# Black-Hole-to-Halo Mass Relation From UNIONS Weak Lensing

Qinxun Li CAS Key Laboratory for Research in Galaxies and Cosmology,
Department of Astronomy,
University of Science and Technology of China, Hefei, Anhui 230026, China
Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah
84102, USA  Martin Kilbinger Université Paris-Saclay, Université Paris Cité,
CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France  Wentao Luo CAS Key Laboratory
for Research in Galaxies and Cosmology, Department of Astronomy,
University of Science and Technology of China, Hefei, Anhui 230026, China  Kai
Wang The Kavli Institute for Astronomy and Astrophysics, Peking University,
Beijing, China  Huiyuan Wang CAS Key Laboratory for Research in Galaxies and
Cosmology, Department of Astronomy,
University of Science and Technology of China, Hefei, Anhui 230026, China
School of Astronomy and Space Science, University of Science and Technology of
China, Hefei 230026, China  Anna Wittje Ruhr University Bochum, Faculty of
Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for
Cosmological Lensing, 44780 Bochum, Germany  Hendrik Hildebrandt Ruhr
University Bochum, Faculty of Physics and Astronomy, Astronomical Institute
(AIRUB), German Centre for Cosmological Lensing, 44780 Bochum, Germany
Ludovic van Waerbeke  Department of Physics and Astronomy, University of
British Columbia, Vancouver, BC V6T 1Z1, Canada  Michael J. Hudson Department
of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada
Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON N2L
3G1, Canada  Perimeter Institute for Theoretical Physics, Waterloo, ON N2L
2Y5, Canada  Samuel Farrens Université Paris-Saclay, Université Paris Cité,
CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France  Tobías I. Liaudat IRFU, CEA,
Université Paris-Saclay, F-91191, Gif-sur-Yvette, France  Huiling Liu CAS Key
Laboratory for Research in Galaxies and Cosmology, Department of Astronomy,
University of Science and Technology of China, Hefei, Anhui 230026, China
Ziwen Zhang CAS Key Laboratory for Research in Galaxies and Cosmology,
Department of Astronomy,
University of Science and Technology of China, Hefei, Anhui 230026, China
Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-
sur-Yvette, France  Qingqing Wang  CAS Key Laboratory for Research in Galaxies
and Cosmology, Department of Astronomy,
University of Science and Technology of China, Hefei, Anhui 230026, China
Elisa Russier  Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM,
91191, Gif-sur-Yvette, France  Lawrence Berkeley National Laboratory, 1
Cyclotron Road, Berkeley, CA 94720, USA  Axel Guinot  Université Paris Cité,
CNRS, Astroparticule et Cosmologie, F-75013 Paris, France  Lucie Baumont
Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-
sur-Yvette, France  Fabian Hervas Peters  Université Paris-Saclay, Université
Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France  Thomas de Boer
Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu
HI 96822  Jiaqi Wang  Department of Astronomy, School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
<EMAIL_ADDRESS>
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###### Abstract

This letter presents, for the first time, direct constraints on the black-
hole-to-halo-mass relation using weak gravitational lensing measurements. We
construct type I and type II Active Galactic Nuclei (AGNs) samples from the
Sloan Digital Sky Survey (SDSS), with a mean redshift of $0.4$ $(0.1)$ for
type I (type II) AGNs. This sample is cross-correlated with weak lensing shear
from the Ultraviolet Near Infrared Northern Survey (UNIONS). We compute the
excess surface mass density of the halos associated with $36,181$ AGNs from
$94,308,561$ lensed galaxies and fit the halo mass in bins of black-hole mass.
We find that more massive AGNs reside in more massive halos. We see no
evidence of dependence on AGN type or redshift in the black-hole-to-halo-mass
relationship when systematic errors in the measured black-hole masses are
included. Our results are consistent with previous measurements for non-AGN
galaxies. At a fixed black-hole mass, our weak-lensing halo masses are
consistent with galaxy rotation curves, but significantly lower than galaxy
clustering measurements. Finally, our results are broadly consistent with
state-of-the-art hydro-dynamical cosmological simulations, providing a new
constraint for black-hole masses in simulations.

Galaxy dark matter halos(1880) — Gravitational lensing(670) — Galaxies(573) —
Supermassive black holes(1663) — Active galactic nuclei(16)

##  1 Introduction

Supermassive black holes (SMBHs), with typical masses of
$10^{6}-10^{10}M_{\odot}$, are among the most mysterious objects in the
Universe. It is widely accepted that most galaxies have an SMBH in their
center (Kormendy & Richstone, 1995). Though the formation and evolution of
SMBHs remain unclear, there is already a large amount of evidence indicating a
coevolution between SMBHs and their host galaxies, (see for a review Kormendy
& Ho, 2013). In addition, galaxy properties are expected and have been shown
to be closely related to their host dark-matter halos, as this is where they
form and evolve (Wechsler & Tinker, 2018, e.g.). These observational results
suggest that a close connection between halos, galaxies, and SMBHs needs to be
established to understand the coevolution of these different classes of
objects (Zhang et al., 2023b, a). The gravitational potential of a halo
determines the accretion of baryons and star formation of galaxies into the
halo. Several mechanisms in galaxies, such as bar instabilities, conduct cold
gas into galaxy centers, feeding the accretion of supermassive black holes.
The energetic feedback of the accretion can push baryons outside the galaxy or
even the halo, which will suppress the SMBH growth and star formation. Such
complex interplay among halos, galaxies, and SMBHs plays a crucial role in
galaxy formation and evolution and is still under exploration.

The first step towards understanding the connection between halos, galaxies,
and SMBHs is to build statistical relationships between these three types of
objects based on observational data. Much effort has been devoted to this
aspect in previous decades. The pioneering work was initiated by Dressler &
Richstone (1988), who noted a positive correlation between the black-hole mass
and the spheroid luminosity. Subsequent studies with more extensive data sets
found a tight correlation between black-hole mass $M_{\bullet}$ and various
galaxy properties, such as bulge mass and stellar velocity dispersion, across
several orders of magnitude (Magorrian et al., 1998; Ferrarese & Merritt,
2000; Gebhardt et al., 2000; Kormendy & Ho, 2013; Saglia et al., 2016). There
are also many studies on the galaxy-halo scaling relations, with the stellar
mass-halo mass relation, see Yang et al. (2008) as a representative example.

The relation between SMBHs and their host halos has yet to be extensively
studied. Ferrarese (2002) used the maximum rotational velocity of late-type
galaxies, $v_{\rm c}$, as a tracer of the halo mass, and the central velocity
dispersion, $\sigma_{*}$, of their bulges as a tracer of the black-hole mass.
This led to the first measurement of the $M_{\bullet}$-$M_{\rm h}$ relation.
This relation for galaxies was further confirmed in larger samples (Baes et
al., 2003; Pizzella et al., 2005; Volonteri et al., 2011) with a similar
method, which used $\sigma_{*}$ and $v_{\rm c}$ as tracers of black-hole and
halo mass. Sabra et al. (2015), Davis et al. (2019), and Marasco et al. (2021)
used direct dynamical black-hole mass instead of $\sigma_{*}$ and found a
correlation between SMBH mass and dynamical halo mass. These works based on
dynamics are limited to small galaxy samples and rely on strong assumptions
about the kinematic state of the gas, and the density profile of the dark
matter halo. There is also evidence for the opposite idea that the SMBH mass
does not correlate with halo mass, as seen in bulgeless galaxies (Kormendy &
Bender, 2011).

Unlike quiescent SMBHs in normal galaxies, Active Galatic Nuclei (AGN) are
SMBHs that are actively accreting matter. The trigger, growth, and feedback of
AGNs are critical issues in the halo-galaxy-SMBH connection. For the
$M_{\bullet}$-$M_{\rm h}$ relation in AGN samples, the halo mass in different
bins of $M_{\bullet}$ is typically inferred from the spatial two-point
correlation function of AGNs together with empirical models such as Halo
Occupation Distribution (HOD) and abundance matching (e.g. Krumpe et al.,
2015; Powell et al., 2018; Shankar et al., 2020; Powell et al., 2022; Krumpe
et al., 2023). Using gas dynamics, the $M_{\bullet}$-$M_{\rm h}$ relation has
been measured from $z=0$ to $z=6$ using reverberation-mapping and virial
black-hole masses (Robinson et al., 2021; Shimasaku & Izumi, 2019). More
massive AGNs were more likely found in more massive halos. However, the
methods used to estimate the halo mass in the works listed above are indirect
and strongly model-dependent.

Gravitational lensing, an effect directly related to the density field, has
been emerging as the most direct and clean method to measure halo mass
(Mandelbaum et al., 2006; Luo et al., 2018). For galaxies, Bandara et al.
(2009) and Zhang et al. (2023c) inferred black-hole masses from the
$M_{\bullet}$-$\sigma_{*}$ relation, and measured halo masses with strong
lensing and weak lensing, respectively. Their results significantly differ
from the $M_{\bullet}$-$M_{\rm h}$ relation from AGN clustering. Previous
weak-lensing studies on AGNs focused on the $M_{*}$-$M_{\rm h}$ relation,
using samples with limited size (Mandelbaum et al., 2009; Leauthaud et al.,
2015; Luo et al., 2022). In this work, we use, for the first time, weak
lensing to constrain the AGN $M_{\bullet}$-$M_{\rm h}$ relation. We utilize
the Sloan Digital Sky Survey (SDSS) AGN sample together with the galaxy shape
catalog derived from the Ultraviolet Near Infrared Optical Northern Survey
(UNIONS) imaging data, achieving a high signal-to-noise ratio measurement of
the $M_{\bullet}$-$M_{\rm h}$ relation.

This paper is organized as follows: Sect. 2 introduces the AGN lens samples
and weak-lensing galaxy shape catalogs, Sect. 3 presents our methodology,
before we show and discuss our results in Sect. 4.

Throughout this work, we assume a Planck18 cosmology (Planck Collaboration et
al., 2020).

##  2 Data

###  2.1 Lens sample

In this work, we construct type I and type II AGN/quasar samples as our lens
samples based on three SDSS spectroscopic cataloguess, described in the
following sections.

Figure 1: Joint and marginalised redshift and black-hole mass distribution of
the type I samples. Low, median, and high black-hole mass bins are shown in
blue, green, and red, respectively. We use the redshift-distribution weights
$w_{\rm l,nofz}$ and the lensing efficiency weights
$(\overline{\Sigma^{-1}_{\textrm{cr}}})^{2}$ for the 1D and 2D distributions.

####  2.1.1 SDSS type I AGNs

Based on the SDSS DR16 quasar catalog (Lyke et al., 2020), Wu & Shen fitted
the spectrum of $750,414$ quasars in the redshift range $0.1<z<6$ and measured
virial black-hole masses. As an update to Shen et al. (2011), they used the
FWHM of H$\beta$, Mg II, and C IV broad emission lines (combined with the
broad-line-region radius inferred from continuum luminosity) for their
estimates. Here, we adopt their black-hole masses based on $H\beta$. The mean
statistical error in $\log M_{\bullet}$ is much smaller than the systematic
error of the virial black-hole mass ($\sim 0.4$ dex, see Shen, 2013).

As a complement to Wu & Shen (2022) at low black-hole masses, we use the AGN
catalog from Liu et al. (2019), a complete AGN sample including both quasars
and Seyfert galaxies from SDSS DR7. Black-hole masses are measured with
H$\alpha$ and H$\beta$, and we adopt the $H\beta$ mass. This catalog contains
$14,584$ AGNs at $z<0.35$.

We merge the two catalogs and remove duplicate objects. As a consistency
check, we compare the fiducial black-hole mass of duplicate objects from both
catalogs and find no significant systematic bias. In this work, we use AGNs
with redshifts $0.05<z<0.6$ in the overlapping sky region between SDSS and
UNIONS, resulting in $14,649$ lenses, three times larger than the sample size
of previous type I AGN weak lensing studies (Luo et al., 2022). We divide the
sample into low ($\log M_{\bullet}/M_{\odot}<7.9$), median ($7.9<\log
M_{\bullet}/M_{\odot}<8.5$) and high ($\log M_{\bullet}/M_{\odot}>8.5$) black-
hole mass bins. We introduce a weight $w_{\rm l,nofz}$ such that the weighted
redshift distributions of the low and median mass bins equal the high-mass
bin. This allows for a fair comparison between the mass bins free of redshift
evolution or selection biases. The weighted distributions of the three bins
are shown in Fig. 1.

####  2.1.2 SDSS type II AGNs

In addition to the type I catalog, we construct a type II AGN sample from the
SDSS DR7 MPA-JHU catalog (Kauffmann et al., 2003; Brinchmann et al., 2004). We
identify galaxies classified as AGNs using the BPT diagram (Baldwin et al.,
1981) within the catalog. We estimate black-hole masses from the velocity
dispersion using the $M_{\bullet}$-$\sigma_{*}$ relation proposed by Saglia et
al. (2016). To correct for the aperture effect of velocity dispersion, we
adopt the method outlined by Cappellari et al. (2006): First, we cross-match
the sample with the NYU-VAGC catalog (Blanton et al., 2005) to obtain the
$r$-band effective radius $R_{\rm e}$. Subsequently, we compute the aperture-
corrected velocity dispersion, $\sigma_{*}$, using the formula
$\sigma_{*}=\sigma_{\rm ap}(R_{\rm e}/R_{\rm ap}/8)^{-0.066}$, where
$\sigma_{\rm ap}$ is the fiber velocity dispersion, and $R_{\rm ap}=3\arcsec$
is the fiber aperture for SDSS spectra. Finally, we restrict the sample to
$21,532$ AGNs within the redshift range $z\in[0,0.2]$, black-hole mass range
$\log M_{\bullet}/M_{\odot}\in[6.67,9.33]$, and falling in the UNIONS
footprint. The type II sample exhibits lower redshifts ($\langle
z\rangle\simeq 0.1$) than the type I sample.

###  2.2 Source sample

The shape catalogs serving as the background source sample in this work are
the v1.3 ShapePipe and v1.0 lensfit catalogs of
UNIONS111https://www.skysurvey.cc/. UNIONS is an ongoing multi-band wide-field
imaging survey conducted with three telescopes (Canada-France-Hawai’i
Telescope for $u$ and $r$ bands, Subaru telescope for $g$ and $z$ bands, and
Pan-STARRS for the $i$ band) in Hawai’i. UNIONS will cover $4,800$ square
degrees of the Northern sky with deep exposures and high-quality images. The
depth (limiting magnitude with point source 5-sigma in a $2\arcsec$ diameter
aperture) reaches $24.3$, $25.2$, $24.9$, $24.3$, and $24.1$ in $u,r,g,i$ and
$z$, respectively.

At the time when the shear catalogs were produced (beginning of 2022), the
survey covered an area of around $3,500$ square degrees in the $r$-band (the
galaxy shapes were measured in this band). We did not have photometric
redshifts for each source galaxy in the catalog at this stage of the UNIONS
processing since the observations and calibration of the multi-band photometry
are still ongoing. Instead, we estimated the overall redshift distribution by
a method based on self-organizing maps. See Appendix A for more details about
photometric redshifts.

The ShapePipe catalog was processed with the ShapePipe software package
(Farrens et al., 2022). It contains $98$ million galaxies over an area of
$3,200$ square degrees effective area. An earlier version of the ShapePipe
catalog was published in Guinot et al. (2022). Some updates in processing were
implemented for the v1.3 shear catalog used here, as follows. First, to model
the PSF, instead of PSFEx (Bertin, 2011) we used MCCD (Liaudat et al., 2021)
that performs a non-parametric Multi-CCD fit of the PSF over the entire focal
plane. Second, we reduced the minimum area to detect an object from $10$ to
$3$ pixels via the SExtractor configuration keyword DETECT_MINAREA = 3. This
leads to a smaller galaxy selection bias on the ensemble shear estimates.
Third, we added the section on the relative size between galaxies,
$T_{\textrm{gal}}$, and the PSF, $T_{\textrm{PSF}}$, as
$T_{\textrm{gal}}/T_{\textrm{PSF}}<3$ to avoid contamination by very diffuse,
mostly low-signal-to-noise objects which tend to be artefacts.

The lensfit shape catalog was created with the THELI processing and lensfit
software (Miller et al., 2007). It contains $109$ million galaxies in a
$2,100$ square degree sky area. The effective area and number density of the
ShapePipe and lensfit catalogs are different due to masking and processing
choices. In the following, we use the more conservative lensfit mask for both
shape catalogs, defining the common UNIONS footprint in which SDSS AGNs are
selected. Both lensfit and ShapePipe catalogs are based on the same image
data.

##  3 Methods

Figure 2: Galaxy-galaxy lensing excess surface mass density of three black-
hole mass bins from ShapePipe. The best-fit HOD models are presented in orange
lines. The baryon contribution, one-halo term of centrals, one-halo term of
satellites, and two-halo term are plotted in green, red dashed, purple dash-
dotted, and brown dotted lines, respectively. The measurements from the
lensfit catalog are similar.

###  3.1 Galaxy-galaxy lensing technique

Galaxy-galaxy lensing denotes the shape distortions of background source
galaxies due to the gravitational field of matter associated with foreground
lens galaxies (see for a review Kilbinger, 2015). The main _physical_ quantity
related to galaxy-galaxy lensing is the excess surface density (ESD),
$\Delta\Sigma$, at a projected distance $R$, defined as the mean surface
density within a disk of radius $R$ minus a boundary term, which is the mean
surface mass at radius $R$,

$\Delta\Sigma(R)=\bar{\Sigma}(<R)-\Sigma(R).$ (1)

The main _observable_ for galaxy-galaxy lensing is the tangential shear,
$\gamma_{\textrm{t}}$, of a source sample induced by a lens at projected
distance $R$. This observable is related to the ESD via

$\Delta\Sigma(R)=\Sigma_{\textrm{cr}}\gamma_{\textrm{t}}(R),$ (2)

where the critical surface mass density $\Sigma_{\textrm{cr}}$ is defined as

$\Sigma_{\textrm{cr}}(z_{\textrm{l}},z_{\textrm{s}})=\frac{c^{2}}{4\pi
G}\frac{d(z_{\textrm{s}})}{d(z_{\textrm{l}})\,d(z_{\textrm{l}},z_{\textrm{s}})}\frac{1}{(1+z_{\textrm{l}})^{2}}.$
(3)

Here, $z_{\textrm{s}}$ ($z_{\textrm{l}}$) is the source (lens) redshift, and
$d(z_{\textrm{s}})$, $d(z_{\textrm{l}})$, and
$d(z_{\textrm{l}},z_{\textrm{s}})$ are the angular diameter distance from the
observer to the source, to the lens, and the lens-source distance,
respectively. The constants are the speed of light $c$ and the Newtonian
gravitational constant $G$.

###  3.2 Estimators

An estimator for the tangential shear of a background source sample around a
lens galaxy population is

$\left\langle\gamma_{\textrm{t}}(R)\right\rangle=\frac{\sum_{\rm ls}w_{\rm
l}w_{\rm s}\epsilon_{\rm t,s}1_{b(R)}(|\vec{r}_{\rm l}-\vec{r}_{\rm
s}|)}{\sum_{\rm ls}w_{\rm l}w_{\rm s}}.$ (4)

This estimator is a weighted sum over the observed tangential ellipticities,
$\epsilon_{\rm t,s}$, of source galaxies around lens galaxies. Source galaxies
have a weight, $w_{\rm s}$, stemming from the galaxy shape estimation that
indicates measurement uncertainties. Lens weights, $w_{\rm l}\equiv w_{\rm
l,nofz}$, are introduced to homogenize the redshift distribution across lens
samples as discussed in Sect. 2.2. The indicator function $1_{S}(x)$ of the
set $S$ is unity if $x\in S$ and zero otherwise. In the above equation, this
function selects galaxy pairs in a bin $b(R)$ around the projected separation
$R$ of the pair; the shape of the function $b$ is chosen to be logarithmic.

Since we do not have photometric redshifts of individual background galaxies
in the shape catalog, we compute an effective surface mass density by
averaging Eq. (3) over the source redshift distribution. Inserting this
effective value into Eq. (2) results in an average excess surface mass
density. Since we cannot select sources to be strictly behind the lens sample,
this leads to a divergence of $\Sigma_{\rm cr}$ when $z_{\textrm{s}}\to
z_{\textrm{l}}$. A practical solution is to compute the inverse effective
critical surface mass density,

$\overline{\Sigma^{-1}_{\textrm{cr}}}(z_{\textrm{l}})=\frac{4\pi
G}{c^{2}}\,d(z_{\textrm{l}})(1+z_{\textrm{l}}^{2})\,\int_{0}^{z_{\textrm{lim}}}\textrm{d}z_{\textrm{s}}\,n(z_{\textrm{s}})\,\frac{\,d(z_{\textrm{l}},z_{\textrm{s}})}{d(z_{\textrm{s}})}.$
(5)

This quantity is the inverse of the critical surface mass density
$\Sigma_{\textrm{cr}}$, Eq. (3), weighted by the source redshift distribution
(see Sect. 2). The effective excess surface mass density is then

$\overline{\Delta\Sigma}(R)=\gamma_{\textrm{t}}\left[\overline{\Sigma^{-1}_{\textrm{cr}}}(z_{\textrm{l}})\right]^{-1}.$
(6)

Using Eq. (5), a first estimator for the excess surface mass density is
readily derived as

$\left\langle\Delta\Sigma(R)\right\rangle^{\prime}=\frac{\sum_{\rm ls}w_{\rm
l}w_{\rm s}\epsilon_{\rm
t,s}\left[\overline{\Sigma^{-1}_{\textrm{cr}}}(z_{\textrm{l}})\right]^{-1}1_{b(R)}(|\vec{r}_{\rm
l}-\vec{r}_{\rm s}|)}{\sum_{\rm ls}w_{\rm l}w_{\rm s}}.$ (7)

When using the effective surface mass density, the weights for a given lens
can be updated by multiplication with the square of the inverse effective
critical surface mass density Eq. (3), to down-weigh lenses with a low lensing
efficiency, $w_{l}\to w_{\rm
l}\left(\overline{\Sigma_{\textrm{cr}}^{-1}}(z_{\rm l})\right)^{2}$. With
this, we write our final estimator of the excess surface mass density as

$\displaystyle\left\langle\Delta\Sigma(R)\right\rangle=\frac{\sum_{\rm
ls}w_{\rm l}\,\overline{\Sigma_{\textrm{cr}}^{-1}}(z_{\rm l})\,w_{\rm
s}\,\epsilon_{\rm t,s}1_{b(R)}(|\vec{r}_{\rm l}-\vec{r}_{\rm s}|)}{\sum_{\rm
ls}w_{\rm l}w_{\rm s}\left(\overline{\Sigma_{\textrm{cr}}^{-1}}(z_{\rm
l})\right)^{2}}.$ (8)

We also conduct a series of systematic tests and apply the boost factor
correction
$\left\langle\Delta\Sigma(R)^{\textrm{c}}\right\rangle=B(R)\left\langle\Delta\Sigma(R)\right\rangle$.
We refer to Appendix B for details.

###  3.3 AGN lens model

Our lens sample contains both central and satellite AGN host galaxies, and we
need to consider contributions to the excess surface density from both. We
adopt a HOD model from Guzik & Seljak (2002) to describe the average ESD
around the AGN sample,

$\Delta\Sigma=\Delta\Sigma_{\rm b}+(1-f_{\rm sat})\Delta\Sigma_{\rm
h,cen}+f_{\rm sat}\Delta\Sigma_{\rm h,sat}+\Delta\Sigma_{2\mathrm{h}},$ (9)

where $f_{\rm sat}$ is the satellite galaxy fraction of the sample, left as a
free parameter. $\Delta\Sigma_{\rm b}$ is the contribution from baryons in the
host galaxy, containing the stellar mass $M_{*}$. $\Delta\Sigma_{\rm h,cen}$
and $\Delta\Sigma_{\rm h,sat}$ are the one-halo terms of the central and
satellite galaxy, respectively. $\Delta\Sigma_{2\mathrm{h}}$ is the two-halo
term. The terms are described in Appendix C in detail.

##  4 Results and Discussion

Figure 3: Black-Hole-to-Halo Mass Relation. In both panels, the dark blue
stars and orange-filled circles are our type I results from the ShapePipe and
lensfit catalogs, respectively. The blank red squares are type I ShapePipe
results with an adjustment on BH mass. Blank green triangles correspond to the
type II results with ShapePipe. Left Panel: Comparison with observational
results from the literature. Brown squares and rhombuses are AGN clustering
results for optically selected and X-ray selected samples from Krumpe et al.
(2015), respectively. Cyan and grey lines show AGN clustering and dynamics
results from Shankar et al. (2020) and Robinson et al. (2021), respectively.
Results for normal galaxies are shown as dashed lines (purple line from
Marasco et al. (2021) and black line from Zhang et al. (2023c)). Right Panel:
Results from EAGLE, TNG, and SIMBA hydro-simulations are shown in cyan, brown,
and purple lines with the $16-84$ percentiles displayed as shaded areas.

###  4.1 AGN black-hole-to-halo mass relation

The measured galaxy-galaxy lensing ESD profiles for the type I sample are
shown in Fig. 2. We measure the ESD with a high signal-to-noise ratio (SNR)
from ShapePipe (lensfit), with values of $24$ ($34$), $26$ ($36$), and $31$
($41$) in the three bins, respectively. Our measurements are well reproduced
by the HOD models with three free parameters. The amplitude of the ESD
increases with $M_{\bullet}$, indicating that more massive SMBHs are situated
in more massive halos. A similar trend is observed in the type II sample.

Fig. 3 shows the $M_{\bullet}$ \- $M_{\textrm{h}}$ relation. Systematic errors
in the shape measurements contribute to the total error budget at a comparable
level to statistical errors, as indicated by the disparity between the type I
ShapePipe and lensfit results. This underscores the robustness of our analysis
across different shape catalogs.

We observe that more massive AGNs inhabit larger dark-matter halos, consistent
with previous findings from dynamical (Robinson et al., 2021) and clustering
analyses (Krumpe et al., 2015; Shankar et al., 2020). Robinson et al. (2021)
employed reverberation mapping to determine black-hole masses and utilized HI
FWHM to estimate halo masses for $24$ local AGNs. Our results are consistent
with Robinson et al. (2021) at low masses but at $\log
M_{\bullet}/M_{\odot}\simeq 8$, we find a lower halo mass. Our measured halo
masses in the high-mass regime for both type I and type II samples are also
systematically lower than the clustering results reported in Krumpe et al.
(2015) and Shankar et al. (2020). The discrepancy between clustering and
lensing halo masses was noticed in previous works (Mandelbaum et al., 2009).
The clustering method leverages the monotonic relation between halo mass and
halo bias, while weak lensing directly probes the matter overdensity around
the tracers. It has long been known that the clustering strength of halos also
depends on their secondary properties, such as the halo structure and the halo
assembly history, which is called the halo assembly bias or the secondary halo
bias (Gao & White, 2007; Wang et al., 2024). Therefore, if the host galaxies
of AGNs prefer to live in dark-matter halos with biased secondary properties,
it will alter the clustering strength without changing the host halo mass,
while lensing is free of this effect.

In our results, both type I and type II samples exhibit similar
$M_{\bullet}$-$M_{\rm h}$ relations despite their distinct classifications and
redshift ranges. At higher black-hole masses, type I AGNs have a lower halo
mass compared to type II. This difference may be interpreted as a systematic
error in virial black-hole mass estimation. Recent spectroscopic
interferometer measurements (GRAVITY Collaboration et al., 2024) of the AGN
size-luminosity relation, upon which the virial mass measurement of type I
AGNs relies, exhibit a significantly lower slope than the one proposed by
Bentz et al. (2006) from reverberation mapping. Our type I black-hole masses
are calculated by Wu & Shen (2022) using the H$\beta$ prescription from
Vestergaard & Peterson (2006), which is based on Bentz et al. (2006). To
account for the new measurements of (GRAVITY Collaboration et al., 2024), we
use their size-luminosity slope to derive a black-hole mass prescription with
the same sample and the same method as Vestergaard & Peterson (2006) and get
the updated relation

$\begin{split}\log\left(\frac{M_{\bullet}}{M_{\odot}}\right)=6.88&+2\log\left(\frac{\text{FWHM}_{\text{H}\beta}}{1000\,\text{km
s}^{-1}}\right)\\\ &+0.30\log\left(\frac{\lambda
L_{\text{5100}}}{10^{44}\,\text{erg s}^{-1}}\right),\end{split}$ (10)

where $\log{\lambda L_{\text{5100}}}$ is the luminosity of the continuum at
$5100\text{\AA}$. We then adjust the average black-hole mass of the type I
sample, as shown in Fig. 3. The black-hole mass of the high-mass bin changes
the most, and moves the black-hole-halo-mass relation closer to the type II
line. In conclusion, we find no evidence of type- or redshift-dependence in
the $M_{\bullet}$-$M_{\rm h}$ relationship.

Furthermore, we compare our results to those of normal (non-AGN) galaxies.
Marasco et al. (2021) measured halo masses through globular cluster dynamics
and galaxy rotation curves in 55 nearby galaxies with directly measured black-
hole masses. Zhang et al. (2023c) used the Dark Energy Camera Legacy Survey
(Dey et al., 2019, DECaLS) shape catalog (Zhang et al., 2022) to measure
galaxy-galaxy lensing of quiescent galaxies for $z<0.2$ for different
$\sigma_{*}$ bins. We plot their result with black-hole masses inferred from
the $M_{\bullet}$-$\sigma_{*}$ relation of Saglia et al. (2016). Compared to
these results, we find that both type I and type II are broadly consistent
with normal galaxies, suggesting no intrinsic difference in the
$M_{\bullet}$-$M_{\rm h}$ relation between non-AGN galaxies and AGNs.

###  4.2 Constraint on black-hole mass in simulations

In state-of-the-art cosmological hydro-dynamical simulations, black-hole
growth fed by gas accretion is a crucial factor in driving AGN feedback,
which, in turn, is a major mechanism to suppress star formation activities in
massive galaxies (Davé et al., 2019). However, these simulations cannot
resolve the detailed accretion process. Instead, empirical subgrid recipes are
employed to model this process, and the free parameters in these recipes need
to be calibrated using observational scaling relations, such as the
$M_{\bullet}$-$M_{*}$ relation (Habouzit et al., 2021).

However, we note that stellar mass itself is subject to several subgrid
processes, including the stellar feedback and the AGN feedback, which makes
the calibration process quite complicated. In contrast, halo masses are
relatively robust and less sensitive to baryonic processes. Therefore, the
black-hole-to-halo mass relation is a better scaling relation to calibration
subgrid parameters in these simulations, and our work takes the first step to
establish this relation in observation.

To compare our measurements with simulations, we used the RefL0100N1054 run
for EAGLE (Schaye et al., 2015), the TNG100 run for IllustrisTNG (Springel et
al., 2017), and the m100n1024 run for SIMBA (Davé et al., 2019). We calculated
AGN luminosity from Eq. 1 in Habouzit et al. (2021) and selected central
subhalos with $L_{\rm BH}/L_{\rm Edd}>0.001$ as the “AGN” sample in the
simulation. We used the snapshot with $z\sim 0.4$, which is the average
redshift of our type I AGN sample. We find no significant evolution between
$z=0.4$ and $z=0.1$ in the three simulations, which is consistent with our
observation. We also compared the “AGN” sample and central galaxy sample in
the simulation and found no statistically significant difference between their
$M_{\bullet}$-$M_{\rm h}$ relations. The AGN $M_{\bullet}$-$M_{\rm h}$
relations from the three simulations are plotted in the right panel of Fig. 3.

Although the three simulations calibrate their models to be in good agreement
with observed relations (McConnell & Ma, 2013; Kormendy & Ho, 2013) between
$M_{\bullet}$ and stellar mass of the galaxy, $M_{*}$, or of the bulge,
$M_{\rm bulge}$, their $M_{\bullet}$-$M_{\rm h}$ relations do not perfectly
match our measurements. The difference among the simulations under similar
calibration clearly reflects how different black-hole accretion and AGN
feedback mechanisms shape the black-hole masses in simulations. The predicted
halo mass from EAGLE is consistent with ours at low masses but is
significantly higher than ours at $\log M_{\bullet}/M_{\odot}>8$. However, TNG
and SIMBA predict lower halo masses at fixed black-hole masses compared to
EAGLE, which are more consistent with our observations (both type I and type
II). Among the three simulations, SIMBA has the $M_{\bullet}$-$M_{\rm h}$
relation that is the closest to ours, with all differences within one sigma.

###  4.3 Future prospects

Current data suffers from a small AGN sample size and limited accuracy of
black-hole mass estimation. Large spectroscopic surveys such as DESI (Levi et
al., 2013) and PFS222https://pfs.ipmu.jp/ will provide larger quasar samples
with reliable virial black-hole mass measurements. Already now integral field
spectroscopy and reverberation mapping observations are improving the virial
black-hole mass measurement accuracy. From the perspective of weak-lensing
data, we will soon get the $4,800$ square degrees shape catalog with
photo-$z$s from the completed UNIONS survey.

Future weak-lensing surveys such as Euclid (Euclid Collaboration et al.,
2020), Rubin-LSST (Željko Ivezić et al., 2019), Roman (Spergel et al., 2015),
CSST (Gong et al., 2019), and WFST (WFST Collaboration et al., 2023), will
provide galaxy samples with accurate shape measurements to higher redshifts,
covering larger sky areas. This will enable us to measure the
$M_{\bullet}$-$M_{\rm h}$ relation with higher accuracy, as well as its
dependency on the host galaxy properties.

We thank Eric Jullo, Shiming Gu, Zheng Zheng, and Chris Miller for their
helpful discussion. The authors thank the support from the National Natural
Science Foundation of China (NO. 12192224, 11833005, 11890693), the National
Key R&D Program of China (2021YFC2203100), the 111 project for ”Observational
and Theoretical Research on Dark Matter and Dark Energy” (B23042), CAS Project
for Young Scientists in Basic Research, Grant No. YSBR-062. H. Hildebrandt is
supported by a DFG Heisenberg grant (Hi 1495/5-1), the DFG Collaborative
Research Center SFB1491, as well as an ERC Consolidator Grant (No. 770935).
Anna Wittje is also supported by SFB1491. We are honored and grateful for the
opportunity of observing the Universe from Maunakea and Haleakala, which both
have cultural, historical and natural significance in Hawai’i. This work is
based on data obtained as part of the Canada-France Imaging Survey, a CFHT
large program of the National Research Council of Canada and the French Centre
National de la Recherche Scientifique. Based on observations obtained with
MegaPrime/MegaCam, a joint project of CFHT and CEA Saclay, at the Canada-
France-Hawaii Telescope (CFHT) which is operated by the National Research
Council (NRC) of Canada, the Institut National des Science de l’Univers (INSU)
of the Centre National de la Recherche Scientifique (CNRS) of France, and the
University of Hawaii. This research used the facilities of the Canadian
Astronomy Data Centre operated by the National Research Council of Canada with
the support of the Canadian Space Agency. This research is based in part on
data collected at Subaru Telescope, which is operated by the National
Astronomical Observatory of Japan. Pan-STARRS is a project of the Institute
for Astronomy of the University of Hawai’i, and is supported by the NASA SSO
Near Earth Observation Program under grants 80NSSC18K0971, NNX14AM74G,
NNX12AR65G, NNX13AQ47G, NNX08AR22G, 80NSSC21K1572 and by the State of Hawai’i.
This work was made possible by utilizing the CANDIDE cluster at the Institut
d’Astrophysique de Paris, which was funded through grants from the PNCG, CNES,
DIM-ACAV, and the Cosmic Dawn Center and maintained by S. Rouberol. This work
was supported in part by the Canadian Advanced Network for Astronomical
Research (CANFAR) and Compute Canada facilities. Funding for the Sloan Digital
Sky Survey has been provided by the Alfred P. Sloan Foundation, the Heising-
Simons Foundation, the National Science Foundation, and the Participating
Institutions. SDSS acknowledges support and resources from the Center for
High-Performance Computing at the University of Utah. The SDSS web site is
www.sdss.org.

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##  Appendix A Estimation of the redshift distribution

From UNIONS $r$-band observations, we follow three steps to estimate the
redshift distribution of our weak-lensing source sample. The first step is
assigning multi-band photometry to UNIONS galaxies. Using the overlap of
UNIONS $r$-band observations with the CFHTLenS (Canada-France-Hawaii Telescope
Lensing Survey; Heymans et al., 2012; Erben et al., 2013) W3 field ($\sim
44.2$ sq. deg), we assign $ugriz$ magnitudes by cross-matching. This can be
done since CFHTLenS has deeper photometry (Hildebrandt et al., 2012) than
UNIONS, basically all CFIS (UNIONS $r$-band) objects are also visible in
CFHTLenS, and the underlying redshift distribution is assumed to be the same
after matching.

We calibrate the redshifts distribution with spectroscopic calibration samples
which are constructed from DEEP2 (DEEP2 Galaxy Redshift Survey; Newman et al.,
2013), VVDS (VIMOS VLT Deep Survey; Le Fèvre et al., 2005), and VIPERS (VIMOS
Public Extragalactic Redshift Survey; Scodeggio et al., 2018). These surveys
are also observed with CFHTLenS $ugriz$ photometry. With the multi-band
photometry of the spectroscopic sample, we then train self-organising maps
(SOM; Kohonen, 1982; Masters et al., 2015) to organise the sample in high
dimensional magnitude space. The SOM splits the matched sample into subsamples
in its so-called SOM cells. The initial SOM cell grid has a resolution of 101
$\times$ 101 cells and is then hierarchically clustered into $5,000$
resolution elements for reliable statistics lateron, shown in Fig. 4. We then
populate the SOM with the UNIONS weak lensing sources with $ugriz$ photometry.

For every SOM cell $i$, a weight $w_{i}^{\mathrm{SOM}}$ is defined, which is
the ratio of the number of UNIONS objects (weighted by their shape weights)
over the number of spectroscopic objects (Wright et al., 2020). Finally, we
get the UNIONS $p(z)$ by re-weighting the spectroscopic redshift distribution
$p^{\rm spec}(z)$ according to the weights $w_{i}^{\mathrm{SOM}}$ in the
$i$’th SOM cells (Wright et al., 2020),

$p(z)=\int w^{\mathrm{SOM}}(z)p^{\rm spec}(z)\mathrm{d}z=\sum
w_{i}^{\mathrm{SOM}}p_{i}^{\mathrm{spec}}(z),$ (A1)

where $p_{i}^{\mathrm{spec}}(z)$ is the histogram of spectroscopic objects per
SOM cell $i$. $p^{\rm Spec}(z)$ and $p(z)$ are shown in Fig. 4.

Figure 4: In panel (a), the trained SOM coloured by the counts of the
spectroscopic sample is shown. In panel (b), the blinded redshift
distributions $p^{\rm spec}(z)$ and $p(z)$ are plotted with green and purple
bars, respectively.

##  Appendix B Systematic Tests for Galaxy-galaxy Lensing Measurements

To validate our galaxy-galaxy lensing measurement, we conducted two null tests
and measured the boost factor.

###  B.1 Cross-shear ($\Delta\Sigma_{\times}$) test

Weak gravitational lensing does not produce shape distortions in the cross
direction, therefore the cross component of the shear $\gamma_{\times}$ or
“cross excess surface density” $\Delta\Sigma_{\times}$ is expected to be zero
in the absence of systematics. Thus, $\Delta\Sigma_{\times}$ can be
interpreted as a null test of systematics in the lensing measurement process.
We measure $\Delta\Sigma_{\times}$ with the same method and sample as for
$\Delta\Sigma$,

$\displaystyle\left\langle\Delta\Sigma(R)^{w}_{\times}\right\rangle$
$\displaystyle=\frac{\sum_{\rm ls}w_{\rm
l}\,\overline{\Sigma_{\textrm{cr}}^{-1}}(z_{\rm l})\,w_{\rm
s}\,\epsilon_{\times,{\rm s}}1_{b(R)}(|\vec{r}_{\rm l}-\vec{r}_{\rm
s}|)}{\sum_{\rm ls}w_{\rm l}w_{\rm
s}\left(\overline{\Sigma_{\textrm{cr}}^{-1}}(z_{\rm l})\right)^{2}}.$ (B1)

The results of the $\Delta\Sigma_{\times}$ test are shown in Fig. 5. All data
points are consistent with zero at three sigma, and $\sim 70\%$ are zero
within one sigma. No evidence is found of any significant systematic errors.

Figure 5: Systematic tests of lensing measurements. The top (bottom) column
shows tests for ShapePipe (lensfit). The left two panels correspond to
$\Delta\Sigma_{\rm x}$. Low-, median-, and high-mass bins of type I sample are
shown in blue circles, green triangles, and red rectangles in each panel. The
middle two panels show the excess surface density and cross excess surface
density around a random sample. The right two panels present the boost factor
of the type I sample. Low-, median-, and high-mass bins are shown in blue,
green, and red points. In all panels, some results are slightly displaced in
the $x$-direction to make the figure clear.

###  B.2 Random lens test

We also measure lensing signals around a random sample as a null test. This
sample is constructed by randomly sampling the SDSS footprint and then
selecting the sub-sample in the sky region overlapping with UNIONS. To match
the redshift distribution, we randomly assign the redshifts of the high-mass
bin lens sample (the other two bins have the same $p(z)$ after weighting) to
the random sample. Our random sample contains $543,402$ “galaxies”.

Following the same procedure as before, we measure both tangential and cross
components with respect to the random sample with the ShapePipe and lensfit
shape catalogs. The results are presented in Fig. 5. The lensing signals are
in good consistency with zero, indicating that systematic errors in the
measurement are not significant.

###  B.3 Boost factor

Galaxy-galaxy lensing signals are diluted by galaxies physically associated
with lens galaxies, whose shapes are not affected by lensing. Since we can not
exclude these galaxies without photo-$z$s in this work, it is important to
quantify this effect. With the same random sample as in the random lens test,
we calculate the boost factor (Hirata et al., 2004), which is defined as
$B(r)=(N_{\rm rs}\sum_{\rm ls}w_{\rm ls})/(N_{\rm ls}\sum_{\rm rs}w_{\rm
rs})$, where $N_{\rm ls}$ and $N_{\rm rs}$ are the number of lens-source pairs
and random-source pairs, respectively, and $w_{\rm ls}$ and $w_{\rm rs}$ are
corresponding lensing weights. The results are shown in Fig. 5. We apply the
boost factor correction to the lensing signals we use in this work.

##  Appendix C Details of the HOD model

###  C.1 Baryonic contribution

For source-lens separations (at the lens redshifts) that are much larger than
the size of a typical galaxy, that galaxy can be considered as a point mass.
The baryonic contribution to the excess surface density, which contains stars,
dust and gas, can then be written as

$\Delta\Sigma_{\textrm{b}}(R)=\frac{M_{*}}{\pi R^{2}}.$ (C1)

###  C.2 One-halo central galaxy contribution

We adopt a Navarro-Frenk-White (NFW) model to describe the density profile of
the host halo for central galaxies,

$\rho(r)=\frac{\rho_{0}}{(r/r_{\textrm{s}})(1+r/r_{\textrm{s}})^{2}},$ (C2)

with $\rho_{0}=200\,\rho_{\textrm{m}}/(3I_{c})$ and
$I_{c}=c^{-3}\int_{0}^{c}\textrm{d}x\,x(1+x)^{-2}$. Here, $\rho_{\textrm{m}}$
is the mean density of the Universe, and $c$ is the halo concentration
parameter, defined as the ratio between the virial radius $r_{200}$ and scale
radius $r_{\textrm{s}}$ of the halo, $c=r_{200}/r_{\textrm{s}}$. Assuming that
the halo center is located at the central galaxy, the excess surface density
$\Delta\Sigma_{\rm NFW}$ of the halo within a disk of radius $R$ is

$\Delta\Sigma_{\textrm{h,cen}}(R)=\Delta\Sigma_{\rm NFW}(R)=\bar{\Sigma}_{\rm
NFW}(<R)-\Sigma_{\rm NFW}(R)=\frac{M_{\rm h}}{2\pi
r_{\textrm{s}}^{2}I}\left[g(R/r_{\textrm{s}})-f(R/r_{\textrm{s}})\right],$
(C3)

where $M_{\rm h}$ is the halo mass. The functions $f(x)$ and $g(x)$ are
defined as

$f(x)=\left\\{\begin{array}[]{lr}\frac{1}{x^{2}-2}\left[1-\frac{\ln[(1+\sqrt{1-x^{2}})/x]}{\sqrt{1-x^{2}}}\right],&x<1\\\
\frac{1}{3},&x=1\\\
\frac{1}{x^{2}-1}\left[1-\frac{\arctan(\sqrt{x^{2}-1})}{\sqrt{x^{2}-1}}\right];&x>1\end{array}\right.,\qquad
g(x)=\left\\{\begin{array}[]{lr}\frac{2}{x^{2}}\left[\ln{x/2}+\frac{\ln[(1+\sqrt{1-x^{2}})/x]}{\sqrt{1-x^{2}}}\right],&x<1\\\
2-2\ln 2,&x=1\\\
\frac{2}{x^{2}}\left[\ln{x/2}+\frac{\arctan(\sqrt{x^{2}-1})}{\sqrt{x^{2}-1}}\right],&x>1\end{array}\right..$
(C4)

###  C.3 One-halo satellite galaxy contribution

We use the NFW model also for the host halo of satellites. Compared to the
central-galaxy term, the satellite halo has a spatial offset. First, the
excesses surface density given the projected distance between the satellite
galaxy and the halo center, $R_{\textrm{sat}}$, is

$\Delta\Sigma_{\rm off}(R|M_{\rm h,sat},R_{\rm
sat})=\frac{1}{2\pi}\int_{0}^{2\pi}\Delta\Sigma_{\rm NFW}\left((R^{2}+R_{\rm
sat}^{2}+2RR_{\rm sat}\cos\theta)^{\frac{1}{2}}|M_{\rm
h,sat}\right)\textrm{d}\theta.$ (C5)

We integrate this equation over the distribution functions of $M_{\rm h,sat}$
and $R_{\rm sat}$ to obtain the effective one-halo satellite term as

$\Delta\Sigma_{\rm h,sat}(R)=\iint\Delta\Sigma_{\rm off}(R|M_{\rm
h,sat},R_{\rm sat})P(R_{\rm sat}|M_{\rm h,sat})P(M_{\rm
h,sat})\textrm{d}R_{\rm sat}\textrm{d}M_{\rm h,sat}$ (C6)

We assume that satellite galaxies follow the spatial distribution of dark
matter, which is the NFW density profile. We set

$P(R_{\rm sat}|M_{\rm h,sat})\propto\Sigma_{\rm NFW}(R_{\rm sat},M_{\rm
h,sat})\cdot 2\pi R_{\rm sat}\textrm{d}R_{\rm sat}=\frac{M_{\rm h,sat}}{2\pi
r_{\textrm{s}}^{2}(M_{\rm h,sat})I}f[R_{\rm sat}/r_{\textrm{s}}(M_{\rm
h,sat})]\cdot 2\pi R_{\rm sat}\textrm{d}R_{\rm sat}.$ (C7)

Following Guzik & Seljak (2002), we use a halo occupation distribution (HOD)
model to infer

$P(M_{\rm h,sat}|M_{\rm sub})\propto P(M_{\rm sub}|M_{\rm h,sat})P(M_{\rm
h,sat})\propto\langle N_{\rm sat}(M_{\rm h,sat})\rangle F_{\rm h}(M_{\rm
h,sat}),$ (C8)

where $F_{\rm h}(M_{\rm h})$ is halo mass function and $\langle N_{\rm
sat}(M_{\rm h})\rangle$ is the halo occupation function of satellite galaxies.
In this work, we use the halo mass function from Tinker et al. (2008) and the
HOD model from Guzik & Seljak (2002).

####  C.3.1 Two-halo term

For the two-halo term, we use the Tinker et al. (2010) halo bias model to
infer the halo-matter correlation function $\xi_{\rm hm}=b_{\rm h}\xi_{\rm
mm}$ based on the dark-matter correlation function $\xi_{\rm mm}$. From that,
we can calculate the surface density as

$\Sigma_{\rm 2h}(R)=b_{\rm h}(M_{\rm h,cen})\times
2\bar{\rho}\int_{R}^{\infty}\xi_{\rm
mm}(r)\frac{r\textrm{d}r}{\sqrt{r^{2}-R^{2}}}.$ (C9)

###  C.4 Lens model validation

To validate the model, we cross-match the type II sample with the Yang et al.
(2007) SDSS group catalog to select a purely central-galaxy subsample.
Subsequently, we measure the ESD for both the type II sample and the central-
galaxy subsample using the ShapePipe catalog. Next, we fit the central-galaxy
subsample lensing signals with the lens model (Sect. 3.3), but set the
satellite fraction to zero. This allows us to make two consistency tests.
First, we compare the contributions of central galaxies from the entire type
II sample by using the best-fit central ESD term to the ESD measured from the
central-galaxy subsample. We find that they are broadly consistent. Second, we
compare the inferred halo masses. They are consistent within one sigma across
all mass ranges considered in this work, indicating that our lens model is
reliable.


between sonar signals bounced off a metal cylinder and those bounced off a
roughly cylindrical rock. It consists of the $208$ data points, divided into
two classes of $111$ and $97$ patterns, each with $60$ features. The applied
pre-processing consists of centering and normalization of the data. The
subsequent features for the $n$-dependent analysis are picked in the order of
descending variance.

This is the main dataset of our work and is used for visualization of the
issues studied. It fits the study particularly well, as the size of the
dataset does not prohibit us from performing extensive simulations for
multiple kernel families and feature map hyperparameters. It is also an almost
balanced data set (imbalance ratio of $1.14$) which makes it suitable for the
analysis using median as a measure of statistical behavior of kernel matrix
entries.

###  F.2 Twonorm

The artificial dataset [35] consisting of $7400$ patterns. The data points are
distributed into two, equal classes with $3700$ points each and with $20$
features. The dataset is too big to perform quantum simulations on it.
Therefore, the dataset was stratified into $74$ balanced subsets, each
consisting of $100$-points, the kernel simulation is performed every subset
separately. Together, the subsets cover the whole dataset. The applied pre-
processing consists of centering and normalization of the data.

###  F.3 Indian Pines

The real-life dataset consisting of a scene AVIRIS sensor [36] over the Indian
Pines test site in North-western Indiana. The scene is taken with the sensor
in range of the wavelengths spanning $0.4–2.510^{-6}m$, which are then
gathered into $200$ features. It covers agriculture areas and serves as a
dataset for multiclass classification problems. In our case, we take only two
classes of pixels: Corn-mintill and Grass-trees. We artificially reduce the
dataset while balancing the classes by removing $100$ points from the Corn-
mintill class. We end up with total number of $1460$ data points, each class
consisting of $730$ points. Again, the applied pre-processing consists of
centering and normalization of the data.


# Measurement of Dielectric Loss in Silicon Nitride at Centimeter and
Millimeter Wavelengths

Z. Pan, P. S. Barry, T. Cecil, C. Albert, A. N. Bender, C. L. Chang, R.
Gualtieri, J. Hood, J. Li, J. Zhang, M. Lisovenko, V. Novosad, G. Wang, and V.
Yefremenko  (Corresponding author: Zhaodi Pan)Z. Pan is with the Argonne
National Laboratory, 9700 South Cass Avenue, Lemont, IL, 60439, USA (email:
panz@anl.gov)P. Barry is with Cardiff University, Cardiff CF10 3AT, UK (email:
barryp2@cardiff.ac.uk)T. Cecil, A. N. Bender, R. Gualtieri, J. Li, J. Zhang,
M. Lisovenko, V. Novosad, G. Wang, and V. Yefremenko are with the Argonne
National Laboratory, 9700 South Cass Avenue, Lemont, IL, 60439, USA (emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>yefremenko@anl.gov)C. Albert was with the the University of Chicago, Chicago,
IL 60637 USA and is now with Caltech, 1200E California Blvd, Pasedena CA 91125
USA (email: calbert@caltech.edu)C. L. Chang is with the Argonne National
Laboratory, Argonne, IL 60439 USA and the University of Chicago, 5640 South
Ellis Avenue, Chicago, IL 60637 USA (email<EMAIL_ADDRESS>J. Hood
is with the University of Chicago, 5640 South Ellis Avenue, IL 60637 USA
(email<EMAIL_ADDRESS>
###### Abstract

This work presents a suite of measurement techniques for characterizing the
dielectric loss tangent across a wide frequency range from $\sim$1 GHz to 150
GHz using the same test chip. In the first method, we fit data from a
microwave resonator at different temperatures to a model that captures the
two-level system (TLS) response to extract and characterize both the real and
imaginary components of the dielectric loss. The inverse of the internal
quality factor is a second measure of the overall loss of the resonator, where
TLS loss through the dielectric material is typically the dominant source. The
third technique is a differential optical measurement at 150 GHz. The same
antenna feeds two microstrip lines with different lengths that terminate in
two microwave kinetic inductance detectors (MKIDs). The difference in the
detector response is used to estimate the loss per unit length of the
microstrip line. Our results suggest a larger loss for SiNx at 150 GHz of
$\boldsymbol{\mathrm{\tan\delta\sim 4\times 10^{-3}}}$ compared to
$\boldsymbol{\mathrm{2.0\times 10^{-3}}}$ and $\boldsymbol{\mathrm{\gtrsim
1\times 10^{-3}}}$ measured at $\sim$1 GHz using the other two methods. These
measurement techniques can be applied to other dielectrics by adjusting the
microstrip lengths to provide enough optical efficiency contrast and other
mm/sub-mm frequency ranges by tuning the antenna and feedhorn accordingly.

###### Index Terms:

Dielectrics loss, silicon nitride (SiNx), millimeter wavelength, two-level
system (TLS), quality factor

##  I Introduction

Dielectric materials are used extensively in mm-wave superconducting detectors
for astrophysical or cosmological surveys. Experiments with photometer arrays
such as CMB-S4 use microstrip lines to transfer the mm-wave signal received at
the antenna to the detectors, where dielectric loss limits the overall optical
efficiency [1]. Dielectric loss is more critical for on-chip spectroscopy,
where it constrains the quality factor of the on-chip filter bank, which
limits the combination of spectral resolving power and the optical efficiency
[2]. The lumped-element MKID architecture typically utilize an interdigitated
capacitor, but adapting to parallel-plate capacitors can significantly reduce
the size of the resonators and increase their packing density. However, one
disadvantage associated with parallel-plate capacitors is that more of the
power is confined to the thin-film dielectric layer; thus, the loss mechanism
in dielectrics will limit detector quality factor and increase the noise. In
quantum information, the dielectric loss can cause decoherence [3]. The
defining roles of low-loss dielectrics in multiple applications motivate this
study.

A few low-loss dielectrics have been developed and are commonly in use,
including $\mathrm{SiO_{2}}$ [4], $\mathrm{SiN_{x}}$ [5], amorphous Si (a-Si)
[6], and silicon carbide [7]. State-of-art loss tangent of
$O(\mathrm{10^{-5}})$ has been achieved for a-Si [6] and silicon carbide [7]
at microwave frequencies. However, past measurements mostly focused on a few
GHz, while applications for mm-wave astrophysical surveys deal with radiation
at millimeter wavelengths. Though a few measurements at mm-wave have been
reported, more data is needed to validate the dielectric performance across
different deposition techniques at mm-wave and cross-compare with measurements
at a few GHz. Given the discrepancy between cm and mm-wavelengths that a few
groups observed for SiNx, hydrogenated amorphous Si and SiC [8, 9, 10], more
work is needed to study the discrepancy and elucidate other potential loss
mechanisms at mm-wavelengths beyond the dielectrics.

This paper presents our recent effort to measure the dielectric loss from
centimeter to millimeter wavelengths. We designed and fabricated a test
structure that can probe dielectric loss using three methods: 1) fitting
frequency vs. temperature response to a TLS model, 2) inverting the quality
factor in the low power limit, and 3) measuring the differential loss for
detectors coupled via different lengths of microstrip transmission lines.
Compared to the Fabry–Pérot method in [10], our method doesn’t require a
coherent source that can sweep in frequency and only needs an internal
blackbody to feed the antenna. The detector chip has a simple design but
offers three different ways to measure the dielectric loss. In a previous
paper [11], we presented a design where the signal from a paddle of the ortho-
mode transducer (OMT) probe is split into two transmission lines with
different lengths and coupled to two detectors. However, the detectors did not
show a noticeable optical response. We then simplified the optical coupling
and fed more power to each detector by differentiating the signal from a pair
of OMT paddles and removing the power split. The new strategy showed the
expected optical coupling level. Here, we report on progress made having
resolved several technological issues since [11]: 1) developed a more robust
electrical contact between the aluminum inductor and niobium capacitor, 2)
solving the step coverage when the transmission line climbs up onto the
capacitor pads, 3) fixed unwanted shorts from niobium etching residuals, and
4) improving parasitic leakage of the optical signal from neighbouring
detectors.

##  II Device Design and Fabrication

Measurement devices optimised for mm-wave were fabricated using a similar
process to that reported in [11] but with a few key differences. Devices start
as a 150 mm silicon wafer with a commercially-grown 450 nm thermal SiO2 and 2
$\mu$m low-pressure chemical-vapor deposited (LPCVD) SiNx bilayer. These
layers form the released membrane for the OMT probes. A 300 nm thick Nb ground
plane is deposited and patterned by fluorine-based reactive ion etch (RIE).
Next, a 500 nm-thick layer of SiNx is deposited for the microstrip dielectric.
The SiNx was deposited by chemical vapor deposition using SiH4 and N2
precursors at a substrate temperature of 275∘C. Following the microstrip
dielectric, the microstrip wiring is fabricated from a 170 nm-thick Nb layer,
along with a 30 nm NbN capping layer and patterned via liftoff. Next, a 50 nm
Al layer is deposited and patterned into inductors via liftoff. Previously we
had deposited the Al inductors before the Nb capacitors. However, due to
issues with poor galvanic contact between the layers, we reversed the steps
and added the NbN capping layer. The NbN limits oxidation of the Nb, making a
galvanic contact to the Al more robust. Additionally, in previous devices, the
Nb wiring layer was etched, which occasionally leads to shorting when the
material is not removed from step sidewalls by the anisotropic RIE. Switching
to using liftoff eliminated this issue. After the wiring layer processing, the
OMT membrane is released by a silicon deep RIE from the backside of the wafer.
We use the same etch step to separate individual chips from the wafer. The
final step is an O2 plasma used to remove a protective resist layer from the
front side of the wafer.

Fig. 1 bottom and top show the chip design and a microscope image of a segment
near a pixel. A coplanar waveguide (CPW) feedline is capacitively coupled to
20 MKIDs detectors, each containing a parallel-plate niobium capacitor and a
meandered aluminum inductor line. We choose parallel-plate capacitors over
interdigitated capacitors (IDCs) due to their higher energy filling factor $F$
that is close to unity. Four of the 20 resonators are coupled to optical
signals from the OMT via hybrid tee and different lengths of microstrip
meander lines. At mm-wave, the dielectric loss can be extracted by comparing
the optical response of detectors connected through varying lengths of
microstrip transmission lines. In [11], we split the signal from one arm of
the OMT into two detectors, which reduced the optical signal level to be
comparable to systematics caused by temperature variation and noise scatter.
To improve this, we increase the signal level by a factor of four by removing
the signal split and differentiating signals from two OMT fins with a hybrid
tee. Other detectors should be dark, meaning they do not receive signals from
the antenna via microstrip. The primary purpose of the dark detectors is to
monitor systematics from temperature variation and optical leakage.

Figure 1: Detector chip design (bottom) and the microscope image for one pixel
(top). A CPW feedline couples to 20 MKIDs, out of which four receive optical
from OMT probes through hybrid tees and different lengths of microstrip
transmission lines. Other detectors do not directly connect to the antenna and
are references for systematics and dark pickup.

##  III Dielectric Loss from TLS Model at 1-2 GHz

The two-level system (TLS) modeling in amorphous solids was first introduced
by [12] and [13], where groups of atoms may tunnel between sites corresponding
to different energy states. The dielectric modification by TLS can be derived
assuming a log-uniform distribution of tunneling states, and the subsequent
shift in resonant frequency for a resonator is given by [14]:

$\frac{f(T)-f_{0}}{f_{0}}=\frac{F\delta_{\mathrm{TLS}}^{0}}{\pi}\left[\operatorname{Re\Psi}\left(\frac{1}{2}-\frac{\hbar
f_{0}}{jk_{B}T}\right)-\log\frac{\hbar f_{0}}{k_{B}T}\right],$ (1)

where $f(T)$ is the resonator frequency at temperature $T$, $f_{0}$ is the
TLS-free resonance frequency, $F$ is the energy filling factor in the
amorphous dielectrics, $\delta_{\mathrm{TLS}}^{0}$ is the intrinsic TLS loss,
$\Psi$ is the digamma function, and $k_{B}$ is the Boltzmann constant.
Typically $F$ and $\delta_{\mathrm{TLS}}^{0}$ are degenerate when fitting
$f(T)$ vs. $T$ to the model, but we eliminated the degeneracy in our system
using parallel-plate capacitors with filling factors $F$ close to one. An
increase in stage temperature can break Cooper pairs and change the kinetic
inductance, resulting in a downshift of the resonance frequency [15]. But this
effect takes over after around 300 mK for aluminum and is beyond the range of
our study.

We measured the resonance frequencies for our resonators using a standard
vector network analyzer (VNA) setup with low-temperature attenuators and
amplifiers and a room-temperature last-stage amplifier. A sample measurement
was shown in Fig. 2 for a detector with SiNx dielectric. We fit for the model
in Eq. (1) to extract $F\delta_{\mathrm{TLS}}^{0}$. The fit result of
$\delta_{\mathrm{TLS}}^{0}$ is $2.2\times 10^{-3}$ for this resonator, which
is representative of detectors on the same chip. Fig. 3 summarizes the
distribution of $F\delta_{\mathrm{TLS}}^{0}$ from two chips, which corresponds
to the two groups in the histogram. $F\delta_{\mathrm{TLS}}^{0}$ within each
chip is uniform within $\pm 10^{-4}$ and the systematic shift between the two
chips is also at the $10^{-4}$ level. To conclude, this measurement yields
$\delta_{\mathrm{TLS}}^{0}=(2.0\pm 0.2)\times 10^{-3}$ for SiNx.

Figure 2: Resonator frequency shift $f(T)$ vs. temperature $T$ and the fit to
a TLS model. The fit $F\delta_{\mathrm{TLS}}^{0}$ is $2.2\times 10^{-3}$ for
this resonator. Figure 3: Histogram showing the measured
$F\delta_{\mathrm{TLS}}^{0}$ for all resonators fitting to the TLS model. The
two groups belong to two different detector chips. Albeit the small
difference, the two chips both show $F\delta_{\mathrm{TLS}}^{0}$ at the
$2\times 10^{-3}$ level. The statistical error within each chip is within
$10^{-4}$.

##  IV Dielectric Loss from $1/Q_{\mathrm{TLS}}$ at 1-2 GHz

We can achieve quality factors at a few hundred thousand for resonators using
interdigitated capacitors but only get quality factors at O($10^{3}$) level
for resonators with parallel-plate capacitors and the same metal targets. This
result suggests that most loss comes from TLS in SiNx dielectric layer and
that the internal quality factor $Q_{i}$ is limited by TLS loss. We can then
use $1/Q_{i}$ to approximate dielectric loss.

However, one caveat for this measurement is that the driving power can
saturate the TLS states and lead to underestimated
$F\delta_{\mathrm{TLS}}^{0}$. From [14], the resonator loss
$\delta_{\mathrm{TLS}}$ depends on power and temperature:

$\delta_{\mathrm{TLS}}=F\delta_{\mathrm{TLS}}^{0}\frac{\mathrm{tanh}(\frac{\hbar\omega}{2k_{B}T})}{\sqrt{1+\frac{\langle
n\rangle}{n_{c}}}},$ (2)

where $\langle n\rangle$ is the average photon number in the resonator, and
$n_{c}$ is the TLS saturation photon number. $\langle n\rangle$ is
proportional to power, so $1/Q_{i}\sim
1/Q_{\mathrm{TLS}}=\delta_{\mathrm{TLS}}\sim(1+P/P_{c})^{-1/2}F\delta_{\mathrm{TLS}}^{0}$,
where $P$ is the power and $P_{c}$ is the power for TLS saturation. We need
the power sufficiently low to use $1/Q_{i}$ as a proxy for
$F\delta_{\mathrm{TLS}}^{0}$.

Figure 4:  Resonator quality factor vs. driving power. Each curve is a
different detector. As the power increases, more TLS states become saturated,
leading to a higher $Q_{i}$. At powers below $-$125 dBm, the VNA data becomes
nosier, making it harder to constrain $Q_{i}$ in the low power limit. Above
$-$105 dBm, the $Q_{i}$ is limited by other loss mechanisms, such as loss in
the metal.  Figure 5: $Q_{i}$ distribution at $-$124 dBm feedline power.

Fig. 4 shows the resonator $Q_{i}$ vs. power at the feedline for all detectors
we measured. We observe an increasing trend of $Q_{i}$ vs. power due to TLS
saturation. At low powers below $-125$ dBm, the VNA loses S/N. As a result, we
cannot probe into the low-power limit and can only obtain a higher limit for
$Q_{i}$ (without TLS saturation) and a lower limit for
$F\delta_{\mathrm{TLS}}^{0}$. The histogram of resonator $Q_{i}$ at $-124$ dBm
is in Fig. 5 and centers around $Q_{i}\sim 1000\pm 100$, corresponding to a
constraint of $\delta_{\mathrm{TLS}}^{0}\mathrm{\gtrsim 1.0\times 10^{-3}}$.
The results are consistent with those from fitting to the TLS model (Section
III) in the same order of magnitude. We tried fitting the $Q_{i}$ dependence
on power to the $(1+P/P_{c})^{-1/2}$ model, but there is strong degeneracy
between saturation power $P_{c}$ and $F\delta_{\mathrm{TLS}}^{0}$. To improve
this method, we can switch to a lower-noise amplifier or increase the
integration time during data collection. We also note that the high-power
microwave $Q_{i}$ of our microstrip with SiNx is lower than that reported by
other groups [16, 17]. The $Q_{i}$ for resonators with aluminum inductors used
in this paper still has a positive slope vs. power at $-$90dBm, indicating
their $Q_{i}$ is still being limited by dielectric loss there. However, these
resonators started to bifurcate at $\sim-$90dBm. For all-Nb resonators with
higher bifurcation power, we noticed that the $Q_{i}$ can reach $10^{5}$ at
$-$70 dBm. For this paper, we are only interested in the low-power limit of
the $Q_{i}$ set by the dielectric loss. The other factors that limit our high-
power $Q_{i}$ are orders of magnitude lower than the TLS loss.

##  V Differential Optical Efficiency at 150 GHz

The third method compares optical efficiency for detectors coupled through
different lengths of microstrip lines, which requests an mm-wave optical
coupling setup. A temperature-controlled cryogenic blackbody provides a method
to sweep the incident power level, and is coupled to the detector sample box
through low-pass metal-mesh filters and infrared blockers (Fig. 6).

Figure 6: Optical testing setup. The cryogenic blackbody is mounted on a
copper plate with four heaters and thermometers for temperature control. An
infrared filter attenuates the power above 1 THz, and two metal-mesh filters
at 4 K and 1 K define a low-pass frequency window below 280 GHz. The detectors
are mounted in a sample box with feedhorns and backshorts. We installed the
optical components in a Bluefors dilution fridge shown on the left. The base
temperature of the fridge is 8 mK. The optical window on the right of the CAD
model came with the fridge and was not used for this paper.

The optical setup we built is compatible with both 150 and 220 GHz optical
detector chips. Fig. 7 shows the detectors and their boxes. The 150 and 220
GHz chips were fabricated and diced from the same silicon wafer.

Figure 7: 150 and 220 GHz sample boxes with detector chips installed. Left of
each subfigure is the box bottom with the detector chip. Right is the box lid
with feedhorns.

To measure the optical response of the detectors, we heat the blackbody to
different temperatures $T_{BB}$ and record the resonator frequency shift
$df/f_{0}$ vs. $T_{BB}$. Here $df$ is the frequency shift, and $f_{0}$ is the
resonant frequency. We took measurements for the 150 GHz device and show the
results in Fig. 8, where we identified three optical detectors with decreasing
optical response corresponding to microstrip meanders lengths of 6.4, 21.3,
and 68.3 mm. The detector with the longest microstrip length of 128.3 mm has
low optical response comparable to other dark detectors, suggesting its
optical power is too weak from too much attenuation. The frequencies for the
dark detectors also shifted slightly due to the chip heating. We keep the
detector stage at a constant $T_{\mathrm{stage}}=$247 mK where
$df/dT_{\mathrm{stage}}\sim 0$ to minimize the detector response to base
temperature variation. However, as the blackbody radiates more power to the
device, the temperature gradient between the chip and the stage PID
thermometer can increase as a function of blackbody temperature, causing
detector frequency shifts. We successfully reduced this effect by fine-tuning
the stage PID temperature such that $df/dT_{\mathrm{stage}}$ is closer to
zero. But since the device temperature ($T_{\mathrm{chip}}$) is not a constant
and depends on the blackbody temperature $T_{BB}$, $df/dT_{\mathrm{stage}}$
cannot always stay zero at varying $T_{BB}$ even if we PID control
$T_{\mathrm{stage}}$ to a constant. We will always have a small amount of
$T_{\mathrm{chip}}$ response, which can be monitored and corrected for using
dark detectors.

Figure 8: Frequency shift vs. cold load temperature. We have four optical
detectors coupled to 6.4, 21.3, 68.3, and 128.3mm meanders. The detector with
the longest length (128.3mm) did not show a clear optical response due to too
much attenuation. Dark detectors showed some response to cold load temperature
due to chip heating.

The next step is to extract the dielectric loss from the measured optical
responses. We use

$\tan\delta=\frac{c}{2\pi
L\sqrt{\xi_{r}}f_{\text{opt}}}\ln\left(\frac{\eta_{\text{long
}}}{\eta_{\text{short }}}\right)$ (3)

to calculate the loss through unit microstrip line length [8]. Here $c$ is the
speed of light, $L$ is the length difference, $f_{\text{opt}}$ is the optical
band central frequency (150 or 220 GHz depending on the chip style),
$\eta_{\text{short}}$($\eta_{\text{long }}$) is the optical efficiency for the
short (long) meander, and permittivity $\xi_{r}=7.0$. The fractional frequency
shift can be decomposed into

$df/f_{0}=P_{\mathrm{opt}}\frac{df/f_{0}}{dn_{\mathrm{qp}}}\frac{dn_{\mathrm{qp}}}{P_{\mathrm{opt}}}=P_{\mathrm{opt}}\frac{df/f_{0}}{dn_{\mathrm{qp}}}\frac{\eta\tau_{\mathrm{qp}}}{\Delta
V_{L}},$ (4)

where $P_{opt}$ is the optical power, $n_{\mathrm{qp}}$ is the quasiparticle
density, $\eta$ is the optical efficiency including line loss and pair-
breaking efficiency, $\tau_{qp}$ is the quasiparticle lifetime, $\Delta$ is
the bandgap, and $V_{L}$ is the inductor volume [18]. We designed our inductor
volume $V_{L}$ to be the same for all detectors. $\tau_{qp}$ and $\Delta$ are
the same for the same wafer. $P_{\mathrm{opt}}$ should be the same for the
same feedhorn, antenna, and blackbody illumination that covers the entire beam
of the detectors. We measured $df/(f_{0}dn_{\mathrm{qp}})$ from resonant
frequency vs. operating temperature. $n_{qp}$ depends on the operating
temperature $n_{\mathrm{qp}}=2N_{0}\sqrt{2\pi
k_{B}T\Delta}\exp\left(-\Delta/k_{B}T\right)$ [19], where $N_{0}$ is the
single spin density of states at the Fermi level. The difference for
$df/(f_{0}dn_{\mathrm{qp}})$ measured within 0.7% for different detectors.
Since all other terms are approximately the same, the $df/f_{0}$ vs. optical
power (scaled with blackbody temperature $T_{BB}$) slopes can be directly
divided to get the optical efficiency ratio, or $\eta_{\text{long
}}/\eta_{\text{short }}$ in Eq. 3. We get $\tan\delta\sim$0.003 from 21.3 mm/
6.4 mm detector responses and 0.005 from 68.3 mm/ 6.4 mm. On average
$\tan\delta\sim(4\pm 1)\times 10^{-3}$. We note that our loss tangent is
larger than reported by [20]. Our group is currently developing SiNx
dielectrics using several deposition methods: chemical vapor deposition (CVD),
ion beam-assisted sputtering (IBAS), and PECVD. We have measured microwave
loss tangents as low as $2x10^{-4}$ [21] in silicon-rich SiNx films deposited
by PECVD. A future paper will present additional details on the dielectric
optimization for mm-wave loss tangent; the focus of this paper is on our
measurement methodology. .

##  VI Measurement systematics

The systematics for our optical measurements include the chip heating effect
discussed in Section IV, optical leakage, and optical coupling variation among
different pixels. We can control chip heating by choosing an appropriate stage
PID temperature where the $df/dT_{\mathrm{stage}}$ responsivity is low and
using the dark detectors to estimate a correction. One concern about our
optical coupling is that different antennas will have slightly different
coupling efficiency if they are misaligned under the feedhorns. A series of
simulations suggest that the optical efficiency variation should be within ten
percent if we can align the antennas and feedhorns within 0.1 mm. Optical
leakage mitigation is a more challenging task [22]. Below we discuss the
leakage measurements and a method for reducing the optical leakage with
absorber chips.

Figure 9: Fractional frequency shift vs. blackbody temperature for dark
detectors with different numbers of feedhorns taped. The response levels for
most dark detectors are proportional to the number of open feedhorns. The
thick line is the average over all detectors (thinner lines).  Figure 10:
Fractional frequency shift vs. blackbody temperature for the dark detectors
without absorber chips, dark detectors with absorber chips, and optical
detectors. The dark pickup is at $\sim$10% of the response level for an
optical detector and can be mitigated by installing absorber chips. The thick
line is the average over all detectors (thinner lines).

We observed dark detector response at 10% compared to optical detectors before
any leakage mitigation. Fig. 9 shows the response level for the dark
detectors. We measured the optical response for detectors on the same device
with different numbers of feedhorns covered over the metallic tape. The
responses of all dark detectors were $\sim$1 ppm/K with all feedhorns open and
uniformly reduced in half after we covered two of the four feedhorns. The
amount of reduction is independent of the detector location on the device.
When we covered all feedhorns, the detector responses were all close to zero,
with a small slope likely caused by chip heating. The measurements are
consistent with a model where optical leakage through the bottom of the
feedhorn randomly scatter within the sample box and can terminate at any
inductors. The leaked photons can break Cooper pairs in the inductor and shift
the resonant frequency. The first data point of the blue line in Fig. 9 was an
outlier caused by stage temperature variation when we started the temperature
PID control.

To address this parasitic optical leakage, we designed and fabricated
dedicated absorbing chips and placed them both above and below the detector
chip to absorb spurious photons. Each absorber chip comprises a piece of
silicon with a tuned absorbing structure that is patterned into an aluminium
thin film. Fig. 10 shows the responses for optical detectors and dark
detectors with and without the absorber chips. The dark response was about 1
ppm/K before adding the absorber chips and was reduced to below measurement
noise level afterward. The nonlinear shift for the green curve at $T_{BB}>25$
K is likely caused by chip heating. A picture of the top absorber device is in
Fig. 11.

Figure 11: An absorber chip is mounted in the box lid for terminating spurious
photons. An additional absorber chip is underneath the detector chip and is
not shown in this picture. The side with absorbing patterns should face the
detector chip for both absorber chips. The other side of the absorber chip has
a metal layer that reflects the radiation to enhance the absorption.

##  VII Conclusions

This paper presents dielectric loss measurements for SiNx using three
different methods covering millimeter to centimeter wavelengths with the same
test structure. At $\sim$1-2 GHz, we measured SiNx loss to be $2\times
10^{-3}$ using the TLS fitting method and $\mathrm{\gtrsim 1.0\times 10^{-3}}$
by inverting $Q_{i}$ at low excitation power. At 150 GHz, we measured the loss
to be $\sim 4\times 10^{-3}$ using the differential optical efficiency method.
The loss at 150 GHz is a factor of $\sim$2 larger compared to the loss at
$\sim$1-2 GHz, which is consistent with NIST measurements [8]. Next, we will
measure dielectric loss for a few more materials, such as SiOx, silicon-rich
SiNx, and amorphous silicon. Early results showed a promising loss tangent of
$2\times 10^{-4}$ at 1 GHz for silicon-rich SiNx. We still need to understand
why the loss tangent at 150 GHz is larger than at $\sim$1 GHz. One possibility
is other loss mechanisms beyond TLS at mm-wave, which can be tested using
samples with lower loss, such as silicon-rich SiNx. We have made 220 GHz
samples and boxes and plan to extend the measurements to 220 GHz.

##  VIII Acknowledgements

Work at Argonne, including the use of the Center for Nanoscale Materials, an
Office of Science user facility, was supported by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences, and Office of High
Energy Physics, under Contract No. DE-AC02-06CH11357. Zhaodi Pan is supported
by ANL under the award LDRD-2021-0186.

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