Present address: ]Institut Laue Langevin, 38000 Grenoble, France

Present address: ]University of Zurich, Switzerland

Present address: ]Massachusetts Institute of Technology, MA, USA

Present address: ]RIKEN, Wako, Saitama 351-0198, Japan

# Search for an interaction mediated by axion-like particles with ultracold
neutrons at the PSI

N. J. Ayres  ETH Zürich, Institute for Particle Physics and Astrophysics,
CH-8093 Zürich, Switzerland  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
T. Bouillaud  Université Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 38026
Grenoble, France  E. Chanel  [  Laboratory for High Energy Physics and Albert
Einstein Center for Fundamental Physics, University of Bern, CH-3012 Bern,
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  C. B. Doorenbos  ETH Zürich, Institute for Particle
Physics and Astrophysics, CH-8093 Zürich, Switzerland  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  P. Flaux  Normandie Université, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC
Caen, 14000 Caen, France  W. C. Griffith  Department of Physics and Astronomy,
University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom  P. G. Harris
Department of Physics and Astronomy, University of Sussex, Falmer, Brighton
BN1 9QH, United Kingdom  N. Hild  ETH Zürich, Institute for Particle Physics
and Astrophysics, CH-8093 Zürich, Switzerland  Paul Scherrer Institut, CH-5232
Villigen PSI, Switzerland  M. Kasprzak  Paul Scherrer Institut, CH-5232
Villigen PSI, Switzerland  K. Kirch  ETH Zürich, Institute for Particle
Physics and Astrophysics, CH-8093 Zürich, Switzerland  Paul Scherrer Institut,
CH-5232 Villigen PSI, Switzerland  V. Kletzl  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  J. Krempel
ETH Zürich, Institute for Particle Physics and Astrophysics, CH-8093 Zürich,
Switzerland  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  M. Rawlik  ETH Zürich, Institute for Particle Physics and
Astrophysics, CH-8093 Zürich, Switzerland  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  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  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  K.
Svirina  Université Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 38026
Grenoble, France  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  S. Touati  Université Grenoble
Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 38026 Grenoble, France  A. Weis
Physics Department, Université de Fribourg, CH-1700 Fribourg, Switzerland  E.
Wursten  [  Instituut voor Kern- en Stralingsfysica, University of Leuven,
B-3001 Leuven, Belgium  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  N. Ziehl  ETH Zürich, Institute for Particle Physics and Astrophysics,
CH-8093 Zürich, Switzerland  G. Zsigmond  Paul Scherrer Institut, CH-5232
Villigen PSI, Switzerland

###### Abstract

We report on a search for a new, short-range, spin-dependent interaction using
a modified version of the experimental apparatus used to measure the permanent
neutron electric dipole moment at the Paul Scherrer Institute. This
interaction, which could be mediated by axion-like particles, concerned the
unpolarized nucleons (protons and neutrons) near the material surfaces of the
apparatus and polarized ultracold neutrons stored in vacuum. The dominant
systematic uncertainty resulting from magnetic-field gradients was controlled
to an unprecedented level of approximately
$4\text{\,}\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$ using an array
of optically-pumped cesium vapor magnetometers and magnetic-field maps
independently recorded using a dedicated measurement device. No signature of a
theoretically predicted new interaction was found, and we set a new limit on
the product of the scalar and the pseudoscalar couplings
$g_{s}g_{p}\lambda^{2}<8.3\times 10^{-28}\,${\mathrm{m}}^{2}$$ (95% C.L.) in a
range of $$5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$<\lambda<$25\text{\,}\mathrm{mm}$$ for the monopole-dipole interaction.
This new result confirms and improves our previous limit by a factor of 2.7
and provides the current tightest limit obtained with free neutrons.

dark matter, axion, axion-like particle, beyond Standard Model physics

##  I Introduction

The extremely successful Standard Model (SM) of particle physics provides
testable experimental predictions usually agreeing with laboratory
measurements and astronomical observations at the highest levels of accuracy
[1, 2]. It is therefore considered as the best theory to describe the
fundamental building blocks of the Universe at current measurement
sensitivities. However, together with the Cosmological Standard Model, it
leaves some phenomena unexplained, e.g., the observed matter-antimatter
imbalance also known as baryon asymmetry of the Universe (BAU) or the nature
of dark matter (DM) and dark energy. Therefore, new physics beyond the SM
(BSM) are needed. Searches for permanent electric dipole moments (EDM) [3, 4,
5], which could provide evidence for BSM CP violation, are playing a vital
role in constraining theoretical models and eventually explaining the BAU [6].
Nevertheless, nonobservation of an EDM constrains the CP-violating phase in
the strong interaction to a value that is particularly small
($\bar{\theta}<${10}^{-10}$$, where $\bar{\theta}$ can take any value between
0 and $2\pi$) [2], constituting another big puzzle known as the strong CP
problem [7, 8, 9].

Although the CP-violating phase of the weak interaction in the SM is not
small, it is insufficient to explain the BAU [2]. In contrast, many BSM
theories predict the existence of new particles. In general, these theories
can be categorized into two broad sectors, i.e., ultraviolet (UV) and infrared
(IR) modifications of the SM. In the UV sector, new particles predicted by
super-symmetric theories [2] are expected to be heavy with masses on the order
of or larger than $100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$. They have been
extensively searched for at the Large Hadron Collider (LHC) in the past
decade, so far without a signal [10, 11]. Theories predicting an extension in
the IR regime anticipate particles with very low masses, often referred to as
weakly interacting sub-eV/slim particles (WISPs) [12], which couple very
weakly to visible matter.

In 1984, Moody and Wilczek [13] proposed to search for a new, short-range,
spin-dependent (SRSD) interaction, which could be mediated by very light,
weakly coupled, spin-0 bosons being well motivated candidates of WISPs. For
spin-0 bosons, only two options exist to couple to fermions, either via a
scalar or a pseudoscalar vertex with the coupling constants $g_{s}$ and
$g_{p}$, respectively. For a fermion-fermion interaction with only one boson
exchange, the scalar and the pseudoscalar vertices permit three distinct
interactions in a $\left({\rm monopole}\right)^{2}$, $\left({\rm
dipole}\right)^{2}$, or monopole-dipole virtual boson fields, involving
$g_{s}\\!^{2}$, $g_{p}\\!^{2}$, or $g_{s}g_{p}$, respectively. One prominent
candidate for the mediator particle of these $\left({\rm
monopole}\right)^{2}$, $\left({\rm dipole}\right)^{2}$, and monopole-dipole
interactions is the axion, which is the pseudo-Goldstone boson [8] arising
from the spontaneous breaking of the Peccei-Quinn symmetry [7] introduced in
1977 to solve the strong CP problem, and is today often referred to as the
“canonical QCD axion.” In general, the mediator particle of these interactions
need not be the canonical QCD axion [14, 15], but may be other hypothetical
bosons. These might be spin-0 axion-like particles (ALPs), which have similar
properties to the canonical QCD axion, or very light spin-1 bosons [16, 17]
coupling via the vector ($g_{v}$) and the axial-vector ($g_{A}$) vertices.
Many experiments world wide [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]
are actively searching for these particles, which are considered promising
candidates as microscopic constituents of DM [30].

###  I.1 The monopole-dipole interaction

Among the three interactions, the monopole-dipole interaction involving
$g_{s}g_{p}$ and violating P and T symmetries as well as combined CP symmetry,
is of particular interest, as the demonstration of CP violation would provide
an evidence to one of the three essential criteria to explain the BAU [31].
The potential generated by the monopole-dipole interaction between a polarized
($\dagger$) and an unpolarized particle can be written as [13, 32]

$V(\boldsymbol{r})=g_{s}g_{p}^{\dagger}\frac{\hbar^{2}}{8\pi
m^{\dagger}}\left({\boldsymbol{\sigma}}^{\dagger}\cdot\hat{\boldsymbol{r}}\right)\left(\frac{1}{r\lambda}+\frac{1}{r^{2}}\right)e^{-r/\lambda},$
(1)

where $m^{\dagger}$ and ${\boldsymbol{\sigma}}^{\dagger}$ are the mass and the
Pauli matrices belonging to the spin of the polarized particle,
$\hat{\boldsymbol{r}}$ is the unit vector along the distance $r$ between the
particles, $\lambda=\hbar/\left(m^{\dagger}c\right)$ is the interaction range,
and $\hbar$ is the reduced Planck’s constant. The unpolarized and the
polarized particles couple to the spin-0 boson via unitless scalar and
pseudoscalar coupling constants $g_{s}$ and $g_{p}^{\dagger}$, respectively.

In Refs. [32, 33], it was proposed that ultracold neutrons (UCNs) can be used
to search for ALPs. We searched for such an interaction with the apparatus
originally built for the search for the electric dipole moment of the neutron
(nEDM) [5], and with which we also set limits for an oscillating nEDM [21]
through the axion-gluon coupling and for neutron to mirror-neutron
oscillations [34] at the UCN source [35, 36] of the Paul Scherrer Institute
(PSI) in Switzerland. In the experiment, polarized UCNs and polarized
${}^{199}\text{Hg}$ atoms populated a vacuum volume between two horizontal
electrodes and an insulator ring. A sketch of the experimental apparatus is
shown in Fig. 2, and details about the spectrometer and the measurement
procedure are described in Sec. II. The SRSD interaction would involve the
polarized UCNs stored in the vessel and unpolarized nucleons (protons and
neutrons) on the electrode surfaces. The measurements were performed by
comparing the Larmor precession frequencies of stored UCNs and
${}^{199}\text{Hg}$ atoms, which served as a cohabiting magnetometer, in a
constant magnetic field $\vec{B}_{0}$. An ALP-mediated SRSD interaction
between vessel materials and trapped particles can be considered as a
pseudomagnetic field $b_{\rm UCN}^{\ast}$ influencing the precession frequency
of UCNs, whereas the effect on the ${}^{199}\text{Hg}$ atoms is negligible as
their mass is much larger ($V(\boldsymbol{r})\propto 1/m^{\dagger}$ in Eq.
(1)). Hence, the ratio of the spin precession frequencies of UCNs and
${}^{199}\text{Hg}$ atoms,

$\mathcal{R}^{\uparrow\downarrow}=\left(\frac{f_{\rm n}}{f_{\rm
Hg}}\right)^{\uparrow\downarrow}=\left|\frac{\gamma_{\rm n}}{\gamma_{\rm
Hg}}\right|\left(1\pm\frac{b_{\rm
UCN}^{\ast}}{\left|\vec{B}_{0}{}\right|}\pm\frac{G_{\rm grav}\left\langle
z\right\rangle}{\left|\vec{B}_{0}\right|}+\delta_{\rm else}\right),$ (2)

is sensitive to this interaction while magnetic-field changes cancel and other
effects corrected for. Here, $\gamma_{\rm n}$ and $\gamma_{\rm Hg}$ are the
gyromagnetic ratios of the neutron and the ${}^{199}\text{Hg}$, respectively,
$b_{\rm UCN}^{\ast}$ is the pseudomagnetic field that would be experienced by
UCNs stored in the apparatus (derived in Sec. I.2), and the $+/-$ signs
correspond to the upward/downward directions of the magnetic field according
to gravity. By measuring $\mathcal{R}$ in opposite directions of
$\vec{B}_{0}$, the magnitude of the pseudomagnetic field can be extracted

$\displaystyle b_{\rm UCN}^{\ast}$
$\displaystyle=\frac{\mathcal{R}^{\uparrow}-\mathcal{R}^{\downarrow}}{\mathcal{R}^{\uparrow}+\mathcal{R}^{\downarrow}}\left|\vec{B}_{0}\right|,$
(3)

and in turn the strength of the interaction $g_{s}g_{p}$ can be deduced. The
dominant systematic effect is due to the vertical center-of-mass offset
$\left\langle z\right\rangle$ between the ${}^{199}\text{Hg}$ atoms and the
UCNs in the presence of an effective magnetic-field gradient $G_{\rm grav}$
[37]. Additional effects $\delta_{\rm else}$ are described in more detail
below.

###  I.2 Derivation of the pseudomagnetic field

The interaction is described by the potential given in Eq. (1), and the
effective interaction generated by one electrode in the apparatus is derived
by integrating over all nucleons from the bulk matter. The corresponding
pseudomagnetic field normal to the electrode surface at a height $d$ is
written as [38]

$\displaystyle b^{\ast}(d)$ $\displaystyle\approx
g_{s}g_{p}^{\dagger}\frac{\hbar
N\lambda}{2\gamma^{\dagger}m^{\dagger}}\left(1-e^{-a/\lambda}\right)e^{-d/\lambda},$
(4)

where $\gamma^{\dagger}$ is the gyromagnetic ratio of the polarized particle,
$N$ is the nucleon density depending on the material of the electrode, and $a$
is the electrode thickness, illustrated schematically in Fig. 1.

Figure 1: Schematic drawing of the interaction between one nucleon within the
electrode (with a thickness $a$ and a radius $R$) and a polarized UCN in the
precession chamber. The pseudomagnetic field results from integration over all
nucleons in the bulk electrode.

In the nEDM apparatus, both the top and the bottom electrodes made of aluminum
contributed to this interaction in opposite directions, pointing from the
electrodes to the UCNs stored in the chamber. We defined $z=0$ at the center
of the precession chamber such that the surfaces of the top and the bottom
electrodes were at $z=+H/2$ and $z=-H/2$, respectively, where
$H=$12\text{\,}\mathrm{c}\mathrm{m}$$ is the chamber height. The total
pseudomagnetic field measured at a vertical coordinate $z$ can be written by
summing up contributions from both electrodes using Eq. (4) as

$b_{\rm
ALP}^{\ast}(z)=g_{s}g_{p}^{\dagger}\frac{\hbar\lambda}{2\gamma^{\dagger}m^{\dagger}}\left(1-e^{-a/\lambda}\right)\left(N_{\rm
bot}e^{-\left(z+H/2\right)/\lambda}-N_{\rm
top}e^{-\left(H/2-z\right)/\lambda}\right),$ (5)

where $N_{\rm bot}$ and $N_{\rm top}$ are the nucleon densities of the bottom
and the top electrodes, respectively.

Since the UCNs have very low kinetic energies, their trajectories are strongly
influenced by gravity. As a result, they were not uniformly distributed within
the precession chamber; instead, the center of mass of the UCNs was shifted to
negative $z$ values. This effectively resulted in a center-of-mass offset
$\left\langle z\right\rangle=$-3.9(3)\text{\,}\mathrm{m}\mathrm{m}$$ [5] with
respect to that of the cohabiting ${}^{199}\text{Hg}$ atoms. A linear
approximation of the normalized vertical-UCN-density function is given as [38]

$\rho_{\rm n}\left(z\right)=\frac{1}{H}\left(1+\frac{12\left\langle
z\right\rangle}{H^{2}}z\right).$ (6)

The setup was sensitive to interactions of short ranges, approximately from
$\mathrm{\SIUnitSymbolMicro m}$ to $\mathrm{mm}$, a similar range as in Refs.
[19, 38], therefore, the UCN-density distribution can be simplified to a
constant density at distances close to the surfaces of the electrodes,
$\rho_{\rm n}\left(-H/2\right)$ and $\rho_{\rm n}\left(+H/2\right)$ [38]. The
effective pseudomagnetic field, defined as pointing upwards with respect to
gravity, experienced by all UCNs within the precession chamber, is solved
analytically by integrating over the chamber height

$\displaystyle b_{\rm UCN}^{\ast}$
$\displaystyle=\int_{\frac{-H}{2}}^{\frac{+H}{2}}b_{\rm
ALP}^{\ast}(z)\rho_{\rm n}\left(z\right)\operatorname{d}\\!z$ (7)
$\displaystyle=g_{s}g_{p}^{\dagger}\frac{\hbar\lambda}{2\gamma^{\dagger}m^{\dagger}}\left(1-e^{-a/\lambda}\right)\int_{\frac{-H}{2}}^{\frac{+H}{2}}\left[N_{\rm
bot}\rho_{\rm
n}\left(\frac{-H}{2}\right)e^{-\left(z+H/2\right)/\lambda}-N_{\rm
top}\rho_{\rm
n}\left(\frac{+H}{2}\right)e^{-\left(H/2-z\right)/\lambda}\right]\operatorname{d}\\!z$
$\displaystyle=g_{s}g_{p}^{\dagger}\frac{\hbar\lambda^{2}\left[H\left(N_{\rm
bot}-N_{\rm top}\right)-6\left\langle z\right\rangle\left(N_{\rm bot}+N_{\rm
top}\right)\right]}{2\gamma^{\dagger}m^{\dagger}H^{2}}\left(1-e^{-a/\lambda}\right)\left(1-e^{-H/\lambda}\right),$

where the top and the bottom electrodes contribute in opposite directions.

##  II Measurement with the nEDM spectrometer

For the measurement, we exchanged the top electrode of the nEDM spectrometer
by one made out of copper to increase the nucleon density, which increased the
sensitivity to this interaction. A sketch of the modified apparatus is shown
in Fig. 2, where no electric field was applied during these measurements. The
nucleon density111The nucleon densities were calculated using the material
densities and the atomic masses obtained from MaTeck’s periodic table of
elements (https://mateck.com/en/, accessed 2023-02-26) and Ref. [39]. for the
top electrode was $N_{\rm top}=N_{\rm Cu}=5.40\times
10^{30}\,${\mathrm{m}}^{-3}$$, whereas the bottom electrode made of aluminum
remained, with $N_{\rm bot}=N_{\rm Al}=1.63\times
10^{30}\,${\mathrm{m}}^{-3}$$. In this way, an asymmetric pseudomagnetic field
$b_{\rm ALP}^{\ast}$ was generated, increasing the sensitivity by a factor of
7.7 compared to that using both electrodes made of aluminum.

Figure 2: Schematic drawing of the modified nEDM apparatus, where the top
electrode was replaced with one made of copper, used to search for a new, ALP-
mediated, SRSD interaction.

Further, we exchanged the ultraviolet light source of the probe beam of the
${}^{199}\text{Hg}$-comagnetometer (HgM) from a mercury discharge lamp [40] to
a locked frequency quadrupled diode laser222TOPTICA Photonics AG. Product
description TA / FA-FHG pro. Accessed 2022-01-11.
http://www.toptica.com/products/tunable-diode-lasers/frequency-converted-
lasers/ta-fhg-pro/ with a wavelength of $253.7\text{\,}\mathrm{n}\mathrm{m}$
[41] to maximize the sensitivity of the HgM readout. The rest of the apparatus
remained unchanged compared to our previous search for an SRSD interaction
mediated by an ALP [38]. Polarized mercury vapor and polarized UCNs precessed
in a cylindrical storage chamber of $H=$12\text{\,}\mathrm{c}\mathrm{m}$$
height. Its side walls were made of normal polystyrene with a radius of
$R=$23.5\text{\,}\mathrm{c}\mathrm{m}$$, and the inside was coated with
deuterated polystyrene [42]. The bottom was closed off by an aluminum
electrode with a central shutter for UCNs and a smaller shutter for mercury
vapor. All inner metal surfaces of the storage cylinder, including the
aluminum and the copper electrode surfaces, were coated with a thin layer
($\sim 1\text{--}$3\text{\,}\mathrm{u}\mathrm{m}$$ thickness) of diamond-like
carbon [43, 44] to improve the coherence and storage times of UCNs and
${}^{199}\text{Hg}$ atoms. Additionally, a total of 15 cesium magnetometers
(CsM), of which seven were installed above and eight below the precession
chamber, were used to monitor the magnetic-field gradient $G_{\rm grav}$ along
the chamber axis [45]. A cosine-theta coil comprising around 50 turns powered
with a current of about $17\text{\,}\mathrm{m}\mathrm{A}$ created a stable and
uniform magnetic field of
$\left|\vec{B}_{0}\right|\approx\pm$1036\text{\,}\mathrm{n}\mathrm{T}$$
vertically across the chamber. Additionally, 30 trimcoils were installed
around the vacuum chamber that could be used to create a certain magnetic-
field configuration if required. Four layers of cylindrical mu-metal shield
and a surrounding field compensation coil system [46] were used to passively
and actively improve the stability of the magnetic field.

In each measurement cycle, polarized UCNs in the magnetic field $\vec{B}_{0}$
were used for Ramsey’s method of separated oscillatory fields [47]. A cycle
started by bringing the polarized UCNs into the precessing chamber, followed
by the filling of the polarized ${}^{199}\text{Hg}$ atoms. When both particle
species were prepared in their initial stages in the storage chamber, two low-
frequency pulses were applied consecutively to the ${}^{199}\text{Hg}$ atoms
and to the UCNs to flip their spins to the transverse plane. These pulses are
called $\pi/2$-pulses. After a free-spin-precession duration of
$\mathcal{T}{}=$180\text{\,}\mathrm{s}$$, a second $\pi/2$-pulse, in phase
with the first one, was applied to the UCNs to further tip there spins for
another $\pi/2$. Afterwards, a spin-sensitive detection system [48, 49]
counted neutrons in spin-up ($N^{\uparrow}$) and spin-down ($N^{\downarrow}$)
states at the end of the cycle, from which the asymmetry
$\mathcal{A}=\left(N^{\uparrow}-N^{\downarrow}\right)/\left(N^{\uparrow}+N^{\downarrow}\right)$
was calculated. The HgM precession frequency was measured using a circularly
polarized, resonant ultraviolet laser beam at
$253.7\text{\,}\mathrm{n}\mathrm{m}$ wavelength, traversing the chamber while
recording the spin-precession-modulated light intensity with a photo-
multiplier tube [41]. A measurement run consisted of approximately ten cycles
with different spin-flipping frequencies $f_{\rm n,RF}$. These frequencies
that were applied as $\pi/2$-pulses were slightly detuned from the resonant
$f_{\rm n}$. The scan of $f_{\rm n,RF}$ throughout every run led to different
asymmetries of the final-state neutrons in each cycle. In combination with the
measured HgM frequency of a single cycle, an interference pattern, Ramsey
pattern, may be plotted by displaying the asymmetry as a function of the
frequency ratio $\mathcal{R}_{\rm RF}=f_{\rm n,RF}/f_{\rm Hg}$ (Fig. 3).

The central Ramsey fringe, approximated well with a cosine function,

$\mathcal{A}=\mathcal{A}_{\rm off}-\alpha\cos\left[2\pi\left(\mathcal{R}_{\rm
RF}-\mathcal{R}\right)\mathcal{T}{}^{\prime}\left\langle f_{\rm
Hg}\right\rangle\right],$ (8)

was fitted to the interference pattern, where $\left\langle f_{\rm
Hg}\right\rangle$ is the average HgM frequency of all cycles within the run.
Three parameters, the asymmetry offset $\mathcal{A}_{\rm off}$, the visibility
(or Ramsey contrast) $\alpha$, and the resonant-frequency ratio
$\mathcal{R}=f_{\rm n}{}/f_{\rm Hg}{}$, were extracted.
$\mathcal{T}{}^{\prime}=\mathcal{T}{}+4\tau_{\rm n}/\pi$ is the effective time
related to the fringe width, where $\tau_{\rm n}=$2\text{\,}\mathrm{s}$$ is
the length of a neutron $\pi/2$-pulse. By taking the ratio of the two
frequencies, we compensated for magnetic-field changes from one cycle to the
next one, which cancel in Eq. (2).

Figure 3: Ramsey pattern: The asymmetry $\mathcal{A}$ is plotted as a function
of the frequency ratio $\mathcal{R}_{\rm RF}$. Blue markers are data from an
ALP measurement run (run 012951), and the red line is the fit to Eq. (8). The
uncertainties on $\mathcal{A}$ are smaller than the marker size. The resonant-
frequency ratio $\mathcal{R}$ is at minimal $\mathcal{A}$.

The difference of resonant-frequency ratios $\mathcal{R}$, taken for direction
inverted magnetic-field configurations permits the extraction of $b_{\rm
UCN}^{\ast}$ using Eq. (3). As $\mathcal{R}$ is a linear function of $G_{\rm
grav}$ (Eq. (2)), we intentionally took data at different vertical magnetic-
field gradients. Each measurement run thus comprised a certain magnetic-field
configuration. The vertical magnetic-field gradient was generated by applying
dedicated currents in a pair of trim coils installed above and below the
vacuum tank [50]. In addition to the high-granularity maps of the magnetic
field, taken using a dedicated measurement device [51], the spatial
distribution of the magnetic field was measured continuously with a sampling
rate of $1\text{\,}\mathrm{H}\mathrm{z}$ with 15 CsM. Using a linear fit of
$G_{\rm grav}$ versus $\mathcal{R}\left(G_{\rm grav}{}\right)$ by correcting
for all known systematic effects of $\mathcal{R}$ (Sec. III.1) and precisely
determining $G_{\rm grav}$ (Sec. III.2), $\mathcal{R}_{0}^{\uparrow}$ and
$\mathcal{R}_{0}^{\downarrow}$, resonant-frequency ratios at $G_{\rm
grav}{}=0$ for both $\vec{B}_{0}$ directions, were precisely determined (Sec.
III.3).

##  III Data analysis

###  III.1 Extraction of the resonant-frequency ratio $\mathcal{R}$

For each measurement run, a resonant $\mathcal{R}$ was obtained by fitting the
Ramsey pattern, Eq. (8). Various statistical uncertainties and systematic
effects influenced the measurement of $\mathcal{R}$. Four effects were
considered for each measurement cycle as stochastic uncertainties. These
include the neutron counting statistics, the uncertainty of the estimated HgM
frequency, the magnetic-field-gradient ($G_{\rm grav}$) drift between cycles,
and the Ramsey-Bloch-Siegert shift [52, 53] induced by the $\pi/2$-pulse of
the HgM onto the neutron spin. The last effect resulted from the fact that the
circularly rotating magnetic field applied to the ${}^{199}\text{Hg}$ atoms
resulted in small random tilts of the neutron spins. Each effect resulted in
an uncertainty on the measured asymmetry $\mathcal{A}$, which was further
propagated to the uncertainty of $\mathcal{R}$ according to Eq. (8) using the
fitted values for $\alpha$ and $\mathcal{R}$. Table 1 shows the average
uncertainties of each effect for all measurement cycles. We calculated the
reduced chi-square $\chi^{2}_{\rm red}$ from the Ramsey fit for all 17 runs
including both directions of $\vec{B}_{0}$, and the mean value was $9.15$.
Assuming pure Poisson statistics, the reduced chi-square $\chi^{2}_{\rm red}$
should be approximately 1. A scaling factor of 2.8, the square-root of the
average $\chi^{2}_{\rm red}$ values excluding one run with $\chi^{2}_{\rm
red}>20$, was applied to the $\mathcal{R}$ errors obtained from the fit of all
runs to account for stochastic errors that were unaccounted for333The reason
of excluding this run was due to the fact that its $\chi^{2}_{\rm red}$ was
almost two times larger than the second largest $\chi^{2}_{\rm red}$ among all
17 runs. However, this run was still included in the final analysis. We
verified that by excluding this specific run, the final result remained
unchanged..

Effect / $1\text{\times}{10}^{-7}$ |  $B_{0}$ up |  $B_{0}$ down
---|---|---
Neutron counts | 1.84 | 2.26
HgM frequency | 0.75 | 0.69
Gradient drift | 0.02 | 0.02
${}^{199}\text{Hg}$ spin-flip pulse | 0.07 | 0.23
Total stochastic effects | 2.02 | 2.41
Table 1: Stochastic uncertainties of $\mathcal{R}$ from all measurement
cycles. The total numbers of neutrons have mean values of approximately 14000
and 10000 for the magnetic field pointing upwards and downwards, respectively.

We recall that the dominant systematic effect is the gravitational shift
$\delta_{\rm grav}$ resulting from the center-of-mass offset $\left\langle
z\right\rangle$ between the UCN and the ${}^{199}\text{Hg}$ ensembles (Eq.
(2)). In the presence of a vertical magnetic-field gradient $G_{\rm grav}$,
both species measure slightly different volume averages of the magnetic field.
The gravitational shift is calculated as

$\displaystyle\delta_{\rm grav}=\frac{{\left\langle B_{z}\right\rangle}_{\rm
n}}{{\left\langle B_{z}\right\rangle}_{\rm Hg}}-1=\pm\frac{G_{\rm
grav}\left\langle z\right\rangle}{\left|\vec{B}_{0}\right|},$ (9)

where

$G_{\rm
grav}{}=G_{1,0}+G_{3,0}\left(\frac{3H^{2}}{20}-\frac{3R^{2}}{4}\right)+G_{5,0}\left(\frac{5R^{4}}{8}-\frac{3R^{2}H^{2}}{8}+\frac{3H^{4}}{112}\right)$
(10)

is the effective magnetic-field gradient parallel to the gravitational
gradient calculated using a polynomial expansion of the magnetic field to
fifth degree [54, 51]. $G_{\ell,m}$ are expansion coefficients of degree $l$
and order $m$ of the harmonic polynomial, whereas
$H=$12\text{\,}\mathrm{c}\mathrm{m}$$ and
$R=$23.5\text{\,}\mathrm{c}\mathrm{m}$$ are the height and the radius of the
precession chamber. The $+/-$ signs in Eq. (9) correspond to $\vec{B}_{0}$ in
upward/downward directions, respectively, with $\left\langle
z\right\rangle<0$. More details on this effect is described in Sec. III.2.

Other known effects $\delta_{\rm else}$ shown in Eq. (2) are summarized as

$\delta_{\rm else}=\delta_{\rm T}+\delta_{\rm Earth}+\delta_{\rm
light}+\delta_{\rm inc}+\delta_{\rm JNN},$ (11)

which are caused by the transverse magnetic-field components, the rotation of
Earth, the UV laser beam for the HgM readout, the incoherent scattering of
UCNs on the ${}^{199}\text{Hg}$ atoms, and magnetic-field fluctuations
resulting from Johnson-Nyquist noise (JNN) [55, 56], respectively. They may be
divided into two categories. On the one hand, constant shifts of the UCN or
HgM frequency lead to a deviation of $\mathcal{R}$ from the ratio of pure
gyromagnetic ratios. These include the first three effects, $\delta_{\rm T}$,
$\delta_{\rm Earth}$, and $\delta_{\rm light}$. On the other hand,
$\delta_{\rm inc}$ and $\delta_{\rm JNN}$ are pure stochastic effects, which
do not shift the mean $\mathcal{R}$ value, but result in an increase of the
measurement uncertainty. There effects are shown in Sec. III.3.

The transverse shift $\delta_{\rm T}$ is a consequence of transverse
components of the magnetic field $B_{\rm T}$. Ultracold neutrons in a magnetic
field of $1\text{\,}\mathrm{\SIUnitSymbolMicro T}$ sample the field in the
adiabatic regime of slow particles in a high magnetic field. The measured mean
frequency $\omega_{\rm n}=\gamma_{\rm
n}\left\langle\sqrt{B_{x}^{2}+B_{y}^{2}+B_{x}^{2}}\right\rangle$ is
proportional to the volume average of the magnetic-field modulus. By contrast,
${}^{199}\text{Hg}$ atoms in the same magnetic field fall into the
nonadiabatic regime of fast particles in a low magnetic field, such that their
spins precess at a mean frequency $\omega_{\rm Hg}=\gamma_{\rm
Hg}\sqrt{\left\langle B_{x}\right\rangle^{2}+\left\langle
B_{y}\right\rangle^{2}+\left\langle B_{z}\right\rangle^{2}}$ given by the
volume average of the vector magnetic field. In the presence of $B_{\rm T}$,
this results in

$\delta_{\rm T}=\frac{\left\langle B_{\rm
T}\\!^{2}\right\rangle}{2\vec{B}_{0}\\!^{2}},$ (12)

with

$\left\langle B_{\rm T}\\!^{2}\right\rangle=\left\langle\Delta
B_{x}^{2}+\Delta B_{y}^{2}\right\rangle$ (13)

being the mean-square transverse magnetic-field components, where $\Delta
B_{x}=B_{x}-\left\langle B_{x}\right\rangle$ and $\Delta
B_{y}=B_{y}-\left\langle B_{y}\right\rangle$. For each run, i.e., one
magnetic-field configuration, we calculated $\left\langle B_{\rm
T}\\!^{2}\right\rangle$ using the field maps [51] and corrected the resonant
$\mathcal{R}$ value obtained from the Ramsey fit (Eq. (8)) by $\delta_{\rm
T}$.

Effectively, given the rotation of Earth, the precession frequencies of UCNs
and ${}^{199}\text{Hg}$ atoms are measured in a rotating frame of reference.
They are a combination of the Larmor frequency in a stationary frame and the
Earth’s rotational frequency, $f_{\rm
Earth}=$11.6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{H}\mathrm{z}$$. The
associated shift in $\mathcal{R}$ was corrected for by calculating

$\delta_{\rm Earth}=\mp\left(\frac{f_{\rm Earth}}{f_{\rm n}}+\frac{f_{\rm
Earth}}{f_{\rm Hg}}\right)\cos\left(\theta_{\rm PSI}\right)=\mp 1.4\times
10^{-6},$ (14)

where $\cos\left(\theta_{\rm PSI}\right)=0.738$ is the cosine of the angle
between the $\vec{B}_{0}$ direction and the rotational axis of the Earth,
corresponding to the latitude of the PSI, and the $-/+$ signs correspond to
the upward/downward directions of $\vec{B}_{0}$, respectively.

The third effect $\delta_{\rm light}$ is related to the UV laser traversing
the precession chamber to read out the HgM signal. This value was not
quantified during the measurement; therefore, we only estimate its effect and
consider it as another contribution to the final uncertainty of $\mathcal{R}$.
Details are given in Sec. III.3.

###  III.2 Determination of the vertical magnetic-field gradient $G_{\rm
grav}$

A total of 15 CsM, which were of scalar-type magnetometers, measured the
magnitudes of the magnetic field above and below the precession chamber during
a measurement, which was used to quantify $G_{\rm grav}$. Because of the
applied $\vec{B}_{0}\approx$1\text{\,}\mathrm{\SIUnitSymbolMicro
T}$\,\hat{e}_{z}$, the transverse fields of order
$1\text{\,}\mathrm{n}\mathrm{T}$ were negligible compared to the vertical
field; hence, the scalar fields measured by the CsM were the effective
vertical-field component $B_{z}$.

The magnetic fields measured by the CsM were described by a polynomial
expansion [54]. The gradients $G_{\ell,m}$ were extracted by fitting the 15
field values measured by the CsM to the $z$ component of the polynomial
expansion

$B_{\rm
CsM}^{i}\left(\boldsymbol{r}^{i}\right)=\sum_{\ell,m}G_{\ell,m}\varPi_{z,\ell,m}\left(\boldsymbol{r}^{i}\right),$
(15)

where $i=1,\,2,\,\ldots,\,15$ is the index of the CsM, $\boldsymbol{r}^{i}$ is
the position of the corresponding CsM, and $\varPi_{z,\ell,m}$ is a function
(or mode) expanded in harmonic polynomials of degree $l$ and order $m$
depending on $\boldsymbol{r}^{i}$.

For each degree $\ell$, the order $m$ runs from $-\ell$ to $+\ell$ for
$\varPi_{z,\ell,m}$, which gives $\left(2\ell+1\right)$ terms. It was observed
that lower-degree fields were subjected to fluctuations; thus, they were
measured online with CsM. Constrained by the number of 15 CsM, the highest
full parametrization one can achieve with this method is of second-degree,
which contains nine free parameters. The contributions of higher-degree fields
were found to be more stable and reproducible. To resolve the problem of
higher-degree fields that were not taken into account, we combined both online
CsM measurements and offline field maps [51] to achieve an improved estimate
of $G_{\rm grav}$. With the field maps, the magnetic fields could be expanded
up to sixth degree for all three components in $x$, $y$, and $z$. The
expression of $G_{\rm grav}$ could thus be expanded to fifth degree (Eq.
(10)). Note that only odd gradients contribute to $G_{\rm grav}$ as the
average-magnetic-field components given by even gradients cancel out for
${}^{199}\text{Hg}$ atoms and UCNs.

Several different methods to combine cycle-by-cycle CsM data with magnetic-
field maps were investigated using synthesized magnetic-field readings, which
worked as follows. We calculated field values at the CsM positions using the
magnetic-field-map data and varying the coefficients $G_{\ell,m}^{\rm syn}$
randomly using a Gaussian distribution (Eq. (17)). In addition, a uniformly
distributed random offset in the range of
$\pm$120\text{\,}\mathrm{p}\mathrm{T}$$ [45] was added at each CsM location,
accounting for possible sensor offsets. In total, 200 random fields were
synthesized and analyzed using eight different methods [57]. The optimal
method was selected by minimizing the difference between the synthesized and
the fitted $G_{\rm grav}$ up to fifth degree using Eq. (10),

$\Delta G_{\rm grav}=G_{\rm grav}^{\rm syn}-G_{\rm grav}^{\rm fit},$ (16)

referred to as the deviation. Each synthesized gradient composing $G_{\rm
grav}^{\rm syn}$ included a random error drawn from a Gaussian distribution,
whose standard deviation was the uncertainty of the map, and was given as

$\begin{gathered}\begin{aligned} G_{\ell,m}^{\rm syn}&=G_{\ell,m}^{\rm
map}+\delta_{G_{\ell,m}}^{\rm map}.\\\ \end{aligned}\end{gathered}$ (17)

The uncertainty of the fitted gradient $\sigma_{G_{\rm grav}}$ was calculated
with error propagation from each gradient-fit error $\sigma_{G_{1,0}^{\rm
fit}}$, $\sigma_{G_{3,0}^{\rm fit}}$, and $\sigma_{G_{5,0}^{\rm fit}}$ in Eq.
(10).

We concluded that the optimal method was achieved by removing the fields
described by higher-degree harmonic polynomials with $\ell=3,\ldots,6$ using
the map gradients and performing a second-degree fit including nine gradients
up to $\ell=2$ to the residual fields. The optimal fit method, even in the
presence of CsM offsets with a standard deviation as large as $\sigma_{B_{\rm
offset}}=$115\text{\,}\mathrm{p}\mathrm{T}$$, estimated the coefficients with
a deviation in the range of $\left|\Delta G_{\rm grav}{}\right|\sim
2\text{--}3\,$\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$$, and with
fit uncertainties of $\sigma_{G_{\rm
grav}}<$3.8\text{\,}\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$$
(upward $\vec{B}_{0}$) and $\sigma_{G_{\rm
grav}}<$4.5\text{\,}\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$$
(downward $\vec{B}_{0}$).

This method was applied to all 17 runs of data, consisting of a total of 170
cycles. The residuals $\Delta B^{i}=B_{\rm low}^{i}-B_{\rm fit}^{i}$ for each
CsM $i$ in each cycle was calculated, where $B_{\rm low}^{i}$ are the field
values used in the fit to the polynomial expansion up to second degree after
removing higher-degree contributions, and $B_{\rm fit}^{i}$ are the calculated
value at each CsM position using the fitted gradients. All $\Delta B^{i}$ were
$<$250\text{\,}\mathrm{p}\mathrm{T}$$, which were below the uncertainties of
the field maps. For each cycle, $G_{\rm grav}$ was calculated using the
expansion up to fifth degree, where $G_{1,0}$ was obtained from the second-
degree-polynomial fit, and $G_{3,0}$ as well as $G_{5,0}$ were taken from the
map values. The average value of the estimated $G_{\rm grav}$ from all cycles
in a run was taken as the vertical gradient of this magnetic-field
configuration.

To correct for potential systematic effects on the calculated effective
gradient $G_{\rm grav}$, we made use of the visibility parabola, which is the
visibility of the Ramsey fringe $\alpha$ plotted as a function of $G_{\rm
grav}$. The parabola reaches its maximum at the minimum vertical magnetic-
field gradient, where gravitationally enhanced depolarization [58, 59] is
negligible. Figure 4 shows the parabola for $\vec{B}_{0}$ pointing upwards
(4a) and downwards (4b) with both reaching a similar maximal visibility. The
parabolas were fitted with a simple parabolic function $\alpha\left(G_{\rm
grav}{}\right)=c(G_{\rm grav}{}-g_{0})^{2}+\alpha_{0}$, where $g_{0}$ is the
expected zero gradient. The maximal visibilities were reached at
$g_{0}^{\uparrow}=-2.2\pm 2.2\leavevmode\nobreak\
$\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$$ and
$g_{0}^{\downarrow}=0.02\pm 3.7\leavevmode\nobreak\
$\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$$ for the upward and the
downward $\vec{B}_{0}$ directions, respectively. The uncertainties on the
fitted parameters were estimated by scaling $\chi^{2}_{\rm
red}=\chi^{2}/\text{d.o.f.}$ to 1 in each parabola fit. To account for this
potential shift, we corrected the effective $G_{\rm grav}$ of each run with
$g_{0}^{\uparrow}=$-2.2\text{\,}\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$$
or
$g_{0}^{\downarrow}=$0.02\text{\,}\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$$.

(a)  (b)

Figure 4: Visibility parabola: Visibility $\alpha$ as a function of vertical
magnetic-field gradient $G_{\rm grav}$ for $\vec{B}_{0}$ pointing (a) upwards
and (b) downwards from all 17 runs of data. The error bar on each data point
is the $1\sigma$ error from the Ramsey fit scaled with a factor of 2.8. The
red line and the shaded band show the fit and the 68% confidence interval
expectation. Both parabolas were fitted simultaneously with a same $c$
parameter.

###  III.3 Crossing-point analysis

Figure 5 shows the $\mathcal{R}$ values, after $\delta_{\rm T}$ and
$\delta_{\rm Earth}$ corrections, as a function of the corrected $G_{\rm
grav}$ (Fig. 4). Red upward triangles and blue downward triangles are runs
with $\vec{B}_{0}$ pointing upwards and downwards, respectively. A linear fit
to the data from all runs with both directions of $\vec{B}_{0}$ ($+/-$
correspond to upward/downward) was applied to

$\mathcal{R}^{\uparrow/\downarrow}=\mathcal{R}_{0}^{\uparrow/\downarrow}\left(1\pm\frac{\left\langle
z\right\rangle}{\left|\vec{B}_{0}{}\right|}G_{\rm grav}^{\uparrow}\right)$
(18)

simultaneously, sharing the parameter $\left\langle z\right\rangle$, while the
$\mathcal{R}_{0}^{\uparrow/\downarrow}$ values were kept separate for both
directions. This is the so-called crossing-point analysis. The best fit was
obtained for

$\displaystyle\mathcal{R}_{0}^{\uparrow}$ $\displaystyle=3.8424563(08),$ (19)
$\displaystyle\mathcal{R}_{0}^{\downarrow}$
$\displaystyle=3.8424622(12),\leavevmode\nobreak\ \text{and}$
$\displaystyle\left\langle z\right\rangle$
$\displaystyle=$-0.43(2)\text{\,}\mathrm{c}\mathrm{m}$,$

with $\chi^{2}_{\rm red}=\chi^{2}/\text{d.o.f.}=31.9/14$. The underestimated
uncertainties caused $\chi^{2}_{\rm red}$ to be larger than 1. The
uncertainties shown in Eq. (19) were corrected for this stochastic error.
Compared to the total statistical errors shown in Tab. 1 summing from all
known effects, these are a factor of 4–5 larger, corresponding to our initial
scaling factor of 2.8 multiplied by the correction factor $\sqrt{31.9/14}$.
The center-of-mass offset $\left\langle z\right\rangle$ was in agreement with
the values found in Ref. [5], $\left\langle
z\right\rangle=$-0.39(3)\text{\,}\mathrm{c}\mathrm{m}$$, and Ref. [54],
$\left\langle z\right\rangle=$-0.38(3)\text{\,}\mathrm{c}\mathrm{m}$$, and the
crossing point was at
$G_{\times}=-1.9(5)\,$\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$$.

Figure 5: Frequency ratio $\mathcal{R}$ as a function of vertical magnetic-
field gradient $G_{\rm grav}$ for all runs with different field
configurations. Red upward and blue downward triangles correspond to
$\vec{B}_{0}$ pointing upwards and downwards, respectively. The error bar for
each $\mathcal{R}$ is the fit error from the Ramsey fit scaled by 2.8. The
error of $G_{\rm grav}$ for each data point with an average value of
$4.05\text{\,}\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$ over all 17
runs was not included in the error of $\mathcal{R}$ individually. Instead, we
considered this effect a global systematic uncertainty to all runs that
resulted in an error contribution to both of the fitted
$\mathcal{R}^{\uparrow/\downarrow}$ values. For this, the error obtained from
each fit of the visibility parabolas was used (vertical magnetic-field
gradient in Tab. 2). Straight lines shown here were obtained using the best-
fit values (Eq. (19)) in the fit equations (Eq. (18)).

In the following paragraphs, we distinguish two different kinds of
uncertainties associated with $\mathcal{R}$ as shown in Eq. (2). The first
kind was systematic effects, leading to a bias, whereas the second kind was
considered purely stochastic, only increasing the measurement uncertainty.

The most important systematic effect of the first kind is the shift
$\delta_{\rm grav}$ induced by a vertical magnetic-field gradient. From the
visibility parabolas (Fig. 4), $g_{0}^{\uparrow/\downarrow}$ were considered
as systematic shifts on $\mathcal{R}$. For all runs, $G_{\rm grav}$ were
corrected for with $g_{0}^{\uparrow/\downarrow}$, and the uncertainties of the
fitted $g_{0}^{\uparrow/\downarrow}$ lead to

$\displaystyle\sigma_{R_{\rm grav}}^{\uparrow}$ $\displaystyle=35\times
10^{-7}\leavevmode\nobreak\ \text{and}$ (20) $\displaystyle\sigma_{R_{\rm
grav}}^{\downarrow}$ $\displaystyle=59\times 10^{-7}.$

The second largest systematic effect $\delta_{\rm T}$ arises from the residual
transverse field. In addition to the shift in $\mathcal{R}$, corrected for
run-by-run, an error on the mean-squared transverse field
$\sigma_{\left\langle B_{\rm T}^{2}\right\rangle}$ was calculated making use
of the concept of reproducibility of the field maps [51], quantifying the
spread in measurements of identical magnetic-field configurations taken over
several years. This results in shifts of

$\displaystyle\mathcal{R}_{\rm T}^{\uparrow}$ $\displaystyle=\left(7.3\pm
4.7\right)\times 10^{-7}\leavevmode\nobreak\ \text{and}$ (21)
$\displaystyle\mathcal{R}_{\rm T}^{\downarrow}$ $\displaystyle=\left(6.4\pm
4.1\right)\times 10^{-7}.$

The third systematic shift $\delta_{\rm light}$ may occur resulting from the
resonant UV laser beam traversing the precession chamber to read out the
${}^{199}\text{Hg}$ spin precession. Two different systematic effects, the
vector and the direct light shifts, were considered. The vector light shift
was measured for our previous experiment [50] using a ${}^{204}\text{Hg}$
discharge lamp as the light source. This shift, which can be interpreted as
the projection of the magnetic field of the photons traversing the precession
chamber onto the $\vec{B}_{0}$-field direction, is magnetic-field direction
dependent. As we exchanged the slightly off-resonant ${}^{204}\text{Hg}$ lamp
with a laser beam resonantly locked to the ${}^{199}\text{Hg}$
$6\,^{1}S_{0}\rightarrow 6\,^{3}P_{1}\leavevmode\nobreak\ F=1/2$ transition,
the shift reduced by a factor of 7.7 [60] to

$\displaystyle\mathcal{R}_{\rm VL}^{\uparrow}$ $\displaystyle=\left(1.5\pm
6.9\right)\times 10^{-7}\leavevmode\nobreak\ \text{and}$ (22)
$\displaystyle\mathcal{R}_{\rm VL}^{\downarrow}$ $\displaystyle=\left(1.2\pm
5.4\right)\times 10^{-7},$

where we kept the original uncertainties as the effect of the laser light
power, which had an impact on the vector light shift, was not quantified. In
addition, the direct light shift accounts for the fact that while the probed
atom is in the $6\,^{3}P_{1}\leavevmode\nobreak\ F=1/2$ state, the spin
precesses at a different frequency. This will lead to a shift proportional to
the light power and was estimated to be about 0.01 ppm [5] in the nEDM
measurement with the same apparatus still using the ${}^{204}\text{Hg}$
discharge lamp. An increase in light power would also result in a decrease of
the transverse depolarization time $T_{2}$ of the mercury precession, which
was not observed. Nevertheless, to account for a possible doubling of the
light intensity because of the change to the resonant laser light, we estimate

$\mathcal{R}_{\rm DL}^{\uparrow/\downarrow}=\left(0.4\pm 0.8\right)\times
10^{-7}.$ (23)

The contributions from both the vector and the direct light shifts are summed
together and shown as the effect from the mercury light in Tab. 2.

Within the medium of spin-polarized ${}^{199}\text{Hg}$ vapor, UCNs are
subjected to incoherent scattering on the ${}^{199}\text{Hg}$ nucleus, which
can be described as a spin-dependent nuclear interaction. This acts as a
pseudomagnetic field, which is proportional to the incoherent scattering
length $\left|b_{\rm inc}\right|=$15.5\text{\,}\mathrm{f}\mathrm{m}$$ [61,
62]. For an imperfect $\pi/2$-pulse of the HgM, a residual polarization along
$\vec{B}_{0}$ creates a pseudomagnetic field, resulting in a frequency shift
of UCNs and consequently a shift $\delta_{\rm inc}$ of $\mathcal{R}$. We
estimated the random fluctuation of ${}^{199}\text{Hg}$ polarization and
quantified the resultant error on $\mathcal{R}$ as $\sigma_{R_{\rm
inc}}^{\uparrow/\downarrow}\leq 5\times 10^{-10}$. This effect is three orders
of magnitude smaller than the light shift $\delta_{\rm light}$; hence, we
consider it negligible.

The precession of spin-polarized particles is affected by magnetic-field
fluctuations resulting from JNN, originating from thermal motion of charge
carriers inside the electrodes. Because of the difference between adiabatic
and nonadiabatic magnetic-field samplings for UCNs and ${}^{199}\text{Hg}$
atoms, the volume-averaged fields sampled by both species are slightly
different. A finite-element analysis was used to simulate temporal and spatial
noise [63] from which we calculated the time-and-volume-averaged magnetic-
field difference sensed by both particle ensembles. As JNN leads to random
magnetic-field fluctuations independent of $\vec{B}_{0}$ polarity, we assume
that this effect $\delta_{\rm JNN}$ only increases the measurement uncertainty
but does not shift the central $\mathcal{R}$ value. The corresponding
uncertainty was estimated to be $\sigma_{R_{\rm
JNN}}^{\uparrow/\downarrow}\leq 1\times 10^{-9}$. As this effect is two orders
of magnitude smaller than $\delta_{\rm light}$, we did not include it in the
error budget.

The analysis of the measured data, including all shifts and uncertainties as
listed in Tab. 2, results in two independent $\mathcal{R}$ values

$\begin{gathered}\begin{aligned} \mathcal{R}^{\uparrow}&=3.8424563(08)_{\rm
stat}(36)_{\rm sys}\enspace\text{and}\\\
\mathcal{R}^{\downarrow}&=3.8424622(12)_{\rm stat}(59)_{\rm
sys}\end{aligned}\end{gathered}$ (24)

at the limit of $G_{\rm grav}{}=0$. Both values are in agreement with our
previous measurement of the neutron to mercury gyromagnetic ratio [50],

$\gamma_{\rm n}{}/\gamma_{\rm Hg}{}=3.8424574(30).$ (25)

Effect / $1\text{\times}{10}^{-7}$ |  $B_{0}$ up |  $B_{0}$ down
---|---|---
Statistics (uncertainty) | $\pm 8$ | $\pm 12$
Vertical magnetic-field gradient | $35\pm 35$ | $-0.3\pm 59$
Residual transverse field | $7.3\pm 4.7$ | $6.4\pm 4.1$
Mercury light | $1.9\pm 6.9$ | $1.6\pm 5.5$
Table 2: Error budget for the overall errors on $\mathcal{R}$ resulting from
statistics and from systematic effects for both $\vec{B}_{0}$ directions. Note
that the effects resulting from the vertical magnetic-field gradient and the
transverse magnetic field were taken into account before the fit for the
crossing point analysis.

##  IV Interpretation of results

According to Eq. (2), $b_{\rm UCN}^{\ast}$ was extracted at the limit of
$G_{\rm grav}{}=0$ after correcting for all systematic effects $\delta_{\rm
else}$,

$b_{\rm
UCN}^{\ast}=\frac{\mathcal{R}^{\uparrow}-\mathcal{R}^{\downarrow}}{\mathcal{R}^{\uparrow}+\mathcal{R}^{\downarrow}}\left|\vec{B}_{0}\right|=$-0.80\text{\,}\mathrm{pT}$,$
(26)

where $\left|\vec{B}_{0}\right|=$1037.19(2)\text{\,}\mathrm{nT}$$ was taken
from the average $\vec{B}_{0}$ value of all runs. The uncertainty of $b_{\rm
UCN}^{\ast}$ was calculated by including uncertainties from
$\mathcal{R}^{\uparrow}$, $\mathcal{R}^{\downarrow}$, and $\vec{B}_{0}{}$,

$\sigma_{b_{\rm
UCN}^{\ast}}=\left\\{\left[\frac{2\mathcal{R}^{\downarrow}\left|\vec{B}_{0}\right|}{\left(\mathcal{R}^{\uparrow}+\mathcal{R}^{\downarrow}\right)^{2}}\sigma_{\mathcal{R}}^{\uparrow}\right]^{2}+\left[\frac{-2\mathcal{R}^{\uparrow}\left|\vec{B}_{0}\right|}{\left(\mathcal{R}^{\uparrow}+\mathcal{R}^{\downarrow}\right)^{2}}\sigma_{\mathcal{R}}^{\downarrow}\right]^{2}+\left[\frac{\mathcal{R}^{\uparrow}-\mathcal{R}^{\downarrow}}{\mathcal{R}^{\uparrow}+\mathcal{R}^{\downarrow}}\sigma_{\vec{B}_{0}{}}\right]^{2}\right\\}^{1/2}=$0.96\text{\,}\mathrm{pT}$.$
(27)

The errors $\sigma_{\mathcal{R}}^{\uparrow}$ and
$\sigma_{\mathcal{R}}^{\downarrow}$ were calculated by summing in quadrature
the statistical error and all systematic effects that contributed to the error
budget of $\mathcal{R}$ (Tab. 2). $\sigma_{\vec{B}_{0}{}}$ was taken from the
larger of the two standard deviations,
$\sigma_{\vec{B}_{0}{}}^{\uparrow}=$17\text{\,}\mathrm{p}\mathrm{T}$$ and
$\sigma_{\vec{B}_{0}{}}^{\downarrow}=$22\text{\,}\mathrm{p}\mathrm{T}$$,
measured within one $\vec{B}_{0}$ direction instead of taking the standard
deviation of all 17 runs. Using Eq. (7), $g_{s}g_{p}$ was derived as

$g_{s}g_{p}=b_{\rm UCN}^{\ast}\frac{2\gamma_{\rm n}m_{\rm
n}H^{2}}{\hbar\lambda^{2}\left[H\left(N_{\rm bot}-N_{\rm
top}\right)-6\left\langle z\right\rangle\left(N_{\rm bot}+N_{\rm
top}\right)\right]}\left(1-e^{\frac{-a}{\lambda}}\right)^{-1}\left(1-e^{\frac{-H}{\lambda}}\right)^{-1}.$
(28)

With the measured $b_{\rm UCN}^{\ast}$ (Eq. (26)) and the estimated error
$\sigma_{b_{\rm UCN}^{\ast}}$ (Eq. (27)), a 95% confidence level limit on
$g_{s}g_{p}$ gives

$g_{s}g_{p}\lambda^{2}<8.3\times 10^{-28}\,${\mathrm{m}}^{2}$,$ (29)

for $$5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$<\lambda<$25\text{\,}\mathrm{mm}$$. On the one hand, the upper limit of
this range was defined as the thickness of the electrodes. Approaching the
upper limit, the last two terms in Eq. (28) depart from 1, and the relation
$g_{s}g_{p}\propto 1/\lambda^{2}$ is not fulfilled anymore, which reduces the
measurement sensitivity on $g_{s}g_{p}$. On the other hand, the lower end of
this range is constrained by the wavelength of UCNs and the surface property
of the electrodes, such as the surface roughness, which was in the range of a
few hundred nm, or the a-few-$\mathrm{u}\mathrm{m}$-thick diamond-like-carbon
coating that has a nucleon density between those of the aluminum and the
copper electrodes.

Figure 6 shows the upper limits of $g_{s}g_{p}$ constrained by the most recent
measurements covering an interaction range of
$$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$<\lambda<$1\text{\,}\mathrm{mm}$$.
The upper horizontal axis displays the corresponding mass of an ALP $m_{\rm
ALP}$, with $\lambda=\hbar/\left(m_{\rm ALP}c\right)$. The figure shows five
measurement results, labeled from A to E. A is the limit obtained from this
work (Eq. (29)), whereas E is obtained from our previous experiment [38]. Both
experiments were based on a clock comparison of precession frequencies between
polarized UCNs and ${}^{199}\text{Hg}$ atoms. An improvement on the
sensitivity by a factor of 2.7 was accomplished, which is the current best
limit of $g_{s}g_{p}$ obtained with UCNs. Additionally, we estimated the
sensitivity of a new experiment, n2EDM, which is currently under construction
at the PSI, to the SRSD interaction. The projected sensitivity is shown as B,
and details of improvements on different statistical and systematic aspects
are given in Sec. V. C is the result based on the comparison of spin-
precession-frequency shifts of cohabiting ${}^{3}{\rm He}$ and ${}^{129}{\rm
Xe}$ atoms, in the presence of an unpolarized mass of BGO (${\rm Bi}_{4}{\rm
Ge}_{3}{\rm O}_{12}$) crystal [64]. D results from the comparison of nuclear-
magnetic-resonance-frequency shifts of cohabiting polarized ${}^{129}{\rm Xe}$
and ${}^{131}{\rm Xe}$ atoms in the presence of a nonmagnetic zirconium rod
[65]. F is the result from the measurement of anomalous spin relaxation of
polarized ${}^{3}{\rm He}$ atoms induced by an additional depolarization
channel, which might be caused by the pseudomagnetic field generated from the
${}^{3}{\rm He}$-cell walls [19].

Figure 6: Upper limits on the $g_{s}g_{p}$-coupling of an interaction mediated
by a spin-0 axion-like particle as a function of the interaction range
$\lambda$ and the mass of an ALP $m_{\rm ALP}$. A: this work (Eq. (29)). B:
outlook on the n2EDM experiment with copper layers (Sec. V). C: clock
comparison of ${}^{3}{\rm He}$ and ${}^{129}{\rm Xe}$ [64]. D: clock
comparison of ${}^{129}{\rm Xe}$ and ${}^{131}{\rm Xe}$ [65]. E: our previous
experiment with UCNs and ${}^{199}\text{Hg}$ atoms [38]. F: ${}^{3}{\rm He}$
depolarization [19].

##  V Prospects for ALP measurements in the n2EDM experiment

A new experiment, n2EDM, to search for a permanent nEDM is currently under
construction at the PSI. The aim is to search for an nEDM with a sensitivity
of about [66, 67]. This will be accomplished by improved statistical
sensitivity and an improved control of systematic effects. The apparatus can
also be used to search for an SRSD interaction mediated by ALPs.

The n2EDM apparatus features two precession chambers mounted on top of each
other. This stack is made of three electrodes from aluminum spaced vertically
by a height of $12\text{\,}\mathrm{c}\mathrm{m}$. In between the electrodes,
cylindrical rings made of isolating polystyrene with a diameter of
$80\text{\,}\mathrm{c}\mathrm{m}$ [67] define the two precession chambers,
increasing the volume of each precession chamber by a factor of three (a
factor of six in total). The double-chamber design allows a simultaneous
measurement in both chambers with opposite electric-field polarities while
being exposed to the same magnetic-field direction. For ALP measurements where
the electric field is not applied, observations simultaneously within both
chamber are discussed. We estimate the improved statistical and systematic
sensitivities with respect to this measurement.

The statistical sensitivity of $\mathcal{R}=f_{\rm n}{}/f_{\rm Hg}{}$ has
contributions from neutron counting statistics and the uncertainty of the
mercury-precession signal. With an optimized connection from the UCN source to
the experiment and the double chamber with a factor of six larger total volume
for UCNs, the projected number of detected neutrons after a free precession
time of $\mathcal{T}=$180\text{\,}\mathrm{s}$$ will increase by a factor of
eight. At the same time, the fringe visibility $\alpha$ slightly improves to
$0.8$ [67]. Together, this will result in an improvement on the sensitivity of
$f_{\rm n}$ by a factor of three, corresponding to a magnetic-field
sensitivity of $110\text{\,}\mathrm{f}\mathrm{T}$. The sensitivity of the
mercury magnetometer is expected to be at least $\sigma_{B_{\rm
Hg}}=$30\text{\,}\mathrm{f}\mathrm{T}$$ per cycle [67]. Hence, we expect a
factor of six in statistical improvement to
$0.2\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$. Recall that unexplained noise
decreased the expected statistical sensitivity by a factor of 4–5 in the
search presented in this article.

The largest uncertainty in the current result stems from the vertical
magnetic-field gradient $G_{\rm grav}$. Assuming that $G_{\rm grav}$ will be
expanded up to fifth degree (Eq. (10)), we expect the uncertainties of
$G_{1,0}$, $G_{3,0}$, and $G_{5,0}$ determined by the HgM, the CsM array, and
a more reproducible magnetic-field maps, respectively, to be [67]

$\displaystyle\sigma_{G_{1,0}}$ $\displaystyle\leq\sqrt{2}\,\sigma_{B_{\rm
Hg}}/H^{\prime}\approx$2.4\text{\,}\mathrm{f}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$,$
(30) $\displaystyle\sigma_{G_{3,0}}$
$\displaystyle\leq$36\text{\times}{10}^{-3}\text{\,}\mathrm{f}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$,\enspace\text{and}$
$\displaystyle\sigma_{G_{5,0}}$
$\displaystyle\leq$20\text{\times}{10}^{-6}\text{\,}\mathrm{f}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}^{5}$,$

where $H^{\prime}=$18\text{\,}\mathrm{c}\mathrm{m}$$ is the distance between
the centers of the upper and the lower precession chambers. Summing up all
contributions, this implies a systematic uncertainty of $\sigma_{G_{\rm
grav}{}}=$51\text{\,}\mathrm{f}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$$ on
the vertical magnetic-field gradient; an improvement by a factor of 80 from
the average uncertainty of
$4.05\text{\,}\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$ observed
here.

The transverse magnetic field was estimated with field maps [51]. In n2EDM,
field maps will be used to measure all higher gradients above $G_{3,0}$, with
a reproducibility requirement matching the previous repeatability [67]. This
results in a tenfold improvement of the uncertainty $\sigma_{\left\langle
B_{\rm T}^{2}\right\rangle}$.

By using a linearly polarized light scheme for reading out the mercury
precession, we can suppress entirely the direct light shift. The vector light
shift can be partially suppressed and characterized in dedicated measurements
and will not significantly contribute to an overall error budget [68].

In summary, this results in a measurement of $\mathcal{R}$ with a statistical
precision of about $\sigma_{\rm stat}^{\rm n2EDM}=$2\text{\times}{10}^{-7}$$
and a systematic precision of about $\sigma_{\rm syst}^{\rm
n2EDM}=$4\text{\times}{10}^{-8}$$, a factor of 15 improvement compared to (25)
[50]. With the estimated improvements on the statistical and, in particular,
on the gradient-induced systematic uncertainties, a factor of three
improvement to $g_{s}g_{p}\lambda^{2}<2.7\times 10^{-28}\,${\mathrm{m}}^{2}$$
is anticipated when using three electrodes all made of aluminum. This might
seem astonishing, considering the 25 times sensitivity gain on the measurement
of the pseudomagnetic field estimated with Eq. (27). However, in the new
experiment, the center-of-mass offsets estimating individually from each
chamber are both $\langle z\rangle\approx$0.1\text{\,}\mathrm{c}\mathrm{m}$$,
which are a factor of four smaller, reducing the sensitivity by a factor of
three.

Note that similar to the current experiment, using an asymmetric nucleon
density between the upper and the lower boundary of each chamber, the
sensitivity can be significantly increased. A possible approach might be
placing copper sheets on the middle and the lower electrodes. With a
$1$-$\mathrm{m}\mathrm{m}$-thick copper sheet, the interaction range up to
$1\text{\times}{10}^{-3}\text{\,}\mathrm{m}$ can be covered. A new upper limit
on the product of the couplings of about $g_{s}g_{p}\lambda^{2}<1.3\times
10^{-29}\,${\mathrm{m}}^{2}$$ (95% C.L.) is then expected (marked as B in Fig.
6), when using copper having a 3.3 times larger nucleon density compared to
aluminum.

##  VI Conclusion

This paper reports on the null result from a search for a hypothetical, short-
range, spin-dependent interaction mediated by axion-like particles. Ultracold
neutrons were stored simultaneously with ${}^{199}\text{Hg}$ atoms in a
cylindrical chamber sandwiched between a copper and an aluminum electrode in a
constant vertical magnetic field in the same apparatus used to measure the
neutron electric dipole moment at the PSI. By measuring the precession-
frequency ratio $\mathcal{R}=f_{\rm n}{}/f_{\rm Hg}{}$ between UCNs and
${}^{199}\text{Hg}$ atoms in opposite magnetic-field directions, we searched
for the SRSD interaction between nucleons of the electrodes and stored UCNs.

Systematic effects from magnetic-field gradients influenced the measurement of
$\mathcal{R}$. The dominant effect arose from the center-of-mass offset
between the two particle species in an effective vertical magnetic-field
gradient $G_{\rm grav}$. As $\mathcal{R}$ is a linear function of $G_{\rm
grav}$, we intentionally applied a vertical magnetic-field gradient in the
measurement and compared $\mathcal{R}$ in different $G_{\rm grav}$ using the
crossing-point analysis. For a better estimation on $G_{\rm grav}$, we
combined both the online CsM data and magnetic-field maps taken at a different
time using a dedicated device for mapping. The optimal method to incorporate
both results was determined using synthesized data. By applying this optimized
method to measurement data, $G_{\rm grav}$ was estimated with an unprecedented
precision of around
$4\text{\,}\mathrm{p}\mathrm{T}\mathrm{/}\mathrm{c}\mathrm{m}$.

By extracting $\mathcal{R}$ at $G_{\rm grav}{}=0$ after correcting for all
known systematic effects, a new limit on the product of the scalar and the
pseudoscaler couplings, corresponding to the monopole-dipole interaction,
gives $g_{s}g_{p}\lambda^{2}<8.3\times 10^{-28}\,${\mathrm{m}}^{2}$$ (95%
C.L.) in an interaction range of $$5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$<\lambda<$25\text{\,}\mathrm{mm}$$. This limit improves our previous
experiment by a factor of 2.7, the best limit obtained with free neutrons.

With the n2EDM apparatus at the PSI, we plan to search for a nonzero nEDM with
a sensitivity of about . By comparing the precession frequencies of
${}^{199}\text{Hg}$ atoms and the UCNs in the new spectrometer, a new, at
least 15 times more accurate measurement of the gyromagnetic ratios
$\gamma_{\rm n}{}/\gamma_{\rm Hg}{}$ becomes possible. Further, a refined
search of ALPs by placing a $1$-$\mathrm{m}\mathrm{m}$-thick copper layer on
the corresponding bottom electrode of each chamber seems attractive. A new
upper limit of $g_{s}g_{p}\lambda^{2}<1.3\times 10^{-29}\,${\mathrm{m}}^{2}$$
(95% C.L.), a factor of 64 better than the result presented here, could be
expected.

##  VII Acknowledgments

We acknowledge the excellent support provided by the PSI technical groups and
by various services of the collaborating universities and research
laboratories. In particular, we acknowledge with gratitude the long term
outstanding technical support by F. Burri and M. Meier. We thank the UCN
source operation group BSQ for their support. We acknowledge financial support
from the Swiss National Science Foundation through projects No. 117696, No.
137664, No. 144473, No. 157079, No. 172626, No. 126562, No. 169596, No. 178951
(all PSI), No. 181996 (Bern), No. 162574, No. 172639 (both ETH). The group
from Jagiellonian University Cracow acknowledges the support from the National
Science Center, Poland, through Grants No. UMO-2015/18/M/ST2/00056, No.
UMO-2020/37/B/ST2/02349, and No. 2018/30/M/ST2/00319, as well as by the
Excellence Initiative – Research University Program at the Jagiellonian
University. This work was supported by the Research Foundation-Flanders (BE)
under Grant No. G.0D04.21N. We acknowledge the support from the DFG (DE) on
PTB core facility center of ultra-low magnetic field KO 5321/3-1 and TR
408/11-1. We acknowledge funding provided by the Institute of Physics Belgrade
through a grant by the Ministry of Education, Science and Technological
Development of the Republic of Serbia. 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.

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