# Optical pumping enhancement of a free-induction-decay magnetometer

Dominic Hunter  Department of Physics, SUPA, University of Strathclyde,
Glasgow G4 0NG, United Kingdom<EMAIL_ADDRESS>Marcin S. Mrozowski
Department of Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, United
Kingdom  Allan McWilliam  Department of Physics, SUPA, University of
Strathclyde, Glasgow G4 0NG, United Kingdom  Stuart J. Ingleby  Department of
Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
Terry E. Dyer  Department of Physics, SUPA, University of Strathclyde, Glasgow
G4 0NG, United Kingdom  Paul F. Griffin  Department of Physics, SUPA,
University of Strathclyde, Glasgow G4 0NG, United Kingdom  Erling Riis
Department of Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, United
Kingdom

###### Abstract

Spin preparation prior to a free-induction-decay (FID) measurement can be
adversely affected by transverse bias fields, particularly in the geophysical
field range. A strategy that enhances the spin polarization accumulated before
readout is demonstrated, by synchronizing optical pumping with a magnetic
field pulse that supersedes any transverse fields by over two order of
magnitude. The pulsed magnetic field is generated along the optical pumping
axis using a compact electromagnetic coil pair encompassing a micro-
electromechanical systems (MEMS) vapor cell. The coils also resistively heat
the cesium (Cs) vapor to the optimal atomic density without spurious magnetic
field contributions as they are rapidly demagnetized to approximately zero
field during spin readout. The demagnetization process is analyzed
electronically, and directly with a FID measurement, to confirm that the
residual magnetic field is minimal during detection. The sensitivity
performance of this technique is compared to existing optical pumping
modalities across a wide magnetic field range. A noise floor sensitivity of
$\mathbf{238}\,$fT$/\boldsymbol{\sqrt{\mathrm{Hz}}}$ was achieved in a field
of approximately $\mathbf{50}\,\boldsymbol{\mu}$T, in close agreement with the
Cramér-Rao lower bound (CRLB) predicted noise density of
$\mathbf{258}\,$fT$/\boldsymbol{\sqrt{\mathrm{Hz}}}$.

††journal: opticajournal

##  1 Introduction

Extensive efforts have been devoted toward developing optically pumped
magnetometers (OPMs) that operate at zero field [1, 2, 3], exploiting the
well-established spin-exchange relaxation-free (SERF) mechanism [4, 5, 6].
Such devices can achieve exceptionally high sensitivity [7], and are therefore
well-suited to applications demanding the capability to resolve fT-level
signals, e.g., magnetoencephalography (MEG) [8]. Sensors operating in the SERF
regime are already commercially available at sensitivities below
$10\,\mathrm{fT/\sqrt{Hz}}$ with a bandwidth of $135\,$Hz [9]. Accordingly,
they offer an attractive alternative to superconducting quantum interference
devices (SQUIDs), particularly in MEG applications, as these compact and
flexible devices can be easily integrated into custom mounting hardware [10].
However, the narrow magnetic resonances essential for high sensitivity
operation in SERF devices also imposes limitations on both sensor bandwidth
and dynamic range. This restricts the implementation of these sensors to low-
field environments that are often conditioned using both passive and active
field compensation techniques [11].
The exceptional sensitivity achievable with SERF operation is attributable to
suppression of spin-exchange collisions that occur between alkali atoms when
operating close to zero field with dense atomic ensembles. Several studies
have been conducted that extend the utility of spin-exchange suppression,
enabling $\mathrm{fT/\sqrt{Hz}}$ sensitivity operation in bias field’s of
several $\mu$T [12, 13, 14]. Such devices provide a framework for unshielded
sensing that could become a valuable resource in many research areas including
geophysical [15], space science [16], GPS-denied navigation [17], and
biomedical applications [18]. These sensors rely on the light-narrowing
phenomenon [19, 20], which requires close to unity spin polarizations to be
generated. Therefore, maintaining sufficient optical pumping efficiency is
crucial in both maximizing signal-to-noise ratio (SNR) and lowering spin-
exchange contributions to the transverse relaxation rate, $\gamma_{2}$.
This study employs an OPM based on the free-induction-decay (FID) measurement
protocol [21, 22, 23]. This modality exhibits a wide and tunable bandwidth
given the flexibility in digital signal processing (DSP) that can be used to
analyse the FID data [23, 24]. A notable benefit of FID-based sensors is their
accuracy, as optical pumping and detection are separated temporally.
Consequently, this allows the polarized spins to precess freely at the Larmor
frequency without being perturbed by intense pumping light, thereby
considerably lowering light-shift systematics compared to continuous-wave (cw)
pumping schemes [25]. Moreover, these sensors are robust as they are commonly
operated in a free-running mode and enable direct Larmor frequency extraction.
Additionally, spin-exchange suppression can be exploited to extend the sensor
dynamic range to bias fields exceeding the Earth’s field through efficient
spin preparation.
The presence of strong magnetic fields can hinder the optical pumping dynamics
as the atoms experience a torque that deflects the generated spin polarization
from the beam propagation (optical pumping) axis. This issue can be
circumvented by, for example, resonantly driving the atoms at the Larmor
frequency by modulating either the amplitude [21] or frequency [26] of the
pump light. Additionally, one can null the external magnetic field
contributions during the spin preparation stage [27]. Although effective, both
these techniques require prerequisite knowledge regarding the field of
interest, and feedback to maintain optimal conditions.
In this work, we exploit a simple and practical approach, hereby referred to
as enhanced optical pumping (EOP), that facilitates efficient spin
polarization buildup throughout a range of bias field conditions. This is
achieved by applying a strong field, $\vec{B_{p}}$, of several mT along the
quantization axis during spin preparation to negate the detrimental effects of
transverse field components. Additionally, the resistive heating produced by
the coils generating $\vec{B_{p}}$ is utilized to elevate the vapor cell
temperature and reach an optimal atomic density. This provides an effective
way of exploiting the FID sensor dead-time whilst simultaneously enhancing the
optical pumping efficiency.
The approximate optimum atomic density occurs when the ratio of $\gamma_{2}$
and SNR is maximized [28]. However, delivering heating power to the vapor cell
can often adversely affect the instrument’s performance. For instance, the
alkali atoms are most commonly heated by passing current through a resistive
element in contact with the cell. Current noise flowing through the heating
element is converted to magnetic fluctuations that can lift the sensor noise
floor. Furthermore, magnetic resonance broadening and additional systematics
can be induced by subsequent stray fields. Oscillating currents are often used
at frequencies far exceeding the atomic bandwidth to alleviate these issues
[29]. However, heating at MHz frequencies far beyond the Larmor precession
rate is often necessary when applied to total-field OPMs [25], which is
typically inefficient. There have been non-electrical heating methods adopted
in the past such as optical heating [30] and hot air systems [31], although
these techniques require a great deal of power and are often restricted to a
laboratory setting. The heating strategy proposed in this work ensures no
current is flowing through the heaters during spin readout, hence is immune to
systematics and magnetic noise contributions that often contaminate other OPM
technologies. This is made possible by demagnetization electronics that enable
the current through the heating coil to be switched from $1.4\,$A to within
$10\,\%$ of the MOSFET leakage current ($<50\,$pA) over a period of
approximately $2500\,$ns.

##  2 Experimental methodology

###  2.1 FID magnetometer setup

Figure 1: (a) Simplified schematic of a FID OPM utilizing a two-color pump-
probe configuration: GT, Glan-Thompson polarizer; NPBS, non-polarizing
beamsplitter; PD, photodiode; M, mirror; PBS, polarizing beamsplitter; DWP,
dual-wavelength waveplate; HWP, half-wave plate; BPF, bandpass filter; WP,
Wollaston prism; BPD, balanced photodetector; PCB, printed circuit board. (b)
Depiction of various optical pumping techniques that can be employed. In the
single-pulse scheme, the atoms are continuously pumped at peak optical power
throughout the period, $T_{p}$. Synchronous pumping modulates the light
resonantly at the Larmor frequency. The EOP scheme applies an additional
longitudinal field, $\vec{B_{p}}$, of approximately 3.7 mT along the $z$-axis,
synchronized with the optical pulse. The vapor cell is positioned between two
PCB coils to generate $\vec{B_{p}}$, which are also used for resistive
heating. Pump light interacting with the atoms is extinguished to
$<\,10\,\mu$W during the readout period, $T_{r}$. The PCB coils are
demagnetized to within $10\,\%$ of the field produced by the MOSFET leakage
current ($|\vec{B_{p}}|\sim 135\,\mathrm{fT}$) at $2.5\,\mu$s.

A simplified schematic of the experimental arrangement is illustrated in Fig.
1(a), showing a two-beam OPM operating in the FID regime. The sensor head
consists of a $3\,$mm thick micro-electromechanical systems (MEMS) Cs vapor
cell with nitrogen (N2) buffer gas [32]. As performed in previous work [26],
the collisional broadening and shift in the optical spectrum measured against
a Cs reference cell determined the internal buffer gas pressure to be
approximately 220 torr.
Optical pumping and detection are performed using two co-propagating probe and
pump beams tuned to the Cs $D_{1}$ and $D_{2}$ transitions, respectively. Both
beam widths were set to a diameter ($1/e^{2}$) of $3.1\,$mm. While the
$895\,$nm probe light remains linearly polarized, the $852\,$nm pump beam
becomes circularly polarized after passing through a dual-wavelength multi-
order waveplate. This enables optical pumping and detection to be optimized
without sacrificing sensitivity, as opposed to launching the beams at a slight
angle relative to one another [12]. A polarizing beamsplitter is used to
combine both beams prior to traversing the waveplate and illuminating the
vapor cell. Overlap between both beams within the interaction region is
optimized based on the maximum FID signal amplitude.
A high power ($\leq 600\,$mW) single-frequency laser is used to optically pump
the alkali spins into a highly polarized state. The pump light is resonant
with the $F=3\longrightarrow F^{\prime}$ hyperfine transition of the Cs MEMS
cell. The transition is collisionally broadened to a full width at half
maximum (FWHM) linewidth of $3.7\,$GHz and shifted by $-1.6\,$GHz due to the
N2 buffer gas. Pumping on this transition provides efficient recycling of
atoms from the $F=3$ ground state so they can subsequently contribute to the
signal [33]. Light narrowing is exploited to partially suppress spin exchange
as most of the atomic population is transferred to the $F=4$ ground state when
pumped optimally, and thus cannot exchange spin due to angular momentum
conservation [20].
The probe laser produces a cw beam that is $20\,$GHz blue-detuned from the
$F=4\rightarrow F^{\prime}=3$ Cs transition. This mitigates broadening of the
magnetic resonance during detection, by reducing residual optical pumping
whilst maintaining an appreciable light-atom interaction strength to maximize
signal amplitude. The Glan-Thompson polarizer purifies the probe beam
polarization and converts polarization noise, e.g., arising from the fiber,
into amplitude noise. A non-polarizing beamsplitter separates the light
equally between a monitor photodiode and the vapor cell. The probe power
($\approx 450\,\mu$W) prior to the vapor cell was actively stabilized, to
within $0.4\,\%$, using an analog PID controller (SRS SIM960) that adjusts the
RF power supplied to an acousto-optic modulator (AOM).
The measurement bias field, $\vec{B_{m}}$, strength and direction can be
controlled by applying currents through a set of three-axis coils that
encapsulate the cell. The experiments performed here used only a single
transverse axis coil to produce a field with magnitude, $B_{y}$, along the
$y$-axis. This was driven by a custom current source with a $\pm\,75\,$mA
range and a noise level considerably below the noise floor of the sensor [34].
Accordingly, the sensor’s dynamic range can be evaluated up to $B_{y}\approx
50\,\mu$T using this coil. The whole assembly is placed inside a three-layer
$\mu$-metal shield that attenuates environmental magnetic variations, and
ambient fields down to nT levels.

###  2.2 Optical pumping schemes

Figure 1(b) shows two optical modulation schemes that can be employed during
the spin preparation period, $T_{p}$, dedicated to optical pumping [26]. The
first method uses single-pulse (SP) modulation where the maximum available
light intensity interacts with the atoms throughout the pumping phase, before
switching to approximately zero during the detection period, $T_{r}$. One can
also optically pump the atoms by resonantly modulating the light intensity at
the Larmor frequency $\omega_{L}=\gamma|\vec{B}|$, where $|\vec{B}|$ is the
magnetic field magnitude and $\gamma$ is the gyromagnetic factor dependent on
the atomic species, i.e., $\sim 3.5\,$Hz/nT for Cs. This technique is known as
synchronous modulation [21]. The peak optical power is $\sim 65\,$mW after the
pump light has traversed beam-conditioning optics, an AOM, and a fiber-
coupling stage. The AOM’s extinction ratio ensures that $<10\,\mu$W of pump
light interacts with the atoms during the off state. A spectral bandpass
filter is placed in front of the balanced photodetector to attenuate $852\,$nm
light whilst allowing $895\,$nm light to pass with $>90\,\%$ transmission,
thus avoiding saturation and lowering optical noise contributed by the pump
light.
This work demonstrates an alternative optical pumping strategy, EOP, which is
exploited by mounting the vapor cell between two compact printed circuit
boards (PCBs) that have a square central aperture for optical access, as seen
in Fig. 1(a). The PCB assembly serves two purposes: to generate a strong
field, $\vec{B_{p}}$, along the beam propagation axis; and maintaining an
optimal atomic density within the vapor cell through resistive heating. The
technique is illustrated in Fig. 1(b) with $|\vec{B_{p}}|$ set to several mT
and applied along the optical pumping ($z$-axis), synchronized with the
optical pulse over the pump period. Subsequently, the PCB coils producing
$\vec{B_{p}}$ are demagnetized by rapidly lowering the current flow to zero
such that only $\vec{B_{m}}$ persists during spin readout. Details regarding
the demagnetization circuitry are discussed in Section 2.3. As the strength of
$\vec{B_{p}}$ supersedes $\vec{B_{m}}$ by at least two orders of magnitude,
the macroscopic spin magnetization is pinned to the $z$-axis. This negates the
adverse impact of any transverse field components, including $\vec{B_{m}}$,
during spin preparation.
The copper tracks on the PCB are printed with a square spiral pattern on both
sides of the two-layer PCB. A via is used to electrically connect both layers.
The compact bi-planar stack can thus generate stronger magnetic fields
compared to a single layer PCB, with the field-to-current ratio theoretically
modelled to be $2.7\,\mu$T/mA at the center of the vapor cell. Additionally,
using multiple layers enables more coil turns in a smaller footprint for
increased heating efficiency. The vapor cell temperature can be controlled by
either adjusting the duty cycle or peak current of the applied current pulse
flowing through the PCB coils. The duty cycle also impacts the time dedicated
to optically pumping the atoms using EOP as the optical and magnetic pulses
are synchronized. The spin polarization was found to be close to saturation
after pumping for $T_{p}\approx 88\,\mu$s. This is equivalent to a $8.8\,\%$
duty cycle for the pump-probe cycle repetition rate, $f_{d}=1\,$kHz, which was
kept consistent in the experiments performed here. A peak current of $1.4\,$A
heats the vapor cell to a temperature of $\mathrm{88\,^{\circ}C}$ providing
the optimal atomic density.
$T_{p}$ was kept consistent when employing EOP or SP modulation to provide a
valid comparison between both techniques. Synchronously pumping the atoms
required longer to reach a steady-state spin polarization compared to EOP,
with $T_{p}$ set to around $286\,\mu$s. This can be attributed to the light
intensity being low for most of the pump phase as the optimal duty cycle of
the square-wave modulation was approximately $30\,$%. As a result, $T_{r}$ was
shorter for synchronous operation since $f_{d}$ was kept constant in each
case. Additionally, gated heating at $0.5\,$Hz was applied to the vapor cell
when utilizing synchronous or SP optical pumping, with measurements conducted
when no current was flowing through the PCB. This was to ensure that magnetic
noise from the heater was not contributing to the noise floor of the sensor
using these techniques.

###  2.3 Demagnetization electronics

Current flowing through the coils cannot appear or disappear instantaneously
due to their inductive nature. The coil acts as an open circuit after being
magnetized and subsequently switched-off. This sudden change in current causes
back EMF to be generated with opposing polarity to the supply voltage until
the current exponentially decays to zero. During this process, the induced
voltage is high ($10\,$s of kV) which can easily damage switching electronics.
A diode can be added in parallel to the coil to clamp this back EMF to the
forward bias voltage of the diode, and allow for safe demagnetization.
However, this approach is relatively slow as the forward bias voltage limits
the maximum energy at which the coil can demagnetize. This process can be sped
up drastically using a Zener diode by exploiting its avalanche breakdown
mechanism to rapidly demagnetize the coil [35]. This clamps the back EMF to
the avalanche breakdown voltage of the Zener, such that more energy is
dissipated at a faster rate leading to more rapid demagnetization.

Figure 2: Simplified schematic of the demagnetization circuitry used to
produce $\vec{B_{p}}$ for EOP, and to drive the resistive heating applied to
the vapor cell. A pair of PCB coils (parasitics represented in light grey)
were found to have an inductance $L_{C}\approx$ 10 $\mu$H, interwinding
capacitance $C_{C}\approx$ 10 pF, and resistance $R_{C}\approx$ 4 $\Omega$.

The circuit designed to rapidly magnetize and demagnetize the PCB coils is
presented in Fig. 2. A P-channel MOSFET ($Q_{1}$) is used as a switch to
control the current flow through the coil. $Q_{1}$ is controlled with a
function generator through a gate driver circuit that allows for a fast
switching rate of the MOSFET, in addition to acting as a buffer. The
demagnetization is managed by diodes $D_{1}$ and $D_{2}$. The transient-
voltage-suppression (TVS) diode, $D_{1}$, works similarly to a Zener diode
although enables more energy dissipation due to its larger area p-n junctions
[36]. The TVS breakdown voltage was selected to be close to the maximum
permitted drain voltage of $Q_{1}$, shortening the demagnetization time while
preventing damage to $Q_{1}$. The fast recovery rectifying diode, $D_{2}$, has
the sole purpose of preventing $D_{1}$ from conducting during the
magnetization process. $R_{1}$ is a 100 $\Omega$ low inductance wirewound
resistor (Vishay WSN) connected in parallel with the coil, and serves as the
final demagnetization device after the induced voltage falls below the $D_{2}$
forward voltage. $R_{1}$ also serves as a dampening device for the
interwinding capacitance $C_{c}$ of the coil. At the end of the
demagnetization process, the current flowing through the coil is equivalent to
the transistor drain leakage. The circuit is powered from a triple output
power supply. The strength of $\vec{B_{p}}$ and the heating power are
controlled by adjusting supply voltage $V_{S}$ using the $2.5\,$A output.

Figure 3: Transient response of the demagnetization circuit derived (a)
experimentally and (b) theoretically. The oscillation in the experimental data
is believed to be an artefact of the oscilloscope’s input capacitance. The
predicted response settles to within 10$\%$ of the final leakage current of
the transistor $I_{DS}$ after approximately 2500 ns from the moment $Q_{1}$
begins to switch off.

Two tests were conducted to analyse the demagnetization circuit’s performance,
including both transient and steady-state responses. These measurements have
to be performed quickly (100s of ns) with very high dynamic range (A to pA),
hence are difficult to perform simultaneously. Accordingly, the transient
response was measured separately using an oscilloscope (Micsig STO1152c),
offering the necessary sample rate and current resolution ($\approx 4$ mA) to
observe the avalanche breakdown clamping mechanism and the bulk of the overall
response. Secondly, a precision source meter unit (SMU) evaluated the steady-
state response, i.e., the leakage current through the drain $I_{DS}$ of
$Q_{1}$.
For the transient response test, a 250 m$\Omega$ shunt resistor was inserted
in series with the coil to monitor its current. The oscilloscope was connected
to the shunt with a short (15 cm) coaxial cable to limit its capacitance. A 1
kHz pulse with a duty cycle of 10 % was used to test the demagnetization. This
was then compared to simulated data with similar components as exact macro-
models were not available. $V_{S}$ was set to 5.8 V and the gate driver
voltage was set to 12 V, to mimic typical experimental operating conditions.
It can be seen in Fig. 3 that the $90\,\%$ to $10\,\%$ experimental
($t_{f}=215$ ns) and simulated ($t_{f}=203$ ns) fall times are in close
agreement. The slight discrepancy most likely arises from the inability to
match the exact macro-models, high variation in the breakdown voltage of the
TVS diode, and other parasitics present.
Next, the leakage current flowing through the transistor’s source and drain,
$I_{DS}$, was estimated during switch off using a high precision SMU due to
the oscilloscope’s limited vertical resolution. Only $Q_{1}$ was tested in
this instance. The SMU was configured to source the voltage $V_{S}$ of
$5.8\,$V through $Q_{1}$. The gate of $Q_{1}$ was driven by a set of two 9 V,
PP3 batteries connected in series, and then through a potentiometer to set the
gate voltage to 12 V. $Q_{1}$ was connected to the SMU through a triaxial
cable (Balden 9922), to limit any leakage current from the cable assembly or
instrument itself [37]. The leakage current of $Q_{1}$ was found to be lower
than 50 pA, which translates to approximately $135\,$fT based on the
theoretically predicted field-to-current ratio of the coils. The response
settles to within $10\,\%$ of this steady state value after approximately 2500
ns from the moment $Q_{1}$ is switched off.

##  3 Results

###  3.1 FID signal analysis

During $T_{r}$, the polarized spins undergo Larmor precession in the presence
of $\vec{B_{m}}$. Consequently, the alkali vapor experiences a modulated
birefringence, detectable through optical rotation in the linearly polarized
probe beam that is monitored by a balanced polarimeter. The optical rotation
angle is proportional to the projection of spin polarization along the
$x$-axis given by,

$M_{x}(t)=M_{0}\,\mathrm{sin}(\omega_{L}\,t+\phi_{0})\,e^{-\gamma_{2}t},$ (1)

where $M_{0}$ is spin polarisation generated through optical pumping and
$\phi_{0}$ is the initial phase. It can be seen that it exhibits a sinusoidal
decay where $\omega_{L}$ corresponds to the precession experienced by the Cs
atoms, and $\gamma_{2}$ is dependent on the intrinsic properties of the vapor
cell and operational systematics.
The photodetector signal is digitized by a data acquisition (DAQ) system based
on a Picoscope (model 5444D) operating with 15-bit voltage resolution at a
sampling rate of $125\,$Mz. The discretized signal can be modelled as,

$S_{n}=A\,\mathrm{sin}\bigg{(}\omega_{L}\,n\,\Delta{t}+\phi_{0}\bigg{)}\,e^{-\gamma_{2}\,n\,\Delta{t}}+\epsilon_{n},$
(2)

where $A$ is the FID amplitude, $n$ is the data point of interest, $\Delta{t}$
is the time interval between adjacent samples, and $\epsilon_{n}$ is the
signal noise. Figure 4(a) depicts examples of FID traces captured by the
polarimeter during readout. Each signal was obtained using the optical pumping
strategies described previously in a field $B_{y}\approx 50\,\mu\mathrm{T}$.
The signal is automatically downsampled by the DAQ device which averages $50$
successive points yielding a final sampling rate of $1/\Delta{t}=2.5\,$MHz.
One can generate a FID signal train by optically pumping the atoms and
measuring the subsequent decay of spin polarization over multiple cycles, as
seen in the inset of Fig. 4(a). This method was used throughout these
experiments to generate a magnetic field time series at a sampling rate,
$f_{d}=1\,$kHz, resulting in a Nyquist limited bandwidth of $500\,$Hz [22].

Figure 4: (a) Polarimeter-detected FID traces acquired using different optical
pumping strategies including EOP, (blue), synchronous (orange), and SP
(yellow). (b) Amplitude-spectral-density (ASD) curves of polarimeter output.
Noise spectra were collected for each optical pumping regime, with no pump
light applied (purple), and without probe light (grey). $\rho_{A}$ is the
average noise density, when no pump light was present, across a $2\,$kHz range
centred at the Larmor frequency $\omega_{L}\approx 2\pi\times 175\,$kHz.

The magnetometer noise budget was assessed by computing the ASD, as shown in
Fig. 4(b) for each optical pumping technique. The ASDs were formulated using
Welch’s method [38] by averaging the discrete Fourier transforms (DFTs)
calculated from 20 subsequent FID traces over the period, $T_{r}$. A Hanning
window was utilized to provide a more accurate determination of the baseline
noise levels. Noise spectra were also collected with no pump light applied to
the atoms and in the absence of probe light, to localize the dominant noise
contributions in the system. These spectra were generated using 20 separate
time domain traces as before, although over the full $1\,$ms window set by
$f_{d}$ as there is no optical pumping in these cases.
Figure 4(b) shows that there is a close match in the baseline noise levels
obtained from the FID spectra and the noise spectrum observed with no optical
pumping, i.e., only probe light is present. This is to be expected as the pump
beam is mostly extinguished by the AOM during readout, and is further
attenuated by the bandpass filter prior to reaching the detector. The spectral
peak for the synchronous case is slightly wider due to the shorter measurement
window as more time is dedicated to optical pumping. The noise density,
$\rho_{A}$, was estimated to be $4\,\mu\mathrm{{V}/\sqrt{Hz}}$, calculated as
the average noise density across a $2\,$kHz range centred at the Larmor
frequency $\omega_{L}\approx 2\pi\times 175\,$kHz. This noise level dictates
the achievable SNR and consequently limits the precision of the Larmor
frequency measurement based on the CRLB condition (see Section 3.2). The
magnetometer is mostly limited by photon shot-noise of the probe light at this
frequency as the technical noise inherent to the detector is less pronounced.
This can be attributed to the effective suppression in common-mode noise
sources facilitated by balanced photodetection e.g., laser intensity or
frequency fluctuations.
The photon shot-noise density can be estimated based on the the detected
optical power, $P_{det}$, as,

$\rho_{sn}=G\,\sqrt{2\,e\,P_{det}\,\mathcal{R}},$ (3)

where $e$ is the electron charge, $\mathcal{R}$ is the detector responsivity,
and $G$ is the amplifier transimpedance gain [14]. With approximately $63\%$
of the probe light reaching the detector, $\rho_{sn}$ was calculated to be
$3.7\,\mu$V$/\sqrt{\mathrm{Hz}}$. This is consistent with the noise density,
$\rho_{A}$, denoted in Fig. 4(b) when added in quadrature with the intrinsic
noise of the detection system which was measured to be
$1.5\,\mu$V$/\sqrt{\mathrm{Hz}}$ at $175\,$kHz.

###  3.2 Sensitivity estimation

The sensitivity performance of a FID-based magnetometer can be assessed using
the CRLB which is a measure of the minimum statistical uncertainty of
determining an unbiased estimator from a signal [39]. Assuming $\epsilon_{n}$
is distributed as white Gaussian noise, the CRLB standard deviation for
extracting $\omega_{L}$ from a FID trace can be calculated as [40],

$\sigma_{CR}\geq\frac{\sqrt{12\,C}}{\gamma\,(A/\rho_{A})\,T_{r}^{3/2}},$ (4)

where $A$ is the FID amplitude, and $\rho_{A}$ is the noise spectral density
at the Larmor frequency. $T_{r}$ is the readout duration, and
$N=T_{r}/\Delta{t}$ is the number of samples in the FID trace. $C$ is a
correction factor accounting for spin depolarization at a rate, $\gamma_{2}$,
and is given by [41],

$C=\frac{N}{12}\frac{(1-z^{2})^{3}(1-z^{2N})}{z^{2}(1-z^{2N})^{2}-N^{2}z^{2N}(1-z^{2})^{2}},$
(5)

where $z=e^{-\gamma_{2}/NT_{r}}$. The correction factor has a lower bound of
unity for an undamped sinusoid. $\sigma_{CR}$ can be converted to a noise
density $\rho_{CR}=\sigma_{CR}/\sqrt{f_{d}/2}$ to provide a sensitivity metric
in units of $\mathrm{T/\sqrt{Hz}}$. This assumes the magnetometer is white
noise limited such that the noise density is flat across all frequencies
within the magnetometer bandwidth.

Figure 5: Magnetic field sensitivity spectra acquired using enhanced (blue), synchronous (orange), and single-pulse (yellow) optical pumping. Each ASD was acquired with $B_{y}\approx 50\,\mu$T. The noise floors (dash-dotted lines) were calculated by averaging the spectra over a $70-500\,$Hz frequency range, whilst avoiding technical noise peaks. The corresponding sensitivity estimations, $\rho_{B}$, are listed in Table 1. Figure 6: FID magnetometer dynamic range comparison for three optical pumping strategies: EOP (blue), synchronous (orange), and SP (yellow). The bias field, $B_{y}$, was varied between $4-50\,\mu$T. (a) SNR based on the fitted FID amplitude, $A$, and the noise density, $\rho_{A}$, at the Larmor frequency of interest. (b) Transverse relaxation rate, $\gamma_{2}$, calculated from a single decay period ($1/e$) measured by fitting each FID trace to the model in Eq. 1. The markers are larger than the associated error bars. (c-e) Noise floor, $\rho_{B}$, estimated from the ASDs (colors), and the associated CRLB noise density, $\rho_{CR}$ (grey). Parameter | EOP | Synchronous | SP | Units
---|---|---|---|---
$\mathbf{B_{y}}$ | 49.5 | 49.7 | 49.8 |  $\mu$T
$\mathbf{T_{p}}$ | 88.4 | 286 | 88.4 |  $\mu$s
$\mathbf{A}$ | 7.29 | 7.02 | 0.86 | V
$\boldsymbol{\gamma_{2}}$ | 1.23 | 1.32 | 1.98 | kHz
$\boldsymbol{\rho_{B}}$ | $238\pm 28.4$ | $368\pm 38.5$ | $3630\pm 400$ |  fT/$\sqrt{\mathrm{Hz}}$
$\boldsymbol{\rho_{CR}}$ | $258\pm 21.7$ | $351\pm 29.5$ | $3130\pm 263$ |  fT/$\sqrt{\mathrm{Hz}}$
Table 1: List of experimental parameters for each implemented optical pumping
strategy: EOP, synchronous, and SP. The amplitude, $A$, decay rate,
$\gamma_{2}$, and magnetic field, $B_{y}$, were determined from a damped
sinusoidal fit. $\rho_{B}$ and $\rho_{CR}$ are the sensitivity estimations
based on the noise floor (see Fig. 5) and CRLB (see Eq. 4) predicted noise
densities, respectively.

An alternative definition of sensitivity is based on the sensor noise floor,
$\rho_{B}$, which can be obtained by monitoring magnetic field fluctuations
recorded by the OPM over a set time interval. A magnetic field time series can
be produced by extracting $\omega_{L}$ from consecutive FID traces in a signal
train over multiple cycles as illustrated in the inset of Fig. 4(a).
Subsequently, the ASD in fT$/\sqrt{\mathrm{Hz}}$ can be generated by applying
the DFT to the time series data, as performed in previous instances. Figure 5
shows a set of magnetic field sensitivity spectra gathered using this approach
for each of the aforementioned optical pumping strategies in a bias field,
$B_{y}\approx 50\,\mu$T. The ASD curves in Fig. 5 were computed in a similar
manner to before by averaging 20 non-overlapping $1\,$s time segments using
Welch’s method. The technical noise peaks observed between the frequencies
$20-60\,$Hz are related to environmental magnetic noise penetrating the three-
layer $\mu$-metal shield encapsulating the vapor cell. Additional noise peaks
at $50\,$Hz, and associated harmonics, originate from the current supply
driving the coil producing the bias field. The noise floor, $\rho_{B}$, and
associated uncertainty were determined by calculating the average and standard
deviation of the noise density over a $70-500\,$Hz frequency range, ignoring
technical noise peaks.
The raw FID traces shown in Fig. 4(a) were processed post-acquisition by
fitting the data to the model given in Eq. 1, providing the amplitude $A$, and
decay rate, $\gamma_{2}$, measurements required to calculate $\rho_{CR}$. The
corresponding sensitivity estimations are listed in Table 1 for each optical
pumping technique along with other relevant experimental parameters. The error
in $\rho_{CR}$ is mostly attributed to the uncertainty in the estimation of
$\rho_{A}$. Evidently, these sensitivity estimations are closely correlated
with $\rho_{B}$ for each optical pumping strategy. This clearly shows that
external magnetic noise contributions, e.g., produced by stray currents in the
heater, are not a limiting factor. Considering the white noise assumption in
determining the CRLB, this further validates that the magnetometer is
predominantly limited by photon shot-noise.

###  3.3 Dynamic range characterization

Dynamic range represents the limits in magnetic field that a magnetometer can
reliably operate within and plays a crucial role in finite-field sensing. Some
OPMs, e.g., SERF systems, can only function close to zero-field requiring
well-conditioned magnetic field environments. In contrast, the FID-based
approach enables operation over a wide range of bias fields as observed in
Fig. 6. This conveys the sensors performance across the range $B_{y}\approx
4-50\,\mu$T and provides a comparison of each optical pumping method discussed
in this work.
Clearly, SP optical pumping performs relatively poorly compared to synchronous
modulation and EOP. This is particularly evident in Fig. 6(a) which shows a
significant degradation in SNR as $B_{y}$ is raised. This is directly related
to a reduction in $A$ as $\rho_{A}$ is consistent for each pumping scheme as
seen in Fig. 4(b). As mentioned previously, this degradation arises from
transverse fields, e.g., $\vec{B_{m}}$, applying a torque on the spin
polarization during optical pumping, and is well described by the Bloch
equation formalism [21]. Figure 6(b) shows the additional impact this has on
$\gamma_{2}$ which experiences a sharp rise as $B_{y}$ is raised. This is to
be expected as the effectiveness of spin-exchange suppression diminishes when
the spins are prepared into a less polarized state.
A more gradual reduction in $A$ is observed at elevated bias fields when
employing either synchronous modulation or EOP, although this is not
immediately apparent from Fig. 6(a) as the SNR stays relatively consistent.
This is because $\rho_{A}$ also reduces as $B_{y}$ is increased; a consequence
of the technical noise present in the detection system being less prominent at
higher Larmor frequencies (see Fig. 4(b)). The $1/f$ noise dependence of
$\rho_{A}$ also explains the drop in SNR at lesser $B_{y}$ values. Slightly
higher signal amplitudes were achieved with synchronous modulation at low bias
fields owing to the longer pumping period used.
It is anticipated that much of the loss in amplitude observed for both EOP and
synchronous modulation at larger bias fields will be attributed to nonlinear
Zeeman (NLZ) splitting. This effect simultaneously broadens and distorts the
magnetic resonance, and is more prevalent at stronger field magnitudes [42].
In the case of EOP, this broadening mechanism will be the main contributor to
the $\approx 80\,$Hz deviation in $\gamma_{2}$ shown in Fig. 6(b), over the
range of fields tested. Further investigation is necessary to fully quantify
the effects NLZ has on the spin dynamics. Broadening due to magnetic field
gradients should also be considered, although the deviation in magnetic field
expected across the beam width based on the coil geometry is $\Delta{B}\approx
5.5\,$nT. Gradient broadening is determined by the spread of precession
frequencies throughout the cell, i.e., $\gamma_{gr}\sim\gamma\,\Delta{B}$
[43], which is around $19\,$Hz in this case.
The integrated area of the magnetic resonance, in the frequency domain, is a
reflection of the spin polarization gained through optical pumping. Thus,
assuming this remains constant, one would expect a lower signal amplitude
given the additional broadening that is induced by the NLZ effect. EOP
demonstrated consistent spin polarization buildup across the full range of
bias fields tested. This is not surprising given that the field strength of
$\vec{B_{p}}$ is at least two orders of magnitude higher than $B_{y}$, thus
the spin polarization will no longer experience a significant torque during
pumping. In contrast, the steady-state spin polarization achieved with
synchronous modulation reduced as a function of $B_{y}$. NLZ splitting
inevitably influences the optical pumping dynamics in this case, as the atoms
become more difficult to resonantly address due to the nonlinearity in the
magnetic sublevel structure. Naturally, this will result in a steeper
amplitude reduction with respect to $B_{y}$ compared to EOP which does not
rely on a resonant driving field. Furthermore, this accounts for the sharper
rise in $\gamma_{2}$ for synchronous pumping as spin-exchange suppression
becomes less effective.
Figures 6(c-e) provide a comparison in sensitivity performance for each of the
optical pumping schemes investigated. Two approaches to sensitivity estimation
were used in accordance with that described in Section 3.2. The first metric
is the CRLB noise density, $\rho_{CR}$, denoted by the gray data points. The
second method estimates the noise floor, $\rho_{B}$, of the ASD computed from
the magnetic field recordings, represented as colored markers. It can be
readily seen that the magnetometer noise floor reaches the CRLB limit across
the full range of bias fields for each pumping technique. This validates that
the heating strategy used in EOP is not lifting the sensor noise level, which
is to be expected since the coil is rapidly demagnetized over a short period
of $200\,$ns. It also verifies that the magnetic field fluctuations produced
by the bias field are well below the noise floor of the sensor. The
sensitivity dependence matches well with expectations in accordance with Eq. 4
when considering the SNR and $\gamma_{2}$ achieved in each case. The best
sensitivity was obtained using EOP as seen in the inset of Fig. 6(c), due to
the improved signal amplitudes and relaxation rates achieved with more
efficient optical pumping, especially at large magnetic field strengths.

###  3.4 Accuracy Considerations

FID-based sensors have a distinct advantage in accuracy compared to other OPM
configurations as they are inherently self-calibrating and do not suffer from
light shifts caused by intense optical pumping. Despite this, systematics are
still present when operating in geophysical magnetic fields due to magnetic
resonance asymmetries [44]. For example, both hyperfine ground states possess
slightly different gyromagnetic factors, and subsequently, their precession
frequencies diverge depending on the magnetic field strength. Moreover, the
magnetic resonance can be distorted owing to NLZ splitting of a single
hyperfine manifold [42]. In both cases, the induced systematics depend on the
atomic distribution among the Zeeman sublevels of both hyperfine levels, which
is sensitive to both spin preparation and subsequent relaxation.
The linearity of the magnetometer response with magnetic field was
characterized for the EOP and SP modulation schemes. The average magnetic
field was monitored over $1\,$s measurement intervals at various coil supply
currents as seen in Fig. 7(a). This shows the residuals obtained from a linear
fit to the recorded magnetic field data. The magnetometer responds relatively
linearly with bias field in the case of EOP as the residuals fluctuate around
zero with no perceivable trends. EOP reliably generates the same polarization
state independent of the applied field strength. Consequently, the atomic
population mainly resides in the $F=4$ ground state such that the spins
precess at a single predominant frequency, minimising systematic effects. The
residuals for SP modulation convey a distinct quadratic behaviour, as a result
of inconsistency in the atomic population distribution achieved through
optical pumping in various bias field conditions. In this case, the Larmor
frequency estimation will be weighted according to the atomic population
occupying each hyperfine state.

Figure 7: (a) Linear fit residuals of magnetic field data recorded at various
bias coil currents, using EOP (blue squares) and SP modulation (yellow
circles). (b) Residual variation in the instantaneous magnetic field recorded
over a FID cycle using EOP and SP modulation as noted in the legend. The data
was processed using a Hilbert transform to calculate the spin precession phase
at each DAQ clock cycle, and hence the instantaneous precession frequency. The
plot above shows the residual variation from a linear phase dependence, and is
an average of 50 consecutive FID pulses.

In the EOP scheme, rapid coil demagnetization is crucial in preventing spin
precession readout from being affected by the magnetic field pulse.
Previously, this was verified electronically by measuring the transient
current response, which demonstrated $\sim 200\,$ns fall time and eventual
decay to around $135\,$fT due to the leakage current of the MOSFET. In order
to observe the transient magnetic response from the magnetometer, an
alternative signal processing strategy was devised based on a Hilbert
Transform. This vastly extends the magnetometer bandwidth enabling resolution
of frequencies up to the Nyquist limit dictated by the sampling rate of the
DAQ system [24]. Generation of the full signal phasor using a dual matched
finite-impulse response filter to perform a Hilbert transform allows
estimation of the signal phase at each DAQ clock cycle, and hence the
instantaneous precession frequency [25]. Consistent variation over time is
observed using the SP scheme, and to a lesser extent, with the EOP scheme, and
the trend can be clarified by averaging over 50 consecutive FID pulses (see
Fig. 7(b)).
Figure 7(b) shows the residual variation in instantaneous magnetic field over
the course of a single FID cycle in a bias field $B_{y}\approx 50\,\mu$T. The
observations for SP optical pumping suggest that the observed systematic
variation of precession frequency during each FID pulse is not a consequence
of demagnetization, as no current was flowing through the PCB coils using this
technique. Instead, the decay of spin polarisation during the FID pulse,
combined with the dependence of NLZ-induced heading error on this
polarisation, is likely to be responsible for this behaviour. Ground-state
spin relaxation creates a time-dependent spin distribution, which varies
consistently during each FID pulse. At high bias fields, the splitting between
the Zeeman sublevels is no longer linear, creating a dependence of the
measured Larmor frequency on the magnetic sublevel populations, which vary
during the FID pulse. One would expect the trend to be more distinctive for
the SP case as the influence of spin-exchange processes results in more
complex evolution of the ground-state spin distriubutions. Significantly, a
lower systematic shift is observed in the EOP case, confirming that the pre-
polarising coil is sufficiently de-energised at the measurement onset.

##  4 Conclusion and Outlook

In conclusion, a more efficient spin preparation scheme for FID atomic
magnetometers was demonstrated, by applying mT-level magnetic field pulses
synchronized with the amplitude modulation of the pump light. A PCB coil pair
encapsulates the miniaturized Cs vapor cell to produce this strong magnetic
field along the optical pumping axis, whilst simultaneously resistively
heating the vapor to the optimal atomic density for maximum sensitivity
performance. No magnetic noise is produced as the coils are only active when
optically pumping the atoms, and are rapidly demagnetized to near zero prior
to spin readout. Tests of the demagnetization circuit’s transient response
yielded a fall time of $213\,$ns, closely matching theoretical predictions.
Furthermore, a digital Hilbert transform applied to individual FID traces
revealed no signature of the demagnetization process; this is noteworthy given
that accuracy is one of the key benefits of the FID modality. The spin
polarization prepared through EOP is consistent over a wide range of bias
fields which aids in maintaining the sensor accuracy. Systematics in the
Larmor frequency measurement induced by heading errors remain stable under
various bias fields. Heading error can be corrected analytically based on the
degree spin polarization [45]; however, this compensation becomes more
straightforward under consistent optical pumping conditions.
An optimal magnetic sensitivity of $238\,\mathrm{fT/\sqrt{Hz}}$ was achieved
in a bias field of $50\,\mu$T using EOP providing an improvement over existing
optical pumping schemes. The sensor noise floor closely matched CRLB noise
density predictions indicating that magnetic noise, e.g., arising from
electrical heating or the bias field, is not a limiting factor. Therefore,
sensitivity improvement can only be achieved by increasing the signal
amplitude, as the magnetometer is mostly photon shot-noise limited especially
at higher Larmor frequencies, or extending the spin coherence time. Optical
rotation experienced by the probe beam could be doubled by placing a reflector
after the vapor cell in a double-pass geometry. A compact device would also
benefit from this geometry due to the reduced standoff distance between the
vapor cell and the signal source of interest. Recent novel fabrications
techniques [32] make it feasible to manufacture thicker MEMS cells that
exhibit higher signal amplitudes and longer spin relaxation times.
Furthermore, the customizable cell geometries improve the available optical
access that can be interrogated. One could also lower the noise density,
$\rho_{A}$, slightly by implementing a faster low-noise DAQ system with higher
bit-resolution, e.g., using field programmable gate arrays (FPGAs).
The EOP strategy is ideally suited to sensor commercialization as it provides
a scalable solution that limits the hardware and software requirements for
robust operation. For example, sensors based on resonantly driven spin
precession require electronic feedback loops to maintain optimal performance
which increases complexity. Furthermore, more elaborate cell heating methods
are required to prevent stray magnetic fields from raising the sensor noise
floor. The magnetometer’s topology could be made more compact and portable by
replacing the pump and probe lasers with a vertical-cavity surface-emitting
laser (VCSEL). The pump light would be frequency modulated in this case, as
opposed to utilizing amplitude modulation with an AOM [26]. The power
consumption of the pre-polarizing field and cell heating in the current
configuration is approximately 0.8 W. This sort of power can be easily
accessible with USB-C [46]. The power draw can be further optimized by the
modification to coil geometry, balancing current draw and field induced. One
could envision a high-performance finite-field sensor with minimal optical
components including: at least one VCSEL for pumping and probing, a quarter-
wave plate, MEMS vapor cell, PCB coils for EOP and heating, and a balanced
detector.

Funding Innovate UK (ISCF-42186).

Acknowledgments AM was supported by a Ph.D. studentship from the Defence
Science and Technology Laboratory (Dstl).

Disclosures The authors declare that there are no conflicts of interest
related to this article.

Data availability Data underlying the results presented in this manuscript are
available in Ref. [47].

## References

  * [1] C. Cohen-Tannoudji, J. Dupont-Roc, S. Haroche, and F. Laloë, “Diverses résonances de croisement de niveaux sur des atomes pompés optiquement en champ nul ii. applications à la mesure de champs faibles,” Revue de Physique Appliquée 5, 102–108 (1970).
  * [2] N. Castagna and A. Weis, “Measurement of longitudinal and transverse spin relaxation rates using the ground-state Hanle effect,” Physical Review A 84, 053421 (2011).
  * [3] R. Jimenez-Martinez, S. Knappe, and J. Kitching, “An optically modulated zero-field atomic magnetometer with suppressed spin-exchange broadening,” Review of Scientific Instruments 85, 045124 (2014).
  * [4] J. Allred, R. Lyman, T. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Physical Review Letters 89, 130801 (2002).
  * [5] V. Shah and M. V. Romalis, “Spin-exchange relaxation-free magnetometry using elliptically polarized light,” Physical Review A 80, 013416 (2009).
  * [6] J. Li, W. Quan, B. Zhou, Z. Wang, J. Lu, Z. Hu, G. Liu, and J. Fang, “Serf atomic magnetometer–recent advances and applications: A review,” IEEE Sensors Journal 18, 8198–8207 (2018).
  * [7] H. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Applied Physics Letters 97, 151110 (2010).
  * [8] E. Boto, S. S. Meyer, V. Shah, O. Alem, S. Knappe, P. Kruger, T. M. Fromhold, M. Lim, P. M. Glover, P. G. Morris, R. Bowtell, G. R. Barnes, and M. J. Brookes, “A new generation of magnetoencephalography: Room temperature measurements using optically-pumped magnetometers,” NeuroImage 149, 404–414 (2017).
  * [9] J. Osborne, J. Orton, O. Alem, and V. Shah, “Fully integrated standalone zero field optically pumped magnetometer for biomagnetism,” in _Steep dispersion engineering and opto-atomic precision metrology XI,_ vol. 10548 (SPIE, 2018), pp. 89–95.
  * [10] E. Boto, N. Holmes, J. Leggett, G. Roberts, V. Shah, S. S. Meyer, L. D. Muñoz, K. J. Mullinger, T. M. Tierney, S. Bestmann, G. R. Barnes, R. Bowtell, and M. J. Brookes, “Moving magnetoencephalography towards real-world applications with a wearable system,” Nature 555, 657–661 (2018).
  * [11] N. Holmes, T. M. Tierney, J. Leggett, E. Boto, S. Mellor, G. Roberts, R. M. Hill, V. Shah, G. R. Barnes, M. J. Brookes, and R. Bowtell, “Balanced, bi-planar magnetic field and field gradient coils for field compensation in wearable magnetoencephalography,” Scientific Reports 9, 14196 (2019).
  * [12] V. Gerginov, S. Krzyzewski, and S. Knappe, “Pulsed operation of a miniature scalar optically pumped magnetometer,” JOSA B 34, 1429–1434 (2017).
  * [13] Y. Guo, S. Wan, X. Sun, and J. Qin, “Compact, high-sensitivity atomic magnetometer utilizing the light-narrowing effect and in-phase excitation,” Applied Optics 58, 734–738 (2019).
  * [14] V. Lucivero, W. Lee, T. Kornack, M. Limes, E. Foley, and M. Romalis, “Femtotesla nearly-quantum-noise-limited pulsed gradiometer at earth-scale fields,” Physical Review Applied 18, L021001 (2022).
  * [15] C. Beggan and M. Musur, “Observation of ionospheric Alfvén resonances at 1–30 Hz and their superposition with the Schumann resonances,” Journal of Geophysical Research: Space Physics 123, 4202–4214 (2018).
  * [16] H. Korth, K. Strohbehn, F. Tejada, A. G. Andreou, J. Kitching, S. Knappe, S. J. Lehtonen, S. M. London, and M. Kafel, “Miniature atomic scalar magnetometer for space based on the rubidium isotope 87Rb,” Journal of Geophysical Research: Space Physics 121, 7870–7880 (2016).
  * [17] A. J. Canciani, “Magnetic navigation on an F-16 aircraft using online calibration,” IEEE Transactions on Aerospace and Electronic Systems 58, 420–434 (2021).
  * [18] M. Limes, E. Foley, T. Kornack, S. Caliga, S. McBride, A. Braun, W. Lee, V. Lucivero, and M. Romalis, “Portable magnetometry for detection of biomagnetism in ambient environments,” Physical Review Applied 14, 011002 (2020).
  * [19] S. Appelt, A. B.-A. Baranga, A. Young, and W. Happer, “Light narrowing of rubidium magnetic-resonance lines in high-pressure optical-pumping cells,” Physical Review A 59, 2078 (1999).
  * [20] T. Scholtes, V. Schultze, R. IJsselsteijn, S. Woetzel, and H.-G. Meyer, “Light-narrowed optically pumped Mx magnetometer with a miniaturized Cs cell,” Physical Review A 84, 043416 (2011).
  * [21] Z. D. Grujić, P. A. Koss, G. Bison, and A. Weis, “A sensitive and accurate atomic magnetometer based on free spin precession,” The European Physical Journal D 69, 1–10 (2015).
  * [22] D. Hunter, R. Jiménez-Martínez, J. Herbsommer, S. Ramaswamy, W. Li, and E. Riis, “Waveform reconstruction with a Cs based free-induction-decay magnetometer,” Optics express 26, 30523–30531 (2018).
  * [23] D. Hunter, T. E. Dyer, and E. Riis, “Accurate optically pumped magnetometer based on Ramsey-style interrogation,” Optics Letters 47, 1230–1233 (2022).
  * [24] N. Wilson, C. Perrella, R. Anderson, A. Luiten, and P. Light, “Wide-bandwidth atomic magnetometry via instantaneous-phase retrieval,” Physical Review Research 2, 013213 (2020).
  * [25] S. Ingleby, P. Griffin, T. Dyer, M. Mrozowski, and E. Riis, “A digital alkali spin maser,” Scientific Reports 12, 1–7 (2022).
  * [26] D. Hunter, S. Piccolomo, J. Pritchard, N. Brockie, T. Dyer, and E. Riis, “Free-induction-decay magnetometer based on a microfabricated Cs vapor cell,” Physical Review Applied 10, 014002 (2018).
  * [27] V. Gerginov, M. Pomponio, and S. Knappe, “Scalar magnetometry below 100 fT/Hz1/2 in a microfabricated cell,” IEEE Sensors Journal 20, 12684–12690 (2020).
  * [28] R. Jiménez-Martínez, W. C. Griffith, Y.-J. Wang, S. Knappe, J. Kitching, K. Smith, and M. D. Prouty, “Sensitivity comparison of Mx and frequency-modulated bell–bloom Cs magnetometers in a microfabricated cell,” IEEE Transactions on Instrumentation and Measurement 59, 372–378 (2009).
  * [29] M. C. Tayler, K. Mouloudakis, R. Zetter, D. Hunter, V. G. Lucivero, S. Bodenstedt, L. Parkkonen, and M. W. Mitchell, “Miniature biplanar coils for alkali-metal-vapor magnetometry,” Physical Review Applied 18, 014036 (2022).
  * [30] D. Sheng, A. R. Perry, S. P. Krzyzewski, S. Geller, J. Kitching, and S. Knappe, “A microfabricated optically-pumped magnetic gradiometer,” Applied Physics Letters 110, 031106 (2017).
  * [31] I. Kominis, T. Kornack, J. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422, 596–599 (2003).
  * [32] S. Dyer, P. Griffin, A. Arnold, F. Mirando, D. Burt, E. Riis, and J. McGilligan, “Micro-machined deep silicon atomic vapor cells,” Journal of Applied Physics 132, 134401 (2022).
  * [33] V. Schultze, T. Scholtes, R. IJsselsteijn, and H.-G. Meyer, “Improving the sensitivity of optically pumped magnetometers by hyperfine repumping,” JOSA B 32, 730–736 (2015).
  * [34] M. S. Mrozowski, I. C. Chalmers, S. J. Ingleby, P. F. Griffin, and E. Riis, “Ultra-low noise, bi-polar, programmable current sources,” Review of Scientific Instruments 94, 014701 (2023).
  * [35] K. G. McKay, “Avalanche breakdown in silicon,” Phys. Rev. 94, 877–884 (1954).
  * [36] K. Walters, “Zeners and transient voltage suppressors: Can either device be used for the same applications?” Tech. Rep. 134, Microsemi (now part of Microchip), One Enterprise, Aliso Viejo, CA 92656 USA (2018).
  * [37] Keithley Instruments, LLC (now part of Tektronix, Inc), Solon, OH, USA, _Low Level Measurements Handbook_ , 7th ed. (2016).
  * [38] P. Welch, “The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Transactions on audio and electroacoustics 15, 70–73 (1967).
  * [39] A. Moschitta and P. Carbone, “Cramér–Rao lower bound for parametric estimation of quantized sinewaves,” IEEE Transactions on Instrumentation and Measurement 56, 975–982 (2007).
  * [40] Y.-X. Yao and S. M. Pandit, “Cramér-Rao lower bounds for a damped sinusoidal process,” IEEE Transactions on Signal Processing 43, 878–885 (1995).
  * [41] C. Gemmel, W. Heil, S. Karpuk, K. Lenz, C. Ludwig, Y. Sobolev, K. Tullney, M. Burghoff, W. Kilian, S. Knappe-Grüneberg, W. Mueller, A. Schnabel, F. Seifert, L. Trahms, and S. Baessler, “Ultra-sensitive magnetometry based on free precession of nuclear spins,” The European Physical Journal D 57, 303–320 (2010).
  * [42] G. Bao, A. Wickenbrock, S. Rochester, W. Zhang, and D. Budker, “Suppression of the nonlinear zeeman effect and heading error in earth-field-range alkali-vapor magnetometers,” Physical Review Letters 120, 033202 (2018).
  * [43] S. J. Seltzer, _Developments in alkali-metal atomic magnetometry_ (Princeton University, 2008).
  * [44] R. Zhang, D. Kanta, A. Wickenbrock, H. Guo, and D. Budker, “Heading-error-free optical atomic magnetometry in the earth-field range,” Physical Review Letters 130, 153601 (2023).
  * [45] W. Lee, V. Lucivero, M. Romalis, M. Limes, E. Foley, and T. Kornack, “Heading errors in all-optical alkali-metal-vapor magnetometers in geomagnetic fields,” Physical Review A 103, 063103 (2021).
  * [46] USB 3.0 Promoter Group, _Universal Serial Bus Type-C Cable and Connector Specification_ (2022). Release 2.2.
  * [47] D. Hunter, M. S. Mrozowski, A. McWilliam, S. J. Ingleby, T. E. Dyer, P. F. Griffin, and E. Riis, Data for: "Optical pumping enhancement of a free-induction-decay magnetometer", University of Strathclyde, https://doi.org/10.15129/49adaba8-c05c-4e18-beca-18f052123984 (2023).

references