[1]Wolfgang Klassen

[1]Physics and Astronomy, University of British Columbia, 6224 Agricultural
Rd, Vancouver, V6T 1Z1, British Columbia, Canada

2]Chemistry, University of British Columbia, 2036 Main Mall, Vancouver, V6T
1Z1, British Columbia, Canada

3]Physics, University of Winnipeg, 515 Portage Avenue, Winnipeg, R3B 2E9,
Manitoba, Canada

4]Physics and Astronomy, University of Manitoba, 30A Sifton Road, Winnipeg,
R3T 2N2, Manitoba, Canada

5]Southwest Sciences, 1570 Pacheco Street, Santa Fe, 87505, New Mexico, United
States

6]Department of Physics and CQIQC, University of Toronto, Toronto, M5S 1A7,
Ontario, Canada

# A magnetically silent optically pumped magnetometer for a neutron electric
dipole moment experiment

<EMAIL_ADDRESS>Shomi Ahmed  Kiera Pond Grehan  Chris Hovde  Kirk W.
Madison  Russell R. Mammei  Jeffery W. Martin  Mark McCrea  Tahereh Mohammadi
Takamasa Momose  Patrick Opsahl  David C.M. Ostapchuk  *  [  [  [  [  [

###### Abstract

We report the performance of a magnetically silent optically pumped cesium
magnetometer with a statistical sensitivity of 3.9 pT and a stability of 90 fT
over 150 seconds of measurement. Optical pumping with coherent, linearly-
polarized, resonant light leads to a relatively long-lived polarized ground
state of the cesium vapour contained in a measurement cell. The state
precesses at its Larmor frequency in the magnetic field to be measured.
Nonlinear magneto-optical rotation then leads to the rotation of the plane of
polarization of a linearly polarized probe laser beam. The rotation angle is
modulated at twice the Larmor frequency. A measurement of this frequency
constitutes an absolute measurement of the magnetic field magnitude. Featuring
purely optical operation, non-magnetic construction, low noise floor, and high
stability, this sensor will be used for the upcoming TUCAN electric dipole
moment experiment and other highly sensitive magnetic applications.

###### keywords:

magnetometer, magnetometry, neutron electric dipole moment, all-optical,
alkali, atomic vapour, magnetically silent, nonlinear magneto-optical
rotation, Faraday rotation, Bell-Bloom, Cesium, free spin precession

##  1 Introduction

Experiments at the frontiers of precision low-energy measurements push the
limits of existing measurement technology. The search for a non-zero neutron
electric dipole moment (nEDM) is pursued by several labs around the world [1,
2, 3, 4, 5, 6]. A non-zero nEDM would have a very high impact on physics
beyond the standard model, and could help explain the strong charge-parity
(CP) problem [7, 8], the baryon asymmetry observed in the universe [9, 10], or
sources of CP violation beyond the standard model [11, 12]. In particular, the
sensitivity of the nEDM to strong sector physics makes this measurement
attractive above other fundamental EDM searches such as the measurement of the
electron EDM, which has been done to very high precision [13].

The current best measurement of the nEDM is $d_{n}=0.0\pm
1.1_{\mathrm{stat}}\pm 0.2_{\mathrm{sys}}\times 10^{-26}$ $e\cdot\mathrm{cm}$
[14], performed at the Paul Scherrer Institute (PSI) by the PSI nEDM
collaboration. The TRIUMF Ultracold Advanced Neutron (TUCAN) collaboration
[15] aims to achieve an experimental sensitivity of $10^{-27}$
$e\cdot\mathrm{cm}$ using a new spallation-driven superfluid helium ultracold
neutron (UCN) source at TRIUMF [16, 17, 18, 19].

Measurements of the nEDM require extremely well measured and controlled
magnetic fields [20, 21, 14]. Cs magnetometry was used in conjunction with Hg
comagnetometry in the latest upper limit paper [14]. The Cs sensors used by
the PSI nEDM collaboration were demonstrated to have sensitivities of 0.75-8
pT/$\sqrt{\mathrm{Hz}}$ and absolute accuracies of 45-90 pT per measurement
[20]. However, due to the method by which these sensors are operated they are
not magnetically silent (see section 2), and can therefore not be used while
neutrons are being measured and thus have limited utility.

Over the past several decades an alternate method [22] of creating and probing
polarized alkali atoms has been developed that uses non-linear magneto-optical
rotation (NMOR), which allows for the development of robust, drift-stable,
completely magnetically silent optical magnetometers [23, 24].

In this work we demonstrate a statistical sensitivity of 3.9 pT and a
stability of 90 fT over 150 seconds of measurement in a configuration that
operates purely optically, using NMOR to both create and probe a polarized
atomic state. This allows precision magnetic field measurements to be done
during the EDM experiment without perturbing the neutrons [25].

##  2 Configuration

Figure 1: The beam geometry for this style of magnetometry. Both the pump and
probe beams are linearly polarized, and the electric field vectors of the two
light fields are aligned. The probe beam propagates through the vapor cell
along the magnetic field vector to be measured, while the pump can be anywhere
in the plane formed by the two beams in this figure. The magnetic field
information is encoded in the time dependence of the polarization angle $\phi$
of the probe beam after it exits the vapour cell

Atomic magnetometry involves interrogating the Larmor frequency of atoms
exposed to a magnetic field. Generally, this involves the creation of a large
coherently precessing population of atoms in some relatively long-lived
polarization state. The polarized atoms then precess in the magnetic field
they experience. The precession is probed optically using resonant light, and
the measured magnetically sensitive quantity is the Larmor frequency of the
polarized atoms [22, 26].

We have chosen a mode of operation which operates similarly to a Bell-Bloom
configuration [27], with some modifications to bring the operation away from
zero field. This follows work done by Higbie et. al. [26] developing NMOR
based alkali magnetometry and also work done by Gruj́ic et. al. [28] in
developing the FSP mode of operation.

In order to illustrate the advantages of this configuration we first present a
brief overview of the sensors used in the PSI nEDM collaboration’s latest
upper limit paper [14], which operate in an $M_{x}$ configuration [20],
similar to some commercially available alkali magnetometers. Circularly
polarized light (CPL) propagates at a 45∘ angle to the measured field,
creating the polarized population along the direction of propagation. A radio
frequency (RF) magnetic field resonant with the polarized state drives the
polarization into the plane transverse to the magnetic field, causing the
state to precess. The absorption of the CPL is proportional to the transverse
magnetization of the atoms, so the Larmor frequency is accessible via
monitoring the intensity of the CPL after it passes through the cell. The same
beam is used for both tasks. In the main configuration used during nEDM
measurements the absorption signal is phase-shifted and fed back to the RF
coils such that the system self-oscillates at the Larmor frequency of the
atoms. This phase is then proportional to the difference between the driving
frequency and the larmor frequency of the atoms. By locking this phase to 0,
the frequency of the RF coil is identified as the Larmor frequency, and the
field is measured.

This configuration is robust and high-bandwidth, but due to the RF it is not
magnetically silent and is subject to two main types of error: errors in
determining the Larmor frequency (i.e. errors due to drifting phase or
electronic changes due to temperature changes) and systematic effects that
change the Larmor frequency itself, making it an inaccurate measure of the
relevant field. It is also subject to potential long-range crosstalk due to
the RF used in the sensors.

The PSI experiment avoided the first kind of error by using a Free Spin
Precession (FSP) mode of operating. Instead of feeding the absorption signal
back to the RF coils, the atoms were allowed to freely precess after the RF
pulse is applied, and the frequency of the resulting oscillating absorption
signal was measured. This is a phase-error-free measurement of the Larmor
frequency, and it was used to verify the offsets present in the main mode of
operation. However, the FSP operation mode was not an option during nEDM
measurement runs because the RF pulses used in this mode were repeated at
close to the Larmor frequency of the mercury comagnetometer and would
potentially cause interference.

PSI carefully studied and controlled the various shifts associated with the
second kind of error: shifts of the Larmor frequency itself. The largest
effect was due to the AC Stark shift, which is caused by the component of CPL
light propagating along the measured field. This was found to cause shifts of
$\pm$10 pT to $\pm$50 pT which were correlated with the light intensity and
frequency [14]. These shifts were only observable using auxiliary offline
measurements, so were not able to be constrained any further than this.

The Bell-Bloom style FSP configuration used in our system also measures the
Larmor frequency of Cs to deduce the magnetic field experienced by the atoms;
however, it uses nonlinear magneto-optical rotation [26, 29] rather than
absorption to both polarize the atoms and interrogate their Larmor frequency.
In this configuration, the electric field vector of both the pump and probe
beams are perpendicular to the magnetic field, as in figure 1. When the pump
beam is turned on, atoms in the beam are driven into a coherent non-
interacting state, polarizing the atoms along the electric field vector of the
pump light. Since we are using linearly polarized light, the polarization is
an aligned state rather than the oriented state used by the $M_{x}$
configuration. Once they are driven into non-interacting states, the atoms
immediately start to precess in the magnetic field, rotating them towards
being interacting states. So as to not de-pump these atoms, the pump beam is
then quickly turned off again. After half of one Larmor period the previously
polarized atoms are again non-interacting, and the pump beam is switched back
on, creating another set of polarized atoms in phase with the previously
created atoms. By repeating this process, a large population of such polarized
atoms can be produced. Once the population is maximized, the pump beam is
switched off. A separate, much weaker probe beam is then passed through the
atoms, propagating parallel to the magnetic field. The polarized state of the
Cs exhibits an axis of birefringence that precesses along with the atoms. This
causes the angle of the linearly polarized probe light, $\phi$, to be
modulated at twice the Larmor frequency of the atoms because of the twofold
symmetry of the aligned state. This modulation is detected via polarimetry
after the probe beam exits the vapour.

The FSP configuration eliminates the issue of RF crosstalk and interference by
eliminating the use of RF entirely. The issue of phase drift is eliminated by
running exclusively in FSP mode, while the issue of the vector light shift is
reduced by using linearly polarized light for the probe beam. Additionally,
servicing all sensors with a single probe laser means any systematic shifts
associated with laser diode intensity or frequency fluctuations should not
impact relative measurements between sensors, meaning gradient extraction is
not affected. These same changes are planned by the PSI collaboration for
their upcoming n2EDM experiment [2] and have been implemented by the panEDM
collaboration as well [1, 30, 31, 32].

##  3 Apparatus

Figure 2: Schematic view of the sensor head, showing the optical components.
The two beams propagate orthogonally to each other, but their electric field
vectors are aligned. The most sensitive measurement axis for the magnetic
field is indicated

Figure 2 shows a schematic view of the optical components of the sensor head,
figure 3 shows a picture of the sensor head in the same orientation, and
figure 4 shows the physical setup and highlights the parts of the beam that
are free space vs contained in fibre optics. The laser diodes (indicated as Cs
D2 laser in figure 4) are held in precision motion stages on an optical
breadboard. Half wave plates and linear polarizers (indicated as $\lambda/2$
and LP respectively) are inserted into this free-space section of beam to
control the polarization with which the beams are launched into the fibres.
After being launched into fibre optics the light is split so that multiple
sensors can run on the same diode. Fibre optics transport the light into the
magnetically shielded environment. After leaving the sensor head the light is
carried back out of the magnetically shielded environment via large diameter
multimode (MM) fibres. The light from the MM fibres is detected via
photodiodes mounted to the polarimetry boards, the signals of which are
digitized by our DAQ system for analysis.

The laser diodes are EYP-DFB-0852-00150-1500-TOC03-0005 from Toptica
Eagleyard. They can be locked using a dichroic atomic vapour laser locking
(DAVLL) system, although the diodes are stable enough to not require locking
for short-term operation, on the order of hours. The diodes are powered by
highly stable diode current supplies from Vescent (D2-005 supply, D2-105-200
controller), and feedback control for the DAVLL is possible with Vescent laser
lock boxes (D2-125 servo). Locking was not needed for these studies, we
verified the laser frequency carefully before and after each measurement to
ensure it was stable by sweeping over the D2 absorption lines and locating our
frequency relative to the absorption peaks. The diodes are collimated into a
$\sim$3 mm2 free-space beam for a short length, into which polarization optics
can be placed to specify launch polarization/intensity. After this short free-
space beam, the lasers are launched into fibre.

All fibre in this system before the sensor is either polarization maintaining
(PM) or polarizing (PZ) fibre. This ensures very clean linear polarization in
the sensor head. The pump laser is launched into the input fibre of a fibre-
coupled acousto-optic modulator (AOM). This is then attached to a $1\times 8$
fibre splitter (also using PM fibre) which splits the pump light equally into
8 pigtailed PM fibres, each of which can provide pump light to a sensor. The
probe light is launched directly into another $1\times 8$ PM splitter. The
overall intensity of the probe beams are controlled by a linear polarizer and
half-wave plate in the free space section of the beam. Individual ports can be
additionally attenuated using an adjustment screw on the body of the splitter.
The free space beams are $\sim$100 mW, typical launch efficiencies with the
current opto-mechanical setup are 20-30%, leading to $\sim$1-2 mW available
for use in the sensor heads after accounting for all losses from splitting and
attenuation. This is sufficient power to optimize the pump light [25] and more
than enough to provide probe power. The pump beams can be globally attenuated
by adjusting the high level of the square wave operating the AOM. The probe
beams are typically $<5~{}\mu$W in the sensor head after being adjusted using
screw-attenuators built into the fibre splitter output ports. $<5~{}\mu$W is
enough to reach shot-noise-limited performance in the photodiodes without
limiting the coherence time of the Cs via de-pumping. The exact power is
optimized to minimize the uncertainty in the frequency fit of the FSP signal.

The magnetically sensitive component of these sensors is an evacuated glass
cell with a small amount of Cs held in a reservoir, attached to the cell via a
capillary tube. This cell was manufactured by Precision Glass Blowing (PGB)
[33]. The cell is treated with an anti-relaxation alkene coating available
from PGB prior to being filled, which greatly extends the polarization
lifetime of the atoms in the cell by reducing polarization-destroying wall
collisions. Light is delivered to the Cs cell where it is collimated into a
free-space beam to pass through the cell and interact with the Cs. After
passing though the Cs cell the probe beam is directed through a polarizing
beamsplitter and the resulting two beams are launched into MM fibre to be
carried out of the magnetically sensitive environment for detection and
analysis. This sensor head was designed and manufactured by Southwest Sciences
[34] in consultation with W. Klassen and J. Martin, and incorporates design
improvements developed in Munich [1, 30, 31, 32].

Figure 3: Cs sensor serial number 19, shown in the same orientation as the
schematic diagram in figure 2

The optical components are held in a plastic clamshell with a very similar
design to the current panEDM Cs design [30], 3D printed in undyed ABS plastic.
Optical components are cemented to one another at the optical interfaces using
RD3-74 epoxy [35]. The epoxy provides the structural connection and maintains
alignment between optical components, with the clamshell providing a mounting
point for the whole sensor. Linearly polarized light from two laser diodes is
delivered to the sensor head via polarizing fibre optics. A gradient-index
(GRIN) lens launches light from the probe fibre to a $\sim$1 mm2 free-space
beam, which is then directed through a clean-up linear polarizer, followed by
the vapour cell via internal reflection through a right-angled prism. This
probe launching assembly is mounted to the main ABS body using a coupling
which constrains the motion of the probe assembly to linear motion in a plane,
which helps with alignment. Linear translation stages are then used to
position the assembly to maximize coupling into the exit fibers prior to
cementing the prism to the face of the vapour cell as the final step in
optical alignment. The beam is then again redirected via a right-angled prism
through a Rochon walk-off polarizing beam splitter which splits the beam into
orthogonal, linearly polarized components. These components are then launched
back into fibre optics: much larger core MM fibres are used on this side so as
to collect the most light possible. The two polarization components are then
used for polarimetry of the probe light. A GRIN lens also launches linearly
polarized pump light into the cell after passing it through a clean-up
polarizer, due to the geometry of our setup it can pass straight through the
cell at 90∘ to the probe beam, since the electric field vectors of the two
beams are aligned. This choice of beam geometry was largely driven by
considering ease of manufacture, launching both beams into the end of the
vapour cell would be difficult to do. Any loss in pumping efficiency due to
the shorter optical path length can be compensated with pump power.

The Rochon prism is oriented with its fast-axis at a 45∘ angle relative to the
probe beam polarization axis so that it evenly splits the unperturbed probe
light into two components of equal power. This means that for small rotations,
the degree of rotation of polarized light is directly proportional to the
difference in power between the two components. The MM fibres are plugged into
a polarimetry board, which amplifies the photo-current difference from the two
photodiodes watching the two fibres. The resulting signal is then captured on
an oscilloscope, or via a 32 channel DAQ system made by D-tAqc [36]. The
D-tAqc system has a bandwidth of 200 kHz with 16 bit resolution. The
polarimetry board differencing circuit has a high-frequency cutoff of around
15 kHz, and is AC coupled since we are not interested in static imbalances in
the two polarization channels. This relatively low frequency cut-off was due
to regulatory requirements imposed on the manufacturer rather than any physics
motivation.

Figure 4: Schematic view of the optics that are not contained in the sensor
head. The free space optics, before the fibres, are bolted to an optical
breadboard. This configuration can operate 8 sensors simultaneously, although
this has not been done due to the space limitations of our magnetically
shielded volumes. Labelled elements are a half-wave plate ($\lambda/2$),
linear polarizer (LP), acousto-optic modulator (AOM), multi-mode optical fibre
(MM), and data-acquisition system (DAQ) Figure 5: The amplified, differential
photo-current corresponding to the degree of rotation of polarized light as a
function of time during the polarization and the free spin precession phases
of the Cs gas. Pumping begins at T=0 on this plot, and stops just after 40 ms
when the signal reaches its maximum. During this time the pump beam is
modulated into a 20% duty cycle square wave at $\sim$7 kHz, approximately the
Larmor frequency of Cs in a 1 $\mu$T field. The remainder of the signal is the
free spin precession portion which is later fit to an exponentially decaying
sine wave. A single instance of this pump-probe cycle is sometimes referred to
as one free spin precession cycle, or one “FSP” cycle Figure 6: The measured
magnetic field (blue line) and measured coil voltage (orange line) versus
time. The voltage across is being monitored by an 8.5 digit multimeter from
Keithley, which is set to an integration time of 4/60 of a second, or 4 power
line cycles, matching the integration time of the FSP signal measuring the
magnetic field

The probe laser frequency is tuned near the Doppler broadened F=4
$\rightarrow$ F′=3,4,5 D2 absorption line of Cs at 852 nm. The pump laser is
tuned to the Doppler broadened F=3 $\rightarrow$ F′=2,3,4 absorption line. The
particular frequencies are chosen to maximize the magnetometry signal. The
pump is adjusted until a maximally polarized ensemble is reached. The probe
adjustment is a trade-off between amplitude and $T_{2}$. Generally a high
amplitude probe signal is correlated with a poor $T_{2}$, and vice versa. The
frequency is adjusted until the area under the decay envelope is maximized,
which results in the best fit uncertainty. This envelope is visible in figure
5. Typically the best probe laser frequency is found around 200 MHz below the
F=4 $\rightarrow$ F′=3,4,5 absorption peak. The best pump laser frequency is
directly at the peak of the F=3 $\rightarrow$ F′=2,3,4 absorption signal,
about 9.2 GHz away from the probe frequency.

The pump is modulated by the AOM at twice the Larmor frequency, $2f_{L}\sim$7
kHz at 1 $\mu$T. The duty cycle and number of cycles are manually adjusted to
maximize the amplitude of the signal at the start of the probe cycle. Since
the pump is shut off during the probe cycle, this is the only relevant measure
of goodness for the pump parameters. A 20% duty cycle is typical.

To operate in 1 $\mu$T we use passive magnetic shielding to reduce both
earth’s field and fluctuations in it. The shield is four nested cylinders of
$\mu$-metal, a highly magnetically permeable alloy of steel. The outer
cylinder is 40 cm long with a 15 cm radius. The innermost volume contains a 3D
printed saddle-wound coil designed to provide a homogeneous transverse field
inside the shield. The volume inside this coil is 20 cm long with radius 3.5
cm. The axial shielding factor has been measured [37] to be 1.4$\times 10^{7}$
with all endcaps installed, and while the transverse shielding factor has not
been measured it is expected to be much higher. Since this inner volume is
quite small, the uniformity is on the order of nT/cm. This is quite non-
uniform, but provides very high AC and DC shielding factors. All sensor tests
are done in this shield.

The current for the transverse coil is provided by a custom made power supply
[38] designed and built by Shomi Ahmed at the University of Winnipeg. The
supply is designed to provide a constant 10 mA and is only adjustable via the
replacement of internal components. The supply has a monitor output that
produces a directly proportional voltage by measuring across a precision 1
$\Omega$ foil resistor. Because of the inability of this power supply to
change its output, a secondary winding is layered in the same wire path to
provide another nearly identical coil so that an additional nT-scale field can
be applied with a second power supply, which only has to supply $\mu$A-scale
current and can be battery operated to reduce noise. This secondary supply was
designed and built at TRIUMF. This additional field allows us to measure the
relative scaling of two sensors by measuring the same nT-scale field step with
both sensors simultaneously.

##  4 Results

Figure 7: Square root of the power spectral density (PSD) of a full second of
the unfiltered $\sim$ 7kHz sinusoid measured by the polarimetry board. The 7
kHz is generated by continuously flashing the pump beam at the Larmor
frequency of the cesium, and the sensor is held at 1 $\mu$T for the test. The
total power of the probe beam was 3.9 $\mu$W, shot noise limit shown as the
dotted line. The average noise power in the frequency band of interest is 1.4
pW/$\sqrt{\mathrm{Hz}}$, and the amplitude of the sinusoid is 3 nW for a
signal to noise ratio (SNR) of 2142 Figure 8: The Allan deviation of the
magnetic field and the coil voltage measurements showing the stability of the
signals at different integration times. The minimum integration time for this
Allan deviation is $\sim$ 350 with a deviation of 3.9 $\pm$ 0.1 pT, in good
agreement with the Cramer-Rao lower bound. The global minimum is 90$\pm$8 fT
at an integration time of 150 seconds

Figure 6 shows a magnetic field measurement with the system. After being split
into individual FSPs the signal is filtered with a windowed-sinc FIR filter,
with a pass-band from 6-8 kHz, transitional bands of 2 kHz on each side, and
$<0.04$ dB of ripple in the pass band. The filtered signal is fit to an
exponentially decaying sine wave

$V(t)=Ae^{-t/T_{2}}\sin(\omega t+\Phi)+\mathrm{Offset},$ (1)

where $A$ is the amplitude, $T_{2}$ is the time constant of the decay
envelope, $\omega$ is the frequency of the precession, and $\Phi$ is the
phase. $\omega$ is directly scaled to be the magnetic field strength via the
gyromagnetic ratio of Cs. While this data was taken, the coil voltage was
being monitored across a precision resistor using a digital multimeter (DMM)
with 8.5 digits of precision. The voltage is scaled to a magnetic field via
the ratio of the averages of the two data sets so they can be directly
compared without a perfectly accurate coil constant. The Allan deviation of
the same data is shown in Figure 8. The voltage monitor from the power supply
is significantly less stable than the sensor readings for most integration
times. We attribute this to environmental effects on both the DMM itself and
the connection to the monitor port. The sensor’s Allan deviation reaches a
minimum of 90$\pm$8 fT at an integration time of 150 seconds, so we can
conclude that our sensors are at least this stable. The minimum integration
time Allan deviation corresponds to the average deviation between successive
points and can be taken as a measurement of sensitivity. In this case the
sensitivity is 3.9 $\pm$ 0.1 pT.

Figure 9: Magnetic field measurements of two sensors (a and b) together with
the difference between them (c) as a function of time. The sensors are
operated simultaneously, and when the field is stepped up and down, the change
in field is observed in each sensor individually. The sensor measurements are
subtracted, as shown in (c), and the difference remains at their combined
noise floor. Because of the relatively inhomogeneous field in the magnetic
shield, the two sensors are operating at different fields and thus are not
able to be perfectly optimized simultaneously given they are driven by the
same optical system with a common pump cycle frequency. The higher noise floor
is the result of this. The spike in the difference signal around 60 seconds
can be associated with the changing magnetic field, and is most likely the
result of the field changing partway through an FSP ringdown

Figure 9 shows the result of operating two such sensors simultaneously. As
they measure the field, the field is stepped up and down. Each sensor clearly
sees this deviation, but the subtraction of the fields, shown in the lower
plot, remains at the noise floor of the subtracted signal. This shows that the
sensors have the same scaling to within the noise of this signal. The overall
noise in the subtracted signal is larger than the signal in Figure 6 for two
reasons: the subtraction of two signals adds their variances, and the variance
of each sensor is slightly higher than the minimum because each is operating
at a slightly different field and thus neither is being pumped at its local
Larmor frequency.

The shield is fairly inhomogeneous due to its small size, so the sensors are
in slightly different fields. Since they are serviced by a single AOM that
cycles the pump light, we must compromise on what frequency at which to pump
both. The subtracted data in the lower plot in figure 9 shows that the sensors
are measuring approximately a 1.35 nT difference between the two sensor
locations.

The sensitivity limit for this system is given by the Cramer-Rao lower bound
(CRLB) for a damped sinusoid, which is given by [39]:

$\sigma_{B}\geq\frac{\sqrt{12\cdot C}}{(A/\rho)\cdot T^{3/2}\cdot\gamma},$ (2)

with

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

where $A$ is the amplitude of the signal, $\rho$ is the average noise density
in the frequency band of interest in units of amplitude per root Hz, $T$ is
the total time of the FSP signal, $\gamma$ is the gyromagnetic ratio of cesium
in radians per second = 7.0196$\cdot 2\pi$, $N$ is the total number of points,
and $z=e^{-\frac{\Delta t}{T_{2}}}$ where $\Delta t$ is the sampling interval
and $T_{2}$ is the FSP time constant as defined in equation 1. $(A/\rho)$ is
the signal-to-noise ratio, written as SNR.

For the data shown in Figure 6 the time constant was 40 ms. With an average
SNR of 2142 the CRLB is 3.81 pT, in good agreement with the data.

##  5 Conclusion and Outlook

We have demonstrated the operation of a drift-stable, magnetically silent
magnetometer with a sensitivity of 3.9 pT and a drift stability of 90 fT over
150 seconds. This meets the requirements of the TUCAN nEDM experiment for
magnetometry [25]. Further studies demonstrating the operation of multiple
sensors, measuring crosstalk, offset, and light shift are being planned for
the completion of the TUCAN magnetically shielded room, currently under
construction at TRIUMF. The final magnetometric array will have 20 sensors
serviced by the same probe laser, eliminating the effects of laser frequency
and intensity variation on differential field measurements.

Acknowledgements

The authors thank Martin Rosner and Georg Bison for helpful communications
during development. The authors also thank all of our TUCAN colleagues for
support and guidance through these studies. W. Klassen wishes to thank Dr.
Derek Fujimoto for help copy-editing the final draft.

## Declarations

This research was supported by the Canada Foundation for Innovation; the
Canada Research Chairs program; the Natural Sciences and Engineering Research
Council of Canada (NSERC) SAPPJ-2016-00024, SAPPJ-2019-00031 and
SAPPJ-2023-00029; Japan Society for the Promotion of Science (JSPS) KAKENHI
grants 18H05230 and 20KK0069; JSPS KENHI grants 17K14307, 19K23442, 20K14487,
21K13940, 22H01236; RCNP CORENet; and the KEK-TRIUMF Exchange Program for
Early Career Researchers (EPECR). Chris Hovde and Southwest Sciences were
contracted for the design and construction of the fiberized sensors.

## References

  *   * Wurm, David et al. [2019] Wurm, David, Beck, Douglas H., Chupp, Tim, Degenkolb, Skyler, Fierlinger, Katharina, Fierlinger, Peter, Filter, Hanno, Ivanov, Sergey, Klau, Christopher, Kreuz, Michael, Lelièvre-Berna, Eddy, Lins, Tobias, Meichelböck, Joachim, Neulinger, Thomas, Paddock, Robert, Röhrer, Florian, Rosner, Martin, Serebrov, Anatolii P., Singh, Jaideep Taggart, Stoepler, Rainer, Stuiber, Stefan, Sturm, Michael, Taubenheim, Bernd, Tonon, Xavier, Tucker, Mark, Grinten, Maurits van der, Zimmer, Oliver: The PanEDM neutron electric dipole moment experiment at the ILL. EPJ Web Conf. 219, 02006 (2019) https://doi.org/10.1051/epjconf/201921902006
  * Abel, C. et al. [2019] Abel, C., Ayres, N. J., Ban, G., Bison, G., Bodek, K., Bondar, V., Chanel, E., Chiu, P.-J., Clement, B., Crawford, C., Daum, M., Emmenegger, S., Flaux, P., Ferraris-Bouchez, L., Griffith, W.C., Grujić, Z.D., Harris, P.G., Heil, W., Hild, N., Kirch, K., Koss, P.A., Kozela, A., Krempel, J., Lauss, B., Lefort, T., Lemière, Y., Leredde, A., Mohanmurthy, P., Naviliat-Cuncic, O., Pais, D., Piegsa, F.M., Pignol, G., Rawlik, M., Rebreyend, D., Ries, D., Roccia, S., Ross, K., Rozpedzik, D., Schmidt-Wellenburg, P., Schnabel, A., Severijns, N., Thorne, J., Virot, R., Voigt, J., Weis, A., Wursten, E., Zejma, J., Zsigmond, G.: The n2EDM experiment at the Paul Scherrer Institute. EPJ Web Conf. 219, 02002 (2019) https://doi.org/10.1051/epjconf/201921902002
  * Ito et al. [2018] Ito, T.M., Adamek, E.R., Callahan, N.B., Choi, J.H., Clayton, S.M., Cude-Woods, C., Currie, S., Ding, X., Fellers, D.E., Geltenbort, P., Lamoreaux, S.K., Liu, C.-Y., MacDonald, S., Makela, M., Morris, C.L., Pattie, R.W., Ramsey, J.C., Salvat, D.J., Saunders, A., Sharapov, E.I., Sjue, S., Sprow, A.P., Tang, Z., Weaver, H.L., Wei, W., Young, A.R.: Performance of the upgraded ultracold neutron source at Los Alamos National Laboratory and its implication for a possible neutron electric dipole moment experiment. Phys. Rev. C 97, 012501 (2018) https://doi.org/10.1103/PhysRevC.97.012501
  * Serebrov et al. [2015] Serebrov, A.P., Kolomenskiy, E.A., Pirozhkov, A.N., Krasnoschekova, I.A., Vassiljev, A.V., Polyushkin, A.O., Lasakov, M.S., Murashkin, A.N., Solovey, V.A., Fomin, A.K., Shoka, I.V., Zherebtsov, O.M., Geltenbort, P., Ivanov, S.N., Zimmer, O., Alexandrov, E.B., Dmitriev, S.P., Dovator, N.A.: New search for the neutron electric dipole moment with ultracold neutrons at ILL. Phys. Rev. C 92, 055501 (2015) https://doi.org/10.1103/PhysRevC.92.055501
  * Ahmed et al. [2019] Ahmed, M.W., Alarcon, R., Aleksandrova, A., Baeßler, S., Barron-Palos, L., Bartoszek, L.M., Beck, D.H., Behzadipour, M., Berkutov, I., Bessuille, J., Blatnik, M., Broering, M., Broussard, L.J., Busch, M., Carr, R., Cianciolo, V., Clayton, S.M., Cooper, M.D., Crawford, C., Currie, S.A., Daurer, C., Dipert, R., Dow, K., Dutta, D., Efremenko, Y., Erickson, C.B., Filippone, B.W., Fomin, N., Gao, H., Golub, R., Gould, C.R., Greene, G., Haase, D.G., Hasell, D., Hawari, A.I., Hayden, M.E., Holley, A., Holt, R.J., Huffman, P.R., Ihloff, E., Imam, S.K., Ito, T.M., Karcz, M., Kelsey, J., Kendellen, D.P., Kim, Y.J., Korobkina, E., Korsch, W., Lamoreaux, S.K., Leggett, E., Leung, K.K.H., Lipman, A., Liu, C.Y., Long, J., MacDonald, S.W.T., Makela, M., Matlashov, A., Maxwell, J.D., Mendenhall, M., Meyer, H.O., Milner, R.G., Mueller, P.E., Nouri, N., O’Shaughnessy, C.M., Osthelder, C., Peng, J.C., Penttila, S.I., Phan, N.S., Plaster, B., Ramsey, J.C., Rao, T.M., Redwine, R.P., Reid, A., Saftah, A., Seidel, G.M., Silvera, I., Slutsky, S., Smith, E., Snow, W.M., Sondheim, W., Sosothikul, S., Stanislaus, T.D.S., Sun, X., Swank, C.M., Tang, Z., Dinani, R.T., Tsentalovich, E., Vidal, C., Wei, W., White, C.R., Williamson, S.E., Yang, L., Yao, W., Young, A.R.: A new cryogenic apparatus to search for the neutron electric dipole moment. JINST 14(11), 11017 (2019) https://doi.org/10.1088/1748-0221/14/11/P11017
  * Itoh et al. [2018] Itoh, S., Nakaji, M., Uchida, Y., Kitaguchi, M., Shimizu, H.M.: Pendellösung interferometry by using pulsed neutrons. NIM-A 908, 78–81 (2018) https://doi.org/10.1016/j.nima.2018.08.043
  * Carena et al. [2019] Carena, M., Liu, D., Liu, J., Shah, N.R., Wagner, C.E.M., Wang, X.-P.: $\nu$ solution to the strong $CP$ problem. Phys. Rev. D 100, 094018 (2019) https://doi.org/10.1103/PhysRevD.100.094018
  * Mimura et al. [2019] Mimura, Y., Mohapatra, R.N., Severson, M.: Grand unified parity solution to the strong $CP$ problem. Phys. Rev. D 99, 115025 (2019) https://doi.org/10.1103/PhysRevD.99.115025
  * Morrissey and Ramsey-Musolf [2012] Morrissey, D.E., Ramsey-Musolf, M.J.: Electroweak baryogenesis. NJP 14(12), 125003 (2012) https://doi.org/10.1088/1367-2630/14/12/125003
  * Bell et al. [2019] Bell, N.F., Corbett, T., Nee, M., Ramsey-Musolf, M.J.: Electric dipole moments from postsphaleron baryogenesis. Phys. Rev. D 99, 015034 (2019) https://doi.org/10.1103/PhysRevD.99.015034
  * Cirigliano et al. [2019] Cirigliano, V., Crivellin, A., Dekens, W., Vries, J., Hoferichter, M., Mereghetti, E.: $CP$ Violation in Higgs-Gauge Interactions: From Tabletop Experiments to the LHC. Phys. Rev. Lett. 123, 051801 (2019) https://doi.org/10.1103/PhysRevLett.123.051801
  * Crivellin and Saturnino [2019] Crivellin, A., Saturnino, F.: Correlating tauonic $B$ decays with the neutron electric dipole moment via a scalar leptoquark. Phys. Rev. D 100, 115014 (2019) https://doi.org/10.1103/PhysRevD.100.115014
  * Andreev et al. [2018] Andreev, V., Ang, D.G., DeMille, D., Doyle, J.M., Gabrielse, G., Haefner, J., Hutzler, N.R., Lasner, Z., Meisenhelder, C., O’Leary, B.R., Panda, C.D., West, A.D., West, E.P., Wu, X., Collaboration, A.C.M.E.: Improved limit on the electric dipole moment of the electron. Nature 562(7727), 355–360 (2018) https://doi.org/10.1038/s41586-018-0599-8
  * Abel et al. [2020] Abel, C., Afach, S., Ayres, N.J., Baker, C.A., Ban, G., Bison, G., Bodek, K., Bondar, V., Burghoff, M., Chanel, E., Chowdhuri, Z., Chiu, P.-J., Clement, B., Crawford, C.B., Daum, M., Emmenegger, S., Ferraris-Bouchez, L., Fertl, M., Flaux, P., Franke, B., Fratangelo, A., Geltenbort, P., Green, K., Griffith, W.C., Grinten, M., Grujić, Z.D., Harris, P.G., Hayen, L., Heil, W., Henneck, R., Hélaine, V., Hild, N., Hodge, Z., Horras, M., Iaydjiev, P., Ivanov, S.N., Kasprzak, M., Kermaidic, Y., Kirch, K., Knecht, A., Knowles, P., Koch, H.-C., Koss, P.A., Komposch, S., Kozela, A., Kraft, A., Krempel, J., Kuźniak, M., Lauss, B., Lefort, T., Lemière, Y., Leredde, A., Mohanmurthy, P., Mtchedlishvili, A., Musgrave, M., Naviliat-Cuncic, O., Pais, D., Piegsa, F.M., Pierre, E., Pignol, G., Plonka-Spehr, C., Prashanth, P.N., Quéméner, G., Rawlik, M., Rebreyend, D., Rienäcker, I., Ries, D., Roccia, S., Rogel, G., Rozpedzik, D., Schnabel, A., Schmidt-Wellenburg, P., Severijns, N., Shiers, D., Tavakoli Dinani, R., Thorne, J.A., Virot, R., Voigt, J., Weis, A., Wursten, E., Wyszynski, G., Zejma, J., Zenner, J., Zsigmond, G.: Measurement of the Permanent Electric Dipole Moment of the Neutron. Phys. Rev. Lett. 124, 081803 (2020) https://doi.org/10.1103/PhysRevLett.124.081803
  * [15] Matsumiya, R., Akatsuka, H., Bidinosti, C.P., Davis, C.A., Franke, B., Fujimoto, D., Gericke, M.T.W., Giampa, P., Golub, R., Hansen-Romu, S., Hatanaka, K., Hayamizu, T., Higuchi, T., Ichikawa, G., Imajo, S., Jamieson, B., Kawasaki, S., Kitaguchi, M., Klassen, W., Klemets, E., Konaka, A., Korkmaz, E., Korobkina, E., Kuchler, F., Lavvaf, M., Lee, L., Lindner, T., Madison, K.W., Makida, Y., Mammei, R., Mammei, J., Martin, J.W., McCrea, M., Miller, E., Mishima, K., Momose, T., Okamura, T., Ong, H.J., Picker, R., Ramsay, W.D., Schreyer, W., Shimizu, H.M., Sidhu, S., Stargardter, S., Tanihata, I., Vanbergen, S., Oers, W.T.H., Watanabe, Y.: The Precision nEDM Measurement with UltraCold Neutrons at TRIUMF. JPS Conf. Proc. https://doi.org/10.7566/JPSCP.37.020701
  * Ahmed et al. [2019a] Ahmed, S., Altiere, E., Andalib, T., Bell, B., Bidinosti, C.P., Cudmore, E., Das, M., Davis, C.A., Franke, B., Gericke, M., Giampa, P., Gnyp, P., Hansen-Romu, S., Hatanaka, K., Hayamizu, T., Jamieson, B., Jones, D., Kawasaki, S., Kikawa, T., Kitaguchi, M., Klassen, W., Konaka, A., Korkmaz, E., Kuchler, F., Lang, M., Lee, L., Lindner, T., Madison, K.W., Makida, Y., Mammei, J., Mammei, R., Martin, J.W., Matsumiya, R., Miller, E., Mishima, K., Momose, T., Okamura, T., Page, S., Picker, R., Pierre, E., Ramsay, W.D., Rebenitsch, L., Rehm, F., Schreyer, W., Shimizu, H.M., Sidhu, S., Sikora, A., Smith, J., Tanihata, I., Thorsteinson, B., Vanbergen, S., Oers, W.T.H., Watanabe, Y.X.: First ultracold neutrons produced at TRIUMF. Phys. Rev. C 99, 025503 (2019) https://doi.org/10.1103/PhysRevC.99.025503
  * Ahmed et al. [2019b] Ahmed, S., Altiere, E., Andalib, T., Barnes, M.J., Bell, B., Bidinosti, C.P., Bylinsky, Y., Chak, J., Das, M., Davis, C.A., Fischer, F., Franke, B., Gericke, M.T.W., Giampa, P., Hahn, M., Hansen-Romu, S., Hatanaka, K., Hayamizu, T., Jamieson, B., Jones, D., Katsika, K., Kawasaki, S., Kikawa, T., Klassen, W., Konaka, A., Korkmaz, E., Kuchler, F., Kurchaninov, L., Lang, M., Lee, L., Lindner, T., Madison, K.W., Mammei, J., Mammei, R., Martin, J.W., Matsumiya, R., Miller, E., Momose, T., Picker, R., Pierre, E., Ramsay, W.D., Rao, Y.-N., Rawnsley, W.R., Rebenitsch, L., Schreyer, W., Sidhu, S., Vanbergen, S., Oers, W.T.H., Watanabe, Y.X., Yosifov, D.: Fast-switching magnet serving a spallation-driven ultracold neutron source. Phys. Rev. Accel. Beams 22, 102401 (2019) https://doi.org/10.1103/PhysRevAccelBeams.22.102401
  * Ahmed et al. [2019c] Ahmed, S., Andalib, T., Barnes, M.J., Bidinosti, C.B., Bylinsky, Y., Chak, J., Das, M., Davis, C.A., Franke, B., Gericke, M.T.W., Giampa, P., Hahn, M., Hansen-Romu, S., Hatanaka, K., Jamieson, B., Jones, D., Katsika, K., Kawasaki, S., Klassen, W., Konaka, A., Korkmaz, E., Kuchler, F., Kurchaninov, L., Lang, M., Lee, L., Lindner, T., Madison, K.W., Mammei, J., Mammei, R., Martin, J.W., Matsumiya, R., Picker, R., Pierre, E., Ramsay, W.D., Rao, Y.-N., Rawnsley, W.R., Rebenitsch, L., Remon, C.A., Schreyer, W., Sikora, A., Sidhu, S., Sonier, J., Thorsteinson, B., Vanbergen, S., van Oers, W.T.H., Watanabe, Y.X., Yosifov, D.: A beamline for fundamental neutron physics at TRIUMF. NIM-A 927, 101–108 (2019) https://doi.org/10.1016/j.nima.2019.01.074
  * Schreyer et al. [2020] Schreyer, W., Davis, C.A., Kawasaki, S., Kikawa, T., Marshall, C., Mishima, K., Okamura, T., Picker, R.: Optimizing neutron moderators for a spallation-driven ultracold-neutron source at TRIUMF. NIM-A 959, 163525 (2020) https://doi.org/10.1016/j.nima.2020.163525
  * Abel et al. [2020] Abel, C., Afach, S., Ayres, N.J., Ban, G., Bison, G., Bodek, K., Bondar, V., Chanel, E., Chiu, P.-J., Crawford, C.B., Chowdhuri, Z., Daum, M., Emmenegger, S., Ferraris-Bouchez, L., Fertl, M., Franke, B., Griffith, W.C., Grujić, Z.D., Hayen, L., Hélaine, V., Hild, N., Kasprzak, M., Kermaidic, Y., Kirch, K., Knowles, P., Koch, H.-C., Komposch, S., Koss, P.A., Kozela, A., Krempel, J., Lauss, B., Lefort, T., Lemière, Y., Leredde, A., Mtchedlishvili, A., Mohanmurthy, P., Musgrave, M., Naviliat-Cuncic, O., Pais, D., Pazgalev, A., Piegsa, F.M., Pierre, E., Pignol, G., Prashanth, P.N., Quéméner, G., Rawlik, M., Rebreyend, D., Ries, D., Roccia, S., Rozpedzik, D., Schmidt-Wellenburg, P., Schnabel, A., Severijns, N., Dinani, R.T., Thorne, J., Weis, A., Wursten, E., Wyszynski, G., Zejma, J., Zsigmond, G.: Optically pumped Cs magnetometers enabling a high-sensitivity search for the neutron electric dipole moment. Phys. Rev. A 101, 053419 (2020) https://doi.org/10.1103/PhysRevA.101.053419
  * Abel et al. [2022] Abel, C., Ayres, N.J., Ban, G., Bison, G., Bodek, K., Bondar, V., Chanel, E., Chiu, P.-J., Clément, B., Crawford, C.B., Daum, M., Emmenegger, S., Ferraris-Bouchez, L., Fertl, M., Flaux, P., Fratangelo, A., Griffith, W.C., Grujić, Z.D., Harris, P.G., Hayen, L., Hild, N., Kasprzak, M., Kirch, K., Knowles, P., Koch, H.-C., Koss, P.A., Kozela, A., Krempel, J., Lauss, B., Lefort, T., Lemière, Y., Mohanmurthy, P., Naviliat-Cuncic, O., Pais, D., Piegsa, F.M., Pignol, G., Prashanth, P.N., Quéméner, G., Rawlik, M., Ries, D., Rebreyend, D., Roccia, S., Rozpedzik, D., Schmidt-Wellenburg, P., Schnabel, A., Severijns, N., Thorne, J.A., Virot, R., Weis, A., Wursten, E., Wyszynski, G., Zejma, J., Zsigmond, G.: Mapping of the magnetic field to correct systematic effects in a neutron electric dipole moment experiment. Phys. Rev. A 106, 032808 (2022) https://doi.org/10.1103/PhysRevA.106.032808
  * Budker et al. [2002] Budker, D., Gawlik, W., Kimball, D.F., Rochester, S.M., Yashchuk, V.V., Weis, A.: Resonant nonlinear magneto-optical effects in atoms. Rev. Mod. Phys. 74, 1153–1201 (2002) https://doi.org/10.1103/RevModPhys.74.1153
  * Pustelny et al. [2008] Pustelny, S., Wojciechowski, A., Gring, M., Kotyrba, M., Zachorowski, J., Gawlik, W.: Magnetometry based on nonlinear magneto-optical rotation with amplitude-modulated light. Journal of Applied Physics 103(6), 063108 (2008) https://doi.org/10.1063/1.2844494 https://pubs.aip.org/aip/jap/article-pdf/doi/10.1063/1.2844494/13946272/063108_1_online.pdf
  * Das [2018] Das, M.: Highly sensitive rb magnetometer for neutron electric dipole moment experiments. Master’s thesis (2018). https://mspace.lib.umanitoba.ca/items/c95adc8a-6ba2-49db-9bbc-7e8e0bad5180
  * Klassen [2020] Klassen, W.: All-optical Cs Magnetometry system for a neutron electric dipole moment experiment. Master’s thesis (2020)
  * Higbie et al. [2006] Higbie, J.M., Corsini, E., Budker, D.: Robust, high-speed, all-optical atomic magnetometer. Rev. Sci. Instrum. 77(11), 113106 (2006) https://doi.org/10.1063/1.2370597 https://pubs.aip.org/aip/rsi/article-pdf/doi/10.1063/1.2370597/8816016/113106_1_online.pdf
  * Bell and Bloom [1961] Bell, W.E., Bloom, A.L.: Optically Driven Spin Precession. Phys. Rev. Lett. 6, 280–281 (1961) https://doi.org/10.1103/PhysRevLett.6.280
  * Grujić et al. [2015] Grujić, Z.D., Koss, P.A., Bison, G., Weis, A.: A sensitive and accurate atomic magnetometer based on free spin precession. EPJD 69(5), 135 (2015) https://doi.org/10.1140/epjd/e2015-50875-3
  * [29] Optical Magnetometry. Cambridge University Press (2013). https://doi.org/10.1017/CBO9780511846380
  * Rosner et al. [2022] Rosner, M., Beck, D., Fierlinger, P., Filter, H., Klau, C., Kuchler, F., Rößner, P., Sturm, M., Wurm, D., Sun, Z.: A highly drift-stable atomic magnetometer for fundamental physics experiments. Appl. Phys. Lett. 120(16), 161102 (2022) https://doi.org/10.1063/5.0083854 https://doi.org/10.1063/5.0083854
  * Rosner [2022] Rosner, M.: Drift-stable Magnetometry for Fundamental Physics Experiments. PhD thesis, Munich, Tech. U. (2022)
  * Sturm [2020] Sturm, M.: A highly drift stable and fully optical Cs atomic magnetometer for a new generation nEDM experiment. PhD thesis, Munich, Tech. U. (2020)
  * Precision Glassblowing of Colorado, Inc. [2024] Precision Glassblowing of Colorado, Inc.: About precision glassblowing (2024). https://precisionglassblowing.com/about-precision-glassblowing/
  * [34] (2016). https://www.swsciences.com/contact/index.html
  * [35] https://www.optical-cement.com/default.htm
  * [36] D-TACQ Solutions Ltd: Intelligent Data Acquisition Boards and Systems. https://www.d-tacq.com/
  * Martin et al. [2015] Martin, J.W., Mammei, R.R., Klassen, W., Cerasani, C., Andalib, T., Bidinosti, C.P., Lang, M., Ostapchuk, D.: Large magnetic shielding factor measured by nonlinear magneto-optical rotation. NIM-A 778, 61–66 (2015) https://doi.org/10.1016/j.nima.2015.01.003
  * Ahmed et al. [2024] Ahmed, S., Jamieson, B., Martin, J.W., Ostapchuk, D.C.M., McCrea, M., Klassen, W.: An Ultra-Stable custom current supply for use in a neutron electric dipole moment experiment. In: 2024 IEEE 22nd Mediterranean Electrotechnical Conference (MELECON) (MELECON 2024), Porto, Portugal, p. 5 (2024)
  * Gemmel et al. [2010] Gemmel, C., Heil, W., Karpuk, S., Lenz, K., Ludwig, C., Sobolev, Y., Tullney, K., Burghoff, M., Kilian, W., Knappe-Grüneberg, S., Müller, W., Schnabel, A., Seifert, F., Trahms, L., Baeßler, S.: Ultra-sensitive magnetometry based on free precession of nuclear spins. EPJD 57(3), 303–320 (2010) https://doi.org/10.1140/epjd/e2010-00044-5