⇦Chinese Physics Letters, 2019, Vol. 36, No. 2, Article code 020701 An Atomic Magnetometer with Spin-Projection Noise Proportional to $\sqrt{{T_2}}$ * Hai-Feng Dong (董海峰)1,2**, Xiao-Fei Wang (王笑菲)1, Ji-Min Li (李继民)1, Jing-Ling Chen (陈静铃)1, Yuan Ren (任元)3 Affiliations 1School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing 100083 2Graduate School of China Academy of Engineering Physics, Beijing 100193 3Department of Aerospace Science and Technology, Space Engineering University, Beijing 101416 Received 12 September 2018, online 22 January 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 51675034 and 61273067, and the Natural Science Foundation of Beijing Municipality under Grant No 7172123.
**Corresponding author. Email: hfdong@buaa.edu.cn Citation Text: Dong H F, Wang X F, Li J M, Chen J L and Ren Y et al 2019 Chin. Phys. Lett. 36 020701    Abstract There is a common sense view for atomic magnetometers that their spin-projection-noises (SPNs) are inversely proportional to $\sqrt{{T_2}}$, where $T_2$ is the transverse relaxation time. We analyze the current atomic magnetometer types and give a counter-example of this common sense, which is the all-optical spin precession modulated three-axis atomic magnetometer proposed by our group in 2015. Unlike the other atomic magnetometers, the SPN of this kind of atomic magnetometers is proportional to $\sqrt{{T_2}}$ due to the fact that the scale factor between $P_x$ and $B$ can be unrelated to the transverse relaxation time $T_2$. We demonstrate this irrelevance experimentally and analyze the SPN theoretically. Using short-pulse ultra-high power laser to fully polarize the atoms, the phenomenon that SPN decreases with $T_2$ may also be demonstrated experimentally and a new tool for researching SPN in atomic magnetometers may be realized. DOI:10.1088/0256-307X/36/2/020701 PACS:07.55.Ge, 32.30.Dx, 42.50.Lc © 2019 Chinese Physics Society Article Text Spin projection noise is one of the fundamental noises of all the atomic magnetometers, which is usually inverse proportional to $\sqrt{{T_2}}$. Not that we said 'usually' here because we have recently found an exception. First, let us see how usual this inverse-proportion property really is. In most cases, the dynamics of the spin polarization in atomic magnetometers can be well-described by the Bloch equations[1-3] $$\begin{align} \frac{d{\boldsymbol P}}{dt}=\gamma {\boldsymbol P}\times {\boldsymbol B}+\frac{{R_{\rm p}}}{q}({\boldsymbol s}-{\boldsymbol P})-\frac{{R_{1}},{R_{2}}}{q}{\boldsymbol P},~~ \tag {1} \end{align} $$ where $\gamma ={{\gamma }^{\rm e}}/q$ is the atomic gyromagnetic ratio, ${{\gamma }^{\rm e}}$ is the gyromagnetic ratio of electron, ${R_{1}}={R_{\rm sd}}+q{R_{\rm wall}}$ is the longitudinal relaxation rate, ${R_{2}}={R_{1}}+q{R_{\rm se}}/{{q}_{\rm se}}+q{R_{\rm gr}}$ is the transverse relaxation rate, ${R_{\rm p}}$ is the pumping rate, $q$ is the slowing-down factor, and ${\boldsymbol B}$ is composed of $B_x$, $B_y$, $B_z$ and $B_1\sin(wt)$. These four components and the scalar $B_0$ can all be the measured magnetic fields. Typically, two different atomic magnetometers can be used to measure $B_1$: the first is a radio-frequency atomic magnetometer (RF AM),[4-6] and the second is a spin-exchange relaxation free atomic magnetometer (SERF AM).[2,3,7-10] Thus, one may not be surprised to see that their optimized scale factor between magnetic field and spin polarization $P_x$ at resonance is the same. The optimized scale factor of the former is close to $\frac{1}{2}\gamma {{T}_{2}}$ considering that $R_2\gg R_1$ in the normal regime. That of the latter is also close to $\frac{1}{2}\gamma {{T}_{2}}$ when magnetic gradient is shielded and compensated, thus ${R_{1}}\approx {R_{2}}$ in the SERF regime.[2] Although the scale factor of RF AM here has the same form with Eq. (1) in Ref. [4] we can obtain it by optimizing the pumping rate instead of supposing that the atoms are nearly completely polarized along $z$ axis. Considering the spin projection noise (SPN) of spin polarization $P_x$ caused by Heisenberg uncertainty $$\begin{align} \delta {{P}_{x}}=2\sqrt{\frac{{{T}_{2}}}{qN}},~~ \tag {2} \end{align} $$ where $q$ is the slowing-down factor, ${{T}_{2}}=q/({R_{\rm p}}+{R_{2}})$ is the transverse relaxation time, and $N$ is the atom numbers. We can deduce both the equivalent magnetic field SPNs of the two atomic magnetometers are $\frac{4}{\gamma \sqrt{qN{{T}_{2}}}}$. It is clear that their SPNs are inversely proportional to $\sqrt{{T_2}}$. Typically, three different magnetometers can be used to measure the scalar magnetic field $B_0$, which are $M_x$,[11] $M_z$,[12] and Bell–Bloom magnetometer.[13-15] For $M_x$ AM, one needs to optimize the oscillation field amplitude $B_1$ and the pumping rate $R_{\rm p}$ at the same time. In the case when $R_2\gg R_1$, the maximum scale factor for measuring $B_0$ using $M_x$ AM is approximately $\frac{1}{2\sqrt{3}}\gamma {{T}_{2}}$. This result is different from Eq. (6) in Ref. [11] where only the field amplitude $B_1$ is optimized. For $M_z$ AM, the $P_z$ signal has an absorption form, and the magnetic field modulation is needed for the synchronous detection. The maximum scale factor to measure $B_0$ using $M_z$ AM is approximately $\frac{1}{4}\gamma {{T}_{2}}$. As far as we know, there are no other forms of optimization result for the scale factor of $M_z$ AM.
Table 1. Scale factors between polarization and magnetic field, and SPNs of several typical atomic magnetometers ($\star$ means the same as the $B_0$-type scalar magnetometry).
AM types Scale factor SPN
$B_1$ RF $\frac{1}{2}\gamma T_2$ $\frac{4}{\gamma \sqrt{qNT_2}}$
SERF $\frac{1}{2}\gamma T_2$ $\frac{4}{\gamma \sqrt{qNT_2}}$
$M_x$ $\frac{1}{2\sqrt{3}}\gamma T_2$ $\frac{4\sqrt{3}}{\gamma \sqrt{qNT_2}}$
$B_0$ $M_z$ $\frac{1}{4}\gamma T_2$ $\frac{8}{\gamma \sqrt{qNT_2}}$
Bell–Bloom $\frac{1}{2}\gamma T_2$ $\frac{4}{\gamma \sqrt{qNT_2}}$
MScM $\star$ $\star$
MRoM $\frac{1}{\sqrt{6}}\gamma T_2$, $\frac{1}{\sqrt{6}}\gamma T_2$, $\frac{1}{2\sqrt{3}}\gamma T_2$ $\frac{2\sqrt{6}}{\gamma \sqrt{qNT_2}}$, $\frac{2\sqrt{6}}{\gamma \sqrt{qNT_2}}$, $\frac{4\sqrt{3}}{\gamma \sqrt{qNT_2}}$
MCoM $\star$ $\star$
$B_x$, $B_y$, $B_z$ MPrM $\star$ $\star$
MCrM $\frac{\sqrt{2}}{4}\gamma T_2$, $\frac{1}{2}\gamma T_2$, $\frac{\sqrt{2}}{4}\gamma T_2$ $\frac{4\sqrt{2}}{\gamma \sqrt{qNT_2}}$, $\frac{4}{\gamma \sqrt{qNT_2}}$, $\frac{4\sqrt{2}}{\gamma \sqrt{qNT_2}}$
MSeM $\frac{3\sqrt{3}}{8}\gamma T_2$, $\frac{3\sqrt{3}}{8}\gamma T_2$, $\frac{9}{64}\gamma T_2$ $\frac{16\sqrt{3}}{9\gamma \sqrt{qNT_2}}$, $\frac{16\sqrt{3}}{9\gamma \sqrt{qNT_2}}$, $\frac{128}{9\gamma \sqrt{qNT_2}}$
SMoM $\star$, $\frac{1}{B_0}$, $\frac{1}{B_0}$ $\star$, $2B_0\frac{\sqrt{T_2}}{qN}$, $2B_0\frac{\sqrt{T_2}}{qN}$
For Bell–Bloom AM, $B_1=0$ and the resonance is realized through pulsed pumping. Only the pumping rate needs to be optimized and the maximum scale factor is $\frac{1}{2}\gamma T_2$. By using these scale factors for $M_x$, $M_z$ and Bell–Bloom magnetometers and Eq. (2), the equivalent magnetic field SPNs of these three magnetometers are deduced to be $\frac{4\sqrt{3}}{\gamma \sqrt{qN{{T}_{2}}}}$, $\frac{8}{\gamma \sqrt{qN{{T}_{2}}}}$ and $\frac{4}{\gamma \sqrt{qN{{T}_{2}}}}$, respectively. It is obvious that all of them are inverse proportional to $\sqrt{{T_2}}$. Note that although $P_z$ is measured in $M_z$ AM, we still use Eq. (2) for the SPN calculation because of the symmetry of the spin projection noise. Finally, let us consider the measurement of magnetic field components; i.e., $B_x$, $B_y$ and $B_z$. There are typically seven methods to realize three-axis vector atomic magnetometry. The first is called magnetic field scanning magnetometry (MScM), which was proposed by Alldredge in the 1960s,[16,17] in which the field along a certain direction is scanned while the scalar field is monitored. The scanning field corresponding to the minimum of the scalar field is opposite to the field in the scanning direction. Because the vector field is measured directly by the scalar magnetometer, the equivalent magnetic field SPN is the same as that of the scalar magnetometer. The second is magnetic field rotation magnetometry (MRoM), which was proposed by Vershovskii et al. in 2004,[18] in which the total field is locked by measuring and compensating the transverse fields. As the measurement of the total field $B_0$ is discussed above, here we focus on the scale factor and SPN of the transverse fields measurement. The transverse fields are measured by adding a rotation magnetic field perpendicular to the total field $B_0$. In this way, the total field will oscillate in the same frequency when there is a transverse field. The amplitude and phase of this oscillation are decided by the transverse field. Thus, one can measure the transverse fields through the measurement of $B_0$'s oscillation. In the real application, $B_1$ is usually much smaller than $B_0$. Here, we just optimize the scale factor theoretically. If the scalar magnetometer has fast response to the direction change, the optimized scale factor happens when the modulation field is equal to $B_0$. It is enhanced by a factor of $\sqrt{2}$ from the $B_0$ measurement. Here, $M_x$ is used more frequently for MRoM than the other $B_0$ magnetometers due to frequency response consideration. Thus, the optimized scale factor of $B_x$, $B_y$ for MRoM is $\frac{1}{\sqrt{6}}\gamma {{T}_{2}}$, and that of $B_z$ is the same as the $M_x$ magnetometer. The third is magnetic field compensation magnetometry (MCoM), which was proposed by Vershovskii in 2006,[19] in which two orthogonal fields are compensated for so that the measured scalar field is equal to the field along direction perpendicular to the previous two fields' directions. By carrying out the compensation successively, one can obtain the three-axis magnetic fields. This is a method with high accuracy due to the orthogonal between the measured field and the compensation field. From the viewpoint of SPN, it is the same as the scalar magnetometer. The fourth is magnetic field projection magnetometry (MPrM), which was proposed by Patton et al. in 2014,[20] which measures the field direction by the projections of three-axis orthogonal fields. As the scalar field is measured directly, the equivalent field SPN is also the same as the scalar magnetometer. The fifth is magnetic field cross-modulation magnetometer (MCrM), which was proposed by Seltzer and Romalis,[21] in which $B_y$ is measured in the same way with the SERF magnetometer mentioned above, $B_x$ and $B_z$ are measured by cross modulation. Here, we focus on the transverse magnetic field $B_x$ and $B_z$. After cross-modulation and lock-in amplifier, the scale factor between output spin polarization and transverse magnetic field is $\frac{R_{\rm p}}{R_{\rm p}+R_2}\frac{B_m}{(\frac{R_{\rm p}+R_2}{\gamma})^2+\frac{B_m^2}{2}}$, which has a maximum value of $\frac{\sqrt{2}}{4}\gamma {{T}_{2}}$. The sixth is magnetic field separate modulation magnetometry (MSeM), in which three different modulation fields are added to three axes separately.[22] We proposed and verified the usage of MSeM in zero field in 2012,[9] and solved the key problems in 2016.[23] In this case, we deduced that the optimized scale factor of three axes are $\frac{3\sqrt{3}}{8}\gamma {{T}_{2}}$, $\frac{3\sqrt{3}}{8}\gamma {{T}_{2}}$, and $\frac{9}{64}\gamma {{T}_{2}}$, respectively.
cpl-36-2-020701-fig1.png
Fig. 1. Schematic diagram of the polarization measurement of spin precession modulated atomic magnetometer.
The seventh vector atomic magnetometer is spin modulation magnetometry (SMoM), which was pioneered by Fairweather and Usher in 1972.[24] In 2015, we proposed a kind of all-optical three-axis SMoM.[25] Furthermore, we recently find that this kind of atomic magnetometers has an abnormal property; i.e., their SPN is proportional to $\sqrt{{{T}_{2}}}$. The detailed sensing mechanism was described in Ref. [25]. Here we give a brief introduction for convenience of reading. The principle of SMoM is shown in Fig. 1. Assuming that the initial magnetic field is parallel to the direction of the probe light ($x$ axis), there is no projection of spin polarization along the probe beam. When a small transverse magnetic field ($B_{\rm t}$) perpendicular to the direction of the probe beam appears, the direction of the total magnetic field is deviated from its initial position. As the deviation angle is small enough, the polarization oscillation amplitude in the direction of the probe beam can be approximately as $\frac{P_0}{B_0}$, which has a maximum value of $\frac{1}{B_0}$ for $P_0=1$. Similar to the measurement of $B_1$ and $B_0$, we can also obtain the SPNs and $B_x$, $B_y$ and $B_z$ using Eq. (2) and the corresponding scale factors. In Table 1, we list all the magnetometers' optimized scale factors and SPNs. It is clearly shown that most of the atomic magnetometers are inverse-proportional to $\sqrt{T_2}$, which is the reason why it becomes a common sense. The only exception is SMoM. The basic reason for this phenomenon is that when all the other atomic magnetometers have optimized scale factors proportional to $T_2$, that of SMoM is unrelated to $T_2$ in the transverse direction. For the main field, it depends on the used scalar magnetometry, which may be Bell–Bloom[15] or $M_x$ magnetometer.[24]
cpl-36-2-020701-fig2.png
Fig. 2. Normalized relation between scale factor, $T_2$ and the cell temperatures. The black square points present the normalized scale factor between $P_x$ and $B$. The red circular points present the normalized transverse relaxation time $T_2$. With the increase of temperature, $T_2$ decreases and the scale factor of SMoM is kept to be constant.
cpl-36-2-020701-fig3.png
Fig. 3. The relationship between probe light power and optical rotation noise. The black square points present the optical rotation noise under variable probe light power. The red line is the fitting curve. Inset: the optical rotation characterization in Ref. [4].
We carry out an experiment to verify this irrelevance. The basic experimental setup is similar to our previous report,[25] and the only difference is that short pulse light is used for the synchronous pumping in this experiment instead of acoustic optical modulation. We change the cell temperature from 50$^{\circ}\!$C to 120$^{\circ}\!$C. The transverse relaxation time $T_2$ is measured by the fitting of the free induced decay signal. Figure 2 shows the normalized transverse relaxation time and scale factor for different temperatures. From Fig. 2, we can see that when $T_2$ is shortened with the increase of the temperature, the scale factor is kept to be constant. As $B_0$ is the same for different temperatures, we know that the spin polarization is also kept constant. We also calibrate the probe noise by blocking the pumping beam, which is shown in Fig. 3. The black square is the optical rotation noise while the red line is photon shot noise calculated by $\delta \phi =\frac{1}{\sqrt{2{\it \Phi} \eta }}$, where ${\it \Phi} $ is the probe beam photons flux, and $\eta=0.9 $ is the photodiode quantum efficiency. The inset in Fig. 3 is the same measurement from Ref. [4] which is given for comparison. From Fig. 3, we can see that the probe noise is limited by photon shot noise, which is also the main noise source of the magnetometer. Thus, we have not observed the SPN and its variation with $T_2$ directly at the current stage. In summary, we have classified atomic magnetometers into three types according to the measured field in the Bloch equation. Their optimized scale factors are analyzed one by one. Together with the variation of spin polarization, we deduce their equivalent magnetic field SPNs, respectively. Our analysis illustrates a general rule for atomic magnetometer's SPN; i.e., the SPN is inversely proportional to $\sqrt{T_2}$ and shows one counter-example at the same time. We analyze the reason for this unusual example and carry out experiments to verify the reason. As the noise is overwhelmed by the photon shot noise, the expected SPN's decrease with the shortening of $T_2$ is not observed directly. Using the high power short pulse pumping source, it is possible to fully polarize the atoms and achieve an atomic magnetometer limited by SPN. This may provide an experimental method for the research of SPNs in the atomic magnetometers. Meanwhile, once the sensitivity is dominated by SPN, this kind of special atomic magnetometers can improve the magnetic field sensitivity by decreasing instead of increasing $T_2$, which is much easier to be realized practically. References Dynamics of Two Overlapping Spin Ensembles Interacting by Spin ExchangeHigh-Sensitivity Atomic Magnetometer Unaffected by Spin-Exchange RelaxationSpin-exchange-relaxation-free magnetometry with Cs vaporTunable Atomic Magnetometer for Detection of Radio-Frequency Magnetic FieldsThrough-barrier electromagnetic imaging with an atomic magnetometerSubfemtotesla radio-frequency atomic magnetometer for detection of nuclear quadrupole resonanceSubpicotesla atomic magnetometry with a microfabricated vapour cellMagnetoencephalography with a two-color pump-probe, fiber-coupled atomic magnetometerThree-dimensional atomic magnetometryA dual-axis, high-sensitivity atomic magnetometerLow-noise high-density alkali-metal scalar magnetometerDouble-Resonance Atomic Magnetometers: From Gas Discharge to Laser PumpingOptical Detection of Magnetic Resonance in Alkali Metal VaporIndirect pumping bell–bloom magnetometerClose-Loop Bell–Bloom Magnetometer with Amplitude ModulationThe technique of scaling indexes K and Q of geomagnetic activityA proposed automatic standard magnetic observatoryThree-component variometer based on a scalar potassium sensorA new method of absolute measurement of the three components of the magnetic fieldAll-Optical Vector Atomic MagnetometerUnshielded three-axis vector operation of a spin-exchange-relaxation-free atomic magnetometerOn the calibration of a vectorial 4He pumped magnetometerSingle-beam three-axis atomic magnetometerA vector rubidium magnetometerThree-axis atomic magnetometer based on spin precession modulation
[1] Kornack T W and Romalis M V 2002 Phys. Rev. Lett. 89 253002
[2] Allred J C, Lyman R N, Kornack T W and Romalis M V 2002 Phys. Rev. Lett. 89 130801
[3] Ledbetter M P et al 2008 Phys. Rev. A 77 033408
[4] Savukov I M, Seltzer S J, Romalis M V and Sauer K L 2005 Phys. Rev. Lett. 95 063004
[5] Deans C, Marmugi L and Renzoni F 2017 Opt. Express 25 17911
[6] Lee S K et al 2006 Appl. Phys. Lett. 89 214106
[7] Shah V, Knappe S, Schwindt P D D and Kitching J 2007 Nat. Photon. 1 649
[8] Johnson C, Schwindt P D D and Weisend M 2010 Appl. Phys. Lett. 97 243703
[9] Dong H F et al 2012 Eur. Phys. J. Appl. Phys. 57 21004
[10] Li R J et al 2017 Chin. Phys. B 26 120702
[11] Smullin S J et al 2009 Phys. Rev. A 80 033420
[12] Schultze V et al 2017 Sensors 17 561
[13] Bell W E and Bloom A L 1957 Phys. Rev. 107 1559
[14] Wang M B, Zhao D F, Zhang G Y and Zhao K F 2017 Chin. Phys. B 26 100701
[15] Huang H C, Dong H F, Hao H J and Hu X Y 2015 Chin. Phys. Lett. 32 098503
[16] Alldredge L R 1960 J. Geophys. Res. 65 3777
[17] Alldredge L R and Ignas S 1964 J. Geophys. Res. 69 1963
[18] Alexandrov E B et al 2004 Meas. Sci. Technol. 15 918
[19] Vershovskii A K 2006 Opt. Spectrosc. 101 309
[20] Patton B, Zhivun E, Hovde D C and Budker D 2014 Phys. Rev. Lett. 113 013001
[21] Seltzer S J and Romalis M V 2004 Appl. Phys. Lett. 85 4804
[22] Gravrand O, Khokhlov A, Mouel J L L and Leger J M 2001 Earth Planets Space 53 949
[23] Huang H C, Dong H F, Chen L and Gao Y 2016 Appl. Phys. Lett. 109 062404
[24] Fairweather A J and Usher M J 1972 J. Phys. E 5 986
[25] Huang H C et al 2015 Appl. Phys. Lett. 107 182403