Atomic and Spin-Based Magnetometry

Magnetometry is the measurement of magnetic flux density $\mathbf B$, its magnitude, a selected component, a spatial derivative, or a spectrum. An atomic magnetometer infers that field from the Zeeman evolution of prepared spins, detected through an optical or electrical signal.

In the simplest picture, a spin with gyromagnetic ratio $\gamma$ precesses at

$$\omega_L = \left| \gamma \right|B.$$

That relation is powerful because frequency and time can be calibrated accurately. It is not the complete measurement equation. Real instruments prepare a distribution of magnetic sublevels, average a nonuniform field over a finite vapor cell or solid-state sensing volume, perturb the spins with pump and probe light, and infer a field through a finite-bandwidth estimator. Nonlinear Zeeman structure, heading-dependent line weights, light shifts, spin-exchange shifts, coil geometry, shield noise, and cross-talk can all bias the result.

There is no universally best magnetometer. A near-zero-field spin-exchange relaxation-free instrument can provide exceptional low-frequency sensitivity but limited dynamic range. A finite-field scalar instrument can track the magnitude of Earth’s field but suffers heading error. A radio-frequency magnetometer is narrowband by design. A solid-state spin sensor can trade ensemble sensitivity for proximity and spatial resolution. The correct architecture follows from the measurand, frequency band, field range, spatial scale, environment, and calibration requirement.

Helpful background. Zeeman structure supplies the atomic energy splittings, and Larmor precession derives the spin motion. This page develops vapor-cell preparation, readout, operating regimes, and calibration. Quantum magnetometry compares information limits and quantum-enhanced protocols across platforms.

Magnetic Fields and Spin Response

Magnetic flux density, not an unspecified field

Atomic magnetic moments couple to magnetic flux density through

$$H_Z = - \boldsymbol\mu \mathbin{\cdot} \mathbf B.$$

The SI unit of $B$ is the tesla:

$$1\ \mathrm G = 10^{-4}\ \mathrm T.$$

Useful submultiples are

$$\begin{aligned} 1\ \mathrm{nT} &= 10^{-9}\ \mathrm T, \\ 1\ \mathrm{pT} &= 10^{-12}\ \mathrm T, \\ 1\ \mathrm{fT} &= 10^{-15}\ \mathrm T. \end{aligned}$$

Magnetic field strength $\mathbf H$ has SI unit $\mathrm{A\,m^{-1}}$ and is not interchangeable with $\mathbf B$ inside magnetized matter. A magnetometer report that says only “magnetic field” should make clear which quantity is intended.

Scalar, vector, gradient, and spectral measurands

A scalar magnetometer aims to estimate

$$B = \left| \mathbf B \right|.$$

A vector magnetometer estimates one or more components in a declared instrument or laboratory frame:

$$\mathbf B = \begin{pmatrix} B_x&B_y&B_z \end{pmatrix}^{\mathsf T}.$$

A gradiometer estimates a difference or derivative, for example

$$G_{zx} = \frac{\partial B_z}{\partial x}.$$

An AC or RF magnetometer may instead estimate a complex Fourier amplitude,

$$\widetilde B(\omega) = \left| \widetilde B(\omega) \right| e^{i\phi_B(\omega)},$$

within a specified bandwidth and phase reference.

These are not interchangeable outputs. A scalar sensor cannot generally recover field direction. A single vector channel does not determine $|\mathbf B|$. A gradiometer rejects only field components that are common under its spatial and transfer-function model.

Reference point and response-weighted field

An ensemble magnetometer occupies a finite cell. Pump intensity, polarization, atom density, diffusion, probe intensity, optical depth, and detection geometry define a weighting function $w(\mathbf r,t)$. For a linear component sensor with local sensitive axis $\widehat{\mathbf e}(\mathbf r,t)$, a useful idealization is

$$B_{\mathrm{eff}}(t) = \frac{ \int_V w(\mathbf r,t) \widehat{\mathbf e}(\mathbf r,t) \mathbin{\cdot} \mathbf B(\mathbf r,t) \,d^3r }{ \int_V w(\mathbf r,t) \,d^3r }.$$

For a fixed axis and a linear spatial field, the weighted centroid of the cell gives a convenient reference point. A more general field model need not associate $B_{\mathrm{eff}}$ with a unique point. In a uniform field the location is immaterial; in a gradient it depends on $w$. A scalar precession signal in a nonuniform field is more complicated because different atoms acquire different phases. The fitted frequency need not equal the simple average of $|\mathbf B|$.

Signed gyromagnetic ratio

Write

$$\boldsymbol\mu = \gamma\mathbf F$$

for an angular momentum $\mathbf F$. The signed Zeeman Hamiltonian is

$$H_Z = - \gamma \mathbf F \mathbin{\cdot} \mathbf B.$$

The sign of $\gamma$ determines energy ordering and precession sense. A positive spectral frequency is

$$\omega_L = \left| \gamma \right|B.$$

Keep these two conventions separate. Discarding the sign of $\gamma$ in a vector or phase-sensitive instrument can reverse a reconstructed component or feedback polarity.

Zeeman Transduction

Weak-field hyperfine response

Within a hyperfine manifold whose $F,m_F$ labels remain valid, the leading energy is

$$E_{F,m_F}(B) \simeq E_F(0) + g_F\mu_Bm_FB.$$

Define

$$\gamma_F = -\frac{g_F\mu_B}{\hbar}.$$

The minus sign follows from $\boldsymbol\mu=\gamma_F\mathbf F$ and $H_Z=-\boldsymbol\mu\cdot\mathbf B$, with $F_z=\hbar m_F$. Some spectroscopy texts instead call the coefficient $g_F\mu_B/\hbar$ the gyromagnetic ratio; that coefficient has the opposite sign to the magnetic-moment convention used here. Adjacent magnetic sublevels have positive angular-frequency separation

$$\omega_{m_F\to m_F+1} \simeq \left| \gamma_F \right|B.$$

For the $^{87}\mathrm{Rb}$ ground-state $F=2$ manifold,

$$\frac{ \left| \gamma_F \right| }{2\pi} \simeq 6.998\ \mathrm{Hz\,nT^{-1}}.$$

A field of $50\ \mu\mathrm T$ therefore corresponds to a frequency near $350\ \mathrm{kHz}$. The numerical conversion applies only after the isotope, hyperfine manifold, field regime, and gyromagnetic convention are specified.

Nonlinear Zeeman structure

At higher precision or field, write

$$E_{F,m_F}(B) = E_F(0) + g_F\mu_Bm_FB + \beta_{F,m_F}B^2 + \cdots.$$

The quadratic term makes adjacent Zeeman transitions inequivalent. A measured optical or magnetic-resonance signal is then a weighted superposition of several frequencies:

$$S(\omega) = \sum_{m_F} A_{m_F} L \left[ \omega-\omega_{m_F}(B) \right].$$

The amplitudes $A_{m_F}$ depend on optical pumping, polarization, beam direction, relaxation, and detection. Rotating the instrument relative to $\mathbf B$ changes those weights, so the fitted line center can change even when $|\mathbf B|$ is constant. This is a principal origin of heading error in scalar alkali-vapor magnetometers.

The Breit–Rabi formula gives the hyperfine crossover model. A precision magnetometer should use it or a validated diagonalization rather than extend one $g_F$ indefinitely.

Frequency does not guarantee zero bias

Frequency readout is attractive because

$$\widehat B = \frac{\widehat\omega_L} {\left|\gamma_{\mathrm{model}}\right|}.$$

The measured $\widehat\omega_L$ can nevertheless include:

• nonlinear Zeeman line pulling;
• spin-exchange and spin-destruction shifts;
• vector ac Stark shifts from pump and probe light;
• Bloch–Siegert shifts from counter-rotating drive components;
• magnetic gradients and diffusion;
• feedback or phase-delay offsets; and
• estimator bias from overlapping, asymmetric, or time-varying lines.

An atomic scalar magnetometer can have excellent long-term scale stability without being calibration-free at the uncertainty level of interest.

Optical Pumping

From photon angular momentum to spin polarization

Circularly polarized resonant light transfers angular momentum to atoms. Repeated absorption and spontaneous-emission cycles redistribute population among Zeeman and hyperfine sublevels. With suitable selection rules, atoms accumulate in a stretched or dark state that no longer absorbs the pump efficiently.

The simple picture requires several qualifications:

• spontaneous emission has several polarization and branching channels;
• off-resonant hyperfine levels can be excited;
• atoms can be pumped into the wrong ground hyperfine manifold;
• a repump laser may be required;
• buffer-gas collisions broaden optical transitions and alter diffusion;
• radiation trapping can reabsorb emitted photons; and
• strong pumping creates spatially varying polarization in an optically thick cell.

The detailed branching structure belongs to Atomic Selection Rules and Spontaneous Emission. Here optical pumping is treated as a preparation and relaxation process in the magnetometer forward model.

Orientation and alignment

An atomic ensemble can carry more structure than a polarization vector. Rank-one orientation distinguishes $+m_F$ from $-m_F$ and behaves like a directed spin polarization. Rank-two alignment describes an axis with no preferred direction. Circular pumping commonly creates orientation; linear pumping can create alignment.

In a linear Zeeman field, a spherical-tensor component with index $q$ acquires phase at $q$ times the signed Larmor rate. Orientation has $q=0,\pm1$; alignment also permits $q=\pm2$. Thus alignment can produce a $2\omega_L$ signal, but not every alignment signal oscillates at that frequency. The observed harmonic depends on preparation and probe geometry. Assigning every optical-rotation resonance to a single classical spin vector can give the wrong gyromagnetic factor or modulation harmonic.

A Bloch-rate model

Let $\mathbf P$ be a normalized spin-polarization vector and $\mathbf s$ the polarization toward which the pump drives it. A useful minimal model is

$$\frac{d\mathbf P}{dt} = \gamma \mathbf P \mathbin{\times} \mathbf B + R \left( \mathbf s-\mathbf P \right) - \Gamma_0\mathbf P,$$

where $R$ is the optical-pumping rate and $\Gamma_0$ collects relaxation without the pump. Define

$$\Gamma = R+\Gamma_0.$$

This equation suppresses hyperfine correlations, higher-rank polarization, diffusion, velocity classes, and nonlinear optical propagation. It remains valuable because it exposes the response, linewidth, and pump-broadening tradeoff.

Take $\mathbf s=s\widehat{\mathbf z}$ and $\mathbf B=B_x\widehat{\mathbf x}$. The steady state is

$$P_z = \frac{ Rs\Gamma }{ \Gamma^2+\gamma^2B_x^2 },$$

and

$$P_y = \frac{ Rs\gamma B_x }{ \Gamma^2+\gamma^2B_x^2 }.$$

$P_z$ is absorptive and even in $B_x$; $P_y$ is dispersive and odd. Near zero field,

$$P_y \simeq \frac{Rs\gamma}{\Gamma^2} B_x.$$

The half-width field scale is

$$B_{1/2} = \frac{\Gamma}{|\gamma|}.$$

Increasing $R$ raises the prepared polarization but also broadens the response. In this minimal model the zero-field slope is proportional to

$$\frac{R}{(R+\Gamma_0)^2},$$

which is maximized at $R=\Gamma_0$. Real optimum power also depends on probe noise, optical depth, spatial pumping, and technical fluctuations.

Sources of Spin Relaxation

The dark relaxation rate can contain:

$$\Gamma_0 = \Gamma_{\mathrm{wall}} + \Gamma_{\mathrm{diff}} + \Gamma_{\mathrm{SD}} + \Gamma_{\mathrm{grad}} + \Gamma_{\mathrm{other}}.$$

Important mechanisms include:

• wall collisions and imperfect anti-relaxation coatings;
• diffusion out of the pumped or probed volume;
• spin-destruction collisions;
• magnetic-field gradients sampled by moving atoms;
• alkali–alkali and alkali–buffer-gas collisions;
• radiation trapping;
• probe-induced pumping and absorption; and
• technical loss from atoms leaving the active region.

$T_1$, $T_2$, and $T_2^*$ need not be equal. A narrow magnetic resonance does not identify the mechanism without power, density, gradient, and geometry studies.

Optical and Electrical Readout

Faraday rotation

A linearly polarized probe can be decomposed into right- and left-circular components. If their refractive indices are $n_+$ and $n_-$, propagation through length $\ell$ rotates the polarization by

$$\theta = \frac{k}{2} \int_0^\ell \operatorname{Re}\!\left[ n_+(z)-n_-(z) \right]dz.$$

The sign assumes the chosen ordering of circular polarizations and viewing direction. If the indices are complex, their imaginary-part difference describes circular dichroism and induced ellipticity, not this rotation angle. Near an atomic transition, the circular birefringence depends on spin polarization and probe detuning. In a linearized model,

$$\theta(t) = \int_V \kappa(\mathbf r) P_{\mathrm{probe}}(\mathbf r,t) \,d^3r + \theta_0,$$

where $\kappa$ includes atom density, optical detuning, line strengths, path length, and spatial probe weighting.

A balanced polarimeter converts small rotation to a difference of detected photon counts. At its balanced operating point, the difference has variance $N$ and rotation slope $2N$, so for $N=\dot N_{\mathrm{ph}}\tau$ detected photons,

$$\delta\theta=\frac{1}{2\sqrt{\dot N_{\mathrm{ph}}\tau}}.$$

For a one-sided white power spectral density per Hz, a rectangular average of duration $\tau$ has variance $S_\theta/(2\tau)$. Hence

$$S_\theta^{1/2}=\frac{1}{\sqrt{2\dot N_{\mathrm{ph}}}}.$$

The averaging coefficient $\delta\theta\sqrt{\tau}$ is smaller by $\sqrt2$. Specifying which quantity is quoted avoids an artificial factor-of-$\sqrt2$ disagreement between sensitivity estimates.

Detector quantum efficiency, optical loss, imbalance, electronic current noise, residual intensity noise, and beam pointing raise the observed noise.

Absorption and fluorescence

Pump transmission can report the projection of $\mathbf P$ along the pump axis. Fluorescence can report state populations or driven resonance. These signals are often simpler optically but can be more sensitive to laser intensity, optical depth, background light, and radiation trapping.

The detector model must include the dependence of signal slope on atom number and optical power. Normalizing by total optical power removes some gain fluctuations but does not remove detuning-dependent optical pumping or detector nonlinearity.

Probe backaction and fictitious magnetic fields

Off-resonant light can shift magnetic sublevels through scalar, vector, and tensor ac Stark effects. The vector part behaves like a polarization-dependent fictitious magnetic field:

$$\mathbf B_{\mathrm{LS}} = \frac{ \boldsymbol\Omega_{\mathrm{LS}} }{\gamma},$$

Here $\boldsymbol\Omega_{\mathrm{LS}}$ is defined by the added term $d\mathbf P/dt=\mathbf P\times\boldsymbol\Omega_{\mathrm{LS}}$. If instead a rotation vector is defined through $\boldsymbol\Omega\times\mathbf P$, its sign is opposite. Residual ellipticity, detuning, and intensity drift can therefore create a magnetic-looking signal.

Increasing probe power lowers photon shot noise but increases pumping, power broadening, light shifts, and quantum backaction. A dark free-precession interval or pulsed probe can separate preparation from measurement, but introduces dead time and estimator requirements.

The minimal Bloch model explains two practical limits. A: with $b=\gamma B_x/\Gamma$ and $P_0=Rs/\Gamma$, the even longitudinal response is $P_z/P_0=1/(1+b^2)$ and the odd transverse response is $P_y/P_0=b/(1+b^2)$. Only the central part of the latter is linear. B: increasing pump rate eventually broadens the response faster than it improves polarization. The normalized small-field slope $4r/(1+r)^2$, where $r=R/\Gamma_0$, peaks at $R=\Gamma_0$. These curves neglect spatial transport, higher spin moments, and probe noise.

Magnetometer Architectures

Zero-field dispersive magnetometer

The steady-state $P_y(B_x)$ response above is the prototype of a near-zero-field vector channel. The instrument is operated near $B_x=B_y=B_z=0$ using magnetic shields, compensation coils, or feedback. A transverse component tips the pumped polarization and produces optical rotation.

Advantages include a direct odd response around zero and no need to track a high carrier frequency. Limitations include a narrow linear range, cross-axis coupling, sensitivity to coil offsets, and ambiguity if several components are large simultaneously. Three-axis operation generally uses orthogonal modulation fields, multiple optical geometries, or feedback that nulls each component.

Driven finite-field resonance

In an Mx-type magnetometer, circular light prepares polarization and an RF field drives transverse spin resonance. In a Bell–Bloom magnetometer, modulated pump light drives spin precession when the modulation frequency matches the Larmor frequency.

A linear complex response near resonance can be written

$$\chi(\omega) = \frac{ \chi_0 }{ \Gamma_2-i(\omega-\omega_L) }.$$

Its phase is

$$\phi(\omega) = \tan^{-1} \left[ \frac{ \omega-\omega_L }{\Gamma_2} \right]$$

up to the instrument’s sign and phase origin. A servo can adjust the drive frequency until the quadrature error vanishes, yielding

$$\widehat B = \frac{ \widehat\omega_{\mathrm{lock}} }{|\gamma|}.$$

The resonant architecture can operate at much larger fields than a zero-field sensor. It inherits RF coil calibration, Bloch–Siegert shifts, drive-phase delays, and line-center systematics.

Free-precession magnetometer

A pulsed instrument optically pumps the ensemble, turns the pump off, and records a free-induction signal:

$$y(t) = A e^{-t/T_2^*} \cos \left( \omega_Lt+\phi \right) + c + \epsilon(t).$$

The dark interval suppresses pump light shifts and permits frequency estimation from phase accumulation. The estimator must handle finite record length, colored noise, multicomponent Zeeman beats, phase transients, and cycle-to-cycle dead time. A high signal-to-noise sinusoid does not justify a single-frequency model when nonlinear Zeeman components are resolved.

RF atomic magnetometer

An RF magnetometer applies a bias field so an atomic Zeeman transition is resonant with the AC field of interest. The signal estimates an amplitude and phase in a narrow band:

$$\widetilde y(\omega) = H_{yB}(\omega) \widetilde B_{\mathrm{RF}}(\omega) + \widetilde n(\omega).$$

Tuning the bias field changes the center frequency. Narrow bandwidth rejects broadband noise but also distorts pulses or modulated signals outside the calibrated response. RF magnetometers are useful for low-field NMR and nuclear quadrupole resonance, but are not broadband DC vector sensors.

Scalar versus vector operation

Scalar operation derives $|\mathbf B|$ from a precession frequency and can be relatively insensitive to gain drift. Vector operation derives components from amplitudes, phases, or feedback currents and requires calibrated axes.

Scalar does not mean orientation independent. Optical pumping and nonlinear Zeeman structure create heading error. Vector does not mean all three components are independently observable. The response matrix must have full rank:

$$\mathbf y = \mathbf G\mathbf B + \mathbf b + \boldsymbol\epsilon.$$

The condition number of $\mathbf G$ determines how calibration and noise are amplified during inversion.

Differential and array operation

For two sensors,

$$\Delta y = y_2-y_1.$$

Let their gains be

$$G_{1,2} = \overline G \mp \frac{\delta G}{2},$$

and their fields

$$B_{1,2} = \overline B \mp \frac{\Delta B}{2}.$$

Then

$$\Delta y = \overline G\,\Delta B + \delta G\,\overline B + \Delta b.$$

Gain mismatch leaks the common field into the gradient channel. Frequency-dependent mismatch, different sensitive volumes, axis misalignment, and modulation-coil cross-talk also limit common-mode rejection.

For baseline vector $\mathbf L$ and a slowly varying component,

$$\Delta B_i \simeq \sum_j \frac{\partial B_i}{\partial x_j} L_j.$$

The effective baseline joins the response-weighted sensor locations, not necessarily the package centers.

Spin-Exchange Relaxation-Free Operation

Why spin exchange normally broadens

Alkali atoms collide and exchange electron-spin character at rate

$$R_{\mathrm{SE}} = n \left\langle \sigma_{\mathrm{SE}}v \right\rangle,$$

where $n$ is alkali density. Spin exchange largely conserves total angular momentum, but transfers atoms among hyperfine manifolds with different precession frequencies. At ordinary field and low polarization, this random switching dephases a coherent ensemble and broadens magnetic resonance.

Heating a cell increases atom number and optical depth, which could improve statistical sensitivity, but it also raises $R_{\mathrm{SE}}$. This collision limit historically prevented arbitrary improvement by density.

Motional narrowing at low field

In the spin-exchange relaxation-free, or SERF, regime,

$$R_{\mathrm{SE}} \gg \omega_L.$$

Each atom changes hyperfine character many times during one precession period. The ensemble evolves with an averaged precession rather than dephasing after every collision. At low polarization, a schematic spin-exchange relaxation rate is

$$\Gamma_{\mathrm{SE}} \sim C(I,P) \frac{ \omega_L^2 }{ R_{\mathrm{SE}} },$$

where $C(I,P)$ depends on nuclear spin, polarization, and the measured mode. This scaling, not a universal coefficient, is the important result. The spin-exchange contribution vanishes quadratically toward zero field.

Hyperfine correlations also slow the electron-spin response. Detailed SERF models introduce a nuclear slowing-down factor and a polarization-dependent effective gyromagnetic ratio. Using the bare electron gyromagnetic ratio in a SERF calibration is incorrect.

What SERF does not remove

SERF suppresses one relaxation mechanism. It does not eliminate:

• spin-destruction collisions;
• wall and diffusion relaxation;
• pump and probe power broadening;
• magnetic gradients;
• light shifts;
• radiation trapping;
• photon shot noise and spin-projection noise;
• Johnson magnetic noise from conducting shields and heaters; or
• drift and cross-talk in field-nulling coils.

The small-field condition also limits dynamic range. A large ambient field can take the sensor out of the SERF regime or rotate polarization beyond the linear response. High-sensitivity SERF measurements commonly require magnetic shielding or active cancellation. This makes them excellent for some biomagnetic and fundamental measurements but does not make them automatic replacements for geomagnetic scalar instruments.

Density and temperature tradeoffs

Increasing temperature raises alkali density and $R_{\mathrm{SE}}$, helping reach the SERF condition. It can also:

• raise spin-destruction rate;
• increase optical depth and pump nonuniformity;
• increase required heater power and thermal gradients;
• bring conductive materials and their Johnson noise closer to the cell;
• accelerate coating degradation; and
• make close placement near biological or chemical samples harder.

“Higher density is better” is therefore no more reliable than “longer interrogation is better” in a clock.

Sensitivity, Bandwidth, and Range

Noise-equivalent magnetic field

Let the measured output have transfer function $H_{yB}(\omega)$ from the declared field component to output. The amplitude spectral density of equivalent input magnetic noise is

$$S_B^{1/2}(\omega) = \frac{ S_y^{1/2}(\omega) }{ \left| H_{yB}(\omega) \right| }.$$

A quoted value such as $10\ \mathrm{fT\,Hz^{-1/2}}$ is incomplete without frequency, bandwidth, operating point, spatial configuration, and whether environmental field noise has been subtracted. A gradiometric noise floor is not the same quantity as a single-channel absolute noise floor.

First-order response and bandwidth

A simple spin response is low-pass. For a complex input proportional to $e^{-i\omega t}$,

$$H_{yB}(\omega) = \frac{ G_0 }{ 1-i\omega/\Gamma_B }.$$

The $-3\ \mathrm{dB}$ bandwidth is

$$f_{3\mathrm{dB}} = \frac{\Gamma_B}{2\pi}.$$

Increasing relaxation or feedback can increase bandwidth, but often reduces open-loop gain or increases noise. Feedback can extend linear range while moving calibration from the atomic slope to coil gain, electronics, delay, and loop stability.

For a transient field, the relevant question is not only the flat-band noise floor. Phase delay and amplitude roll-off must be deconvolved or included in the signal model.

Spin-projection-noise baseline

For $N$ independent polarized spins with coherent interrogation time $T_2$, an order-of-magnitude continuous sensitivity is

$$\eta_B^{(\mathrm{SPN})} \equiv \delta B\sqrt{\tau} \sim \frac{ 1 }{ |\gamma| \sqrt{NT_2} }.$$

Order-one factors depend on spin, polarization, duty cycle, measured quadrature, and estimator. Spin exchange creates correlations that can modify simple independent-spin spectra. Entanglement or squeezing can improve the projection-noise term, but cannot remove a light-shift drift or coil-calibration error.

Photon shot noise and backaction

For optical rotation with slope $d\theta/dB$, the equivalent input one-sided amplitude spectral density is

$$\bigl[S_B^{(\mathrm{PSN})}\bigr]^{1/2} =\frac{S_\theta^{1/2}}{|d\theta/dB|}.$$

To compare with the averaging coefficient $\eta_B=\delta B\sqrt\tau$ defined above, assume white noise over the bandwidth of a rectangular average. Its response is $H_\tau(f)=\sin(\pi f\tau)/(\pi f\tau)$, so

$$\operatorname{Var}(\overline B_\tau) =\int_0^\infty S_B(f)|H_\tau(f)|^2\,df =\frac{S_B}{2\tau}, \qquad \eta_B^{(\mathrm{PSN})} =\frac{S_\theta^{1/2}}{\sqrt2\,|d\theta/dB|}.$$

The spectral noise level is therefore $\sqrt2$ times the averaging coefficient in this convention. Their units are dimensionally equivalent, so units alone do not identify which number is quoted. For colored noise or a different averaging filter, use the integral with its actual spectrum and response instead of this white-noise conversion.

More probe photons reduce $S_\theta^{1/2}$ but perturb the atoms more strongly. The optimum balances photon shot noise, spin noise, power broadening, and light shifts under the actual optical-depth model.

Technical noise can enter through:

• laser intensity, frequency, polarization, and pointing;
• photodetector electronics and digitization;
• heater current and modulation;
• coil-current noise;
• mechanical motion through magnetic gradients;
• shield vibration and remanence;
• nearby electronics and mains harmonics; and
• data-acquisition clock jitter.

A noise budget should measure transfer coefficients rather than merely list possible sources.

Dynamic range and linearity

For the zero-field dispersive model, the characteristic linear scale is

$$B_{1/2} = \frac{\Gamma}{|\gamma|}.$$

The exact acceptable range depends on the allowed nonlinearity. If the full response is

$$P_y(B) = \frac{ A B }{ B_{1/2}^2+B^2 },$$

then the ratio to its linear approximation is

$$\frac{ P_y(B) }{ P_y^{(\mathrm{lin})}(B) } = \frac{ 1 }{ 1+(B/B_{1/2})^2 }.$$

Keeping gain error below $1\%$ requires approximately

$$|B| \lesssim 0.10 B_{1/2}.$$

Sensitivity, bandwidth, and linear dynamic range are coupled through $\Gamma$. Quoting only the best noise floor conceals this design tradeoff.

Stability and drift

Allan deviation or overlapping Allan deviation can characterize a stationary field estimate. The result combines sensor drift and real field variation unless a stable source or differential channel separates them. Pump detuning, cell temperature, coil offset, shield magnetization, and laser polarization can create long-term floors.

A flat long-term scalar frequency record does not prove vector-axis stability. A vector sensor can rotate physically while maintaining its component gain.

Calibration and Systematic Effects

Measurement equation

A component channel can be represented as

$$y(t) = y_0 + h \mathbin{*} \left[ B_{\mathrm m} + B_{\mathrm{coil}} + B_{\mathrm{LS}} + B_{\mathrm{contact}} + B_{\mathrm{other}} \right](t) + n(t),$$

where $h$ is the impulse response and $*$ denotes convolution. The target measurand $B_{\mathrm m}$ is accompanied by:

• applied calibration and feedback fields;
• pump and probe light shifts;
• contact or magnetization fields from nearby polarized matter; and
• other physical or instrumental nuisance fields.

The inverse estimator must use the same axis, sign, delay, and bandwidth conventions as the forward model.

Coil calibration

For a calibration coil,

$$B_{\mathrm{coil}} = C_I I,$$

where $C_I$ is the coil factor and $I$ the current. The standard uncertainty is

$$u^2(B_{\mathrm{coil}}) ={} I^2u^2(C_I) + C_I^2u^2(I) + 2C_II \operatorname{cov}(C_I,I).$$

$C_I$ depends on winding geometry, sensor location and orientation, nearby magnetic material, shield response, frequency, and current return path. Calculating an ideal Helmholtz factor is not enough when the coil is inside a high-permeability shield.

Useful calibration routes include:

• dimensional and current calibration of a characterized coil;
• comparison with an independently calibrated magnetometer;
• frequency calibration using a validated gyromagnetic model;
• mechanical rotation in a known field;
• injected AC fields across the measurement bandwidth; and
• reversal of current and sensor orientation.

The routes can share common inputs and should not be assumed independent.

Axis and response-matrix calibration

For a vector instrument,

$$\mathbf y = \mathbf G \left( \mathbf B+\mathbf B_0 \right) + \mathbf b.$$

The matrix $\mathbf G$ contains gains, nonorthogonality, and cross-axis response. Calibrate it by applying several noncoplanar fields whose amplitudes and directions are known. A least-squares fit should include coil-field covariance and residual background.

Sensor axes can vary with pump and probe alignment, modulation phase, and operating point. Package fiducials are not necessarily magnetic axes.

Heading error

A scalar heading test rotates the sensor relative to a field of constant magnitude. The heading error is

$$\Delta B_{\mathrm{head}} \left( \widehat{\mathbf B} \right) = \widehat B_{\mathrm{reported}} \left( \widehat{\mathbf B} \right) - B_{\mathrm{true}}.$$

Sources include:

• nonlinear Zeeman splitting and orientation-dependent line weights;
• unresolved contributions from both hyperfine manifolds;
• light shifts whose direction follows the optical axis;
• RF polarization and Bloch–Siegert shifts;
• cell and coil anisotropy;
• dead zones where pumping or detection vanishes; and
• fitting one line to a changing multicomponent spectrum.

Testing only two opposite headings can miss even angular harmonics. Characterization should cover the solid angle relevant to operation and repeat at several fields, temperatures, and pump powers.

Light-shift and power extrapolation

Reverse pump helicity, change detuning, vary optical power, and introduce dark evolution to separate light shifts. A linear extrapolation to zero power is credible only if the measured range is linear and the zero-power intercept refers to the same atomic state and estimator.

Pump and probe changes can alter polarization, line weights, temperature, and signal-to-noise ratio at the same time. A global model with these nuisance changes is stronger than a one-variable correction.

Gradients, motion, and diffusion

Atoms moving through a nonuniform field accumulate different phases. In a ballistic or diffusive cell, this produces relaxation, line distortion, and a response-weighted location. Motion of the whole sensor through a static gradient produces a time-dependent field:

$$\delta B(t) \simeq \boldsymbol\nabla B \mathbin{\cdot} \delta\mathbf r(t).$$

This is central in wearable biomagnetism and mobile geomagnetic sensing. An accelerometer or optical tracker can aid artifact modelling, but requires calibrated time alignment and a measured field map.

Magnetic shields and Johnson noise

High-permeability shields reduce ambient field and gradient but introduce:

• remanent field and hysteresis;
• temperature-dependent permeability;
• magnetic viscosity and slow relaxation;
• vibration-to-field coupling;
• Johnson-current magnetic noise from conductive layers; and
• altered calibration-coil geometry.

Demagnetization procedures and waiting time should be documented. Environmental noise measured outside the sensor is not automatically the noise inside the shield, and vice versa.

Array cross-talk

Each OPM in an array can produce modulation, compensation, heater, and feedback fields seen by neighboring sensors. Optical beams can also leak between channels. A linear array model is

$$\mathbf y = \mathbf G \left( \mathbf B_{\mathrm{sig}} + \mathbf C\mathbf u \right) + \mathbf b + \boldsymbol\epsilon,$$

where $\mathbf u$ contains all channel control currents and $\mathbf C$ is the coil cross-talk matrix. Measure $\mathbf C$ with each channel driven separately and test whether it depends on frequency, orientation, or shield state.

Uncertainty and covariance

Let the reported field be

$$\widehat B = f \left( \widehat y, \widehat{\mathbf c}, \widehat{\mathbf x} \right),$$

where $\mathbf c$ contains calibration parameters and $\mathbf x$ environmental inputs. First-order propagation gives

$$u^2(B) = \mathbf J \boldsymbol\Sigma \mathbf J^{\mathsf T},$$

where

$$\mathbf J = \frac{\partial f} {\partial (y,\mathbf c,\mathbf x)}.$$

Covariance is common in arrays because channels share lasers, shields, clocks, field coils, and environmental models. Treating every channel and systematic as independent can substantially understate uncertainty in a source reconstruction or gradient.

Measuring the Frequency Response

A field calibration at one frequency determines only one point on the response curve. Inject small sinusoidal fields across the desired band, measure gain and phase, and repeat near the largest expected field offset. For example, a sensor with a 100 Hz first-order bandwidth transmits a 100 Hz input with $1/\sqrt2$ of its low-frequency gain and a 45° phase lag. Using its DC gain would therefore underestimate that signal’s amplitude.

Reversals test different effects. Reversing coil current isolates terms odd in that current; rotating the sensor changes heading-dependent line weights. Reversing pump helicity can reverse both the useful response and a vector light shift, so that reversal alone cannot separate them.

Applications

Biomagnetism

Magnetocardiography and magnetoencephalography detect fields generated by bioelectric currents. OPMs can operate without a cryogenic dewar and can be placed close to the body. The benefit is application specific: sensor noise, standoff distance, channel count, bandwidth, source geometry, shielding, and motion artifacts all matter.

Wearable MEG illustrates the full measurement problem. Movement changes the sensor’s location and orientation in a residual field and gradient, often producing artifacts much larger than the neural signal. Active field nulling, accurate sensor tracking, array calibration, reference channels, and source-reconstruction covariance are part of the instrument, not optional post-processing.

A magnetic map is not a direct image of neural current. The inverse source problem is nonunique and requires anatomical, geometrical, and statistical assumptions.

Low-field NMR and MRI

Atomic magnetometers can detect nuclear free-induction fields where inductive pickup coils become inefficient. Prepolarization, encoding, and detection can occur in separate regions. Near zero field, spectra are often governed by spin–spin $J$ couplings rather than high-field chemical shifts.

The atomic sensor can perturb the sample through bias and modulation coils, and polarized noble gases can produce enhanced contact fields in alkali vapor. The sample–sensor distance, thermal isolation, field homogeneity, and transfer geometry belong in the sensitivity claim.

Geophysics and navigation

Earth-field scalar magnetometers are used for surveys, observatories, space measurements, and anomaly detection. Relevant requirements include:

• operation at fields of order tens of microtesla;
• low heading error over vehicle attitude;
• large dynamic range and fast recovery;
• stable frequency estimation under vibration;
• platform magnetic cleanliness;
• calibrated location and timing; and
• separation of temporal geomagnetic variation from spatial anomalies.

A shielded SERF noise record near zero field does not establish these properties. Finite-field scalar and free-precession architectures are often better matched to geomagnetic operation.

RF sensing and communication

An RF atomic magnetometer can be tuned to a carrier or resonance and detect amplitude, phase, and modulation. Narrowband gain can make weak signals visible, including those used in low-frequency communication or nuclear quadrupole resonance.

Performance should be reported as a transfer function, noise spectrum, linearity, and intermodulation response. A strong nearby carrier can shift or saturate the atomic resonance even when the desired sideband is weak.

Materials and magnetic imaging

Atomic-vapor cells, cold-atom clouds, and solid-state spins can map fields from magnetized rocks, currents, magnetic particles, superconductors, and mesoscopic samples. Spatial resolution is set by sensor–sample distance, sensitive-volume size, field propagation, and inversion, not detector pixel size alone.

NV-center magnetometry can place spins nanometres to micrometres from a sample and operate over broad conditions. Vapor-cell magnetometers can provide excellent field sensitivity over larger volumes. These are different points in a sensitivity–bandwidth–distance–resolution space, not a single ranking.

Fundamental and comagnetometer measurements

Two spin species can occupy nearly the same volume. A weighted difference of their precession frequencies can reject ordinary magnetic-field noise:

$$\Omega_{\mathrm{co}} = \omega_1 - \frac{\gamma_1}{\gamma_2} \omega_2.$$

Here $\omega_i$ are signed coefficients chosen consistently so that the common magnetic contribution is $\omega_i=\gamma_iB$; substituting two positive frequency magnitudes with signed $\gamma_i$ can spoil the cancellation. The cancellation also requires identical spatial weighting and well-known gyromagnetic ratios. Species-dependent gradients, contact fields, wall shifts, light shifts, and backaction remain. Such comagnetometers can test rotations, symmetry-violating interactions, or exotic spin couplings, but the interpretation belongs to a declared physical model.

Choosing a platform

Requirement	Often suitable	Principal cautions
shielded low-frequency field component	SERF OPM	narrow range, residual fields, light shifts
Earth’s-field magnitude	scalar driven or free-precession OPM	heading error, nonlinear Zeeman structure
narrowband RF field	tuned RF atomic magnetometer	bandwidth, drive shifts, saturation
near-body multichannel biomagnetism	compact OPM array	motion, cross-talk, thermal and geometric constraints
nanoscale proximity	defect-spin magnetometer	surface noise, readout contrast, calibration
common-mode rejection	atomic gradiometer or array	gain and phase mismatch, baseline definition
field-independent frequency comparison	comagnetometer	species-dependent weighting and contact shifts

Common Pitfalls

A scalar sensor still has heading error. Its leading energy splitting depends on field magnitude, but optical pumping, nonlinear Zeeman structure, and detection weights depend on orientation.

A noise floor is not the total uncertainty. Light shifts, axis errors, calibration, and spatial averaging can bias a stable measurement. A differential noise floor can also conceal noise shared by both channels.

SERF suppresses one relaxation mechanism. Spin destruction, diffusion, light shifts, and magnetic noise remain. The fast-exchange, small-field conditions must hold for the measured mode.

Motion changes the field sampled by the sensor. A quiet field map can produce a large time-dependent signal when the sensor moves through its gradient or rotates its sensitive axis.

Related Reading

• Magnetic resonance develops driven and pulsed spectroscopy, line shapes, and relaxation.
• NV-center sensing treats defect-spin optical readout and nanoscale field measurements.
• Tests of fundamental symmetries explains comagnetometer mismatch and electric-field reversal channels.

References

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• W. Happer, “Optical pumping,” Reviews of Modern Physics 44, 169–249 (1972).
• W. Happer and H. Tang, “Spin-exchange shift and narrowing of magnetic resonance lines in optically pumped alkali vapors,” Physical Review Letters 31, 273–276 (1973).
• W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Physical Review A 16, 1877–1891 (1977).
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Exercises

1. Zeeman scale and frequency uncertainty

Use

$$\frac{|\gamma_F|}{2\pi} = 6.998\ \mathrm{Hz\,nT^{-1}}$$

for the $^{87}\mathrm{Rb}$ $F=2$ manifold.

1. Find the Larmor frequency at $B=50.000\ \mu\mathrm T$.
2. A fit gives $f_L=349\,902.0\ \mathrm{Hz}$ with standard uncertainty $0.8\ \mathrm{Hz}$. Ignoring uncertainty in $\gamma_F$, find $\widehat B$ and its standard uncertainty.
3. If an unmodelled heading shift changes the fitted frequency by $3.5\ \mathrm{Hz}$, express the magnetic bias in nT.

Solution

Convert

$$50.000\ \mu\mathrm T = 50\,000\ \mathrm{nT}.$$

The frequency is

$$\begin{aligned} f_L &= \left( 6.998\ \mathrm{Hz\,nT^{-1}} \right) \left( 50\,000\ \mathrm{nT} \right) \\ &= 349\,900\ \mathrm{Hz}. \end{aligned}$$

The fitted field is

$$\widehat B = \frac{349\,902.0}{6.998}\ \mathrm{nT} \simeq 50\,000.29\ \mathrm{nT} = 50.00029\ \mu\mathrm T.$$

The frequency-fit contribution is

$$u(B) = \frac{0.8}{6.998}\ \mathrm{nT} \simeq 0.114\ \mathrm{nT}.$$

The heading bias is

$$\Delta B_{\mathrm{head}} = \frac{3.5}{6.998}\ \mathrm{nT} \simeq 0.50\ \mathrm{nT}.$$

It is more than four times the statistical standard uncertainty, showing why a precise frequency fit is not the whole uncertainty budget.

2. Zero-field Bloch response

Starting from

$$\frac{d\mathbf P}{dt} = \gamma\mathbf P\times B_x\widehat{\mathbf x} + Rs\widehat{\mathbf z} - \Gamma\mathbf P,$$

derive the steady-state $P_y$ and $P_z$. Find the field half-width and the zero-field slope. For

$$\frac{|\gamma|}{2\pi} = 7.0\ \mathrm{Hz\,nT^{-1}}, \qquad \Gamma=2\pi(100)\ \mathrm{s^{-1}},$$

find $B_{1/2}$.

Solution

For $\mathbf B=B_x\widehat{\mathbf x}$,

$$\mathbf P\times\mathbf B = \begin{pmatrix} 0\\ P_zB_x\\ -P_yB_x \end{pmatrix}.$$

The steady $y$ and $z$ equations are

$$0 = \gamma B_xP_z-\Gamma P_y,$$

and

$$0 = -\gamma B_xP_y+Rs-\Gamma P_z.$$

The first gives

$$P_y = \frac{\gamma B_x}{\Gamma}P_z.$$

Substitution gives

$$P_z = \frac{ Rs\Gamma }{ \Gamma^2+\gamma^2B_x^2 },$$

and

$$P_y = \frac{ Rs\gamma B_x }{ \Gamma^2+\gamma^2B_x^2 }.$$

The half-width is

$$B_{1/2} = \frac{\Gamma}{|\gamma|},$$

and the zero-field slope is

$$\left. \frac{dP_y}{dB_x} \right|_{0} = \frac{Rs\gamma}{\Gamma^2}.$$

Numerically, the factors of $2\pi$ cancel:

$$B_{1/2} = \frac{ 100\ \mathrm{Hz} }{ 7.0\ \mathrm{Hz\,nT^{-1}} } \simeq 14.3\ \mathrm{nT}.$$

The useful range for less than $1\%$ gain error would be only about $1.4\ \mathrm{nT}$ in the simple Lorentzian model.

3. Projection-noise baseline

An ensemble contains $N=1.0\times10^{11}$ effectively measured atoms with $T_2=10\ \mathrm{ms}$ and

$$|\gamma| = 2\pi \left( 7.0\times10^9 \right) \mathrm{rad\,s^{-1}\,T^{-1}}.$$

Use the stated estimate $\eta_B^{(\mathrm{SPN})} \sim1/(|\gamma|\sqrt{NT_2})$ for the averaging coefficient $\delta B\sqrt\tau$.

1. Estimate $\eta_B^{(\mathrm{SPN})}$ in $\mathrm{fT}\sqrt{\mathrm s}$.
2. If contrast and duty cycle together worsen this coefficient by a factor $3$, find the adjusted coefficient.
3. An observed flat one-sided amplitude spectral density is $12\ \mathrm{fT}/\sqrt{\mathrm{Hz}}$. Convert it to the same averaging convention and compare with the adjusted baseline. Assume rectangular averages long compared with the sensor correlation time.

Solution

The gyromagnetic magnitude is $4.398\times10^{10}\, \mathrm{rad\,s^{-1}\,T^{-1}}$ and $\sqrt{NT_2}=3.162\times10^4\ \mathrm{s^{1/2}}$. Thus

$$\eta_B^{(\mathrm{SPN})} \sim\frac{1}{(4.398\times10^{10})(3.162\times10^4)} \simeq0.72\ \mathrm{fT}\sqrt{\mathrm s}, \qquad \eta_B^{(\mathrm{adj})} \simeq2.16\ \mathrm{fT}\sqrt{\mathrm s}.$$

For the specified one-sided spectrum, $\operatorname{Var}(\overline B_\tau)=S_B/(2\tau)$. Consequently,

$$\eta_B^{(\mathrm{obs})} =\frac{12}{\sqrt2}\ \mathrm{fT}\sqrt{\mathrm s} \simeq8.49\ \mathrm{fT}\sqrt{\mathrm s}, \qquad \frac{\eta_B^{(\mathrm{obs})}}{\eta_B^{(\mathrm{adj})}} \simeq\frac{8.49}{2.16}\simeq3.9.$$

Equivalently, convert the adjusted baseline to a one-sided spectral level $\sqrt2(2.16)\simeq3.05\ \mathrm{fT}/\sqrt{\mathrm{Hz}}$ and compare $12/3.05$. Directly dividing $12$ by $2.16$ mixes conventions.

The projection-noise estimate remains an order-of-magnitude baseline; the spectral conversion does not fix its omitted spin, polarization, correlation, and estimator factors. A quantitative quantum-noise claim requires those factors as well as the measured noise spectrum and filter.

4. Coil calibration with covariance

A coil has

$$C_I = (50.00\pm0.10)\ \mathrm{nT\,mA^{-1}},$$

and is driven at

$$I = (2.000\pm0.004)\ \mathrm{mA}.$$

The correlation coefficient between fitted $C_I$ and current calibration is $\rho=-0.40$.

1. Find the applied field.
2. Find its standard uncertainty including covariance.
3. Compare with the uncertainty obtained by assuming independence.

Solution

The field is

$$B = C_II = (50.00)(2.000)\ \mathrm{nT} = 100.00\ \mathrm{nT}.$$

The covariance is

$$\operatorname{cov}(C_I,I) = \rho\,u(C_I)u(I) = (-0.40)(0.10)(0.004) = -1.6\times10^{-4}\ \mathrm{nT}.$$

Using units consistent with $C_I I$,

$$\begin{aligned} u^2(B) ={}& I^2u^2(C_I) + C_I^2u^2(I) + 2C_II\operatorname{cov}(C_I,I) \\ =& (2.000)^2(0.10)^2 + (50.00)^2(0.004)^2 + 2(50.00)(2.000) (-1.6\times10^{-4}) \\ =& 0.040+0.040-0.032 \\ =& 0.048\ \mathrm{nT^2}. \end{aligned}$$

Therefore

$$u(B) \simeq 0.219\ \mathrm{nT}.$$

Ignoring covariance gives

$$u_{\rho=0}(B) = \sqrt{0.080}\ \mathrm{nT} \simeq 0.283\ \mathrm{nT}.$$

The negative covariance reduces the propagated uncertainty. It must be supported by the joint calibration fit rather than chosen to improve the budget.

5. SERF criterion and narrowing

An alkali vapor has spin-exchange rate

$$R_{\mathrm{SE}} = 2.0\times10^5\ \mathrm{s^{-1}}$$

and effective

$$\frac{|\gamma|}{2\pi} = 7.0\ \mathrm{Hz\,nT^{-1}}.$$

Use the schematic estimate

$$\Gamma_{\mathrm{SE}} \simeq \frac{\omega_L^2}{R_{\mathrm{SE}}}$$

with coefficient one.

1. Evaluate $\omega_L/R_{\mathrm{SE}}$ and $\Gamma_{\mathrm{SE}}$ at $B=1.0\ \mathrm{nT}$.
2. Repeat at $B=1.0\ \mu\mathrm T$.
3. Explain the domain of this estimate.

Solution

At $1.0\ \mathrm{nT}$,

$$\omega_L = 2\pi(7.0) \simeq 44.0\ \mathrm{s^{-1}}.$$

Thus

$$\frac{\omega_L}{R_{\mathrm{SE}}} \simeq 2.2\times10^{-4} \ll 1,$$

and

$$\Gamma_{\mathrm{SE}} \simeq \frac{(44.0)^2}{2.0\times10^5} \simeq 9.7\times10^{-3}\ \mathrm{s^{-1}}.$$

At $1.0\ \mu\mathrm T=1000\ \mathrm{nT}$,

$$\omega_L \simeq 4.40\times10^4\ \mathrm{s^{-1}},$$

so

$$\frac{\omega_L}{R_{\mathrm{SE}}} \simeq 0.22.$$

The fast-exchange separation is much weaker, and the schematic expression would give

$$\Gamma_{\mathrm{SE}} \simeq \frac{(4.40\times10^4)^2}{2.0\times10^5} \simeq 9.7\times10^3\ \mathrm{s^{-1}}.$$

The quadratic estimate is an asymptotic low-field scaling with a species-, polarization-, and mode-dependent coefficient. It should not be used quantitatively when $\omega_L/R_{\mathrm{SE}}$ is no longer small; the coupled hyperfine spin-exchange dynamics must then be solved.

6. Gradiometer gain mismatch

Two OPM channels have mean gain $\overline G=1.00\ \mathrm{V\,nT^{-1}}$ and relative mismatch

$$\frac{\delta G}{\overline G} = 3.0\times10^{-4}.$$

They measure a common background $\overline B=40\,000\ \mathrm{nT}$ and a true differential signal $\Delta B=0.50\ \mathrm{nT}$.

1. Find the desired differential output.
2. Find the common-field leakage.
3. What relative matching is required to keep leakage below $0.05\ \mathrm{nT}$ equivalent field?

Solution

The desired output is

$$\Delta y_{\mathrm{sig}} = \overline G\Delta B = 0.50\ \mathrm V.$$

The gain difference is

$$\delta G = (3.0\times10^{-4}) (1.00\ \mathrm{V\,nT^{-1}}) = 3.0\times10^{-4}\ \mathrm{V\,nT^{-1}}.$$

The leaked output is

$$\Delta y_{\mathrm{leak}} = \delta G\overline B = (3.0\times10^{-4})(40\,000) = 12\ \mathrm V.$$

This is equivalent to $12\ \mathrm{nT}$ at the mean gain, far larger than the signal. To keep the equivalent leakage below $0.05\ \mathrm{nT}$,

$$\frac{\delta G}{\overline G} < \frac{0.05}{40\,000} = 1.25\times10^{-6}.$$

In practice a geomagnetic gradiometer often estimates and subtracts channel gains continuously or operates scalar frequency channels. Axis and frequency-response mismatch remain.

7. Heading-parity model

A scalar magnetometer has heading error

$$\Delta B(\theta) = a_0 + a_1\cos\theta + b_1\sin\theta + a_2\cos2\theta.$$

Measurements at $\theta=0,\pi/2,\pi,3\pi/2$ give, in nT,

$$\begin{array}{c|rrrr} \theta &0&\pi/2&\pi&3\pi/2\\ \hline \Delta B &1.8&-0.4&-0.2&0.8 \end{array}.$$

Find $a_0$, $a_1$, $b_1$, and $a_2$. Why would measurements only at $0$ and $\pi$ be insufficient?

Solution

The four equations are

$$\begin{aligned} 1.8 &= a_0+a_1+a_2, \\ -0.4 &= a_0+b_1-a_2, \\ -0.2 &= a_0-a_1+a_2, \\ 0.8 &= a_0-b_1-a_2. \end{aligned}$$

Adding all four gives

$$4a_0 = 2.0,$$

so

$$a_0 = 0.50\ \mathrm{nT}.$$

The opposite-heading differences give

$$a_1 = \frac{1.8-(-0.2)}{2} = 1.00\ \mathrm{nT},$$

and

$$b_1 = \frac{-0.4-0.8}{2} = -0.60\ \mathrm{nT}.$$

Using the first equation,

$$a_2 = 1.8-0.50-1.00 = 0.30\ \mathrm{nT}.$$

Measurements only at $0$ and $\pi$ determine $a_1$ and the combination $a_0+a_2$, but cannot separate the constant and even second harmonic. They also provide no information about $b_1$. Opposite-heading reversal is not a complete heading characterization.

8. Motion through a residual field

A sensor with linear range $\pm1.5\ \mathrm{nT}$ moves by $0.20\ \mathrm m$ along a residual gradient of $12\ \mathrm{nT\,m^{-1}}$. It also rotates its sensitive axis by $5^\circ$ toward a transverse residual field of $10\ \mathrm{nT}$.

1. Estimate the separate translation and rotation signals.
2. Can the nominal range accommodate their worst-case sum?
3. If the translation correction uses a position estimate uncertain by $0.5\ \mathrm{mm}$, compare its resulting uncertainty with the noise of a sensor having a flat one-sided amplitude spectral density of $10\ \mathrm{fT}/\sqrt{\mathrm{Hz}}$, averaged for one second. Assume the bandwidth is much greater than 1 Hz and the gradient is exact.

Solution

Translation produces

$$\Delta B_{\rm trans}=12(0.20)=2.4\ \mathrm{nT}.$$

For an initially perpendicular field and axis, rotation produces

$$\Delta B_{\rm rot}=10\sin5^\circ\simeq0.872\ \mathrm{nT}.$$

Their sum can reach $3.27\ \mathrm{nT}$; translation alone already exceeds the linear range. Calibration cannot recover a signal recorded in saturation, so the residual field and gradient must be reduced or the range increased before taking data.

The position uncertainty contributes

$$u(B_{\rm trans})=(12)(0.0005)\ \mathrm{nT}=6.0\ \mathrm{pT}.$$

The ideal one-second white-noise average has uncertainty

$$\sigma_B=\sqrt{\frac{S_B}{2\tau}} =\frac{10}{\sqrt2}\ \mathrm{fT}\simeq7.1\ \mathrm{fT}.$$

The position contribution is about 850 times larger. A low intrinsic noise floor helps only after motion, field maps, axis rotation, and their uncertainties are controlled. Actual sensor filtering changes the averaging formula when its bandwidth is comparable with $1/\tau$.