#  Quantum transport observed in films of magnetic topological semimetal EuSb2

Mizuki Ohno  Department of Applied Physics, University of Tokyo, Tokyo
113-8656, Japan  Quantum-Phase Electronics Center (QPEC), University of Tokyo,
Tokyo 113-8656, Japan  Masaki Uchida<EMAIL_ADDRESS>Department of
Applied Physics, University of Tokyo, Tokyo 113-8656, Japan  Quantum-Phase
Electronics Center (QPEC), University of Tokyo, Tokyo 113-8656, Japan  PRESTO,
Japan Science and Technology Agency (JST), Tokyo 102-0076, Japan  Department
of Physics, Tokyo Institute of Technology, Tokyo 152-8550, Japan  Ryosuke
Kurihara  The Institute for Solid State Physics (ISSP), The University of
Tokyo, Kashiwa, Chiba 277-8581, Japan  RIKEN Center for Emergent Matter
Science (CEMS), Wako 351-0198, Japan  Susumu Minami  Department of Physics,
University of Tokyo, Tokyo 113-8656, Japan  Yusuke Nakazawa  Department of
Applied Physics, University of Tokyo, Tokyo 113-8656, Japan  Quantum-Phase
Electronics Center (QPEC), University of Tokyo, Tokyo 113-8656, Japan  Shin
Sato  Department of Applied Physics, University of Tokyo, Tokyo 113-8656,
Japan  Quantum-Phase Electronics Center (QPEC), University of Tokyo, Tokyo
113-8656, Japan  Markus Kriener  RIKEN Center for Emergent Matter Science
(CEMS), Wako 351-0198, Japan  Motoaki Hirayama  Department of Applied Physics,
University of Tokyo, Tokyo 113-8656, Japan  RIKEN Center for Emergent Matter
Science (CEMS), Wako 351-0198, Japan  Atsushi Miyake  The Institute for Solid
State Physics (ISSP), The University of Tokyo, Kashiwa, Chiba 277-8581, Japan
Yasujiro Taguchi  RIKEN Center for Emergent Matter Science (CEMS), Wako
351-0198, Japan  Ryotaro Arita  Department of Applied Physics, University of
Tokyo, Tokyo 113-8656, Japan  RIKEN Center for Emergent Matter Science (CEMS),
Wako 351-0198, Japan  Masashi Tokunaga  The Institute for Solid State Physics
(ISSP), The University of Tokyo, Kashiwa, Chiba 277-8581, Japan  RIKEN Center
for Emergent Matter Science (CEMS), Wako 351-0198, Japan  Masashi Kawasaki
Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
Quantum-Phase Electronics Center (QPEC), University of Tokyo, Tokyo 113-8656,
Japan  RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan

###### Abstract

We report fabrication of EuSb2 single-crystalline films and investigation of
their quantum transport. First-principles calculations demonstrate that EuSb2
is a magnetic topological nodal-line semimetal protected by nonsymmorphic
symmetry. Observed Shubnikov-de Haas oscillations with multiple frequency
components exhibit small effective masses and two-dimensional field-angle
dependence even in a 250 nm thick film, further suggesting possible
contributions of surface states. This finding of the high-mobility magnetic
topological semimetal will trigger further investigation of exotic quantum
transport phenomena by controlling magnetic order in topological semimetal
films.

##  I INTRODUCTION

Topological semimetals are characterized by nontrivial band crossings at nodal
points or nodal lines in the momentum space Fang et al. (2016); Burkov et al.
(2011); Lu and Shen (2017); Wang et al. (2017a); Armitage et al. (2018). In
topological nodal-point semimetals, Dirac or Weyl semimetals, these gapless
points give rise to exotic quantum transport phenomena Lu and Shen (2017);
Wang et al. (2017a). Under magnetic fields, for example, Fermi-arcs of the
surface state merge with the bulk state and form so-called Weyl orbits Potter
et al. (2014); Wang et al. (2017b), as recently observed in its quantum Hall
state Zhang et al. (2017); Lin et al. (2019); Zhang et al. (2019); Nishihaya
et al. (2019).

Topological nodal line semi-metals (TNLSMs) are characterized by the Zak phase
Zak (1989) in loops encircling the nodal lines. It has been theoretically
suggested that this topological feature appears in bulk Oroszlány et al.
(2018); Yang et al. (2018) and surface Chan et al. (2016); Bian et al. (2016);
Yamakage et al. (2016); Hirayama et al. (2017, 2018) quantum oscillations. In
this context, it is important to prepare high-mobility films of TNLSMs. On the
other hand, many compounds which possess nodal lines without spin-orbit
coupling (SOC) turn into topological insulators or nodal point semimetals when
considering SOC Fang et al. (2015); Kobayashi et al. (2017). In order to
protect these nodal lines from gapping by strong SOC, nonsymmorphic symmetry
is required Fang et al. (2015); Kobayashi et al. (2017); Zhao and Schnyder
(2016).

In general, time-reversal symmetry breaking destroys the robustness of nodal
lines Okugawa and Murakami (2017), while they remain stable only when the
magnetic ordering still respects space group symmetries Bzdušek et al. (2016);
Wang (2017). Therefore, there are very few reports about intrinsic magnetic
TNLSMs under SOC (single-layer GdAg2 Ormaza et al. (2016); Feng et al. (2019)
and GdSbTe Hosen et al. (2018); Sankar et al. (2019)). Many of the nodes are
slightly gapped on the entire lines (Fe3GeTe2 Chen et al. (2013); Kim et al.
(2018) and Fe${}_{3}X$ ($X=$Ga, Al) Sakai et al. (2020)) or except the Weyl
points (Co3Sn2S2 Liu et al. (2018) and Co2MnGa Chang et al. (2017); Sakai et
al. (2018); Belopolski et al. (2019)) under SOC. Moreover, quantum
oscillations have not been observed in all of these compounds. Hence, to
identify new magnetic TNLSMs is strongly called for exploring unique quantum
transport phenomena.

CaSb2 is a novel TNLSM where nodal lines are protected by two-fold screw
symmetry along $b$ axis even in the presence of SOC Funada et al. (2019a).
Recently, superconductivity has been also observed in polycrystalline samples
Ikeda et al. (2020). In EuSb2, on the other hand, large magnetic moments are
expected to be introduced while keeping the unique crystal structure. Although
antiferromagnetic (AFM) ordering of nearly Eu2+ spins has been investigated in
the 1980s Hulliger and Schmelczer (1978); Niggli and Hulliger (1984),
electronic structures and transport properties have long been unknown for
EuSb2. Here we demonstrate that EuSb2 is a new magnetic TNLSM protected by
nonsymmorphic symmetry of the space group $P2_{1}$/$m$. We fabricate single-
crystalline EuSb2 films by molecular beam epitaxy and investigate their
magnetotransport. Observed Shubnikov-de Haas (SdH) oscillations with multiple
frequency components indicate small effective masses and two-dimensional (2D)
field-angle dependence characteristics in topological semimetals.

##  II EXPERIMENTAL METHODS

EuSb2 epitaxial films were grown on single-crystalline (11$\bar{2}$0) Al2O3
substrates in an Epiquest RC1100 chamber. The molecular beams were
simultaneously provided from conventional Knudsen cells containing 3N Eu and
6N Sb. The growth temperature was set at 850 ∘C, and the beam equivalent
pressures, measured by an ionization gauge, were set to 1${\times}$10-5 Pa for
Eu and 4.0${\times}$10-4 Pa for Sb, considering the Eu-Sb binary phase diagram
Abdusalyamova and Vasilyeva (2011). The film thicknesses were typically set at
250 nm for structural characterization and magnetotransport measurements, and
750 nm for magnetization measurements. The growth rate was about 0.3 Å/s.
High-field resistivity and magnetization measurements were performed at the
International MegaGauss Science Laboratory in the Institute for Solid State
Physics at the University of Tokyo. Resistivity up to 58 T was measured by a
standard four-probe method for 200 $\mu$m-width multi-terminal Hall bars,
using a non-destructive pulsed magnet Uchida et al. (2017). Temperature
dependence was also measured using a Quantum Design Physical Properties
Measurement System. Magnetization up to 27 T was measured by a conventional
induction method with coaxial pickup coil Yoshida et al. (2009). Temperature
dependence was also measured using a superconducting quantum interference
device magnetometer in a Quantum Design Magnetic Property Measurement System.
First-principles calculations of the band structure with SOC were performed by
using the VASP package Kresse and Furthmüller (1996a, b); Kresse and Joubert
(1999), with the experimentally determined lattice constants Hulliger and
Schmelczer (1978). The generalized gradient approximation of Perdew-Burke-
Ernzerhof was adopted for the exchange-correlation functional Perdew et al.
(1996). $16\times 16\times 10$ $k$-point mesh with Monkhorst-Pack scheme
Monkhorst and Pack (1976) was used for the Brillouin zone sampling of the
primitive cell and Gaussian smearing with a width of $0.02$ eV was applied.

##  III RESULTS AND DISCUSSION

Figure 1:  (a) Crystal structure of EuSb2 with AFM ordering of Eu2+ spins at
zero field. The solid lines indicate the unit cell, which is doubled along the
$a$ axis by the AFM ordering. The screw symmetry that protects the nodal lines
is represented in combination with rotational ($C_{2}$) and translational
($T$) operations. (b) Brillouin zone of EuSb2. The $\Gamma$-Z line is parallel
to the $k_{y}$ axis. (c) Band structure of EuSb2 calculated for the AFM phase
including SOC. Band crossings at the high-symmetry points, which are protected
by the two-fold screw symmetry, are denoted by blue arrows. (d) Line nodes
appearing on the $k_{y}=\pi/b$ plane in the presence of SOC.

Figure 1(a) shows the crystal structure of EuSb2. This structure is
characterized by Sb zigzag chains running parallel to the $b$ axis and mirror
planes in the $ac$ plane. This Sb chain forms Sbnn- and the valence of Eu
becomes divalent (Eu2+). These Eu spin moments are oriented along the [010]
direction and antiferromagnetically ordered with a propagation vector of
$(1/2,0,0)$, corresponding to the magnetic space group $P_{a}2_{1}/m$ Niggli
and Hulliger (1984). Therefore, this AFM ordering does not break the two-fold
screw symmetry ($2_{1}/m$) which protects the nodal lines on the $k_{y}=\pi/b$
plane. It is thus expected that the nodal lines stably exist in the AFM phase.

The Brillouin zone and band structure calculated for the AFM phase are
presented in Figs. 1(b) and 1(c). While flat bands located between -0.3 and
-0.7 eV are attributed to the Eu 4$f$ orbitals, dispersive bands crossing
$E_{\mathrm{F}}$ originate from the Sb 5$p$ ones, resembling TNLSM CaSb2.
There are band crossing points at the high-symmetry points Z, A, E, and D as
in CaSb2 Funada et al. (2019a), and which are connected by nodal lines on the
$k_{y}=\pi/b$ plane, as shown in Fig. 1(d). This indicates that EuSb2 is a
magnetic TNLSM even in the presence of SOC.

Figure 2:  (a) XRD ${\theta}$-${2\theta}$ scan of a EuSb2 film grown on Al2O3
(11$\bar{2}$0) substrate. Al2O3 substrate peaks are marked with an asterisk.
Epitaxial relation between (b) the EuSb2 film and (c) the Al2O3 substrate
viewed along the out-of-plane direction. (d) Cross-sectional image of the
EuSb2 film, taken by high-angle annular dark-field scanning transmission
electron microscopy. Energy dispersive X-ray spectrometry map for Eu $L$ (e)
and Sb $L$ (f) edges in the boxed region in (d). The Sb zigzag chains along
the $b$ axis are atomically resolved.

X-ray diffraction (XRD) ${\theta}$-${2\theta}$ scan in Fig. 2(a) reveals sharp
reflections from the (001) EuSb2 lattice plane without any impurity phases.
Figures. 2(b) and 2(c) show the epitaxial relation viewed along the out-of-
plane $c^{\ast}$ axis, with a mismatch of 0.7 % between the $c$ axis of the
(11$\bar{2}$0) Al2O3 plane and the $b$ axis of the (001) EuSb2 plane. The $c$
axis of EuSb2 is tilted by approximately $13$ deg. from the $c^{\ast}$ axis
perpendicular to the $a$-$b$ plane. Figure 2(d) shows a cross-section image of
the EuSb2 film, taken by high-angle annular dark-field scanning transmission
electron microscopy. The periodic arrangement of EuSb2 is clearly confirmed
which corresponds to the EuSb2 crystal structure determined by single-crystal
XRD Hulliger and Schmelczer (1978). Elemental maps are taken by energy
dispersive X-ray spectrometry as shown in Figs. 2(e) and 2(f). The Sb zigzag
chains along the $b$ axis are atomically resolved. Further XRD
characterization was also performed to confirm high crystallinity and in-plane
orientation (see Supplementary Materials Supplement ).

Figure 3:  (a) Temperature dependence of the resistivity
${\rho_{\mathrm{xx}}}$ measured at zero field. The inset shows temperature
dependence of the magnetization $M$, taken with applying an out-of-plane
magnetic field of $1.0$ T. Both ${\rho_{\mathrm{xx}}}$ and $M$ exhibit a clear
kink at the Néel temperature ${T_{\mathrm{N}}}$ of $26.2$ K, as marked by a
circle and a square respectively. (b) ${\rho_{\mathrm{xx}}}$ taken with
sweeping the out-of-plane magnetic field at various temperatures. The inset
shows second derivative of $\rho_{\mathrm{xx}}$, where the saturation field is
more clearly confirmed as marked by a triangle. (c) Magnetization curve taken
at various temperatures, where the saturation field is marked by a diamond.
(d) Magnetic phase diagram obtained for out-of-plane magnetic fields
($B\parallel c^{\ast}$). The dashed curve represents the boundary between the
AFM and FM phases. As shown in the inset, the antiferromagnetically ordered
Eu2+ spins along the $b$ axis at zero field are gradually canted toward the
$c^{\ast}$ axis upon increasing the magnetic field.

Figure 3(a) summarizes temperature dependence of the resistivity. The
resistivity exhibits metallic behavior down to $2$ K with a residual
resistivity ratio
(RRR$=\rho_{\mathrm{xx}}(300\,\mathrm{K})/\rho_{\mathrm{xx}}(2\,\mathrm{K})$)
of $26$, reflecting high quality of the film. A clear kink observed at 26.2 K
corresponds to the Néel temperature ($T_{\mathrm{N}}$) Hulliger and Schmelczer
(1978), which is also seen in temperature dependence of the magnetization
(inset). The resistivity decreases rapidly with decreasing temperature below
$T_{\mathrm{N}}$ probably due to suppression of the magnetic fluctuations.
Figure 3(b) presents out-of-plane magnetic field sweeps of the resistivity up
to 58 T at various temperatures. The resistivity at the base temperature of
1.4 K exhibits a cusp-like behavior at 14.1 T, which is more clearly seen in
its second derivative (inset). This saturation field shifts to the lower
values with increasing temperature and then disappears above
${T_{\mathrm{N}}}$. Figure 3(c) shows magnetization curves measured at various
temperatures. The magnetization taken at $1.4$ K increases approximately
linearly with field and then saturates at about 7$\mu_{B}$, indicating
transition from the AFM phase at low fields to the forced ferromagnetic (FM)
one. This behavior is consistent with previous magnetization measurements
Hulliger and Schmelczer (1978). Below ${T_{\mathrm{N}}}$, the saturation field
shifts to lower values with increasing temperature, as confirmed in the
resistivity change. Above ${T_{\mathrm{N}}}$, the kink indicating saturation
disappears and the magnetization curve follows the Brillouin function.
Therefore, the kink indicates the phase transition from AFM to FM phase, where
Eu2+ $S=7/2$ spins are polarized along the ${c^{\ast}}$ direction by the
magnetic field.

Figure 3(d) summarizes a $B$-$T$ phase diagram as determined by the above
transport and magnetization measurements. Upon increasing the out-of-plane
magnetic field, the Eu2+ spin moments, which are initially oriented along the
$b$ axis at zero field, are gradually canted toward the $c^{\ast}$ axis
without spin-flop transition. For the in-plane field, on the other band, the
magnetization below ${T_{\mathrm{N}}}$ exhibits a spin-flop transition at
$2.4$ T, corresponding to reorientation of the spin moments perpendicular to
the applied field (see Supplementary Materials Supplement ). EuSb2 exhibits
large positive magnetoresistance both in the AFM and FM phases, similar to
nonmagnetic TNLSMs (such as ZiSiS Sankar et al. (2017) and CaSb2 Funada et al.
(2019a)). In the FM phase, the Eu2+ localized spins are polarized along the
magnetic-field direction and electron scattering by the localized spins is
expected to be further suppressed.

Figure 4:  (a) Temperature dependence of the SdH oscillation
$\Delta\rho_{\mathrm{xx}}$, plotted against $1/B$ after subtracting a smooth
background from $\rho_{\mathrm{xx}}$. (b) FFT spectra of the SdH oscillation
$\Delta\rho_{\mathrm{xx}}$ for the same set of temperatures. As shown in the
inset, temperature dependence of the FFT amplitudes of $\alpha,\beta$, and
$\gamma$ peaks is well fitted by the Lifshitz-Kosevich formula. (c)
Magnetoresistance $(\rho_{\mathrm{xx}}(B)-\rho_{\mathrm{xx}}(0$
$\mathrm{T}))/\rho_{\mathrm{xx}}(0$ $\mathrm{T})$ taken with rotating the
magnetic field direction at $4.2$ K. The field angle $\theta$ is measured from
the normal direction. (d) Field angle dependence of the oscillation
frequencies obtained from the FFT spectra at $4.2$ K. All the peaks follow a
${\cos\theta}^{-1}$ dependence (dotted curves) rather than being constant
(dashed curves).

The SdH oscillations appear at high fields below 40 K, where the Eu2+ spin
moments are oriented along the out-of-plane direction and band structures for
the FM phase are expected to be realized (see Supplementary Materials
Supplement ). Figure 4(a) plots the oscillatory component of the resistivity
$\Delta\rho(B)$ as a function of $1/B$ involved at various temperatures.
Apparently, there is more than one type of frequency. As shown in Fig. 4(b),
the fast Fourier transform (FFT) of $\Delta\rho$ yields three major
frequencies at $B_{\mathrm{F}}^{\alpha}=247$ T, $B_{\mathrm{F}}^{\beta}=372$
T, and $B_{\mathrm{F}}^{\gamma}=583$ T and their harmonics. The obtained
frequencies suggest the presence of three Fermi pockets ($\alpha$, $\beta$,
$\gamma$) perpendicular to the $c^{\ast}$ axis. As summarized in Table 1, the
Fermi surface cross-section areas $A_{\mathrm{F}}$ perpendicular to the
applied magnetic field are calculated to be $2.3$, $3.6$, and $5.6\times
10^{-2}$ Å-2, using the Lifshitz-Onsager relation,
$B_{\mathrm{F}}=\frac{\hbar}{2\pi e}A_{\mathrm{F}}$ with the electron charge
$e$ and the reduced Planck constant $\hbar$. These cross-sectional areas are
typically small, taking only $\sim 2$ % of the entire BZ on the
$k_{x}$-$k_{y}$ plane. The FFT amplitudes of these three peaks gradually
decrease with elevating temperature. As confirmed in the inset, their
temperature dependence can be suitably fitted using the standard Lifshitz-
Kosevich formula with thermal damping factors

$\displaystyle\Delta
R(T)\propto\frac{2\pi^{2}k_{B}T/\hbar\omega_{c}}{\sinh\left(2\pi^{2}k_{B}T/\hbar\omega_{c}\right)},$
(1)

where $k_{B}$ is Boltzman constant and $\omega_{c}=\frac{eB}{m^{\ast}}$ the
cyclotron frequency. The effective masses $m^{\ast}$ are estimated as small as
about 0.11$m_{0}$ for all three peaks with the free electron mass $m_{0}$, as
listed in Table 1. Importantly, this value is comparable to or even smaller
than one reported in many other topological semimetals, e.g., 0.049$m_{0}$
(bulk) Uchida et al. (2017) and 0.27$m_{0}$ (surface) Nishihaya et al. (2019)
in Cd3As2, 0.07$m_{0}$ (bulk) Zhang et al. (2017b) and 0.50$m_{0}$ (surface)
Nair et al. (2020) in TaAs, and 0.04-0.18$m_{0}$ (bulk) Matusiak et al. (2017)
in ZrSiS. The Fermi velocity
$v_{\mathrm{F}}=\frac{\hbar}{m^{\ast}}\sqrt{\frac{A_{\mathrm{F}}}{m^{\ast}}}$
is also estimated as large as $\sim 4\times 10^{5}$ ms-1, similar to other
topological semimetals Uchida et al. (2017); Nishihaya et al. (2019); Zhang et
al. (2017b); Nair et al. (2020); Matusiak et al. (2017). More detailed
analysis of the SdH oscillations indicates that the mobility is estimated as
high as about $200$ cm2V-1s-1 (for details see Supplementary Materials
Supplement ).

Table 1: Parameters for the three Fermi pockets extracted from the SdH oscillations. carrier |  $B_{\mathrm{F}}$(T) |  $A_{\mathrm{F}}$ (Å-2) |  $m^{\ast}$ ($m_{0}$) |  $v_{\mathrm{F}}$ ($10^{5}$ ms-1)
---|---|---|---|---
$\alpha$ | 247 | 0.023 | 0.109 | 3.66
$\beta$ | 372 | 0.036 | 0.113 | 4.39
$\gamma$ | 566 | 0.056 | 0.117 | 5.24

Figure 4(c) shows field-angle dependence of the SdH oscillations taken at
$4.2$ K. The magnetoresistance is significantly suppressed with rotation of
the field direction from the out-of-plane ($\theta=0^{\circ}$, $B\parallel
c^{\ast}$) to the in-plane ($\theta=90^{\circ}$, $B\parallel a$-$b$)
direction. In addition, peaks and valleys of the oscillations shift to higher
fields upon increasing $\theta$. As confirmed in Fig. 4(d), although the
oscillation amplitude becomes small and it is difficult to follow the
frequency at higher angles ($\theta>40^{\circ}$), all frequencies extracted
from the FFT spectra can be fitted to $1/\cos\theta$, indicating that the
quantum transport observed in these EuSb2 films originates from the 2D
electronic structure.

One possible origin of the 2D field-angle dependence is that EuSb2 hosts a 2D
bulk Fermi surface structure. However, most of the calculated Fermi surfaces
are three-dimensional (3D), except only one 2D-like Fermi surface with low
Fermi velocity (for details see Supplementary Materials Supplement ). Another
possible origin is the quantum confinement effect. When the film thickness is
comparable to the de Broglie wavelength of the carriers, the 3D bulk state is
confined to form a 2D quantum-well state. However, the film thickness of 250
nm is considered large enough to maintain the 3D state, because even in Dirac
semimetal Cd3As2 with extremely low carrier density and large Fermi velocity
exhibits a 3D state above 100 nm film thickness Uchida et al. (2017). A third
possibility is surface states of EuSb2. By fitting $\Delta\rho_{\mathrm{xx}}$
using the LK formula, we obtain nontrivial Berry Phases for the $\beta$ and
$\gamma$ pockets (for details see Supplementary Materials Supplement ). These
bands may be non-trivial surface states, which are likely to be protected by
the combination of two-fold screw and time reversal symmetry. Another
possibility is that they are trivial surface states formed on the $(001)$
polar surface, which can also obtain non-trivial phases with the Rashba
splitting. In any case, it is unobvious why the 2D conduction state is much
more dominant than the 3D state and further studies are needed to determine
its origin.

##  IV CONCLUSION

In summary, we have fabricated single-crystalline EuSb2 films by molecular
beam epitaxy and have studied quantum transport at high fields. First-
principles calculations have demonstrated that EuSb2 hosts topological nodal
lines protected by nonsymmorphic symmetry, which remains preserved even under
AFM ordering. Effective masses extracted from multiple SdH oscillations are
fairly small, which is characteristic of topological semimetals. The 2D field-
angle dependence of the SdH oscillations suggests the possibility of surface
quantum transport in EuSb2. Our finding of the new magnetic TNLSM and
observation of quantum transport will stimulate further investigations of
exotic quantum transport as represented by Weyl orbit and topological phase
transitions in magnetic topological semimetals.

##  V ACKNOWLEDGMENTS

We thank M.-T. Huebsch and T. Yu for fruitful discussions. This work was
supported by JST PRESTO No. JPMJPR18L2 and CREST Grant No. JPMJCR16F1, Japan
and by Grant-in-Aids for Scientific Research on Innovative Areas No. 19H05825,
Scientific Research (S) No. 16H06345, and Scientific Research (B) No.
JP18H01866 from MEXT, Japan.

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# Near-Quantum-limited Haloscope Detection of Dark Photon Dark Matter Enhanced
by a High-Q Superconducting Cavity

Runqi Kang  CAS Key Laboratory of Microscale Magnetic Resonance and School of
Physical Sciences, University of Science and Technology of China, Hefei
230026, China  CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China
Hefei National Laboratory, Hefei 230088, China  Man Jiao  CAS Key Laboratory
of Microscale Magnetic Resonance and School of Physical Sciences, University
of Science and Technology of China, Hefei 230026, China  Institute of Quantum
Sensing and School of Physics, Zhejiang University, Hangzhou 310027, China  Yu
Tong  CAS Key Laboratory of Microscale Magnetic Resonance and School of
Physical Sciences, University of Science and Technology of China, Hefei
230026, China  CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China
Yang Liu  International Quantum Academy, Shenzhen 518048, China  Youpeng Zhong
Shenzhen Institute for Quantum Science and Engineering, Southern University of
Science and Technology, Shenzhen, China  International Quantum Academy,
Shenzhen 518048, China  Yi-Fu Cai  Deep Space Exploration Laboratory/School of
Physical Sciences, University of Science and Technology of China, Hefei, Anhui
230026, China  CAS Key Laboratory for Researches in Galaxies and
Cosmology/Department of Astronomy, School of Astronomy and Space Science,
University of Science and Technology of China, Hefei, Anhui 230026, China
Jingwei Zhou<EMAIL_ADDRESS>CAS Key Laboratory of Microscale Magnetic
Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Hefei National Laboratory, Hefei 230088, China
Xing Rong<EMAIL_ADDRESS>CAS Key Laboratory of Microscale Magnetic
Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Hefei National Laboratory, Hefei 230088, China
Jiangfeng Du<EMAIL_ADDRESS>CAS Key Laboratory of Microscale Magnetic
Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Hefei National Laboratory, Hefei 230088, China
Institute of Quantum Sensing and School of Physics, Zhejiang University,
Hangzhou 310027, China

###### Abstract

We report new experimental results on the search for dark photons based on a
near-quantum-limited haloscope equipped with a superconducting cavity. The
loaded quality factor of the superconducting cavity is $6\times 10^{5}$, so
that the expected signal from dark photon dark matter can be enhanced by more
than one order compared to a copper cavity. A Josephson parametric amplifier
with a near-quantum-limited noise temperature has been utilized to minimize
the noise during the search. Furthermore, a digital acquisition card based on
field programmable gate arrays has been utilized to maximize data collection
efficiency with a duty cycle being 100$\%$. This work has established the most
stringent constraints on dark photons at around 26.965 $\mu$eV. In the future,
our apparatus can be extended to search for other dark matter candidates, such
as axions and axion-like particles, and scrutinize new physics beyond the
Standard Model.

Figure 1:  Simplified diagram of the experimental apparatus. The green box
stands for the detection part. It includes a cylindrical superconducting
cavity of high quality factor and an output line with an amplifier chain for
the measurement of the signal. $A^{\prime}$ and $A$ refer to the dark photon
and the photon, respectively. The blue box stands for the calibration part. It
includes three input lines and a 50 Ohm load. The inset is a photograph of the
cavity. Its inner side is finely polished for a high Q factor. MW signals are
coupled to the transmission lines through two coaxial probes. The two probes
are fabricated on the opposite sides of the cavity to avoid their direct
coupling.

Dark matter is one of the most enigmatic and salient topics in modern physics
[1, 2, 3, 4, 5]. Although there are abundant phenomena that point to dark
matter, its existence yet remains a mystery [6, 7, 8, 9]. The dark photon is
an important candidate for dark matter, and also a possible portal that
connects the Standard Model sector and the dark sector. It is claimed to be
helpful to explain many unsolved problems in physics, such as the velocity
discrepancy of galaxies [6], the cosmic ray anomaly [10, 11], the muon
anomalous magnetic moment [12, 13] and the W-boson mass anomaly [14, 15, 16].

The dark photon is a spin-1 massive boson that rises from a slight extension
of the Standard Model [17, 18, 19]. It couples to the ordinary photon via the
following Lagrangian:

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}^{2}-\frac{1}{4}V_{\mu\nu}^{2}-\frac{m_{A^{\prime}}^{2}}{2}A^{\prime}_{\mu}A^{\prime\mu}+\frac{\chi}{2}F_{\mu\nu}V^{\mu\nu},$
(1)

where $A_{\mu}$ and $A^{\prime}_{\mu}$ are the gauge fields of the ordinary
photon and the dark photon, respectively,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and
$V_{\mu\nu}=\partial_{\mu}A^{\prime}_{\nu}-\partial_{\nu}A^{\prime}_{\mu}$ are
the corresponding electromagnetic field tensors, $m_{A^{\prime}}$ is the mass
of the dark photon, and $\chi$ is the kinetic mixing, through which the dark
photon interacts with the ordinary photon. The dark photon mass,
$m_{A^{\prime}}$, is a free parameter that can take a wide range of values,
while the kinetic mixing $\chi$ can be very small. These two features render
the search for the dark photon an extremely challenging task.

The haloscope is one of the most widely used methods for dark photon
searching. It is based on the fact that dark photons can convert to ordinary
photons in a microwave cavity due to the kinetic mixing term [20, 21, 22]. The
power of the dark photon signal is

$P_{\rm
s}(f)=\frac{m_{A^{\prime}}\rho_{A^{\prime}}}{\hbar}\chi^{2}VC\frac{Q_{L}Q_{a}}{Q_{L}+Q_{a}}\frac{\beta}{1+\beta}L(f,f_{c},Q_{L}),$
(2)

where $\rho_{A^{\prime}}=0.45$ GeV/cm3 is the density of dark matter [23],
$\beta$ is the coupling coefficient between the cavity and the output port,
$f_{c}$, $V$, $C$ are the resonant frequency, the volume and the filling
factor of the cavity, respectively, $Q_{L}$ and $Q_{a}$ are the quality factor
of the cavity and dark photons, respectively, and
$L(f,f_{c},Q_{L})=[1+4(Q_{L}(f-f_{c})/f_{c})^{2}]^{-1}$ is the Lorentzian line
shape. Many groups across the world have been working on searching for dark
photons. Projects such as MADMAX [24], QUAX [25, 26] and ADMX [27] use
cavities to conduct resonant detections. The SQuAD experiment [28] utilizes
qubits to enhance the sensitivity. Experiments like FUNK [29], BREAD [30],
DOSUE [31], QUALIPHIDE [32], BRASS-p [33], and Dark SRF [34] utilize the non-
resonant methods to achieve large bandwidths. The signal-to-noise ratio (SNR)
of a haloscope system is typically limited by factors such as the quality
factor and volume of the cavity, the noise temperature and the data
acquisition efficiency. Plentiful efforts has been made to improve the SNR of
the detection systems. For example, HAYSTAC employed JPAs to minimize the
noise [61], while QUAX used a superconductive cavity with a high quality
factor to increase the potential dark matter signal [36]. Groups like ADMX and
CAPP fabricated cavities with large volumes to enhance the potential signal
[27, 37]. In order to further improve the searching ability, it is desirable
to simultaneously push these key parameters to their limits. However, this is
practically challenging due to factors such as frequency drifts, difficulties
in fabrication of the high-Q, large-volume cavity, dead time of electronic
devices, and compatibility issues.

Here we focus on a promising region of the dark photon mass, which is in the
order of tens of $\mathsf{\mu}$eV [38, 39, 40]. In order to improve the SNR, a
superconducting cavity with a high quality factor and a large volume was
utilized to maximize the potential dark photon signal. To minimize the noise,
a Josephson Parametric Amplifier (JPA) with a near-quantum-limited noise
temperature was installed. Additionally, we employed a homemade DAQ card based
on the Field Programmable Gate Array (FPGA) to achieve a 100$\%$ duty cycle
[41]. The hitherto highest sensitivity to $\chi$ is achieved in a 0.4 neV
region around 26.96514 $\mu$eV. While the constraints on $\chi$ have never
touched the $10^{-16}$ line in prior searching efforts, this work sets an
upper limit of $\chi$ of $6.0\times 10^{-16}$ at 90$\%$ confidence level at
the center of the region.

Figure 2:  Calibration results. (a) Reflection coefficients of Port 1 of the
cavity. The resonant frequency of the cavity, $f_{c}$, is 6.52014 GHz. (b) The
relation between the noise power and the temperature of the 50 Ohm load. The
noise temperature of the HEMT chain $T^{\prime}_{\rm n}$ is 1.9 K. (c)
Calibration result of the gain of the JPA. The red and yellow lines refer to
the transmission coefficients S21 from the Bypass line to the Output line when
the JPA is on and off, respectively. Their difference is plotted as the blue
line. The gain of the JPA at $f_{c}$ is $G_{\rm J}=$20 dB.

Figure 1 shows the simplified diagram of the experimental setup (see more
details in Sec. I in [44]), which consisted of two main parts: the detection
part for searching dark photons and the calibration part for calibrating
significant parameters of the detection part. A temperature as low as 21 mK
and extremely stable was achieved with a dilution refrigerator.

The detection part is presented in the green box in Fig. 1. It includes a
superconducting cavity, an amplification chain, and a DAQ card. The
cylindrical superconducting cavity working at the TM010 mode performed as an
antenna to collect the dark photon signal. Dark photons inside the cavity
would continuously oscillate into ordinary photons due to the kinetic mixing
and then generate a microwave (mw) signal. The inset of Fig. 1 presents the
picture of the superconducting cavity. The cavity was made of 6061 aluminum
alloy, whose critical temperature is around 1 K. The inner side of the cavity
was finely polished to enhance the Q factor. The diameter and length are 35.31
mm and 150.00 mm, respectively. The electric field of the TM010 mode is along
the cylindrical axis of the cavity, thus the filling factor can be expressed
as

$C=\frac{|\int dV\mathbf{E}|^{2}}{V\int
dV|\mathbf{E}|^{2}}\langle\cos^{2}\theta\rangle,$ (3)

where $\theta$ stands for the angle between the polarization direction of the
dark photon field and the electric field. In the random polarization scenario,
$\langle\cos\theta\rangle$ equals to 1/3 [22, 42]. The simulation result shows
that the filling factor $C$ is 0.23. There were two ports in the cavity: the
strong one (Port 1) for readout and the weak one (Port 2) for injection of a
simulated dark photon signal. The coupling from the cavity to the transmission
lines was achieved through coaxial probes. In order to avoid direct coupling
between the two probes, they were fabricated on the opposite sides of the
cavity. The amplification chain was connected to the Port 1. A JPA accompanied
by a circulator was taken as the first-stage amplifier since its noise
temperature approaches the quantum limit $T_{n}=\hbar\omega/k_{B}$ [43]. The
signal was further amplified by High-Electron-Mobility Transistor (HEMT)
amplifiers to a level where the electrical noise doesn’t dominate, and readout
by a homemade DAQ card based on FPGAs (see more details in Sec. II in [44]).
The parallel sampling module and Fast-Fourier-Transform (FFT) modules of the
DAQ card allowed for simultaneous data collection and frequency spectrum
calculation with a 100$\%$ duty cycle.

The calibration part is indicated by the blue box in Fig. 1. It aimed to
measure significant parameters of the detection part, including the resonant
frequency of the cavity and the noise temperature of the HEMTs. The Strong
line allowed for the measurement of the resonant frequency of the cavity. The
50 Ohm load and a heater (not shown) played the role of a temperature-variable
black radiation source, and a low-temperature switch supported convenient
shift between the cavity and the 50 Ohm load. They were used to measure the
noise temperature of the HEMTs through the Y-factor method. The Bypass line
was used to measure the temperature noise and the gain of the JPA. The Weak
line was used for injection of a simulated dark photon signal verification of
the detection system (see more details in Sec. III in [44]).

To locate the resonant frequency $f_{c}$ of the cavity, the reflection
coefficient of Port 1 of the cavity was measured, as presented in Fig 2(a).
The result is fitted to the equation [45]

$S_{11}=\alpha
e^{i\psi}[1-\frac{2\beta_{1}/(1+\beta_{1})e^{i\phi}}{1+2iQ_{L}(f/f_{c}-1)}],$
(4)

where $\alpha$ and $\psi$ are the power loss and the phase shift during
transmission, respectively, and the parameter $\phi$ describes the impedance
mismatch, which makes the spectrum asymmetric. The fitting gives $f_{c}=$
6.52014 GHz.

During the calibration of the HEMTs, the pumping of the JPA was off, and the
switch was thrown to the 50 Ohm load. The thermal noise power of the system
can be expressed as

$P_{\rm t}=G_{\rm H}B[hf_{c}(\frac{1}{e^{hf_{c}/k_{\rm B}T^{\prime}_{\rm
n}}-1}+\frac{1}{2})+k_{\rm B}T_{\rm b}],$ (5)

where $G_{\rm H}$ is the gain of the HEMT chain, $k_{\rm B}$ is the Boltzmann
constant, $B$ is the bin width of the spectrum, $T^{\prime}_{\rm n}$ is the
noise temperature of the HEMT chain, and $T_{\rm b}$ is the physical
temperature of the 50 Ohm load. By varying $T_{\rm b}$ and fitting the
temperature dependence of $P_{\rm t}$, as plotted in Fig 2(b), $G_{\rm
H}=76.9\pm 0.3$ dB and $T^{\prime}_{\rm n}=1.9\pm 0.1$ K are obtained.

The transmission coefficients from the Bypass line to the Output line as the
pumping of the JPA was on and off are shown as the red and yellow lines in
Fig. 2(c), respectively. The blue line refers to their difference, i.e., the
gain of the JPA, which is 20 dB at the resonant frequency of the cavity. The
noise temperature of the JPA $T_{\rm n}$ is calculated to be $297\pm 22$ mK
from the formula:

$T_{\rm n}=\frac{1}{G_{\rm J}}\frac{P_{\rm J\\_on}}{P_{\rm
J\\_off}}T^{\prime}_{\rm n},$ (6)

where $G_{\rm J}$ is the gain of the JPA, $P_{\rm J\\_on}$ and $P_{\rm
J\\_off}$ are the thermal noise power measured from the Output line with the
JPA on and off, respectively. Since the bandwidth of the JPA is much broader
than that of the cavity, its gain and noise temperature can be considered
uniform in the detection region.

During the detection of dark photons, all the inputs except the pumping of the
JPA were off. The data collection lasted for six hours. Forty thousand raw
spectra have been obtained. The frequency bin width $B$ of each spectrum was
0.48 kHz. These raw spectra were divided into 100 subruns, each containing 400
raw spectra. The average of the first subrun is shown as the blue dots in Fig
3 (a). The line shape is right the resonant dip of the cavity.

Figure 3:  (a) The power spectrum of the first subrun. The blue dots are the
measured powers averaged from 400 raw spectra. Two consecutive fittings were
used to remove the baseline due to the cavity and the mw devices. The product
of the fitting results is shown as the orange line. (b) The normalized
residuals $\Delta$ and their distribution. The normalized residuals obtained
from subrun 1 and subrun 2-99 are plotted as the blue dots and the gray dots,
respectively. The red bars are the counts of data while the red line is the
fitting result which is pretty close to the unit Gaussian distribution.

The spectra are firstly fitted to equation 4 (Fit1) to remove the baseline due
to the cavity. Then, the fluctuations in the transmission efficiency of the
devices are removed by applying a sum-of-sines fitting (Fit2). Meanwhile, the
values of the coupling coefficient $\beta_{1}$ and the quality factor of the
cavity $Q_{L}$ during each subrun are obtained from the fitting results. The
product of the two fitting results of the first subrun is shown as the orange
line in Fig 3 (a), with $\beta_{1}=0.4$ and $Q_{L}=6.1\times 10^{5}$. The blue
dots in Fig 3 (b) depict the standardized power residuals $\Delta$ of subrun
1,

$\Delta=\frac{\Delta_{\rm s}}{\sigma_{\rm s}},$ (7)

where $\Delta_{\rm s}$ is the normalized power excess obtained from the
baseline removing operation, and $\sigma_{\rm s}$ is the standard deviation of
the normalized power excess (see more details in Sec. IV in [44]). The gray
dots depict the standardized residuals of subrun 2-100.

A power excess over five times the standard deviation is required for a data
point to be claimed as a candidate signal. To check the presence of any
candidate signal in our results, the standardized power residuals are counted,
as illustrated by the red bars in Fig. 3 (b). The red line is the fitting
result with the Gaussian distribution N$(\mu,\sigma^{\prime 2})$, where $\mu$
is the mean value of the data and $\sigma^{\prime}$ is the standard deviation.
The distribution of the standardized power residuals fits well with the unit
Gaussian distribution, with $\mu=0.009\pm 0.010$ and $\sigma^{\prime}=1.009\pm
0.010$. No power excess over 5$\sigma$ is observed. Therefore, our results
provide an exclusion of dark photons in the detection bandwidth.

The data analysis that provides the constraints of the kinetic mixing $\chi$
follows the method developed in the ADMX experiment [47] (see more details in
Sec. IV, V in [44]). Figure 4 shows the upper bounds on the kinetic mixing in
the range of dark photon masses from 26.96493 $\mu$eV to 26.96534 $\mu$eV with
a confidence level of 90$\%$. The red line refers to the upper bounds of the
kinetic mixing in the random polarization scenario. An upper bound of
$\chi<6.0\times 10^{-16}$ is achieved at 26.96514 $\mu$eV, which is more than
three orders of magnitude more stringent than the bound established by the
QUALIPHIDE experiment. For a linear polarization scenario, the constraints
depend on the angle between the polarization direction of the dark photon
field and the electric field [22]. Although the polarization of dark photons
is unknown, the best constraints and the worst constraints can be obtained by
going over all the possible polarization directions, as the shallow pink and
the dark pink region in Fig. 4 shows.

Figure 4:  Upper limits on the kinetic mixing between dark photons and
ordinary photons. The red line refers to the upper bounds of the kinetic
mixing in the random polarization scenario set by this work. The shallow and
dark pink region refer to the excluded parameter space in the linear
polarization scenario in the best and worst condition, respectively. The
x-ticks at the top are the frequency biases from the central value, 6.52014
GHz, while the x-ticks at the bottom are the corresponding dark photon masses.
The blue, shallow green, and deep green regions refer to the previous results
of cosmological observations [38], the SHUKET [48] and the QUALIPHIDE [32]
experiments, respectively. Data adapted from [49].

In summary, we report an experimental search for dark photons, which is a
possible candidate for dark matter. Our results improve the constraints of the
kinetic mixing by more than three orders of magnitude. This work has shown an
outstanding $\frac{Q_{L}Q_{a}}{Q_{L}+Q_{a}}VC\frac{\beta}{1+\beta}$ that is
comparable with existing resonant haloscopes. For example, we have achieved an
improvement of around 2 times comparing with QUAX [26] and HAYSTAC [61].
However, this work is at a preliminary stage because of the lacking of a
tuning system. There is a large room for improvement of
$\frac{Q_{L}Q_{a}}{Q_{L}+Q_{a}}V^{2}C^{2}\frac{\beta^{2}}{(1+\beta)^{2}}$,
which is more concerned in terms of scan rate. For future dark matter search
experiments, new tunable cavities with larger mode volume are under
construction. The sensitivity of our apparatus can be further improved by
taking advantage of the single mw photon counting technique [50, 51, 52]. The
searching efficiency can be multiplied by combining a series of cavities with
different resonant frequencies. The experimental setup can also be used to
test other predictions arising from new physics beyond the Standard Model,
such as the existence of axions and axion-like particles, etc., with further
upgrades [53, 54]. In the future, it is practicable to examine the
Kim–Shifman–Vainshtein–Zakharov model with the present setup, a
superconducting cavity with through holes and 6 T magnet [55, 56]. The
sensitivity required by the Dine-Fischler-Srednicki-Zhitnitsky model is also
within reach [57, 58].

The authors are grateful to the anonymous referees for their valuable
comments. This work was supported by the Innovation Program for Quantum
Science and Technology (2021ZD0302200), the Chinese Academy of Sciences (No.
GJJSTD20200001), the National Key R&D Program of China (Grant
No.2021YFB3202800, No. 2018YFA0306600, No. 2021YFC2203100), Anhui Initiative
in Quantum Information Technologies (Grant No. AHY050000), NSFC (12150010,
12205290, 12261160569, 12261131497). Y. F. C. and M. J. thank the Fundamental
Research Funds for Central Universities. Y.F.C. is supported in part by CAS
Young Interdisciplinary Innovation Team (JCTD-2022-20), 111 Project (B23042).
M. J. Thanks is supported in part by China Postdoctoral Science Foundation
(2022TQ0330). This work was partially carried out at the USTC Center for Micro
and Nanoscale Research and Fabrication.

##  Appendix A The Schematic Diagram of the Experimental Apparatus

Figure 5:  Schematic diagram of the experimental apparatus.

The schematic diagram of the experimental apparatus is shown in Fig. 5. The
whole setup was housed in a multistage cryogenic system. Each horizontal
dashed line in Fig. 5 refers to a cold flange. A temperature as low as 21 mK
was achieved in the last stage with a dilution refrigerator.

In the detection part, two input lines (AC and DC) were used for the pumping
of the JPA. A bias current of a few milliamperes was injected from the DC
line, while a mw signal with twice the frequency as the working point was
injected from the AC line. The signal from the cavity was first reflected and
amplified by the JPA, and then further amplified by a cascade of HEMTs. Then,
the amplified signal was mixed with a local oscillator signal and down-
converted to 1 MHz. The down-converted signal was finally acquired by a
homemade DAQ card based on FPGAs.

Three input lines were employed in the calibration part. By measuring the
transmission coefficients from the Strong and the Bypass line to the Output
line 1, $S_{11}$ of the superconducting cavity and the gain curve of the JPA
could be obtained, respectively. The Weak line was used for the injection of a
simulated dark photon signal. It was accompanied by another Output line for
the measurement of the reflection coefficients of Port 2.

Each input line was carefully attenuated so that the noise was suppressed to a
negligible extent.

##  Appendix B The Data Acquisition Card

Figure 6:  Simplified schematic diagram of the DAQ card.

A data acquisition (DAQ) card based on the Field Programmable Gate Array
(FPGA) has been developed to achieve a 100$\%$ duty cycle. Figure. 6 presents
the simplified schematic diagram of the DAQ card. The down-converted input
digital signals are processed by an N-stage digital Low-Pass Filter (LPF) to
suppress the noise and the aliasing, and an N-stage downsampling is used to
improve the frequency resolution, where N is a configurable parameter. In this
work, N was set to 6. In order to realize a 100$\%$ real-time data utilization
ratio for the data flow, a Double-DataRate-III (DDR3) memory is used to
provide sufficient on-board data storage capacity. The time-domain data are
temporarily stored in the DDR3 before being sent to the FFT module. The high
parallel processing capability of the FPGA allows the collection of data and
the calculation of frequency spectra to be conducted simultaneously. The
output spectra are averaged in an accumulation module and the average result
is sent to the host computer through a high-speed universal serial bus (USB)
port. More details about the DAQ card can be seen in [41].

To test the electrical noise of the DAQ card, the output spectrum without any
input signal was measured, as shown in Fig. 7. The outputs $y$ of the DAQ card
are values in the unit of $1/$Hz1/2 that are proportional to the actual input
voltages. A noise floor in the frequency range from 0 MHz to 5 MHz is
observed. The sharp decrease of the electrical noise beyond 5 MHz is caused by
the multi-stage LPF. Actually, only a narrow region around 1 MHz is concerned
since it corresponds to the frequency band that is sensitive to the dark
photon. The electrical noise floor in this region is 6.6 Hz-1/2.

Figure 7:  Electronic noise power spectrum of the DAQ card. The sensitive
region is indicated by the red ribbon.

In order to map the outputs of the DAQ card to the actual power, the card is
calibrated by inputting mw signals with known power levels. Fig. 8 shows the
relationship between the output values and the input signals with power levels
ranging from -100 to -8 dBm. The result is fitted to the equation

$10\lg[(y^{2}-y_{0}{{}^{2}})B]=10\lg\xi+\widetilde{P}_{\rm in},$ (8)

which is equivalent to

$(y^{2}-y_{0}{{}^{2}})B=\xi P_{\rm in},$ (9)

where $y_{0}$ is the electrical noise, $B$ is the bin width of the spectrum,
$\xi$ is the conversion coefficient, $P_{\rm in}$ and $\widetilde{P}_{\rm in}$
are the power of the input signal in the unit of mW and dBm, respectively. The
fitting result gives $\xi=10^{15.32}$ mW-1. Note that the relation is no
longer linear when the input power exceeds -10 dBm, since the power is so high
that the DAQ card will be saturated.

Figure 8:  Relationship between the output values of the DAQ card and the
input powers.

##  Appendix C Verification of the detection system

To demonstrate the capability of our system in detecting a potential weak dark
photon signal, the cavity was designed to have an extra port (Port 2) for
injection of a simulated dark photon signal. The coupling coefficient of the
Port 2 was adjusted to a very small value. Figure 9 shows the reflection
coefficient of the Port 2. The fitting gives $\beta_{2}=0.0043$.

Figure 9: The reflection coefficient of the Port 2 of the cavity. The coupling
coefficient $\beta_{2}$ is 0.0043.

A pure-tone mw signal with the power being -43 dBm was synthesized to simulate
the dark photon signal. The power was further attenuated to -181 dBm, i.e.,
one zeptowatt, corresponding to a dark photon signal with $\chi=6\times
10^{-15}$. The blue and orange lines in Fig. 10 represent the power spectrum
averaged over one minute at the Port 1 of the cavity and the baseline of the
spectrum, respectively. By subtracting the baseline from the power spectrum,
the power residual can be obtained, as shown by the gray line. We do observe a
simulated dark photon signal, whose amplitude is consistent with the input
signal. The signal-to-noise ratio (SNR) is 58.7.

Figure 10:  The result of the simulated dark photon experiment. The blue line
is the output power spectrum at the Port 1 of the cavity. The orange line is
the baseline of the spectrum. Their difference is plotted as the gray line.

##  Appendix D Data analysis

Figure 11:  The average power spectrum of subrun 1. The blue dots stand for
the mean value of 400 raw spectra, and the orange line stands for the fitting
to the $S_{11}$ function of the cavity.  Figure 12:  Fluctuations of
experimental parameters during the data collection. (a) The quality factor.
(b) The coupling coefficient. (c) The resonant frequency.

After the six-hour data collection, forty thousand raw spectra have been
obtained. In order to suppress the noise, the raw spectra are divided into 100
subruns. Each subrun contains 400 neighbouring raw spectra, and the average of
them is calculated. The average result of the first subrun is shown as the
blue dots in Fig. 11.

The curve is due to the cavity. Since there was a static temperature gradient
between the cavity and the environment, the power spectrum can be expressed as

$P_{\rm n}\varpropto T_{\rm n}+T_{\rm e}+(T_{\rm c}-T_{\rm e})|S_{11}|^{2},$
(10)

where $T_{\rm e}$ is the temperature of the environment, $T_{\rm c}$ is the
temperature of the cavity, and $S_{11}$ is the reflection coefficient of the
cavity (Equation 4 in the main text). The fitting result Fit1 is shown as the
orange line in Fig. 11. The quality factor $Q_{\rm L}$, the coupling
coefficient $\beta_{1}$, and the resonant frequency $f_{\rm c}$ for each
subrun are also accessible through the fitting results. Their fluctuations
during the data collection are shown in Fig. 12.

Figure 13:  The normalized power spectrum of subrun 1. The blue dots stand for the power spectra divided by the cavity baseline and the orange line stands for the fitting to a sum of sinusoidal functions.  Table 1: Fitting parameters in Fig. 13 $k$ |  $C_{k}$ |  $\nu_{k}$ |  $\xi_{k}$
---|---|---|---
0 |  $1.0$ |  - |  -
1 |  $0.2\times 10^{-3}$ |  3.6 |  3.0
2 |  $1.2\times 10^{-3}$ |  10.0 |  2.9
3 |  $0.1\times 10^{-3}$ |  2.8 |  3.0
4 |  $-1.3\times 10^{-3}$ |  9.1 |  2.9

Every averaged power spectrum is divided by the cavity baseline $\rm Fit_{1}$,

$\Delta_{\rm n}=\frac{P_{\rm n}}{\rm Fit_{1}}.$ (11)

Figure 13 shows the normalized power spectrum $\Delta_{\rm n}$ of the first
subrun. The fluctuations in the normalized power spectrum are caused by the
variation in transmission efficiency of the mw devices. They are removed by
fitting the normalized power spectrum to a sum of sines,

${\rm Fit_{2}}=C_{0}+\sum_{k=1}^{4}C_{k}\sin(\frac{f}{\nu_{k}}+\xi_{k}),$ (12)

as the orange line in Fig. 13 shows. The fitting parameters in this step are
shown in Table 1. By dividing the normalized power spectrum by this fitting
result and subtracting the central value of 1 from the quotient, we obtain the
normalized power excess

$\Delta_{\rm s}=\frac{\Delta_{\rm n}}{\rm Fit_{2}}-1,$ (13)

where $\rm Fit_{2}$ is the fitting result of the variation of transmission
efficiency. The same operations are also performed on the rescaled power
spectra of subrun 2-100. The blue dots in Fig. 14 show the normalized power
excess obtained from all subruns. The standard deviation is taken as the
statistical error, as the shallow red ribbon shows.

Figure 14:  The normalized power excess of all subruns. The shallow red ribbon
stands for the statistical error.  Figure 15:  (a) The expected line shape of
dark photons. (b) Normalized power spectra with simulated dark photon signals
added to the original spectrum.

The frequency distribution of the dark photon can be described by the Maxwell-
Boltzmann distribution,

$\mathcal{F}(f)=2\sqrt{\frac{f-f_{\rm a}}{\pi}}[\frac{3}{\eta f_{\rm
a}\langle(v_{\rm a}/c)^{2}\rangle}]^{\frac{3}{2}}\exp\frac{-3(f-f_{\rm
a})}{\eta f_{\rm a}\langle(v_{\rm a}/c)^{2}\rangle},$ (14)

where $f_{\rm a}$ is the frequency of the photon that couples to the dark
photon, $c$ is the velocity of light, $v_{\rm a}$ is the velocity of the dark
photon, and $\eta$ is a correction factor introduced by the complex motion of
the lab frame. Here $\langle v_{\rm a}^{2}\rangle=1.5\langle v_{\rm
a}\rangle^{2}=(270\rm km/s)^{2}$ and $\eta=1.7$ is adopted [59, 60, 61]. For
the dark photon in the frequency region around 6.52 GHz, the quality factor
$Q_{a}$ and linewidth $\Delta f_{a}$ are $1\times 10^{6}$ and 6.52 kHz,
respectively. Figure 15(a) shows the expected line shape of the dark photon
signal. Given that the fitting results give $Q_{\rm L}<Q_{\rm a}$ and $\Delta
f_{a}/{\nu_{k}}<\pi$, and that the line shape of dark photons (Maxwell-
Boltzmann) is quite different from that of the cavity (Lorentzian), if a dark
photon signal existed, it would survive in the baseline removing operations
and show up in the normalized power excess spectrum as a narrow peak.

To further demonstrate this point, we performed the first baseline removal
operation with simulated dark photon signals added to the original spectrum.
Figure 15(b) shows 5 examples of the results after the first baseline removal.
The strength of the signals is set to 5 times of our final constraints at the
corresponding frequencies. The black line refers to the original data (no dark
photon signal), while the colored lines refer to the data with dark photon
signals at different frequencies. The ribbons of corresponding colors indicate
the positions of the dark photon signals. The signals are obvious, especially
for the sharp rising edges. We have also compared the amplitude of the signals
after the baseline removal operation and the expected values. The correction
coefficient is 1.1$\pm$0.2.

The following analysis employs the method developed in the ADMX experiment
[2022CervanteS3].

The normalized power excess indicates the power fluctuation due to the thermal
noise. It is in the unit of $k_{\rm B}T_{\rm n}B$, where $k_{\rm B}$ is the
Boltzmann constant, $T_{\rm n}$ is the noise temperature, and $B$ is the
frequency bin width of the spectra. In order to relate the normalized power
excess to the real power, each data point is multiplied by $k_{\rm B}T_{\rm
n}B$. Since the power of the expected dark photon signal varies depending on
its central frequency, the normalized power excess is rescaled to the dark
photon signal with $\chi=1$. The rescaled power excess

$\Delta_{\rm r}=\Delta_{\rm s}\frac{k_{\rm B}T_{\rm n}B}{P_{\rm s}(\chi=1)}$
(15)

is plotted in Fig. 16, where $P_{\rm s}(\chi)$ is the expected dark photon
signal with a specific value of $\chi$. As a result, the expected dark photon
signal would have the same amplitude regardless of its position on the
spectrum.

Figure 16:  The rescaled power excess of all subruns. The shallow red ribbon
stands for the statistical error.

Since the linewidth of the dark photon is 6.52 kHz, the expected dark photon
signal would spread over about 15 frequency bins. As a result, the SNR would
be reduced by about 15 times. Since the amplitude of the expected dark photon
signal is already normalized in the previous step, a convolution between the
rescaled power excess and the line shape of the dark photon could consolidate
the expected dark photon signal into a single bin. Thus, the SNR would be
enhanced. The convolution operation leads to the spectrum of the filtered
power excess $\Delta_{\rm f}$, as Fig. 17 shows.

Figure 17:  The filtered power excess of all subruns. The shallow red ribbon
stands for the statistical error.

The 100 filtered power excess spectra obtained individually from each subrun
are averaged to further reduce the noise. The result $\Delta_{\rm c}$ is shown
in Fig. 18. The value of each point is averaged from 100 filtered power excess
values on the corresponding frequency bin. The shallow red ribbon refers to
the statistical errors $\sigma_{\rm c}$. The systematic errors are now
introduced into the analysis. The mean value and relative uncertainties of the
experimental parameters are shown in Table 2. Since the uncertainty introduced
by the resonant frequency $f_{\rm c}$ is too small, this term is neglected in
the analysis. The systematic errors are combined with the statistical errors
as $\sigma^{\prime}_{c}=\sqrt{\sigma_{c}^{2}+\Delta_{c}^{2}(\sigma_{Q_{\rm
L}}^{2}+\sigma_{\beta}^{2}+\sigma_{V}^{2}+\sigma_{T_{\rm n}}^{2})}$, where
$\sigma_{Q_{\rm L}}$, $\sigma_{\beta}$, $\sigma_{V}$, $\sigma_{T_{\rm n}}$ are
the relative uncertainties of $Q_{\rm L}$, $\beta$, $V$ and $T_{\rm n}$,
respectively. The contribution of the systematic errors to the total errors is
shown as the deep red ribbons in Fig. 18.

Table 2: Summary of the mean values and the relative systematic uncertainties of experimental parameters. Parameter | Value | relative uncertainty
---|---|---
$Q^{\prime}_{\rm L}=Q_{L}Q_{a}/(Q_{L}+Q_{a})$ | $3.8\times 10^{5}$ | $9.3\times 10^{-4}$
$\beta$ | 0.4 | $9.7\times 10^{-4}$
$f_{\rm c}$ |  $6.5$ GHz | $2.1\times 10^{-9}$
$V$ |  $1.5\times 10^{-1}$ L | $5.7\times 10^{-3}$
$T_{\rm n}$ |  $3.0\times 10^{-1}$ K | $7.4\times 10^{-2}$
Figure 18:  The average power excess. The value of each data point is the mean
of 100 filtered power excess values on the corresponding frequency bin. The
shallow red ribbon stands for the statistical error. The deep red ribbon
stands for the systematic error introduced by the uncertainties of $Q_{\rm
L}$, $\beta$, $V$, and $T_{\rm n}$.

##  Appendix E Bayesian analysis

Since no candidate dark photon signal has been discovered, as Fig. 3 (b) in
the main text shows, Bayesian analysis is employed to provide constraints of
the kinetic mixing $\chi$ of the dark photon. For a given dark photon signal
$P_{\rm s}$, the measured power excess $\widetilde{\Delta}_{\rm c}$ would
distribute around $P_{\rm s}$ in a Gaussian form:

$p(\widetilde{\Delta}_{\rm c}|P_{\rm s})=\frac{1}{\sqrt{2\pi}\sigma_{\rm
c}}\exp[-\frac{(\widetilde{\Delta}_{\rm c}-P_{\rm s})^{2}}{2\sigma_{\rm
c}^{2}}].$ (16)

Note that the dark photon signal $P_{\rm s}$ is in the unit of $P_{\rm
s}(\chi=1)$, so it can be replaced by $\chi^{2}$:

$p(\widetilde{\Delta}_{\rm c}|\chi^{2})=\frac{1}{\sqrt{2\pi}\sigma_{\rm
c}}\exp[-\frac{(\widetilde{\Delta}_{\rm c}-\chi^{2})^{2}}{2\sigma_{\rm
c}^{2}}].$ (17)

However, we are actually in an opposite condition where the power excess
$\Delta_{\rm c}$ is a known parameter while $P_{\rm s}$ is unknown. Therefore,
Bayes’ theorem is employed to estimate the distribution of $P_{\rm s}$:

$p(\chi^{2}|\Delta_{\rm c})=p(\widetilde{\Delta}_{\rm c}=\Delta_{\rm
c}|\chi^{2})\frac{p(\chi^{2})}{p(\widetilde{\Delta}_{\rm c}=\Delta_{\rm c})},$
(18)

where $p(\chi^{2})$ and $p(\widetilde{\Delta}_{\rm c})$ are the probability
density function of $\chi^{2}$ and $\widetilde{\Delta}_{\rm c}$ respectively.
The distribution of $\chi^{2}$ should be uniform on the interval
$[0,\chi^{2}_{\rm Q}]$, because $\chi^{2}<0$ is impossible while
$\chi^{2}>\chi^{2}_{\rm Q}=3\times 10^{-24}$ have been excluded by the
QUALIPHIDE experiment [32]. Since $\widetilde{\Delta}_{\rm c}$ has a fixed
value from the measurement, the distribution of $\widetilde{\Delta}_{\rm c}$
should be a delta function, $p(\widetilde{\Delta}_{\rm
c})=\delta(\widetilde{\Delta}_{\rm c}-\Delta_{\rm c})$. Here we replace the
delta function with the limit of a finite function,

$\displaystyle\delta(x)$ $\displaystyle=\lim_{\kappa\to\infty}\xi(x;\kappa),$
(19) $\displaystyle\xi(x;\kappa)$
$\displaystyle=\left\\{\begin{array}[]{lr}1/\kappa,&0<x<\kappa\\\ 0,&\rm
otherwise\\\ \end{array}\right.$

in order to avoid the problem of divergence.

Consequently, the conditional distribution of $\chi^{2}$ can be simplified to

$\displaystyle p(\chi^{2}|\Delta_{\rm c})$
$\displaystyle=\frac{p(\widetilde{\Delta}_{\rm c}=\Delta_{\rm
c}|\chi^{2})p(\chi^{2})/p(\widetilde{\Delta}_{\rm c}=\Delta_{\rm c})}{\int
p(\widetilde{\Delta}_{\rm c}=\Delta_{\rm
c}|\chi^{2})p(\chi^{2})/p(\widetilde{\Delta}_{\rm c}=\Delta_{\rm
c})\,d\chi^{2}}$ (20) $\displaystyle=\frac{p(\Delta_{\rm
c}|\chi^{2})}{\int_{0}^{\chi^{2}_{\rm Q}}p(\Delta_{\rm
c}|\chi^{2})\,d\chi^{2}}$

Finally, by solving the equation

$\displaystyle\int_{0}^{\chi_{90\%}^{2}}p(\chi^{2}|\Delta_{\rm
c})d\chi^{2}=90\%$ (21)

for each frequency bin, the constraints of the kinetic mixing with a
confidence level of $90\%$ are obtained, as Fig. 4 (a) in the main text shows.

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