resonance linewidth in an applied DC magnetic field by using an all-optical atomic magnetometer. The density measurement results showed that the experimental results the theoretical predictions had a good consistency in the investigated temperature range from 413 K to 463 K, while, the experimental results were approximately 1.5 ∼ 2 times less than the theoretical predictions estimated from the saturated vapor pressure curve. These deviations were mainly induced by the radiative heat transfer efficiency, which inevitably leaded to a lower temperature in cell than the setting temperature.

© 2016 Optical Society of America

1. Introduction

With the increasing demands for magnetic field detecting with higher sensitivity [1–4], optically pumped atomic magnetometer is beginning to display its ultra-high sensitivity to a weak magnetic field [5–8]. This is strongly motivated due to numerous current and potential future applications. Among such researches, an on-site accurate monitoring of the alkali atomic density number is probably the most invaluable as the density directly affects the magnetometer’s sensitivity.

Currently, the laser absorption spectroscopy [9] and Faraday rotation [10] methods are two main approaches to monitor the atomic density number in cell. With the laser absorption spectroscopy method, Y. Ito et al. investigated the densities of pure potassium (K) and pure rubidium (Rb) atoms. The results showed that the experimental values were approximately 10 times less than those obtained from the saturated vapor pressure at 453 K [11]. The accuracy of laser absorption spectroscopy method is limited by the broadened and distorted Lorentzian line shape. With regard to the Faraday rotation method, E. Vliegen et al. studied the measurement of atomic density number using the Faraday rotation of linearly-polarized light. The results illustrated that the laser absorption spectroscopy and Faraday rotation methods had the similar measurement accuracy [10]. Yet, this method required applying a strong magnetic field (∼ 1.2 T), which could magnetize the atomic magnetometer’s magnetic shields.

In this work, based on analyzing the features of the all-optical atomic magnetometer operated in an applied DC magnetic field, we present an on-site atomic density number monitoring method, which is based on the magnetic resonance linewidth of the magnetometer and can measure more accurately atomic density number than the laser absorption spectroscopy and Faraday rotation methods. This monitoring method has the advantages of high-accuracy and convenience. Moreover, it realizes the function of real-time monitoring of the atomic density number during the magnetometer’s work process.

2. Principle and method

The outermost layer of the alkali metal atom has a single valence electron, and it can be polarized by a circularly-polarized light. Through optical pumping, the angular momentum can be transferred from the circularly-polarized pump light to the ensemble of the atoms. A linearly-polarized probing light propagates perpendicular to the pumping direction. A magnetic field perpendicular to the plane composed by the pump beam and the probe beam can rotate the direction of the electron spins by a small angle, and this small angle can be monitored by the probe beam. Usually, the density matrix (DM) equation can be used to accurately described the dynamics of the all-optical atomic magnetometer [12, 13],

dρdt=[H,ρ]ih¯+RSE[φ(1+4SS)ρ]+Rrel[φρ]+ROP[φ(1+2sS)ρ]+D2ρ,
where ρ is the density matrix, which has dimension of the number of hyperfine states, [H, ρ] is the commutator, [H, ρ] = ρH; φ = ρ/4 + S · ρS is the purely number part of the density matrix, S is the atomic spin vector, and Rrel and ROP are the relaxation rate and the optical pumping rate, respectively; s is the photon spin vector, D is the diffusion constant, and RSE is the spin exchange relaxation, which is directly proportional to the atomic density number only closely related to the temperature. The first term in Eq. (1) is the evolution due to the free-atom Hamiltonian; the second term expresses the spin exchange collisions between alkali atoms; the third term describes the spin relaxation of electrons; the fourth term characterizes optical pumping effect; the last term gives the spin diffusion collisions of electrons.

In an applied magnetic field, since the spin exchange (SE) collisions predominantly dominate in the polarization lifetime, the SE collisions between two alkali metal atoms are typically the dominant due to its high rate among these collisions, much larger than any other spin relaxation mechanism. Hence, the magnetometer’s magnetic resonance linewidth (i. e. here the magnetic resonance linewidth is well described by a simple Lorentzian with the full width at half maximum (FWHM)) is determined by the SE relaxation [14].

In the regime of low polarization, the magnetic resonance linewidth ΔυB depends on the magnetic field and the SE relaxation can be expressed as [12, 15, 16]

ΔυB=1π{[(2I+1)2+2]RSE3(2I+1)2gS2μB2B2h¯2(2I+1)22gSμBBRSEih¯(2I+1)2+[(2I+1)2+2]2RSE29(2I+1)4iγeq|B|},
where I is the nuclear spin quantum number, is the reduced Planck constant, gB is the electron’s g factor, μB is the Bohr magneton, B is the magnetic field, γe is the gyromagnetic ratio of the electron, γe = 28 Hz/nT, and q is the nuclear slowing down factor. For the all-optical atomic magnetometer operated in the limit of low polarization, q = 4 [16].

Figure 1 shows that the magnetic resonance linewidth calculated from Eq. (2) for the potassium atoms as a function of the magnetic field amplitude. At a certain magnetic field, the elevation of the temperature enlarges the magnetic resonance linewidth. At a certain temperature, the magnetic resonance linewidth increases with the enhancive magnetic field; while, when the magnetic field exceeds to a certain value, the magnetic resonance linewidth is almost unchanged (i.e. the magnetic resonance linewidth reaches a steady state). During our experiments, the working temperature is 463 K. According to Fig. 1, below 463 K, it is well known that the magnetic resonance linewidth increases monotonically with the strength of magnetic field then achieves steady state after 2700 nT, which is also suitable for RSE. In order to obtain a steady value of the RSE below 463 K, the amplitude of the applied magnetic field is 2700 nT.

 figure: Fig. 1

Fig. 1 Magnetic resonance linewidth Δυ estimated from Eq. (2) for the all-optical potassium atomic magnetometer at different temperatures.

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The magnetic resonance linewidth in the presence of the spin exchange and spin destruction relaxation collisions can also be given by [17]

ΔυSE=1πT2=ROP4π+RSERSDROPπG(ω0,RSE),
G(ω0,RSE)=Re[RSE+4iω02/πυHF5RSE+8iω02/πυHF],
where RSD is the spin destruction relaxation rate, ω0 is the Zeeman transition frequency, and υHF is the ground state hyperfine splitting. In the low polarization limit, P ≪ 1, the magnetic resonance linewidth is only subject to the SE collisions,
ΔυSE=RSE8π,
and RSE expressed as
RSE=nσSE8KBTπM,
where n is the density number of the alkali metal atoms, σSE is the SE cross section (σSE = 1.8 × 10−14 cm2 for the potassium atoms [18]), KB is the Boltzmann constant, T is the temperature in Kelvin, and M is the reduced mass of two alkali metal atoms A and B, 1M=1mA+1mB. For 8KBT/πM, its value changes small as temperature changing.

Substituting Eq. (6) into Eq. (5) and simplifying, the density number of the alkali metal atoms can be computed by

n=8πΔυSEσSE8πMKBT,
Equation (7) shows that the atomic density number is proportional to the magnetic resonance linewidth.

In order to verify the validity of the magnetic resonance linewidth method measuring the atomic density number, a comparison of theoretical simulation between the saturated vapor pressure and the magnetic resonance linewidth method is carried out. Generally, the atomic density number in cell can be estimated via the saturated vapor pressure,

n=1T1021.866+AB/T,
where A and B are the parameters, for potassium in the liquid phase, A = 4.402, B = 4453 [19].

Figure 2 shows that the atomic density numbers estimated from the saturated vapor pressure curve and the magnetic resonance linewidth method present in this paper. These two methods fit very well with each other. Therefore, magnetic resonance linewidth method could be used to obtain the atomic density number.

 figure: Fig. 2

Fig. 2 Theoretical predictions of the atomic density numbers determining by the saturated vapor pressure and the magnetic resonance linewidth method presented in this paper, respectively. Solid blue line is the atomic density number obtained by saturated vapor pressure and red crosses are that obtained by the method presented in this paper.

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3. Experimental setup

The verification experiment was carried out with an all-optical atom magnetometer. The experimental apparatus is shown schematically in Fig. 3. A cubical alkali metal cell with a 25 mm length is made from aluminosilicate glass. The cell is filled with a droplet of K atoms, 700 torr (1 torr=133.322 Pa) 4He and 30 torr N2 for quenching. It is placed in the center of an oven and can be heated by an alternating current electrical heating device. A Pt1000 temperature sensor pasted on the alkali metal cell, where does not affect the pump and probe beams passing, is used to monitor the setting temperature. The outer layer of the oven is a three-dimensional Helmholtz coil, whose y direction connects with an output port of lock-in amplifier (Zurich Instrument HF2LI) and x and z directions join with a function generator. Such a triaxial coil can be utilized to produce uniform magnetic fields. A five nested cylindrical magnetic shields are applied to isolate the external magnetic field, and the shielding factor is approximately 105.

 figure: Fig. 3

Fig. 3 Schematic of the experimental device. The pump and probe beams propagate along the z and x directions, respectively.

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In our experimental apparatus, we use two distributed-feedback (DFB) lasers to pump the atom spin and detect the optical rotation, respectively. Generally, the wavelength of the pump beam is tuned to the D1 line of K atoms (770.108 nm); the probe beam detunes 0.5 nm from the D2 line due to the maximal signal amplitude. As for the pumping light path, a quarter-wave plate placed behind a polarizer is used to insure that the pump beam is a circularly-polarized light after passing through it. With regard to the probe light path, after the cell the probe beam passes through a polarizing beam splitter (PBS) set at 45° to the initial polarization. Then, the two probe beams divided by the PBS are simultaneously detected by a balanced photodetector. The pump and probe beams are perpendicular to each other in the center of the alkali metal cell. To increase the intersection area, the pump beam and the probe beam are expanded by two beam expanders, respectively.

In order to avoid the effect of optical pumping rate, the beam intensity was attenuated to approximately 32 μW, the alkali metal cell was heated from 413 K to 463 K. An approximately 2700 nT magnetic field was applied to the z direction, the sine waves with 3 nT amplitude swept in the y direction. The amplitude of output signal f(ν) and the sweep frequency can be fitted as a Lorentzian line shape,

f(ν)=a(νν0)2+(ΔυSE/2)2+c,
where both a and c are the fitting coefficients, ν is the frequency, and ν0 is the center frequency.

4. Experimental results and discussion

Figure 4 shows the experimental results of the magnetic resonance linewidth. With increasing temperature, the magnetic resonance linewidth of the all-optical atomic magnetometer broadened. Figure 4 (b) illuminates that the experimental results and the theoretical predictions followed the similar trends, however, these experimental results were less than those obtained from the theory, i.e. there were some deviations between the experiment and theory. Such deviations augmented with increasing temperature.

 figure: Fig. 4

Fig. 4 Experimental results of the magnetic resonance linewidth measurement in a 2700 nT magnetic field. (a) Fitting results of the magnetic resonance linewidth at the different temperatures. (b) Comparison between the experimental results and the theoretical predications.

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Table 1 shows the experimental results of the atomic density number. The atomic density number increased due to the increasing temperature. At 413 K, the experimental result was approximately 1.5 times less than that obtained from the saturated vapor pressure. While, at 463 K, the deviation between the experimental result and the theoretical prediction was about 2 times. There are two main reasons inducing the discrepancies in atomic density number between the experimental results and the theoretical predictions. One is the temperature differences exist in the setting and actual temperatures due to the radiative heat transfer efficiency. Such efficiency inevitably leads to a lower temperature in cell than the setting temperature. Therefore, the higher the temperature is, the larger the temperature difference exists due to the radiative heat transfer efficiency. In fact, the actual temperature inside cell cannot be measured due to the sealed cell. However, according to the theoretical analysis, the saturated vapor pressure curve is in good agreement with the magnetic resonance linewidth method. Thus, after the atomic density number is obtained by the magnetic resonance linewidth method, the actual temperature inside the cell can be derived reversely. The results show that the actual temperatures were almost 2.2% lower than the setting temperatures, as shown in Table 1. The other one is the fitting errors from test results, which are caused by the small deviation between the ideal Lorentzian line shape and the experimental results. In future, in order to improve the system performance and obtain a more accurate measurement results, the weights of these two reasons will be intensively studied.

Tables Icon

Table 1. Experimental results of the atomic density number.

Compared with the laser absorption spectroscopy reported in [11], the density measurement accuracy of the magnetic resonance linewidth method presented in this paper has been improved 5 times at 453 K.

5. Conclusion

In this paper, we present an atomic density number monitoring method on site directly utilizing the all-optical atomic magnetometer. The atomic density numbers could be estimated by measuring magnetic resonance linewidth in an applied DC magnetic field. This monitoring method coincided very well with the density saturated vapor pressure curve in theory. The experimental results showed that the maximal and minimal deviations between the experimental results and the theoretical predictions were approximately 1.5 times and 2 times, respectively. The magnetic resonance linewidth method presented in this paper can be used in all optical magnetometers to monitoring the alkali metal density number on site.

Funding

This work was supported in part by the Major Program of the National Natural Science Foundation of China (Grant No.61227902), Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant No. CXLX13_083), and the Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBJJ1573).

References and links

1. E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, “Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields,” Appl. Phys. Lett. 105(19), 192406 (2014). [CrossRef]  

2. A. Wickenbrock, S. Jurgilas, A. Dow, L. Marmugi, and F. Renzoni, “Magnetic induction tomography using an all-optical 87Rb atomic magnetometer,” Opt. Lett. 39(22), 6367–6370 (2014). [CrossRef]   [PubMed]  

3. N. Behbood, F. M. Ciurana, G. Colangelo, M. Napolitano, M. W. Mitchell, and R. J. Sewell, “Real-time vector field tracking with a cold-atom magnetometer,” Appl. Phys. Lett. 102(17), 173504 (2013). [CrossRef]  

4. J. Kitching, S. Knappe, and E. A. Donley, “Atomic sensors - a review,” IEEE Sensors J. 11(9), 1749–1758 (2011). [CrossRef]  

5. S. Zou, H. Zhang, X. Y. Chen, and W. Quan, “Magnetization produced by spin-polarized xenon-129 gas detected by using all-optical atomic magnetometer,” J. Korean Phys. Soc. 66(6), 887–893 (2015). [CrossRef]  

6. I. Savukov, T. Karaulanov, and M. G. Boshier, “Ultra-sensitive high-density Rb-87 radio-frequency magnetometer,” Appl. Phys. Lett. 104(2), 023504 (2014). [CrossRef]  

7. A. Wickenbrock, F. Tricot, and F. Renzoni, “Magnetic induction measurements using an all-optical 87Rb atomic magnetometer,” Appl. Phys. Lett. 103(24), 243503 (2013). [CrossRef]  

8. I. Savukov and T. Karaulanov, “Magnetic-resonance imaging of the human brain with an atomic magnetometer,” Appl. Phys. Lett. 103(4), 043703 (2013). [CrossRef]  

9. M. W. Millard, P. P. Yaney, B. N. Ganguly, and C. A. DeJoseph, “Diode laser absorption measurements of metastable helium in glow discharges,” Plasma Sources Sci. Technol. 7(3), 389–394 (1998). [CrossRef]  

10. E. Vliegen, S. Kadlecek, L. W. Anderson, T. G. Walker, C. J. Erickson, and W. Happer, “Faraday rotation density measurements of optically thick alkali metal vapors,” Nucl. Instrum. Methods Phys. Res. Sect. A 460(2001), 444–450 (2001). [CrossRef]  

11. Y. Ito, H. Ohnishi, K. Kamada, and T. Kobayashi, “Development of an optically pumped atomic magnetometer using a K-Rb hybrid cell and its application to magnetocardiography,” AIP Adv. 2(3), 032127 (2012). [CrossRef]  

12. W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A 16(5), 1877–1891 (1977). [CrossRef]  

13. S. Appelt, A. B. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, “Theory of spin-exchange optical pumping of 3He and 129Xe,” Phys. Rev. A 58(2), 1412–1439 (1998). [CrossRef]  

14. W. Happer, “Optical pumping,” Rev. Mod. Phys. 44(2), 169–249 (1972). [CrossRef]  

15. A. B. Baranga, S. Appelt, C. J. Erickson, A. R. Young, and W. Happer, “Alkali-metal-atom polarization imaging in high-pressure optical-pumping cells,” Phys. Rev. A 58(3), 2282–2294 (1998). [CrossRef]  

16. S. J. Seltzer, “Developments in alkali-metal atomic magnetometry,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, USA, (2008).

17. I. M. Savukov, S. J. Seltzer, M.V. Romalis, and K. L. Sauer, “Tunable atomic magnetometer for detection of radio-frequency magnetic fields,” Phys. Rev. Lett. 95(6), 063004 (2005). [CrossRef]   [PubMed]  

18. N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for 133Cs,87Rb,85Rb,39K, and 23Na,” Phys. Rev. 184(1), 102–118 (1969). [CrossRef]  

19. C. B. Alcock, V. P. Itkin, and M. K. Horrigan, “Vapour pressure equations for the metallic elements: 298–2500K,” Scan. Metall. Quart. 23(3), 309–313 (1984). [CrossRef]  

References

  • View by:

  1. E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, “Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields,” Appl. Phys. Lett. 105(19), 192406 (2014).
    [Crossref]
  2. A. Wickenbrock, S. Jurgilas, A. Dow, L. Marmugi, and F. Renzoni, “Magnetic induction tomography using an all-optical 87Rb atomic magnetometer,” Opt. Lett. 39(22), 6367–6370 (2014).
    [Crossref] [PubMed]
  3. N. Behbood, F. M. Ciurana, G. Colangelo, M. Napolitano, M. W. Mitchell, and R. J. Sewell, “Real-time vector field tracking with a cold-atom magnetometer,” Appl. Phys. Lett. 102(17), 173504 (2013).
    [Crossref]
  4. J. Kitching, S. Knappe, and E. A. Donley, “Atomic sensors - a review,” IEEE Sensors J. 11(9), 1749–1758 (2011).
    [Crossref]
  5. S. Zou, H. Zhang, X. Y. Chen, and W. Quan, “Magnetization produced by spin-polarized xenon-129 gas detected by using all-optical atomic magnetometer,” J. Korean Phys. Soc. 66(6), 887–893 (2015).
    [Crossref]
  6. I. Savukov, T. Karaulanov, and M. G. Boshier, “Ultra-sensitive high-density Rb-87 radio-frequency magnetometer,” Appl. Phys. Lett. 104(2), 023504 (2014).
    [Crossref]
  7. A. Wickenbrock, F. Tricot, and F. Renzoni, “Magnetic induction measurements using an all-optical 87Rb atomic magnetometer,” Appl. Phys. Lett. 103(24), 243503 (2013).
    [Crossref]
  8. I. Savukov and T. Karaulanov, “Magnetic-resonance imaging of the human brain with an atomic magnetometer,” Appl. Phys. Lett. 103(4), 043703 (2013).
    [Crossref]
  9. M. W. Millard, P. P. Yaney, B. N. Ganguly, and C. A. DeJoseph, “Diode laser absorption measurements of metastable helium in glow discharges,” Plasma Sources Sci. Technol. 7(3), 389–394 (1998).
    [Crossref]
  10. E. Vliegen, S. Kadlecek, L. W. Anderson, T. G. Walker, C. J. Erickson, and W. Happer, “Faraday rotation density measurements of optically thick alkali metal vapors,” Nucl. Instrum. Methods Phys. Res. Sect. A 460(2001), 444–450 (2001).
    [Crossref]
  11. Y. Ito, H. Ohnishi, K. Kamada, and T. Kobayashi, “Development of an optically pumped atomic magnetometer using a K-Rb hybrid cell and its application to magnetocardiography,” AIP Adv. 2(3), 032127 (2012).
    [Crossref]
  12. W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A 16(5), 1877–1891 (1977).
    [Crossref]
  13. S. Appelt, A. B. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, “Theory of spin-exchange optical pumping of 3He and 129Xe,” Phys. Rev. A 58(2), 1412–1439 (1998).
    [Crossref]
  14. W. Happer, “Optical pumping,” Rev. Mod. Phys. 44(2), 169–249 (1972).
    [Crossref]
  15. A. B. Baranga, S. Appelt, C. J. Erickson, A. R. Young, and W. Happer, “Alkali-metal-atom polarization imaging in high-pressure optical-pumping cells,” Phys. Rev. A 58(3), 2282–2294 (1998).
    [Crossref]
  16. S. J. Seltzer, “Developments in alkali-metal atomic magnetometry,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, USA, (2008).
  17. I. M. Savukov, S. J. Seltzer, M.V. Romalis, and K. L. Sauer, “Tunable atomic magnetometer for detection of radio-frequency magnetic fields,” Phys. Rev. Lett. 95(6), 063004 (2005).
    [Crossref] [PubMed]
  18. N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for 133Cs,87Rb,85Rb,39K, and 23Na,” Phys. Rev. 184(1), 102–118 (1969).
    [Crossref]
  19. C. B. Alcock, V. P. Itkin, and M. K. Horrigan, “Vapour pressure equations for the metallic elements: 298–2500K,” Scan. Metall. Quart. 23(3), 309–313 (1984).
    [Crossref]

2015 (1)

S. Zou, H. Zhang, X. Y. Chen, and W. Quan, “Magnetization produced by spin-polarized xenon-129 gas detected by using all-optical atomic magnetometer,” J. Korean Phys. Soc. 66(6), 887–893 (2015).
[Crossref]

2014 (3)

I. Savukov, T. Karaulanov, and M. G. Boshier, “Ultra-sensitive high-density Rb-87 radio-frequency magnetometer,” Appl. Phys. Lett. 104(2), 023504 (2014).
[Crossref]

E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, “Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields,” Appl. Phys. Lett. 105(19), 192406 (2014).
[Crossref]

A. Wickenbrock, S. Jurgilas, A. Dow, L. Marmugi, and F. Renzoni, “Magnetic induction tomography using an all-optical 87Rb atomic magnetometer,” Opt. Lett. 39(22), 6367–6370 (2014).
[Crossref] [PubMed]

2013 (3)

N. Behbood, F. M. Ciurana, G. Colangelo, M. Napolitano, M. W. Mitchell, and R. J. Sewell, “Real-time vector field tracking with a cold-atom magnetometer,” Appl. Phys. Lett. 102(17), 173504 (2013).
[Crossref]

A. Wickenbrock, F. Tricot, and F. Renzoni, “Magnetic induction measurements using an all-optical 87Rb atomic magnetometer,” Appl. Phys. Lett. 103(24), 243503 (2013).
[Crossref]

I. Savukov and T. Karaulanov, “Magnetic-resonance imaging of the human brain with an atomic magnetometer,” Appl. Phys. Lett. 103(4), 043703 (2013).
[Crossref]

2012 (1)

Y. Ito, H. Ohnishi, K. Kamada, and T. Kobayashi, “Development of an optically pumped atomic magnetometer using a K-Rb hybrid cell and its application to magnetocardiography,” AIP Adv. 2(3), 032127 (2012).
[Crossref]

2011 (1)

J. Kitching, S. Knappe, and E. A. Donley, “Atomic sensors - a review,” IEEE Sensors J. 11(9), 1749–1758 (2011).
[Crossref]

2005 (1)

I. M. Savukov, S. J. Seltzer, M.V. Romalis, and K. L. Sauer, “Tunable atomic magnetometer for detection of radio-frequency magnetic fields,” Phys. Rev. Lett. 95(6), 063004 (2005).
[Crossref] [PubMed]

2001 (1)

E. Vliegen, S. Kadlecek, L. W. Anderson, T. G. Walker, C. J. Erickson, and W. Happer, “Faraday rotation density measurements of optically thick alkali metal vapors,” Nucl. Instrum. Methods Phys. Res. Sect. A 460(2001), 444–450 (2001).
[Crossref]

1998 (3)

A. B. Baranga, S. Appelt, C. J. Erickson, A. R. Young, and W. Happer, “Alkali-metal-atom polarization imaging in high-pressure optical-pumping cells,” Phys. Rev. A 58(3), 2282–2294 (1998).
[Crossref]

S. Appelt, A. B. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, “Theory of spin-exchange optical pumping of 3He and 129Xe,” Phys. Rev. A 58(2), 1412–1439 (1998).
[Crossref]

M. W. Millard, P. P. Yaney, B. N. Ganguly, and C. A. DeJoseph, “Diode laser absorption measurements of metastable helium in glow discharges,” Plasma Sources Sci. Technol. 7(3), 389–394 (1998).
[Crossref]

1984 (1)

C. B. Alcock, V. P. Itkin, and M. K. Horrigan, “Vapour pressure equations for the metallic elements: 298–2500K,” Scan. Metall. Quart. 23(3), 309–313 (1984).
[Crossref]

1977 (1)

W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A 16(5), 1877–1891 (1977).
[Crossref]

1972 (1)

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44(2), 169–249 (1972).
[Crossref]

1969 (1)

N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for 133Cs,87Rb,85Rb,39K, and 23Na,” Phys. Rev. 184(1), 102–118 (1969).
[Crossref]

Alcock, C. B.

C. B. Alcock, V. P. Itkin, and M. K. Horrigan, “Vapour pressure equations for the metallic elements: 298–2500K,” Scan. Metall. Quart. 23(3), 309–313 (1984).
[Crossref]

Anderson, L. W.

E. Vliegen, S. Kadlecek, L. W. Anderson, T. G. Walker, C. J. Erickson, and W. Happer, “Faraday rotation density measurements of optically thick alkali metal vapors,” Nucl. Instrum. Methods Phys. Res. Sect. A 460(2001), 444–450 (2001).
[Crossref]

Appelt, S.

S. Appelt, A. B. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, “Theory of spin-exchange optical pumping of 3He and 129Xe,” Phys. Rev. A 58(2), 1412–1439 (1998).
[Crossref]

A. B. Baranga, S. Appelt, C. J. Erickson, A. R. Young, and W. Happer, “Alkali-metal-atom polarization imaging in high-pressure optical-pumping cells,” Phys. Rev. A 58(3), 2282–2294 (1998).
[Crossref]

Baranga, A. B.

A. B. Baranga, S. Appelt, C. J. Erickson, A. R. Young, and W. Happer, “Alkali-metal-atom polarization imaging in high-pressure optical-pumping cells,” Phys. Rev. A 58(3), 2282–2294 (1998).
[Crossref]

S. Appelt, A. B. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, “Theory of spin-exchange optical pumping of 3He and 129Xe,” Phys. Rev. A 58(2), 1412–1439 (1998).
[Crossref]

Behbood, N.

N. Behbood, F. M. Ciurana, G. Colangelo, M. Napolitano, M. W. Mitchell, and R. J. Sewell, “Real-time vector field tracking with a cold-atom magnetometer,” Appl. Phys. Lett. 102(17), 173504 (2013).
[Crossref]

Boshier, M. G.

I. Savukov, T. Karaulanov, and M. G. Boshier, “Ultra-sensitive high-density Rb-87 radio-frequency magnetometer,” Appl. Phys. Lett. 104(2), 023504 (2014).
[Crossref]

Budker, D.

E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, “Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields,” Appl. Phys. Lett. 105(19), 192406 (2014).
[Crossref]

Chen, X. Y.

S. Zou, H. Zhang, X. Y. Chen, and W. Quan, “Magnetization produced by spin-polarized xenon-129 gas detected by using all-optical atomic magnetometer,” J. Korean Phys. Soc. 66(6), 887–893 (2015).
[Crossref]

Ciurana, F. M.

N. Behbood, F. M. Ciurana, G. Colangelo, M. Napolitano, M. W. Mitchell, and R. J. Sewell, “Real-time vector field tracking with a cold-atom magnetometer,” Appl. Phys. Lett. 102(17), 173504 (2013).
[Crossref]

Colangelo, G.

N. Behbood, F. M. Ciurana, G. Colangelo, M. Napolitano, M. W. Mitchell, and R. J. Sewell, “Real-time vector field tracking with a cold-atom magnetometer,” Appl. Phys. Lett. 102(17), 173504 (2013).
[Crossref]

DeJoseph, C. A.

M. W. Millard, P. P. Yaney, B. N. Ganguly, and C. A. DeJoseph, “Diode laser absorption measurements of metastable helium in glow discharges,” Plasma Sources Sci. Technol. 7(3), 389–394 (1998).
[Crossref]

Donley, E. A.

J. Kitching, S. Knappe, and E. A. Donley, “Atomic sensors - a review,” IEEE Sensors J. 11(9), 1749–1758 (2011).
[Crossref]

Dow, A.

Erickson, C. J.

E. Vliegen, S. Kadlecek, L. W. Anderson, T. G. Walker, C. J. Erickson, and W. Happer, “Faraday rotation density measurements of optically thick alkali metal vapors,” Nucl. Instrum. Methods Phys. Res. Sect. A 460(2001), 444–450 (2001).
[Crossref]

S. Appelt, A. B. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, “Theory of spin-exchange optical pumping of 3He and 129Xe,” Phys. Rev. A 58(2), 1412–1439 (1998).
[Crossref]

A. B. Baranga, S. Appelt, C. J. Erickson, A. R. Young, and W. Happer, “Alkali-metal-atom polarization imaging in high-pressure optical-pumping cells,” Phys. Rev. A 58(3), 2282–2294 (1998).
[Crossref]

Ganguly, B. N.

M. W. Millard, P. P. Yaney, B. N. Ganguly, and C. A. DeJoseph, “Diode laser absorption measurements of metastable helium in glow discharges,” Plasma Sources Sci. Technol. 7(3), 389–394 (1998).
[Crossref]

Happer, W.

E. Vliegen, S. Kadlecek, L. W. Anderson, T. G. Walker, C. J. Erickson, and W. Happer, “Faraday rotation density measurements of optically thick alkali metal vapors,” Nucl. Instrum. Methods Phys. Res. Sect. A 460(2001), 444–450 (2001).
[Crossref]

S. Appelt, A. B. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, “Theory of spin-exchange optical pumping of 3He and 129Xe,” Phys. Rev. A 58(2), 1412–1439 (1998).
[Crossref]

A. B. Baranga, S. Appelt, C. J. Erickson, A. R. Young, and W. Happer, “Alkali-metal-atom polarization imaging in high-pressure optical-pumping cells,” Phys. Rev. A 58(3), 2282–2294 (1998).
[Crossref]

W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A 16(5), 1877–1891 (1977).
[Crossref]

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44(2), 169–249 (1972).
[Crossref]

Horrigan, M. K.

C. B. Alcock, V. P. Itkin, and M. K. Horrigan, “Vapour pressure equations for the metallic elements: 298–2500K,” Scan. Metall. Quart. 23(3), 309–313 (1984).
[Crossref]

Itkin, V. P.

C. B. Alcock, V. P. Itkin, and M. K. Horrigan, “Vapour pressure equations for the metallic elements: 298–2500K,” Scan. Metall. Quart. 23(3), 309–313 (1984).
[Crossref]

Ito, Y.

Y. Ito, H. Ohnishi, K. Kamada, and T. Kobayashi, “Development of an optically pumped atomic magnetometer using a K-Rb hybrid cell and its application to magnetocardiography,” AIP Adv. 2(3), 032127 (2012).
[Crossref]

Jurgilas, S.

Kadlecek, S.

E. Vliegen, S. Kadlecek, L. W. Anderson, T. G. Walker, C. J. Erickson, and W. Happer, “Faraday rotation density measurements of optically thick alkali metal vapors,” Nucl. Instrum. Methods Phys. Res. Sect. A 460(2001), 444–450 (2001).
[Crossref]

Kamada, K.

Y. Ito, H. Ohnishi, K. Kamada, and T. Kobayashi, “Development of an optically pumped atomic magnetometer using a K-Rb hybrid cell and its application to magnetocardiography,” AIP Adv. 2(3), 032127 (2012).
[Crossref]

Karaulanov, T.

I. Savukov, T. Karaulanov, and M. G. Boshier, “Ultra-sensitive high-density Rb-87 radio-frequency magnetometer,” Appl. Phys. Lett. 104(2), 023504 (2014).
[Crossref]

I. Savukov and T. Karaulanov, “Magnetic-resonance imaging of the human brain with an atomic magnetometer,” Appl. Phys. Lett. 103(4), 043703 (2013).
[Crossref]

Kitching, J.

J. Kitching, S. Knappe, and E. A. Donley, “Atomic sensors - a review,” IEEE Sensors J. 11(9), 1749–1758 (2011).
[Crossref]

Knappe, S.

J. Kitching, S. Knappe, and E. A. Donley, “Atomic sensors - a review,” IEEE Sensors J. 11(9), 1749–1758 (2011).
[Crossref]

Kobayashi, T.

Y. Ito, H. Ohnishi, K. Kamada, and T. Kobayashi, “Development of an optically pumped atomic magnetometer using a K-Rb hybrid cell and its application to magnetocardiography,” AIP Adv. 2(3), 032127 (2012).
[Crossref]

Marmugi, L.

Millard, M. W.

M. W. Millard, P. P. Yaney, B. N. Ganguly, and C. A. DeJoseph, “Diode laser absorption measurements of metastable helium in glow discharges,” Plasma Sources Sci. Technol. 7(3), 389–394 (1998).
[Crossref]

Mitchell, M. W.

N. Behbood, F. M. Ciurana, G. Colangelo, M. Napolitano, M. W. Mitchell, and R. J. Sewell, “Real-time vector field tracking with a cold-atom magnetometer,” Appl. Phys. Lett. 102(17), 173504 (2013).
[Crossref]

Napolitano, M.

N. Behbood, F. M. Ciurana, G. Colangelo, M. Napolitano, M. W. Mitchell, and R. J. Sewell, “Real-time vector field tracking with a cold-atom magnetometer,” Appl. Phys. Lett. 102(17), 173504 (2013).
[Crossref]

Ohnishi, H.

Y. Ito, H. Ohnishi, K. Kamada, and T. Kobayashi, “Development of an optically pumped atomic magnetometer using a K-Rb hybrid cell and its application to magnetocardiography,” AIP Adv. 2(3), 032127 (2012).
[Crossref]

Patton, B.

E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, “Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields,” Appl. Phys. Lett. 105(19), 192406 (2014).
[Crossref]

Quan, W.

S. Zou, H. Zhang, X. Y. Chen, and W. Quan, “Magnetization produced by spin-polarized xenon-129 gas detected by using all-optical atomic magnetometer,” J. Korean Phys. Soc. 66(6), 887–893 (2015).
[Crossref]

Renzoni, F.

A. Wickenbrock, S. Jurgilas, A. Dow, L. Marmugi, and F. Renzoni, “Magnetic induction tomography using an all-optical 87Rb atomic magnetometer,” Opt. Lett. 39(22), 6367–6370 (2014).
[Crossref] [PubMed]

A. Wickenbrock, F. Tricot, and F. Renzoni, “Magnetic induction measurements using an all-optical 87Rb atomic magnetometer,” Appl. Phys. Lett. 103(24), 243503 (2013).
[Crossref]

Ressler, N. W.

N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for 133Cs,87Rb,85Rb,39K, and 23Na,” Phys. Rev. 184(1), 102–118 (1969).
[Crossref]

Romalis, M. V.

S. Appelt, A. B. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, “Theory of spin-exchange optical pumping of 3He and 129Xe,” Phys. Rev. A 58(2), 1412–1439 (1998).
[Crossref]

Romalis, M.V.

I. M. Savukov, S. J. Seltzer, M.V. Romalis, and K. L. Sauer, “Tunable atomic magnetometer for detection of radio-frequency magnetic fields,” Phys. Rev. Lett. 95(6), 063004 (2005).
[Crossref] [PubMed]

Sands, R. H.

N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for 133Cs,87Rb,85Rb,39K, and 23Na,” Phys. Rev. 184(1), 102–118 (1969).
[Crossref]

Sauer, K. L.

I. M. Savukov, S. J. Seltzer, M.V. Romalis, and K. L. Sauer, “Tunable atomic magnetometer for detection of radio-frequency magnetic fields,” Phys. Rev. Lett. 95(6), 063004 (2005).
[Crossref] [PubMed]

Savukov, I.

I. Savukov, T. Karaulanov, and M. G. Boshier, “Ultra-sensitive high-density Rb-87 radio-frequency magnetometer,” Appl. Phys. Lett. 104(2), 023504 (2014).
[Crossref]

I. Savukov and T. Karaulanov, “Magnetic-resonance imaging of the human brain with an atomic magnetometer,” Appl. Phys. Lett. 103(4), 043703 (2013).
[Crossref]

Savukov, I. M.

I. M. Savukov, S. J. Seltzer, M.V. Romalis, and K. L. Sauer, “Tunable atomic magnetometer for detection of radio-frequency magnetic fields,” Phys. Rev. Lett. 95(6), 063004 (2005).
[Crossref] [PubMed]

Seltzer, S. J.

I. M. Savukov, S. J. Seltzer, M.V. Romalis, and K. L. Sauer, “Tunable atomic magnetometer for detection of radio-frequency magnetic fields,” Phys. Rev. Lett. 95(6), 063004 (2005).
[Crossref] [PubMed]

S. J. Seltzer, “Developments in alkali-metal atomic magnetometry,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, USA, (2008).

Sewell, R. J.

N. Behbood, F. M. Ciurana, G. Colangelo, M. Napolitano, M. W. Mitchell, and R. J. Sewell, “Real-time vector field tracking with a cold-atom magnetometer,” Appl. Phys. Lett. 102(17), 173504 (2013).
[Crossref]

Stark, T. E.

N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for 133Cs,87Rb,85Rb,39K, and 23Na,” Phys. Rev. 184(1), 102–118 (1969).
[Crossref]

Tam, A. C.

W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A 16(5), 1877–1891 (1977).
[Crossref]

Tricot, F.

A. Wickenbrock, F. Tricot, and F. Renzoni, “Magnetic induction measurements using an all-optical 87Rb atomic magnetometer,” Appl. Phys. Lett. 103(24), 243503 (2013).
[Crossref]

Vliegen, E.

E. Vliegen, S. Kadlecek, L. W. Anderson, T. G. Walker, C. J. Erickson, and W. Happer, “Faraday rotation density measurements of optically thick alkali metal vapors,” Nucl. Instrum. Methods Phys. Res. Sect. A 460(2001), 444–450 (2001).
[Crossref]

Walker, T. G.

E. Vliegen, S. Kadlecek, L. W. Anderson, T. G. Walker, C. J. Erickson, and W. Happer, “Faraday rotation density measurements of optically thick alkali metal vapors,” Nucl. Instrum. Methods Phys. Res. Sect. A 460(2001), 444–450 (2001).
[Crossref]

Wickenbrock, A.

A. Wickenbrock, S. Jurgilas, A. Dow, L. Marmugi, and F. Renzoni, “Magnetic induction tomography using an all-optical 87Rb atomic magnetometer,” Opt. Lett. 39(22), 6367–6370 (2014).
[Crossref] [PubMed]

E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, “Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields,” Appl. Phys. Lett. 105(19), 192406 (2014).
[Crossref]

A. Wickenbrock, F. Tricot, and F. Renzoni, “Magnetic induction measurements using an all-optical 87Rb atomic magnetometer,” Appl. Phys. Lett. 103(24), 243503 (2013).
[Crossref]

Yaney, P. P.

M. W. Millard, P. P. Yaney, B. N. Ganguly, and C. A. DeJoseph, “Diode laser absorption measurements of metastable helium in glow discharges,” Plasma Sources Sci. Technol. 7(3), 389–394 (1998).
[Crossref]

Young, A. R.

S. Appelt, A. B. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, “Theory of spin-exchange optical pumping of 3He and 129Xe,” Phys. Rev. A 58(2), 1412–1439 (1998).
[Crossref]

A. B. Baranga, S. Appelt, C. J. Erickson, A. R. Young, and W. Happer, “Alkali-metal-atom polarization imaging in high-pressure optical-pumping cells,” Phys. Rev. A 58(3), 2282–2294 (1998).
[Crossref]

Zhang, H.

S. Zou, H. Zhang, X. Y. Chen, and W. Quan, “Magnetization produced by spin-polarized xenon-129 gas detected by using all-optical atomic magnetometer,” J. Korean Phys. Soc. 66(6), 887–893 (2015).
[Crossref]

Zhivun, E.

E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, “Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields,” Appl. Phys. Lett. 105(19), 192406 (2014).
[Crossref]

Zou, S.

S. Zou, H. Zhang, X. Y. Chen, and W. Quan, “Magnetization produced by spin-polarized xenon-129 gas detected by using all-optical atomic magnetometer,” J. Korean Phys. Soc. 66(6), 887–893 (2015).
[Crossref]

AIP Adv. (1)

Y. Ito, H. Ohnishi, K. Kamada, and T. Kobayashi, “Development of an optically pumped atomic magnetometer using a K-Rb hybrid cell and its application to magnetocardiography,” AIP Adv. 2(3), 032127 (2012).
[Crossref]

Appl. Phys. Lett. (5)

E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, “Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields,” Appl. Phys. Lett. 105(19), 192406 (2014).
[Crossref]

N. Behbood, F. M. Ciurana, G. Colangelo, M. Napolitano, M. W. Mitchell, and R. J. Sewell, “Real-time vector field tracking with a cold-atom magnetometer,” Appl. Phys. Lett. 102(17), 173504 (2013).
[Crossref]

I. Savukov, T. Karaulanov, and M. G. Boshier, “Ultra-sensitive high-density Rb-87 radio-frequency magnetometer,” Appl. Phys. Lett. 104(2), 023504 (2014).
[Crossref]

A. Wickenbrock, F. Tricot, and F. Renzoni, “Magnetic induction measurements using an all-optical 87Rb atomic magnetometer,” Appl. Phys. Lett. 103(24), 243503 (2013).
[Crossref]

I. Savukov and T. Karaulanov, “Magnetic-resonance imaging of the human brain with an atomic magnetometer,” Appl. Phys. Lett. 103(4), 043703 (2013).
[Crossref]

IEEE Sensors J. (1)

J. Kitching, S. Knappe, and E. A. Donley, “Atomic sensors - a review,” IEEE Sensors J. 11(9), 1749–1758 (2011).
[Crossref]

J. Korean Phys. Soc. (1)

S. Zou, H. Zhang, X. Y. Chen, and W. Quan, “Magnetization produced by spin-polarized xenon-129 gas detected by using all-optical atomic magnetometer,” J. Korean Phys. Soc. 66(6), 887–893 (2015).
[Crossref]

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

E. Vliegen, S. Kadlecek, L. W. Anderson, T. G. Walker, C. J. Erickson, and W. Happer, “Faraday rotation density measurements of optically thick alkali metal vapors,” Nucl. Instrum. Methods Phys. Res. Sect. A 460(2001), 444–450 (2001).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (1)

N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for 133Cs,87Rb,85Rb,39K, and 23Na,” Phys. Rev. 184(1), 102–118 (1969).
[Crossref]

Phys. Rev. A (3)

A. B. Baranga, S. Appelt, C. J. Erickson, A. R. Young, and W. Happer, “Alkali-metal-atom polarization imaging in high-pressure optical-pumping cells,” Phys. Rev. A 58(3), 2282–2294 (1998).
[Crossref]

W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A 16(5), 1877–1891 (1977).
[Crossref]

S. Appelt, A. B. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, “Theory of spin-exchange optical pumping of 3He and 129Xe,” Phys. Rev. A 58(2), 1412–1439 (1998).
[Crossref]

Phys. Rev. Lett. (1)

I. M. Savukov, S. J. Seltzer, M.V. Romalis, and K. L. Sauer, “Tunable atomic magnetometer for detection of radio-frequency magnetic fields,” Phys. Rev. Lett. 95(6), 063004 (2005).
[Crossref] [PubMed]

Plasma Sources Sci. Technol. (1)

M. W. Millard, P. P. Yaney, B. N. Ganguly, and C. A. DeJoseph, “Diode laser absorption measurements of metastable helium in glow discharges,” Plasma Sources Sci. Technol. 7(3), 389–394 (1998).
[Crossref]

Rev. Mod. Phys. (1)

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44(2), 169–249 (1972).
[Crossref]

Scan. Metall. Quart. (1)

C. B. Alcock, V. P. Itkin, and M. K. Horrigan, “Vapour pressure equations for the metallic elements: 298–2500K,” Scan. Metall. Quart. 23(3), 309–313 (1984).
[Crossref]

Other (1)

S. J. Seltzer, “Developments in alkali-metal atomic magnetometry,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, USA, (2008).

Cited By

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Figures (4)

Fig. 1
Fig. 1 Magnetic resonance linewidth Δυ estimated from Eq. (2) for the all-optical potassium atomic magnetometer at different temperatures.
Fig. 2
Fig. 2 Theoretical predictions of the atomic density numbers determining by the saturated vapor pressure and the magnetic resonance linewidth method presented in this paper, respectively. Solid blue line is the atomic density number obtained by saturated vapor pressure and red crosses are that obtained by the method presented in this paper.
Fig. 3
Fig. 3 Schematic of the experimental device. The pump and probe beams propagate along the z and x directions, respectively.
Fig. 4
Fig. 4 Experimental results of the magnetic resonance linewidth measurement in a 2700 nT magnetic field. (a) Fitting results of the magnetic resonance linewidth at the different temperatures. (b) Comparison between the experimental results and the theoretical predications.

Tables (1)

Tables Icon

Table 1 Experimental results of the atomic density number.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

d ρ d t = [ H , ρ ] i h ¯ + R SE [ φ ( 1 + 4 S S ) ρ ] + R rel [ φ ρ ] + R OP [ φ ( 1 + 2 s S ) ρ ] + D 2 ρ ,
Δ υ B = 1 π { [ ( 2 I + 1 ) 2 + 2 ] R SE 3 ( 2 I + 1 ) 2 g S 2 μ B 2 B 2 h ¯ 2 ( 2 I + 1 ) 2 2 g S μ B B R SE i h ¯ ( 2 I + 1 ) 2 + [ ( 2 I + 1 ) 2 + 2 ] 2 R SE 2 9 ( 2 I + 1 ) 4 i γ e q | B | } ,
Δ υ SE = 1 π T 2 = R OP 4 π + R SE R SD R OP π G ( ω 0 , R SE ) ,
G ( ω 0 , R SE ) = Re [ R SE + 4 i ω 0 2 / π υ HF 5 R SE + 8 i ω 0 2 / π υ HF ] ,
Δ υ SE = R SE 8 π ,
R SE = n σ SE 8 K B T π M ,
n = 8 π Δ υ SE σ SE 8 π M K B T ,
n = 1 T 10 21.866 + A B / T ,
f ( ν ) = a ( ν ν 0 ) 2 + ( Δ υ SE / 2 ) 2 + c ,

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