Present address: ]Fraunhofer-Institut für Physikalische Messtechnik IPM, 79110
Freiburg i. Breisgau, Germany

Present address: ]University of Chicago, 5801 S Ellis Ave, Chicago, IL 60637,
USA.

# Johnson-Nyquist Noise Effects in Neutron Electric-Dipole-Moment Experiments

N. J. Ayres  ETH Zürich, Institute for Particle Physics and Astrophysics,
CH-8093 Zürich, Switzerland  G. Ban  Normandie Université, ENSICAEN, UNICAEN,
CNRS/IN2P3, LPC Caen, 14000 Caen, France  G. Bison  Paul Scherrer Institut,
CH-5232 Villigen PSI, Switzerland  K. Bodek  Marian Smoluchowski Institute of
Physics, Jagiellonian University, 30-348 Cracow, Poland  V. Bondar  ETH
Zürich, Institute for Particle Physics and Astrophysics, CH-8093 Zürich,
Switzerland  P.-J. Chiu<EMAIL_ADDRESS>ETH Zürich, Institute for
Particle Physics and Astrophysics, CH-8093 Zürich, Switzerland  Paul Scherrer
Institut, CH-5232 Villigen PSI, Switzerland  B. Clement  Université Grenoble
Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 38026 Grenoble, France  C. B. Crawford
Department of Physics and Astronomy, University of Kentucky, Lexington,
Kentucky, 40506, USA  M. Daum  Paul Scherrer Institut, CH-5232 Villigen PSI,
Switzerland  S. Emmenegger  ETH Zürich, Institute for Particle Physics and
Astrophysics, CH-8093 Zürich, Switzerland  M. Fertl  Institute of Physics,
Johannes Gutenberg University Mainz, 55128 Mainz, Germany  A. Fratangelo
Laboratory for High Energy Physics and Albert Einstein Center for Fundamental
Physics, University of Bern, CH-3012 Bern, Switzerland  W. C. Griffith
Department of Physics and Astronomy, University of Sussex, Falmer, Brighton
BN1 9QH, United Kingdom  Z. D. Grujić  Institute of Physics Belgrade,
University of Belgrade, 11080 Belgrade, Serbia  P. G. Harris  Department of
Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, United
Kingdom  K. Kirch  ETH Zürich, Institute for Particle Physics and
Astrophysics, CH-8093 Zürich, Switzerland  Paul Scherrer Institut, CH-5232
Villigen PSI, Switzerland  P. A. Koss  [  Instituut voor Kern- en
Stralingsfysica, University of Leuven, B-3001 Leuven, Belgium  B. Lauss  Paul
Scherrer Institut, CH-5232 Villigen PSI, Switzerland  T. Lefort  Normandie
Université, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France  P.
Mohanmurthy  [  ETH Zürich, Institute for Particle Physics and Astrophysics,
CH-8093 Zürich, Switzerland  Paul Scherrer Institut, CH-5232 Villigen PSI,
Switzerland  O. Naviliat-Cuncic  Normandie Université, ENSICAEN, UNICAEN,
CNRS/IN2P3, LPC Caen, 14000 Caen, France  D. Pais  ETH Zürich, Institute for
Particle Physics and Astrophysics, CH-8093 Zürich, Switzerland  Paul Scherrer
Institut, CH-5232 Villigen PSI, Switzerland  F. M. Piegsa  Laboratory for High
Energy Physics and Albert Einstein Center for Fundamental Physics, University
of Bern, CH-3012 Bern, Switzerland  G. Pignol  Université Grenoble Alpes,
CNRS, Grenoble INP, LPSC-IN2P3, 38026 Grenoble, France  D. Rebreyend
Université Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 38026 Grenoble,
France  I. Rienäcker  ETH Zürich, Institute for Particle Physics and
Astrophysics, CH-8093 Zürich, Switzerland  Paul Scherrer Institut, CH-5232
Villigen PSI, Switzerland  D. Ries  Department of Chemistry - TRIGA site,
Johannes Gutenberg University Mainz, 55128 Mainz, Germany  S. Roccia
Université Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 38026 Grenoble,
France  Institut Laue-Langevin, CS 20156 F-38042 Grenoble Cedex 9, France  K.
U. Ross  Department of Chemistry - TRIGA site, Johannes Gutenberg University
Mainz, 55128 Mainz, Germany  D. Rozpedzik  Marian Smoluchowski Institute of
Physics, Jagiellonian University, 30-348 Cracow, Poland  P. Schmidt-Wellenburg
<EMAIL_ADDRESS>Paul Scherrer Institut, CH-5232 Villigen
PSI, Switzerland  A. Schnabel  Physikalisch-Technische Bundesanstalt, D-10587
Berlin, Germany  N. Severijns  Instituut voor Kern- en Stralingsfysica,
University of Leuven, B-3001 Leuven, Belgium  B. Shen  Department of Chemistry
- TRIGA site, Johannes Gutenberg University Mainz, 55128 Mainz, Germany  R.
Tavakoli Dinani  Instituut voor Kern- en Stralingsfysica, University of
Leuven, B-3001 Leuven, Belgium  J. A. Thorne  Laboratory for High Energy
Physics and Albert Einstein Center for Fundamental Physics, University of
Bern, CH-3012 Bern, Switzerland  R. Virot  Université Grenoble Alpes, CNRS,
Grenoble INP, LPSC-IN2P3, 38026 Grenoble, France  N. Yazdandoost  Department
of Chemistry - TRIGA site, Johannes Gutenberg University Mainz, 55128 Mainz,
Germany  J. Zejma  Marian Smoluchowski Institute of Physics, Jagiellonian
University, 30-348 Cracow, Poland  G. Zsigmond  Paul Scherrer Institut,
CH-5232 Villigen PSI, Switzerland

###### Abstract

Magnetic Johnson-Nyquist noise (JNN) originating from metal electrodes, used
to create a static electric field in neutron electric-dipole-moment (nEDM)
experiments, may limit the sensitivity of measurements. We present here the
first dedicated study on JNN applied to a large-scale long-measurement-time
experiment with the implementation of a co-magnetometry. In this study, we
derive surface- and volume-averaged root-mean-square normal noise amplitudes
at a certain frequency bandwidth for a cylindrical geometry. In addition, we
model the source of noise as a finite number of current dipoles and
demonstrate a method to simulate temporal and three-dimensional spatial
dependencies of JNN. The calculations are applied to estimate the impact of
JNN on measurements with the new apparatus, n2EDM, at the Paul Scherrer
Institute. We demonstrate that the performances of the optically pumped 133Cs
magnetometers and 199Hg co-magnetometers, which will be used in the apparatus,
are not limited by JNN. Further, we find that in measurements deploying a co-
magnetometer system, the impact of JNN is negligible for nEDM searches down to
a sensitivity of $4\,\times\,10^{-28}\,e\cdot{\rm cm}$ in a single
measurement; therefore, the use of economically and mechanically favored solid
aluminum electrodes is possible.

##  I Introduction

The search for a permanent electric dipole moment of the neutron (nEDM) has
been an important topic in fundamental physics research since the 1950s Smith
_et al._ (1957). These experiments have been carried out by comparing the
Larmor precession frequencies of neutron spins ($f_{\rm{n}}$) under static
uniform parallel and anti-parallel electric and magnetic fields, using the
Ramsey technique of separated oscillating fields Ramsey (1950); Purcell and
Ramsey (1950). For an accurate and precise measurement of $f_{\rm{n}}$,
controlling the stability and uniformity as well as reducing the noise of the
magnetic field in the apparatus are of paramount importance. A potential
source of magnetic-field disturbance is Johnson-Nyquist noise (JNN) Johnson
(1928); Nyquist (1928), originating from the thermal motion of electrons in
metal components within the experimental apparatus.

Johnson-Nyquist noise was originally observed as a “random variation of
potential between the ends of a conductor” Johnson (1928). The same underlying
effect, random thermal motion of charge carriers, also results in fluctuation
of the electromagnetic field near a conductor. Magnetic JNN is relevant in
various research domains, all related to measurements with highest precision.
The first published numerical analysis of JNN came from the bio-magnetic
measurements Varpula and Poutanen (1984) using superconducting quantum
interference devices (SQUIDs). Johnson-Nyquist noise often exceeds the
intrinsic noise of modern high-sensitivity detectors such as SQUIDs Varpula
and Poutanen (1984); Clem (1987); Nenonen  _et al._ (1996); Körber (2018) and
high-density alkali atomic magnetometers Allred  _et al._ (2002). More
recently, it was observed that these magnetic-near-field fluctuations induce
spin-flip processes, and in turn are a crucial element of decoherence in
magnetic traps which limits the trapping lifetime of atoms Jones  _et al._
(2003); Harber  _et al._ (2003); Lin  _et al._ (2004); Rekdal  _et al._
(2004); Henkel (2005); Emmert  _et al._ (2009). In addition, relaxations of
spin states in the presence of magnetic fluctuations were also studied in the
context of magnetic resonance force microscopy and quantum computation Sidles
_et al._ (2003). In EDM measurements, following the requirement of sensitivity
enhancement, potential constraints from JNN received extensive attentions in
the last few decades Lamoreaux (1999); C. T. Munger, Jr. (2005); Amini  _et
al._ (2007); Rabey  _et al._ (2016). Johnson-Nyquist noise from metal parts in
high precision experiments impose a limit on the measurement sensitivity.
Quantifying the impact of JNN in the design of the n2EDM experiment Abel  _et
al._ (2019a); Ayres  _et al._ (2021) to search for an nEDM at the Paul
Scherrer Institute (PSI) in Villigen, Switzerland, with the PSI ultracold-
neutron (UCN) source Lauss (2014); Bison  _et al._ (2020) motivates the
presented research.

As in the past, the challenges in measuring nEDM, $d_{\rm n}$, are the
increase of the statistical sensitivity and the corresponding control of
systematic effects. New sources of UCN worldwide Bison  _et al._ (2017); Ito
_et al._ (2018) improve the statistical sensitivity, $\sigma_{d_{\rm
n}}\propto N^{-1/2}$, by increasing the number of neutrons available after
storage, $N$. Nevertheless, owing to systematic effects, e.g. random frequency
shifts due to possible magnetic-field noise or drifts, the improvement in pure
counting statistics might be compromised. This study investigates the impact
of JNN on nEDM experiments, focusing especially on the n2EDM spectrometer Abel
_et al._ (2019a); Ayres  _et al._ (2021) currently under construction at the
PSI.

The n2EDM apparatus features two cylindrical storage chambers, $12$ cm in
height and $40$ cm in radius, stacked vertically. The two chambers share the
central plane, an electrode to which a high voltage of up to 200 kV can be
applied. The top and bottom of the cylinder pair are closed with grounded
plates. Figure 1 shows a simplified drawing of the central storage chambers
with the components relevant to this paper, whereas detailed descriptions and
schematics of the experimental apparatus can be found in Figs. 1 and 2 of Ref.
Abel  _et al._ (2019a). The three electrodes are made of aluminum plates,
whose surfaces pointing towards the inside of the UCN storage chambers are
coated with diamond-like carbon (DLC) Atchison  _et al._ (2005, 2006) in order
to optimize the UCN reflection properties.

Thermal motion of charge carriers in the bulk aluminum results in magnetic
JNN, which might affect the sensitivity of the magnetometers in its vicinity.
In the following, we investigate the effects of JNN on the cesium
magnetometers (CsM) Weis and Wynands (2005); Groeger  _et al._ (2006), glass
bulbs filled with saturated 133Cs vapor positioned around the precession
chambers, the UCN and the mercury co-magnetometers (HgM), polarized 199Hg
atoms occupying the same volumes as the UCN in the chambers and read out by
resonant light beams Green  _et al._ (1998); Baker  _et al._ (2014); Ban  _et
al._ (2018).

Figure 1: Cross-sectional drawing of the central storage chambers of the n2EDM
apparatus located inside the vacuum tank. Included are only components
relevant to this study. For a better visibility, the support structure of the
Cs magnetometers (CsM) and the top UCN shutter are omitted. The electrodes are
separated by polystyrene insulating rings.

##  II Spectrum of Magnetic Johnson-Nyquist Noise

The relevant magnetic JNN spectrum was first analytically derived for research
in biophysical applications Varpula and Poutanen (1984); Nenonen  _et al._
(1996). There it is shown that for an infinite-slab conductor of thickness $a$
and conductivity $\sigma$ at a temperature $T$, the normal component of the
amplitude spectral density (ASD) found in a distance $d$ to the conductor
surface within a finite frequency interval $\Delta f$, with the $z$-axis
defined to be perpendicular to the conductor surface, can be written as
Varpula and Poutanen (1984)

$\displaystyle\mathcal{B}_{z}(d,f)$
$\displaystyle=\sqrt{\overline{B_{z}(d,f)^{2}}/\Delta f}$ (1)
$\displaystyle=\mu_{0}\sqrt{\frac{2\sigma k_{{\rm
B}}T}{\pi}}\left[\int_{0}^{\infty}R(\rho,a,\sigma,f)e^{-2\rho
d}\rho\operatorname{d}\\!\rho\right]^{1/2},$

where $k_{{\rm B}}$ is the Boltzmann constant, $\rho$ is the radial component
of the infinite conductor, and $B_{z}(d,f)$ is the magnetic-field amplitude
normal to the surface at a given frequency $f$ 111Compare to Eq. (38) of Ref.
Varpula and Poutanen (1984), where variables $z$, distance, and $t$,
thickness, were replaced with $d$ and $a$ in our paper to avoid confusion with
other variables used in the remainder of the article.. $R(\rho,a,\sigma,f)$ in
Eq. (1) is a function of conductor properties, $a$ and $\sigma$, at a given
frequency $f$, with the original expression defined in Eq. (39) of Ref.
Varpula and Poutanen (1984). For the horizontal components, due to symmetry
and Maxwell’s equations, one can infer that Varpula and Poutanen (1984)

$\mathcal{B}_{x}(d,f)=\mathcal{B}_{y}(d,f)=\dfrac{1}{\sqrt{2}}\mathcal{B}_{z}(d,f).$
(2)

All three components of the noise spectrum depend on the normal distance
between the point of interest and the surface of the infinite slab. Figure 2
shows the normal ASD of a 2.5-cm-thick aluminum infinite-slab conductor,
$\sigma=3.77\times 10^{7}$ S/m, under 20 $\mathrm{\SIUnitSymbolCelsius}$ at
various distances, using Eq. (1). The spectral line flattens at low
frequencies towards a constant value equal to the root-mean-square (RMS) limit
for $f\rightarrow 0$. With increasing frequency, the noise amplitude decreases
due to self-damping of high-frequency noise within the conductor slab.

Figure 2: Normal amplitude spectral density of a 2.5-cm-thick aluminum
($\sigma=3.77\times 10^{7}$ S/m) infinite-slab conductor at various distances
under 20 $\mathrm{\SIUnitSymbolCelsius}$.

To verify these noise spectra, measurements of the magnetic-field noise
created by an aluminum sheet were carried out in the magnetically shielded
room BMSR-2 at the Physikalisch-Technische Bundesanstalt (PTB), Berlin,
Germany Bork  _et al._ (2001), using the 304-channel SQUID-vector-magnetometer
system Thiel  _et al._ (2005). The system is based on nineteen modules placed
on a hexagonal grid with each comprising of sixteen SQUIDs placed at various
vertical planes. A total of 304 SQUIDs each with a 7 mm effective pickup-coil
diameter permit to calculate all three vector components of the magnetic field
on three measurement planes. The noise of a sheet made of 99.5% aluminum with
dimensions of 1.3 m $\times$ 1 m $\times$ 0.5 mm was measured. In the first
measurement, the sheet was placed to touch the flat bottom of the dewar
cryostat, resulting in a minimal distance of 27.5 mm between sample and
SQUIDs, due to the cold-warm distance of the dewar. Another measurement was
carried out by adding a wood plate in between aluminum sheet and dewar to
provide an additional 7.5 mm distance to the dewar bottom. In each
measurement, seven SQUIDs at different heights from two central modules were
used, with four measuring the vertical field component w.r.t. the probe and
three measuring the horizontal field component.

Figures 3(a) and 3(b) are the combined results from the two measurements, with
or without the wood plate, with the former showing the vertical field
component and the latter displaying the horizontal component. The spectra are
results from 300 s measurements averaged over 5 s samples. The background
noise measured without aluminum sheet was subtracted. The increase in noise
below $2\text{\,}\mathrm{f}\mathrm{T}\mathrm{/}\sqrt{Hz}$ is due to the SQUID
white noise. Mechanical vibrations influence the measurement between 5-25 Hz
at the level of $10\text{\,}\mathrm{f}\mathrm{T}\mathrm{/}\sqrt{Hz}$.
Comparing between the measured spectra and the theoretical ASD calculated with
Eqs. (1) and (2), with $\sigma=3.77\times 10^{7}$ S/m and $T=22$
$\mathrm{\SIUnitSymbolCelsius}$, discrepancies up to $\sim 15\%$ are found.
The uncertainties on $\sigma$ and $T$ are not enough to explain the
differences. Nenonen, Montonen and Katila Nenonen  _et al._ (1996) pointed out
that correlation of JNN within pickup coils with finite surface areas should
be taken into account if the pickup-coil diameter is larger than the
measurement distance (see Sec. III.2 for details). In our measurements, the
ratios between the pickup-coil diameter and the distances were all smaller
than one third; thus, the correlation is considered negligible (see Fig. 4).
Additionally, considering the dimensions of the aluminum sheet, the pick-up
coils and the measurement separations, we can approximate the aluminum sheet
as an infinite conductor.

Comparisons between theoretical calculations and noise measurements on copper
Varpula and Poutanen (1984) and aluminum Nenonen  _et al._ (1996) conductors
using first-order gradiometers have been performed in bio-magnetic researches.
It is reported in Ref. Varpula and Poutanen (1984) that the agreement lies
within uncertainties of $\pm 15\%$. Our measurements performed with a
magnetometer confirm the theoretical spectrum in a non-differential manner to
a fairly good level, and verify the relation between transverse and normal
noise components (Eq. (2)). The reason for the small disagreement is still
unclear; nonetheless, by using the theoretical ASD, we guarantee that the
noise will not be underestimated.

(a)

(b)

Figure 3: (a) Vertical and (b) horizontal magnetic-field noise component of a
0.5-mm-thick aluminum sheet measured by the SQUID system at PTB w.r.t. the
theoretical ASD at various distances. The spectra were averaged over 5 s
samples from 300 s measurements. The ASD decreases with increasing distance.

For JNN studies in EDM experiments, a principle simplification can be made
assuming that the frequency of the fluctuating field is low enough such that
the eddy currents generated in the bulk material can be neglected. This is
stated as the static approximation by Lamoreaux Lamoreaux (1999), which is
valid when the thickness of the conductor is smaller than the skin depth

$\lambda=\sqrt{\frac{1}{\pi\mu\sigma f}},$ (3)

where $f$ is the fluctuation frequency. In the context of EDM experiments,
this corresponds to approximately the reciprocal of the spin-coherence time,
$T_{2}$. In the n2EDM experiment, the free-spin-precession period for a single
measurement will be $\Delta t\sim$ 200 s, approximately two times the spin-
coherence time of mercury and a fraction of the spin-coherence time of
neutrons. Hence, we assume $f^{-1}=\Delta t\sim$200\text{\,}\mathrm{s}$\approx
T_{2,{\rm Hg}}$ will be the free-spin-precession period for a single
measurement. At 5 mHz, $\lambda\sim$ 116 cm, so the low-frequency assumption
can be applied safely for conductors with a thickness of less than 10 cm.

##  III Magnetic-field fluctuation observed by field-sensing particles

During an nEDM-measurement cycle, 199Hg atoms occupying the same volumes as
UCN, are used as co-habiting magnetometers (HgM) Green  _et al._ (1998); Baker
_et al._ (2014); Ban  _et al._ (2018). As the HgM and the UCN measure the
magnetic field simultaneously, the ratio of the two precession frequencies,
$f_{{\rm n}}/f_{{\rm Hg}}$, is robust against magnetic-field changes.
Nonetheless, the two spin species sample the magnetic field differently. The
UCN sample the field adiabatically and have a negative center-of-mass offset
whereas the 199Hg atoms sample the field non-adiabatically Abel  _et al._
(2019b). For a nominal field of
$B_{0}=1$$\mathrm{\SIUnitSymbolMicro}\mathrm{T}$, we investigate the degree to
which the effects of JNN can be controlled when taking the frequency ratio of
two ensembles within one precession chamber.

###  III.1 Analytical derivation of spatial properties

In the first step to calculate the RMS magnetic-field noise sensed by the
particles, it is useful to derive the spatial correlation of JNN at different
locations within the volume. For this purpose we calculated the magnetic noise
originating from thermal noise currents by dividing a volume conductor into
infinitesimal cuboidal elements, $\Delta V=\Delta x\Delta y\Delta z$, similar
to the seminal calculation in Refs. Varpula and Poutanen (1984); Nenonen  _et
al._ (1996). There an equivalent current dipole for the volume element is
defined, whose component $P_{\alpha}=I_{\alpha}\Delta\alpha$ $(\alpha=x,y,z)$
in the direction $\alpha$ is the product of this short-circuit current and the
finite size of the element. Following this concept, the source of thermal
magnetic noise is represented by a great number of randomly oriented current
dipoles on the surface of the conductor.

We consider an infinite conductor and assume its surface is an $x-y$ plane on
the reference of the vertical coordinate $z=0$. A current dipole element on an
infinitesimal surface area $\operatorname{d}\\!s$ located at
$(\boldsymbol{x},0)$, where $\boldsymbol{x}$ now denotes a two-dimensional
vector on the $x-y$ plane, is written as
$\boldsymbol{I}(\boldsymbol{x})\operatorname{d}\\!s$, where the $z=0$
component is omitted for simplicity. The magnetic field created by this dipole
at a point $(\boldsymbol{r},z)$ can be calculated, according to the Biot-
Savart law, as

$\operatorname{d}\\!\boldsymbol{B}=\frac{\mu_{0}}{4\pi}\frac{\boldsymbol{I}(\boldsymbol{x})\operatorname{d}\\!s\times\boldsymbol{k}}{d^{2}},$
(4)

where

$\boldsymbol{k}=\frac{\boldsymbol{r}-\boldsymbol{x}}{d}+\frac{z}{d}\,\hat{e_{z}}$
(5)

is the unit directional vector pointing from $(\boldsymbol{x},0)$ to
$(\boldsymbol{r},z)$, and

$d=\sqrt{{\left|\boldsymbol{r}-\boldsymbol{x}\right|}^{2}+z^{2}}$ (6)

is the distance between the dipole and the observation point. Now, we obtain
the normal component $\hat{e}_{z}$ of the field

$\displaystyle\operatorname{d}\\!B_{z}(\boldsymbol{r},z)$
$\displaystyle=\operatorname{d}\\!\boldsymbol{B}\cdot\hat{e}_{z}$
$\displaystyle=\frac{\mu_{0}}{4\pi}\left(\boldsymbol{I}(\boldsymbol{x})\operatorname{d}\\!s\times\frac{\boldsymbol{r}-\boldsymbol{x}}{d^{3}}\right)\cdot\hat{e}_{z}$
$\displaystyle=\frac{\mu_{0}}{4\pi}\left(\hat{e}_{z}\times\boldsymbol{I}(\boldsymbol{x})\operatorname{d}\\!s\right)\cdot\frac{\boldsymbol{r}-\boldsymbol{x}}{d^{3}}$
$\displaystyle=\frac{\mu_{0}}{4\pi}\boldsymbol{F}(\boldsymbol{r}-\boldsymbol{x},z)\cdot\boldsymbol{\mathcal{I}}(\boldsymbol{x})\operatorname{d}\\!s,$
(7)

with

$\boldsymbol{F}(\boldsymbol{x},z)\equiv\frac{\boldsymbol{x}}{\left(\left|\boldsymbol{x}\right|^{2}+z^{2}\right)^{3/2}}$
(8)

and

$\boldsymbol{\mathcal{I}}(\boldsymbol{x})\equiv\hat{e}_{z}\times\boldsymbol{I}(\boldsymbol{x})$
(9)

being the rotated current component transformed from the triple product.

####  III.1.1 Variance of a disk-averaged field

Consider a disk parallel to the conductor, which has a radius $R$ and is
located at a normal distance $z$ above the conductor. The average normal
magnetic field over this disk generated by thermal noise in a finite element
$\operatorname{d}\\!s$ from the conductor can be written as

$\displaystyle\operatorname{d}\\!\bar{B}_{z}(R,z)$
$\displaystyle\equiv\int_{S_{R}}\frac{\operatorname{d}\\!^{2}r}{\pi
R^{2}}\,\operatorname{d}\\!B_{z}(\boldsymbol{r},z)$
$\displaystyle=\frac{\mu_{0}}{4\pi}\int_{S_{R}}\frac{\operatorname{d}\\!^{2}r}{\pi
R^{2}}\,\boldsymbol{F}(\boldsymbol{r}-\boldsymbol{x},z)\cdot\boldsymbol{\mathcal{I}}(\boldsymbol{x})\operatorname{d}\\!s$
$\displaystyle=\frac{\mu_{0}}{4\pi}\,\boldsymbol{\mathcal{I}}(\boldsymbol{x})\operatorname{d}\\!s\cdot\bar{\boldsymbol{M}}(\boldsymbol{x},R,z),$
(10)

where

$\bar{\boldsymbol{M}}(\boldsymbol{x},R,z)\equiv\int_{S_{R}}\frac{\operatorname{d}\\!^{2}r}{\pi
R^{2}}\boldsymbol{F}(\boldsymbol{r}-\boldsymbol{x},z)$ (11)

is the average over the disk. For an infinite conductor, we integrate over all
dipoles

$\bar{B}_{z}(R,z)=\int\operatorname{d}\\!\bar{B}_{z}(R,z).$ (12)

The variance of this surface average is then calculated as

$\begin{split}&\left\langle\bar{B}_{z}(R,z)^{2}\right\rangle=\left(\frac{\mu_{0}}{4\pi}\right)^{2}\int\operatorname{d}\\!s\int\operatorname{d}\\!s^{\prime}\\\
&\qquad\left\langle\left(\boldsymbol{\mathcal{I}}(\boldsymbol{x})\cdot\bar{\boldsymbol{M}}(\boldsymbol{x},R,z)\right)\left(\boldsymbol{\mathcal{I}}(\boldsymbol{x^{\prime}})\cdot\bar{\boldsymbol{M}}(\boldsymbol{x^{\prime}},R,z)\right)\right\rangle.\end{split}$
(13)

As shown in Eq. (1) in Ref. Varpula and Poutanen (1984), based on Nyquist’s
theorem,

$\left\langle\boldsymbol{\mathcal{I}}(\boldsymbol{x})\boldsymbol{\mathcal{I}}(\boldsymbol{x^{\prime}})\right\rangle=4\sigma
k_{{\rm B}}T\Delta fa\delta(\boldsymbol{x}-\boldsymbol{x^{\prime}}),$ (14)

where the conductor properties are identical to those indicated in Eq. (1).
With the change of variables and further derivations, the variance of the
surface-averaged field can be expressed as

$\displaystyle\left\langle\bar{B}_{z}(R,z)^{2}\right\rangle$
$\displaystyle=\frac{\mathcal{C}\pi}{2z^{2}}\mathfrak{I}\left(\frac{R}{z}\right)$
$\displaystyle=\left\langle
B_{z}(\boldsymbol{r},z)^{2}\right\rangle\mathfrak{I}\left(\frac{R}{z}\right)$
$\displaystyle=\left\langle
B_{z}(0,z)^{2}\right\rangle\mathfrak{I}\left(\frac{R}{z}\right),$ (15)

normalized to the variance of the single-point field at a random location of
distance $z$, $\left\langle B(\boldsymbol{r},z)^{2}\right\rangle=\left\langle
B(0,z)^{2}\right\rangle$, with

$\mathcal{C}\equiv\left(\frac{\mu_{0}}{4\pi}\right)^{2}4\sigma k_{{\rm
B}}T\Delta fa.$ (16)

$\mathfrak{I}(R/z)$ is an integration over three two-dimensional vectors
calculated as

$\displaystyle\mathfrak{I}(\xi)$
$\displaystyle\equiv\frac{2}{\pi^{3}}\xi^{4}\int_{S_{1}}\operatorname{d}\\!^{2}u\int_{S_{1}}\operatorname{d}\\!^{2}v\int\operatorname{d}\\!^{2}X$
$\displaystyle\qquad\frac{\boldsymbol{u}-\boldsymbol{X}}{\left(\xi^{2}\left|\boldsymbol{u}-\boldsymbol{X}\right|^{2}+1\right)^{3/2}}\cdot\frac{\boldsymbol{v}-\boldsymbol{X}}{\left(\xi^{2}\left|\boldsymbol{v}-\boldsymbol{X}\right|^{2}+1\right)^{3/2}}.$
(17)

$\boldsymbol{u}$ and $\boldsymbol{v}$ are two observation points on the disk,
where an integration over a unit circle $S_{1}$ is performed, and
$\boldsymbol{X}$ is a dipole on the conductor integrated from zero to
infinity.

####  III.1.2 Variance of a cylinder-averaged field

In our case, we are interested in the average field observed within a cylinder
of radius $R$ and height $H$ on the surface of the conductor,

$\operatorname{d}\\!\bar{\bar{B}}_{z}(R,H)\equiv\int_{0}^{H}\frac{\operatorname{d}\\!z}{H}\int_{S_{R}}\frac{\operatorname{d}\\!^{2}r}{\pi
R^{2}}\,\operatorname{d}\\!B_{z}(\boldsymbol{r},z).$ (18)

Physically, direct contact between a dipole and an observation point will
result in a divergent magnetic field; hence, we regularize the integration by
starting from a small distance $h$ ($h\ll H,R$) to the conductor surface

$\displaystyle\operatorname{d}\\!\bar{\bar{B}}_{z}(R,H)$
$\displaystyle\approx\operatorname{d}\\!\bar{\bar{B}}_{z}(R,H,h)$
$\displaystyle=\int_{h}^{h+H}\frac{\operatorname{d}\\!z}{H}\int_{S_{R}}\frac{\operatorname{d}\\!^{2}r}{\pi
R^{2}}\,\operatorname{d}\\!B_{z}(\boldsymbol{r},z)$
$\displaystyle=\frac{\mu_{0}}{4\pi}\int_{h}^{h+H}\frac{\operatorname{d}\\!z}{H}\int_{S_{R}}\frac{\operatorname{d}\\!^{2}r}{\pi
R^{2}}\boldsymbol{F}(\boldsymbol{r}-\boldsymbol{x},z)\cdot\boldsymbol{\mathcal{I}}(\boldsymbol{x})\operatorname{d}\\!s$
$\displaystyle=\frac{\mu_{0}}{4\pi}\boldsymbol{\mathcal{I}}(\boldsymbol{x})\operatorname{d}\\!s\cdot\bar{\bar{\boldsymbol{M}}}(\boldsymbol{x},R,H,h),$
(19)

with

$\bar{\bar{\boldsymbol{M}}}(\boldsymbol{x},R,H,h)\equiv\int_{S_{R}}\frac{\operatorname{d}\\!^{2}r}{\pi
R^{2}}\bar{\boldsymbol{F}}(\boldsymbol{r}-\boldsymbol{x},H,h)$ (20)

and

$\bar{\boldsymbol{F}}(\boldsymbol{x},H,h)\equiv\int_{h}^{h+H}\frac{\operatorname{d}\\!z}{H}\boldsymbol{F}(\boldsymbol{x},z).$
(21)

Similarly, the contributions of all dipoles are integrated over,
$\bar{\bar{B}}_{z}(R,H,h)=\int\operatorname{d}\\!\bar{\bar{B}}_{z}(R,H,h)$,
and the variance of the volume-averaged field can be carried out as

$\displaystyle\left\langle\bar{\bar{B}}_{z}(R,H,h)^{2}\right\rangle=\frac{\mathcal{C}}{\pi^{2}R^{2}}\mathfrak{J}\left(\frac{H}{R},\frac{h}{R}\right),$
(22)

where

$\displaystyle\mathfrak{J}\left(\eta,\zeta\right)$
$\displaystyle\equiv\frac{1}{\eta^{2}}\int_{S_{1}}\operatorname{d}\\!^{2}u\int_{S_{1}}\operatorname{d}\\!^{2}v\int\operatorname{d}\\!^{2}X$
$\displaystyle\qquad\left\\{\frac{\left(\eta+\zeta\right)\left(\boldsymbol{u}-\boldsymbol{X}\right)}{\left|\boldsymbol{u}-\boldsymbol{X}\right|^{2}\left[\left|\boldsymbol{u}-\boldsymbol{X}\right|^{2}+\left(\eta+\zeta\right)^{2}\right]^{1/2}}\right.-$
$\displaystyle\qquad\qquad\left.\frac{\zeta\left(\boldsymbol{u}-\boldsymbol{X}\right)}{\left|\boldsymbol{u}-\boldsymbol{X}\right|^{2}\left(\left|\boldsymbol{u}-\boldsymbol{X}\right|^{2}+\zeta^{2}\right)^{1/2}}\right\\}\cdot$
$\displaystyle\qquad\left\\{\frac{\left(\eta+\zeta\right)\left(\boldsymbol{v}-\boldsymbol{X}\right)}{\left|\boldsymbol{v}-\boldsymbol{X}\right|^{2}\left[\left|\boldsymbol{v}-\boldsymbol{X}\right|^{2}+\left(\eta+\zeta\right)^{2}\right]^{1/2}}\right.-$
$\displaystyle\qquad\qquad\left.\frac{\zeta\left(\boldsymbol{v}-\boldsymbol{X}\right)}{\left|\boldsymbol{v}-\boldsymbol{X}\right|^{2}\left(\left|\boldsymbol{v}-\boldsymbol{X}\right|^{2}+\zeta^{2}\right)^{1/2}}\right\\}.$
(23)

At the limit of $H\rightarrow 0$, the variance of the volume average reduces
to the variance of the disk average at a distance $h$, which gives

$\left\langle\bar{\bar{B}}_{z}(R,H\rightarrow
0,h)^{2}\right\rangle\approx\left\langle\bar{B}_{z}(R,h)^{2}\right\rangle.$
(24)

###  III.2 Analytical derivation computed with Monte Carlo integration

The variances of surface and volume averages are important for practical
purposes. To calculate the corresponding results, integrals in Eqs. (17) and
(23) were computed using the method of Monte Carlo integration Caflisch
(1998).

As described in Ref. Nenonen  _et al._ (1996), SQUID detectors used to measure
magnetic fields have pickup coils with finite surface areas within which the
correlation of JNN should be taken into account. Nenonen et al. Nenonen  _et
al._ (1996) calculated the magnetic noise observed by a single circular coil
of diameter $d$ parallel to a conducting slab at a distance $z$, $B_{n,z}^{\rm
coil}$, and plotted the ratio to the single-point spectral density,
$B_{n,z}^{\rm coil}/B_{n,z}$, as a function of $d/z$, shown in Fig. 4 of Ref.
Nenonen  _et al._ (1996). $B_{n,z}^{\rm coil}$ and $B_{n,z}$ are the notations
used in the original paper of Nenonen et al., where $n$ in the subscript
stands for JNN. They are equivalent to $\bar{B}_{z}(R,z)$ and
$B_{z}(\boldsymbol{r},z)$ in our study, respectively. The ratio $B_{n,z}^{\rm
coil}/B_{n,z}$ is the same as $\mathfrak{I}\left(R/z\right)^{1/2}$ in Eq.
(15). We computed this integral $\mathfrak{I}\left(R/z\right)$ and compared it
to the calculation made by Nenonen et al. shown in Fig. 4. The black solid
line in the graph is the result from Ref. Nenonen  _et al._ (1996). The red
dashed line is our result using the Monte Carlo integration, averaged over
thirty random numerical solutions. The other colored points were calculated
with a numerical finite-element method which will be explained in Secs. III.3
and III.4. All methods agree with one another, and the remaining small
deviations are inconsequential for our pragmatic intent.

Figure 4: Root-mean-square normal noise averaged over a finite area w.r.t. a
random single point $(\boldsymbol{r},z)$. Comparison among results obtained
with Monte Carlo integration based on analytical derivation (red dashed line),
calculation shown in Ref. Nenonen  _et al._ (1996) (black solid line) and
numerical finite-element method computed at various distances (colored
points).

As described in Eq. (24), with reduction of cylinder height, the volume
variance converges to the surface variance. Figure 5 displays both
$\left\langle\bar{\bar{B}}_{z}(R,H\rightarrow 0,h)^{2}\right\rangle^{1/2}$
(red, square) and $\left\langle\bar{B}_{z}(R,h)^{2}\right\rangle^{1/2}$ (blue,
circular) for various chamber radii. Integrations were performed using ten to
fifty random solutions in the Monte Carlo method for various radii, where the
mean values are plotted with their standard errors shown as error bars. Filled
data points were computed with a regularization distance
$h=$2.5\text{\,}\mathrm{m}\mathrm{m}$$, whereas open points were calculated
with $h=$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Both methods agree with
each other which confirms the validity of the convergence of the volume
calculation to the surface solution in the limit of zero chamber height
($H\rightarrow 0$).

Figure 5: Comparison of RMS normal noise amplitude at a frequency bandwidth
$\Delta f=1/(2\\!\times\\!200\,{\rm s})$ between surface average (blue,
circular) and volume average with an infinitesimal cylinder height (red,
square). Two regularization distances, $h=$ $2.5\text{\,}\mathrm{m}\mathrm{m}$
or $10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$, are shown as filled or
open data points, respectively.

By integrating over a larger cylinder height, $H$, the volume-averaged JNN
decreases as a result of averaging over uncorrelated noise at relatively
larger distances. Results of
$\left\langle\bar{\bar{B}}_{z}(R,H,h)^{2}\right\rangle^{1/2}$ with various $H$
and $R$ are displayed in Fig. 6. Again, filled and open points were computed
with $h=$ $2.5\text{\,}\mathrm{m}\mathrm{m}$ and
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$, respectively.

Figure 6: Volume-averaged RMS normal noise amplitude at a frequency bandwidth
$\Delta f=1/(2\\!\times\\!200\,{\rm s})$ with various cylinder dimensions. Two
regularization distances, $h=$ $2.5\text{\,}\mathrm{m}\mathrm{m}$ or
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$, are shown as filled or
open data points, respectively.

From Figs. 5 and 6, one can see that the larger the cylinder volume ($R$ or
$H$), the smaller the influence of the regularization distance, $h$.

###  III.3 Finite-element method with discrete dipoles

To estimate the JNN originating from the electrodes, instead of infinite
slabs, conductors of finite size need to be considered. Lee and Romalis Lee
and Romalis (2008) calculated magnetic noise from conducting objects of simple
geometries. The JNN calculation for a thin circular planar conductor is shown
in Tab. VI of Ref. Lee and Romalis (2008), which accords with the geometry of
the electrodes and was used in our study. To estimate an upper limit of the
average-field difference observed by UCN and HgM in the presence of JNN, it is
sufficient to apply the static approximation, where only a white-noise
spectrum at the limit of $f\rightarrow 0$ should be considered, as shown in
Fig. 2. The $z$ component of the ASD measured at a normal distance $d$
generated by a thin film of radius $R$, thickness $a$ and conductivity
$\sigma$ at a temperature $T$ is Lee and Romalis (2008) 222Compare to the
original expression shown in Ref. Lee and Romalis (2008), where variables $a$,
distance, $t$, thickness, and $r$, radius, were replaced with $d$, $a$ and $R$
in our paper to avoid confusion and keep consistency with other variables used
in the remainder of the article.

$\frac{1}{\sqrt{8\pi}}\frac{\mu_{0}\sqrt{k_{{\rm B}}T\sigma
a}}{d}\frac{1}{1+\frac{d^{2}}{R^{2}}}=:\mathcal{B}_{z}^{{\rm
thin}}(d,f\rightarrow 0).$ (25)

However, for a calculation of the magnetic field sampled by particles within
the chamber, the JNN spectrum shown above, Eq. (25), which depends only on the
normal distance between the source of noise and the observation point need to
be replaced with a time-dependent three-dimensional magnetic-noise source. For
this reason, we used a supplementary method by considering a finite number of
random magnetic dipoles on the surface of the conductors as noise sources.

Although the n2EDM apparatus consists of a double-chamber, to study the
possible cancellation of field fluctuation deploying a co-magnetometer system,
we considered only the field measurements taken place in one precession
chamber. In general, only the contribution from the two electrodes defining
top and bottom of the relevant chamber needs to be considered, as the effect
from metal plates further away is exponentially suppressed. Random time series
were generated for each discrete dipole on both electrodes. The superposition
of all these time series and the applied field $B_{0}$ at discrete positions
within the volume of the chamber permits to calculate the distinct magnetic
field $\boldsymbol{B}(\boldsymbol{r},t)$ for any time and locations.

Following the idea of equivalent current dipoles introduced initially in Refs.
Varpula and Poutanen (1984); Nenonen  _et al._ (1996) described in Sec. III.1,
we divided the surface of the electrodes into triangular areas. The motivation
of using triangular grids instead of common quadrilateral meshing methods will
be explained later. For each triangular element, three noise-current sources
located at the center of the triangle and oriented along the three Cartesian
coordinates representing the normal component of the three directions were
created. For a sampling period of $\Delta t$, the magnetic field created by a
number of $n_{\rm dip}$ dipoles and measured at position $\boldsymbol{r}$ can
be calculated by the discrete Biot-Savart law,

$\boldsymbol{B}(\boldsymbol{r},\Delta
t)=\frac{\mu_{0}}{4\pi}\sum_{i=1}^{n_{\rm
dip}}\sum_{\alpha=x,y,z}\frac{I_{\alpha,i}(\Delta
t)\operatorname{d}\\!\boldsymbol{l}\times(\boldsymbol{r}-\boldsymbol{r_{i}^{\prime}})}{|\boldsymbol{r}-\boldsymbol{r_{i}^{\prime}}|^{3}},$
(26)

where $\boldsymbol{r_{i}^{\prime}}$ and $I_{\alpha,i}(\Delta t)$ are the
position and current of an individual dipole in direction $\alpha$, and the
unit-length vector $\operatorname{d}\\!\boldsymbol{l}$ was defined to be the
average side length of the triangles.

The white-noise ASD of the thermal current is Varpula and Poutanen (1984);
Nenonen  _et al._ (1996); Lee and Romalis (2008)

$\mathcal{I}=\sqrt{k_{\rm B}T\sigma a},$ (27)

having a unit of A/$\sqrt{\rm Hz}$. By using the power spectral density (PSD =
ASD2), the variance of the dipole-current time series can be calculated as

$\displaystyle\sigma^{2}(I(\Delta t))$ $\displaystyle=2\,\mathcal{I}^{2}\Delta
f_{\text{\tiny{\rm BW}}}$ $\displaystyle=2k_{\rm B}T\sigma a\Delta
f_{\text{\tiny{\rm BW}}},$ (28)

where $\Delta f_{\text{\tiny{\rm BW}}}$ is the bandwidth, corresponding to
$1/(2\Delta t)$ with $\Delta t$ being the average observation time. The
current $I_{\alpha,i}(\Delta t)$ in Eq. (26) is a random current drawn from a
Gaussian distribution with the defined variance, $\sigma^{2}(I(\Delta t))$.

The surface of the aluminum electrode was divided into approximately 1500
finite surface elements, whose average side length was about 28 mm. The
electrode-division layout was optimized to provide a theoretically compatible
noise spectrum and to be computationally efficient. A dipole with three time-
averaged current components lies at the center of each surface element. The
current components were randomly created from a Gaussian distribution with a
defined variance using $\Delta t=200$ s, which is the free-spin-precession
period of one measurement. Figure 7 shows the time-domain magnetic-field
distribution along a horizontal cut line, $x=-40\dots 40$ cm within the
diameter of the chamber, with one random current-dipole set, where (a)-(d)
indicate field distributions at various distances from the electrode. The
shorter the distance to the source, the larger the amplitude of the field.
Note that also the fluctuation of the field is larger in close vicinity to the
source. This validates the argument in the beginning of this section, that a
normal-distance dependent JNN spectrum, Eq. (25), is inadequate for the
purpose of calculating the impact of JNN on the sensitivity of field-sensing
spin-$\frac{1}{2}$ particles, since the spatial correlation between adjacent
observation points is not considered in the spectral-density formulation.

Figure 7: Normal component of time-domain magnetic-field distribution along a
horizontal cut line at various distances. This is an example created from one
random current-dipole set.

This finite-element method was used to calculate the time-and-volume-averaged
magnetic field observed by field-sensing particles within the chamber over one
measurement, in the presence of JNN. Monte Carlo simulations matching
experimental results Zsigmond (2018) show that the whole volume of the chamber
is sampled isotropically during one 200 s measurement in the case of a large
number of particles; therefore, it is sufficient to divide the chamber into
equally-sized finite volumes, and calculate the magnetic fields observed at
the center of each of these rectangular cuboids. A good balance between
numerical accuracy and computational efficiency was reached with a size of
$10\\!\times\\!10\\!\times 5$ mm3 for these voxels. The reason for using a
smaller vertical dimension was due to the fact that, according to the noise
spectrum, JNN is normal-distance dependent; hence, a transverse separation
between two observation points has fewer impacts than a vertical separation.
This was confirmed by using voxels with a size of $5\\!\times\\!5\\!\times 5$
mm3 whose result was comparable to that calculated with a
$10\text{\,}\mathrm{m}\mathrm{m}$ transverse dimension. In addition, the
volume that was divided has a six times larger diameter than height; to
partition the chamber in both transverse and vertical directions into numbers
of units with the same order of magnitude, we decided to use a voxel with a
smaller vertical size. Voxels with a size of $10\\!\times\\!10\\!\times 2$ mm3
were also studied which gave a negligible difference. As a result, for a
better computational efficiency, a voxel size of $10\\!\times\\!10\\!\times 5$
mm3 was selected as the optimal size for chamber partition and used for the
results presented below.

The reason for using triangular meshes for the conductor is to avoid aligned
patterns between the conductor grids and the voxels within the chamber volume.
In a preliminary study, we found that an alignment between the two meshing
patterns could result in artifacts that computed extremely large magnetic-
field values due to minimum distances between the noise sources and the
observation points. This should be avoided and was resolved by implementing
different meshing geometries for the conductors and the chamber where
alignments could be well reduced.

###  III.4 Comparison between the analytical derivation and the finite-element
method

With the finite-element method, the magnetic noise averaged over a disk or a
cylinder can be easily estimated. With the optimal voxel height of
$5\text{\,}\mathrm{m}\mathrm{m}$, the n2EDM precession chamber was divided
into 24 layers each consists of 5024 pieces. First, we calculated the average
field over different numbers of adjacent voxels on the same layer,
corresponding to radii ranging from $10\dots 35$ mm, w.r.t. a central piece,
where 100 random central pieces were selected. Results from four different
layers, with distances of $7.5\dots 22.5$ mm, are shown in distinct colors and
shapes in Fig. 4. The error bar on each point is the standard deviation of
these 100 randomly chosen central pieces. We confirmed that the surface-
averaged magnetic-field noise computed with the finite-element method is in
good agreement with both the analytical derivation and the calculation
performed in Ref. Nenonen  _et al._ (1996). Next, to calculate a cylinder
average with the finite-element method compatible with the analytical
description, we considered only half of the precession chamber and one
electrode. The half-chamber volume average of normal magnetic field generated
by magnetic dipoles on this electrode, $\left\langle B_{z}\right\rangle$, was
computed and shown in Fig. 8. Each entry in the histogram is the result of one
simulated cycle. For one finite-element calculation, i.e. one simulated cycle,
approximately 1500 dipoles were created on the conductor using three random
noise currents at a bandwidth of $\Delta f_{\text{\tiny{\rm
BW}}}=1/(2\\!\times\\!200\,{\rm s})$. A total of more than 3000 random
configurations were generated to accumulate statistics. The standard deviation
of these random solutions is $\sigma_{Bz}=$3.060\pm
0.037\text{\,}\mathrm{f}\mathrm{T}$$. The uncertainty is the standard error of
$\sigma_{Bz}$ estimated theoretically with
$SE(\sigma_{Bz})=\sigma_{Bz}/\sqrt{2S-2}$, where $S=3450$ is the number of
simulations. On the other hand, using the analytical formula of volume
variance, Eq. (22), and replacing the infinite conductor with a finite
conductor of $R=$40\text{\,}\mathrm{c}\mathrm{m}$$, the standard deviation of
the volume average with twenty random solutions for the Monte Carlo
integration is

$\left\langle\bar{\bar{B}}_{z}(40\,{\rm cm},6\,{\rm cm},2.5\,{\rm
mm})^{2}\right\rangle^{1/2}=2.805\pm 0.005\,{\rm fT},$ (29)

where the error is the statistical error of the Monte Carlo sample. The
results from the two methods agree within half a femtotesla. The small
deviation is negligible for our purpose and confirms the use of voxels for
volume-average calculation with the finite-element method.

Figure 8: Volume average of normal JNN component $\left\langle
B_{z}\right\rangle$ over half of the n2EDM chamber calculated with the
numerical finite-element method.

##  IV Effects on the n2EDM experiment

###  IV.1 Magnetic fields observed by UCN and HgM

Due to the difference in the velocity spectrum and the Larmor precession
frequency, UCN and HgM sample the volume differently under a nominal
$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{T}$ $B_{0}$ field. Much faster
thermal 199Hg atoms fall into the non-adiabatic regime. The spins precess
under a vectorial volume average of the field; hence, the average magnetic
field observed by 199Hg atoms is calculated as

$\begin{split}\left\langle
B_{\text{\tiny{Hg}}}\right\rangle&=\left|\left\langle\boldsymbol{B}\right\rangle\right|\\\
&=\sqrt{\left\langle B_{x}\right\rangle^{2}+\left\langle
B_{y}\right\rangle^{2}+\left\langle B_{0}+B_{z}\right\rangle^{2}}.\end{split}$
(30)

By contrast, due to the much smaller velocity and larger precession frequency,
UCN sample the volume in the adiabatic regime, such that their spins precess
under the volume average of the modulus of the field. In addition, taking into
account the negative center-of-mass offset $\left\langle z\right\rangle$ of
the ensemble of UCN, the average field sampled by UCN is

$\begin{split}\left\langle
B_{\text{\tiny{UCN}}}\right\rangle&=\left\langle\left|\boldsymbol{B}\right|\rho_{\text{{\tiny
UCN}}}(z)\right\rangle\\\
&=\left\langle\sqrt{B_{x}\\!^{2}+B_{y}\\!^{2}+\left(B_{0}+B_{z}\right)^{2}}\rho_{\text{{\tiny
UCN}}}(z)\right\rangle,\end{split}$ (31)

where

$\rho_{\text{{\tiny UCN}}}(z)=\frac{1}{H}\left(1+\frac{12\left\langle
z\right\rangle}{H^{2}}z\right)$ (32)

is the normalized vertical UCN density function.

To estimate the time-and-volume average of the magnetic fields over one
precession chamber sandwiched between two electrodes, the finite-element
method was employed. Figure 9 shows the average magnetic fields observed by
UCN and HgM over one simulated cycle, calculated with $\left\langle
z\right\rangle=-4.1$ mm for Eq. (32). This offset value was obtained using a
Monte Carlo simulation Zsigmond (2018) and is in agreement with the offset
obtained in Ref. Abel  _et al._ (2020). Each entry in the histogram in Fig. 9
is the result of one $\Delta t=200$ s time average simulated with
approximately 1500 dipoles created on both electrodes. The average magnetic
fields were computed with Eqs. (30) and (31). Histograms for UCN and 199Hg
atoms are shown in Figs. 9(a) and 9(b), respectively. The standard deviation
of these distributions, $\sigma\left(\left\langle
B_{\text{\tiny{UCN}}}\right\rangle\right)=$3.781\pm
0.046\text{\,}\mathrm{f}\mathrm{T}$\approx\sigma\left(\left\langle
B_{\text{\tiny{Hg}}}\right\rangle\right)=$3.777\pm
0.046\text{\,}\mathrm{f}\mathrm{T}$$, are comparable within the statistical
error, confirming the naive hypothesis that they similarly sense the effects
from JNN. In addition, this is an order of magnitude lower than the
sensitivity requirement, 30 fT per measurement cycle, for the HgM in the n2EDM
experiment Ayres  _et al._ (2021), indicating that the performance of HgM will
not be limited by JNN.

(a)

(b)

Figure 9: Deviations of time-and-volume-averaged field to the nominal constant
$B_{0}$ magnetic field sampled by (a) UCN and (b) 199Hg atoms.

Next, we studied the influence of JNN on the ratio of precession frequencies
of the two spin-$\frac{1}{2}$ ensembles by looking at the difference of the
average magnetic fields (see Fig. 10). The standard deviation of the
differences of average magnetic fields is $\sigma\left(\left\langle
B_{\text{\tiny{UCN}}}\right\rangle-\left\langle
B_{\text{\tiny{Hg}}}\right\rangle\right)\sim 0.1\,{\rm fT}$. The sensitivity
of an nEDM measurement depends on the uncertainty of the magnetic-field
measurement. By using a mercury co-magnetometer, the effect from JNN is
reduced to $\sigma_{\text{\tiny{\rm JNN}}}\sim 0.1\,{\rm fT}$ per measurement,
and induces an uncertainty on the neutron EDM of

$\sigma_{d_{\rm n}}=\frac{\hbar}{2E}\gamma_{\textrm{n}}\sigma_{\text{\tiny{\rm
JNN}}}=4\times 10^{-28}~{}e\\!\cdot\\!{\rm cm},$ (33)

assuming an electric field $E=15$
$\mathrm{k}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{m}$ and $\gamma_{\rm
n}/2\pi=29.16$ $\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{T}$ is the
gyro-magnetic ratio of the neutron. The experiment will consist of a total of
$M$ $200\text{\,}\mathrm{s}$ long measurement cycles to improve the
statistical sensitivity. Note that the uncertainty on $\sigma_{d_{\rm n}}$
calculated for one cycle in Eq. (33) scales statistically with $M^{-1/2}$.
Results shown in Figs. 9 and 10 had been crosschecked with another 2000 random
configurations which showed similar results, confirming the negligibility of
the statistical error arising from the sampling size.

Figure 10: Difference between the average fields sampled by UCN and mercury
ensembles, $\left\langle B_{\text{\tiny{UCN}}}\right\rangle-\left\langle
B_{\text{\tiny{Hg}}}\right\rangle$.

###  IV.2 Magnetic field measured by CsM

The design of the n2EDM experiment Ayres  _et al._ (2021) deploys more than
100 cesium magnetometers (CsM) being installed above and below the precession-
chamber stack in order to provide essential information about the homogeneity
and stability of the magnetic field. They are arranged radially in groups of
four on vertical modules. Each CsM contains a glass bulb filled with saturated
vapor pressure of 133Cs atoms. They are operated as Bell-Bloom type Bell and
Bloom (1961) magnetometers. Tensor-polarization (alignment) is created by
amplitude-modulated linearly-polarized laser light that traverses the bulb, at
a frequency roughly matched with the 133Cs Larmor precession frequency,
similar as in Refs. Grujić  _et al._ (2015); Afach  _et al._ (2015). Once the
atomic vapor is spin aligned, the light intensity is reduced and kept
constant. As the spin-polarized atoms precess under the influence of
$\boldsymbol{B}$ with a frequency proportional to the magnitude of the
magnetic field, the intensity of transmitted light is periodically modulated
by precessing atoms and detected by a photodiode.

Consider a CsM with a radius of 1.5 cm placed above the top-most electrode.
Polarized 133Cs atoms at different locations within the bulb are exposed to
magnetic noise from the electrode which decreases with distance according to
Eq. (1). The finite-element method introduced in Sec. III.3 was employed to
calculate the average magnetic field measured by 133Cs atoms in the presence
of JNN. For a CsM bulb, the measurement time of the magnetic field is $\delta
t=70$ ms, which is roughly two times the spin-coherence time of 133Cs atoms.
The skin depth at $14\text{\,}\mathrm{H}\mathrm{z}$ is 2.2 cm which is outside
of the thickness range in which the static approximation is valid.
Nonetheless, the static approximation can be used to obtain an upper limit for
the field fluctuation.

In this case, only the closest electrode which was relevant to a specific CsM
was considered. Similarly, the noise source was represented by a number of
dipoles lying on the surface of the electrode, each with three random noise
currents at the bandwidth of $\Delta f_{\text{\tiny{\rm
BW}}}=1/(2\\!\times\\!70\,{\rm ms})$. The bulb was divided into about 14000
voxels of size 1 mm3, much smaller than the voxel size used for chamber
division due to the orders of magnitude smaller volume. The 133Cs atoms sense
the field in the same way as the 199Hg atoms in the precession chamber. Hence,
the average magnetic field observed by a CsM was calculated by averaging over
the fields in all voxels using Eq. (30). More than 3000 random dipole sets
were simulated. The average magnetic fields from JNN for four CsM bulbs placed
on one module with different distances to the electrode were simulated. The
corresponding standard deviations of the time-and-volume-averaged fields at
these positions are shown as orange points in Fig. 11 whose statistical errors
are three orders of magnitude smaller.

In a perfectly spherical CsM bulb, the 133Cs atoms are uniformly distributed
over the volume. Due to the fast movement of 133Cs atoms, the average magnetic
field over the sphere is sampled homogeneously and its value is equal to the
field measured at the center based on the mean-value theorem Griffiths (2017);
Jackson (1998), assuming all sources are outside the sphere. The RMS magnetic
noise can be estimated by the noise observed at the center of the bulb within
a time span $\delta t$. At a distance $d$ measured to the center of the bulb,
the RMS magnetic noise is

$B^{\rm CsM}_{i}(d,\delta t)=\left\\{\int_{0}^{\frac{1}{2\delta
t}}\mathcal{B}_{i}\left(d,f\right)^{2}\operatorname{d}\\!f\right\\}^{1/2},$
(34)

with $i$ being $x,y$ or $z$. In the presence of an applied
$B_{0}\\!\parallel\\!B_{z}$ field of about
$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{T}$, the lateral components
$B_{x},B_{y}\ll B_{0}$ of JNN are quadratically suppressed, hence negligible.
For this reason, we only take the vertical component into account. The normal
RMS magnetic noise estimated at the center of the CsM, $B^{\rm
CsM}_{z}(d,\delta t)$, as a function of distance, is also displayed in Fig 11.

In the figure, both methods deliver similar results with small differences
which can be understood by the following explanations. Equation (34) is the
frequency-bandwidth integrated RMS noise generated by an infinite slab,
whereas the finite-element method took a finite size of the electrode and only
the low-frequency noise was considered. Therefore, the results calculated from
the finite-element method will in principle be larger due to the use of static
approximation which is true for three of the cases. As for the result
calculated at $d=16.3$ cm, the finite-element method computed a smaller value.
For this specific case, the CsM is placed at
$R=$55\text{\,}\mathrm{c}\mathrm{m}$$, which is larger than the electrode
radius; hence, the effect of noise from the electrode will be smaller compared
to the theoretical calculation which used an infinite conductor. In general,
this method provides a sufficiently precise estimation of the impact of JNN on
the measurements by the CsM.

Figure 11: Comparison between RMS normal noise amplitude integrated over 70
ms, calculated from the noise spectrum (blue line), and the average-field
noise measured by a CsM estimated with the finite-element method (orange
points). The statistical errors on the results calculated with the finite-
element method are smaller than the marker size.

The sensitivity goal for n2EDM translates to a maximum RMS noise of 2.7 pT in
70 ms for the CsM Ayres  _et al._ (2021). The upper limits of the noise for
CsM at various distances all lie below the sensitivity limit. In addition, for
an nEDM-measurement cycle of 200 s $\gg$ 70 ms, the magnetic noise seen by a
CsM will be averaged out to a much lower value; hence, we confirm that JNN
from the electrodes is negligible for the design and placement of all CsM
within the experiment.

##  V Conclusion

This paper reports on a finite-element study of Johnson-Nyquist noise (JNN)
originating from the bulk metal electrodes in the n2EDM experiment being
constructed by the nEDM collaboration at PSI. In the first part, we revisited
the theoretical noise spectra Varpula and Poutanen (1984); Nenonen  _et al._
(1996), and compared them to the measurements on a thin aluminum sheet using a
superconducting quantum interference device (SQUID).

Next, we derived for a given frequency bandwidth expressions for the root-
mean-square normal noise amplitudes of averages over a two-dimensional disk
and a cylinder of finite volume. These are important in understanding the
spatial correlation of JNN and are necessary for practical purposes. Numerical
results from the analytical derivation were computed with the method of Monte
Carlo integration and demonstrate good agreement with the calculation
performed in the literature Nenonen  _et al._ (1996).

Using a discretization of the electrodes into a finite number of magnetic
white-noise dipoles, we calculated temporal and spatial magnetic fields
generated by JNN. By averaging these magnetic fields over time and volume, we
obtained the mean magnetic field sensed by precessing ultracold neutrons (UCN)
and mercury (199Hg) atoms. The standard deviation of more than 3000 randomly
produced configurations for UCN and mercury is approximately 3.8 fT, which we
consider as small enough for next-generation neutron electric-dipole-moment
(nEDM) searches. With the same method, we found that for the cesium (133Cs)
vapor magnetometers, the maximum RMS noise observed within a measurement time
of 70 ms is approximately 0.6 pT, which lies below the sensitivity goal of 2.7
pT for n2EDM. Thus, we confirm that the precision of the cesium magnetometers
will not be constrained by JNN from the aluminum electrodes.

Additionally, by computing the average-field difference observed by UCN and
mercury, we found that the noise is sensed highly correlated and mostly
cancels out by using a co-magnetometer to normalize the UCN measurements. That
is, the impact of JNN is negligible for nEDM searches down to a sensitivity of
$4\,\times\,10^{-28}\,e\cdot{\rm cm}$ for a single 200 s measurement. Assuming
a projected experiment of 500 days with $\sim$ 280 cycles per day, this
results in a factor of 374 smaller limit, which is sufficiently small for our
planned nEDM search using a co-magnetometer concept.

##  VI Acknowledgments

We would like to thank A. Crivellin and M. Spira for helpful discussions. We
are grateful for the technical support from P. Hömmen and R. Körber with the
material measurements in BMSR-2, PTB, Berlin. The material measurements inside
BMSR-2 were supported by the Core Facility “Metrology of Ultra-Low Magnetic
Fields” at PTB funded by Deutsche Forshungsgemeinschaft (DFG) through funding
codes: DFG KO 5321/3-1 and TR408/11-1. The swiss members acknowledge the
financial support from the Swiss National Science Foundation through projects
157079, 163413, 169596, 188700 (all PSI), 181996 (Bern), 172639 (ETH), and
FLARE20FL21-186179. This work has also been supported by the Cluster of
Excellence “Precision Physics, Fundamental Interactions, and Structure of
Matter” (PRISMA + EXC2118/1) funded by DFG within the German Excellence
Strategy (Project ID 39083149) from Johannes Gutenberg University Mainz. This
work is also supported by Sigma Xi grants # G2017100190747806 and #
G2019100190747806, and by the award of the Swiss Government Excellence
Scholarships (SERI-FCS) # 2015.0594. The group from Jagellionian University
Cracow acknowledges the support from National Science Centre, Poland, through
grants No. 2018/30/M/ST2/00319, and No. 2020/37/B/ST2/02349. The group from
University of Leuven acknowledges supports from the Fund for Scientific
Research Flanders (FWO).

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