1.   Introduction

Advances in precise magnetic field measurement technology have had a profound impact on many fields, such as brain magnetism in biomedicine.^[1] and heart magnet^[2,3] Imaging technology uses magnetic field measurements to detect movement inside the body; in the military and defense fields^[4] magnetic anomaly detection^[5] pair of high precision navigation^[6,7] and Space Technology^[8] is crucial; in basic science research, ultra-sensitive weak magnetic measurement techniques are opening up new areas of research, such as dark matter, dark energy, and other unknown interactions and new physical phenomena that may exist in the universe.^[9–12].The progress and development of magnetic field precision measurement technology not only promote engineering applications for the benefit of mankind, but also expand the boundaries of science and lay the foundation for future scientific discoveries.

The development of magnetic sensors with ultra-high sensitivity is very important for biomedicine, geospace, new physics and other fields.As a key indicator to measure the performance of magnetic sensors, magnetic sensitivity is mainly limited by thermal noise, quantum measurement noise and other noises.^[13].The thermal noise originated from the random thermal motion can be limited by the fluctuation-dissipation theorem, and the quantum measurement noise originated from the uncertainty principle defines the standard quantum limit of magnetic sensitivity^[14](standward quantum limit, SQL).Recently, the literature [15] defines a new energy resolution limit (ERL):$S_{{\mathrm{B}}}V\geqslant2\mu_{0}\hbar$, where$S_{{\mathrm{B}}}\left(\omega\right)$ is the power spectrum density (PSD) of magnetic field noise;$\mu_{0}, \hbar$ is the vacuum permeability and the reduced Planck constant;V represents the spatial volume of the sensor.ERL gives the weakest magnetic signal that can be detected by a magnetic sensor in a finite volume, which has been effectively verified on a variety of technology platforms and is regarded as a performance benchmark for evaluating the sensitivity of magnetic sensors: most magnetic sensors (such as SQUID, atomic sensors, solid-state spin systems) satisfy this relationship and are only close to ERL in the best case.

There is a huge demand for ultra-high sensitivity magnetic sensors in basic research and engineering applications, and the sensitivity of magnetic sensors is often limited by ERL, which makes these demands difficult to meet.Magnetic sensors beyond ERL are of great significance to basic research, and can promote experimental progress and achieve breakthrough results.Taking the new interaction as an example, the essence of the experimental detection of the new interaction is to measure the pseudo-magnetic field generated by it, and the detection accuracy depends on the magnetic sensitivity of the sensor and the distance between the sensor and the nucleon source.Specifically, the improvement of magnetic sensitivity helps to improve the detection accuracy of new interactions, while ERL shows that the improvement of magnetic sensitivity means a larger space size of the magnetic sensor, which will increase the distance between the sensor and the nuclear source, thus limiting the measurement accuracy of new interactions.In a sense, ERL limits the upper limit of the measurement accuracy of new interactions, and ultra-high sensitivity magnetic sensors beyond ERL are of great significance for such research work, which can promote the progress of experimental detection and achieve breakthrough results.

ERL is traditionally considered to be the upper limit of the achievable sensitivity of magnetic sensors, but it is essentially an empirical summary of experimental results, not from first principles, and the realization of magnetic sensors beyond ERL does not violate the basic principles.Recent studies have shown that magnetic sensors based on ferromagnetic-mechanical systems are expected to surpass ERL^[16] in sensitivity, and ferromagnetic-mechanical systems can be used for magnetic field measurement by measuring their mechanical response to magnetic signals.For example, ferromagnets can be connected to mechanical oscillators such as cantilevers for magnetic field measurement, such as magnetic force microscopes and magnetic resonance force microscopes; suspended ferromagnetic-mechanical systems can eliminate clamping dissipation and achieve higher quality factors, thus achieving sensitivity^[17]. Beyond ERL.Up to now, there are two schemes^[16,18] for the suspension of ferromagnetic torsional oscillator (FMTO) and ferromagnetic gyroscope (FMG), and their sensitivity can exceed ERL by five orders of magnitude under the influence of basic noise.This breakthrough not only improves the sensitivity and resolution of magnetic field measurement, but also broadens the boundaries of physics research, which has a far-reaching impact on new interactions, dark energy, dark matter and other basic physics fields.

In this paper, a magnetic sensor scheme based on a suspended ferromagnetic pendulum is proposed. The ferromagnetic pendulum converts the magnetic signal into a torque signal to drive the rotation of the vibrator, and the motion of the vibrator is measured by optical method to extract the magnetic signal.Different from the scheme in reference [16], the resonance frequency of FMTO can be controlled by adjusting the bias magnetic field, so this scheme is more flexible and controllable.In this paper, the mechanical properties and magnetic sensitivity of FMTO system are systematically analyzed. The magnetic sensitivity of FMTO system can be several orders of magnitude ahead of ERL. The application of high sensitivity FMTO magnetic sensor in the detection of new interactions is expected to push the lower limit of coupling constants of new interactions to 2-9 orders of magnitude.

2.   FMTO kinetic model

The dynamical evolution of a ferromagnet in a uniform magnetic field${{\boldsymbol{B}}}_{{\mathrm{Bias}}}$ is described by the following equation^[19]:

$$\begin{split} \dot{{{\boldsymbol{J}}}} ={{\boldsymbol{\mu}}}\times{{\boldsymbol{B}}}_{{\mathrm{Bias}}},~~~~ \dot{{{\boldsymbol{S}}}} ={{{{\boldsymbol{\varOmega}}}}}\times{{\boldsymbol{S}}}, \end{split}$$

(1)

${{\boldsymbol{J}}}={{\boldsymbol{L}}} +{{\boldsymbol{S}}}$ is the total angular momentum of the ferromagnetic sphere,${{\boldsymbol{L}}}=I{{{{\boldsymbol{\varOmega}}}}}$ is the classical rigid-body angular momentum of the sphere,${{\boldsymbol{S}}}$ is the intrinsic spin angular momentum,${{\boldsymbol{\mu}}}=\gamma_{0}{{\boldsymbol{S}}}$ is the magnetic moment,$I=2 mR^{2}/5$ represents the moment of inertia of the sphere with radiusR and massm,${{{{\boldsymbol{\varOmega}}}}}$ is the angular velocity of the sphere, and$\gamma_{0}$ is the electron gyromagnetic ratio.

There are two modes of motion, torsion and precession, in the evolution of ferromagnetism described by (1).Which motion mode is dominant depends on the relative magnitude of its rigid body angular momentum and spin angular momentum: when the rigid body angular momentum is much greater than the spin angular momentum, as shown in the left figure ofFig. 1, the ferromagnet performs a torsional pendulum motion at an angular frequency of$\omega_{{\mathrm{T}}}=\sqrt{\mu B_{{\mathrm{Bias}}}/I}$, which is called a ferromagnetic torsional pendulum oscillator, that is, FMTO; when the spin angle momentum is much greater than the rigid body angle momentum, as shown in the right figure ofFig. 1, the ferromagnet precesses around the magnetic field gyroscope at a Larmor frequency of$\omega_{{\mathrm{L}}}=\gamma_{0}B_{{\mathrm{Bias}}}$.It is pointed out that both FMTO and FMG have great potential in the field of magnetic field measurement. The difference between them is that the ferromagnetic radius of FMG system is smaller and the magnetic field is weaker, while FMTO requires a larger ferromagnetic radius or a stronger external field.

The ratio of the rigid body angular momentum to the spin angular momentum of a ferromagnet is

$$\begin{aligned} \dfrac{L}{S} = \dfrac{I\omega_{{\mathrm{T}}}}{S}=\sqrt{2/5}\gamma_{0}\rho_{{\mathrm{m}}}^{1/2}\left(\rho_{{\mathrm{e}}}\mu_{{\mathrm{B}}}\right) ^{-1/2}RB^{1/2}_{\rm Bias}, \end{aligned}$$

(2)

式中$\omega_{{\mathrm{T}}}$为扭摆频率; $\rho_{{\mathrm{m}}},\; \rho_{{\mathrm{e}}}$分别代表质量密度和自旋密度. (2)式指出: 半径或外磁场较大时, 刚体角动量远大于自旋角动量, 铁磁体以扭摆运动为主; 半径和外磁场较小时, 自旋角动量远大于刚体角动量, 铁磁体以拉莫尔进动为主. 图2展示了半径和磁场对铁磁运动模式的影响: 粉色和青色区域分别对应$S\gg L$和$L \gg S$两种情况(取10作为临界判据, 如红色虚线代表$S/L=10$, 青色虚线代表$L/S=10$). 粉色区域内自旋角动量是刚体角动量的10倍以上, 铁磁表现为以陀螺进动为主的FMG, 青色区域内刚体角动量远大于自旋角动量, 铁磁体表现为以扭摆运动为主的FMTO.

对刚体角动量和自旋角动量的初步分析发现: 自旋角动量取决于磁矩, 正比于半径的三次方, 刚体角动量作为转动惯量与角速度之积则正比于半径的四次方. 这意味着宏观尺寸的铁磁体通常表现为以扭摆为主的FMTO, 此时(1)式退化为典型的受迫阻尼谐振子^[21], 其运动方程由下式描述:

$$\begin{aligned} I\ddot{\theta}\left( t\right) + I\gamma\dot{\theta}\left( t\right) + k\theta\left( t\right) =\tau_{{\mathrm{tot}}}\left( t\right), \end{aligned}$$

(3)

式中$\theta$为偏转角; $k=\mu B_{{\mathrm{Bias}}}$为等效弹性系数; $\gamma= \omega_{0}/Q$为耗散项, 其中$\omega_{0}=\sqrt{k/I}$为共振角频率, Q为品质因子; 右边$\tau_{{\mathrm{tot}}}\left(t\right)$代表施加在振子上的总力矩. 对(3)式进行傅里叶变换可得表示在频域上的稳态解:

$$\begin{aligned} \theta\left( \omega\right) =\chi\left( \omega\right) \tau_{{\mathrm{tot}}}\left( \omega\right) /I, \end{aligned}$$

(4)

式中$\theta\left(\omega\right), \tau_{{\mathrm{tot}}}\left(\omega\right)$分别为$\theta\left(t\right), \tau_{{\mathrm{tot}}}\left(t\right)$的频谱; $\chi\left(\omega\right)$通常被称为振子的机械敏感系数:

$$\begin{aligned} \chi\left( \omega\right) =1/\left( \omega_{0}^{2}-\omega^{2}+ {\mathrm{i}}\omega\omega_{0}/Q\right) . \end{aligned}$$

(5)

以上分析仅针对FMTO的动力学模型, 并未涉及其结构设计和物理实现, 实验中可通过超导磁悬浮^[22,23]、抗磁悬浮^[24]等多种方式^[25–27]实现铁磁振子的悬浮. 在水平面内两个正交方向上放置两对亥姆霍兹线圈, 其中一对线圈通直流电以产生均匀偏置磁场${{\boldsymbol{B}}}_{{\mathrm{Bias}}}$用以控制磁矩指向调节共振频率等, 另一线圈则通以较弱的交流信号以产生交变磁信号${{\boldsymbol{B}}}_{{\mathrm{sig}}}\left(t\right)$. 铁磁体通过${{\boldsymbol{\tau}}}_{{\mathrm{sig}}}\left(t\right) ={{\boldsymbol{\mu}}}\times{{\boldsymbol{B}}}_{{\mathrm{sig}}}\left(t\right)$将磁信号转化为驱动FMTO运动的力矩信号; 测量FMTO对磁信号的动力学响应(功率谱、噪声本底等), 当由磁信号驱动所产生的运动高于噪声时, 即可从噪声中分辨出磁信号.

3.   FMTO噪声与灵敏度分析

磁灵敏度作为评估磁传感器性能的最关键指标, 定义为最小可分辨磁场与测量时间平方根的乘积:

$$\begin{aligned} \eta_{{{B}}}=\delta B\sqrt{T_{{\mathrm{mea}}}}, \end{aligned}$$

(6)

其中$\eta_{{{B}}}$为磁灵敏度, $T_{{\mathrm{mea}}}$为测量时间, $\delta B$为最小可分辨磁场. 若采用${\mathrm{SNR}} \geqslant 1$, 即信噪比大于等于1作为最小可分辨的临界判据^[28], 则可以磁噪声的标准差$\Delta B$作为最小可分辨磁场$\delta B$, 此时磁传感器的灵敏度为

$$\begin{aligned} \eta_{{{B}}}=\Delta B\sqrt{T_{{\mathrm{mea}}}}=\sqrt{S_{B_{{\mathrm{tot}}}}}, \end{aligned}$$

(7)

式中$S_{B_{{\mathrm{tot}}}}$的单位为${\rm{T^{2}/Hz}}$代表磁噪声功率谱. 注意对噪声功率谱开根号即为噪声本底(noise floor), 故(7)式表明磁传感器灵敏度等于其磁噪声本底(magnetic noise floor)$\sqrt{S_{B_{{\mathrm{tot}}}}}$.

本文主要考虑热噪声、量子测量噪声等基本噪声对灵敏度的影响. 热噪声是与频率无关的高斯白噪声, 以随机涨落的力矩进入力学系统驱动振子运动, 热涨落力矩的功率谱由涨落耗散定理给出^[29]:

$$\begin{aligned} S_{\tau_{{\mathrm{th}}}}=4k_{{\mathrm{B}}}TI\omega_{0}/Q, \end{aligned}$$

(8)

式中$k_{{\mathrm{B}}}$为玻尔兹曼常数, T为环境温度. 在FMTO系统中, 力矩与磁信号间存在转化关系${{\boldsymbol{\tau}}}={{\boldsymbol{\mu}}} \times {{\boldsymbol{B}}}$, 故热噪声对磁噪声的贡献为

$$\begin{aligned} S_{B_{{\mathrm{th}}}}=S_{\tau_{{\mathrm{th}}}}/\mu^{2}, \end{aligned}$$

(9)

式中$\mu$为振子磁矩, $S_{B_{{\mathrm{th}}}}$为由热噪贡献的磁噪声功率谱.

基础噪声之一的量子测量源自测不准原理, 测量过程会反馈给系统一定的反作用力, 从而引入额外的测量噪声. 若振子偏转角测量噪声功率谱为$S_{\theta}$, 则测量过程反馈给力学系统的反作用力矩功率谱(back-action torque)为$S_{\tau_{{\mathrm{BA}}}}$, 测量噪声与反作用力矩二者的功率谱应满足测量噪声与反作用力矩间满足海森伯测不准原理:

$$\begin{aligned} S_{\theta}S_{\tau_{{\mathrm{BA}}}}\geqslant\hbar^{2}. \end{aligned}$$

(10)

注意到偏转角$\theta$与力矩${{\boldsymbol{\tau}}}$通过(4)式相联系, 可将测量噪声$S_{\theta}$转化为等效的总力矩噪声$S_{\tau_{\theta}}$:

$$\begin{aligned} S_{\tau_{\theta}}=S_{\theta}I^{2}\left\vert \chi\left( \omega\right) \right\vert ^{-2}. \end{aligned}$$

(11)

测量过程引入总力矩噪声包括$S_{\tau_{\theta}}$和$S_{\tau_{{\mathrm{BA}}}}$两部分:

$$\begin{aligned} S_{\tau_{{\mathrm{m}}}}=S_{\tau_{\theta}} + S_{\tau_{{\mathrm{BA}}}}\geqslant2\hbar I\left\vert \chi\left( \omega\right) \right\vert ^{-1}, \end{aligned}$$

(12)

力矩噪声$S_{\tau_{{\mathrm{m}}}}$在$S_{\tau_{{\mathrm{BA}}}}=S_{\theta}I^{2}\left\vert \chi\left(\omega\right) \right\vert ^{-2}$处的最小值$S_{\tau_{{\mathrm{SQL}}}}$被命名为标准量子极限, 通过(4)式将力矩噪声转化为磁噪声, 可得磁噪声的标准量子极限为^[30]

$$\begin{split} S_{B_{{\mathrm{SQL}}}}=\;&2\hbar I\left\vert \chi\left( \omega\right) \right\vert^{-1}\mu^{-2}\\ \approx\;&\left\{\begin{aligned} & 2\hbar I\omega_{0}^{2}\mu^{-2}, && \omega\ll\omega_{0},\\ & 2\hbar I\omega_{0}^{2}\mu^{-2}Q^{-1}, && \omega\approx\omega_{0}. \end{aligned} \right. \end{split}$$

(13)

综上所述, 包含热噪声、量子测量噪声的基础噪声的等效磁噪声功率谱为

$$\begin{aligned} S_{B_{{\mathrm{tot}}}}=S_{B_{{\mathrm{th}}}} + S_{B_{{\mathrm{SQL}}}}. \end{aligned}$$

(14)

将(14)式代入(7)式即可求得系统的磁灵敏度. 上述关于热噪声、量子噪声和灵敏度的理论分析并非仅局限于FMTO系统, 而是对基于阻尼谐振子模型(铁磁-力学系统)的磁传感器均有效.

在更详细的讨论噪声对磁传感器性能的影响前有必要进一步明晰灵敏度和噪声本底的差别: 如(7)式所述, 若取信噪比大于等于1作为磁场可分辨的标准, 则只要信号高于噪声即意味着信号是可分辨的, 故此时传感器灵敏度等于其噪声本底, 二者在数值和表达式上完全一致. 在实际讨论中非常容易混淆灵敏度与噪声本底, 引起进一步的误解, 如降低噪声本底意味着数值的减小, 而灵敏度降低则意味着数值的增大. 在接下来的讨论中热灵敏度、SQL灵敏度、ERL灵敏度这些概念用于表述仅考虑系统热噪声、SQL噪声、ERL噪声下的测量灵敏度 ¹.

首先分析ERL对磁灵敏度的限制, 并将其与FMTO系统在量子测量噪声和热噪声下的磁灵敏度进行比较. ERL将磁灵敏度与传感器体积进行关联, 指出磁传感器的磁噪声功率谱满足$S_{{\mathrm{B}}}V\geqslant 2\mu_{0}\hbar$, 故ERL给出的磁灵敏度上限为

$$\begin{aligned} \eta_{B_{{\mathrm{ERL}}}} =\sqrt{3/2}\pi^{-1/2}\hbar^{1/2}\mu_{0}^{1/2}R^{-3/2}. \end{aligned}$$

(15)

由(15)式可知, ERL灵敏度仅与传感器尺寸相关, 即磁灵敏度正比于$R^{-3/2}$, 随振子半径的增加而提高.

进一步考虑由${\rm{NdFeB}}$材料制成的球形磁体, 以超导磁悬浮、抗磁悬浮等机制悬浮在均匀磁场中构成FMTO系统, 则FMTO系统的力学参数如下:

$$\begin{split} I & =2mR^{2}/5=8\pi\rho_{{\mathrm{m}}}R^{5}/15, \\ k & =\mu B_{{\mathrm{Bias}}}=4\pi\rho_{{\mathrm{e}}}\mu_{{\mathrm{B}}}R^{3}B_{{\mathrm{Bias}}}/3, \\ \omega_{0} & =\sqrt{5/2}\left( \rho_{{\mathrm{e}}}\mu_{{\mathrm{B}}}\right) ^{1/2}B_{{\mathrm{Bias}}} ^{1/2}\rho_{{\mathrm{m}}}^{-1/2}R^{-1}, \end{split}$$

(16)

式中$\rho_{{\mathrm{m}}}=7430\ {\rm{kg/m^{3}}}$为NdFeB材料的质量密度, $\rho_{{\mathrm{e}}}=6\times10^{28}\ {\rm{m^{-3}}}$为NdFeB关联电子自旋密度 ². (16)式指出FMTO共振频率与半径成反比且正比于偏置磁场的1/2次方, 这意味着宏观FMTO系统尤其适合低频弱磁信号的精密测量.

FMTO系统在标准量子极限下的灵敏度由(7)式和(13)式给出, 代入(16)式中的FMTO参数后可得

$$\begin{split} &\eta_{B_{{\mathrm{SQL}}}}=\\ &\left\{ \begin{array}{ll} \sqrt{\dfrac{3\hbar}{2\pi}}\left( \rho_{{\mathrm{e}}}\mu_{{\mathrm{B}}}\right) ^{-\tfrac12}B_{{\mathrm{Bias}}} ^{\tfrac12}R^{-\tfrac32}, & \omega \ll \omega_{0}, \\ \sqrt{\dfrac{3\hbar}{2\pi}}\left( \rho_{{\mathrm{e}}}\mu_{{\mathrm{B}}}\right) ^{-\tfrac12}B_{{\mathrm{Bias}}} ^{\tfrac12}R^{-\tfrac32}Q^{-\tfrac12}, & \omega \approx \omega_{0}. \end{array} \right. \end{split}$$

(17)

标准量子极限下的灵敏度$\eta_{{\mathrm{SQL}}}$正比于$B^{1/2}_{{\mathrm{Bias}}}R^{-3/2}$, 这意味着增加振子半径、减小外磁场均有助于减小量子测量噪声提高灵敏度. 由于量子测量噪声由反作用力矩与测量噪声构成, 故标准量子极限下的磁灵敏度依赖频率, 且其灵敏度在$\omega\approx\omega_{0}$的共振情况下提升$Q^{1/2}$倍.

热噪声对半径为R、处于偏置磁场$B_{{\mathrm{Bias}}}$中的FMTO磁灵敏度的影响由(7)式与(9)式给出:

$$\begin{aligned} \eta_{B_{{\mathrm{th}}}}=C_{0}\left( k_{{\mathrm{B}}}T\right) ^{1/2}\left( \rho_{{\mathrm{e}}}\mu _{{\mathrm{B}}}\right) ^{-3/4}\rho_{{\mathrm{m}}}^{1/4}Q^{-1/2}B_{{\mathrm{Bias}}}^{1/4}R^{-1}, \end{aligned}$$

(18)

式中常数$C_{0}=\left(2/5\right)^{1/4}\left(3/\pi\right)^{1/2}$. 由热噪声引入的涨落力矩为高斯白噪声, 其功率谱均匀分布在各频段, 故热噪声影响下的磁灵敏度与测量频率无关.

热噪声对磁灵敏度的影响与测量频率无关, 但量子测量噪声、测量噪声、振动噪声等在共振测量时对磁灵敏度影响最小. 在实际的科研和工程应用中, 通常使磁传感器工作在谐振情况下, 故磁灵敏度与共振频率$\omega_{0}$的关系非常重要, 由(16)式与(18)式可得

$$\begin{split} \eta_{B_{{\mathrm{th}}}}=\;& C_{1}\left( k_{{\mathrm{B}}}T\right) ^{1/2}\left( \rho_{{\mathrm{e}}}\mu _{{\mathrm{B}}}\right) ^{-5/4}\rho_{{\mathrm{m}}}^{3/4} \\ & \times Q^{-1/2}B_{{\mathrm{Bias}}}^{-1/4}\omega_{0}, \end{split}$$

(19)

式中常数$C_{1}=\left(2/5\right)^{3/4}\left(3/\pi\right) ^{1/2}$. FMTO共振频率$\omega_{0}$由偏置磁场$B_{{\mathrm{Bias}}}$和振子半径R决定, 故(18)式与(19)式可通过(16)式中$\omega_{0}$的表达式互相转换, 两者分别描述了FMTO在热噪声影响下的磁灵敏度与偏置磁场和共振频率的关系. (18)式指出增大半径或减小偏置磁场均有助于提高磁灵敏度, (19)式则表明磁灵敏度随共振频率降低而提高, 故FMTO尤为适合低频磁场的精密测量.

应注意到(18)式仅代表处于磁场$B_{{\mathrm{Bias}}}$中FMTO系统的磁灵敏度, 并非所有FMTO系统的磁灵敏度极限. 根据(18)式, 减小偏置磁场$B_{{\mathrm{Bias}}}$即可进一步提高磁灵敏度, 但这种方法面临技术和原理的双重限制: 技术挑战来自磁噪声、$1/f$噪声(减小偏置磁场将降低FMTO频率, 则$1/f$噪声成为主要噪声)等; 原理上的限制源于FMTO要求振子刚体角动量远大于其自旋角动量, 这在实质上限制了偏置磁场的下限. 偏置磁场取下限时FMTO所达到的磁灵敏度即为所有FMTO系统的磁灵敏度极限.

为研究所有FMTO系统热噪声下的磁灵敏度极限, 定义$\varepsilon_{{\mathrm{r}}}\equiv L/S$为刚体角动量与自旋角动量之比, 基于(2)式有

$$\begin{aligned} B_{{\mathrm{Bias}}}=\frac{5\varepsilon_{{\mathrm{r}}}^{2}\rho_{{\mathrm{e}}}\hbar}{4\rho_{{\mathrm{m}}}R^{2}\gamma_{0} }. \end{aligned}$$

(20)

将磁偏置磁场代入(18)式, 可得FMTO所能达到的磁灵敏度:

$$\begin{split} \eta_{B_{{\mathrm{th}}}}=\;&5^{1/4}C_{0}\varepsilon_{{\mathrm{r}}}^{1/2}\left( k_{{\mathrm{B}}}T\right) ^{1/2} \left( \rho_{{\mathrm{e}}}\mu_{{\mathrm{B}}}\right) ^{-1/2} \\ &\times \gamma_{0}^{-1/2}Q^{-1/2}R^{-3/2}, \end{split}$$

(21)

FMTO要求$\varepsilon_{{\mathrm{r}}} = L/S\gg1$, 故$\varepsilon_{{\mathrm{r}}}$的下限给出FMTO系统的极限磁灵敏度, 通常要求刚体角动量比自旋角动量大1—2个量级以上, 即$\varepsilon_{{\mathrm{r}}} > 10$.

图3选择半径$R=30\; {\text{μm}}$的钕铁硼(NdFeB)作为铁磁振子, 通过悬浮系统消除夹持耗散使其品质因子$Q = 10^{7}$^[17], 图3(a)和图3(b)分别为FMTO各种磁噪声本底的频率分布, 以及铁磁半径对其磁噪声本底的影响.

图3(a)中偏置磁场$B_{{\mathrm{Bias}}}=1\;\text{μ} {\rm{T}}$, FMTO共振频率$f_{0}=73.15\ {\rm{Hz}}$, 则刚体角动量比自旋角动量高两个量级($L/S\approx382$), 振子扭摆模式的运动占绝对主导. 环境温度分别取$50\ {\rm{mK}}$和$4.2\ {\rm{K}}$, 则分析结果显示: 亚共振情况下的SQL与$50\ {\rm{mK}}$温度下的热噪声较为接近; 共振情况下SQL噪声比$4.2\ {\rm{K}}$, $50\ {\rm{mK}}$温度下的热噪声分别低3和4个量级. 热噪声和SQL噪声均远低于ERL噪声, 低温$4.2\ {\rm{K}}$, $50\ {\rm{mK}}$下的热噪声比ERL低$2—4$个量级. 在实际测量中, SQL相较热噪声几乎可忽略, FMTO系统的磁灵敏度主要取决于热噪声.

图3(b)重点研究不同FMTO噪声与半径的关系: 红色实线代表(21)式中FMTO热噪声极限(采用$L/S\geqslant 10$作为FMTO的临界条件), 粉色阴影区域为FMTO所能达到的磁灵敏度范围, 粉色实线代表(18)式中特定偏置磁场(见右下角坐标)中的FMTO磁灵敏度; FMTO在$4.2\ {\rm{K}}$时热噪声极限比ERL低3个量级, 若环境温度下降至$50\ {\rm{mK}}$则其热噪声极限比ERL低4个量级. 需要强调的是红实线仅代表FMTO原理上的热噪声极限, 实验中在半径较大时很难达到该极限, 如对厘米级别的FMTO系统, 欲达到其热噪声极限则要求偏置磁场稳定在${\rm{fT}}$量级, 这在地表磁噪声环境和现有磁屏蔽技术下是几乎不可能的. 地表环境中配合良好磁屏蔽, 可将剩磁下降至${\rm{nT}}$以下, 此时根据图3(b)中粉色实线, FMTO系统仍可在米以下尺寸领先ERL, 这充分验证了铁磁-力学系统在磁场精密测量应用中的巨大潜力. 噪声与半径的幂律具有非常重要的意义, 根据(15)式、(17)式、(21)式可知FMTO系统的SQL (橙色实线), ERL, 热噪声极限均正比于$R^{-3/2}$, 但保持偏置磁场不变情况下FMTO的热噪声比于$R^{-1}$.

FMTO超越ERL主要归功于以下因素: 1)铁磁材料极高的自旋密度: 根据(18)式和(17)式热噪和SQL的磁噪声本底分别正比于$\rho_{{\mathrm{e}}}^{-3/4}$和$\rho_{{\mathrm{e}}}^{-1/2}$, 故铁磁材料极高的自旋密度能有效克服噪声提高灵敏度; 2)自旋-晶格相互作用^[31]: 铁磁内部所有关联电子自旋均平行于磁矩方向, 故可通过自旋-晶格相互作用快速平均掉热噪声和量子测量噪声, 即作用于单个电子的噪声会通过自旋-晶格相互作用由整个铁磁体承担, 这一机制使其能快速平均单自旋的噪声从而降低磁噪声提高灵敏度; 3)悬浮系统极低的耗散^[17]: 对热噪声, 涨落耗散定理指出Q的提高能有效降低热涨落力矩功率谱; 对测量噪声, 力学系统对信号的响应在共振情况下放大Q倍, 测量噪声的影响被降低$Q^{1/2}$倍. 综上, 减小耗散提高品质因子Q能显著降低热噪和量子测量噪声的影响; 悬浮力学系统在消除夹持耗散后能够取得极高的Q值, 据报道基于超导磁悬浮的铁磁能获得高达$10^{6}—10^{7}$的品质因子, 这也是FMTO系统能够超越ERL的重要原因.

4.   新相互作用探测

超高灵敏度磁场测量最有前途的应用场景之一是通过对赝磁场的探测来探索新相互作用. 若在极化电子自旋和移动的非极化核子之间由未知玻色子传递自旋和速度相关的新相互作用^[32], 则电子与核子间的势能可由下式描述 ³:

$$\begin{split} V_{{\mathrm{int}}}\left( {{\boldsymbol{r}}}\right) =\;&-f^{4 + 5}\frac{\hbar^{2}}{8\pi m_{{\mathrm{e}}}c}\left[{{\boldsymbol{\sigma}}}\cdot\left( {{\boldsymbol{v}}}\times\hat{{\boldsymbol{r}}}\right) \right]\\ &\times\left( \frac{1}{\lambda r} + \frac{1}{r^{2}}\right) {\mathrm{e}}^{-r/\lambda}, \end{split}$$

(22)

其中$m_{{\mathrm{e}}}$和c分别为电子质量和真空光速, ${{\boldsymbol{\sigma}}}$为泡利矢量算符, ${{\boldsymbol{v}}}$是自旋电子与核子间的相对速度, $\hat{{\boldsymbol{r}}}={{\boldsymbol{r}}}/r$则代表电子与核子间的单位矢量, ${{\boldsymbol{r}}}$是电子与核子间的相对位移矢量, $\lambda=\hbar/m_{{\mathrm{b}}}c$是质量为$m_{{\mathrm{b}}}$的未知玻色子的康普顿波长, $f^{4 + 5}=g_{{\mathrm{s}}}^{{\mathrm{e}}}g_{{\mathrm{s}}}^{{\mathrm{N}}}$是无量纲的耦合常数.

新相互作用势能项$V_{{\mathrm{int}}}\left({{\boldsymbol{r}}}\right)$可看作电子磁矩与赝磁场的相互作用——$V_{{\mathrm{int}}}\left({{\boldsymbol{r}}}\right) ={{\boldsymbol{\mu}}}_{{\mathrm{B}}}\cdot{{\boldsymbol{B}}}^{{\mathrm{pse}}}= \gamma_{0}\hbar{{\boldsymbol{\sigma}}}\cdot {{\boldsymbol{B}}}^{{\mathrm{pse}}}/2$, 故电子所受新相互作用等价为运动核子产生赝磁场${{\boldsymbol{B}}}^{{\mathrm{pse}}}$作用于电子. 有限体积核子以速度${{\boldsymbol{v}}}$运动时所产生的赝磁场为^[35]

$${{\boldsymbol{B}}}^{{\mathrm{pse}}} = f^{4+5} C\int_{V}{\mathrm{d}} V \bigg[ ({{\boldsymbol{v}}}\times{{\boldsymbol{r}}}) \Big( \frac{1}{\lambda r} + \frac{1}{r^{2}}\Big) {\mathrm{e}}^{-r/\lambda}\bigg],$$

(23)

式中$C=\hbar\rho_{{\mathrm{N}}}/\left(4\pi m_{{\mathrm{e}}}c\gamma_{0}\right)$, $\rho_{{\mathrm{N}}}$为核子数密度.

新相互作用探测的实验装置图如图4(a)所示. 立体角为$\varTheta$的球壳状BGO晶体作为核子源, 其内外表面半径分别为$l_{0},\; l_{1}$, 核子以振幅$A_{n}$和频率$\omega_{n}$垂直纸面振动, 产生竖直方向的赝磁场, 磁传感器放置于球壳状核子源的球心位置用于探测赝磁场. 假设振子半径和振幅远小于$l_{0}$, 即$\varepsilon_{A} = A_{n}/l_{0}\ll 1, \varepsilon_{R}=R/l_{0}\ll 1$, 则此时核子源的速度振幅为$v_{n}= \omega_{n}A_{n}=\varepsilon_{A}\omega_{n}l_{0}$. 积分(23)式可得其产生赝磁场:

$$\begin{aligned} {{\boldsymbol{B}}}^{{\mathrm{pse}}}=\varepsilon_{A}\varTheta CC_{\lambda}\left( l_{0}\right)f^{4 + 5}\omega_{n}\lambda^{2}, \end{aligned}$$

(24)

式中$C_{\lambda}(l_{0})=(l_{0}^{2}/\lambda^{2} + 2 l_{0}/\lambda){\mathrm{e}}^{-l_{0}/\lambda}$是关于$\lambda, \;l_{0}$的无量纲函数. 注意分析发现球壳状核子源中距离内表面在$7\lambda$以上部分对赝磁场的贡献仅在千分之一量级, 这意味着满足$l_{1}-l_{0}>7\lambda$的条件时可将积分上限$l_{1}$近似为正无穷, 计算中使用了这一条件.

若采用灵敏度为$\eta_{{{B}}}\left(\omega\right)$的磁传感器对赝磁场进行持续时间为$T_{{\mathrm{mea}}}$的测量, 可给出耦合常数$f^{4 + 5}$精度为

$$\Delta f^{4 + 5} = \frac{\Delta B}{\left\vert \partial B^{{\mathrm{pse}}}/\partial f^{4 + 5}\right\vert } = \frac{\eta_{{{B}}}\left( \omega\right) T_{{\mathrm{mea}}}^{-1/2} }{\varepsilon_{A}\varTheta CC_{\lambda} (l_{0}) \omega_{n} \lambda^{2}},$$

(25)

$\Delta B=\eta_{{{B}}}(f)T_{{\mathrm{mea}}}^{-1/2}$为磁传感器的最小可分辨磁场, 将磁传感器的灵敏度代入(25)式即可得到新相互作用的探测精度.

若使用FMTO磁传感器探测新相互作用, 则可通过控制偏置磁场$B_{{\mathrm{Bias}}}$调节FMTO共振频率等于核子振动频率$(\omega_{0}=\omega_{n})$, 使FMTO系统工作在共振情况下以获得最佳灵敏度, 将FMTO和ERL磁传感器的灵敏度代入(25)式可得

$$\begin{aligned} \Delta f_{{\mathrm{FMTO}}}^{4 + 5} & =\frac{\left( 5/2\right) ^{1/4}C_{1}\left(k_{{\mathrm{B}}}T\right) ^{1/2}\rho_{{\mathrm{m}}}^{1/2}}{\varepsilon_{A}\varTheta CC_{\lambda}\left(l_{0}\right) \rho_{{\mathrm{e}}}\mu_{{\mathrm{B}}}T_{{\mathrm{mea}}}^{1/2}Q^{1/2}R^{1/2}\lambda^{2}\omega_{n}^{1/2}}, \end{aligned}$$

(26)

$$\begin{aligned} \Delta f_{{\mathrm{ERL}}}^{4 + 5} & =\frac{\sqrt{3\hbar\mu_{0}/2\pi}}{\varepsilon_{A}\varTheta CC_{\lambda}\left( l_{0}\right) T_{{\mathrm{mea}}}^{1/2}R^{3/2}\lambda^{2}\omega_{n}}. \end{aligned}$$

(27)

(24)式表明传感器与核子源间距$l_{0}$对赝磁场的场强至关重要, 优化此参数$l_{0}$能够有效提高探测精度. 分析赝磁场表达式(24)式中$C_{\lambda}(l_{0})$的性质可发现, $l_{0}=\sqrt{2}\lambda$处赝磁场场强最大, $l_{0}\gg\sqrt{2}\lambda$时赝磁场随间距呈指数衰减. 这意味着探测康普顿波长为$\lambda$的未知玻色子传递的新相互作用时, 设置间距$l_{0}=\sqrt{2}\lambda$时能获得最佳探测精度. 为进一步系统分析FMTO和ERL在新相互作用领域的潜力 ⁴, 可以假设实验条件总是满足$l_{0}=\sqrt{2}\lambda$, 此时$R= l_{0}\varepsilon_{R}=\sqrt{2}\lambda\varepsilon_{R}$且$C_{\lambda}\left(l_{0}\right)=2\left(1 + \sqrt{2}\right) {\mathrm{e}}^{-\sqrt{2}}\approx1.174$, 将其代入到(26)式和(27)式即可得到参数优化后的探测结果.

图4(b)展示了ERL和FMTO磁传感器探测新相互作用的精度, 以及与现有工作的对比. 黑色实线${\rm{I}} —{\rm{V}}$为现有实验的测量结果, ${\rm{VI}}— {\rm{IX}}$为FMTO系统和ERL系统的理论结果(参数$\varepsilon_{R}= 1/5, \varepsilon_{A}= 1/5,\; T=50\; {\rm{mK}}$, 核子源振动频率为$10\ {\rm{Hz}}$即$\omega_{n}= 20\pi \ {\rm{rad/s}}$), 红色实线VI和青色实线VII对应$30\ {\text{μm}}$半径的FMTO和ERL磁传感器在固定间距$l_{0}= 150\ {\text{μm}}$时的探测结果(见(26)式); 红色虚线VIII和青色虚线IX为FMTO和ERL在最优情况下所能取得的探测结果(核子源与磁传感器间距$l_{0}$总是取$l_{0}=\sqrt{2}\lambda$的最优值以保证在康普顿波长为$\lambda$时取得最优结果, 见(27)式). 图4(b)显示, 已有实验的精度距离ERL仍有较大差距, 仅在微米和毫米量级逼近ERL极限.

对比图4(b)中曲线VIII和IX可知, FMTO磁传感器在新相互作用探测精度上领先ERL高达7个量级, 但ERL在康普顿波长$\lambda$接近$0.1\ {\rm{m}}$时将超越FMTO系统. 然而目前实验在$\lambda$较小时探测精度较差, 故FMTO针对$\lambda$较小时的新相互作用探测拥有极大潜力, (26)式指出$\Delta f_{{\mathrm{FMTO}}}^{4 + 5}\propto\rho_{{\mathrm{e}}}^{-5/4}$, 即铁磁材料极高的关联电子自旋密度是FMTO超越ERL的重要因素.

5.   总结与展望

本文提出基于悬浮铁磁扭摆振子的磁传感器方案, 深入分析了该系统的动力学响应、基础噪声和探测性能. 研究显示: 铁磁材料极高的自旋密度及其较强的自旋-晶格相互作用, 能够帮助FMTO降低磁噪声并快速平均掉热噪声和量子测量噪声, 悬浮系统则能够消除FMTO的夹持耗散实现极高的品质因子, 使FMTO能够进一步谐振放大信号, 有效克服测量噪声和环境噪声的影响, 最终达到超越ERL的磁灵敏度. 此外, FMTO磁传感器在探测新型相互作用方面展现出巨大潜力, 其能够在探测精度上领先ERL最高5个量级, 领先现有实验2—9个量级, 这对于未来的科学研究和技术应用具有重要意义, 不仅推动了磁传感器技术的发展, 也为新相互作用的探测提供了新的思路和工具.