11institutetext:  University of Sussex, Brighton BN1 9RH, United
Kingdom22institutetext: Paul Scherrer Institute, 5232 Villigen,
Switzerland33institutetext: Marian Smoluchowski Institute of Physics,
Jagiellonian University, 30-348 Kraków, Poland44institutetext: Institute for
Nuclear and Radiation Physics, KU Leuven, 3001 Heverlee, Belgium
55institutetext: Institute for Particle Physics and Astrophysics, ETH Zürich,
8093 Zürich, Switzerland 66institutetext: LPC Caen, ENSICAEN, Normandie
Université, CNRS/IN2P3, 14000 Caen, France77institutetext: Laboratoire de
Physique Subatomique et de Cosmologie, 38000 Grenoble, France88institutetext:
University of Fribourg, 1700 Fribourg, Switzerland 99institutetext: Henryk
Niedwodniczański Institute of Nuclear Physics, 31-342 Kraków,
Poland1010institutetext: Laboratory for High Energy Physics and Albert
Einstein Center for Fundamental Physics, University of Bern, 3012 Bern,
Switzerland 1111institutetext: Institut für Kernchemie, Johannes Gutenberg-
Universität, 55128 Mainz, Germany1212institutetext: CSNSM, Université Paris
Sud, CNRS/IN2P3, Université Paris Saclay, 91405 Orsay-Campus,
France1313institutetext: Physikalisch Technische Bundesanstalt, 10587 Berlin,
Germany

# Statistical sensitivity of the nEDM apparatus at PSI to $n-n^{\prime}$
oscillations

C. Abel 11 N. J. Ayres 11 G. Bison 22 K. Bodek 33 V. Bondar 44 P.-J. Chiu 2255
M. Daum 22 S. Emmenegger 55 P. Flaux 66 L. Ferraris-Bouchez 77 W. C. Griffith
11 Z. D. Grujić 88 N. Hild 2255 K. Kirch 2255 P. A. Koss 44 A. Kozela 99 J.
Krempel 55 B. Lauss 22 T. Lefort 66 A. Leredde 77 P. Mohanmurthy 2255
<EMAIL_ADDRESS>O. Naviliat-Cuncic now at Michigan State University,
East-Lansing, MI 48824, USA66 D. Pais 2255 F. M. Piegsa 1010 G. Pignol 77 M.
Rawlik 55 D. Rebreyend 77 D. Ries 1111 S. Roccia present address: Institut
Laue Langevin, 38000 Grenoble, France1212 D. Rozpedzik 33 P. Schmidt-
Wellenburg 22 A. Schnabel 1313 N. Severijns 44 J. Thorne 1010 R. Virot 77 J.
Zejma 33 G. Zsigmond 22<EMAIL_ADDRESS>
###### Abstract

The neutron and its hypothetical mirror counterpart, a sterile state
degenerate in mass, could spontaneously mix in a process much faster than the
neutron $\beta$-decay. Two groups have performed a series of experiments in
search of neutron - mirror-neutron ($n-n^{\prime}$) oscillations. They
reported no evidence, thereby setting stringent limits on the oscillation time
$\tau_{nn^{\prime}}$. Later, these data sets have been further analyzed by
Berezhiani et al.(2009-2017), and signals, compatible with $n-n^{\prime}$
oscillations in the presence of mirror magnetic fields, have been reported.
The Neutron Electric Dipole Moment Collaboration based at the Paul Scherrer
Institute performed a new series of experiments to further test these signals.
In this paper, we describe and motivate our choice of run configurations with
an optimal filling time of $29\leavevmode\nobreak\ $s, storage times of
$180\leavevmode\nobreak\ $s and $380\leavevmode\nobreak\ $s, and applied
magnetic fields of $10\leavevmode\nobreak\ \mu$T and $20\leavevmode\nobreak\
\mu$T. The choice of these run configurations ensures a reliable overlap in
settings with the previous efforts and also improves the sensitivity to test
the signals. We also elaborate on the technique of normalizing the neutron
counts, making such a counting experiment at the ultra-cold neutron source at
the Paul Scherrer Institute possible. Furthermore, the magnetic field
characterization to meet the requirements of this $n-n^{\prime}$ oscillation
search is demonstrated. Finally, we show that this effort has a statistical
sensitivity to $n-n^{\prime}$ oscillations comparable to the current leading
constraints for $B^{\prime}=0$.

##  1 Introduction

In the paper in which Lee and Yang proposed parity violation [6-1] , they also
noted that it may be resolved by the introduction of a parity conjugated copy
of standard model (SM) particles - SM′. Kobzarev, Okun and Pomeranchuk further
developed this idea and formalized mirror matter [1] . They also introduced
mirror photons, _i.e._ mirror magnetic fields which were not defined in Ref.
[6-1] . Foot and Volkas showed that with the introduction of mirror matter,
parity and time reversal symmetries could be restored in the global sense
[6-1-1] ; [6-1-2] .

Mixing of SM′ and SM particles could provide answers for several long-standing
issues in physics today. Mirror matter could act as a dark matter candidate
[6-2-0] ; [6-2-1] ; [6-2-2] ; [6-2-3] ; [6-2-4] ; [6-2-5] ; [6-2-6] . It could
provide a mechanism to help solve the sterile neutrino anomaly [6-3-1] ;
[6-3-2] ; [6-3-3] . Mirror matter could also provide an additional channel of
CP violation through mixing of SM and SM′ particles, thus helping to explain
baryogenesis [6-4] . It could provide a mechanism to relax the Greisen-
Zatsepin-Kuzmin (GZK) limit through $n-n^{\prime}$ oscillations [6-5-1] ;
[6-5-2] . Baryogenesis also requires baryon number violation. Neutron -
mirror-neutron oscillations are one such process [6-5-3] . The corresponding
oscillation time may be smaller than the neutron $\beta-$decay lifetime
[6-5-4] . A historical overview of the physics of mirror matter can be found
in Ref. [6-6] .

It was shown in Ref. [4]  that neutrons may mix with mirror-neutrons with an
interaction Hamiltonian:

$\displaystyle\mathcal{H}=\begin{bmatrix}\mu_{n}\bm{B\cdot\sigma}&\hbar/\tau_{nn^{\prime}}\\\
\hbar/\tau_{nn^{\prime}}&\mu_{n}\bm{B^{\prime}\cdot\sigma}\end{bmatrix},$ (1)

where $\tau_{nn^{\prime}}$ is the $n-n^{\prime}$ oscillation time, and ${\bf
B}$, ${\bf B^{{}^{\prime}}}$ are the magnetic and mirror magnetic fields,
respectively. Eq. 1 shows that applying a magnetic field (${\bf
B^{\prime}}\neq{\bf B}$) lifts the degeneracy between neutron and mirror-
neutron states, thereby suppressing the oscillation between the states. From
Eq. 1 it follows that there can be two general techniques for searching for
such oscillations:

  1. 1.

ultra-cold neutron (UCN) storage experiments, where one searches for a
magnetic field dependence of the storage curve [2] ; [3] ; [5] ,

  2. 2.

regeneration experiments, where a âparticle through a wallâ measurement is
performed to look for cold neutrons regenerating after crossing a barrier in
‘mirror state’ [5-1] ; [5-2] .

Our attempt to search for $n-n^{\prime}$ oscillations employs the UCN storage
technique. Counting the number of stored neutrons as a function of time
($N(t_{s})$) is usually referred to as the storage curve. We performed an
experiment where the storage curves of neutrons were compared with magnetic
field turned on and off. Neutron - mirror-neutron oscillation, depending on
the magnetic field(s) applied, may add an additional loss channel to the
storage curve:

$\displaystyle N_{\\{0,B\\}}(t_{s})$ $\displaystyle=$ $\displaystyle
N_{\\{0,B\\}}(t_{s}=0)$
$\displaystyle\times\exp{-\left(R+R^{nn^{\prime}}_{\\{0,B\\}}\right)t_{s}}.$

Here, $N_{\\{0,B\\}}(t_{s}=0)$ are the initial numbers of neutrons stored in
the chamber and $N_{\\{0,B\\}}(t_{s})$ denotes the UCN data points as a
function of storage time $t_{s}$, when the magnetic field is switched off or
switched on, respectively. The value $R$ refers to the sum of rates of known
loss channels as up-scattering, absorption, and $\beta-$decay of neutrons in
the chamber, which are all independent of the applied magnetic field.
$R^{nn^{\prime}}_{\\{0,B\\}}$ is the loss channel added due to $n-n^{\prime}$
oscillations when the magnetic field applied is zero or when it is different
from zero. Note that when the applied magnetic field, $B$, is zero, there
could still be a non-zero mirror magnetic field, $B^{\prime}\neq 0$. The
additional loss channel due to $n-n^{\prime}$ oscillation can be isolated as
derived in section 3 of [4] . By taking the ratio of UCN counts with magnetic
field turned off ($B_{0}\sim 0$) and on ($B_{0}>0$) and the decay rate
formulas in [4]  follows:

$\displaystyle E_{B}\left(t_{s}\right)$ $\displaystyle=$
$\displaystyle\frac{N_{0}\left(t_{s}\right)}{N_{B}\left(t_{s}\right)}-1$ (3)
$\displaystyle=$
$\displaystyle\exp{-\left(R^{nn^{\prime}}_{0}-R^{nn^{\prime}}_{B}\right)t_{s}}-1$
(4) $\displaystyle=$
$\displaystyle\frac{t_{s}}{\left<t_{f}\right>}\frac{\eta^{2}\left(3-\eta^{2}\right)}{2\tau_{nn^{\prime}}^{2}\omega^{\prime
2}\left(1-\eta^{2}\right)^{2}},$ (5)

where $\omega^{(^{\prime})}=45.81\leavevmode\nobreak\
(\mu\text{T.s})^{-1}\cdot B^{(^{\prime})}$, $\eta=B/B^{\prime}$, and
$\left<t_{f}\right>$ is the mean time between two consecutive wall collisions
of the UCN within the storage chamber. Eq. 5 is valid in the range
$\omega^{\prime}\left<t_{f}\right>\gg 1$ [4] . In the case when no mirror
magnetic field is assumed, Eq. 4 reduces to:

$\displaystyle E_{0}(t_{s})$ $\displaystyle\approx$
$\displaystyle\frac{\left<t^{2}_{f}\right>}{\left<t_{f}\right>}\frac{t_{s}}{\tau^{2}_{nn^{\prime}}}.$
(6)

Eq. 6 is valid in the range $\omega\left<t_{f}\right>\ll 1$. Eqs. 5 and 6 form
the “ratio channel” of $n-n^{\prime}$ oscillations. To summarize, $E_{B}$
denotes the generalized case when $B^{\prime}\neq 0$, whereas $E_{0}$ denotes
the special case when we assume $B^{\prime}=0$.

In Ref. [4] , Berezhiani showed that the asymmetry between storage curves is
also sensitive to $n-n^{\prime}$ oscillations when magnetic fields of opposing
directions are applied:

$\displaystyle A_{B}\left(t_{s}\right)$ $\displaystyle=$
$\displaystyle\frac{\left(N_{\bf
B}\left(t_{s}\right)-N_{\bf{-B}}\left(t_{s}\right)\right)}{\left(N_{\bf
B}\left(t_{s}\right)+N_{\bf{-B}}\left(t_{s}\right)\right)}$ (7)
$\displaystyle=$
$\displaystyle-\frac{t_{s}}{\left<t_{f}\right>}\frac{\eta^{3}\cos\left(\beta\right)}{\tau^{2}_{nn^{\prime}}\omega^{2}\left(1-\eta^{2}\right)^{2}}.$

Here the new variable $\beta$ is the angle between ${\bf B}$ and ${\bf
B^{\prime}}$. Eq. 7 forms the “asymmetry channel” of $n-n^{\prime}$
oscillations. Particularly, the asymmetry channel is sensitive to the
direction of the mirror magnetic field _w.r.t_ the applied magnetic field.

If there are no $n-n^{\prime}$ oscillations, the measured values of $E_{B}$,
$E_{0}$, $A_{B}$ would all be zero. This defines the null hypothesis. In order
to measure $n-n^{\prime}$ oscillations, searches for deviations from the null-
hypothesis using both the UCN storage technique, as well as the regeneration
technique, have been attempted.

The first experiments in search of $n-n^{\prime}$ oscillations were performed
using the ratio channel. They set the limits
$\tau_{nn^{\prime}}>103\leavevmode\nobreak\ \mbox{s (95 \% C.L.)}$ [2]  and
$\tau_{nn^{\prime}}>414\leavevmode\nobreak\ \mbox{s (90 \% C.L.)}$ [3] ,
respectively. The current leading limit on the oscillation time under the
assumption the mirror magnetic field, $B^{\prime}=0$ is
$\tau_{nn^{\prime}}>448\leavevmode\nobreak\ \mbox{s (90 \% C.L.)}$ [3-2] . The
current leading constraint for the case of $B^{\prime}\neq 0$ is
$\tau_{nn^{\prime}}>12\leavevmode\nobreak\ \mbox{s (95\% C.L.)
over}\leavevmode\nobreak\ (0<B^{\prime}<12.5\leavevmode\nobreak\ \mu$T$)$ [5]
. Reanalysis of experiments in Refs. [2] ; [3] ; [5]  by Berezhiani et al. in
[4] ; [4-2] ; [6]  showed signals in the asymmetry channel in the region
$6<B^{\prime}<40\leavevmode\nobreak\ \mu$T, which could not be excluded by any
experiment thus far. Testing these signals was the primary motivation for the
new side-project of the Neutron Electric Dipole Moment (nEDM) Collaboration
[6-0]  searching for neutron - mirror-neutron oscillations with the apparatus
at the Paul Scherrer Institute (PSI).

Our search made use of the PSI UCN source [7-0-1-1] ; [7-0-1-2] ; [7-0-1-3] ;
[7-0-1-4] ; [8-0]  and a re-purposed nEDM apparatus [10-2] ; [7-1-1-1]  for
this experiment. At PSI neutrons are produced from a proton driven spallation
source. These spallation neutrons are then moderated in heavy water and
successively down-scattered in solid deuterium to UCNs. In the PSI nEDM
apparatus, a switch directs the neutrons to and from the storage chamber. The
neutrons can be stored under the influence of magnetic (and electric) fields.
The storage chamber is enclosed in a 4-layer $\mu$-metal shield which is
housed inside an active magnetic-field compensation system [7-0-2] . A cycle
usually involves letting the UCNs from the source fill the storage chamber
after passing through the appropriately configured switch. The UCN shutter at
the bottom of the storage chamber then closes and the storage of UCNs begins.
During this time, the rest of the UCN exiting from the source are guided to
the detectors and are used as monitor counts. After a period of storage, the
UCN shutter below the storage chamber is opened and the neutrons are spin-
analyzed in the U-shaped Simultaneous Spin Analyzer (USSA) [7-1-1]  before
being simultaneously detected by a pair of neutron detectors (called NANOSC-A
and B), each detecting a single spin state [7-1-2] . A schematic diagram of
the PSI nEDM apparatus can be found in Ref. [7-1-1-1] . The cycle schedule is
handled by a slow-control software called ‘micro-timer’. In the mirror-neutron
search, we used unpolarized neutrons, thus spin-analysis was not required.
Patterns of different magnetic fields were applied after every 4 cycles,
referred to in the following as a ‘dwell’.

##  2 Determining the effective storage time

The emptying phase is the period of time when the UCNs in the nEDM storage
chamber are allowed to drop down into the detector systems and be detected
after a period of storage. During the emptying phase, the UCN shutter at the
bottom of the nEDM storage chamber is opened and the switch is set to the
empty position. During this phase, time-of-arrival spectra are recorded in the
UCN detectors. A set of emptying time-constants $\tau_{\text{emp}}$ can be
extracted by fitting these spectra (also called ‘emptying curves’) in the
time-of-arrival histograms with an exponential-decay function. These fits
yield cycle-by-cycle values for $\tau_{\text{emp}}$ as a function of micro-
timer storage time. These values averaged over all cycles for a given storage
time are shown in Fig. 1. The corresponding errors on the mean do not reflect
the large scatter of the central values shown in Fig. 1. This is caused by the
fluctuations in the motion of the switch, and in the opening and closing of
the UCN shutter at the bottom of the UCN chamber. Therefore in Fig. 1 the
standard deviations are shown as error bars.

Studying the emptying time-constant also allows us to understand the effective
storage time of neutrons which is different from the micro-timer storage time
for the following reasons. The neutrons experience the same magnetic field,
which is present during storage, also during both the filling and the emptying
phases when neutrons could oscillate into their mirror counterparts as well.
While the detected neutrons are recorded with a time tag (with a precision of
$1$ ns), it is hard to precisely say when the neutrons left the nEDM storage
chamber once the UCN shutter at its bottom was opened. Also, it is not
possible to tell the precise time at which a neutron entered the nEDM storage
chamber during the filling phase. This makes the micro-timer storage time
$t^{*}_{s}$ not equal to the precise time period during which they were
present in the chamber. The micro-timer storage time starts with the end of
the filling phase and ends at the beginning of the emptying phase, and it is
indicated by $t^{*}_{s}$ here.

The analysis using Eqs. 5, 6 and 7 relies on the mean time (the average over
UCN trajectories) the UCN have spent in the magnetic field. As explained
above, this should also include the mean time of emptying and filling. The
mean emptying time is given by the time-constant $\tau_{\text{emp}}$ since the
time-spectra read by the detectors have a shape of an exponential decay. The
chamber is also filled through the same opening, and same vertical guide
(narrower than the beam-line guides), as it is emptied. Thus, for a given
energy spectrum, the mean filling time will be approximately equal to the mean
emptying time. The optimal filling time, a different quantity discussed in the
next section, is about three times larger. Only the UCNs that survived the
storage phase and were counted are relevant. These surviving UCNs would thus
have similar filling and emptying time constants, owing to the fact that the
surviving UCNs would have had similar energy spectra during both the filling
and emptying phases. Consequently, the effective time of storage can be
obtained by adding an additional time period, $2\tau_{\text{emp}}$,
representing contributions of the mean times of filling and emptying, to the
micro-timer storage time as follows:

$\displaystyle t_{s}$ $\displaystyle=$ $\displaystyle
t^{*}_{s}+2\tau_{\text{emp}}.$ (8)

Figure 1: The emptying time constant of UCNs as a function of micro-timer
storage time. The solid line indicates a linear fit. The dashed lines show the
case for a $\pm 1\sigma$ deviation of the constant term.

The emptying time is a function of UCN velocity. Furthermore, we cannot
exclude that the energy spectra of the neutrons after storage could change
when varying the storage time because faster UCN are lost at a higher rate
than slower ones. Taking the data shown in Fig. 1, we can assume that if there
is a dependence of $\tau_{\text{emp}}$ on the storage time, this is linear to
a first approximation. Henceforth, the effective ‘storage-time’ will imply
adding to the micro-timer storage-time $2\tau_{\text{emp}}$ from the linear
fit, and the associated fit error.

##  3 Optimizing the filling time

Figure 2: Plot showing the variation of total neutron counts after a fixed
period of storage _w.r.t._ the filling time in seconds, along with a parabolic
fit to the data-set. The red data set, with squares for data points and dashed
line fit, corresponds to $t^{*}_{s}=90\leavevmode\nobreak\ $s, whereas the
blue data set, with dots for data points and solid line fit, corresponds with
$t^{*}_{s}=180\leavevmode\nobreak\ $s.

The filling time is defined as the time interval after a proton beam pulse
during which the UCN shutter at the bottom of the nEDM storage chamber is
open, while the UCN switch is set to the filling position. As UCN were only
produced during a short period of $7.5\leavevmode\nobreak\ $s in the source,
the number of neutrons delivered to the storage cell decreases once production
stops. At the same time UCNs also leave the nEDM cell during the same filling
period, or some of them are lost on the walls, similarly as during storage.
The optimal filling time is the moment when filling and loss rates are equal.
As mentioned earlier, the storage time determines the energy spectrum of those
detected UCN which have been filled.

We studied the optimal filling time by varying it while keeping the storage
time constant. Figure 2 shows two curves corresponding to the case where the
micro-timer storage time was held fixed at $90$ s and $180$ s respectively,
while the filling time was scanned. A parabolic fit to the data yields an
optimal filling time:

$\displaystyle
t^{\text{optimal}}_{\text{filling}}(t^{*}_{s}=180\leavevmode\nobreak\
\text{s})$ $\displaystyle=$ $\displaystyle 29.4\pm 0.1\leavevmode\nobreak\
\text{s},$ (9) $\displaystyle
t^{\text{optimal}}_{\text{filling}}(t^{*}_{s}=90\leavevmode\nobreak\
\text{s})$ $\displaystyle=$ $\displaystyle 29.3\pm 0.1\leavevmode\nobreak\
\text{s}.$ (10)

We notice that within the error bars, the optimal filling time is independent
of the UCN storage time. We used $29\leavevmode\nobreak\ $s as the filling
time for all the runs in this effort.

Figure 3: (Top) Plot showing UCN monitor counts in purple (in units of ten
thousands, corresponding to the vertical axis on the right hand side), and
emptying counts in red (in units of thousands, corresponding to the vertical
axis on the left hand side), both averaged over a dwell of 4 cycles over the
entire data taking stretch of the $n-n^{\prime}$ oscillation search. (Bottom)
Ratio between the emptying and monitor counts as a function of time, for each
dwell of 4 cycles. In both plots, yellow regions labeled - ‘$t_{s}-$Scans’
refer to a time period when the stored UCN energy spectra were estimated. The
pink regions labeled ‘$B\sim 10\leavevmode\nobreak\ \mu$T’ and green regions
labeled ‘$B\sim 20\leavevmode\nobreak\ \mu$T’ refer to runs which were
collected by applying the corresponding indicated magnetic field. The large
steps in the normalized counts correspond to switching between the micro-timer
storage times, 180 s and 380 s. The vertical orange dotted lines indicate when
the solid deuterium moderator was reconditioned [8-0] , to recover the maximal
neutron flux from the UCN source.

##  4 Normalizing emptying counts with monitor counts

So far, a common feature of sD2 based UCN sources at neutron spallation
targets is that the maximum intensity, after each proton pulse, decays over
timescales of the order of days [8-0] , as shown in Fig. 3 (top). In order to
use Eqs. 5 and 7, one has to assume that the number of neutrons that were
filled until the beginning of the storage time is stable. If that is not the
case, the counts have to be normalized by a reliable method. This allows us to
use counts such as $N_{0}(t_{s})$, $N_{\bf B}(t_{s})$, and $N_{-{\bf
B}}(t_{s})$ from different cycles. Other experiments searching for mirror-
neutrons based on UCN counting, such as Ban et al. (2007) [2] , Serebrov et
al. (2008) [3] , and Altarev et al. (2009) [5] , used neutrons from the PF2
source at the Institute Laue-Langevin (ILL) Steyerl86 , where the initial
neutron counts did not decay over time. The initial number of neutrons in the
ILL turbine-based experiments only fluctuated by about $2$% over time
correlating to the reactor performance. Here, we present our method to correct
for both fluctuations and drifts in the initial number of neutrons stored in
the chamber.

Just after filling the chamber with UCN and closing the shutter, during the
storage phase, the neutrons still emerging from the source are directly guided
to the UCN detectors for a constant time period. These neutrons are referred
to as monitor counts. We use monitor counts to normalize the emptying counts
as shown in Fig. 3 (bottom). Eqs. 5, 6, and 7 use the number of neutrons
counted after storage under various magnetic field configurations. We shall
henceforth replace bare counts such as $N_{0}(t_{s})$, $N_{\bf B}(t_{s})$, and
$N_{-{\bf B}}(t_{s})$ with normalized counts $n_{0}(t_{s})$, $n_{\bf
B}(t_{s})$, and $n_{-{\bf B}}(t_{s})$ as follows:

$\displaystyle n_{\\{0,{\bf B},-{\bf B}\\}}(t_{s})$ $\displaystyle=$
$\displaystyle\frac{N^{\text{emptying}}_{\\{0,{\bf B},-{\bf
B}\\}}(t_{s})}{N^{\text{monitor}}_{\\{0,{\bf B},-{\bf B}\\}}(t_{s})}.$ (11)

Using Eq. 11 transforms Eqs. 5 and 7 as follows:

$\displaystyle E_{B}$ $\displaystyle=$
$\displaystyle\frac{n_{0}\left(t_{s}\right)}{n_{\bf B}\left(t_{s}\right)}-1,$
(12) $\displaystyle A_{B}$ $\displaystyle=$ $\displaystyle\frac{\left(n_{\bf
B}\left(t_{s}\right)-n_{\bf{-B}}\left(t_{s}\right)\right)}{\left(n_{\bf
B}\left(t_{s}\right)+n_{\bf{-B}}\left(t_{s}\right)\right)}.$ (13)

The monitor counts are typically of the order of a million, far exceeding the
emptying counts which are of the order of few tens of thousands. Thus, the
uncertainty on the ratio of emptying and monitor counts is mostly dependent on
the uncertainty coming from the emptying counts. Studying the uncertainty on
the ratio will also help us understand if the monitor counts are an accurate
normalizing number.

Figure 3 (top) shows monitor and emptying counts averaged over a single dwell,
of 4 cycles. Figure 3 (bottom) shows the corresponding ratio between emptying
and monitor counts. In the following paragraphs we analyze the relative spread
of the normalized counts and the contribution of the Poisson statistical
uncertainty of the associated emptying counts. In Fig. 3 (bottom), for each of
the magnetic field series (in pink regions labeled ‘$B\sim
10\leavevmode\nobreak\ \mu$T’, and green regions labeled ‘$B\sim
20\leavevmode\nobreak\ \mu$T’, respectively), we see two distinct levels. The
higher level corresponds to $180$ s of micro-timer storage, and the lower
level corresponds to $380$ s of micro-timer storage. The distribution of the
residuals, after subtracting the mean value from the ratio between the UCN
counts and their corresponding monitor counts,
$(n-\left<n\right>)/\left<n\right>$, is shown in Fig. 4 for each of the four
series along with a best fit with a Gaussian distribution. Building histograms
from normalized residuals allows us to study the relative width parameters of
the corresponding distributions (_w.r.t_ $\left<n\right>$). The final analysis
will focus instead on the distributions of $E_{B}$ and $A_{B}$.

The aim of the present study was to check if the errors propagated from the
neutron counts are compatible with the data scatter, and whether the
histograms are normal distributions. For each histogram in Fig. 4, we give the
average counts per cycle $\left<N^{\text{emptying}}\right>$ and the simple
mean of the normalized counts (indiscernible from the weighted mean because of
the large counts). The following width parameters were calculated for
comparison: (i) standard deviation, $s$ of $\left<n\right>$ quantifying the
scatter of the central values, (ii) $\sigma_{\text{Poisson}}$, the width
parameter of the distribution calculated from counting statistics, (iii) width
parameter of the best fit with a normal distribution, $\sigma_{\text{Gaussian-
fit}}$ along with the associated $\chi^{2}/ndf$. The corresponding values
inserted in Fig. 4 were normalized with $\left<n\right>$ for easier
interpretation.

Thus for a series of data points $\\{n_{i},\sigma_{n_{i}}\\}$ we have:

$\displaystyle\left<n\right>$ $\displaystyle=$
$\displaystyle\sum^{\\#}_{i=1}\frac{n_{i}}{\\#},$ (14) $\displaystyle s$
$\displaystyle=$
$\displaystyle\sqrt{\sum^{\\#}_{i=1}\frac{1}{(\\#-1)}\left(n_{i}-\left<n\right>\right)^{2}},$
(15) $\displaystyle\sigma_{\text{Poisson}}$ $\displaystyle=$
$\displaystyle\sigma_{\left<n\right>}\sqrt{\\#}$ (16) $\displaystyle=$
$\displaystyle\sqrt{\frac{\\#}{\sum^{\\#}_{i=1}(1/\sigma^{2}_{n_{i}})}}.$

In the case of $\sigma_{\text{Poisson}}$ in Eq. 16, the errors associated
($\sigma_{n_{i}}$) with each ratio
$n_{i}=N_{i}^{\text{emptying}}/N_{i}^{\text{monitor}}$, came directly from
Poisson statistic uncertainties:

$\displaystyle\left(\frac{\sigma_{n_{i}}}{n_{i}}\right)^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{\sqrt{N^{\text{emptying}}}}{N^{\text{emptying}}}\right)^{2}+\left(\frac{\sqrt{N^{\text{monitor}}}}{N^{\text{monitor}}}\right)^{2}.$
(17)

We found that $s/\left<n\right>\cong\sigma_{\text{Gaussian-fit}}$ with a
relatively good $\chi^{2}$, indicating a normal distribution. Note that the
Poisson-propagated widths in Fig. 4 are dominated by errors arising from
emptying counts. The Poisson propagated widths, normalized with
$\left<n\right>$, are around a factor of two narrower compared to the widths
which one would obtain from $1/\sqrt{\left<N^{\text{emptying}}\right>}$ alone.
For example, the Poisson-propagated width in Fig. 4 (top-left) is close to
$\sqrt{\left<N^{\text{emptying}}\right>}/\left<N^{\text{emptying}}\right>\div
2\approx 0.6\%\div 2\approx 0.3\%$ (_w.r.t_
$\left<N^{\text{emptying}}\right>$). Here, a factor of approximately 2 arises
because we are considering ratios, ‘$n$’, which are from emptying and monitor
counts from the average of 4 cycles in a dwell.

Figure 4: Histograms of normalized residuals of the ratio between emptying and
monitor counts from Fig. 3 (bottom), along with the corresponding best fit
with a Gaussian distribution. In clockwise order: top-left corresponds to
dwells with run identifiers of $t^{*}_{s}=180\leavevmode\nobreak\ \text{s
and}\leavevmode\nobreak\ B_{0}\leavevmode\nobreak\ \sim 10\leavevmode\nobreak\
\mu\text{T}$; top-right corresponds to dwells with run identifiers of
$t^{*}_{s}=380\leavevmode\nobreak\ \text{s and}\leavevmode\nobreak\
B_{0}\leavevmode\nobreak\ \sim 10\leavevmode\nobreak\ \mu\text{T}$; bottom-
right corresponds to dwells with run identifiers of
$t^{*}_{s}=380\leavevmode\nobreak\ \text{s and}\leavevmode\nobreak\ B_{0}\sim
20\leavevmode\nobreak\ \mu\text{T}$; bottom-left corresponds to dwells with
run identifiers of $t^{*}_{s}=180\leavevmode\nobreak\ \text{s
and}\leavevmode\nobreak\ B_{0}\sim 20\leavevmode\nobreak\ \mu\text{T}$. The
extracted values are explained in the text.

In all the four series of measurements in Fig. 4, the scatter of the central
values expressed by $s$ is larger than $\sigma_{\text{Poisson}}$. This means
that the Poisson statistics from the counts alone does not explain the width
of these distributions. Indeed, we observed a correlation between the position
parameter of the UCN switch and $N^{\text{emptying}}$. The corresponding
difference, in quadrature, between $\sigma_{\text{Poisson}}$ and $s$, is
consistently $\sim 0.3\%$ for all the four sets of runs shown in Fig. 4. Based
on this histogram analysis, we decided not to use Poisson propagated errors
later in the evaluation of the $E_{B}$ and $A_{B}$ distributions. Instead we
will use the scatter of the center values.

##  5 Magnetic field calibration using atomic magnetometers

During the storage phase, the neutrons are exposed to a magnetic field
$B_{0}$. The PSI nEDM experiment ran with a routine magnetic field of
$1\leavevmode\nobreak\ \mu$T and associated linear spatial gradient
$|g_{z}|<40$ pT/cm. A $B_{0}$ coil current of $\sim 17\leavevmode\nobreak\ $mA
is required to produce a $1\leavevmode\nobreak\ \mu$T field. From Refs.[4] ;
[4-2] ; [6]  it is clear that we need a magnetic field up to
$20\leavevmode\nobreak\ \mu$T for a $n-n^{\prime}$ oscillation search.
$B_{0}=20\leavevmode\nobreak\ \mu$T required a coil current of $\sim
340\leavevmode\nobreak\ $mA. We limited the maximum applied magnetic field to
$B_{0}\leq 20\leavevmode\nobreak\ \mu$T, in order to be able to effectively
degauss the passive $\mu$-metal magnetic shield.

In the $n-n^{\prime}$ oscillation search, patterns of magnetic field such as
[$0\uparrow 0\downarrow 0\downarrow 0\uparrow 0\downarrow 0\uparrow 0\uparrow
0\downarrow$] (where the arrow shows the direction of the $B_{0}$ vector
_w.r.t._ to the vertical direction) were applied. This magnetic field pattern
allows to measure both the ratio and asymmetry values of $n-n^{\prime}$
oscillations while compensating for drifts in magnetic field.

The requirement for precision on the magnetic field comes from the condition
for the zero mirror field case $\omega\left<t_{f}\right>\ll 1$ (where
$\omega=45.81\leavevmode\nobreak\ (\mu\text{T.s})^{-1}\cdot B$, and $B$ is the
applied magnetic field) under which Eq. 6 is valid. Assuming a maximum value
of $\left<t_{f}\right>\sim(0.074\pm 0.004)\leavevmode\nobreak\ $s obtained
from the free flight time in the nEDM chamber in simulations with MCUCN [9-0]
, we require a precision on the magnetic field better than $\sim
0.27\leavevmode\nobreak\ \mu$T (at 1 $\sigma$ C.L.) [9]  or $\sim
0.26\leavevmode\nobreak\ \mu$T (at 95% C.L.). The power source has a precision
better than $\sim 0.2\leavevmode\nobreak\ \%$ (relative to
$17\leavevmode\nobreak\ $mA which is used to create a $1\leavevmode\nobreak\
\mu$T $B_{0}$ field), and results in a precision of better than $\sim
20\leavevmode\nobreak\ n$T. This is well within the $\sim
0.26\leavevmode\nobreak\ \mu$T requirement. In order to measure a magnetic
field with a precision better than $\sim 0.26\leavevmode\nobreak\ \mu$T, we do
not need to run the 199Hg co-magnetometer during every cycle. Using a
nanoampere meter to measure the current supplied to the $B_{0}$ coil is
sufficient to measure the $B_{0}$ magnetic field. The analogous condition
under which Eq. 5 is valid, _i.e._ $\omega^{\prime}\left<t_{f}\right>\gg 1$,
imposes a lower limit on the validity of the constraints on
$\tau_{nn^{\prime}}$ from this measurement. Assuming a minimum value of
$\left<t_{f}\right>\sim(0.061\pm 0.003)\leavevmode\nobreak\ $s based on
simulations, our results are valid in the range
$B^{\prime}>0.36\leavevmode\nobreak\ \mu$T (at 1 $\sigma$ C.L.) or
$B^{\prime}>0.38\leavevmode\nobreak\ \mu$T (at 95% C.L.). The requirement of
precision on the magnetic field better than $\sim 0.26\leavevmode\nobreak\
\mu$T also sets the requirement on the linear spatial gradient
$|g_{z}|<21.67\leavevmode\nobreak\ $nT/cm.

The magnetic field, $B_{0}$ can be obtained using the value of the applied
coil current. In order to calibrate the current source and to characterize the
current and the resulting $B_{0}$ magnetic field, we used the 199Hg co-
magnetometer [9-1-1] ; [9-1-2] ; [9-1-3]  and 133Cs magnetometers [11-1] ;
[11-2] .

Figure 5: Plot showing the $B_{0}$ magnetic field measured by 199Hg (indicated
by blue dots) and 133Cs (indicated by green squares) magnetometers as a
function of $B_{0}$ coil current.

For the calibration of the dependence of $B_{0}$ on the coil current, we
measured the precession frequencies of 199Hg and 133Cs magnetometers. We know
the gyromagnetic ratios of atomic 199Hg and 133Cs to be
$\gamma_{{}^{199}\text{Hg}}/2\pi=7.5901152(62)\leavevmode\nobreak\
\text{MHz/T}$ [10]  and
$\gamma_{{}^{133}\text{Cs}}/2\pi=3.49862111(39)\leavevmode\nobreak\
\text{GHz/T}$ [11-2] . The fit in Fig. 5 gives a consistent relationship for
the two magnetometer data, between $B_{0}$ and the coil current:

$\displaystyle\frac{B_{0}}{\mu\text{T}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 0.05996\leavevmode\nobreak\ (1)\leavevmode\nobreak\
\frac{I_{0}}{\text{mA}}\leavevmode\nobreak\ +\leavevmode\nobreak\
0.0016\leavevmode\nobreak\ (1).$ (18)

Figure 6: Decay in the magnitude of $B_{0}$ magnetic field when the current
through the $B_{0}$ coil is turned off, as measured by the 199Hg magnetometer.
The line and the shaded region indicate an exponential decay fit and its
$1\sigma$ uncertainty, respectively.

Since we did not degauss the $\mu$-metal shield while the magnetic field
patterns were applied, we also studied how the $B_{0}$ magnetic field behaved
when it was ramped down in magnitude. When the current to the $B_{0}$ coil was
switched off for the next storage phase of the neutrons in the chamber, it is
vital that the magnetic field is smaller than $0.26\leavevmode\nobreak\ \mu$T
so that the condition of $\omega\left<t_{f}\right>\ll 1$ holds true. In order
to measure the residual magnetic field when the current to the $B_{0}$ coil
was switched off, we ramped the magnetic field to $20\leavevmode\nobreak\
\mu$T, the maximum field used in the measurements, held this field for
$500\leavevmode\nobreak\ $s, and then switched off the current. This was
repeated with reversed polarity. The results are plotted in Fig. 6. The
average magnetic field felt by the neutrons during storage when the $B_{0}$
coil current is off can be obtained by averaging the exponential decay curve
in Fig. 6 in the appropriate time interval. For $300$ s long cycles with
micro-timer storage time of $180\leavevmode\nobreak\ $s, storage occurs in the
time interval of $[30,210]\leavevmode\nobreak\ $s, whereas for $500$ s long
cycles with micro-timer storage time of $380\leavevmode\nobreak\ $s, storage
occurs in the time interval of $[30,410]\leavevmode\nobreak\ $s. The average
“zero” magnetic field experienced by the neutrons during storage is well below
the $0.26\leavevmode\nobreak\ \mu$T requirement for both $300$ s and $500$ s
long cycles, and given by:

$\displaystyle\left<B_{0}(t^{*}_{s})\right>^{B_{0}\sim
0}_{t^{*}_{s}=[30,210]\leavevmode\nobreak\ \text{s}}=\left(18.98\pm
0.53\right)\leavevmode\nobreak\ n\text{T}$ (19)
$\displaystyle\left<B_{0}(t^{*}_{s})\right>^{B_{0}\sim
0}_{t^{*}_{s}=[30,410]\leavevmode\nobreak\ \text{s}}=\left(18.58\pm
0.73\right)\leavevmode\nobreak\ n\text{T}.$ (20)

##  6 Statistical sensitivity to $n-n^{\prime}$ oscillations

The sensitivity of an experiment searching for $n-n^{\prime}$ oscillations is
represented by the time variable, $\tau_{nn^{\prime}}$ in Eq. 1. The
quantities from which $\tau_{nn^{\prime}}$, or a constraint on it, can be
obtained appear in Eqs. 5, 6, and 7. In the case where we assume
$B^{\prime}=0$, the oscillation time is purely dependent on the time of
storage $t_{s}$, the mean time-of-flight between two consecutive wall
collisions $\left<t_{f}\right>$, and the neutron count ratio $E_{B}+1$. When
we assume the mirror-magnetic field experienced by neutrons to be non-zero
(${\bf B^{\prime}}\neq 0$), the sensitivity, also depends on the applied
magnetic field ${\bf B}$. The sensitivity rises dramatically when
$\bm{B^{\prime}}=\bm{B}$, _i.e._ when the neutron and mirror-neutron states
are degenerate.

Note that in the ratio channel under both conditions of ${\bf B^{\prime}}=0$
and ${\bf B^{\prime}}\neq 0$, as well as in the asymmetry channel under the
condition of ${\bf B^{\prime}}\neq 0$, the oscillation time
$\tau_{nn^{\prime}}$ depends on $1/\sqrt{A_{B}}$ or $1/\sqrt{E_{B}}$. If no
deviations from zero are observed in either $E_{B}$ or $A_{B}$, then the
constraint on the oscillation time depends on the uncertainty associated with
$E_{B}$ and $A_{B}$. If one assumes similar individual neutron counts involved
in the computation of $E_{B}$ and $A_{B}$, _i.e._ $N_{0}\sim N_{B}\sim
N_{-B}\approx N$, then the uncertainty on $E_{B}$ and $A_{B}$ is given by
$2/\sqrt{N}$ and $4/\sqrt{N}$ respectively. Along with Eqs. 5 and 6 it follows
that:

$\displaystyle\zeta_{\tau^{{\bf B^{\prime}=0}}_{nn^{\prime}}}$
$\displaystyle\propto$
$\displaystyle\sqrt[4]{N}\cdot\sqrt{\frac{t_{s}\left<t^{2}_{f}\right>}{\left<t_{f}\right>}}\qquad\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{[9]}{\@@citephrase{(}}{\@@citephrase{)}}}}$ (21)
$\displaystyle\zeta_{\tau^{{\bf B^{\prime}\neq 0}}_{nn^{\prime}}}$
$\displaystyle\propto$
$\displaystyle\sqrt[4]{N}\cdot\sqrt{\frac{t_{s}}{\left<t_{f}\right>}}\cdot
f(\eta),$ (22)

where $\zeta$ is the maximum constraint on oscillation time
$\tau_{nn^{\prime}}$, and $f(\eta)$ is a function in units of time. The above
equations also imply that an uncertainty on the $\left<t_{f}\right>$ and the
$\left<t^{2}_{f}\right>$ distributions, due to uncertainty on the energy
spectrum, will affect the final sensitivity by error propagation. This will be
treated carefully in the analysis. Here, we consider the $\left<t_{f}\right>$
variables as fixed and thus constrain ourselves to the statistical sensitivity
coming from the spread in UCN counts.

Null-hypothesis constraints can directly be estimated from Eqs. 21 and 22,
imposing a lower limit on $\tau_{nn^{\prime}}$ with given statistical
confidence, for example $1\sigma\sim 68.3\%$ C.L.. Note that in order to
increase the sensitivity, _i.e._ increase the lower bound on
$\tau_{nn^{\prime}}$, we could:

  1. 1.

increase the storage time of the UCNs

  2. 2.

increase the number of neutrons surviving after a storage time of $t_{s}$

  3. 3.

in the case of $\bm{B^{\prime}}=0$, use lower-energy UCNs so that the free
flight-time is as high as possible (from Eq. 21); in the case of
$\bm{B^{\prime}}\neq 0$ use high energy UCNs so that $\left<t_{f}\right>$ is
as low as possible (from Eq. 22).

Usually, the sensitivity is a compromise between the above three items, as:

  1. 1.

since the sensitivity always goes as $\sqrt[4]{N}\cdot\sqrt{t_{s}}$ and
neutron counts decay exponentially with increase in storage time, there will
be an optimal $t_{s}$,

  2. 2.

the mean energy of the UCNs decreases with an increase in storage time,
resulting in an increase of $\left<t_{f}\right>$.

Figure 7: Number of neutrons as a function of storage time in the PSI nEDM
apparatus compared to the storage curve of Ref. [2]  at the ILL. Solid blue
curve, with blue dots for data points, indicates the storage curve measured at
PSI for unpolarized UCNs. The dashed red curve, with red squares for data
points, shows the storage curve measured at ILL for unpolarized UCNs.

The energy spectra of the UCNs at a specific storage time is a given. However
we can optimize the time of storage $t_{s}$ in view of achieving the best
sensitivity, by studying the decay curve of the neutrons in the chamber. In
Fig. 7, the decay curve is shown for unpolarized neutrons in the PSI nEDM
apparatus, compared to the measurement at ILL [2] . At ILL the measurements
routinely counted around $\sim 17,000$ neutrons after $180$ s of storage,
whilst at PSI we measured on average over $\sim 27,000$ neutrons after $180$ s
of storage. The number of cycles scales linearly with the number of neutrons
counted, thus an increase in the number of cycles has a similar effect as that
shown by neutron counts in Eq. 22 ($\propto\sqrt[4]{N}$).

Figure 8: Plot showing the length of the predicted run time it takes to
achieve a sensitivity in $\tau^{\bm{B^{\prime}}\neq 0}_{nn^{\prime}}$ (at 95%
C.L.) _w.r.t._ storage time. Here, the blue dotted curve indicates a goal of
$\tau_{nn^{\prime}}\geq 12\leavevmode\nobreak\ $s, the orange dot-dashed curve
corresponds to a goal of $\tau_{nn^{\prime}}\geq 24\leavevmode\nobreak\ $s,
and the green curve shows the time required to achieve a goal of
$\tau_{nn^{\prime}}\geq 36\leavevmode\nobreak\ $s.

Since this effort is aimed at testing the signals reported in the asymmetry
channel (Eq. 13) in Refs. [2] ; [3] ; [5] , we can estimate the sensitivity of
the PSI nEDM apparatus to $\tau_{nn^{\prime}}$ by scaling up the constraint of
$\tau_{nn^{\prime}}>12\leavevmode\nobreak\ \text{s}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ (0\leq B^{\prime}\leq 12.5\mu$T at 95% C.L.) in
Ref. [5] , by using Eqs. 21 and 22. The PSI nEDM measurement used a micro-
timer storage time of $180$ s with each cycle being $300$ s long. Thus an
overhead of $120$ s was reasonable to accommodate filling UCNs into the
storage chamber and counting the UCNs after storage for time $t_{s}$. An
overhead of $120$ s along with the storage time $t_{s}$ allows us to calculate
the number of cycles per day which then scales linearly with the number of
neutrons counted. Along with the decay curve in Fig. 7 and Eq. 22, by scaling
up the published constraint of Ref. [5] , we can plot (Fig. 8) the time it
takes to achieve a sensitivity in $\tau^{\bm{B^{\prime}}\neq 0}_{nn^{\prime}}$
(at 95% C.L.), _viz._ $\tau^{\bm{B^{\prime}}\neq
0}_{nn^{\prime}}\geq\\{12,24,36\\}$ s as a function of storage time. From Fig.
8 we conclude that the shortest time it takes to achieve a certain sensitivity
is always $t^{*}_{s}=380$ s. Therefore we chose to operate our experiment with
$380$ s of micro-timer storage time.

Previous experiments were performed with $\sim 180\leavevmode\nobreak\ $s of
storage time [5] ; [2] . We performed a part of this effort using this storage
time, to ensure that we can compare old simulations of $\left<t_{f}\right>$
with recent ones. These values will be used in the analysis even though the
storage chamber has since been renewed, and the energy spectrum from the PSI
UCN source is not identical to the one at ILL.

Figure 9: Plot showing the sensitivity to
$\tau^{\bm{B^{\prime}}=0}_{nn^{\prime}}$ ($90\%$ C.L.) achievable as a
function of total running time of the experiment. The solid blue line marks
the sensitivity projection for the PSI nEDM apparatus using 500 s long cycles
(with $t^{*}_{s}=380\leavevmode\nobreak\ $s). The dashed lines indicate the
constraints imposed by older experiments: the flat dashed red curve shows the
constraint from Ref. [3] , the flat dot-dashed gray line indicates a
constraint which is twice that of the constraint in Ref. [3] , and the flat
solid green line indicates the constraint from Ref. [2] .

In Fig. 9 we plotted the sensitivity of the PSI nEDM apparatus to
$\tau^{\bm{B^{\prime}}=0}_{nn^{\prime}}$ by scaling up the constraint of
$\tau_{nn^{\prime}}>103$ s (at 95% C.L.) in Ref. [2] . For this we used Eq.
21, and assumed a micro-timer storage time of $380$ s, with a total overhead
of $120$ s, resulting in $500$ s cycles. A running time of the order of a
month with the PSI nEDM apparatus is sufficient to achieve a sensitivity
comparable to the current leading constraint under the assumption of
$\bm{B^{\prime}}=0$ [3] . However, Fig. 9 also makes it clear that it is
unreasonable to schedule for an effective sensitivity a factor two better than
the current leading constraints in the PSI nEDM apparatus.

The sensitivity evaluations in Fig. 8 and 9 do not include the contributions
from the uncertainty on the energy spectra, affecting $\left<t_{f}\right>$.

##  7 Conclusion

The measurements by the PSI nEDM collaboration searching for $n-n^{\prime}$
oscillations were performed using storage times
$t^{*}_{s}=\\{180,380\\}\leavevmode\nobreak\ $s corresponding to total cycle
times of $t_{t}=\\{300,500\\}\leavevmode\nobreak\ $s, and applied magnetic
fields $B_{0}=\\{10,20\\}\leavevmode\nobreak\ \mu$T. These together provided
four different configurations for the experiment cycles. While the storage
time, $t^{*}_{s}=380\leavevmode\nobreak\ $s is statistically optimized for
greatest sensitivity, $t^{*}_{s}=180\leavevmode\nobreak\ $s provides ample
performance overlap with the previous generation of experiments reported in
Refs. [2] ; [5] . The $B_{0}=\\{10,20\\}\leavevmode\nobreak\ \mu$T
configurations provide ample coverage of the relevant parameter space to test
the signals reported in Refs. [4] ; [4-2] ; [6] . UCN storage curve data were
collected by applying magnetic field patterns of [$0\uparrow 0\downarrow
0\downarrow 0\uparrow 0\downarrow 0\uparrow 0\uparrow 0\downarrow$] for
$\gtrsim 2000$ dwells over 42 days. The long run schedule also allows us to
study periodic behavior, which may be interpreted in the $n-n^{\prime}$
oscillation framework. This provided a sensitivity to $n-n^{\prime}$
oscillations comparable to the leading constraints for $B^{\prime}=0$. The
final analysis will pay special attention to the uncertainty in the energy
spectra of UCNs. This PSI nEDM effort also marks the first dedicated
simultaneous search for $n-n^{\prime}$ oscillations in both the asymmetry and
ratio channels.

## Acknowledgments

The authors thank the exceptional support provided by Michael Meier, Fritz
Burri and the BSQ group at PSI. LPC and LPSC groups are supported by ANR grant
# ANR-09-BLAN-0046. University of Sussex group is supported by STFC grants #
ST/N504452/1, ST/M003426/1, and ST/N000307/1. University of Sussex group is
also supported by their School of Mathematical and Physical Sciences. PSI
group is supported by SNSF grants # 200020-137664, # 200021-117696, #
200020-144473, # 200021-126562, # 200020-163413 and # 200021-157079. ETHZ is
supported SNSF grant # 200020-172639. University of Fribourg group is
supported by SNSF grant # 200020-140421. The Polish groups are supported by
National Science Center grant # 2015/18/M/ST2/00056. For the KU Leuven group,
this work is also partly supported by Project GOA/2010/10 and Fund for
Scientific Research in Flanders (FWO). One of the authors, P. M. would like to
acknowledge support from the SERI-FCS award # 2015.0594 and Sigma Xi grant #
G2017100190747806. We also acknowledge the grid computing resource provided by
PL-GRID [12] . We would also like to thank Zurab Berezhiani for discussions
and theoretical guidance.

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