# A Novel Nuclear Emulsion Detector for Measurement of Quantum States of
Ultracold Neutrons in the Earth’s Gravitational Field

Naoto Muto  Hartmut Abele  Tomoko Ariga  Joachim Bosina  Masahiro Hino
Katsuya Hirota  Go Ichikawa  Tobias Jenke  Hiroaki Kawahara  Shinsuke Kawasaki
Masaaki Kitaguchi  Jakob Micko  Kenji Mishima  Naotaka Naganawa  Mitsuhiro
Nakamura  Stéphanie Roccia  Osamu Sato  René I. P. Sedmik  Yoshichika Seki
Hirohiko M. Shimizu  Satomi Tada  Atsuhiro Umemoto

###### Abstract

Hypothetical short-range interactions could be detected by measuring the
wavefunctions of gravitationally bound ultracold neutrons on a mirror in the
Earth’s gravitational field. Searches for them with higher sensitivity require
detectors with higher spatial resolution. We developed and have been improving
an ultracold neutron detector with a high spatial resolution, which consists
of a Si substrate, a thin converter layer including 10B4C, and a layer of
fine-grained nuclear emulsion. Its resolution was estimated to be less than
$100\text{\,}\mathrm{nm}$ by fitting tracks of either 7Li nuclei or
$\alpha$-particles, which were created when neutrons interacted with the 10B4C
layer. For actual measurements of the spatial distributions, the following two
improvements were made. The first improvement was to establish a method to
align microscopic images with high accuracy within a wide region of
$65\text{\times}0.2\text{\,}\mathrm{mm}$. We created reference marks of
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter with an interval of
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$, respectively, on the Si substrate
by electron beam lithography and realized a position accuracy of less than
$30\text{\,}\mathrm{nm}$. The second improvement was to build a holder for the
detector that could maintain the atmospheric pressure around the nuclear
emulsion to utilize it under vacuum during exposure to ultracold neutrons.

The intrinsic resolution of the improved detector was estimated to be better
than $0.56(8)\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by evaluating the blur of
a transmission image of a gadolinium grating taken by cold neutrons. The
evaluation included the precision of the gadolinium grating. A test exposure
was conducted to obtain the spatial distribution of ultracold neutrons in the
quantized states on a mirror in the Earth’s gravitational field. The
distribution was obtained, fitted with the theoretical curve, and turned out
to be reasonable for UCNs in quantized states when we considered a blurring of
$6.9\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The blurring was well explained
as a result of neutron refraction due to the large surface roughness on the
upstream side of the Si substrate. By using a double-side-polished Si
substrate, a resolution of less than $0.56\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ is expected to be achieved for UCNs.

##  1 Introduction

Hypothetical fields manifesting as dark energy or dark matter may also lead to
interactions causing effective deviations from Newtonian gravity at short
distances. In recent years, various experiments have been searching for such
interactions [1]. Some of them have been searching hypothetical fields by
measuring the spatial distributions of quantized states of ultracold neutrons
(UCNs) bound by the Earth’s gravitational potential on mirrors and comparing
those distributions to theoretical curves including only Newtonian gravity [2,
3, 4, 5]. Those distributions have characteristic structures in the scale of
ten micrometers, and therefore they require detectors with a high spatial
resolution of not more than a few micrometers. The sensitivity of those
experiments depends on the spatial resolution of the detectors and the
experimental setup including mirrors.

So far, experiments studying gravitational levels have been using spatial
resolution detectors made of CR39, originally proposed in [2], or a pixelated
Si detector equipped with a Ni-coated cylindrical glass rod to expand the
images [4]. Measurements using CR39 detectors with 235-U-converter are
presented in [6, 7]. The qBounce collaborations has been using detectors with
10-B-converter [3]. Its handling and optical readout is presented in [8]. Both
concepts reach a spatial resolution in the order of 1–2
$\mathrm{\SIUnitSymbolMicro m}$. The Si detector with the cylindrical rod
reached a resolution of $0.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

To further increase the sensitivity, a detector with a higher spatial
resolution is necessary. We have developed such a detector based on a fine-
grained nuclear emulsion. The resolution obtained is superior by more than one
order of magnitude.

Nuclear emulsions consist of silver halide crystals dispersed in gelatin.
Particles passing through the emulsion leave a linear series of silver grains
(a track) that can be detected optically. The high spatial resolution of these
tracks has led to important results in nuclear and particle physics, such as
the discovery of $\pi$ mesons [9], studies on double hypernuclei [10, 11, 12],
and the first detection of tau neutrino interactions [13]. Recently, a fine-
grained nuclear emulsion was developed for use in experiments searching for
dark matter [14]. With a crystal size of $40\text{\,}\mathrm{nm}$ and a
density of 13 crystals per $\mathrm{\SIUnitSymbolMicro m}$, our emulsion
detectors are 6 times denser and have 5 times smaller crystals than a normal
emulsion, thereby improving the spatial resolution. In addition, no track is
formed by minimum ionizing particles or electrons produced by $\gamma$-rays
occurring as background in neutron experiments because silver halide crystals
have small sizes and have not been sensitized.

Emulsion detectors can be sensitive to cold neutrons or UCNs when using a
10B4C conversion layer sputtered onto a Si substrate, followed by NiC, and C
layers [15]. The emulsion is applied above the C top layer (Figure 1). NiC and
C act as stabilizers for 10B4C and the emulsion, respectively. The thickness
of 10B4C, NiC, and C are $200\text{\,}\mathrm{nm}$, $60\text{\,}\mathrm{nm}$,
and $20\text{\,}\mathrm{nm}$, respectively. When neutrons interact with the
10B4C layer, 7Li nuclei and $\alpha$-particles are emitted back-to-back by
neutron capture reactions and either of them passes through the emulsion
layer. The detection efficiency is estimated to be $40\text{\,}\mathrm{\char
37\relax}$ for neutron velocities around
$10\text{\,}\mathrm{m}\text{/}\mathrm{s}$. The emulsion is capable of
recording tracks of 7Li nuclei with a range of
$2.7(4)\text{\,}\mathrm{\SIUnitSymbolMicro m}$ or $\alpha$-particles with a
range of $5.2(4)\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the emulsion,
calculated by SRIM-2013 based on the composition of the recent fine-grained
nuclear emulsion. Tracks from the absorption points of neutrons were observed
under an optical microscope with an epi-illumination system [16]. The
wavelength of the illumination was $455\text{\,}\mathrm{nm}$, and the
microscope was equipped with a CMOS image sensor. We used an oil immersion
lens with magnification $100\times$, and a numerical aperture of 1.45. As a
feature of the detector, both real and mirror images of the tracks are
obtained because the surface of the C layer in the substrate functions as an
optical mirror (Figure 2).

Figure 1: Schematic cross-sectional view of our high-resolution nuclear
emulsion detector for cold neutrons and UCNs. Neutrons enter into the detector
from the left. A 10B4C layer converts the neutrons to $\alpha$-particles and
7Li particles, emitted back-to-back. Both of these create a track in the
emulsion on the right side.

The coordinates of the absorption points are obtained by the following eight
steps. First, tomographic images are acquired by taking images at different
heights in 63 steps distributed equally over the thickness of the emulsion
layer. Each image has a field of view of
$113\text{\times}113\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and the resolution
is $2048\times 2048$, resulting in a pixel size of
$55\text{\times}55\text{\,}\mathrm{nm}$. The thickness of each imaging layer
(corresponding to the vertical resolution) is
$0.37\text{\,}\mathrm{\SIUnitSymbolMicro m}$ prior to shrinking of the
emulsion during development. Second, pixels with low brightness values
corresponding to silver particles are detected in the acquired images, and
those pixels are recognized as a part of grains. This detection is performed
twice: once for images that only contain real images and again for images
containing the mirror images of the tracks. Third, grains in the lowermost 1.5
layers above the base layer are selected. These grains correspond to the
starting points of tracks in the emulsion. Fourth, pairs of two grains
corresponding to the starting and ending point of original tracks whose
distance is consistent with the theoretical track length (more than
$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and less than
$7.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$) are assigned. If the density of
original tracks is high, the starting or ending points of two or more tracks
may be mixed during the selection process. In this case, there are no silver
particles on the three-dimensional straight line connecting the starting and
ending points, which means that the total brightness value of the pixels on
this line is higher than it should be for a real track. Fifth, to reject these
pairs, if the distance between the starting points or ending points of
different pairs is less than $0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$, the
pairs with the higher total brightness value are excluded. Sixth, for the
remaining starting/ending point pairs having proper distances, tracks are
reconstructed from the grains lying on the connecting line between the two
points. These are called subsequently ‘reconstructed tracks’. Seventh, the
coordinates of the point of the boundary between real and mirror images at the
surface of the C layer, and the slopes of each reconstructed track are
obtained by fitting both images of each track with symmetrical lines. Finally,
absorption points of neutrons are identified by extrapolating the
reconstructed track of the real images to the middle of 10B4C layer using the
fitting results. The spatial resolution of this detector was calculated from
the data of Reference [15] to be less than $100\text{\,}\mathrm{nm}$. Note
that we only consider tracks with angles $\theta\leq$
$0.9\text{\,}\mathrm{rad}$ between the track and the normal line to the
surface of the C layer, as for tracks at larger angles, the error increases
significantly.

Figure 2: Track of an $\alpha$-particle visible in an optical microscope image
(left), and the reconstructed track in the $x$-$z$ plane (right). Pixels with
low brightness correspond to silver particles, which are recognized as grains.
Right: Black dots show the grain coordinates reconstructed from the figure on
the left. The blue line shows a reconstructed track obtained from fitting the
grains. The red dot shows the coordinates of the point of the boundary between
real and mirror images on the surface of the C layer ($z$ = 0) resulting from
the line fits.

In order to utilize the detector in the measurement of the spatial
distribution of UCNs, two key issues had to be resolved. The first issue was
to establish a method to ensure that high spatial accuracy and resolution in a
detection area of $65\text{\times}0.2\text{\,}\mathrm{mm}$, where a field of
view of the microscope is
$113\text{\times}113\text{\,}\mathrm{\SIUnitSymbolMicro m}$. This cannot be
guaranteed by simply stitching images from different positions, as offsets
accumulate and lead to increasing errors. Therefore, we created microscopic
marks on the Si substrate that serve as independent reference points to align
the optical images. We demonstrated this method to work and the achievement of
high spatial resolution by an analysis of the transmission image of a
gadolinium grating taken by cold neutrons at the Japan Proton Accelerator
Research Complex (J-PARC). The second issue was to find a method allowing us
to utilize the emulsion detector in a vacuum of $1\text{\,}\mathrm{Pa}$ or
less environment. This was necessary, as measurements with UCNs have to be
done under vacuum to prevent scattering of neutrons by compounds in the air.
Therefore, we developed a holder that allows maintaining the volume around the
emulsion at atmospheric pressure. Finally, a test exposure to UCNs was
conducted at Institut Laue-Langevin to obtain the spatial distribution.

##  2 Improvements of the detector

In order to perform an actual measurement of the spatial distribution of UCNs,
we assembled a new emulsion detector (see Figure 3) implementing all of the
improvements mentioned above. We created circular reference marks of
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter with an interval of
$d_{1}=$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$d_{2}=$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, respectively, over an
area of $65\text{\times}0.2\text{\,}\mathrm{mm}$ on the Si substrate using an
electron beam lithography system (JBX-6300FS). Absolute coordinate values of
the centers of the marks were determined with an accuracy better than
$30\text{\,}\mathrm{nm}$ thanks to laser interferometers used to determine the
position during the marking process. The depth of the marks was
$500\text{\,}\mathrm{nm}$. The marks were successfully recognized as black
circles under the microscope as shown in Figure 4. These marks make it
possible to define the entire analysis area in a unified coordinate system.

Figure 3: Detector used in the measurement of the spatial distribution of
quantized UCNs. ($a$) – ($c$) indicate the following regions: ($a$) nuclear
emulsion, ($b$) Si substrate and 10B4C-NiC-C layer, ($c$) plain Si substrate.
Figure 4: Tracks and reference marks on an optical microscope image. Reference
marks have been highlighted by red squares. There are two different types of
markings. Mark4 in the figure are $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in
diameter and lie on a grid with $500\text{\,}\mathrm{\SIUnitSymbolMicro m}$
spacing. All other marks (Mark1, Mark2, Mark3) in the figure are
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in diameter and lie on a grid with
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ spacing. 7Li nuclei and
$\alpha$-particles tracks are indicated by blue ellipses. Green circles
indicate stains or tracks at large angles $\theta>$0.9\text{\,}\mathrm{rad}$$,
which are excluded from the analysis (see main text).

The coordinate values of neutron absorption points were converted from the
coordinate system of each image to the unified coordinate system defined by
the marks using an affine transformation as follows,

$\begin{pmatrix}X_{i}\\\ Y_{i}\\\ 1\\\ \end{pmatrix}=\begin{pmatrix}A&B&C\\\
D&E&F\\\ 0&0&1\\\ \end{pmatrix}\begin{pmatrix}x_{i}\\\ y_{i}\\\ 1\\\
\end{pmatrix},$ (2.1)

where the values of the $i$-th absorption point in the coordinate system of
the image are ($x_{i}$ $\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$,
$y_{i}$ $\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$), those in the
unified coordinate system are ($X_{i}$ $\mathrm{\SIUnitSymbolMicro m}$,
$Y_{i}$ $\mathrm{\SIUnitSymbolMicro m}$) as shown in Figure 5, and $A$, $B$,
$C$, $D$, $E$, and $F$ are affine transformation parameters. The parameters
were acquired by the following formula using the information of three marks
among four in each view:

$\begin{pmatrix}A_{j}\\\ B_{j}\\\ C_{j}\\\
\end{pmatrix}=\begin{pmatrix}x^{\prime}_{j}&y^{\prime}_{j}&1\\\
x^{\prime}_{k}&y^{\prime}_{k}&1\\\ x^{\prime}_{l}&y^{\prime}_{l}&1\\\
\end{pmatrix}^{-1}\begin{pmatrix}d_{1}M^{\prime}_{j}\\\ d_{1}M^{\prime}_{k}\\\
d_{1}M^{\prime}_{l}\\\ \end{pmatrix},$ (2.2) $\begin{pmatrix}D_{j}\\\ E_{j}\\\
F_{j}\\\ \end{pmatrix}=\begin{pmatrix}x^{\prime}_{j}&y^{\prime}_{j}&1\\\
x^{\prime}_{k}&y^{\prime}_{k}&1\\\ x^{\prime}_{l}&y^{\prime}_{l}&1\\\
\end{pmatrix}^{-1}\begin{pmatrix}d_{1}N^{\prime}_{j}\\\ d_{1}N^{\prime}_{k}\\\
d_{1}N^{\prime}_{l}\\\ \end{pmatrix},$ (2.3)

where ($j$, $k$, $l$) are (1, 2, 3), (2, 3, 4), (3, 4, 1), or (4, 1, 2), and
($x^{\prime}_{j}$, $y^{\prime}_{j}$) are the center coordinates of the
($M^{\prime}_{j}$, $N^{\prime}_{j}$)-th mark in ($x$, $y$) directions,
respectively, with respect to the origin of the unified coordinate system
shown in Figure 5. $A$, $B$, $C$, $D$, $E$, and $F$ are determined by the
average of the $A_{j}$, $B_{j}$, $C_{j}$, $D_{j}$, $E_{j}$, and $F_{j}$
obtained for all ($j$, $k$, $l$) pairs. We obtained the coordinate values of
the marks in two steps. First, the edge of the mark was acquired using a Canny
edge detector [17]. Then, the coordinates of the marks were obtained using a
Hough transform [18] of the detected edge assuming that marks were circular.
To avoid errors caused by distortion due to the microscope lens, the region
inside the marks ($50\text{\times}50\text{\,}\mathrm{\SIUnitSymbolMicro m}$)
was set to the center of each tomographic image and the region was called
subsequently ‘view’. Tomographic images were acquired with an interval of
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the $x$-$y$ plane. If deformed
markers are created, the respective views are excluded from the analysis since
the center coordinates of marks cannot be determined by a Hough transform.

Figure 5: Schematic view of the image and unified coordinate systems. The
black circles indicate reference marks. The blue dot represents an absorption
point in both coordinate systems, whose values were
$(x_{i}\,$\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}\mathrm{s}$,\,y_{i}\,$\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}\mathrm{s}$)$
and $(X_{i}\,$\mathrm{\SIUnitSymbolMicro
m}$,\,Y_{i}\,$\mathrm{\SIUnitSymbolMicro m}$)$. The coordinates of the marks
are
$(x^{\prime}\,$\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$,\,y^{\prime}\,$\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$)$,
and the subscripts 1, 2, 3, and 4 refer to the upper left, upper right, lower
left, and lower right of each view, respectively. $M$ and $N$ are the lattice
indices in $x$ and $y$ directions, respectively, starting from the mark on the
origin of the unified coordinate system. $M^{\prime}_{1}$ or $M^{\prime}_{3}$
corresponds to $M$. $M^{\prime}_{2}$ or $M^{\prime}_{4}$ corresponds to $M$+1.
$N^{\prime}_{3}$ or $N^{\prime}_{4}$ corresponds to $N$. $N^{\prime}_{1}$ or
$N^{\prime}_{2}$ corresponds to $N$+1.

The error of the transformation from the image coordinate system to the
unified coordinate system depends on the error of the center coordinates of
the marks. The error was estimated by measuring the distance between adjacent
marks $d^{\prime}_{1}$ $\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$
and those between diagonally adjacent marks $d^{\prime}_{2}$
$\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$ were calculated from
$(x^{\prime}\,$\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$,\,y^{\prime}\,$\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$)$
(see Figure 5). Then,
$d^{\prime}_{1}\,$\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$$ and
$d^{\prime}_{2}\,$\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$$ were
translated to $d_{1}\,$\mathrm{\SIUnitSymbolMicro m}$$ and
$d_{2}\,$\mathrm{\SIUnitSymbolMicro m}$$, respectively, by using the fact that
one pixel is equal to $55\text{\,}\mathrm{nm}$. $d{{}_{1}}$ was measured for
400 marks and $d{{}_{2}}$ for 200 marks, resulting in average values
$d_{1}=$50.0010(19)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$d_{2}=$70.7046(29)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and standard
deviations $\sigma_{d_{1}}=$38.3(14)\text{\,}\mathrm{nm}$$ and
$\sigma_{d_{2}}=$40.7(20)\text{\,}\mathrm{nm}$$, obtained by fitting a
Gaussian function to the histograms, respectively. Assuming the errors of the
single mark coordinates to be equal, we divide the errors in
$d_{1}\,$\mathrm{\SIUnitSymbolMicro m}$$ and
$d_{2}\,$\mathrm{\SIUnitSymbolMicro m}$$, by $\sqrt{2}$ to estimate the errors
of the mark center coordinates. Conservatively, using the larger value
$\sigma_{d_{2}}$, we thus obtain the standard deviation of the center
coordinates of the marks to be $28.8(14)\text{\,}\mathrm{nm}$. Note that this
standard deviation includes not only the limited accuracy of determining the
position of the marks during the electron beam lithography but also the
accuracy of reading the coordinate values of the marks. The given value
($28.8(14)\text{\,}\mathrm{nm}$) is consistent with the specified accuracy
(less than $30\text{\,}\mathrm{nm}$) of our electron beam lithography
equipment.

The second issue solved by the new detector relates to measurements in a
vacuum environment. If the nuclear emulsion would be exposed to a vacuum, the
contained water would disperse quickly, causing cracking and peeling of the
emulsion surface. The torsion would further lead to an increase in silver
grains not related to tracks that would be recognized as noise (fog) in the
image. To operate the emulsion in a vacuum nonetheless, we developed a holder
to maintain the volume around it at atmospheric pressure (Figure 6). The
central part of the top cover was hollowed out in an area of
$10\text{\times}80\text{\,}\mathrm{mm}$ to prevent scattering of incident
ultracold neutrons at the holder. The nuclear emulsion gel was applied to the
central area of $7\text{\times}65\text{\,}\mathrm{mm}$ to prevent contact with
the holder. We conducted a durability test by placing a nuclear emulsion in
this holder under a vacuum of $3\text{\times}{10}^{-3}\text{\,}\mathrm{Pa}$
for two days, corresponding to a typical duration for an exposure with
ultracold neutrons. Subsequently, the state of the nuclear emulsion was
confirmed with an optical microscope after development. Neither cracks nor the
increase of fog was observed on the surface of it, which confirms that the
holder serves its purpose.

Figure 6: Holder allowing in-vacuum use of the emulsion detector. (left:
photograph of the holder, right: schematic view of its cross section). The
central part of the top cover of the holder was hollowed out to allow direct
exposure of the surface of Si to vacuum. The yellow part of the schematic view
shows the nuclear emulsion layer; the volume around it is kept at atmospheric
pressure.

##  3 Measurement of the resolution using a gadolinium grating

The spatial resolution of the improved detector was evaluated by acquiring a
transmission image of a neutron-absorbing Gd grating that has a periodic
structure along the $X$-axis direction (see Figure 7) and creates a sharp
shadow on the detector. Subsequently, the edges of the shadow were numerically
analyzed by fitting the profile with an error function. This experiment was
conducted at BL05 [19, 20] of the Materials and Life Science Experimental
Facility in J-PARC using cold neutrons of approximately
$1000\text{\,}\mathrm{m}\text{/}\mathrm{s}$. The
$12\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick grating was created by vapor
deposition of Gd from an oblique angle on a Si mold with a comb-like structure
of period of $9.00\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (measured by
confocal microscopy), and the opening of the Gd grating was
$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ [21]. Figure 8 shows an electron
micrograph of the cross section of a similar Gd grating made by the same
process. A nuclear emulsion wrapped with two aluminum foils with a total
thickness of $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for light shielding
was placed behind the Gd grating. The detector was then exposed through the Gd
grating for $2.0\text{\times}{10}^{4}\text{\,}\mathrm{s}$ using cold neutrons
(Figure 7) with a beam flux of
$7.6\text{\times}{10}^{4}\text{\,}\mathrm{n}\mathrm{e}\mathrm{u}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}\mathrm{/}\mathrm{s}$,
as measured with a 3He proportional counter. We estimated the transmittance of
the $12\text{\,}\mathrm{\SIUnitSymbolMicro m}$ Gd layer to be
$2\text{\,}\mathrm{\char 37\relax}$, which can be ignored. Due to the
geometric view factor given by the Gd grating, one-third of the total beam
reached the detector. From the total number of impeding neutrons, and the
track count on the detector we determined a detection efficiency of
$0.50\text{\,}\mathrm{\char 37\relax}$. This corresponds well to the
theoretical estimate $0.65\text{\,}\mathrm{\char 37\relax}$ (corresponding to
$3.3\text{\times}{10}^{2}$ tracks per
$100\text{\times}100\text{\,}\mathrm{\SIUnitSymbolMicro m}$) based on the
thickness of the 10B4C layer. The beam divergence angle was
$5.6\text{\times}{10}^{-2}\text{\,}\mathrm{mrad}$ and
$0.56\text{\,}\mathrm{mrad}$ in the $X$-axis and $Y$-axis direction,
respectively.

Figure 7: Setup for the measurement of cold neutron detection resolution. The
Gd grating is located on the downstream side of a Si-slab placed in front of
the detector. A Si plate was inserted to separate the emulsion layer from the
Al foil to avoid the chemical effects of Al on the emulsion. Figure 8:
Electron micrograph showing a cross section of a Gd grating that was
manufactured by the same process as that used in this experiment. The
thickness of Gd in (1) is not sufficient to prevent cold neutrons from
passing. We used the transmission image of the edge (2) for the estimation of
the resolution of the detector.

Figure 10 shows an image of the detector taken with an optical microscope
after development, in which the striped distribution of tracks can clearly be
seen. We acquired the absorption points using the analysis method explained in
Section 1. The reconstructed tracks were classified into four types: ‘Track’,
‘Track+Track’, ‘Track+Dust/Fog’, and ‘Dust/Fog’ as shown in Figure 9. ‘Track’
means that all grains belong to an original track. ‘Track+Track’ is an
erroneously reconstructed track that consists of grains originally belonging
to different tracks. ‘Track+Dust/Fog’ means that some grains of the track are
fog or dust origin. ‘Dust/Fog’ means that all grains are dust or fog origin.
The types other than ‘Track’ were treated as noise. Since the grains of a
track that was categorized as noise are not distributed on a straight line, it
is considered that the $\chi^{2}/\mathrm{ndf}$ of the corresponding line fit
is increased with respect to a regular track. Therefore, the noise
contamination rate can be reduced by not using reconstructed tracks with a
large value of $\chi^{2}/\mathrm{ndf}$ in the analysis. $\chi^{2}$ was
acquired by the following formula,

$\chi^{2}=\sum_{i}\left(\left(\frac{\delta R_{i}}{\delta
a\sqrt{1+\left(\frac{\delta b}{\delta
a}\right)^{2}\tan^{2}{\theta}}}\right)^{2}+\left(\frac{\delta L_{i}}{\delta
a}\right)^{2}\right),$ (3.1)

where

$\begin{pmatrix}\delta R_{i}\\\ \delta
L_{i}\end{pmatrix}=\begin{pmatrix}\cos{\phi}&&\sin{\phi}\\\
-\sin{\phi}&&\cos{\phi}\end{pmatrix}\begin{pmatrix}x_{t}(z_{i})-x_{i}\\\
y_{t}(z_{i})-y_{i}\end{pmatrix}.$ (3.2)

Here, $\theta$ and $\phi$ are the angle of a track in a spherical coordinate
system, the coordinates of the $i$-th grain are ($x_{i}$, $y_{i}$, $z_{i}$),
those of the reconstructed track at $z_{i}$ are ($x_{t}(z_{i})$,
$y_{t}(z_{i})$), $\delta a=$57\text{\,}\mathrm{nm}$$ is the standard deviation
in the $\delta L_{i}$ direction , and $\delta b=$300\text{\,}\mathrm{nm}$$ is
the standard deviation in the $z$ direction. In order to systematically remove
noise from our data, we introduce the empirical criterion
$\chi^{2}/\mathrm{ndf}<2$. This cutoff value was determined by analyzing the
typical $\chi^{2}/\mathrm{ndf}$ for reconstructed tracks belonging to the four
categories described above (see also Figure 9). For this purpose, we randomly
selected 20 reconstructed tracks and classified them according to their
respective $\chi^{2}/\mathrm{ndf}$ as shown in Table 1. The number of survived
tracks after applying the cutoff was 3022.

Figure 9: Classification of the reconstructed tracks. The reconstructed tracks
were classified into four types. Red lines show reconstructed tracks. Black
dots show silver particles. White dots of ‘Track+Track’ show silver particles
on a track originally belonging to a different track than that of black dots.
Blue ellipses of ‘Track+Dust/Fog’ or ‘Dust/Fog’ show fog or dust. Table 1:
Classification of 20 randomly selected reconstructed tracks according to their
$\chi^{2}/\mathrm{ndf}$, used to empirically set the threshold for acceptance
$\chi^{2}/\mathrm{ndf}<2$. These tracks were visually checked and classified
as described in the main text.
$\chi^{2}/\mathrm{ndf}$ | 0.00 – 0.67 | 0.67 – 1.33 | 1.33 – 2.00 | 2.00 – 2.67
---|---|---|---|---
Total events | 939 | 1267 | 816 | 529
Track | 17 | 16 | 14 | 5
Track+Track | 1 | 2 | 3 | 10
Track+Dust/Fog | 2 | 0 | 2 | 5
Dust/Fog | 0 | 2 | 1 | 0
Checked tracks | 20 | 20 | 20 | 20
Figure 10: Transmission image of Gd grating. Figure 11: Distribution of the
length of survived tracks in the analysis. The figure shows the length from
the absorption points to the ending points of the grain, assuming the
absorption reaction happens in the middle of the thickness of 10B4C layer. The
two peaks of the distribution correspond to ranges of tracks of 7Li nuclei and
$\alpha$-particles with theoretical center values of
$2.7(4)\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$5.2(4)\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the emulsion, calculated by
SRIM-2013, respectively. Slight differences from the calculated values are
explained with the error of the shrinkage rate of the emulsion layer during
the development.  Figure 12: Distribution of absorption points of neutrons
that passed through the Gd grating.

The distribution of the length of survived tracks used in the analysis is
shown in Figure 11. The analyzed region was
$500\text{\times}500\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and the
coordinates of the absorption points obtained in each view was converted into
the unified coordinate system using Equation 2.1. Figure 12 shows a plot of
the absorption points. A striped pattern corresponding to that of the
structure of the Gd grating can be observed in the plot, as expected. In order
to remove an overall tilt angle $\alpha$ and determine the period $d$ of the
pattern, we use the Rayleigh test function $R(\alpha,d)$ [22, 23]:

$R(\alpha,d)=\frac{2}{N}\left(\left(\sum_{i}\sin\left(\frac{2\pi
x_{i}(\alpha)}{d}\right)\right)^{2}+\left(\sum_{i}\cos\left(\frac{2\pi
x_{i}(\alpha)}{d}\right)\right)^{2}\right)\,,$ (3.3)

where

$x_{i}=X^{\prime}_{i}\cos\alpha+Y^{\prime}_{i}\sin\alpha+x_{m}\,,$ (3.4)

and

$\begin{pmatrix}X^{\prime}_{i}\\\ Y^{\prime}_{i}\\\
\end{pmatrix}=\begin{pmatrix}X_{i}-x_{m}\\\ Y_{i}-y_{m}\\\ \end{pmatrix}.$
(3.5)

Here, $N$ is the total number of absorption points, ($X_{i}$, $Y_{i}$) are the
coordinates in the unified coordinate system [c.f. Equation 2.1], and
($x_{m}$, $y_{m}$) are the average values of the coordinates of all absorption
points. Optimum values $\alpha_{m}$ =
$-1.623(30)\text{\times}{10}^{-2}\text{\,}\mathrm{rad}$ and $d_{m}$ =
$8.9997(26)\text{\,}\mathrm{\SIUnitSymbolMicro m}$ were obtained by maximizing
$R(\alpha=\alpha_{m},d=d_{m})$ numerically. The detected period is consistent
with the designed value of the Gd grating,
$9.00\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

Figure 13:  Results of the Rayleigh test. The color coding expresses the value
of the Rayleigh test function $R(\alpha,d)$. The brightest $R$ point indicates
the optimum values of $\alpha$ and $d$.

Subsequently, the tilt was removed from the data by rotating coordinates by
$-\alpha_{m}$ using Equation 3.4. The histogram of the distribution of the
absorption points folded with the period of $d_{m}$, i.e., $x_{i}$ modulo
$d_{m}$, is given in Figure 14. A peak corresponding to the opening of the Gd
grating was observed from 3.5 to $7.0\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
In addition, it was understood that the part from 4 to
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ corresponded to (1) in Figure 8
because the number of events from 4 to $5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ of the horizontal axis in Figure 14 was smaller than the number of events
from 5 to $6\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The transmission image
corresponding to (1) is more blurred than that corresponding to (2) since the
thickness of Gd in (1) is not sufficient. Therefore, we did not use the left
side of the transmission image but the right side of it. We fitted the right
edge of the data in Figure 14 with the error function:

$f(x)=A\left(\frac{1}{\sqrt{2\pi}\sigma}\int_{x}^{\infty}\exp{\left(-\frac{(t-\mu)^{2}}{2\sigma^{2}}\right)}\,\mathrm{d}t\right)+B,$
(3.6)

where $\mu$ $\mathrm{\SIUnitSymbolMicro m}$ is the coordinate of the edge of a
stripe on the detector, $\sigma$ $\mathrm{\SIUnitSymbolMicro m}$ is the
effective blurring of the distribution, $A$ is a scaling factor of the
distribution, and $B$ is the amount of background noise. All of the mentioned
variables were used as free parameters of the fit. Since the cross section of
the sputtered Gd layer is not a perfect rectangle as shown in Figure 8, it is
assumed that $\sigma$ becomes larger than a certain value. Assuming that all
Gd shapes are the same as in Figure 8, the thickness distribution can be
obtained from the figure. The lower limit of $\sigma$ was
$0.14\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by fitting the transmission image
estimated from the thickness distribution with Equation 3.6. The fitting
result is shown in Figure 15. We obtain a spread of
$\sigma=$0.56(8)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, which includes the
effects of the intrinsic resolution of the detector, the accuracy of the
connecting of views using the reference marks, the spread of the beam, the
shape of the Gd grating, and the variation in the amount of Gd on each Si
mold. It is therefore reasonable to state that the intrinsic resolution of the
detector is expected to be better than the given value. The divergence angle
of the beam in the $x$-axis direction (horizontal axis in Figure 14) was
obtained by rotating the original divergence angle by $-\alpha_{m}$. The
maximum value was $6.5\text{\times}{10}^{-2}\text{\,}\mathrm{mrad}$. The
distance between the Gd grating and the 10B4C layer of the detector was
$900\text{\,}\mathrm{\SIUnitSymbolMicro m}$; hence, the spread of the
transmission image due to the divergence angle was $60\text{\,}\mathrm{nm}$.
It was considered that probably the variation in the amount of Gd on each Si
mold was dominant for the value of the resolution.

Figure 14: Distribution of $x_{i}$ modulo $d_{m}$ in the Gd transmission
measurement. The resolution of the detector was evaluated using $x_{i}\text{
modulo }d_{m}>$5.0\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ whose region
corresponded to the sharp edge of Gd grating. Figure 15: Fit result of the
absorption points around the Gd grating edge. The blue histogram data is an
excerpt of the data in Figure 14 for $x_{i}\text{ modulo
}d_{m}>$5.0\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The red line shows the
result of the least squares fit of Equation 3.6 to the data.

##  4 Measurement of the spatial distribution of quantized UCNs

As a demonstration of our emulsion detector, we attempted to measure the
spatial distribution of UCNs that take quantized states in the potential well
created between the Pseudo-Fermi potential of a horizontal flat mirror and the
gravitational potential of the Earth, a system commonly known as ‘quantum
bouncer’ [24]. This experiment was conducted at PF2 of the Institut Laue-
Langevin. The experimental setup is shown in Figure 16. A combination of a
flat mirror with a rough absorber/scatterer on top, separated by precision
spacers of nominal thickness $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$
creates a selector for vertical energy states of neutrons [25]. At the exit of
the selector, neutrons arrive in one of the lowest few gravitational quantum
states [26]. The mirror, absorber/scatterer and vacuum chamber in Figure 16
were taken from the qBounce experiment [3]. Both the flat mirror and the rough
absorber/scatterer are made of borosilicate glass. The size of the mirror was
$15\text{\,}\mathrm{cm}$ and $20\text{\,}\mathrm{cm}$ in the $Z$\- and
$X$-axes, respectively. The size of the absorber/scatterer was
$15\text{\,}\mathrm{cm}$ and $10\text{\,}\mathrm{cm}$ in the $Z$-axis and the
$X$-axis directions, respectively. All parts of the setup were installed in a
vacuum chamber, and the detector was exposed with UCNs for 16 hours in a
vacuum of $1\text{\,}\mathrm{Pa}$ or less, where the scattering by air can be
neglected. The beam flux was estimated to be
$0.21\text{\,}\mathrm{n}\mathrm{e}\mathrm{u}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{s}$
for the region exposed to neutrons from an actual measurement using a gas
detector with a 10B layer [26]. The velocity distribution of the UCNs was
measured using an aperture system as shown in Figure 17. Two plates of borated
Al create a slit at horizontal and vertical distances $w$ and $h$,
respectively. The slit together with the gap between the absorber/scatterer
and flat mirror define a flight parabola that selects the horizontal velocity
$v$ according to

$v=\sqrt{\frac{gw^{2}}{2h}}\,,$ (4.1)

where $g$ is the gravitational acceleration. The velocity distribution was
measured by varying $h$ and counting the admitted UCNs by a gas detector as
shown in the left of Figure 17. The result of this measurement is plotted on
the right of Figure 17 and shows an almost Gaussian distribution with a mean
of $9.53\text{\,}\mathrm{m}\text{/}\mathrm{s}$ and a standard deviation of
$2.18\text{\,}\mathrm{m}\text{/}\mathrm{s}$.

Figure 16: Setup for obtaining the quantized spatial distribution of UCNs. The
figures on the left and right show the schematic view of the setup of the
measurement of coordinates and velocity of UCNs, respectively. The distance
from the exit of the mirror and absorber/scatterer to 10B4C layer is
$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Figure 17: Measured velocity
distribution at the UCN beam of the PF2 installation. The counts of UCNs of 4
– $11\text{\,}\mathrm{m}\text{/}\mathrm{s}$ were measured using a gas detector
with a 10B layer.

We reconstructed the tracks and classified them as explained in Section 3. The
result of the classification and the number of total events are summarized in
Table 2. The reason for the noise contamination rate here being lower than in
the analysis of Gd grating is that the tracks are less likely to overlap in
this measurement due to the smaller density of tracks on the detector.

Figure 18 shows the distribution of the scanned area and the absorption points
(red) of neutrons. The area shown in light blue in Figure 18 shows the scanned
region; green indicates areas that were excluded due to difficulties in the
recognition of marks due to deformed reference marks or the presence of large
dust particles on the surface of the C layer. The scanned area was
$65\text{\,}\mathrm{mm}$ and $0.2\text{\,}\mathrm{mm}$ in the directions of
the $X$\- and $Y$-axes, respectively. The excluded area was
$2.7\text{\,}\mathrm{\char 37\relax}$ of the scanned area. We restricted our
analysis to absorption points inside the purple lines, containing a total of
1821 tracks.

Table 2: Classification of 20 randomly selected reconstructed tracks depending
on their $\chi^{2}/\mathrm{ndf}$. These tracks were visually checked and
classified as described in the main text.
$\chi^{2}/\mathrm{ndf}$ | 0.00 – 0.67 | 0.67 – 1.33 | 1.33 – 2.00 | 2.00 – 2.67
---|---|---|---|---
Total events | 729 | 830 | 314 | 175
Track | 20 | 20 | 19 | 14
Track+Track | 0 | 0 | 0 | 0
Track+Dust/Fog | 0 | 0 | 0 | 1
Dust/Fog | 0 | 0 | 1 | 5
Checked tracks | 20 | 20 | 20 | 20
Figure 18: Scanned area and the distribution of absorption points. The length
of the vertical axis is 1/100 of the length of the horizontal axis.

The obtained line of absorption points was slightly tilted with respect to the
horizontal axis as shown in Figure 18, due to the setting accuracy of the
detector. From the least squares fit of the coordinates to a linear function,
we obtained a rotation of $9.146(17)\text{\,}\mathrm{mrad}$ and a small
vertical offset that both were removed from the data. We created a histogram
of the data along the vertical $y$ direction shown in Figure 19, where the
effect of the excluded area was considered. We compared the obtained
distribution with the theoretical curve to verify the precision of the
measurement and estimate the ratio of each quantum state. Neutrons bound by
the pseudo-Fermi potential $V_{F}$ of a horizontal surface from below and the
gravitational potential $mgy$ rising linearly with the distance $y$ above the
surface obey the Schrödinger equation [24]

$E\psi(y)=\left(-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
y^{2}}+mgy+V_{F}\Theta(y_{s}-y)\right)\psi(y),$ (4.2)

where $\psi$ is the wave function, $y_{s}$ is the vertical coordinate of the
mirror surface, $\Theta$ is the Heaviside step function, $m$ is the mass of a
neutron, $E$ is the energy of the system, $\hbar$ is the reduced Planck
constant, and $g$ is the gravitational acceleration. Since $V_{F}$ is several
orders of magnitude larger than the neutron’s kinetic energy in the vertical
direction, we can safely assume the boundary condition

$\psi(y=y_{s})=0,$ (4.3)

for which neutrons cannot enter the mirror. The same is true for the rough
surface of the absorber/scatterer. However, due to the stochastic roughness of
this surface, an effective gap size $d_{e}$ has to be considered at this
boundary,

$\psi(y=y_{s}+d_{e})=0.$ (4.4)

When Equation 4.2 is solved under the boundary conditions of Equation 4.3 and
4.4, the eigenfunction of the $n$-th state can be written as,

$\psi_{n}(y)=C_{n}\mathrm{Ai}\left(\frac{y}{y_{0}}-\frac{E_{n}}{E_{0}}\right)+D_{n}\mathrm{Bi}\left(\frac{y}{y_{0}}-\frac{E_{n}}{E_{0}}\right),$
(4.5)

where $y_{0}$ and $E_{0}$ are the characteristic scale and energy for the
gravitational quantum states, respectively;

$y_{0}=\left(\frac{\hbar^{2}}{2m^{2}g}\right)^{\frac{1}{3}}\approx$5.9\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ (4.6)

and

$E_{0}=mgy_{0}\approx$0.60\text{\,}\mathrm{peV}$.$ (4.7)

Here, Ai and Bi denote the Airy functions, $C_{n}$ and $D_{n}$ are
normalization constants, and $E_{n}$ is the energy of the $n$-th state.

For the evaluation of our experiment, we also need to consider the time
evolution during free fall because the distance from the exit of the
absorber/scatterer to the 10B4C layer of the detector was
$\ell_{0}=$1.5\text{\,}\mathrm{mm}$$ (see Figure 16). Suppose that the $n$-th
wave function at the exit of the mirror and absorber/scatterer is the initial
state $\psi_{n}(y,t=0)$, then the wave function reaching the 10B4C layer can
be computed from [27],

$\psi_{n}(y,t)=\int
K(y,y^{\prime},t)\psi_{n}(y^{\prime},t=0)\,\mathrm{d}y^{\prime},$ (4.8)

where

$K(y,y^{\prime},t)=\sqrt{\frac{m}{2\pi\hbar
t}}\exp{\left(\frac{\mathrm{i}}{\hbar}S(y,y^{\prime},t)\right)}$ (4.9)

and

$S(y,y^{\prime},t)=\frac{m}{2t}\left(y^{\prime}-y-\frac{1}{2}gt^{2}\right)^{2}+mgyt-\frac{1}{6}mg^{2}t^{2}.$
(4.10)

Here, $t=l_{0}/v$, $v$ is the horizontal velocity of the neutrons.

In order to compare our theoretical prediction based on Equation 4.8 with our
measurements, we used a model which was given by an ideal function with
infinitely high spatial resolution. However, the fit resulted in an extremely
small $p$-value of $5.4\text{\times}{10}^{-17}$, indicating a low statistical
probability that the model represents the data well and the necessity to add
the effect of the blurring to the model. Therefore, we used a model given by
the following function which was a convolution of the previous model and
Gaussian function:

$f(y)=A\sum_{n=1}^{N}\left(\frac{P_{n}}{\sqrt{2\pi}\sigma}\int_{-3\sigma}^{3\sigma}|\psi_{n}(y-y_{s}-y^{\prime\prime},t)|^{2}\,\exp{\left(-\frac{y^{\prime\prime
2}}{2\sigma^{2}}\right)}\,\mathrm{d}y^{\prime\prime}\right)+B,$ (4.11)

where $N$ is the maximum number of the quantum state, $A$ is the magnification
of the distribution, $B$ is the amount of background noise, $\sigma$
represents the blurring of the distribution, and $P_{n}$ is the relative
probability of quantum state $n$ obeying $\sum_{n=1}^{N}P_{n}=1$. In Equation
4.11, the effects of interference terms are ignored, which is reasoned by
these terms becoming extremely small on average due to the random phase of
each neutron.

This model was then used in the least squares fits to the histogrammized data
from our measurement. We used $A$, $B$, $y_{s}$, $\sigma$, and $P_{n}$ ($n$ =
1, 2, …, $N$-1) as free parameters. $N$ and $d_{e}$ were varied by hand but
kept fixed for each fit. The true value of the effective gap $d_{e}$ was
assumed to lie between $26\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$36\text{\,}\mathrm{\SIUnitSymbolMicro m}$, considering the measured value of
the gap and the effect of the roughness. We performed fits with the values
$d_{e}=26,\,31,\,\text{ and }$36\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. For
$N$, we initially restricted ourselves to the most probable cases $N=3$ and
$N=4$. Generally, for $N=4$, the probability $P_{4}$ of the state $n=4$
becomes almost zero (except for the case of
$d_{e}=$26\text{\,}\mathrm{\SIUnitSymbolMicro m}$$). In the latter case,
$P_{4}$ was the largest of all states, which we consider an artifact caused by
fixing $d_{e}$ to a value far from the real one. Therefore, the number of
states was fixed to be $N=3$ in all subsequent fits.

Figure 19: Fit result using Equation 4.11, which includes the effect of
blurring. The effective gap was fixed to
$31\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and the maximum number of the
quantum state was 3. The blue histogram shows the distribution of obtained
absorption points. The red line shows the best fit result obtained with model
$f(y)$. Dashed lines of green, purple, and light blue show the probability
distributions of the ground state, the first excited state, and the second
excited state, respectively. Figure 20: Relative probabilities of each quantum
state for the fit result shown in Figure 19.

Figure 19 and Figure 20 show the fit results and the contributing probability
distributions for each quantum state using the model $f(y)$ for fixed
effective gap of $d_{e}=$31\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The best
fit with $p$-value 0.70 was obtained for
$\sigma=$6.9\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, indicating a valid
result. The accuracy of determining the probability of each quantum state was
lowered due to this blurring.

The cause of the difference in resolution between cold neutrons (used for the
measurements with the Gd grating) and UCNs was considered to be the effect of
neutron refraction due to the large surface roughness of the upstream (beam
side) of the Si substrate in front of the emulsion layer. The refraction was
negligible at the downstream side (emulsion side) of Si substrate as the
surface roughness was polished. To evaluate this effect, we measured the
height of the roughness $z(x,\,y)$ of the surface of the Si substrate in a
$560\text{\times}560\text{\,}\mathrm{\SIUnitSymbolMicro m}$ region with an
interval of $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$, where $z$ is a
coordinate value in the direction of the thickness of the substrate, using a
3D Optical Profiler (Zygo NewView 6000). From the results, we obtained a set
of values,

$g(x,x^{\prime},y)=\langle(z(x,y)-z(x^{\prime},y))^{2}\rangle,$ (4.12)

as discussed in Reference [28]. For the estimation of the blur described
below, we followed the procedure outlined in Reference [29]. Then, the set was
transformed to $g^{\prime}(r)$, where $r=\sqrt{x^{2}-x^{\prime 2}}$ is the
absolute distance between two measurement points. Finally, $g^{\prime}(r)$ was
fitted by the following function,

$f(r)=2b^{2}\left(1-\exp{\left(-\left(\frac{r}{\sqrt{2}w}\right)^{2}\right)}\right)\,,$
(4.13)

where $b$ and $w$ are the standard deviation of the surface roughness and the
correlation length, respectively. From the fit results, we obtained
$b=$0.293(6)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$w=$3.44(20)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. For these values, the
classical refraction effect is dominant for UCNs. Assuming that the neutrons
hit the silicon surface at normal incidence, the refraction angle can be
computed:

$\theta=\frac{b}{\sqrt{2}w}\frac{k_{s}^{2}}{k_{n}^{2}},$ (4.14)

Using the values $k_{n}=$0.15\text{\,}{\mathrm{nm}}^{-1}$$ for the wavenumber
of the neutrons for a mean velocity
$v=$9.5\text{\,}\mathrm{m}\text{/}\mathrm{s}$$, and
$k_{s}=$0.051\text{\,}{\mathrm{nm}}^{-1}$$ for the critical wavenumber of Si,
and we arrive at $\theta=$6.9\text{\,}\mathrm{mrad}$$. The resulting blur
(spread of the waveform) for our Si substrate having a thickness of
$0.5\text{\,}\mathrm{mm}$ is $3.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$. In
the case of the resolution measurement using the transmission image of the Gd
grating, the typical velocity of cold neutrons used in the measurement was
$1000\text{\,}\mathrm{m}\text{/}\mathrm{s}$ and the wavenumber was
$16\text{\,}{\mathrm{nm}}^{-1}$; hence, the refraction angle was
$0.62\text{\,}\mathrm{\SIUnitSymbolMicro rad}$ and the blur was
$0.31\text{\,}\mathrm{nm}$. The magnitude of the latter blur was negligible
compared to the resolution of the detector. Thus, the observed difference of
the measured resolution between cold neutrons and UCNs can be explained
quantitatively.

In order to estimate the blur in the case of a double-side-polished Si
substrate, we measured $b$ and $w$ of the polished surface of its substrate in
the same manner as described above. As a result, we obtained
$0.60(4)\text{\,}\mathrm{nm}$ and $42(8)\text{\,}\mathrm{\SIUnitSymbolMicro
m}$, respectively. In case $b$ is smaller than the UCN wavelength, the surface
scattering will follow the coherent scattering described by the micro-
roughness model [29]. In this model, $6.9\text{\,}\mathrm{\char 37\relax}$ of
the total UCNs would diffuse and the most likely scattering angle was
$150\text{\,}\mathrm{\SIUnitSymbolMicro rad}$ if a non-divergent beam is used
under the same conditions as in the UCN experiment described above. It
corresponds to $75\text{\,}\mathrm{nm}$ for $0.5\text{\,}\mathrm{mm}$ thick
Si. Since this is small enough compared to the characteristic scale of $y_{0}$
in Equation 4.6, it is expected that UCNs in the quantized states can be
clearly observed by using the emulsion detector with a double-side-polished Si
substrate.

##  5 Conclusion

Different experiments use the system of a quantum bouncer, realized by the
quantized states of cold neutrons or UCNs on a flat mirror in the Earth’s
gravitational field, to search for non-Newtonian interactions. For this
purpose, neutron detectors with a high spatial resolution are used. In this
study, we improved upon a detector so that it could measure the spatial
distribution of UCNs with a high spatial resolution using a fine-grained
nuclear emulsion. Two tasks were accomplished with the detector to conduct
such measurements. The first task involved establishing a method to ensure
that the high spatial accuracy and resolution can be achieved within a wide
region of $65\text{\times}0.2\text{\,}\mathrm{mm}$, for which reference marks
were created using electron beam lithography. The second task involved the
establishment of a method for the emulsion to work in vacuum, for which a
holder was developed to maintain the volume around the emulsion at atmospheric
pressure.

A transmission image of a Gd grating was acquired by cold neutron detection at
BL05 of the J-PARC Material and Life Science Experimental Facility in Japan,
and the resolution of the improved detector was evaluated by fitting the edge
portion of the image with an error function. The results demonstrate that the
intrinsic resolution of the detector was better than
$0.56(8)\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for cold neutrons.

A test exposure was conducted to obtain a spatial distribution of UCNs in
quantized states on a mirror in the Earth’s gravitational field at PF2 of the
Institut Laue-Langevin. The distribution was obtained, fitted with the
theoretical curve, and turned out to be reasonable for UCNs in quantized
states when we considered a blurring of
$6.9\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The blurring was well explained
as a result of neutron refraction due to the large surface roughness on the
upstream side of the Si substrate. By using a double-side-polished Si
substrate, a resolution of less than $0.56\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ is expected to be achieved for UCNs.

## Acknowledgments

We thank T. Naka for providing us fine-grained nuclear emulsion gel and advice
regarding its use. We are grateful to T. Shinohara, T. Samoto, and A. Momose
for his help in using the $9.00\text{\,}\mathrm{\SIUnitSymbolMicro m}$ Gd
grating and its electron micrograph. We also thank P. Geltenbort for
constructive advice and T. Brenner for his tremendous support on the
experiment at Institut Laue-Langevin. The sputtering of the converter layer
and test experiments for the development of the detector was conducted under
the support of common use programs of the Institute for Integrated Radiation
and Nuclear Science, Kyoto University. The reference marks were developed and
created at Nagoya University Research Facility for Advanced Science and
Technology with the assistance of T. Kato, D. Oshima and M. Sakashita. This
research was supported by JSPS KAKENHI Grant Number 18H05210. The neutron
experiment at the Materials and Life Science Experimental Facility of the
J-PARC was performed under user programs (Proposal No. 2017B0336, 2018A0263,
and 2019A0227) and S-type project of KEK (Proposal No. 2019S03).

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