# Radioactive Decay of Specific Heavy Elements as an Energy Source for Late-
Time Kilonovae and Potential JWST Observations

Meng-Hua Chen1,2 and En-Wei Liang2
1Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing
100871, China
2Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical
Science and Technology, Guangxi University, Nanning 530004, China
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Revealing the temporal evolution of individual heavy elements synthesized in
the merger ejecta from binary neutron star mergers not only improves our
understanding of the origin of heavy elements beyond iron but also clarifies
the energy sources of kilonovae. In this work, we present a comprehensive
analysis of the temporal evolution of the energy fraction of each nuclide
based on the $r$-process nucleosynthesis simulations. The heavy elements
dominating the kilonova emission within $\sim 100$ days are identified,
including 127Sb, 128Sb, 129Sb, 130Sb, 129Te, 132I, 222Rn, 223Ra, 224Ra, and
225Ac. It is found that the late-time kilonova light curve ($t\gtrsim 20$
days) is highly sensitive to the presence of the heavy element 225Ac (with a
half-life of 10.0 days). Our analysis shows that the James Webb Space
Telescope (JWST), with its high sensitivity in the near-infrared band, is a
powerful instrument for the identification of these specific heavy elements.

###### keywords:

nuclear reactions, nucleosynthesis, abundances—stars: neutron—binaries: close

††pubyear: 2023††pagerange: Radioactive Decay of Specific Heavy Elements as an
Energy Source for Late-Time Kilonovae and Potential JWST
Observations–Radioactive Decay of Specific Heavy Elements as an Energy Source
for Late-Time Kilonovae and Potential JWST Observations

##  1 Introduction

The mergers of binary neutron stars or neutron star-black hole binaries have
long been speculated to be a promising site for the production of heavy
elements beyond iron via the $r$-process nucleosynthesis in the universe
(Lattimer & Schramm, 1974). The radioactive decay of newly synthesized
$r$-process nuclei in the merger ejecta produces an electromagnetic transient
in the ultraviolet/optical/near-infrared wavelengths, known as a kilonova (Li
& Paczyński, 1998; Metzger et al., 2010; Korobkin et al., 2012; Barnes &
Kasen, 2013; Kasen et al., 2013; Barnes et al., 2016; see Metzger, 2019 for a
recent review). The radioactive decay of unstable nuclei may also produces a
gamma-ray transient accompanying the kilonova (Hotokezaka et al., 2016; Li,
2019; Chen et al., 2021, 2022) and a MeV neutrino flash (Chen et al., 2023).
The solid detection of the kilonova AT2017gfo, associated with gravitational
wave event GW170817, provides strong evidence to support this speculation
(Abbott et al., 2017a, b, c).

The luminosity and color evolution of AT2017gfo are well consistent with the
radioactively powered kilonova model, suggesting that such mergers could be
the major sites for $r$-process nucleosynthesis (Cowperthwaite et al., 2017;
Drout et al., 2017; Evans et al., 2017; Kasliwal et al., 2017; Nicholl et al.,
2017; Pian et al., 2017; Smartt et al., 2017; Tanvir et al., 2017). The
analysis of the kilonova AT2017gfo reveals that the ejected material has two
distinct components: an early “blue” kilonova and a later “red” kilonova
(Kasen et al., 2017; Kilpatrick et al., 2017; Nicholl et al., 2017; Villar et
al., 2017). The blue kilonova is interpreted as a signature of light
$r$-process nuclei with atomic mass number $A\lesssim 140$, while the red
kilonova is commonly believed to be powered by a lanthanide-rich ejecta with
high opacity ($\kappa\sim 10$ cm2 g-1). It is worth noting that all of these
studies have considered the radioactive decay of heavy elements as the energy
source powering the kilonova transient. However, the late-time kilonova
emission may also be attributed to the energy injection from a long-lived
remnant neutron star (Yu et al., 2018; Ren et al., 2019).

Determining the nuclide composition of the merger ejecta, especially the
nulides located around the $r$-process peaks and the lanthanide elements, is
crucial for understanding the origin of heavy elements and the energy sources
of kilonovae. Several research groups have been investigating the connection
between the heavy elements and kilonova light curves (Metzger et al., 2010;
Lippuner & Roberts, 2015; Wanajo, 2018; Watson et al., 2019; Wu et al., 2019;
Kasliwal et al., 2022). Metzger et al. (2010) were pioneers in estimating the
radioactive heating rates and predicting kilonova light curves resulting from
neutron star mergers by using $r$-process nucleosynthesis simulations.
Lippuner & Roberts (2015) conducted a parameter study of $r$-process
calculations and found that the ejected materials are lanthanide-free for high
electron fractions ($Y_{\rm e}\gtrsim$ 0.25). Zhu et al. (2018) included the
contribution from fission fragments to calculate the kilonova light curves
from neutron star mergers. Wanajo (2018) focused on calculating the
contributions from individual $r$-process elements and found that the heavy
elements around the second $r$-process peaks dominate the radioactive heating
rates. The impact of nuclear physics uncertainties on $r$-process abundances
patterns and kilonova emission was investigated by Barnes et al. (2021).
Moreover, Wu et al. (2019) suggested that heavy elements can be distinguished
from kilonova light curves because the radioactive power is often dominated by
a few nuclei, which can result in identifiable features in the light curves,
such as bump-like or exponential decay-like features in the bolometric light
curve. This distinctiveness could potentially be utilized to infer the yield
of heavy elements in future kilonova observations (Wu et al., 2019; Kasliwal
et al., 2022).

Given its high sensitivity in the near-infrared band, the James Webb Space
Telescope (JWST) is expected to be a powerful instrument for follow-up
observations and providing a comprehensive view of the temporal and spectral
evolution of kilonovae from early to late-time phase (Bartos et al., 2016).
Such observations offer an opportunity to reveal the production of individual
heavy elements. In this paper, we study this issue by analyzing the temporal
evolution of energy fraction of each nuclide based on detailed $r$-process
nucleosynthesis simulations. We investigate the heavy nuclei that dominate the
kilonova emission within $\sim 100$ days, and find out distinguished features
in their bolometric light curves. We further explore whether these feature are
detectable using the JWST telescope. The paper is organized as follows. In
Section 2, we describe the methodology used to calculate the kilonova light
curves. Our results are presented in Section 3. Conclusions and discussions
are given in Section 4.

##  2 Methods

###  2.1 Nucleosynthesis

We employ the nuclear reaction network code SkyNet(Lippuner & Roberts, 2015,
2017) to perform simulations of $r$-process nucleosynthesis. SkyNet evolves
the abundances of 7843 nuclide species, ranging from free neutrons (atomic
number $Z=0$) and protons ($Z=1$) to 337Cn ($Z=112$), and includes over 140
000 nuclear reactions. The nuclear reaction rates used in SkyNet are taken
from the JINA REACLIB database (Cyburt et al., 2010). In our calculations, we
use the latest nuclear database, including nuclear mass data from AME2020
(Wang et al., 2021) and radioactive decay data from NUBASE2020 (Kondev et al.,
2021). For nuclide species without experimental data, we use the theoretical
data derived from the Finite-Range Droplet Model (Möller et al., 2016).
Neutron capture rates are calculated using the corresponding nuclear mass data
through the nuclear reaction code TALYS (Goriely et al., 2008). Nuclear
fission processes are calculated using fission barriers from Möller et al.
(2015) and double Gaussian fission fragment distributions from Kodama &
Takahashi (1975).

In our $r$-process nucleosynthesis simulations, we use the parameterized
trajectory provided by Lippuner & Roberts (2015). The initial conditions are
consistent with the results of numerical relativity simulations given by
Radice et al. (2018). We adopt the results of $(1.40+1.40)M_{\odot}$ binary
system, which is simulated using three distinct equations of state:
BHBlp_M140140_LK, DD2_M140140_LK, and LS220_M140140_LK. The initial electron
fractions for these binary system are $Y_{\rm e}=$ 0.15, 0.17, 0.14, and the
initial specific entropies are $s=$ 18, 22, 16 $k_{\rm B}$/baryon. Based on
the numerical relativity simulations conducted by Radice et al. (2018), the
expansion timescale $\tau$ can be estimated as $\tau\approx cRe/3v_{R}$, where
$e\simeq 2.718$ is the Euler’s number and $v_{R}$ corresponds to the velocity
measured on a sphere with a coordinate radius of $R=300G/c^{2}M_{\odot}\simeq
443$ km. Thus the expansion timescales of these binary system are 7.87 ms,
6.08 ms, and 7.87 ms, repectively. Additionally, we consider the result of the
unequal mass binary $(1.40+1.20)M_{\odot}$, labeled LS220_M140120_M0. This
system has an initial electron fraction of $Y_{\rm e}=0.18$, an initial
entropy of $s=14~{}k_{\rm B}$/baryon-1, and an expansion timescale of
$\tau=8.92$ ms). In the calculations of radioactive power, we use the recent
nuclear data from the Evaluated Nuclear Data File library (EDNF/B-VIII.0,
Brown et al., 2018). We note that we do not include the extra radioactive
energy resulting from fission fragments since the EDNF library contains very
few radiation data for fission processes. Zhu et al. (2018) have identified
the specific fission reactions responsible for the radioactive power by using
a double Gaussian fragment distribution from Kodama & Takahashi (1975).

###  2.2 Radioactive Heating Rate

The effective heating rate of merger ejecta is obtained by summing the
contributions of all decay products, including $\alpha$-particles,
$\beta$-particles, and $\gamma$-rays, i.e.,

$\dot{Q}(t)=\sum_{\rm p}f_{\rm p}(t)\dot{q}_{\rm p}(t),~{}~{}~{}{\rm
p}\in\\{\alpha,\beta,\gamma\\},$ (1)

where $f_{\rm p}(t)$ is the thermalization efficiency. The radioactive heating
rate $\dot{q}_{\rm p}(t)$ is given by the sum of the energy released by all
nuclides in the ejecta,

$\dot{q}_{\rm p}(t)=N_{\rm A}\sum_{i}\lambda_{i}Y_{i}(t)E_{\rm
p},~{}~{}~{}{\rm p}\in\\{\alpha,\beta,\gamma\\}$ (2)

where $N_{\rm A}$ is the Avogadro’s number, $\lambda_{i}$ is the decay rate of
the $i$th nuclide, $Y_{i}(t)$ is the corresponding abundance, and $E_{\rm p}$
is the energy released by individual nuclear reaction. The decay rate is given
by $\lambda_{i}=\ln 2/T_{1/2,i}$ with $T_{1/2,i}$ being the half-life of the
$i$th nuclide. The thermalization efficiencies for each decay product are
estimated by following the analytic methods presented in Kasen & Barnes
(2019).
For $\beta$-particles,

$f_{\beta}(t)=(1+t/t_{\beta})^{-1},$ (3)

where $t_{\beta}$ is the characteristic timescale at which thermalization
starts to become inefficient,

$t_{\beta}=12.9~{}\left[\frac{M_{\rm
ej}}{0.01M_{\odot}}\right]^{2/3}\left[\frac{v_{\rm
ej}}{0.2c}\right]^{-2}\zeta^{2/3}~{}{\rm days},$ (4)

where $M_{\rm ej}$ is the total ejecta mass, $v_{\rm ej}$ is the
characteristic ejecta velocity, and $\zeta$ is a constant defined to close to
unity (we adopt $\zeta=1$).
For $\alpha$-particles,

$f_{\alpha}(t)=(1+t/t_{\alpha})^{-1.5},$ (5)

where the thermalization timescale of $\alpha$-particles is roughly
$t_{\alpha}\approx 3t_{\beta}$.
For $\gamma$-rays,

$f_{\gamma}(t)=1-\exp\left[-\frac{t_{\gamma}^{2}}{t^{2}}\right],$ (6)

where the thermalization timescale $t_{\gamma}$ is given by

$t_{\gamma}=0.3~{}\left[\frac{M_{\rm
ej}}{0.01M_{\odot}}\right]^{1/2}~{}\left[\frac{v_{\rm
ej}}{0.2c}\right]^{-1}~{}{\rm days}.$ (7)

In this paper, we adopt an ejecta mass of $M_{\rm ej}=0.04M_{\odot}$ and an
ejecta velocity of $v_{\rm ej}=0.15c$. These values are consistent with the
estimations derived from the red kilonova component observed in AT2017gfo, as
reported by Kasen et al. (2017).

###  2.3 Kilonova Model

We assume a spherical merger ejecta as that adopted by Metzger (2019) . The
merger ejecta is divided into $N$ ($N=100$) layers and each layer evolves
nearly independently (see also Chen et al., 2021, 2022). The expansion
velocity of the $n$th layer is given as $v_{n}=v_{0}+n\Delta v$, where
$n=1,2,\ldots,100$, $\Delta v=0.001c$, and $v_{0}$ is a free parameter. The
radius of the $n$th layer at time $t$ is given by $R_{n}=v_{n}t$ and its mass
can be estimated by $m_{n}=\int_{R_{n}}^{R_{n+1}}4\pi\rho r^{2}dr$. The merger
ejecta is assumed to have a power-law density profile (Nagakura et al., 2014)

$\rho(v_{n},t)=\rho_{0}(t)\left(\frac{v_{n}}{v_{0}}\right)^{-3},$ (8)

where $\rho_{0}(t)$ is the density of the innermost layer. For a given ejecta
mass $M_{\rm ej}$, $\rho_{0}(t)$ can be derived from

$M_{\rm ej}=\int_{R_{\rm 1}}^{R_{\rm N}}4\pi\rho(v_{n},t)r^{2}dr.$ (9)

The characteristic ejecta velocity can be estimated as

$v_{\rm ej}=(2E_{\rm kin}/M_{\rm ej})^{1/2},$ (10)

where $E_{\rm kin}=\sum_{n}m_{n}v_{n}^{2}/2$ is the sum of the kinetic energy
of all layers. The velocity of the innermost layer, $v_{0}$, is determined by
the characteristic ejecta velocity $v_{\rm ej}$ through the equation:

$\frac{1}{2}M_{\rm ej}v_{\rm ej}^{2}=\sum_{n}\frac{1}{2}m_{n}(v_{0}+n\Delta
v)^{2}.$ (11)

Following Metzger (2019), the thermal energy of each layer evolves according
to

$\frac{dE_{n}}{dt}=-\frac{E_{n}}{R_{n}}\frac{dR_{n}}{dt}-L_{n}+\dot{Q}(t)m_{n},$
(12)

where $E_{n}$ is the internal energy of the $n$th layer, $L_{n}$ is the
radiation luminosity, and $\dot{Q}(t)$ is the $r$-process heating rate. The
radiation luminosity of each layer can be estimated by

$L_{n}=\frac{E_{n}}{t_{{\rm lc},n}+t_{{\rm d},n}},$ (13)

where $t_{{\rm lc},n}=v_{n}t/c$ is the light crossing time and $t_{{\rm
d},n}=\tau_{n}v_{n}t/c$ is the photon diffusion timescale. The optical depth
from the $n$th layer to the observer $\tau_{n}$ is given by

$\tau_{n}=\int_{R_{n}}^{\infty}\rho(v_{n},t)\kappa dr,$ (14)

where $\kappa$ is the ejecta opacity. We set $\kappa=10$ cm2 g-1 in our
calculation, which is a reasonable approximation for a lanthanide-rich ejecta
(Tanaka et al., 2018). The total bolometric luminosity of the merger ejecta
can be obtained by summarizing the contributions of all layers,

$L_{\rm bol}=\sum_{n}L_{n}.$ (15)

The effective temperature of kilonova emission can be determined by

$T_{\rm eff}=\left(\frac{L_{\rm bol}}{4\pi\sigma_{\rm SB}R_{\rm
ph}^{2}}\right)^{1/4},$ (16)

where $\sigma_{\rm SB}$ is the Stephan-Boltzmann constant. The radius of the
photosphere $R_{\rm ph}$ is defined by the layer at which the optical depth is
unity, i.e.,

$\tau_{\rm ph}=\int_{R_{\rm ph}}^{\infty}\rho(v_{n},t)\kappa dr=1.$ (17)

The flux density of the kilonova emission at frequency $\nu$ can be given by

$F_{\nu}(t)=\frac{2\pi h\nu^{3}}{c^{2}}\frac{1}{{\rm exp}({h\nu/kT_{\rm
eff}})-1}\frac{R_{\rm ph}^{2}}{D_{L}^{2}},$ (18)

where $h$ is the Planck constant, $k$ is the Boltzmann constant, and $D_{L}$
is the luminosity distance. Then, the monochromatic AB magnitude is given by

$M_{\nu}(t)=-2.5\log_{10}\left(\frac{F_{\nu}(t)}{3631~{}{\rm Jy}}\right).$
(19)

##  3 Results

Figure 1: Abundance patterns for $r$-process nucleosynthesis simulations. The
initial conditions are consistent with the results of numerical relativity
simulations (Radice et al., 2018).  Figure 2: Effective heating rates produced
by the radioactive decay of $r$-process elements. Figure 3: The radioactive
energy generated by $\alpha$-particles, $\beta$-particles, and $\gamma$-rays,
respectively.

Figure 1 shows the abundance patterns obtained from $r$-process
nucleosynthesis simulations. It is found that the merger ejecta is
sufficiently neutron rich, enabling the production of a wide range of heavy
elements from the second to third $r$-process peak. The $r$-process
simulations calculated with various astrophysical trajectories all produce a
rather broad distribution of heavy elements, and the overall abundance
patterns are in agreement with the observed solar $r$-process abundances,
particularly for atomic mass numbers $A\geq 120$.

Based on the $r$-process nucleosynthesis simulations, we calculate the
effective heating rate produced by the radioactive decay of newly synthesized
heavy elements in neutron star mergers, as shown in Figure 2. It is shown that
the discrepancies in effective heating rates across different equations of
state are modest. Figure 3 shows the decay products responsible for effective
heating rate. The total heating rate mainly comes from the radioactive energy
of $\alpha$-particles, $\beta$-particles, and $\gamma$-rays. It is shown that
$\beta$-particles and $\gamma$-rays play a crucial role in the heating rate
during the early stage ($t\lesssim 10$ days), while $\alpha$-particles
dominate the nuclear heating at later times. This is caused by the fact that
the lifetimes of $\alpha$-decay nuclei are typically longer than those of
$\beta$-decay nuclei (Hotokezaka & Nakar, 2020).

Figure 4: The energy fraction of the dominant nuclides that contribute
significantly to the total energy generation within 100 days for
$\beta$-particles (top panel), $\gamma$-rays (middle panel), and
$\alpha$-particles (bottom panel), respectively. Here we take BHBlp_M140140_LK
as a representative case for our analysis.

To investigate the nuclides that make dominant contribution to the radioactive
heating rate, we calculate the energy generation of each nuclide. Given the
insignificant discrepancies in radioactive heating rates when using different
astrophysical trajectories, we take BHBlp_M140140_LK as the representative
case for our analysis. In the top panel of Figure 4, we show the energy
fraction of the dominant nuclides that contribute significantly to the total
energy generation of $\beta$-particles between 0.01 day and 100 days. It is
clearly shown that radioactive heating rate is powered by several dominant
nuclides with different lifetimes. Each dominant nuclide has a characteristic
time when it plays a crucial role in the radioactive heating of the ejecta.
For example, the radioactive decay of 129Te generates over one-third of the
total $\beta$-particles energy at $\sim 0.35$ day. Therefore, the kilonova
emission at $\sim 0.35$ day may sensitive to 129Te since it produces the most
significant energy at that moment. This suggests that it may be possible to
distinguish the dominant nuclides from kilonova light curves when the
contributions of other nuclei are relatively smaller or subdominant. It should
be noted that the contributing nuclides may depend on the expansion models and
nuclear ingredients adopted.

The dominant nuclides for the total $\gamma$-rays energy are also identified,
as shown in the middle panel of Figure 4. It is found that the top
contributing nuclides of the total $\gamma$-rays energy are almost common
among all four models studied, including 127Sb, 128Sb, 129Sb, 130Sb, and 132I.
These top five nuclides contribute to more than 20% of the total $\gamma$-rays
energy at their characteristic time. For nuclides with characteristic time
$t_{\rm c}\gtrsim 1$ day, their $\gamma$-rays may escape from ejecta without
thermalization and form a $\gamma$-ray transient, since merger ejecta becomes
transparent to $\gamma$-rays at about a day after merger. Hence, the
radioactive nuclei of 128Sb, 129Sb, and 130Sb with characteristic times
$t_{\rm c}\lesssim 1$ day could be distinguished from kilonova light curves,
but 127Sb and 132I with characteristic times $t_{\rm c}\gtrsim 1$ day could be
directly identified by MeV gamma-ray detectors. It should be noted that the
identification of gamma-ray lines produced by radioactive decay of heavy
elements may be possible only for quite rare, nearby events (Hotokezaka et
al., 2016; Li, 2019; Chen et al., 2021, 2022).

The energy fraction of the dominant $\alpha$-decay chains that contribute
significantly to the total heating rate is shown in the bottom panel of Figure
4. These $\alpha$-decay chains are responsible for the late-time heating rate
due to they liberate more energy per decay and thermalize with greater
efficiency than $\beta$-decay products. For instance, the $\alpha$-decay 225Ac
(with an $\alpha$-decay half-life of $10.0$ days, originating from the
$\beta$-decay of 225Ra with a $\beta$-decay half-life of $14.9$ days) releases
19.32 MeV, which is significantly higher than the energy released by the
$\beta$-decay process. Since no other radioactive nuclei can release a
comparable amount of energy within this timescale, 225Ac is the dominant
contributor to the late-time kilonova emission. Furthermore, the initial
conditions for $r$-process calculations provided by numerical relativity
simulations (Radice et al., 2018) are neutron-rich enough to produce a
significant amount of 225Ac. This suggests that precise measurement of future
kilonova late-time light curves could possibly be used to infer the presence
of 225Ac.

Figure 5: The sensitivity of kilonova light curve to the dominant nuclides.
Here $L_{\rm bol}$ represents the bolometric luminosity of kilonova emission,
and $\Delta L$ represents the increase in luminosity caused by the doubling of
the abundances of specific heavy elements.

To explore the sensitivity of kilonova emission to the dominant nuclides, we
calculate the ratio of luminosity increment ($\Delta L$) caused by the
doubling the abundances of specific heavy elements, as shown in Figure 5. It
is found that the early-time ($t\lesssim 10$ days) kilonova light curve
exhibits weak dependence on dominant nuclides, making it challenging to
identify heavy elements contributing to the early-time light curve. However,
at later times ($t\gtrsim 20$ days), kilonova emission becomes inceasingly
sensitive to 225Ac, indicating that the late-time kilonova light curve
strongly depends on the yield of 225Ac. This feature can provide an important
diagnostic for the nuclide composition in neutron star mergers. If the late-
time observation up to 100 days after merger can be detected with sensitive
instruments like the JWST, 225Ac could be distinguished from the bolometric
light curve.

Figure 6: K-band light curves with and without the contributions from 223Ra
(top panel) or 225Ac (bottom panel) for a merger event assumed to be at a
distance of $40$ Mpc, similar to that of GW170817/AT2017gfo. The observed data
of AT2017gfo are obtainted from the combined data set (Villar et al., 2017).
Blue solid lines represent the kilonova light curves obtained from $r$-process
nucleosynthesis simulations, while red solid lines represent the light curves
where the abundances of specific nuclides have been multiplied by a factor of
2.

The kilonova AT2017gfo provides a solid case for studying the nulide
composition of merger ejecta, and observations in the near-infrared band have
been obtained by the Gemini Observatory and the Hubble Space Telescope, as
shown in Figure 6. The near-infrared light curves can be fit by the kilonova
model except for the data at $\sim 20$ days (blue solid lines). As discussed
earlier (see Figure 5), at around $20$ days, the kilonova emission is
dominated by two $\alpha$-decay nuclei, 223Ra (with an $\alpha$-decay half-
life of $11.43$ days) and 225Ac. In simulations of $r$-process
nucleosynthesis, the resulting abundances of 223Ra and 225Ac are $1.9\times
10^{-5}$ and $2.3\times 10^{-5}$, respectively. The gray dashed lines in
Figure 6 show the scenario where there is no contribution from either 223Ra or
225Ac, resulting in the kilonova light curve in the near-infrared band dimming
by almost one or two magnitudes at $\sim 20$ days after the merger,
respectively. To explain the late-time near-infrared data of the kilonova
AT2017gfo, the required yield of 223Ra or 225Ac in the merger ejecta must be
multiplied by at least a factor of $\sim 2$ to produce the observed bolometric
luminosity (red solid lines). This suggests that the late-time feature in the
kilonova AT2017gfo at $\sim 20$ days indicates the production of 223Ra or
225Ac at an abundance level of $\sim 10^{-5}$, which is consistent with the
results of Wu et al. (2019). However, distinguishing the contribution of 225Ac
from that of 223Ra is challenging due to their similar contributions at $\sim
20$ days. Further observations in the near-infrared band are necessary to
confirm the individual heavy elements responsible for the late-time kilonova
light curves. Note that we employ the assumption of local thermal equilibrium
in our analysis, which is reasonable for modeling kilonova light curve during
the early stages. However, the kilonova emission transitions into a non-local
thermal equilibrium in the later stages (Pognan et al., 2022; Pognan et al.,
2023; Hotokezaka et al., 2023).

Figure 7: Multi-band light curves for neutron star mergers. The ejecta mass is
set to be $0.04M_{\odot}$ and the velocity is set to be $0.15c$, which are
consistent with those inferred from the red kilonova component of AT2017gfo
(Kasen et al., 2017). The merger is assumed to occur at a distance of 200 Mpc.
The dashed lines represent the approximate observation limits of the NIRCam
instruments with a $10^{4}$ s exposure time.

The JWST is expected to be a powerful instrument for detecting and studying
kilonovae, particularly in the near-infrared band. In Figure 7, we compare the
resulting multi-band light curves to the sensitivity of JWST. Here the total
ejecta mass is set to be $0.04M_{\odot}$ and the characteristic ejecta
velocity is set to be $0.15c$, which are consistent with those inferred from
the red kilonova component of AT2017gfo (Kasen et al., 2017). The merger is
assumed to occur at a distance of $200$ Mpc, which represents the average
reach of the Advanced LIGO/Virgo gravitational wave detectors during the
fourth observing run. Adopting the limits of NIRCam’s wide-band filters from
the JWST website111https://jwst-docs.stsci.edu/jwst-near-infrared-
camera/nircam-performance/nircam-sensitivity, it is shown that kilonovae can
be detectable by JWST up to $\sim 50$ days. If a merger event occurs at a
closer distance, such as the GW170817/AT2017gfo event at $40$ Mpc, JWST could
potentially detect the corresponding kilonova up to $\sim 100$ days.
Therefore, the heavy element 225Ac that make a dominant contribution to the
late-time kilonova can be distinguished in the future observations with JWST.

Figure 8: Temporal evolution of the abundance of the spontaneous fission
nucleus 254Cf.

We should emphasize that our calculations do not included the extra
radioactive energy generated by fission fragments due to limited radiation
data available for fission processes in the EDNF library. Zhu et al. (2018)
suggested that the presence of the spontaneous fission nucleus 254Cf could
potentially impact the late-time kilonova light curves, competing with the
$\alpha$-decay of 225Ac. Figure 8 shows the temporal evolution of the
abundance of spontaneous fission nucleus 254Cf. The resulting abundance of
254Cf in our $r$-process simulations spans the range of $\sim 10^{-7}$ to
$10^{-9}$. It is found that a lower electron fraction leads to a higher
production of 254Cf. By taking into account the initial conditions of
$r$-process simulations presented in Radice et al. (2018), the abundance of
254Cf is consistently lower than the value ($\sim 10^{-6}$) reported by Zhu et
al. (2018) by at least one order of magnitude. This is reasonable because the
initial conditions (electron fraction $Y_{\rm e}\simeq 0.15$) provided by
Radice et al. (2018) are neutron-rich enough to generate actinides but not
sufficient to produce a significant amount of 254Cf (Wu & Banerjee, 2022).
Therefore, in our calculations, the contribution of the spontaneous fission
nucleus 254Cf to the overall radioactive heating rates is relatively small
compared to that of $\alpha$-decay chains. We also note that the choices of
fission barriers and nuclear mass models may influence the production of 254Cf
in $r$-process simulations (Goriely, 2015; Barnes et al., 2016; Wanajo, 2018;
Vassh et al., 2019; Zhu et al., 2021).

##  4 Conclusions and Discussions

Based on simulations of $r$-process nucleosynthesis, we have studied the
energy fractions of dominant nuclides that contribute significantly to the
energy generation of $\alpha$-particles, $\beta$-particles, and $\gamma$-rays.
We have identified the heavy elements that dominate the kilonova emission
within $\sim 100$ days, including 127Sb, 128Sb, 129Sb, 130Sb, 129Te, and 132I.
The $\alpha$-decay chains responsible for the late-time radioactive heating
rate are also identified, including 222Rn, 223Ra, 224Ra, and 225Ac. These
nuclide species could be distinguished from the kilonova light curves in
future observation with sensitive instruments such as JWST. It is found that
the late-time kilonova light curve ($t\gtrsim 20$ days) is highly sensitive to
the yield of 225Ac (with an $\alpha$-decay half-life of $10.0$ days,
originating from the $\beta$-decay of 225Ra with a $\beta$-decay half-life of
$14.9$ days), making it the most promising heavy elements to be identified in
upcoming kilonova observations. Our results are consistent with previous
studies (Metzger et al., 2010; Lippuner & Roberts, 2015; Wanajo, 2018; Wu et
al., 2019). We have also derived the characteristic time of dominant nuclides
when they play a crucial role in the radioactive heating rate, which provides
clues for identifying specific heavy elements in the late-time photometry.

The neutron star merger rate inferred from the LIGO/Virgo gravitational wave
detectors is $R_{\rm BNS}=1540^{+3200}_{-1220}$ Gpc-3 yr-1 (Abbott et al.,
2017a), and thus JWST is expected to detect $\sim 52^{+107}_{-41}$ kilonova
events per year following gravitational wave triggers. Recent results from the
third observing run of LIGO/Virgo suggest a lower merger rate density of
$R_{\rm BNS}=320^{+490}_{-240}$ Gpc-3 yr-1 (Abbott et al., 2021), which is
broadly consistent with the predictions obtained from population synthesis
models (Belczynski et al., 2020). Kilonovae are also expected to be discovered
following short duration gamma-ray bursts (GRBs), which are commonly believed
to originate from compact binary mergers (Tanvir et al., 2013; Abbott et al.,
2017c). Rastinejad et al. (2021) have demonstrated that JWST is capable of
detecting kilonova events following short GRBs at redshifts $z\lesssim 1$,
which increases the expected rate of detectable kilonovae. The LIGO/Virgo
network can provide a medium localization accuracy of 10 deg2.To cover a
localized sky area of 10 deg2 uniformly, JWST would require $\sim 50$ hours of
total exposure time due to its limited field of view (9.7 arcmin2). However,
Bartos et al. (2016) proposed that the total search time could be reduced to
12.6 hours by focusing on known galaxies within 200 Mpc, since most neutron
star mergers are expected to occur in or near galaxies. The search time could
be further decreased by concentrating on galaxies with high star-formation
rates that are more likely to host kilonovae. Therefore, the JWST will be able
to rapidly locate kilonovae, enabling us to search for specific heavy elements
in the early or late phases of the kilonova. If the radioactive energy from
fission fragments is included in the heating rate calculations, the resulting
kilonova would be much brighter than the one without fission process.
Consequently, it would be more easier to detect the kilonova in future
observations.

## Acknowledgements

We thank Li-Xin Li for fruitful discussion and an anonymous referee for
helpful comments. This work was supported by the National Natural Science
Foundation of China (Grant No. 12133003).

ORCID iDs

Meng-Hua Chen: https://orcid.org/0000-0001-8406-8683

En-Wei Liang: https://orcid.org/0000-0002-7044-733X

## Data Availability

The data underlying this paper can be shared on reasonable request to the
corresponding author.

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# A simple imaging solution for chip-scale laser cooling

A. Bregazzi  SUPA and Department of Physics, University of Strathclyde, G4
0NG, United Kingdom  P. F. Griffin  SUPA and Department of Physics, University
of Strathclyde, G4 0NG, United Kingdom  A. S. Arnold  SUPA and Department of
Physics, University of Strathclyde, G4 0NG, United Kingdom  D. P. Burt  Kelvin
Nanotechnology, University of Glasgow, G12 8LS, United Kingdom  G. Martinez
University of Colorado, Department of Physics, Boulder, Colorado 80309, USA
National Institute of Standards and Technology, Boulder, Colorado 80305, USA
R. Boudot  FEMTO-ST, CNRS, 26 Chemin de l‘Epitaphe, 25030 Besançon, France  J.
Kitching  National Institute of Standards and Technology, Boulder, Colorado
80305, USA  E. Riis  SUPA and Department of Physics, University of
Strathclyde, G4 0NG, United Kingdom  J. P. McGilligan  SUPA and Department of
Physics, University of Strathclyde, G4 0NG, United Kingdom  Kelvin
Nanotechnology, University of Glasgow, G12 8LS, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

We demonstrate a simple stacked scheme that enables absorption imaging through
a hole in the surface of a grating magneto-optical trap (GMOT) chip, placed
immediately below a micro-fabricated vacuum cell. The imaging scheme is
capable of overcoming the reduced optical access and surface scatter that is
associated with this chip-scale platform, while further permitting both
trapping and imaging of the atoms from a single incident laser beam. The
through-hole imaging is used to characterise the impact of the reduced optical
overlap volume of the GMOT in the chip-scale cell, with an outlook to an
optimised atom number in low volume systems.

††preprint: AIP/123-QED

Micro-fabricated physics packages based on the measurement of thermal atomic
ensembles have been integrated in metrological instruments, ranging from
magnetometers to interferometers and clocks Knappe  _et al._ (2004); Kitching
(2018); Carlé  _et al._ (2021). Most such instruments rely on spin transitions
in atoms, and require the presence of buffer gas mixtures or cell wall
coatings to reduce the relaxation rate due to wall collisions. For clocks, the
presence of these buffer gases causes temperature dependent frequency offsets
and require careful temperature stabilisation to achieve good medium- and
long-term stability Kozlova, Guérandel, and de Clercq (2011); Hasegawa  _et
al._ (2011); Seltzer and Romalis (2009). The use of cold atom ensembles avoids
these difficulties and can result in an increased interaction time and
improved absolute accuracy compared to their thermal atom counterparts.

While the transition to cold atoms offers clear advantages over thermal atom
packages Eckel  _et al._ (2018), the additional experimental size and
complexity associated with laser cooling has limited the deployability and
application range of cold atom devices Rushton, Aldous, and Himsworth (2014);
Marlow and Scherer (2021); Muquans . Significant efforts have been made on the
miniaturisation of cold atom components to facilitate portability for next-
generation atomic sensors Burrow  _et al._ (2021); Little  _et al._ (2021);
Nichols  _et al._ (2020); Nshii  _et al._ (2013); Ravenhall, Yuen, and Foot
(2021); Kang  _et al._ (2019); McGehee  _et al._ (2021); Sitaram  _et al._
(2020a). However, the fabrication complexity of many of these components
remains unfavourable for mass production, preventing their adoption in
commercial applications.

Recent work has demonstrated a dramatic reduction of the cooling platform to
the chip-scale by combining an anodically bonded glass-silicon-glass vapour
cell with a diffractive optical element that redirects a single incident beam
into the components required for laser cooling McGilligan  _et al._ (2020).
However, such systems have demonstrated that the detection of cold atoms is
made difficult by the reduced optical access of the cell and light scattering
from the grating and cell surfaces. As such, the authors required adopting a
non-trivial two-photon spectroscopy scheme for improved detection, greatly
increasing both the size and complexity of the optical system McGilligan  _et
al._ (2020); Sheludko  _et al._ (2008).

In this letter we demonstrate a simple imaging solution for chip-scale laser
cooling platforms. The stackable structure of the apparatus provides
simplicity in alignment, as well as enabling future scalability for device
mass production. A hole is laser cut in the centre of the grating chip to
enable on-axis absorption imaging from the cooling beam without significantly
degrading atom trapping. Although grating chips with a central hole have been
used to reduce surface reflections Imhof  _et al._ (2017), as a source for
cold electrons Franssen  _et al._ (2019), and for Zeeman slowing with an
alkali Barker  _et al._ (2019) and alkaline-earth metal source Sitaram  _et
al._ (2020b), the impact of the reduced grating surface area on the MOT number
has not yet been quantified or used as an imaging axis. The ability to measure
the trapped atom number in such a chip-scale system is used to characterise
the impact of the 3 mm internal height constraint on the optical overlap
volume, with an outlook to an optimised atom number.

A simplified schematic for the cooling and imaging of 87Rb is shown in Fig 1.
The incident light is derived from a single volume-Bragg-grating laser (VBG)
frequency stabilised using saturated absorption spectroscopy. A double-pass
acousto-optic modulator (AOM) shifts the light frequency to be approximately 8
MHz red detuned from the $780$ nm $D_{2}$ $F=2\rightarrow F^{\prime}=3$
cycling transition, while also enabling frequency and intensity control for
the imaging process. Re-pumping is achieved by modulating a free-space
electro-optic modulator (EOM) at 6.5 GHz to generate 5 $\%$ sidebands on the
carrier frequency. The light is then fibre coupled into a single-mode,
polarization maintaining fibre and passed to the cooling platform.

From the fibre, we expand an $\approx$30 mW beam to a $1/e^{2}$ radius of
$\approx 1.6$ cm to flatten the intensity distribution at the grating surface.
The trap beam is then circularly polarised with a quarter wave-plate and
aligned onto the grating chip, mounted externally to the actively pumped chip-
scale cell. A pair of anti-Helmholtz coils are used to produce a trapping
field within the cell volume, with an axial gradient of $\approx$1.5 mT/cm (15
G/cm).

Figure 1:  Experimental set-up. CCD: charge coupled device, AHC: anti-
Helmholtz coils, $\lambda/4$: quarter-wave plate. (a): Image of the grating
chip with a central hole removed. (b): An on-axis, through-hole absorption
image of $\approx$ 2$\times$105 atoms within the chip-scale platform. This
image was taken with a peak beam intensity of 115 $\mu$W/cm2 at zero detuning.

The cell, shown in Fig. 1, is composed of a $3$ mm thick silicon wafer,
sandwiched between two anodically bonded aluminosilicate glass wafers of 0.7
mm thickness Dellis  _et al._ (2016). The cell was fabricated with a square
central region of dimensions 2.5 cm$\times$2.5 cm, to enable cooling with a 2
cm wide grating chip. The upper window of the cell is mechanically drilled
with a 5 mm hole to enable active pumping from an ion pump through a copper
pinch-off tube, adhered to the upper cell surface. Square cavities are cut in
the silicon walls to house non-evaporable getters (NEGs) for future passive
pumping measurements.

The grating chip, shown in Fig. 1 (a), has a 1100 nm period in a tri-segment
geometry. The central region of the chip is laser cut to provide a through-
hole axis for absorption imaging. This central region of the grating plays a
limited role in forming the trap overlap volume while also providing a
unfavourable zeroth order reflection, such that its removal has a minimal
impact on atom number. A two-lens imaging telescope and camera are placed
behind the grating hole to detect atomic absorption from the GMOT.

While extracting atom number with the through-hole system, fluorescence
measurements from the MOT loading curves were simultaneously measured using a
separate detection telescope with a spatially selective focal plane (described
in Ref. Boudot  _et al._ (2020)) at an angle of around 30° to the grating
surface (not shown in Fig. 1). The loading curves were used for the extraction
of background pressure at the MOT location using the relation between MOT
lifetime and background rubidium pressure Arpornthip, Sackett, and Hughes
(2012); Moore  _et al._ (2015); Burrow  _et al._ (2021), while also checking
the validity of the through-hole measured atom number. We note that the atom
number measured from fluorescence imaging at this angle well matched the atom
number extracted from conventional orthogonal fluorescence imaging.
Additionally, an average ratio of 1.6$\pm$0.4 was observed between the
calculated atom number from the through-hole absorption and orthogonal
fluorescence during our data sets. However, the alignment of the angled
fluorescence imaging axis was complicated due to the surface scatter and
diffracted orders further restricting the available imaging angles and
contributing to an increased noise level on the atom number extraction. The
complexity in aligning the camera at a position and angle that minimises the
impact of surface scatter does not meet the needs of a simple, mass-producible
device.

The experimental procedure is initiated by turning the trap coils on, followed
by a 500 ms loading time at $8$ mW/cm2 peak intensity. The 500 ms load time
enables resolving the vacuum pressure down to 1.4 $\times$10-6 Pa
(1.4$\times$10-8 mbar) McGilligan  _et al._ (2017). The trap coils are then
turned off while concurrently decreasing the incident beam below saturation to
around 115 $\mu$W/cm2 and bringing the frequency on resonance. This serves the
dual purpose of reducing image distortion due to diffraction effects within
the atom cloud while also maximising the signal contrast. We note that the
measured atom number from this method did not differ with the addition of a
static magnetic field along the imaging axis. An initial image $I_{1}$ is
taken with the MOT present for an exposure time of 25 $\mu$s. We then wait 100
ms such that the trapped atoms are no longer present before taking a second
image $I_{2}$. Finally the trap beam is turned off with an extinction ratio of
62 dB and a dark background image $I_{3}$ is acquired. These three images are
then processed in terms of optical density (OD) using the equation
$\rm{OD}=\rm{ln}\Big{(}\frac{\textit{I}_{1}-\textit{I}_{3}}{\textit{I}_{2}-\textit{I}_{3}}\Big{)}$.
An objective lens of focal length $f_{1}=15$ mm and an imaging lens $f_{2}=30$
mm, arranged in $2f_{1}:2f_{2}$ configuration, is used to image the atoms,
with the objective placed directly behind the hole laser cut in the centre of
the grating. This arrangement of lenses provides an improved signal-to-noise
ratio for data extraction Smith  _et al._ (2011).

In the present work, the stretched state saturation intensity
$I_{\rm{sat}}$=1.67 mW/cm2 was used for calculating the trapped atom number
from the fluorescence and absorption methods, as is consistent with previous
publications of our group Nshii  _et al._ (2013). Additionally, we found good
agreement between the calculated atom numbers from the two imaging methods
with the stretched state saturation intensity, compared to the average of all
polarizations and magnetic sub-levels, $I_{\rm{sat}}$=3.57 mW/cm2. A
saturation parameter, $S=\frac{I}{I_{\rm{sat}}}$, sums over the intensities of
the single input trap beam and the three diffracted beams from the grating to
account for the total intensity the atoms experience. This inclusion modifies
$S$ such that
$S=(1+n\eta_{1}\sec\theta)\frac{I_{\rm{im}}}{I_{\rm{sat}}}\approx\frac{2.4I_{\rm{im}}}{I_{\rm{sat}}}$,
where $I_{\rm{im}}$ is the imaging light intensity, $n$ is the number of
diffracted first orders interacting with the atoms, $\eta_{1}$ is the
efficiency of the first diffracted order, and $\theta$ is the angle of
diffraction. An example of the obtained MOT image is shown in Fig. 1 (b),
where 2$\times 10^{5}$ atoms are trapped in the chip-scale cell.

Our investigation of the impact of removing the central region of the grating
chip was initially carried out in a conventional glass vacuum chamber ($2.5$
cm$\times 2.5$ cm$\times 10$ cm with 1 mm thick glass walls) using
fluorescence imaging, with the results shown in Fig. 2. We fabricated 5
identical grating chips, and laser cut the central regions for a hole size
ranging from 1 mm to 5 mm. For each subsequent measurement of the atom number,
the grating chip was carefully implemented immediately below the same vacuum
cell, under the same conditions as the previous grating chip. We found that a
hole diameter up to 3 mm did not significantly degrade the measured atom
number, shown in Fig. 2 (a), due to the minimal impact that such a hole size
has on the total optical overlap volume, shown in Fig. 2 (b). The theoretical
optical overlap volume is numerically integrated for a 2 cm beam diameter,
incident upon a 1100 nm period tri-segmented grating chip with a central hole
of varying diameter. As the grating hole was increased to 4 mm, the atom
number decreases, reducing to the point that no MOT was detected for a 5 mm
hole diameter, even with additional alignment of the MOT parameters. The atom
number drops faster than would be expected from the decrease in the overlap
volume, likely due to the additional sensitivity of the radiation pressure
balance from the grating chip. This is emphasized by the red curve shown in
Fig. 2 (a), where the theoretically calculated overlap volume is scaled to the
atom number through $N\propto V^{1.2}$ and normalised to the measured atom
number for the 1 mm hole. For a grating hole diameter of 4 mm, a factor of
$\sim$2 difference between the expected and measured atom number is observed.
However, the increased hole diameter aids an improved numerical aperture, also
plotted in Fig. 2 (b) for a MOT position of 2 mm above the grating surface,
such that a larger hole diameter is favourable for imaging with the small atom
numbers that have been observed in passively pumped vacuum cells Boudot  _et
al._ (2020). Taking this into account, a 3 mm hole diameter was selected for
through-hole imaging.

Figure 2:  Critical parameters as a function of the grating centre hole
diameter for an incident 2 cm beam diameter. (a): The measured atom number
from orthogonal fluorescence imaging in the conventional glass cell. The
measured error bars are smaller than the shown data points. The red curve
shows the normalized $N\propto V^{1.2}$ for the overlap volume shown in (b).
(b): Theoretically calculated optical overlap volume and numerical aperture
from the grating chip assuming that the grating hole is the aperture stop of
the imaging system.

Once the through-hole imaging was well aligned and in focus, the glass cell
was replaced by the actively pumped chip-scale vacuum cell. The chip-scale
cell was pumped down to an initial pressure of 10-6 Pa (10-8 mbar), measured
using an ion pump. The validity of the ion pump reading was later verified
with the pressure calculated from MOT loading curves. Following pump down, a
resistively heated dispenser within the larger vacuum chamber was used to
provide a moderate rubidium density. With the grating overlap volume aligned
within the chip-scale cell, an image of the cold atoms was extracted from the
through-hole scheme, highlighted in Fig. 1. The initial atom number
measurement yielded a total of 2$\times 10^{5}$ atoms, which was an order of
magnitude lower than what had been observed for the glass cell.

To investigate the cause of the reduced atom number, the diameter of the
incident cooling light onto the grating surface, $d$, was reduced while
measuring the trapped atom number for both the glass and chip-scale cells.
This measurement provides an insight to the impact of the reduced vacuum
volume of the chip-scale cell on the grating chip’s optical overlap volume,
and therefore trapped atom number. During this process a comparable background
vapour density of $\approx 3.1\times 10^{6}$ cm-3 and total pressure of
$\approx 3\times 10^{-5}$ Pa ($\approx 3\times 10^{-7}$ mbar) was maintained
in both vacuum systems. The resulting data set is shown in Fig. 3 (a) in blue
and red for the glass and chip-scale cells, respectively.

At the largest values of $d$ an order of magnitude difference in the atom
number is observed between the glass and chip-scale cell, with the glass cell
data reaching a maximum around $10^{6}$ atoms, in line with previous atom
numbers measured with a similar incident intensity McGilligan  _et al._
(2015). Concurrently, the red data set extracted from the chip-scale cell
reaches a maximum around 105 atoms. When $d$ is reduced from this maximum atom
number, there is initially no impact on the optical overlap volume, since the
majority of the overlap volume is outside the cell’s internal thickness and
therefore cannot play a role in the cooling process. As $d$ is decreased below
$\approx 17$ mm, the overlap volume is reduced below the internal height of
the chip-scale cell, contributing to a degradation of the atom number.

The impact of the reduced optical overlap volume is emphasised by the
theoretical atom number expected from the grating MOT as a function of the
incident beam diameter, shown with solid curves in Fig. 3 (a). This atom
number, $N$, is derived from a numerical integration of the optical overlap
volume, $V$, for a $1100$ nm period linear grating, where
$N\,\propto\,V^{1.2}\,\propto\,d^{3.6}$ Nshii  _et al._ (2013). The red curve
represents the overlap volume from the grating chip with a 3 mm diameter hole
at its centre. A vertical restriction is placed on the integration to account
for the 3 mm thickness of the silicon frame. The overlap volume is then
converted to a normalised atom number with a maximum corresponding to the
largest measured atom number from the glass cell. A total potential overlap
volume of $0.23$ cm3 is possible for a 2 cm diameter incident upon a grating
chip with a 3 mm diameter hole, held 1 mm from the cell window. However, when
the grating is mounted similarly for the chip-scale cell, only 21$\%$ of the
relative overlap volume coincides within the cell vacuum volume. This process
is illustrated in Fig. 3 (b)-(d), where the numerically calculated geometry of
the optical overlap volume is shown for different values of $d$. The region of
the overlap volume occupied by the 3 mm tall chip-scale cell is highlighted to
show the clipping of the overlap volume as a function of $d$.

The dashed black line in Fig. 3 shows a $d^{3.6}$ scaling of the atom number,
expected for a constant intensity of the incident beam Lindquist, Stephens,
and Wieman (1992); Gibble, Kasapi, and Chu (1992) that expands in three-
dimensions. Previous studies have demonstrated that significantly smaller
overlap volumes for 6-beam MOTs Hoth, Donley, and Kitching (2013) and pyramid
MOTs Pollock  _et al._ (2011) are reduced to a $d^{6}$ scaling, as the atom
number is then limited by the stopping distance of the overlap volume rather
than the cooling light intensity that dominates the process for larger beam
diameters. We note that the measured atom number in the chip-scale cell is
lower than the theoretical $V^{1.2}$, shown as a solid red curve. The cause of
this is currently unknown and will be investigated in future studies.

When limited by atom shot noise, the stability of cold atom sensors scales
inversely with the square-root of the atom number. In this case, an order of
magnitude improvement in the chip-scale cell would be favourable. Expanding
the overlap volume simulation to a 5 mm thick silicon frame enables a $71\%$
overlap between the optical overlap volume and the vacuum volume of the chip-
scale cell when the grating is held 1 mm from the cell lower window. Using
identical scaling parameters to the 3 mm chip-scale cell curve shown in Fig. 3
(a) results in an achievable atom number of 106 atoms for a 2 cm incident beam
diameter in this thicker cell geometry. Thus, we plan to fabricate thicker
silicon cells in the near-future to enable a wider range of applications for
this chip-scale laser cooling platform. Additionally, a steeper angle of
diffraction with a two-dimensional grating geometry would provide an overlap
volume that is largest at the chip surface for an improved atom number within
the reduced vacuum volume compared to the tri-segmented chip used here. The
improved atom number will enable studies into passive pumping with NEGs in an
aluminosilicate based chip-scale cell Boudot  _et al._ (2020).

Figure 3: (a): Atom number measured from absorption imaging as a function of
the incident beam diameter, $d$. Blue/red data points represent measurements
made in a glass/chip-scale cell. Error bars were calculated from 5 subsequent
atom number measurements at each beam diameter. Solid curves are theoretical
expectations of the atom number, determined from numerical integration of the
optical overlap volume from the grating chip as a function of $d$. The
normalised theoretical atom number is pinned to the maximum measured atom
number for the glass cell. The red line shows an overlap volume that is
restricted to a 3 mm height due to the vacuum volume of the chip-scale cell.
The dashed line represents a $d^{3.6}$ fit to the atom number in the glass
cell. (b)-(d) illustrate the optical overlap volume of the grating chip as a
function of $d$ with a highlighted 3 mm height of the chip-scale cell. The
grey line highlights the grating chip position.

In conclusion, we have demonstrated a simple imaging solution for atom number
extraction in a chip-scale cooling platform. The laser cut hole in the grating
chip has shown a negligible atom number degradation, while enabling a
sufficient numerical aperture for imaging small atom numbers in future
experiments. We have shown that the 3 mm height of the micro-fabricated cell
reduces the achievable overlap volume, and hence atom number, when coupled
with a micro-fabricated grating chip. However, moving to the machining of
thicker silicon wafers will enable a work around to bring this platform to a
competitive stance for quantum technologies. In addition it may also be
advantageous to apply an anti-reflection coating to the cell windows during
fabrication due to the increased etaloning observed in the through-hole
images. While the effect of these fringes on the images was minimal,
especially for larger MOTs, and can be mitigated through alignment there are
also several fringe removal algorithms that can be applied in post-processing
to further reduce their impact Niu  _et al._ (2018); Xiong, Long, and Parker
(2020).

###### Acknowledgements.

The authors acknowledge funding from Defence Security and Technology
Laboratory, Engineering and Physical Sciences Research Council (EP/T001046/1),
and Defence and Security Accelerator. A. B. was supported by a PhD studentship
from the Defence Science and Technology Laboratory (Dstl). The authors
gratefully acknowledge W. R. McGehee and D. S. Barker (NIST) for the careful
reading of the manuscript before submission. Additionally, the authors would
like to thank B. Lewis and M. Himsworth for useful conversations.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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