REVIEW OF SCIENTIFIC INSTRUMENTS 86, 043104 (2015)
A fast determination method for transverse relaxation of spin-exchange-relaxation-free magnetometer Jixi Lu,a) Zheng Qian, and Jiancheng Fang School of Instrument Science and Opto-Electronics Engineering, Beihang University, Beijing 100191, China
(Received 6 January 2015; accepted 21 March 2015; published online 6 April 2015) We propose a fast and accurate determination method for transverse relaxation of the spin-exchangerelaxation-free (SERF) magnetometer. This method is based on the measurement of magnetic resonance linewidth via a chirped magnetic field excitation and the amplitude spectrum analysis. Compared with the frequency sweeping via separate sinusoidal excitation, our method can realize linewidth determination within only few seconds and meanwhile obtain good frequency resolution. Therefore, it can avoid the drift error in long term measurement and improve the accuracy of the determination. As the magnetic resonance frequency of the SERF magnetometer is very low, we include the effect of the negative resonance frequency caused by the chirp and achieve the coefficient of determination of the fitting results better than 0.998 with 95% confidence bounds to the theoretical equation. The experimental results are in good agreement with our theoretical analysis. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4916681]
I. INTRODUCTION
High sensitivity optical atomic magnetometers have been widely used in many fields.1 The fundamental sensitivity limit of atomic magnetometers is due to their shot-noise,2 1 δB = √ , (1) γ nT2Vt where n is the density of atoms, γ is the gyromagnetic ratio, T2 is the transverse relaxation time, V is the cell volume, and t is the measurement time. The most sensitive atomic magnetometer at present is the spin-exchange-relaxation-free (SERF) magnetometer,3 in which spin-exchange relaxation is eliminated by operating in a low magnetic field and at a high alkali-metal density.4 In the SERF regime, T2 lengthens substantially and the intrinsic sensitivity improves greatly. Precise determination of the transverse relaxation time (or the transverse relaxation rate) is important for studying various relaxations and parameters of the SERF magnetometer.5,6 Magnetic resonance linewidth is often used to get transverse relaxation time. By using the synchronous pumping technique4 or the RF magnetic field excitation technique,6 the resonance curve can be measured to obtain the linewidth. The synchronous pumping technique modulates the pump rate, whereas the RF magnetic field excitation technique applies the rotating magnetic field to get the resonance curve. For both techniques, we must apply excitations at different frequencies and then measure the response at the corresponding frequencies. Therefore, the determination process will take a lot of time to get good frequency resolution, especially in the low frequency range such as for the SERF magnetometer. In long term measurement process, the low frequency a)[email protected]
0034-6748/2015/86(4)/043104/5/$30.00
drift (caused by laser intensity and polarization drift, cell, and environmental temperature fluctuation) will generate determination error. To eliminate the drift error, high performance devices or closed-loop control can be applied to the experiment system. Shortening the duration of determination process can avoid the drift error or reduce the demands on the experimental setup. Chirp is a signal in which the instantaneous frequency varies linearly with time. It is commonly used in sonar and radar applications,7,8 as well as stimulus response analysis, such as calibrating frequency response of magnetometers.9,10 In this paper, we propose a magnetic resonance linewidth determination method based on a chirped magnetic field excitation and the amplitude spectrum analysis. In the experiment, a chirped magnetic field whose bandwidth covered the range around Larmor frequency was applied to the SERF magnetometer. We took the chirp signal and the response signal of the magnetometer and then got the amplitude frequency response function through amplitude spectrum analysis. The frequency response function was then fitted to Lorentzian to obtain the linewidth. The whole process took only few seconds which can be shortened further under the condition of sufficient timebandwidth product of the chirp. In scalar magnetometers, the resonance frequency is often much higher than its linewidth, and thus, a RF oscillating cosine magnetic field can substitute the rotating magnetic field. However, for the SERF magnetometer whose resonance frequency is very low, the effect of the negative resonance frequency should not be ignored.11 Therefore, in data analysis, we include the effect of the resonance at the negative resonance frequency caused by the chirped magnetic field signal in Lorentzian curve fitting. Consequently, we achieve the coefficient of determination of the fits better than 0.998 with 95% confidence bounds. Finally, we demonstrate that our determination results of transverse relaxation are in good agreement with the theoretical analysis.
86, 043104-1
© 2015 AIP Publishing LLC
043104-2
Lu, Qian, and Fang
II. PRINCIPLE A. Chirp
A chirp signal is characterized by a linear increase or decrease of the instantaneous frequency with time from a starting frequency to a final frequency. Its time domain function can be defined as ( ) ( f 1 − f 0) y(t) = A sin 2π f 0 + t t , (2) 2τ where A is the amplitude, f 0 is the starting frequency, f 1 is the final frequency, t is the instantaneous time, and τ is the duration time. The bandwidth of the chirp signal is given by BW = f 1 − f 0 and the time-bandwidth product is τ × BW . When τ × BW ≥ 1, the amplitude spectrum can be roughly simplified as Eq. (3), which is approximately flat over its bandwidth. When the time-bandwidth product increases, the spectrum shape will become smoother, ( ) f − ( f 1 + f 0) /2 2τ rect . (3) Schirp( f ) = A BW BW Through the FFT analysis, the magnitude spectrum of linear chirp can be seen explicitly. In Fig. 1, we plot the magnitude spectrum of a simulated chirp signal, where A = 1, f 0 = 5 Hz, f 1 = 45 Hz, and τ = 10 Hz. The chirp combines a fast frequency sweeping with good frequency resolution. In the experiment described in Sec. III, we apply a chirped magnetic field to stimulate the SERF magnetometer. The chirp bandwidth is chosen to cover the Larmor frequency. B. Bloch equations
The behavior of spins can be fully described by a density matrix equation.12 However, the description can be simplified when the spin-exchange rate is much faster than the rate of Larmor precession. In this case, the density matrix assumes a spin-temperature distribution and the electron spin S can be described by a Bloch equation as follows:6,13 1 1 S d e S= γ B × S + Rs − , (4) dt q(P) 2 T2 where γ e is electron gyromagnetic ratio, B is the applied magnetic field, R is the pumping rate, s is the optical pumping
FIG. 1. Magnitude spectrum of simulated chirp signal.
Rev. Sci. Instrum. 86, 043104 (2015)
vector whose magnitude is the degree of circular polarization, T2 is the transverse relaxation time, and q(P) is slowing-down factor, which for nuclear spin I = 3/2 is12 q(P) =
6 + 2P2 . 1 + P2
(5)
In the low polarization limit, q(0) = 6. The transverse relaxation time is 1 1 1 = RSD + + + R + Rpr, T2 TD T2SE
(6)
where RSD is the spin-destruction relaxation rate, 1/TD is the relaxation rate due to diffusion, 1/T2SE is the spin-exchange relaxation rate, R is optical pump rate, and Rpr is the rate of depolarization due to the probe beam. For low polarizations and small magnetic fields, spin-exchange relaxation is quadratic in the magnetic field,14 1 (2I + 1)2 1 2 = ω T − q(0)2, (7) SE 0 2 2q(0)2 T2SE where ω0 is the Larmor frequency and TSE is the spin-exchange time. In the present experimental setup (see Fig. 3), optical pumping is along the z axis and the probe beam is along the x axis. The direction vertical to the x − z plane is the y axis. If an oscillating magnetic field B1 cos (ωt) yˆ is applied in the y direction, which can be decomposed into two counterpropagating rotating fields as iB ′ iωt Bˆ1 = e + e−i wt , 2 the response of the magnetometer is15 S0γ e B ′ ∆ω cos(ωt) + (ω − ω0) sin(ωt) Sx = 2q(P) (∆ω)2 + (ω − ω0)2 ∆ω cos(−ωt) + (ω + ω0) sin(−ωt) , + (∆ω)2 + (ω + ω0)2
(8)
(9)
where S0 = sRT2, ω0 = γ e Bz /q(P), and ∆ω = (q (P) T2)−1 and ∆ω is the magnetic resonance linewidth, which is the half width at half maximum (HWHM) of the magnetic resonance curve. The magnetic field response Sx is the sum of two Lorentzian curves centered at ±ω0 and both of the Lorentzian curves have in-phase and out-phase components. We apply a chirped magnetic field whose bandwidth covers ω0 and then takes the response Sx of the magnetometer. The amplitude frequency response function is Smag(ω) , A(ω) = (10) Schirp(ω) where Schirp(ω) and Smag(ω) are the amplitude spectra of the chirp signal and the magnetometer response signal, respectively. We use the amplitude frequency response function to fit the quadrature sum of the in- and out-of-phase optical rotation signals (without considering the scale factor between them),
043104-3
Lu, Qian, and Fang
Rev. Sci. Instrum. 86, 043104 (2015)
which is Sx(R) =
)2 ( S0γ e B ′ ∆ω ∆ω + 2q(P) ∆ω2 + (ω − ω0)2 ∆ω2 + (ω + ω0)2 ) 2 1/2 ( ω + ω0 ω − ω0 + . (11) + ∆ω2 + (ω − ω0)2 ∆ω2 + (ω + ω0)2
If the resonance frequency ω0 is much larger than the linewidth ∆ω, the counter-propagating response is not obvious, and a cosine field can substitute the rotation field with no obvious effect; Eq. (11) can be simplified as Sx(R) =
S0γ e B ′ 1 . 2q(P) ∆ω2 + (ω − ω0)2
(12)
However, for the SERF magnetometer, whose magnetic resonance frequency is always very low, the −ω0 effect must be considered. In Fig. 2, we plot the quadrature sum of the in and out-of-phase optical rotation signals at several different magnetic resonance frequencies using Eqs. (11) and (12). It shows that the −ω0 effect is significant in the low frequency range and less obvious as the magnetic resonance frequency increases.
III. EXPERIMENTAL SETUP AND PROCEDURE
The experimental setup is shown in Fig. 3. A inner diameter 9 mm spherical glass cell, containing a droplet of K metal, is filled with about 2000 Torr 4He as buffer gas and 55 Torr N2 as quenching gas. It was placed in an oven and heated to 170 oC by an electronic heater using ac currents at 100 kHz. The density of K vapor is about 3.7 × 1013 cm−3 from saturated vapor density equation.16 We used three nested cylindrical magnetic shields to attenuate outside magnetic field to below 1 nT. The diameter of inner magnetic shield is 40 cm. A triaxial coil, driven by NI PXIe-6368, controlled the three components of the magnetic field. Non-magnetic and weakly magnetic material components were selected carefully to be installed in the magnetic shield, which ensured the magnetic field around the glass cell
FIG. 3. Experimental setup of the K SERF magnetometer (not to scale).
to be about 2 nT. The remaining magnetic field was mainly due to the magnetism of photoelectric detector (Newfocus Model 2307). All of the components in the shield were installed on an epoxy board. K atoms were pumped by circularly polarized laser propagating in the z direction tuned to the center of K D1 line with diameter of 4 mm. The x component of K spin polarization Sx was measured using linearly polarized laser with diameter of 2 mm tuned about 120 GHz from the center of K D2 line. The optical rotation angle which is proportional to Sx was analyzed after the cell by a balanced polarimeter. Both light sources were Newfocus ECDL (external cavity diode laser) laser systems with optical fiber-coupled output. The fiber polarization changed at low frequency due to environmental temperature fluctuation resulting in slow intensity fluctuation of the polarization laser about 5%. The three components of the magnetic field were zeroed before the experiment to exclude the effect of residual magnetic field in the shield. Next, in different bias fields Bz , a small transverse chirped magnetic field was applied in the y direction. The response of the magnetometer was acquired using Model 2307 and a NI PXI-4461 DAQ card. After the data acquisition and amplitude spectrum analysis, we obtained the amplitude frequency response function using Eq. (10). Finally, the amplitude frequency response function was fitted to Eq. (11) by the least square method to get the linewidth ∆ω and the resonance frequency ω0. IV. EXPERIMENTAL RESULTS AND DISCUSSION
FIG. 2. Comparison of the quadrature sum of the in- and out-of-phase optical rotation signals obtained from Eqs. (11) and (12). The resonance curves have the same linewidth ∆ω = 2π × 3 Hz and different resonance frequencies ω 0 (2π × 5 Hz, 2π × 15 Hz, 2π × 25 Hz, and 2π × 35 Hz).
In the experiment, in order to eliminate the broadening due to the pump and probe beams as much as possible, the pump and probe powers were set to be low (Ipump ≈ 0.15 mW/cm2 and Ipr ≈ 0.5 mW/cm2). The magnitude of the chirped magnetic field was set to be 0.4 nT to ensure that the spin-exchange relaxation can be ignored, and meanwhile to get enough signal strength. In Fig. 4, we show the frequency response function (Eq. (10)) for several different values of the bias magnetic field Bz , in which the data were fitted to Eq. (11). The fitting results are in good agreement with the theoretical equation.
043104-4
Lu, Qian, and Fang
Rev. Sci. Instrum. 86, 043104 (2015)
FIG. 4. The frequency response function of the chirp in the y direction for several different values of the bias magnetic field B z (2 nT, 6 nT, 10 nT, and 14 nT). The measured data (black dots) were fitted to Eq. (11) (red line). The pump and probe intensities: Ipump ≈ 0.15 mW/cm2 and Ipr ≈ 0.5 mW/cm2. The magnitude of the chirped magnetic field is 0.4 nT.
FIG. 5. Linewidth as a function of bias magnetic field B z . The fit based on Eq. (7) gives a limiting low-frequency linewidth of 2.9 Hz and a spin exchange rate R SE ≈ 4.8 × 104 s−1.
and low pump and probe intensity, and The bandwidth of the chirp was set to be about ten times larger than the respective linewidth. The duration time was 10 s for each measurement, which can also be shortened under the condition of sufficient time-bandwidth product. The frequency resolution is 0.1 Hz, which is determined by the measurement time and sample rate according to FFT. In Table I, we present the fitting results in different Bz . The coefficient of determination of the fits for all measurements is better than 0.998 with 95% confidence bounds. In Fig. 5, we show the linewidth as a function of the bias field Bz . Overlaying the data is a fit based on Eq. (7), which gives a limiting low-frequency linewidth of 2.9 Hz and a spin-exchange rate RSE ≈ 4.8 × 104 s−1. From limiting lowfrequency linewidth, we can obtain the transverse relaxation rate due to spin-destruction and diffusion as 109 s−1. For n = 3.7 × 1013 cm−3, the expected spin exchange rate is 4.6 × 104 s−1,17 which is close to the fitting result. To compare the experimental results with the expected values, we calculated the relaxation values using Eq. (6), where the spin-exchange relaxation, pump rate, and relaxation due to the probe beam can be ignored under the condition of zero field TABLE I. Fitting results of resonance frequency and linewidth for different values of B z . B z (nT)
f 0 (Hz)
△υ (Hz)
R-square
2 6 10 14 18 22 26 30 34 38 42 46
9.423 27.95 46.79 65.42 84.16 103.0 121.7 140.4 159.3 178.2 197.1 216.1
3.003 3.126 3.403 3.826 4.423 5.165 6.081 7.017 8.402 9.616 11.21 13.05
0.9982 0.9981 0.9984 0.9989 0.9990 0.9992 0.9988 0.9991 0.9992 0.9985 0.9984 0.9985
SD RSD = nσKSD ν¯ + nσHe ν¯ + nσNSD2 ν, ¯ ( ) π 2 1 = q(P)D , TD a
(13) (14)
where σ SD are the spin-destruction cross sections, ν¯ is the thermal velocity, D is the diffusion constant of alkali atom within the buffer gas, and a is the radius of the spherical cell. SD Here, σKSD = 1 × 10−18 cm2, σHe = 5 × 10−25 cm2, σNSD2 = 7.9 −23 2 17 2 × 10 cm , and D = 0.24 cm /s (for the temperature and the buffer gas content of this cell).18 Consequently, we can get that RSD = 22 s−1 and 1/TD = 69 s−1, yielding the total relaxation due to spin-destruction and diffusion is 91 s−1, which is in reasonable agreement with the experimentally determined value (109 s−1). V. CONCLUSION
We demonstrated a fast and accurate determination method for transverse relaxation of the SERF magnetometer. This method realizes magnetic resonance linewidth determination by a chirped magnetic field excitation and amplitude spectrum analysis method. The advantage of this method is that the measurement time can be shortened to several seconds, and meanwhile, good frequency resolution can be achieved. Therefore, it can avoid the drift error and improve the accuracy, and meanwhile not increase the demands on experimental setup. In data analysis, we considered the effect of the resonance at −ω0 caused by the calibration chirp signal which should not be ignored for the SERF magnetometer in the low frequency range. Finally, we calculated the relaxation values and compared them with experimentally determined result, which was found to be in good agreement with our theoretical analysis. ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of China (No. 61227902).
043104-5 1D.
Lu, Qian, and Fang
Budker and M. V. Romalis, Nat. Phys. 3, 227 (2007).
2D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and
A. Weis, Rev. Mod. Phys. 74, 1153 (2002). 3H. B. Dang, A. C. Maloof, and M. V. Romalis, Appl. Phys. Lett. 97, 151110-
1 (2010). C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, Phys. Rev. Lett. 89, 130801 (2002). 5I. M. Savukov and M. V. Romalis, Phys. Rev. A 71, 023405 (2005). 6M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V. Romalis, Phys. Rev. A 77, 033408 (2008). 7R. K. Raney, H. Runge, R. Bamler, I. G. Cumming, and F. H. Wong, IEEE Trans. Geosci. Remote Sens. 32, 786 (1994). 8L. R. LeBlanc, L. Mayer, M. Rufino, S. G. Schock, and J. King, J. Acoust. Soc. Am. 91, 107 (1992). 4J.
Rev. Sci. Instrum. 86, 043104 (2015) 9P.
D. D. Schwindt, B. Lindseth, S. Knappe, V. Shah, J. Kitching, and L.-A. Liew, Appl. Phys. Lett. 90, 081102 (2007). 10R. Wyllie, Ph.D. thesis, University of Wisconsin-Madison, Wisconsin, 2012. 11D. Budker and D. F. J. Kimball, Optical Magnetometry (Cambridge University Press, Cambridge, 2013), p. 95. 12S. Appelt, A. B.-A. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, Phys. Rev. A 58, 1412 (1998). 13S. J. Seltzer and M. V. Romalis, Appl. Phys. Lett. 85, 4804 (2004). 14W. Happer and A. C. Tam, Phys. Rev. A 16, 1877 (1977). 15S. J. Seltzer, Ph.D. thesis, Princeton University, Princeton, 2008. 16C. B. Alcock, V. P. Itkin, and M. K. Horrigan, Can. Metall. Q. 23, 309 (1984). 17S. Kadlecek, L. W. Anderson, and T. Walker, Nucl. Instrum. Methods Phys. Res., Sect. A 402, 208 (1998). 18F. A. Franz and C. Volk, Phys. Rev. A 26, 85 (1982).
Review of Scientific Instruments is copyrighted by AIP Publishing LLC (AIP). Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. For more information, see http://publishing.aip.org/authors/rights-and-permissions.
A fast determination method for transverse relaxation of spin-exchange-relaxation-free magnetometer Jixi Lu,a) Zheng Qian, and Jiancheng Fang School of Instrument Science and Opto-Electronics Engineering, Beihang University, Beijing 100191, China
(Received 6 January 2015; accepted 21 March 2015; published online 6 April 2015) We propose a fast and accurate determination method for transverse relaxation of the spin-exchangerelaxation-free (SERF) magnetometer. This method is based on the measurement of magnetic resonance linewidth via a chirped magnetic field excitation and the amplitude spectrum analysis. Compared with the frequency sweeping via separate sinusoidal excitation, our method can realize linewidth determination within only few seconds and meanwhile obtain good frequency resolution. Therefore, it can avoid the drift error in long term measurement and improve the accuracy of the determination. As the magnetic resonance frequency of the SERF magnetometer is very low, we include the effect of the negative resonance frequency caused by the chirp and achieve the coefficient of determination of the fitting results better than 0.998 with 95% confidence bounds to the theoretical equation. The experimental results are in good agreement with our theoretical analysis. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4916681]
I. INTRODUCTION
High sensitivity optical atomic magnetometers have been widely used in many fields.1 The fundamental sensitivity limit of atomic magnetometers is due to their shot-noise,2 1 δB = √ , (1) γ nT2Vt where n is the density of atoms, γ is the gyromagnetic ratio, T2 is the transverse relaxation time, V is the cell volume, and t is the measurement time. The most sensitive atomic magnetometer at present is the spin-exchange-relaxation-free (SERF) magnetometer,3 in which spin-exchange relaxation is eliminated by operating in a low magnetic field and at a high alkali-metal density.4 In the SERF regime, T2 lengthens substantially and the intrinsic sensitivity improves greatly. Precise determination of the transverse relaxation time (or the transverse relaxation rate) is important for studying various relaxations and parameters of the SERF magnetometer.5,6 Magnetic resonance linewidth is often used to get transverse relaxation time. By using the synchronous pumping technique4 or the RF magnetic field excitation technique,6 the resonance curve can be measured to obtain the linewidth. The synchronous pumping technique modulates the pump rate, whereas the RF magnetic field excitation technique applies the rotating magnetic field to get the resonance curve. For both techniques, we must apply excitations at different frequencies and then measure the response at the corresponding frequencies. Therefore, the determination process will take a lot of time to get good frequency resolution, especially in the low frequency range such as for the SERF magnetometer. In long term measurement process, the low frequency a)[email protected]
0034-6748/2015/86(4)/043104/5/$30.00
drift (caused by laser intensity and polarization drift, cell, and environmental temperature fluctuation) will generate determination error. To eliminate the drift error, high performance devices or closed-loop control can be applied to the experiment system. Shortening the duration of determination process can avoid the drift error or reduce the demands on the experimental setup. Chirp is a signal in which the instantaneous frequency varies linearly with time. It is commonly used in sonar and radar applications,7,8 as well as stimulus response analysis, such as calibrating frequency response of magnetometers.9,10 In this paper, we propose a magnetic resonance linewidth determination method based on a chirped magnetic field excitation and the amplitude spectrum analysis. In the experiment, a chirped magnetic field whose bandwidth covered the range around Larmor frequency was applied to the SERF magnetometer. We took the chirp signal and the response signal of the magnetometer and then got the amplitude frequency response function through amplitude spectrum analysis. The frequency response function was then fitted to Lorentzian to obtain the linewidth. The whole process took only few seconds which can be shortened further under the condition of sufficient timebandwidth product of the chirp. In scalar magnetometers, the resonance frequency is often much higher than its linewidth, and thus, a RF oscillating cosine magnetic field can substitute the rotating magnetic field. However, for the SERF magnetometer whose resonance frequency is very low, the effect of the negative resonance frequency should not be ignored.11 Therefore, in data analysis, we include the effect of the resonance at the negative resonance frequency caused by the chirped magnetic field signal in Lorentzian curve fitting. Consequently, we achieve the coefficient of determination of the fits better than 0.998 with 95% confidence bounds. Finally, we demonstrate that our determination results of transverse relaxation are in good agreement with the theoretical analysis.
86, 043104-1
© 2015 AIP Publishing LLC
043104-2
Lu, Qian, and Fang
II. PRINCIPLE A. Chirp
A chirp signal is characterized by a linear increase or decrease of the instantaneous frequency with time from a starting frequency to a final frequency. Its time domain function can be defined as ( ) ( f 1 − f 0) y(t) = A sin 2π f 0 + t t , (2) 2τ where A is the amplitude, f 0 is the starting frequency, f 1 is the final frequency, t is the instantaneous time, and τ is the duration time. The bandwidth of the chirp signal is given by BW = f 1 − f 0 and the time-bandwidth product is τ × BW . When τ × BW ≥ 1, the amplitude spectrum can be roughly simplified as Eq. (3), which is approximately flat over its bandwidth. When the time-bandwidth product increases, the spectrum shape will become smoother, ( ) f − ( f 1 + f 0) /2 2τ rect . (3) Schirp( f ) = A BW BW Through the FFT analysis, the magnitude spectrum of linear chirp can be seen explicitly. In Fig. 1, we plot the magnitude spectrum of a simulated chirp signal, where A = 1, f 0 = 5 Hz, f 1 = 45 Hz, and τ = 10 Hz. The chirp combines a fast frequency sweeping with good frequency resolution. In the experiment described in Sec. III, we apply a chirped magnetic field to stimulate the SERF magnetometer. The chirp bandwidth is chosen to cover the Larmor frequency. B. Bloch equations
The behavior of spins can be fully described by a density matrix equation.12 However, the description can be simplified when the spin-exchange rate is much faster than the rate of Larmor precession. In this case, the density matrix assumes a spin-temperature distribution and the electron spin S can be described by a Bloch equation as follows:6,13 1 1 S d e S= γ B × S + Rs − , (4) dt q(P) 2 T2 where γ e is electron gyromagnetic ratio, B is the applied magnetic field, R is the pumping rate, s is the optical pumping
FIG. 1. Magnitude spectrum of simulated chirp signal.
Rev. Sci. Instrum. 86, 043104 (2015)
vector whose magnitude is the degree of circular polarization, T2 is the transverse relaxation time, and q(P) is slowing-down factor, which for nuclear spin I = 3/2 is12 q(P) =
6 + 2P2 . 1 + P2
(5)
In the low polarization limit, q(0) = 6. The transverse relaxation time is 1 1 1 = RSD + + + R + Rpr, T2 TD T2SE
(6)
where RSD is the spin-destruction relaxation rate, 1/TD is the relaxation rate due to diffusion, 1/T2SE is the spin-exchange relaxation rate, R is optical pump rate, and Rpr is the rate of depolarization due to the probe beam. For low polarizations and small magnetic fields, spin-exchange relaxation is quadratic in the magnetic field,14 1 (2I + 1)2 1 2 = ω T − q(0)2, (7) SE 0 2 2q(0)2 T2SE where ω0 is the Larmor frequency and TSE is the spin-exchange time. In the present experimental setup (see Fig. 3), optical pumping is along the z axis and the probe beam is along the x axis. The direction vertical to the x − z plane is the y axis. If an oscillating magnetic field B1 cos (ωt) yˆ is applied in the y direction, which can be decomposed into two counterpropagating rotating fields as iB ′ iωt Bˆ1 = e + e−i wt , 2 the response of the magnetometer is15 S0γ e B ′ ∆ω cos(ωt) + (ω − ω0) sin(ωt) Sx = 2q(P) (∆ω)2 + (ω − ω0)2 ∆ω cos(−ωt) + (ω + ω0) sin(−ωt) , + (∆ω)2 + (ω + ω0)2
(8)
(9)
where S0 = sRT2, ω0 = γ e Bz /q(P), and ∆ω = (q (P) T2)−1 and ∆ω is the magnetic resonance linewidth, which is the half width at half maximum (HWHM) of the magnetic resonance curve. The magnetic field response Sx is the sum of two Lorentzian curves centered at ±ω0 and both of the Lorentzian curves have in-phase and out-phase components. We apply a chirped magnetic field whose bandwidth covers ω0 and then takes the response Sx of the magnetometer. The amplitude frequency response function is Smag(ω) , A(ω) = (10) Schirp(ω) where Schirp(ω) and Smag(ω) are the amplitude spectra of the chirp signal and the magnetometer response signal, respectively. We use the amplitude frequency response function to fit the quadrature sum of the in- and out-of-phase optical rotation signals (without considering the scale factor between them),
043104-3
Lu, Qian, and Fang
Rev. Sci. Instrum. 86, 043104 (2015)
which is Sx(R) =
)2 ( S0γ e B ′ ∆ω ∆ω + 2q(P) ∆ω2 + (ω − ω0)2 ∆ω2 + (ω + ω0)2 ) 2 1/2 ( ω + ω0 ω − ω0 + . (11) + ∆ω2 + (ω − ω0)2 ∆ω2 + (ω + ω0)2
If the resonance frequency ω0 is much larger than the linewidth ∆ω, the counter-propagating response is not obvious, and a cosine field can substitute the rotation field with no obvious effect; Eq. (11) can be simplified as Sx(R) =
S0γ e B ′ 1 . 2q(P) ∆ω2 + (ω − ω0)2
(12)
However, for the SERF magnetometer, whose magnetic resonance frequency is always very low, the −ω0 effect must be considered. In Fig. 2, we plot the quadrature sum of the in and out-of-phase optical rotation signals at several different magnetic resonance frequencies using Eqs. (11) and (12). It shows that the −ω0 effect is significant in the low frequency range and less obvious as the magnetic resonance frequency increases.
III. EXPERIMENTAL SETUP AND PROCEDURE
The experimental setup is shown in Fig. 3. A inner diameter 9 mm spherical glass cell, containing a droplet of K metal, is filled with about 2000 Torr 4He as buffer gas and 55 Torr N2 as quenching gas. It was placed in an oven and heated to 170 oC by an electronic heater using ac currents at 100 kHz. The density of K vapor is about 3.7 × 1013 cm−3 from saturated vapor density equation.16 We used three nested cylindrical magnetic shields to attenuate outside magnetic field to below 1 nT. The diameter of inner magnetic shield is 40 cm. A triaxial coil, driven by NI PXIe-6368, controlled the three components of the magnetic field. Non-magnetic and weakly magnetic material components were selected carefully to be installed in the magnetic shield, which ensured the magnetic field around the glass cell
FIG. 3. Experimental setup of the K SERF magnetometer (not to scale).
to be about 2 nT. The remaining magnetic field was mainly due to the magnetism of photoelectric detector (Newfocus Model 2307). All of the components in the shield were installed on an epoxy board. K atoms were pumped by circularly polarized laser propagating in the z direction tuned to the center of K D1 line with diameter of 4 mm. The x component of K spin polarization Sx was measured using linearly polarized laser with diameter of 2 mm tuned about 120 GHz from the center of K D2 line. The optical rotation angle which is proportional to Sx was analyzed after the cell by a balanced polarimeter. Both light sources were Newfocus ECDL (external cavity diode laser) laser systems with optical fiber-coupled output. The fiber polarization changed at low frequency due to environmental temperature fluctuation resulting in slow intensity fluctuation of the polarization laser about 5%. The three components of the magnetic field were zeroed before the experiment to exclude the effect of residual magnetic field in the shield. Next, in different bias fields Bz , a small transverse chirped magnetic field was applied in the y direction. The response of the magnetometer was acquired using Model 2307 and a NI PXI-4461 DAQ card. After the data acquisition and amplitude spectrum analysis, we obtained the amplitude frequency response function using Eq. (10). Finally, the amplitude frequency response function was fitted to Eq. (11) by the least square method to get the linewidth ∆ω and the resonance frequency ω0. IV. EXPERIMENTAL RESULTS AND DISCUSSION
FIG. 2. Comparison of the quadrature sum of the in- and out-of-phase optical rotation signals obtained from Eqs. (11) and (12). The resonance curves have the same linewidth ∆ω = 2π × 3 Hz and different resonance frequencies ω 0 (2π × 5 Hz, 2π × 15 Hz, 2π × 25 Hz, and 2π × 35 Hz).
In the experiment, in order to eliminate the broadening due to the pump and probe beams as much as possible, the pump and probe powers were set to be low (Ipump ≈ 0.15 mW/cm2 and Ipr ≈ 0.5 mW/cm2). The magnitude of the chirped magnetic field was set to be 0.4 nT to ensure that the spin-exchange relaxation can be ignored, and meanwhile to get enough signal strength. In Fig. 4, we show the frequency response function (Eq. (10)) for several different values of the bias magnetic field Bz , in which the data were fitted to Eq. (11). The fitting results are in good agreement with the theoretical equation.
043104-4
Lu, Qian, and Fang
Rev. Sci. Instrum. 86, 043104 (2015)
FIG. 4. The frequency response function of the chirp in the y direction for several different values of the bias magnetic field B z (2 nT, 6 nT, 10 nT, and 14 nT). The measured data (black dots) were fitted to Eq. (11) (red line). The pump and probe intensities: Ipump ≈ 0.15 mW/cm2 and Ipr ≈ 0.5 mW/cm2. The magnitude of the chirped magnetic field is 0.4 nT.
FIG. 5. Linewidth as a function of bias magnetic field B z . The fit based on Eq. (7) gives a limiting low-frequency linewidth of 2.9 Hz and a spin exchange rate R SE ≈ 4.8 × 104 s−1.
and low pump and probe intensity, and The bandwidth of the chirp was set to be about ten times larger than the respective linewidth. The duration time was 10 s for each measurement, which can also be shortened under the condition of sufficient time-bandwidth product. The frequency resolution is 0.1 Hz, which is determined by the measurement time and sample rate according to FFT. In Table I, we present the fitting results in different Bz . The coefficient of determination of the fits for all measurements is better than 0.998 with 95% confidence bounds. In Fig. 5, we show the linewidth as a function of the bias field Bz . Overlaying the data is a fit based on Eq. (7), which gives a limiting low-frequency linewidth of 2.9 Hz and a spin-exchange rate RSE ≈ 4.8 × 104 s−1. From limiting lowfrequency linewidth, we can obtain the transverse relaxation rate due to spin-destruction and diffusion as 109 s−1. For n = 3.7 × 1013 cm−3, the expected spin exchange rate is 4.6 × 104 s−1,17 which is close to the fitting result. To compare the experimental results with the expected values, we calculated the relaxation values using Eq. (6), where the spin-exchange relaxation, pump rate, and relaxation due to the probe beam can be ignored under the condition of zero field TABLE I. Fitting results of resonance frequency and linewidth for different values of B z . B z (nT)
f 0 (Hz)
△υ (Hz)
R-square
2 6 10 14 18 22 26 30 34 38 42 46
9.423 27.95 46.79 65.42 84.16 103.0 121.7 140.4 159.3 178.2 197.1 216.1
3.003 3.126 3.403 3.826 4.423 5.165 6.081 7.017 8.402 9.616 11.21 13.05
0.9982 0.9981 0.9984 0.9989 0.9990 0.9992 0.9988 0.9991 0.9992 0.9985 0.9984 0.9985
SD RSD = nσKSD ν¯ + nσHe ν¯ + nσNSD2 ν, ¯ ( ) π 2 1 = q(P)D , TD a
(13) (14)
where σ SD are the spin-destruction cross sections, ν¯ is the thermal velocity, D is the diffusion constant of alkali atom within the buffer gas, and a is the radius of the spherical cell. SD Here, σKSD = 1 × 10−18 cm2, σHe = 5 × 10−25 cm2, σNSD2 = 7.9 −23 2 17 2 × 10 cm , and D = 0.24 cm /s (for the temperature and the buffer gas content of this cell).18 Consequently, we can get that RSD = 22 s−1 and 1/TD = 69 s−1, yielding the total relaxation due to spin-destruction and diffusion is 91 s−1, which is in reasonable agreement with the experimentally determined value (109 s−1). V. CONCLUSION
We demonstrated a fast and accurate determination method for transverse relaxation of the SERF magnetometer. This method realizes magnetic resonance linewidth determination by a chirped magnetic field excitation and amplitude spectrum analysis method. The advantage of this method is that the measurement time can be shortened to several seconds, and meanwhile, good frequency resolution can be achieved. Therefore, it can avoid the drift error and improve the accuracy, and meanwhile not increase the demands on experimental setup. In data analysis, we considered the effect of the resonance at −ω0 caused by the calibration chirp signal which should not be ignored for the SERF magnetometer in the low frequency range. Finally, we calculated the relaxation values and compared them with experimentally determined result, which was found to be in good agreement with our theoretical analysis. ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of China (No. 61227902).
043104-5 1D.
Lu, Qian, and Fang
Budker and M. V. Romalis, Nat. Phys. 3, 227 (2007).
2D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and
A. Weis, Rev. Mod. Phys. 74, 1153 (2002). 3H. B. Dang, A. C. Maloof, and M. V. Romalis, Appl. Phys. Lett. 97, 151110-
1 (2010). C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, Phys. Rev. Lett. 89, 130801 (2002). 5I. M. Savukov and M. V. Romalis, Phys. Rev. A 71, 023405 (2005). 6M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V. Romalis, Phys. Rev. A 77, 033408 (2008). 7R. K. Raney, H. Runge, R. Bamler, I. G. Cumming, and F. H. Wong, IEEE Trans. Geosci. Remote Sens. 32, 786 (1994). 8L. R. LeBlanc, L. Mayer, M. Rufino, S. G. Schock, and J. King, J. Acoust. Soc. Am. 91, 107 (1992). 4J.
Rev. Sci. Instrum. 86, 043104 (2015) 9P.
D. D. Schwindt, B. Lindseth, S. Knappe, V. Shah, J. Kitching, and L.-A. Liew, Appl. Phys. Lett. 90, 081102 (2007). 10R. Wyllie, Ph.D. thesis, University of Wisconsin-Madison, Wisconsin, 2012. 11D. Budker and D. F. J. Kimball, Optical Magnetometry (Cambridge University Press, Cambridge, 2013), p. 95. 12S. Appelt, A. B.-A. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, Phys. Rev. A 58, 1412 (1998). 13S. J. Seltzer and M. V. Romalis, Appl. Phys. Lett. 85, 4804 (2004). 14W. Happer and A. C. Tam, Phys. Rev. A 16, 1877 (1977). 15S. J. Seltzer, Ph.D. thesis, Princeton University, Princeton, 2008. 16C. B. Alcock, V. P. Itkin, and M. K. Horrigan, Can. Metall. Q. 23, 309 (1984). 17S. Kadlecek, L. W. Anderson, and T. Walker, Nucl. Instrum. Methods Phys. Res., Sect. A 402, 208 (1998). 18F. A. Franz and C. Volk, Phys. Rev. A 26, 85 (1982).
Review of Scientific Instruments is copyrighted by AIP Publishing LLC (AIP). Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. For more information, see http://publishing.aip.org/authors/rights-and-permissions.
