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Search for an interaction mediated by axion-like particles with ultracold
neutrons at the PSI
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New J. Phys. 25 (2023) 103012 https://doi.org/10.1088/1367-2630/acfdc3
PAPER
Search for an interaction mediated by axion-like particles with
OPEN ACCESS ultracold neutrons at the PSI
RECEIVED
2 April 2023 N J Ayres1, G Bison2, K Bodek3, V Bondar1, T Bouillaud4, E Chanel5,13, P-J Chiu1,2,14,∗, B Clement4,
REVISED C B Crawford6, M Daum2, C B Doorenbos1,2, S Emmenegger1, M Fertl7, P Flaux8, W C Griffith9,
15 August 2023
P G Harris9, N Hild1,2, M Kasprzak2, K Kirch1,2, V Kletzl1,2, P A Koss10, J Krempel1, B Lauss2, T Lefort8,
ACCEPTED FOR PUBLICATION P Mohanmurthy1,2,15, O Naviliat-Cuncic8, D Pais1,227 September 2023 , F M Piegsa
5, G Pignol4, M Rawlik1, I Rienäcker1,2,
D Ries11, S Roccia4, D Rozpedzik3, P Schmidt-Wellenburg2,∗, N Severijns10, B Shen11, K Svirina4,
PUBLISHED
R Tavakoli Dinani10, J A Thorne5, S Touati4, A Weis12, E Wursten10,16 119 October 2023 , N Yazdandoost , J Zejma3, N Ziehl1
and G Zsigmond2
Original Content from 1 ETH Zürich, Institute for Particle Physics and Astrophysics, CH-8093 Zürich, Switzerland
this work may be used 2
under the terms of the Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
3
Creative Commons Marian Smoluchowski Institute of Physics, Jagiellonian University, 30-348 Cracow, Poland
Attribution 4.0 licence. 4 Université Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 38026 Grenoble, France
5
Any further distribution Laboratory for High Energy Physics and Albert Einstein Center for Fundamental Physics, University of Bern, CH-3012 Bern,
of this work must Switzerland
maintain attribution to 6 Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, United States of America
the author(s) and the title 7
of the work, journal Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany
8
citation and DOI. Normandie Université, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France
9 Department of Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
10 Instituut voor Kern- en Stralingsfysica, University of Leuven, B-3001 Leuven, Belgium
11 Department of Chemistry—TRIGA site, Johannes Gutenberg University Mainz, 55128 Mainz, Germany
12 Physics Department, Université de Fribourg, CH-1700 Fribourg, Switzerland
13 Present address: Institut Laue Langevin, 38000 Grenoble, France.
14 Present address: Physik-Institut, University of Zurich, CH-8057, Switzerland.
15 Present address: Massachusetts Institute of Technology, MA, United States of America.
16 Present address: RIKEN, Ulmer Fundamental Symmetries Laboratory, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan.
∗ Authors to whom any correspondence should be addressed.
E-mail: pin-jung.chiu@physik.uzh.ch and philipp.schmidt-wellenburg@psi.ch
Keywords: dark matter, axion, axion-like particle, beyond Standard Model physics
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 pT cm−1 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 gsgpλ2 < 8.3× 10−28m2 (95% C.L.) in a range of 5µm
< λ < 25mm 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.
1. 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
© 2023 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft
New J. Phys. 25 (2023) 103012 N J Ayres et al
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 [3].
In the 1967 seminal article ‘Violation of CP invariance, C asymmetry, and baryon asymmetry of the
Universe’ by A. Sakharov, the violation of the combined symmetry of charge conjugation and parity (CP) is
identified as one of the three necessary criteria for the creation of the observed BAU [4] from initial
symmetric conditions. In the SM, CP violation (CPV) may occur in the weak and the strong sectors. In the
weak sector, it appears in the complex phase of the Cabibbo–Kobayashi–Maskawa [5] or the
Pontecorvo–Maki–Nakagawa–Sakata [6] matrices describing the mixing of quarks or leptons in their mass
and flavor states, respectively. In the quark sector, CPV has been observed and precisely measured in
kaon [7–10] and B-meson [11, 12] decays. Although the CPV phase of the CKMmatrix is nearly
maximal [13], it is insufficient to generate the observed BAU [14].
In the strong sector, CPV appears as vacuum polarization term in the Lagrangian of quantum
chromodynamics (QCDs) [15, 16], giving rise to an electric dipole moment (EDM) of the neutron on the
order of θ̄× 10−16 e · cm, where θ̄ can take any value between 0 and 2π. Nevertheless, non-observation of a
neutron EDM constrains the CP-violating phase in the strong interaction to a value that is particularly small,
θ̄ < 10−10, constituting a puzzle known as the strong CP-problem [17].
A possible solution to the strong CP-problem was proposed by Peccei and Quinn in 1977 by introducing
an additional chiral U(1)PQ symmetry [18, 19]. The spontaneous breaking of this symmetry results in a
pseudo-Goldstone boson [17] to which one often refers to as the ‘canonical QCD axion.’ In 1984, Moody and
Wilczek [20] proposed to search for a new, short-range, spin-dependent (SRSD) interaction, which could be
mediated by very light, weakly coupled, spin-0 bosons, similar to the canonical axion. For spin-0 bosons,
only two options exist to couple to fermions, either via a scalar or a pseudoscalar vertex with the coupling
constants gs and gp, respectively. For a fermion-fermion interaction with only one boson exchange, the scalar
and the pseudoscalar vertices permit three distinct interactions with (monopole 2) , (dipole 2) , or
monopole-dipole virtual boson fields, involving g 2s , g 2p , or gsgp, respectively. One prominent candidate for
the mediator particle of these (monopole 2) , dipole 2( ) , and monopole-dipole interactions is the canonical
axion, but may be other hypothetical bosons [21, 22], such as spin-0 axion-like particles (ALPs), which have
similar properties to the canonical axion, or very light spin-1 bosons [23, 24] coupling via the vector (gv) and
the axial-vector (gA) vertices. These low-mass particles, which couple very weakly to visible matter, are often
referred to as weakly interacting sub-eV/slim particles (WISPs) [25].
The canonical QCD axion and additional ALPs are predicted in many BSM theories including
super-symmetric extensions and string theory. They are considered promising candidates as microscopic
constituents of DM [26, 27]. For the canonical axion, their allowed mass range is 10−13–10−2 eV, constrained
by an assumption on the Peccei–Quinn symmetry breaking scale (lower bound) and stellar evolution
arguments (upper bound) [28], and the original Peccei-Quinn symmetry sets a fixed relation between its
mass and its coupling to the SM particles. On the contrary, there is no a priori relation between mass and
coupling strength for ALPs, making the potential parameter space broader [28, 29]. Many experiments world
wide are actively searching for these particles using resonant cavity haloscopes [30, 31] or large-scale
helioscopes [32] to detect WISPs based on cosmological and astrophysical sources. Additionally,
laboratory-based experiments employing nuclear magnetic resonant precision magnetometers searching for
EDM interactions with ALPs [33, 34], a broadband/resonant approach with a toroidal magnet looking for an
axion-induced magnetic field [35], or employing search methods based on the concept of ‘light shining
through a wall’ [36, 37] or the search for a ‘fifth force’ [38, 39], have also been looking for WISPs in the past
few decades. To date, no evidence of the canonical QCD axion or any ALPs has been observed.
1.1. The monopole–dipole interaction
Among the three interactions, the monopole-dipole interaction involving gsgp and violating P and
T symmetries as well as combined CP symmetry, is of particular interest, as the demonstration of CPV
would provide an evidence to one of the three essential criteria to explain the BAU [4]. The potential
generated by the monopole–dipole interaction between a polarized (†) and an unpolarized particle can be
written as [20, 40]
h̄2 ( )( )1 1
V(r) = gsg
†
p † σ
† · r̂ + e−r/λ, (1)
8πm rλ r2
wherem† and σ† are the mass and the Pauli matrices belongin(g to)the spin of the polarized particle, r̂ is the
unit vector along the distance r between the particles, λ= h̄/ m†c is the interaction range, and h̄ 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 and g†s p , respectively.
2
New J. Phys. 25 (2023) 103012 N J Ayres et al
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. Reproduced with permission from [49].
In [40, 41], 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 EDM of the neutron
(nEDM) [42], and with which we also set limits for an oscillating nEDM [43] through the axion-gluon
coupling and for neutron to mirror-neutron oscillations [44] at the UCN source [45, 46] of the Paul Scherrer
Institute (PSI) in Switzerland. In the experiment, polarized UCNs and polarized 199Hg atoms populated a
vacuum volume between two horizontal electrodes and an insulator ring. A sketch of the experimental
apparatus is shown in figure 2, and details about the spectrometer and the measurement procedure are
described in section 2. 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 199Hg atoms, which served as a
cohabiting magnetometer, in a constant magnetic field B⃗0. An ALP-mediated SRSD interaction between
vessel materials and trapped particles can be considered as a pseudomagnetic field b∗UCN influencing the
precession frequency of UCNs, whereas the effect on the 199Hg atoms is negligible as their mass is much
larger (V(r)∝ 1/m† in equation (1)). Hence, the ratio of the spin precession frequencies of UCNs and
199Hg atoms,
( )  
f ↑/↓ ∣∣∣∣ ∣γ ∗R↑/↓ n n= = ∣∣∣ b1± ∣∣∣UCN∣∣∣ Gg∣∣∣rav ⟨∣∣z⟩± + δelse , (2)fHg γHg B⃗0 B⃗0∣
is sensitive to this interaction while magnetic-field changes cancel and other effects corrected for. Here, γn
and γHg are the gyromagnetic ratios of the neutron and the 199Hg, respectively, b∗UCN is the pseudomagnetic
field that would be experienced by UCNs stored in the apparatus (derived in section 1.2), and the+/− signs
correspond to the upward/downward directions of the magnetic field according to gravity. By measuringR
in opposite directions of B⃗0, the magnitude of the pseudomagnetic field can be extracted
R↑∗ −R↓
∣∣ ∣∣
bUCN = ↑ ↓ ∣B⃗0∣ , (3)R +R
and in turn the strength of the interaction gsgp can be deduced. The dominant systematic effect is due to the
vertical center-of-mass offset ⟨z⟩ between the 199Hg atoms and the UCNs in the presence of an effective
magnetic-field gradient Ggrav [47]. Additional effects δelse are described in more detail below.
1.2. Derivation of the pseudomagnetic field
The interaction is described by the potential given in equation (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 elec(trode surfa)ce at a height d is written as [48]h̄Nλ
b∗ (d)≈ g g† −a/λ −d/λs p † † 1− e e , (4)2γ m
where γ† 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 figure 1.
3
New J. Phys. 25 (2023) 103012 N J Ayres et al
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 cm 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 equation (4) as
∗ † h̄λ
( )( )
b (z) = g g 1− e−a/λ N e−(z+H/2)/λ −N e−(H/2−z)/λALP s p † † bot top , (5)2γ m
where Nbot and Ntop 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 ⟨z⟩=−3.9(3)
mm [42] with respect to that of the cohabiting 199Hg atoms. A linear approximation of the normalized
vertical-UCN-density function is given as [48] ( )
1 12⟨z⟩
ρn (z) = 1+ z . (6)
H H2
The setup was sensitive to interactions of short ranges, approximately from µm to mm, a similar range as in
[48, 50]; therefore, the UCN-density distribution can be simplified to a constant density at distances close to
the surfaces of the electrodes, ρn (−H/2) and ρn (+H/2) [48]. 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
ˆ +H
∗ 2bUCN = b
∗
ALP (z)ρn (z)dz
−H
2 ( )ˆ +H [ ( ) ( ) ]h̄λ 2
= g g† 1− e−a/λ −H −(z+H/2)/λ +H −(H/2−z)/λ (7)s p † [ † ( ) N(botρn )]e −Ntopρn e dz2γ m −H 2 22
† h̄λ
2 H Nbot −Ntop − 6⟨z⟩ Nbot +N ( )( )top
= g g 1− e−a/λ 1− e−H/λs p ,2γ†m†H2
where the top and the bottom electrodes contribute in opposite directions.
2. 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 figure 2, where no electric field was applied during these measurements. The nucleon
density17 for the top electrode was N = N = 5.40× 1030m−3top Cu , whereas the bottom electrode made of
aluminum remained, with Nbot = NAl = 1.63× 1030m−3. In this way, an asymmetric pseudomagnetic field
b∗ALP was generated, increasing the sensitivity by a factor of 7.7 compared to that using both electrodes made
of aluminum.
Further, we exchanged the ultraviolet light source of the probe beam of the 199Hg-comagnetometer
(HgM) from a mercury discharge lamp [52] to a locked frequency quadrupled diode laser18 with a
wavelength of 253.7 nm [53] 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 [48].
Polarized mercury vapor and polarized UCNs precessed in a cylindrical storage chamber of H= 12 cm
height. Its side walls were made of normal polystyrene with a radius of R= 23.5 cm, and the inside was
coated with deuterated polystyrene [54]. 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 (∼1–3µm
thickness) of diamond-like carbon [55, 56] to improve the coherence and storage times of UCNs and 199Hg
17 The 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 26 February 2023) and [51].
18 TOPTICA Photonics AG. Product description TA / FA-FHGpro. Accessed 11 January 2022. http://www.toptica.com/products/tunable-
diode-lasers/frequency-converted-lasers/ta-fhg-pro/.
4
New J. Phys. 25 (2023) 103012 N J Ayres et al
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. Reprinted (figure) with permission from [42], Copyright (2020) by
the American Physical Society.
atoms. Additionally, a total of 15 cesium magnetometers (CsM), of which seven were installed above and
eight below the precession chamber, were used∣to m∣ onitor the magnetic-field gradient Ggrav along thechamber axis [57]. A cosine-theta coil compris∣∣ing∣∣around 50 turns powered with a current of about 17mAcreated a stable and uniform magnetic field of B⃗0 ≈±1036 nT 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 [58] were used to passively and actively improve the stability of the magnetic field.
In each measurement cycle, polarized UCNs in the magnetic field B⃗0 were used for Ramsey’s method of
separated oscillatory fields [59]. A cycle started by bringing the polarized UCNs into the precessing chamber,
followed by the filling of the polarized 199Hg atoms. When both particle species were prepared in their initial
stages in the storage chamber, two low-frequency pulses were applied consecutively to the 199Hg atoms and to
the UCNs to flip their spins to the transverse plane. These pulses are called π/2-pulses. After a
free-spin-precession duration of T = 180 s, a second π/2-pulse, in phase with the first one, was applied to
the UCNs to further tip their spins for another π/2. Afterwards, a spin-sensitive detection system [60, 61]
counted neutrons( in spin-u)p ((N↑) and s)pin-down (N↓) states at the end of the cycle, from which the
asymmetryA= N↑ −N↓ / N↑ +N↓ was calculated. The HgM precession frequency was measured using
a circularly polarized, resonant ultraviolet laser beam at 253.7 nm wavelength, traversing the chamber while
recording the spin-precession-modulated light intensity with a photo-multiplier tube [53]. A measurement
run consisted of approximately ten cycles with different spin-flipping frequencies fn,RF. These frequencies
that were applied as π/2-pulses were slightly detuned from the resonant fn. The scan of fn,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 ratioRRF = fn,RF/fHg (figure 3).
The central Ramsey fringe, approximated well[with a cosine funct〈ion,〉]
A=Aoff −〈αco〉s 2π (R ′RF −R)T fHg , (8)
was fitted to the interference pattern, where fHg is the average HgM frequency of all cycles within the run.
Three parameters, the asymmetry offsetAoff, the visibility (or Ramsey contrast) α, and the
resonant-frequency ratioR= fn/fHg, were extracted. T ′ = T + 4τn/π is the effective time related to the
fringe width, where τn = 2 s is the length of a neutron π/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 equation (2).
The difference of resonant-frequency ratiosR, taken for direction-inverted magnetic-field
configurations permits the extraction of b∗UCN using equation (3). AsR is a linear function of Ggrav
(equation (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
5
New J. Phys. 25 (2023) 103012 N J Ayres et al
Figure 3. Ramsey pattern: The asymmetryA is plotted as a function of the frequency ratioRRF. Blue markers are data from an
ALP measurement run (run 012 951), and the red line is the fit to equation (8). The uncertainties onA are smaller than the
marker size. The resonant-frequency ratioR is at minimalA. Reproduced with permission from [49].
tank [62]. In addition to the high-granularity maps of the magnetic field, taken using a dedicated
measurement device [63], the spatial distribution of the magnetic field(was m)easured continuously with a
sampling rate of 1Hz with 15 CsM. Using a linear fit of Ggrav versusR Ggrav by correcting for all known
systematic effects ofR (section 3.1) and precisely determining Ggrav (section 3.2),R↑0 andR
↓
0 ,
resonant-frequency ratios at Ggrav = 0 for both B⃗0 directions, were precisely determined (section 3.3).
3. Data analysis
3.1. Extraction of the resonant-frequency ratioR
For each measurement run, a resonantR was obtained by fitting the Ramsey pattern, equation (8). Various
statistical uncertainties and systematic effects influenced the measurement ofR. 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 (Ggrav) drift between cycles, and
the Ramsey–Bloch–Siegert shift [64, 65] induced by the π/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 199Hg atoms resulted
in small random tilts of the neutron spins. Each effect resulted in an uncertainty on the measured asymmetry
A, which was further propagated to the uncertainty ofR according to equation (8) using the fitted values for
α andR. Table 1 shows the average uncertainties of each effect for all measurement cycles. We calculated the
reduced chi-square χ2red from the Ramsey fit for all 17 runs including both directions of B⃗0, and the mean
value was 9.15. Assuming pure Poisson statistics, the reduced chi-square χ2red should be approximately 1. A
scaling factor of 2.8, the square-root of the average χ2 2red values excluding one run with χred > 20, was applied
to theR errors obtained from the fit of all runs to account for stochastic errors that were unaccounted for19.
We recall that the dominant systematic effect is the gravitational shift δgrav resulting from the
center-of-mass offset ⟨z⟩ between the UCN and the 199Hg ensembles (equation (2)). In the presence of a
vertical magnetic-field gradient Ggrav, both species measure slightly different volume averages of the
magnetic field. The gravitational shift is calculated as
⟨Bz⟩n Ggrav ⟨z⟩δgrav = − 1=± ∣∣∣ ∣∣∣ , (9)⟨Bz⟩Hg B⃗0
19 The reason of excluding this run was due to the fact that its χ2red was almost two times larger than the second largest χ
2
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.
6
New J. Phys. 25 (2023) 103012 N J Ayres et al
Table 1. Stochastic uncertainties ofR from all measurement cycles. The total numbers of neutrons have mean values of approximately
14 000 and 10 000 for the magnetic field pointing upwards and downwards, respectively.
Effect / 1× 10−7 B0 up B0 down
Neutron counts 1.84 2.26
HgM frequency 0.75 0.69
Gradient drift 0.02 0.02
199Hg spin-flip pulse 0.07 0.23
Total stochastic effects 2.02 2.41
where ( ) ( )
3H2 − 3R
2 5R4 2 2 4
Ggrav = G1,0 +G3,0 +G5,0 −
3R H 3H
+ (10)
20 4 8 8 112
is the effective magnetic-field gradient parallel to the gravitational gradient calculated using a polynomial
expansion of the magnetic field to fifth degree [63, 66]. Gℓ,m are expansion coefficients of degree l and
orderm of the harmonic polynomial, whereas H= 12 cm and R= 23.5 cm are the height and the radius of
the precession chamber. The+/− signs in equation (9) correspond to B⃗0 in upward/downward directions,
respectively, with ⟨z⟩< 0. More details on this effect is described in section 3.2.
Other known effects δelse shown in equation (2) are summarized as
δelse = δT + δEarth + δlight + δinc + δJNN, (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 199Hg atoms, and magnetic-field fluctuations
resulting from Johnson–Nyquist noise (JNN) [67, 68], respectively. They may be divided into two categories.
On the one hand, constant shifts of the UCN or HgM frequency lead to a deviation ofR from the ratio of
pure gyromagnetic ratios. These include the first three effects, δT, δEarth, and δlight. On the other hand, δinc
and δJNN are pure stochastic effects, which do not shift the meanR value, but result in an increase of the
measurement uncertainty. There effects are shown in section 3.3.
The transverse shift δT is a conse〈q√uence of transverse components of the magnetic field BT. UCNs in amagnetic field of 1µT sample the field in the adiabat〉ic regime of slow particles in a high magnetic field. The
measured mean frequency ωn = γ B2n x +B
2
y +B
2
z is proportional to the volume average of the
magnetic-f√ield modulus. By contrast,
199Hg atoms in the same magnetic field fall into the nonadiabatic
regime of fast particle〈s in〉a low magnetic field, such that their spins precess at a mean frequency2 2 2
ωHg = γHg ⟨Bx⟩ + By + ⟨Bz⟩ given by the volume average of the vector magnetic field. In the presence
of BT, this results in 〈 〉
B 2T
δT = , (12)
2B⃗ 20
with 〈 〉 〈 〉
B 2 = ∆B2T x +∆B
2
y (13)
being the me〈an-〉square transverse magnetic-field components, where∆Bx = Bx −⟨B〈x⟩ an〉d
∆By = By − By . For each run, i.e. one magnetic-field configuration, we calculated B 2T using the field
maps [63] and corrected the resonantR value obtained from the Ramsey fit (equation (8)) by δT.
Effectively, given the rotation of Earth, the precession frequencies of UCNs and 199Hg 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, fEarth =(11.6µHz. Th)e associated shift inR was corrected for by calculating
∓ fEarth fEarthδEarth = + cos(θPSI) =∓1.4× 10−6, (14)
fn fHg
where cos(θPSI) = 0.738 is the cosine of the angle between the B⃗0 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 B⃗0, respectively.
The third effect δlight 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 ofR. Details are given in section 3.3.
7
New J. Phys. 25 (2023) 103012 N J Ayres et al
3.2. Determination of the vertical magnetic-field gradientGgrav
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 Ggrav. Because
of the applied B⃗0 ≈ 1µT êz, the transverse fields of order 1 nT were negligible compared to the vertical field;
hence, the scalar fields measured by the CsM were the effective vertical-field component Bz.
The magnetic fields measured by the CsM were described by a polynomial expansion [66]. The gradients
Gℓ,m were extracted by fitting the 15 field values measured by the CsM to the z component of the polynomial
expansion
( ) ∑ ( )
Bi iCsM r = G
i
ℓ,mΠz,ℓ,m r , (15)
ℓ,m
where i = 1, 2, . . . , 15 is the index of the CsM, ri is the position of the corresponding CsM, andΠz,ℓ,m is a
function (or mode) expanded in harmonic polynomials of degree l and orderm depending on ri.
For each degree ℓ, the orderm runs from−ℓ to+ℓ forΠz,ℓ,m, which gives (2ℓ+ 1) 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 [63] to achieve an improved
estimate of Ggrav. 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 Ggrav could thus be expanded to fifth degree (equation (10)).
Note that only odd gradients contribute to Ggrav as the average-magnetic-field components given by even
gradients cancel out for 199Hg atoms and UCNs.
Several 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 Gsynℓ,m randomly using a Gaussian distribution
(equation (17)). In addition, a uniformly distributed random offset in the range of±120 pT [57] was added
at each CsM location, accounting for possible sensor offsets. In total, 200 random fields were synthesized and
analyzed using eight different methods [49]. The optimal method was selected by minimizing the difference
between the synthesized and the fitted Ggrav up to fifth degree using equation (10),
∆Ggrav = G
syn fit
grav −Ggrav, (16)
referred to as the deviation. Each synthesized gradient composing Gsyngrav included a random error drawn from
a Gaussian distribution, whose standard deviation was the uncertainty of the map, and was given as
Gsyn Gmap mapℓ,m = ℓ,m + δG . (17)ℓ,m
The uncertainty of the fitted gradient σGgrav was calculated with error propagation from each gradient-fit
error σGfit , σGfit , and σGfit in equation (10).1,0 3,0 5,0
We concluded that the optimal method was achieved by removing the fields described by higher-degree
harmonic polynomials with ℓ= 3, . . . ,6 using the map gradients and performing a second-degree fit
including nine gradients up to ℓ= 2 to the residual fields. The optimal fit method, even in the presence of
CsM offsets wit∣∣h a stan∣∣dard deviation as large as σBoffset = 115 pT, estimated the coefficients with a deviationin the range of ∆Ggrav ∼ 2–3 pT cm−1, and with fit uncertainties of σ < 3.8 pT cm−1Ggrav (upward B⃗0) and
σGgrav < 4.5 pT cm
−1 (downward B⃗0).
This method was applied to all 17 runs of data, consisting of a total of 170 cycles. The residuals
∆Bi = Bi −Bilow fit for each CsM i in each cycle was calculated, where Bilow are the field values used in the fit to
the polynomial expansion up to second degree after removing higher-degree contributions, and Bifit are the
calculated values at each CsM position using the fitted gradients. All∆Bi were< 250 pT, which were below
the uncertainties of the field maps. For each cycle, Ggrav was calculated using the expansion up to fifth degree,
where G1,0 was obtained from the second-degree-polynomial fit, and G3,0 as well as G5,0 were taken from the
map values. The average value of the estimated Ggrav 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 Ggrav, we made use of the
visibility parabola, which is the visibility of the Ramsey fringe α plotted as a function of Ggrav. The parabola
reaches its maximum at the minimum vertical magnetic-field gradient, where gravitationally enhanced
8
New J. Phys. 25 (2023) 103012 N J Ayres et al
Figure 4. Visibility parabola: visibility α as a function of vertical magnetic-field gradient Ggrav for B⃗0 pointing (a) upwards and (b)
downwards from all 17 runs of data. The error bar on each data point is the 1σ 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. Reproduced with permission from [49].
depolarization [69, 70] is neg(ligibl)e. Figure 4 shows the parabola for B⃗0 pointing upwards (figure 4(a)) anddownwards (figure 4(b)) with both reaching a similar maximal visibility. The parabolas were fitted with a
simple parabolic function α G 2grav = c(Ggrav − g0) +α0, where g0 is the expected zero gradient. The
maximal visibilities were reached at g↑ =−2.2± 2.2 pT cm−1 and g↓ = 0.02± 3.7 pT cm−10 0 for the upward
and the downward B⃗0 directions, respectively. The uncertainties on the fitted parameters were estimated by
scaling χ2 2red = χ /d.o.f. to 1 in each parabola fit. To account for this potential shift, we corrected the effective
G of each run with g↑ =−2.2 pT cm−1 or g↓ = 0.02 pT cm−1grav 0 0 .
3.3. Crossing-point analysis
Figure 5 shows theR values, after δT and δEarth corrections, as a function of the corrected Ggrav (figure 4).
Red upward triangles and blue downward triangles are runs with B⃗0 pointing upwards and downwards,
respectively. A linear fit to the data from all runs with both directions of B⃗0 (+/− correspond to
upward/downward) was applied to
 
R↑/↓ R↑/↓ ⟨z⟩= 0 1± ∣∣∣ ∣∣∣G↑ grav (18)B⃗0
simultaneously, sharing the parameter ⟨z⟩, while theR↑/↓0 values were kept separate for both directions. This
is the so-called crossing-point analysis. The best fit was obtained for
R↑0 = 3.8424563(08) ,
R↓0 = 3.8424622(12) , and (19)
⟨z⟩=−0.43(2) cm,
with χ2 2red = χ /d.o.f.= 31.9/14. The underestimated uncertainties caused χ
2
red to be larger than 1. The
uncertainties shown in equation (19) were corrected for this stocha√stic error. Compared to the total statisticalerrors shown in table 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 31.9/14. The center-of-mass offset ⟨z⟩
was in agreement with the values found in [42], ⟨z⟩=−0.39(3) cm, and [66], ⟨z⟩=−0.38(3) cm, and the
crossing point was at G× =−1.9(5) pT cm−1.
In the following paragraphs, we distinguish two different kinds of uncertainties associated withR as
shown in equation (2). The first kind was systematic effects, leading to a bias, whereas the second kind was
considered purely stochastic, only increasing the measurement uncertainty.
9
New J. Phys. 25 (2023) 103012 N J Ayres et al
Figure 5. Frequency ratioR as a function of vertical magnetic-field gradient Ggrav for all runs with different field configurations.
Red upward and blue downward triangles correspond to B⃗0 pointing upwards and downwards, respectively. The error bar for each
R is the fit error from the Ramsey fit scaled by 2.8. The error of Ggrav for each data point with an average value of 4.05 pT cm−1
over all 17 runs was not included in the error ofR individually. Instead, we considered this effect a global systematic uncertainty
to all runs that resulted in an error contribution to both of the fittedR↑/↓ values. For this, the error obtained from each fit of the
visibility parabolas was used (vertical magnetic-field gradient in table 2). Straight lines shown here were obtained using the
best-fit values (equation (19)) in the fit equations (equation (18)). Reproduced with permission from [49].
The most important systematic effect of the first kind is the shift δgrav induced by a vertical magnetic-field
gradient. From the visibility parabolas (figure 4), g↑/↓0 were considered as systematic shifts onR. For all runs,
Ggrav were corrected for with g
↑/↓
0 , and the uncertainties of the fitted g
↑/↓
0 lead to
σ↑R = 35× 10
−7 and
grav
↓ (20)σR = 59× 10
−7.
grav
The second largest systematic effect δT arises from the residual transverse field. In addition to the shift in
R, corrected for run-by-run, an error on the mean-squared transverse field σ⟨B2⟩ was calculated making useT
of the concept of reproducibility of the field maps [63], quantifying the spread in measurements of identical
magnetic-field configurations taken over several years. This results in shifts of
R↑T = (7.3± 4.7)× 10
−7 and
(21)
R↓T = (6.4± 4.1)× 10
−7.
The third systematic shift δlight may occur resulting from the resonant UV laser beam traversing the
precession chamber to read out the 199Hg 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 [62]
using a 204Hg 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 B⃗0-field direction, is magnetic-field
direction dependent. As we exchanged the slightly off-resonant 204Hg lamp with a laser beam resonantly
locked to the 199Hg 6 1S 30 → 6 P1 F= 1/2 transition, the shift reduced by a factor of 7.7 [71] to
R↑ = (1.5± 6.9)× 10−7VL and
R↓
(22)
VL = (1.2± 5.4)× 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 3P1 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 [42] in the nEDMmeasurement
with the same apparatus still using the 204Hg discharge lamp. An increase in light power would also result in
a decrease of the transverse depolarization time T2 of the mercury precession, which was not observed.
10
New J. Phys. 25 (2023) 103012 N J Ayres et al
Table 2. Error budget for the overall errors onR resulting from statistics and from systematic effects for both B⃗0 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.
Effect / 1× 10−7 B0 up B0 down
Statistics (uncertainty) ±8 ±12
Vertical magnetic-field gradient 35± 35 −0.3± 59
Residual transverse field 7.3± 4.7 6.4± 4.1
Mercury light 1.9± 6.9 1.6± 5.5
Nevertheless, to account for a possible doubling of the light intensity because of the change to the resonant
laser light, we estimate
R↑/↓DL = (0.4± 0.8)× 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 table 2.
Within the medium of spin-polarized 199Hg vapor, UCNs are subjected to incoherent scattering on the
199Hg 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 |binc|= 15.5 fm [72, 73]. For
an imperfect π/2-pulse of the HgM, a residual polarization along B⃗0 creates a pseudomagnetic field,
resulting in a frequency shift of UCNs and consequently a shift δinc ofR. We estimated the random
fluctuation of 199Hg polarization and quantified the resultant error onR as ↑/↓σ −10R ⩽ 5× 10 . This effect isinc
three orders of magnitude smaller than the light shift δlight; 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 199Hg atoms, the volume-averaged fields
sampled by both species are slightly different. A finite-element analysis was used to simulate temporal and
spatial noise [74] 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 B⃗0 polarity,
we assume that this effect δJNN only increases the measurement uncertainty but does not shift the centralR
value. The corresponding uncertainty was estimated to be ↑/↓σ ⩽ 1× 10−9R . As this effect is two orders ofJNN
magnitude smaller than δlight, we did not include it in the error budget.
The analysis of the measured data, including all shifts and uncertainties as listed in table 2, results in two
independentR values
R↑ = 3.8424563(08)stat (36)sys and
↓ (24)R = 3.8424622(12)stat (59)sys
at the limit of Ggrav = 0. Both values are in agreement with our previous measurement of the neutron to
mercury gyromagnetic ratio [62],
γn/γHg = 3.8424574(30) . (25)
4. Interpretation of results
According to equation (2), b∗UCN was extracted at the limit of Ggrav = 0 after correcting for all systematic
effects δelse,
R↑ −R↓ ∣∗ ∣∣ ∣∣bUCN = ↑ ↓ B⃗0∣=−0.80pT, (26)∣∣ ∣∣ R +R
where ∣B⃗0∣= 1037.19(2) nT was taken from the average B⃗0 value of all runs. The uncertainty of b∗UCN was
calculated by including uncertainties fromR
↑,R↓, and B⃗0,
 2R↓ ∣
∣∣ ∣∣∣ 2 
∣
 − R↑ ∣∣ ∣
∣ 2
B⃗ 2 B⃗ [ ]21/2
 0 ↑ 0
∣  R↑↓ −R↓ σb∗ = 2σR + σ + σ = 0.96pT. (27)UCN (R↑ +R↓) (R↑ +R↓ 2 R) R↑ +R↓ B⃗0 
11
New J. Phys. 25 (2023) 103012 N J Ayres et al
Figure 6. Upper limits on the gsgp-coupling of an interaction mediated by a spin-0 axion-like particle as a function of the
interaction range λ and the mass of an ALPmALP. A: this work (equation (29)). B: outlook on the n2EDM experiment with
copper layers (section 5). C: clock comparison of 3He and 129Xe [75]. D: clock comparison of 129Xe and 131Xe [76]. E: our
previous experiment with UCNs and 199Hg atoms [48]. F: 3He depolarization [50]. Reproduced with permission from [49].
The errors σ↑R and σ
↓
R were calculated by summing in quadrature the statistical error and all systematic
effects that contributed to the error budget ofR (table 2). σB⃗ was taken from the larger of the two standard0
deviations, σ↑ = 17 pT and σ↓ = 22 pT, measured within one B⃗0 direction instead of taking the standardB⃗0 B⃗0
deviation of all 17 runs. Using equation (7), gsgp was derived as
gsgp = b
∗
UCN [ ( 2γ )m H2 ( )] ( )−1( )−1n n 1− e−a/λ 1− e−H/λ . (28)h̄λ2 H Nbot −Ntop − 6⟨z⟩ Nbot +Ntop
With the measured b∗UCN (equation (26)) and the estimated error σb∗ (equation (27)), a 95% confidenceUCN
level limit on gsgp gives
g g λ2 < 8.3× 10−28m2s p , (29)
for 5µm< λ < 25mm. 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 equation (28) depart from 1, and the relation
gsgp ∝ 1/λ2 is not fulfilled anymore, which reduces the measurement sensitivity on gsgp. 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
diamond-like-carbon coating that has a thickness of a few µm and a nucleon density between those of the
aluminum and the copper electrodes.
Figure 6 shows the upper limits of gsgp constrained by the most recent measurements covering an
interaction range of 1µm< λ < 1mm. The upper horizontal axis displays the corresponding mass of an
ALPmALP, with λ= h̄/(mALPc). The figure shows five measurement results, labeled from A to E. A is the
limit obtained from this work (equation (29)), whereas E is obtained from our previous experiment [48].
Both experiments were based on a clock comparison of precession frequencies between polarized UCNs and
199Hg atoms. An improvement in the sensitivity by a factor of 2.7 was accomplished, which is the current best
limit of gsgp 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 section 5. C is the
result based on the comparison of spin-precession-frequency shifts of cohabiting 3He and 129Xe atoms, in the
presence of an unpolarized mass of BGO (Bi4Ge3O12) crystal [75]. D results from the comparison of
nuclear-magnetic-resonance-frequency shifts of cohabiting polarized 129Xe and 131Xe atoms in the presence
of a nonmagnetic zirconium rod [76]. F is the result from the measurement of anomalous spin relaxation of
polarized 3He atoms induced by an additional depolarization channel, which might be caused by the
pseudomagnetic field generated from the 3He-cell walls [50].
12
New J. Phys. 25 (2023) 103012 N J Ayres et al
5. Prospects for ALPmeasurements 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 1× 10−27 e · cm [77, 78]. 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 cm. In between the electrodes,
cylindrical rings made of isolating polystyrene with a diameter of 80 cm [78] 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. Without applying an electric field, it
may be used for a refined search for a SRSD interaction with improved statistical and systematic sensitivities.
In the following paragraphs, we estimate possible improvements in statistical and systematic sensitivity with
respect to the above presented measurement.
The statistical sensitivity ofR= fn/fHg 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 T = 180 s will increase by a factor of eight. At the same
time, the fringe visibility α slightly improves to 0.8 [78]. Together, this will result in an improvement on the
sensitivity of fn by a factor of three, corresponding to a magnetic-field sensitivity of 110 fT. The sensitivity of
the mercury magnetometer is expected to be at least σBHg = 30 fT per cycle [78]. Hence, we expect a factor of
six in statistical improvement to 0.2 ppm. 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 Ggrav.
Assuming that Ggrav will be expanded up to fifth degree (equation (10)), we expect the uncertainties of G1,0,
G3,0, and G5,0 determined by the HgM, the CsM array, and a more reproducible magnetic-field maps,
respectively, to be [78]
√
σG1,0 ⩽ 2σ /H ′ ≈ 2.4 fTcm−1BHg ,
σ ⩽ 36× 10−3 fTcm−3G3 0 , and (30),
σG5 0 ⩽ 20× 10−6 fTcm−5,,
where H ′ = 18 cm is the distance between the centers of the upper and the lower precession chambers.
Summing up all contributions, this implies a systematic uncertainty of σGgrav ≈ 50 fT cm−1 on the vertical
magnetic-field gradient; an improvement by a factor of 80 from the average uncertainty of about 4 pT cm−1
presented in this article.
The transverse magnetic field was estimated with field maps [63]. In n2EDM, field maps will be used to
measure all higher gradients above G3,0, with a reproducibility requirement matching the previous
repeatability [78]. This results in a tenfold improvement of the uncertainty σ⟨B2T⟩.
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 [79].
In summary, assuming perfect scaling of various aspects described above to hold without introducing
new uncertainties, this results in a measurement ofR with a statistical precision of about σn2EDM −7stat = 2× 10
and a systematic precision of about σn2EDMsyst = 4× 10−8, more than one order of magnitude improvement
compared to (25) [62]. With the estimated improvements on the statistical and, in particular, on
the gradient-induced systematic uncertainties, we anticipate a factor of three improvement to
gsgpλ2 < 3× 10−28m2 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
equation (27). However, in the new experiment, the center-of-mass offsets individually estimated for each
chamber are both ⟨z⟩ ≈ 0.1 cm, 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 thin copper sheets on the middle and the lower electrodes. With a 1mm-thick copper sheet,
the interaction range up to 1× 10−3m can be covered. In case of a null result, the prospected sensitivity
would result in an upper limit on the product of the couplings of better than g g λ2 < 1× 10−29s p m2 (95%
C.L., marked as B in figure 6) when using copper having a 3.3 times larger nucleon density compared to
aluminum.
13
New J. Phys. 25 (2023) 103012 N J Ayres et al
6. Conclusion
This paper reports on the null result from a search for a hypothetical, SRSD interaction mediated by ALPs.
UCNs were stored simultaneously with 199Hg 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 EDM at the PSI. By measuring the precession-frequency ratioR= fn/fHg between UCNs and 199Hg
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 ofR. The dominant effect
arose from the center-of-mass offset between the two particle species in an effective vertical magnetic-field
gradient Ggrav. AsR is a linear function of Ggrav, we intentionally applied a vertical magnetic-field gradient in
the measurement and comparedR in different Ggrav using the crossing-point analysis. For a better
estimation on Ggrav, 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, Ggrav was estimated with an
unprecedented precision of 4.05 pT cm−1.
By extractingR at Ggrav = 0 after correcting for all known systematic effects, a new limit on the product
of the scalar and the pseudoscalar couplings, corresponding to the monopole-dipole interaction, gives
g g λ2s p < 8.3× 10−28m2 (95% C.L.) in an interaction range of 5µm< λ < 25mm. 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
1× 10−27 e · cm . By comparing the precession frequencies of 199Hg atoms and the UCNs in the new
spectrometer, a new, more than one order of magnitude more accurate measurement of the gyromagnetic
ratios γn/γHg becomes possible. Further, a refined search of ALPs by placing a 1mm-thick copper layer on
the corresponding bottom electrode of each chamber seems attractive. A new upper limit of
g g λ2 < 1× 10−29s p m2 (95% C.L.), a factor of 80 better than the result presented here, could be expected.
Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that
is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are
available upon reasonable request from the authors.
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 MMeier. We thank the UCN source operation group BSQ
for their support. We acknowledge financial support from the Swiss National Science Foundation through
Project Nos. 117696, 137664, 144473, 157079, 172626, 126562, 169596, 178951 (all PSI), 181996 (Bern),
162574 and 172639 (both ETH). The group from Jagiellonian University Cracow acknowledges the support
from the National Science Center, Poland, through Grant Nos. UMO-2015/18/M/ST2/00056,
UMO-2020/37/B/ST2/02349 and 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.0D04.21 N. We acknowledge the support from the DFG (DE)
on PTB core facility center of ultra-low magnetic field KO5321/3-1 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.
ORCID iDs
K Bodek https://orcid.org/0000-0002-2866-2837
P-J Chiu https://orcid.org/0000-0002-3772-0090
M Fertl https://orcid.org/0000-0002-1925-2553
P Mohanmurthy https://orcid.org/0000-0002-7573-7010
P Schmidt-Wellenburg https://orcid.org/0000-0001-5474-672X
14
New J. Phys. 25 (2023) 103012 N J Ayres et al
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